LMFDB - Embedded newform 4012.2.a.f.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4012,2,Mod(1,4012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4012, 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("4012.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4012 = 2^{2} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4012.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:	$32.0359812909$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.321.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 1 $ x^3 - x^2 - 4*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.69963$ of defining polynomial
Character	$\chi$	$=$	4012.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +4.39926 q^{5} +1.00000 q^{7} -2.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +4.39926 q^{5} +1.00000 q^{7} -2.00000 q^{9} -2.00000 q^{11} -3.39926 q^{13} -4.39926 q^{15} +1.00000 q^{17} -6.39926 q^{19} -1.00000 q^{21} +0.222528 q^{23} +14.3535 q^{25} +5.00000 q^{27} +2.39926 q^{29} -6.57598 q^{31} +2.00000 q^{33} +4.39926 q^{35} -9.97524 q^{37} +3.39926 q^{39} -6.39926 q^{41} -10.5760 q^{43} -8.79851 q^{45} -4.22253 q^{47} -6.00000 q^{49} -1.00000 q^{51} +13.3535 q^{53} -8.79851 q^{55} +6.39926 q^{57} -1.00000 q^{59} +1.39926 q^{61} -2.00000 q^{63} -14.9542 q^{65} -2.44506 q^{67} -0.222528 q^{69} +6.35346 q^{71} +2.00000 q^{73} -14.3535 q^{75} -2.00000 q^{77} -5.00000 q^{79} +1.00000 q^{81} -13.9752 q^{83} +4.39926 q^{85} -2.39926 q^{87} -4.57598 q^{89} -3.39926 q^{91} +6.57598 q^{93} -28.1520 q^{95} -15.7527 q^{97} +4.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + q^{5} + 3 q^{7} - 6 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 + q^5 + 3 * q^7 - 6 * q^9	$ 3 q - 3 q^{3} + q^{5} + 3 q^{7} - 6 q^{9} - 6 q^{11} + 2 q^{13} - q^{15} + 3 q^{17} - 7 q^{19} - 3 q^{21} + 20 q^{25} + 15 q^{27} - 5 q^{29} + 4 q^{31} + 6 q^{33} + q^{35} + 6 q^{37} - 2 q^{39} - 7 q^{41} - 8 q^{43} - 2 q^{45} - 12 q^{47} - 18 q^{49} - 3 q^{51} + 17 q^{53} - 2 q^{55} + 7 q^{57} - 3 q^{59} - 8 q^{61} - 6 q^{63} - 34 q^{65} - 6 q^{67} - 4 q^{71} + 6 q^{73} - 20 q^{75} - 6 q^{77} - 15 q^{79} + 3 q^{81} - 6 q^{83} + q^{85} + 5 q^{87} + 10 q^{89} + 2 q^{91} - 4 q^{93} - 37 q^{95} - 12 q^{97} + 12 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + q^5 + 3 * q^7 - 6 * q^9 - 6 * q^11 + 2 * q^13 - q^15 + 3 * q^17 - 7 * q^19 - 3 * q^21 + 20 * q^25 + 15 * q^27 - 5 * q^29 + 4 * q^31 + 6 * q^33 + q^35 + 6 * q^37 - 2 * q^39 - 7 * q^41 - 8 * q^43 - 2 * q^45 - 12 * q^47 - 18 * q^49 - 3 * q^51 + 17 * q^53 - 2 * q^55 + 7 * q^57 - 3 * q^59 - 8 * q^61 - 6 * q^63 - 34 * q^65 - 6 * q^67 - 4 * q^71 + 6 * q^73 - 20 * q^75 - 6 * q^77 - 15 * q^79 + 3 * q^81 - 6 * q^83 + q^85 + 5 * q^87 + 10 * q^89 + 2 * q^91 - 4 * q^93 - 37 * q^95 - 12 * q^97 + 12 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350	−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	0	0
$5$	4.39926	1.96741	0.983704	−	0.179798i	$-0.0575443\pi$
$5$	4.39926	1.96741	0.983704	+	0.179798i	$0.0575443\pi$
$6$	0	0
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	−3.39926	−0.942784	−0.471392	−	0.881924i	$-0.656248\pi$
$13$	−3.39926	−0.942784	−0.471392	+	0.881924i	$0.656248\pi$
$14$	0	0
$15$	−4.39926	−1.13588
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−6.39926	−1.46809	−0.734045	−	0.679101i	$-0.762370\pi$
$19$	−6.39926	−1.46809	−0.734045	+	0.679101i	$0.762370\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	0.222528	0.0464004	0.0232002	−	0.999731i	$-0.492614\pi$
$23$	0.222528	0.0464004	0.0232002	+	0.999731i	$0.492614\pi$
$24$	0	0
$25$	14.3535	2.87069
$26$	0	0
$27$	5.00000	0.962250
$28$	0	0
$29$	2.39926	0.445531	0.222765	−	0.974872i	$-0.428492\pi$
$29$	2.39926	0.445531	0.222765	+	0.974872i	$0.428492\pi$
$30$	0	0
$31$	−6.57598	−1.18108	−0.590541	−	0.807008i	$-0.701086\pi$
$31$	−6.57598	−1.18108	−0.590541	+	0.807008i	$0.701086\pi$
$32$	0	0
$33$	2.00000	0.348155
$34$	0	0
$35$	4.39926	0.743610
$36$	0	0
$37$	−9.97524	−1.63992	−0.819960	−	0.572421i	$-0.806004\pi$
$37$	−9.97524	−1.63992	−0.819960	+	0.572421i	$0.806004\pi$
$38$	0	0
$39$	3.39926	0.544317
$40$	0	0
$41$	−6.39926	−0.999396	−0.499698	−	0.866200i	$-0.666556\pi$
$41$	−6.39926	−0.999396	−0.499698	+	0.866200i	$0.666556\pi$
$42$	0	0
$43$	−10.5760	−1.61282	−0.806411	−	0.591355i	$-0.798593\pi$
$43$	−10.5760	−1.61282	−0.806411	+	0.591355i	$0.798593\pi$
$44$	0	0
$45$	−8.79851	−1.31160
$46$	0	0
$47$	−4.22253	−0.615919	−0.307960	−	0.951399i	$-0.599646\pi$
$47$	−4.22253	−0.615919	−0.307960	+	0.951399i	$0.599646\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	−1.00000	−0.140028
$52$	0	0
$53$	13.3535	1.83424	0.917119	−	0.398613i	$-0.130508\pi$
$53$	13.3535	1.83424	0.917119	+	0.398613i	$0.130508\pi$
$54$	0	0
$55$	−8.79851	−1.18639
$56$	0	0
$57$	6.39926	0.847602
$58$	0	0
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	1.39926	0.179156	0.0895782	−	0.995980i	$-0.471448\pi$
$61$	1.39926	0.179156	0.0895782	+	0.995980i	$0.471448\pi$
$62$	0	0
$63$	−2.00000	−0.251976
$64$	0	0
$65$	−14.9542	−1.85484
$66$	0	0
$67$	−2.44506	−0.298711	−0.149356	−	0.988784i	$-0.547720\pi$
$67$	−2.44506	−0.298711	−0.149356	+	0.988784i	$0.547720\pi$
$68$	0	0
$69$	−0.222528	−0.0267893
$70$	0	0
$71$	6.35346	0.754017	0.377008	−	0.926210i	$-0.376953\pi$
$71$	6.35346	0.754017	0.377008	+	0.926210i	$0.376953\pi$
$72$	0	0
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	0	0
$75$	−14.3535	−1.65739
$76$	0	0
$77$	−2.00000	−0.227921
$78$	0	0
$79$	−5.00000	−0.562544	−0.281272	−	0.959628i	$-0.590756\pi$
$79$	−5.00000	−0.562544	−0.281272	+	0.959628i	$0.590756\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−13.9752	−1.53398	−0.766991	−	0.641658i	$-0.778247\pi$
$83$	−13.9752	−1.53398	−0.766991	+	0.641658i	$0.778247\pi$
$84$	0	0
$85$	4.39926	0.477166
$86$	0	0
$87$	−2.39926	−0.257227
$88$	0	0
$89$	−4.57598	−0.485053	−0.242527	−	0.970145i	$-0.577976\pi$
$89$	−4.57598	−0.485053	−0.242527	+	0.970145i	$0.577976\pi$
$90$	0	0
$91$	−3.39926	−0.356339
$92$	0	0
$93$	6.57598	0.681898
$94$	0	0
$95$	−28.1520	−2.88833
$96$	0	0
$97$	−15.7527	−1.59945	−0.799723	−	0.600369i	$-0.795020\pi$
$97$	−15.7527	−1.59945	−0.799723	+	0.600369i	$0.795020\pi$
$98$	0	0
$99$	4.00000	0.402015
$100$	0	0
$101$	−2.37822	−0.236641	−0.118321	−	0.992975i	$-0.537751\pi$
$101$	−2.37822	−0.236641	−0.118321	+	0.992975i	$0.537751\pi$
$102$	0	0
$103$	−11.8443	−1.16705	−0.583527	−	0.812093i	$-0.698328\pi$
$103$	−11.8443	−1.16705	−0.583527	+	0.812093i	$0.698328\pi$
$104$	0	0
$105$	−4.39926	−0.429323
$106$	0	0
$107$	9.79851	0.947258	0.473629	−	0.880724i	$-0.342944\pi$
$107$	9.79851	0.947258	0.473629	+	0.880724i	$0.342944\pi$
$108$	0	0
$109$	6.44506	0.617324	0.308662	−	0.951172i	$-0.400119\pi$
$109$	6.44506	0.617324	0.308662	+	0.951172i	$0.400119\pi$
$110$	0	0
$111$	9.97524	0.946808
$112$	0	0
$113$	3.84431	0.361643	0.180821	−	0.983516i	$-0.442124\pi$
$113$	3.84431	0.361643	0.180821	+	0.983516i	$0.442124\pi$
$114$	0	0
$115$	0.978959	0.0912884
$116$	0	0
$117$	6.79851	0.628523
$118$	0	0
$119$	1.00000	0.0916698
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	6.39926	0.577002
$124$	0	0
$125$	41.1483	3.68041
$126$	0	0
$127$	19.1978	1.70353	0.851763	−	0.523927i	$-0.175533\pi$
$127$	19.1978	1.70353	0.851763	+	0.523927i	$0.175533\pi$
$128$	0	0
$129$	10.5760	0.931163
$130$	0	0
$131$	5.15197	0.450130	0.225065	−	0.974344i	$-0.427741\pi$
$131$	5.15197	0.450130	0.225065	+	0.974344i	$0.427741\pi$
$132$	0	0
$133$	−6.39926	−0.554886
$134$	0	0
$135$	21.9963	1.89314
$136$	0	0
$137$	13.7985	1.17889	0.589443	−	0.807810i	$-0.299347\pi$
$137$	13.7985	1.17889	0.589443	+	0.807810i	$0.299347\pi$
$138$	0	0
$139$	6.35346	0.538893	0.269447	−	0.963015i	$-0.413159\pi$
$139$	6.35346	0.538893	0.269447	+	0.963015i	$0.413159\pi$
$140$	0	0
$141$	4.22253	0.355601
$142$	0	0
$143$	6.79851	0.568520
$144$	0	0
$145$	10.5549	0.876540
$146$	0	0
$147$	6.00000	0.494872
$148$	0	0
$149$	−9.39926	−0.770017	−0.385009	−	0.922913i	$-0.625802\pi$
$149$	−9.39926	−0.770017	−0.385009	+	0.922913i	$0.625802\pi$
$150$	0	0
$151$	−6.82327	−0.555270	−0.277635	−	0.960687i	$-0.589551\pi$
$151$	−6.82327	−0.555270	−0.277635	+	0.960687i	$0.589551\pi$
$152$	0	0
$153$	−2.00000	−0.161690
$154$	0	0
$155$	−28.9294	−2.32367
$156$	0	0
$157$	−13.7527	−1.09759	−0.548793	−	0.835958i	$-0.684912\pi$
$157$	−13.7527	−1.09759	−0.548793	+	0.835958i	$0.684912\pi$
$158$	0	0
$159$	−13.3535	−1.05900
$160$	0	0
$161$	0.222528	0.0175377
$162$	0	0
$163$	11.2436	0.880664	0.440332	−	0.897835i	$-0.354861\pi$
$163$	11.2436	0.880664	0.440332	+	0.897835i	$0.354861\pi$
$164$	0	0
$165$	8.79851	0.684963
$166$	0	0
$167$	−3.44506	−0.266586	−0.133293	−	0.991077i	$-0.542555\pi$
$167$	−3.44506	−0.266586	−0.133293	+	0.991077i	$0.542555\pi$
$168$	0	0
$169$	−1.44506	−0.111158
$170$	0	0
$171$	12.7985	0.978727
$172$	0	0
$173$	12.9542	0.984890	0.492445	−	0.870344i	$-0.336103\pi$
$173$	12.9542	0.984890	0.492445	+	0.870344i	$0.336103\pi$
$174$	0	0
$175$	14.3535	1.08502
$176$	0	0
$177$	1.00000	0.0751646
$178$	0	0
$179$	−4.60074	−0.343876	−0.171938	−	0.985108i	$-0.555003\pi$
$179$	−4.60074	−0.343876	−0.171938	+	0.985108i	$0.555003\pi$
$180$	0	0
$181$	19.1062	1.42015	0.710075	−	0.704126i	$-0.248661\pi$
$181$	19.1062	1.42015	0.710075	+	0.704126i	$0.248661\pi$
$182$	0	0
$183$	−1.39926	−0.103436
$184$	0	0
$185$	−43.8836	−3.22639
$186$	0	0
$187$	−2.00000	−0.146254
$188$	0	0
$189$	5.00000	0.363696
$190$	0	0
$191$	7.37450	0.533600	0.266800	−	0.963752i	$-0.414034\pi$
$191$	7.37450	0.533600	0.266800	+	0.963752i	$0.414034\pi$
$192$	0	0
$193$	11.1978	0.806033	0.403017	−	0.915193i	$-0.367962\pi$
$193$	11.1978	0.806033	0.403017	+	0.915193i	$0.367962\pi$
$194$	0	0
$195$	14.9542	1.07089
$196$	0	0
$197$	7.90840	0.563450	0.281725	−	0.959495i	$-0.409093\pi$
$197$	7.90840	0.563450	0.281725	+	0.959495i	$0.409093\pi$
$198$	0	0
$199$	−17.3535	−1.23015	−0.615077	−	0.788467i	$-0.710875\pi$
$199$	−17.3535	−1.23015	−0.615077	+	0.788467i	$0.710875\pi$
$200$	0	0
$201$	2.44506	0.172461
$202$	0	0
$203$	2.39926	0.168395
$204$	0	0
$205$	−28.1520	−1.96622
$206$	0	0
$207$	−0.445057	−0.0309336
$208$	0	0
$209$	12.7985	0.885292
$210$	0	0
$211$	−9.02104	−0.621034	−0.310517	−	0.950568i	$-0.600502\pi$
$211$	−9.02104	−0.621034	−0.310517	+	0.950568i	$0.600502\pi$
$212$	0	0
$213$	−6.35346	−0.435332
$214$	0	0
$215$	−46.5265	−3.17308
$216$	0	0
$217$	−6.57598	−0.446407
$218$	0	0
$219$	−2.00000	−0.135147
$220$	0	0
$221$	−3.39926	−0.228659
$222$	0	0
$223$	1.15197	0.0771415	0.0385708	−	0.999256i	$-0.487719\pi$
$223$	1.15197	0.0771415	0.0385708	+	0.999256i	$0.487719\pi$
$224$	0	0
$225$	−28.7069	−1.91379
$226$	0	0
$227$	21.6218	1.43509	0.717544	−	0.696513i	$-0.245266\pi$
$227$	21.6218	1.43509	0.717544	+	0.696513i	$0.245266\pi$
$228$	0	0
$229$	−19.5302	−1.29059	−0.645295	−	0.763933i	$-0.723266\pi$
$229$	−19.5302	−1.29059	−0.645295	+	0.763933i	$0.723266\pi$
$230$	0	0
$231$	2.00000	0.131590
$232$	0	0
$233$	12.7738	0.836836	0.418418	−	0.908254i	$-0.362585\pi$
$233$	12.7738	0.836836	0.418418	+	0.908254i	$0.362585\pi$
$234$	0	0
$235$	−18.5760	−1.21176
$236$	0	0
$237$	5.00000	0.324785
$238$	0	0
$239$	−8.39926	−0.543303	−0.271651	−	0.962396i	$-0.587570\pi$
$239$	−8.39926	−0.543303	−0.271651	+	0.962396i	$0.587570\pi$
$240$	0	0
$241$	−27.1978	−1.75196	−0.875981	−	0.482345i	$-0.839785\pi$
$241$	−27.1978	−1.75196	−0.875981	+	0.482345i	$0.839785\pi$
$242$	0	0
$243$	−16.0000	−1.02640
$244$	0	0
$245$	−26.3955	−1.68635
$246$	0	0
$247$	21.7527	1.38409
$248$	0	0
$249$	13.9752	0.885645
$250$	0	0
$251$	−17.5512	−1.10782	−0.553912	−	0.832575i	$-0.686866\pi$
$251$	−17.5512	−1.10782	−0.553912	+	0.832575i	$0.686866\pi$
$252$	0	0
$253$	−0.445057	−0.0279805
$254$	0	0
$255$	−4.39926	−0.275492
$256$	0	0
$257$	−8.15197	−0.508506	−0.254253	−	0.967138i	$-0.581830\pi$
$257$	−8.15197	−0.508506	−0.254253	+	0.967138i	$0.581830\pi$
$258$	0	0
$259$	−9.97524	−0.619831
$260$	0	0
$261$	−4.79851	−0.297020
$262$	0	0
$263$	−4.39926	−0.271270	−0.135635	−	0.990759i	$-0.543307\pi$
$263$	−4.39926	−0.271270	−0.135635	+	0.990759i	$0.543307\pi$
$264$	0	0
$265$	58.7453	3.60869
$266$	0	0
$267$	4.57598	0.280046
$268$	0	0
$269$	10.7317	0.654322	0.327161	−	0.944969i	$-0.393908\pi$
$269$	10.7317	0.654322	0.327161	+	0.944969i	$0.393908\pi$
$270$	0	0
$271$	−13.5091	−0.820622	−0.410311	−	0.911946i	$-0.634580\pi$
$271$	−13.5091	−0.820622	−0.410311	+	0.911946i	$0.634580\pi$
$272$	0	0
$273$	3.39926	0.205732
$274$	0	0
$275$	−28.7069	−1.73109
$276$	0	0
$277$	5.60074	0.336516	0.168258	−	0.985743i	$-0.446186\pi$
$277$	5.60074	0.336516	0.168258	+	0.985743i	$0.446186\pi$
$278$	0	0
$279$	13.1520	0.787388
$280$	0	0
$281$	30.0604	1.79325	0.896626	−	0.442789i	$-0.146011\pi$
$281$	30.0604	1.79325	0.896626	+	0.442789i	$0.146011\pi$
$282$	0	0
$283$	−13.3077	−0.791058	−0.395529	−	0.918453i	$-0.629439\pi$
$283$	−13.3077	−0.791058	−0.395529	+	0.918453i	$0.629439\pi$
$284$	0	0
$285$	28.1520	1.66758
$286$	0	0
$287$	−6.39926	−0.377736
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	15.7527	0.923440
$292$	0	0
$293$	−29.0421	−1.69666	−0.848328	−	0.529471i	$-0.822391\pi$
$293$	−29.0421	−1.69666	−0.848328	+	0.529471i	$0.822391\pi$
$294$	0	0
$295$	−4.39926	−0.256135
$296$	0	0
$297$	−10.0000	−0.580259
$298$	0	0
$299$	−0.756431	−0.0437455
$300$	0	0
$301$	−10.5760	−0.609590
$302$	0	0
$303$	2.37822	0.136625
$304$	0	0
$305$	6.15569	0.352474
$306$	0	0
$307$	5.24729	0.299479	0.149739	−	0.988726i	$-0.452157\pi$
$307$	5.24729	0.299479	0.149739	+	0.988726i	$0.452157\pi$
$308$	0	0
$309$	11.8443	0.673799
$310$	0	0
$311$	−15.0000	−0.850572	−0.425286	−	0.905059i	$-0.639826\pi$
$311$	−15.0000	−0.850572	−0.425286	+	0.905059i	$0.639826\pi$
$312$	0	0
$313$	6.60074	0.373096	0.186548	−	0.982446i	$-0.440270\pi$
$313$	6.60074	0.373096	0.186548	+	0.982446i	$0.440270\pi$
$314$	0	0
$315$	−8.79851	−0.495740
$316$	0	0
$317$	13.2436	0.743833	0.371916	−	0.928266i	$-0.378701\pi$
$317$	13.2436	0.743833	0.371916	+	0.928266i	$0.378701\pi$
$318$	0	0
$319$	−4.79851	−0.268665
$320$	0	0
$321$	−9.79851	−0.546900
$322$	0	0
$323$	−6.39926	−0.356064
$324$	0	0
$325$	−48.7911	−2.70644
$326$	0	0
$327$	−6.44506	−0.356412
$328$	0	0
$329$	−4.22253	−0.232796
$330$	0	0
$331$	−26.3993	−1.45103	−0.725517	−	0.688204i	$-0.758399\pi$
$331$	−26.3993	−1.45103	−0.725517	+	0.688204i	$0.758399\pi$
$332$	0	0
$333$	19.9505	1.09328
$334$	0	0
$335$	−10.7564	−0.587687
$336$	0	0
$337$	23.9505	1.30467	0.652333	−	0.757933i	$-0.273790\pi$
$337$	23.9505	1.30467	0.652333	+	0.757933i	$0.273790\pi$
$338$	0	0
$339$	−3.84431	−0.208794
$340$	0	0
$341$	13.1520	0.712219
$342$	0	0
$343$	−13.0000	−0.701934
$344$	0	0
$345$	−0.978959	−0.0527054
$346$	0	0
$347$	11.8836	0.637947	0.318974	−	0.947764i	$-0.396662\pi$
$347$	11.8836	0.637947	0.318974	+	0.947764i	$0.396662\pi$
$348$	0	0
$349$	−3.62178	−0.193870	−0.0969348	−	0.995291i	$-0.530904\pi$
$349$	−3.62178	−0.193870	−0.0969348	+	0.995291i	$0.530904\pi$
$350$	0	0
$351$	−16.9963	−0.907194
$352$	0	0
$353$	−16.7738	−0.892777	−0.446388	−	0.894839i	$-0.647290\pi$
$353$	−16.7738	−0.892777	−0.446388	+	0.894839i	$0.647290\pi$
$354$	0	0
$355$	27.9505	1.48346
$356$	0	0
$357$	−1.00000	−0.0529256
$358$	0	0
$359$	−0.0458003	−0.00241725	−0.00120862	−	0.999999i	$-0.500385\pi$
$359$	−0.0458003	−0.00241725	−0.00120862	+	0.999999i	$0.500385\pi$
$360$	0	0
$361$	21.9505	1.15529
$362$	0	0
$363$	7.00000	0.367405
$364$	0	0
$365$	8.79851	0.460535
$366$	0	0
$367$	−31.8836	−1.66431	−0.832156	−	0.554541i	$-0.812894\pi$
$367$	−31.8836	−1.66431	−0.832156	+	0.554541i	$0.812894\pi$
$368$	0	0
$369$	12.7985	0.666264
$370$	0	0
$371$	13.3535	0.693277
$372$	0	0
$373$	−12.3535	−0.639638	−0.319819	−	0.947479i	$-0.603622\pi$
$373$	−12.3535	−0.639638	−0.319819	+	0.947479i	$0.603622\pi$
$374$	0	0
$375$	−41.1483	−2.12489
$376$	0	0
$377$	−8.15569	−0.420039
$378$	0	0
$379$	−28.5970	−1.46893	−0.734465	−	0.678646i	$-0.762567\pi$
$379$	−28.5970	−1.46893	−0.734465	+	0.678646i	$0.762567\pi$
$380$	0	0
$381$	−19.1978	−0.983531
$382$	0	0
$383$	−13.9084	−0.710686	−0.355343	−	0.934736i	$-0.615636\pi$
$383$	−13.9084	−0.710686	−0.355343	+	0.934736i	$0.615636\pi$
$384$	0	0
$385$	−8.79851	−0.448414
$386$	0	0
$387$	21.1520	1.07521
$388$	0	0
$389$	11.1520	0.565427	0.282714	−	0.959204i	$-0.408765\pi$
$389$	11.1520	0.565427	0.282714	+	0.959204i	$0.408765\pi$
$390$	0	0
$391$	0.222528	0.0112537
$392$	0	0
$393$	−5.15197	−0.259882
$394$	0	0
$395$	−21.9963	−1.10675
$396$	0	0
$397$	0.692344	0.0347478	0.0173739	−	0.999849i	$-0.494469\pi$
$397$	0.692344	0.0347478	0.0173739	+	0.999849i	$0.494469\pi$
$398$	0	0
$399$	6.39926	0.320364
$400$	0	0
$401$	−5.04580	−0.251975	−0.125988	−	0.992032i	$-0.540210\pi$
$401$	−5.04580	−0.251975	−0.125988	+	0.992032i	$0.540210\pi$
$402$	0	0
$403$	22.3535	1.11350
$404$	0	0
$405$	4.39926	0.218601
$406$	0	0
$407$	19.9505	0.988909
$408$	0	0
$409$	9.04580	0.447286	0.223643	−	0.974671i	$-0.428205\pi$
$409$	9.04580	0.447286	0.223643	+	0.974671i	$0.428205\pi$
$410$	0	0
$411$	−13.7985	−0.680630
$412$	0	0
$413$	−1.00000	−0.0492068
$414$	0	0
$415$	−61.4807	−3.01797
$416$	0	0
$417$	−6.35346	−0.311130
$418$	0	0
$419$	−10.2473	−0.500613	−0.250306	−	0.968167i	$-0.580531\pi$
$419$	−10.2473	−0.500613	−0.250306	+	0.968167i	$0.580531\pi$
$420$	0	0
$421$	−12.4844	−0.608452	−0.304226	−	0.952600i	$-0.598398\pi$
$421$	−12.4844	−0.608452	−0.304226	+	0.952600i	$0.598398\pi$
$422$	0	0
$423$	8.44506	0.410613
$424$	0	0
$425$	14.3535	0.696245
$426$	0	0
$427$	1.39926	0.0677148
$428$	0	0
$429$	−6.79851	−0.328235
$430$	0	0
$431$	23.5549	1.13460	0.567301	−	0.823511i	$-0.307988\pi$
$431$	23.5549	1.13460	0.567301	+	0.823511i	$0.307988\pi$
$432$	0	0
$433$	0.151969	0.00730314	0.00365157	−	0.999993i	$-0.498838\pi$
$433$	0.151969	0.00730314	0.00365157	+	0.999993i	$0.498838\pi$
$434$	0	0
$435$	−10.5549	−0.506071
$436$	0	0
$437$	−1.42402	−0.0681199
$438$	0	0
$439$	−14.3535	−0.685053	−0.342527	−	0.939508i	$-0.611283\pi$
$439$	−14.3535	−0.685053	−0.342527	+	0.939508i	$0.611283\pi$
$440$	0	0
$441$	12.0000	0.571429
$442$	0	0
$443$	−8.17301	−0.388311	−0.194156	−	0.980971i	$-0.562197\pi$
$443$	−8.17301	−0.388311	−0.194156	+	0.980971i	$0.562197\pi$
$444$	0	0
$445$	−20.1309	−0.954297
$446$	0	0
$447$	9.39926	0.444570
$448$	0	0
$449$	20.3077	0.958378	0.479189	−	0.877712i	$-0.340931\pi$
$449$	20.3077	0.958378	0.479189	+	0.877712i	$0.340931\pi$
$450$	0	0
$451$	12.7985	0.602658
$452$	0	0
$453$	6.82327	0.320585
$454$	0	0
$455$	−14.9542	−0.701064
$456$	0	0
$457$	42.3708	1.98202	0.991011	−	0.133783i	$-0.0427124\pi$
$457$	42.3708	1.98202	0.991011	+	0.133783i	$0.0427124\pi$
$458$	0	0
$459$	5.00000	0.233380
$460$	0	0
$461$	8.84803	0.412094	0.206047	−	0.978542i	$-0.433940\pi$
$461$	8.84803	0.412094	0.206047	+	0.978542i	$0.433940\pi$
$462$	0	0
$463$	−35.4166	−1.64595	−0.822974	−	0.568079i	$-0.807687\pi$
$463$	−35.4166	−1.64595	−0.822974	+	0.568079i	$0.807687\pi$
$464$	0	0
$465$	28.9294	1.34157
$466$	0	0
$467$	16.7985	0.777342	0.388671	−	0.921377i	$-0.372934\pi$
$467$	16.7985	0.777342	0.388671	+	0.921377i	$0.372934\pi$
$468$	0	0
$469$	−2.44506	−0.112902
$470$	0	0
$471$	13.7527	0.633692
$472$	0	0
$473$	21.1520	0.972569
$474$	0	0
$475$	−91.8514	−4.21443
$476$	0	0
$477$	−26.7069	−1.22283
$478$	0	0
$479$	18.7490	0.856663	0.428332	−	0.903622i	$-0.359101\pi$
$479$	18.7490	0.856663	0.428332	+	0.903622i	$0.359101\pi$
$480$	0	0
$481$	33.9084	1.54609
$482$	0	0
$483$	−0.222528	−0.0101254
$484$	0	0
$485$	−69.3002	−3.14676
$486$	0	0
$487$	13.2619	0.600952	0.300476	−	0.953789i	$-0.402854\pi$
$487$	13.2619	0.600952	0.300476	+	0.953789i	$0.402854\pi$
$488$	0	0
$489$	−11.2436	−0.508452
$490$	0	0
$491$	41.1483	1.85699	0.928497	−	0.371339i	$-0.121101\pi$
$491$	41.1483	1.85699	0.928497	+	0.371339i	$0.121101\pi$
$492$	0	0
$493$	2.39926	0.108057
$494$	0	0
$495$	17.5970	0.790927
$496$	0	0
$497$	6.35346	0.284991
$498$	0	0
$499$	3.44506	0.154222	0.0771110	−	0.997023i	$-0.475430\pi$
$499$	3.44506	0.154222	0.0771110	+	0.997023i	$0.475430\pi$
$500$	0	0
$501$	3.44506	0.153914
$502$	0	0
$503$	1.12721	0.0502598	0.0251299	−	0.999684i	$-0.492000\pi$
$503$	1.12721	0.0502598	0.0251299	+	0.999684i	$0.492000\pi$
$504$	0	0
$505$	−10.4624	−0.465570
$506$	0	0
$507$	1.44506	0.0641772
$508$	0	0
$509$	24.9963	1.10794	0.553970	−	0.832536i	$-0.313112\pi$
$509$	24.9963	1.10794	0.553970	+	0.832536i	$0.313112\pi$
$510$	0	0
$511$	2.00000	0.0884748
$512$	0	0
$513$	−31.9963	−1.41267
$514$	0	0
$515$	−52.1062	−2.29607
$516$	0	0
$517$	8.44506	0.371413
$518$	0	0
$519$	−12.9542	−0.568626
$520$	0	0
$521$	−7.15197	−0.313333	−0.156667	−	0.987652i	$-0.550075\pi$
$521$	−7.15197	−0.313333	−0.156667	+	0.987652i	$0.550075\pi$
$522$	0	0
$523$	−21.2894	−0.930919	−0.465460	−	0.885069i	$-0.654111\pi$
$523$	−21.2894	−0.930919	−0.465460	+	0.885069i	$0.654111\pi$
$524$	0	0
$525$	−14.3535	−0.626436
$526$	0	0
$527$	−6.57598	−0.286454
$528$	0	0
$529$	−22.9505	−0.997847
$530$	0	0
$531$	2.00000	0.0867926
$532$	0	0
$533$	21.7527	0.942215
$534$	0	0
$535$	43.1062	1.86364
$536$	0	0
$537$	4.60074	0.198537
$538$	0	0
$539$	12.0000	0.516877
$540$	0	0
$541$	−29.6364	−1.27417	−0.637083	−	0.770795i	$-0.719859\pi$
$541$	−29.6364	−1.27417	−0.637083	+	0.770795i	$0.719859\pi$
$542$	0	0
$543$	−19.1062	−0.819924
$544$	0	0
$545$	28.3535	1.21453
$546$	0	0
$547$	4.00000	0.171028	0.0855138	−	0.996337i	$-0.472747\pi$
$547$	4.00000	0.171028	0.0855138	+	0.996337i	$0.472747\pi$
$548$	0	0
$549$	−2.79851	−0.119438
$550$	0	0
$551$	−15.3535	−0.654079
$552$	0	0
$553$	−5.00000	−0.212622
$554$	0	0
$555$	43.8836	1.86276
$556$	0	0
$557$	−9.39554	−0.398102	−0.199051	−	0.979989i	$-0.563786\pi$
$557$	−9.39554	−0.398102	−0.199051	+	0.979989i	$0.563786\pi$
$558$	0	0
$559$	35.9505	1.52054
$560$	0	0
$561$	2.00000	0.0844401
$562$	0	0
$563$	13.1520	0.554289	0.277145	−	0.960828i	$-0.410612\pi$
$563$	13.1520	0.554289	0.277145	+	0.960828i	$0.410612\pi$
$564$	0	0
$565$	16.9121	0.711498
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−11.8836	−0.498188	−0.249094	−	0.968479i	$-0.580133\pi$
$569$	−11.8836	−0.498188	−0.249094	+	0.968479i	$0.580133\pi$
$570$	0	0
$571$	43.7920	1.83264	0.916320	−	0.400447i	$-0.131145\pi$
$571$	43.7920	1.83264	0.916320	+	0.400447i	$0.131145\pi$
$572$	0	0
$573$	−7.37450	−0.308074
$574$	0	0
$575$	3.19405	0.133201
$576$	0	0
$577$	25.0000	1.04076	0.520382	−	0.853934i	$-0.325790\pi$
$577$	25.0000	1.04076	0.520382	+	0.853934i	$0.325790\pi$
$578$	0	0
$579$	−11.1978	−0.465363
$580$	0	0
$581$	−13.9752	−0.579791
$582$	0	0
$583$	−26.7069	−1.10609
$584$	0	0
$585$	29.9084	1.23656
$586$	0	0
$587$	21.4661	0.886001	0.443000	−	0.896521i	$-0.353914\pi$
$587$	21.4661	0.886001	0.443000	+	0.896521i	$0.353914\pi$
$588$	0	0
$589$	42.0814	1.73393
$590$	0	0
$591$	−7.90840	−0.325308
$592$	0	0
$593$	6.46334	0.265418	0.132709	−	0.991155i	$-0.457632\pi$
$593$	6.46334	0.265418	0.132709	+	0.991155i	$0.457632\pi$
$594$	0	0
$595$	4.39926	0.180352
$596$	0	0
$597$	17.3535	0.710230
$598$	0	0
$599$	10.3077	0.421159	0.210580	−	0.977577i	$-0.432465\pi$
$599$	10.3077	0.421159	0.210580	+	0.977577i	$0.432465\pi$
$600$	0	0
$601$	33.2857	1.35775	0.678875	−	0.734254i	$-0.262468\pi$
$601$	33.2857	1.35775	0.678875	+	0.734254i	$0.262468\pi$
$602$	0	0
$603$	4.89011	0.199141
$604$	0	0
$605$	−30.7948	−1.25199
$606$	0	0
$607$	31.4871	1.27802	0.639012	−	0.769197i	$-0.279344\pi$
$607$	31.4871	1.27802	0.639012	+	0.769197i	$0.279344\pi$
$608$	0	0
$609$	−2.39926	−0.0972228
$610$	0	0
$611$	14.3535	0.580679
$612$	0	0
$613$	−38.1730	−1.54179	−0.770897	−	0.636960i	$-0.780192\pi$
$613$	−38.1730	−1.54179	−0.770897	+	0.636960i	$0.780192\pi$
$614$	0	0
$615$	28.1520	1.13520
$616$	0	0
$617$	−26.3077	−1.05911	−0.529553	−	0.848277i	$-0.677640\pi$
$617$	−26.3077	−1.05911	−0.529553	+	0.848277i	$0.677640\pi$
$618$	0	0
$619$	20.2436	0.813658	0.406829	−	0.913504i	$-0.366635\pi$
$619$	20.2436	0.813658	0.406829	+	0.913504i	$0.366635\pi$
$620$	0	0
$621$	1.11264	0.0446488
$622$	0	0
$623$	−4.57598	−0.183333
$624$	0	0
$625$	109.254	4.37018
$626$	0	0
$627$	−12.7985	−0.511123
$628$	0	0
$629$	−9.97524	−0.397739
$630$	0	0
$631$	−24.2619	−0.965849	−0.482925	−	0.875662i	$-0.660425\pi$
$631$	−24.2619	−0.965849	−0.482925	+	0.875662i	$0.660425\pi$
$632$	0	0
$633$	9.02104	0.358554
$634$	0	0
$635$	84.4559	3.35153
$636$	0	0
$637$	20.3955	0.808101
$638$	0	0
$639$	−12.7069	−0.502678
$640$	0	0
$641$	−5.50542	−0.217451	−0.108726	−	0.994072i	$-0.534677\pi$
$641$	−5.50542	−0.217451	−0.108726	+	0.994072i	$0.534677\pi$
$642$	0	0
$643$	34.4559	1.35881	0.679404	−	0.733764i	$-0.262238\pi$
$643$	34.4559	1.35881	0.679404	+	0.733764i	$0.262238\pi$
$644$	0	0
$645$	46.5265	1.83198
$646$	0	0
$647$	−38.4413	−1.51128	−0.755642	−	0.654984i	$-0.772675\pi$
$647$	−38.4413	−1.51128	−0.755642	+	0.654984i	$0.772675\pi$
$648$	0	0
$649$	2.00000	0.0785069
$650$	0	0
$651$	6.57598	0.257733
$652$	0	0
$653$	−41.5933	−1.62767	−0.813836	−	0.581095i	$-0.802625\pi$
$653$	−41.5933	−1.62767	−0.813836	+	0.581095i	$0.802625\pi$
$654$	0	0
$655$	22.6648	0.885588
$656$	0	0
$657$	−4.00000	−0.156055
$658$	0	0
$659$	31.6639	1.23345	0.616725	−	0.787179i	$-0.288459\pi$
$659$	31.6639	1.23345	0.616725	+	0.787179i	$0.288459\pi$
$660$	0	0
$661$	−31.7490	−1.23489	−0.617446	−	0.786613i	$-0.711833\pi$
$661$	−31.7490	−1.23489	−0.617446	+	0.786613i	$0.711833\pi$
$662$	0	0
$663$	3.39926	0.132016
$664$	0	0
$665$	−28.1520	−1.09169
$666$	0	0
$667$	0.533902	0.0206728
$668$	0	0
$669$	−1.15197	−0.0445377
$670$	0	0
$671$	−2.79851	−0.108035
$672$	0	0
$673$	−7.24081	−0.279113	−0.139556	−	0.990214i	$-0.544568\pi$
$673$	−7.24081	−0.279113	−0.139556	+	0.990214i	$0.544568\pi$
$674$	0	0
$675$	71.7673	2.76232
$676$	0	0
$677$	−8.09160	−0.310985	−0.155493	−	0.987837i	$-0.549697\pi$
$677$	−8.09160	−0.310985	−0.155493	+	0.987837i	$0.549697\pi$
$678$	0	0
$679$	−15.7527	−0.604534
$680$	0	0
$681$	−21.6218	−0.828549
$682$	0	0
$683$	−40.4596	−1.54814	−0.774072	−	0.633097i	$-0.781783\pi$
$683$	−40.4596	−1.54814	−0.774072	+	0.633097i	$0.781783\pi$
$684$	0	0
$685$	60.7032	2.31935
$686$	0	0
$687$	19.5302	0.745123
$688$	0	0
$689$	−45.3918	−1.72929
$690$	0	0
$691$	34.0814	1.29652	0.648259	−	0.761420i	$-0.275497\pi$
$691$	34.0814	1.29652	0.648259	+	0.761420i	$0.275497\pi$
$692$	0	0
$693$	4.00000	0.151947
$694$	0	0
$695$	27.9505	1.06022
$696$	0	0
$697$	−6.39926	−0.242389
$698$	0	0
$699$	−12.7738	−0.483148
$700$	0	0
$701$	−15.2829	−0.577227	−0.288614	−	0.957446i	$-0.593194\pi$
$701$	−15.2829	−0.577227	−0.288614	+	0.957446i	$0.593194\pi$
$702$	0	0
$703$	63.8341	2.40755
$704$	0	0
$705$	18.5760	0.699612
$706$	0	0
$707$	−2.37822	−0.0894420
$708$	0	0
$709$	29.9542	1.12495	0.562477	−	0.826813i	$-0.309849\pi$
$709$	29.9542	1.12495	0.562477	+	0.826813i	$0.309849\pi$
$710$	0	0
$711$	10.0000	0.375029
$712$	0	0
$713$	−1.46334	−0.0548026
$714$	0	0
$715$	29.9084	1.11851
$716$	0	0
$717$	8.39926	0.313676
$718$	0	0
$719$	47.3918	1.76742	0.883708	−	0.468038i	$-0.155039\pi$
$719$	47.3918	1.76742	0.883708	+	0.468038i	$0.155039\pi$
$720$	0	0
$721$	−11.8443	−0.441105
$722$	0	0
$723$	27.1978	1.01150
$724$	0	0
$725$	34.4376	1.27898
$726$	0	0
$727$	−9.90840	−0.367482	−0.183741	−	0.982975i	$-0.558821\pi$
$727$	−9.90840	−0.367482	−0.183741	+	0.982975i	$0.558821\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	−10.5760	−0.391167
$732$	0	0
$733$	17.0604	0.630139	0.315070	−	0.949069i	$-0.397972\pi$
$733$	17.0604	0.630139	0.315070	+	0.949069i	$0.397972\pi$
$734$	0	0
$735$	26.3955	0.973614
$736$	0	0
$737$	4.89011	0.180130
$738$	0	0
$739$	11.5329	0.424246	0.212123	−	0.977243i	$-0.431962\pi$
$739$	11.5329	0.424246	0.212123	+	0.977243i	$0.431962\pi$
$740$	0	0
$741$	−21.7527	−0.799106
$742$	0	0
$743$	−50.3039	−1.84547	−0.922736	−	0.385432i	$-0.874052\pi$
$743$	−50.3039	−1.84547	−0.922736	+	0.385432i	$0.874052\pi$
$744$	0	0
$745$	−41.3497	−1.51494
$746$	0	0
$747$	27.9505	1.02265
$748$	0	0
$749$	9.79851	0.358030
$750$	0	0
$751$	−12.8480	−0.468831	−0.234416	−	0.972136i	$-0.575318\pi$
$751$	−12.8480	−0.468831	−0.234416	+	0.972136i	$0.575318\pi$
$752$	0	0
$753$	17.5512	0.639602
$754$	0	0
$755$	−30.0173	−1.09244
$756$	0	0
$757$	−42.2436	−1.53537	−0.767684	−	0.640828i	$-0.778591\pi$
$757$	−42.2436	−1.53537	−0.767684	+	0.640828i	$0.778591\pi$
$758$	0	0
$759$	0.445057	0.0161545
$760$	0	0
$761$	−23.2619	−0.843242	−0.421621	−	0.906772i	$-0.638539\pi$
$761$	−23.2619	−0.843242	−0.421621	+	0.906772i	$0.638539\pi$
$762$	0	0
$763$	6.44506	0.233327
$764$	0	0
$765$	−8.79851	−0.318111
$766$	0	0
$767$	3.39926	0.122740
$768$	0	0
$769$	13.3993	0.483190	0.241595	−	0.970377i	$-0.422330\pi$
$769$	13.3993	0.483190	0.241595	+	0.970377i	$0.422330\pi$
$770$	0	0
$771$	8.15197	0.293586
$772$	0	0
$773$	−33.4386	−1.20270	−0.601351	−	0.798985i	$-0.705371\pi$
$773$	−33.4386	−1.20270	−0.601351	+	0.798985i	$0.705371\pi$
$774$	0	0
$775$	−94.3881	−3.39052
$776$	0	0
$777$	9.97524	0.357860
$778$	0	0
$779$	40.9505	1.46720
$780$	0	0
$781$	−12.7069	−0.454689
$782$	0	0
$783$	11.9963	0.428712
$784$	0	0
$785$	−60.5017	−2.15940
$786$	0	0
$787$	−25.9084	−0.923535	−0.461767	−	0.887001i	$-0.652785\pi$
$787$	−25.9084	−0.923535	−0.461767	+	0.887001i	$0.652785\pi$
$788$	0	0
$789$	4.39926	0.156618
$790$	0	0
$791$	3.84431	0.136688
$792$	0	0
$793$	−4.75643	−0.168906
$794$	0	0
$795$	−58.7453	−2.08348
$796$	0	0
$797$	36.1483	1.28044	0.640218	−	0.768193i	$-0.278844\pi$
$797$	36.1483	1.28044	0.640218	+	0.768193i	$0.278844\pi$
$798$	0	0
$799$	−4.22253	−0.149382
$800$	0	0
$801$	9.15197	0.323369
$802$	0	0
$803$	−4.00000	−0.141157
$804$	0	0
$805$	0.978959	0.0345038
$806$	0	0
$807$	−10.7317	−0.377773
$808$	0	0
$809$	−5.08513	−0.178784	−0.0893918	−	0.995997i	$-0.528492\pi$
$809$	−5.08513	−0.178784	−0.0893918	+	0.995997i	$0.528492\pi$
$810$	0	0
$811$	−44.2792	−1.55485	−0.777426	−	0.628974i	$-0.783475\pi$
$811$	−44.2792	−1.55485	−0.777426	+	0.628974i	$0.783475\pi$
$812$	0	0
$813$	13.5091	0.473786
$814$	0	0
$815$	49.4633	1.73263
$816$	0	0
$817$	67.6784	2.36777
$818$	0	0
$819$	6.79851	0.237559
$820$	0	0
$821$	26.3287	0.918878	0.459439	−	0.888209i	$-0.348051\pi$
$821$	26.3287	0.918878	0.459439	+	0.888209i	$0.348051\pi$
$822$	0	0
$823$	16.2473	0.566345	0.283172	−	0.959069i	$-0.408613\pi$
$823$	16.2473	0.566345	0.283172	+	0.959069i	$0.408613\pi$
$824$	0	0
$825$	28.7069	0.999446
$826$	0	0
$827$	−19.2436	−0.669164	−0.334582	−	0.942367i	$-0.608595\pi$
$827$	−19.2436	−0.669164	−0.334582	+	0.942367i	$0.608595\pi$
$828$	0	0
$829$	49.6574	1.72467	0.862336	−	0.506336i	$-0.169000\pi$
$829$	49.6574	1.72467	0.862336	+	0.506336i	$0.169000\pi$
$830$	0	0
$831$	−5.60074	−0.194288
$832$	0	0
$833$	−6.00000	−0.207888
$834$	0	0
$835$	−15.1557	−0.524484
$836$	0	0
$837$	−32.8799	−1.13650
$838$	0	0
$839$	−7.57970	−0.261680	−0.130840	−	0.991403i	$-0.541767\pi$
$839$	−7.57970	−0.261680	−0.130840	+	0.991403i	$0.541767\pi$
$840$	0	0
$841$	−23.2436	−0.801502
$842$	0	0
$843$	−30.0604	−1.03533
$844$	0	0
$845$	−6.35717	−0.218693
$846$	0	0
$847$	−7.00000	−0.240523
$848$	0	0
$849$	13.3077	0.456718
$850$	0	0
$851$	−2.21977	−0.0760929
$852$	0	0
$853$	−29.9468	−1.02536	−0.512679	−	0.858580i	$-0.671347\pi$
$853$	−29.9468	−1.02536	−0.512679	+	0.858580i	$0.671347\pi$
$854$	0	0
$855$	56.3039	1.92555
$856$	0	0
$857$	1.07056	0.0365696	0.0182848	−	0.999833i	$-0.494179\pi$
$857$	1.07056	0.0365696	0.0182848	+	0.999833i	$0.494179\pi$
$858$	0	0
$859$	11.9112	0.406403	0.203202	−	0.979137i	$-0.434865\pi$
$859$	11.9112	0.406403	0.203202	+	0.979137i	$0.434865\pi$
$860$	0	0
$861$	6.39926	0.218086
$862$	0	0
$863$	31.9084	1.08617	0.543087	−	0.839676i	$-0.317255\pi$
$863$	31.9084	1.08617	0.543087	+	0.839676i	$0.317255\pi$
$864$	0	0
$865$	56.9888	1.93768
$866$	0	0
$867$	−1.00000	−0.0339618
$868$	0	0
$869$	10.0000	0.339227
$870$	0	0
$871$	8.31137	0.281620
$872$	0	0
$873$	31.5054	1.06630
$874$	0	0
$875$	41.1483	1.39106
$876$	0	0
$877$	−6.44134	−0.217509	−0.108754	−	0.994069i	$-0.534686\pi$
$877$	−6.44134	−0.217509	−0.108754	+	0.994069i	$0.534686\pi$
$878$	0	0
$879$	29.0421	0.979565
$880$	0	0
$881$	−33.6639	−1.13416	−0.567082	−	0.823661i	$-0.691928\pi$
$881$	−33.6639	−1.13416	−0.567082	+	0.823661i	$0.691928\pi$
$882$	0	0
$883$	−47.4596	−1.59714	−0.798572	−	0.601900i	$-0.794411\pi$
$883$	−47.4596	−1.59714	−0.798572	+	0.601900i	$0.794411\pi$
$884$	0	0
$885$	4.39926	0.147879
$886$	0	0
$887$	−18.8406	−0.632605	−0.316303	−	0.948658i	$-0.602441\pi$
$887$	−18.8406	−0.632605	−0.316303	+	0.948658i	$0.602441\pi$
$888$	0	0
$889$	19.1978	0.643873
$890$	0	0
$891$	−2.00000	−0.0670025
$892$	0	0
$893$	27.0210	0.904225
$894$	0	0
$895$	−20.2399	−0.676544
$896$	0	0
$897$	0.756431	0.0252565
$898$	0	0
$899$	−15.7775	−0.526208
$900$	0	0
$901$	13.3535	0.444868
$902$	0	0
$903$	10.5760	0.351947
$904$	0	0
$905$	84.0529	2.79401
$906$	0	0
$907$	9.40297	0.312221	0.156110	−	0.987740i	$-0.450104\pi$
$907$	9.40297	0.312221	0.156110	+	0.987740i	$0.450104\pi$
$908$	0	0
$909$	4.75643	0.157761
$910$	0	0
$911$	30.7307	1.01815	0.509077	−	0.860721i	$-0.329987\pi$
$911$	30.7307	1.01815	0.509077	+	0.860721i	$0.329987\pi$
$912$	0	0
$913$	27.9505	0.925026
$914$	0	0
$915$	−6.15569	−0.203501
$916$	0	0
$917$	5.15197	0.170133
$918$	0	0
$919$	47.8836	1.57954	0.789768	−	0.613406i	$-0.210201\pi$
$919$	47.8836	1.57954	0.789768	+	0.613406i	$0.210201\pi$
$920$	0	0
$921$	−5.24729	−0.172904
$922$	0	0
$923$	−21.5970	−0.710875
$924$	0	0
$925$	−143.179	−4.70770
$926$	0	0
$927$	23.6886	0.778037
$928$	0	0
$929$	−49.9505	−1.63882	−0.819411	−	0.573206i	$-0.805699\pi$
$929$	−49.9505	−1.63882	−0.819411	+	0.573206i	$0.805699\pi$
$930$	0	0
$931$	38.3955	1.25836
$932$	0	0
$933$	15.0000	0.491078
$934$	0	0
$935$	−8.79851	−0.287742
$936$	0	0
$937$	48.7911	1.59393	0.796967	−	0.604022i	$-0.206436\pi$
$937$	48.7911	1.59393	0.796967	+	0.604022i	$0.206436\pi$
$938$	0	0
$939$	−6.60074	−0.215407
$940$	0	0
$941$	58.8552	1.91862	0.959312	−	0.282349i	$-0.0911136\pi$
$941$	58.8552	1.91862	0.959312	+	0.282349i	$0.0911136\pi$
$942$	0	0
$943$	−1.42402	−0.0463723
$944$	0	0
$945$	21.9963	0.715539
$946$	0	0
$947$	−33.3955	−1.08521	−0.542605	−	0.839988i	$-0.682562\pi$
$947$	−33.3955	−1.08521	−0.542605	+	0.839988i	$0.682562\pi$
$948$	0	0
$949$	−6.79851	−0.220689
$950$	0	0
$951$	−13.2436	−0.429452
$952$	0	0
$953$	−5.73814	−0.185877	−0.0929384	−	0.995672i	$-0.529626\pi$
$953$	−5.73814	−0.185877	−0.0929384	+	0.995672i	$0.529626\pi$
$954$	0	0
$955$	32.4423	1.04981
$956$	0	0
$957$	4.79851	0.155114
$958$	0	0
$959$	13.7985	0.445577
$960$	0	0
$961$	12.2436	0.394954
$962$	0	0
$963$	−19.5970	−0.631505
$964$	0	0
$965$	49.2619	1.58580
$966$	0	0
$967$	20.0668	0.645306	0.322653	−	0.946517i	$-0.395425\pi$
$967$	20.0668	0.645306	0.322653	+	0.946517i	$0.395425\pi$
$968$	0	0
$969$	6.39926	0.205574
$970$	0	0
$971$	28.3077	0.908436	0.454218	−	0.890891i	$-0.349919\pi$
$971$	28.3077	0.908436	0.454218	+	0.890891i	$0.349919\pi$
$972$	0	0
$973$	6.35346	0.203682
$974$	0	0
$975$	48.7911	1.56256
$976$	0	0
$977$	−51.0631	−1.63365	−0.816827	−	0.576883i	$-0.804269\pi$
$977$	−51.0631	−1.63365	−0.816827	+	0.576883i	$0.804269\pi$
$978$	0	0
$979$	9.15197	0.292498
$980$	0	0
$981$	−12.8901	−0.411550
$982$	0	0
$983$	30.8406	0.983662	0.491831	−	0.870691i	$-0.336328\pi$
$983$	30.8406	0.983662	0.491831	+	0.870691i	$0.336328\pi$
$984$	0	0
$985$	34.7911	1.10854
$986$	0	0
$987$	4.22253	0.134405
$988$	0	0
$989$	−2.35346	−0.0748355
$990$	0	0
$991$	32.4203	1.02986	0.514932	−	0.857231i	$-0.327817\pi$
$991$	32.4203	1.02986	0.514932	+	0.857231i	$0.327817\pi$
$992$	0	0
$993$	26.3993	0.837755
$994$	0	0
$995$	−76.3423	−2.42021
$996$	0	0
$997$	47.1903	1.49453	0.747266	−	0.664525i	$-0.231366\pi$
$997$	47.1903	1.49453	0.747266	+	0.664525i	$0.231366\pi$
$998$	0	0
$999$	−49.8762	−1.57801

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.a.f.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.a.f.1.3	✓	3	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:	\( N \)	\(=\)	\( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4012.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +4.39926 q^{5} +1.00000 q^{7} -2.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +4.39926 q^{5} +1.00000 q^{7} -2.00000 q^{9} -2.00000 q^{11} -3.39926 q^{13} -4.39926 q^{15} +1.00000 q^{17} -6.39926 q^{19} -1.00000 q^{21} +0.222528 q^{23} +14.3535 q^{25} +5.00000 q^{27} +2.39926 q^{29} -6.57598 q^{31} +2.00000 q^{33} +4.39926 q^{35} -9.97524 q^{37} +3.39926 q^{39} -6.39926 q^{41} -10.5760 q^{43} -8.79851 q^{45} -4.22253 q^{47} -6.00000 q^{49} -1.00000 q^{51} +13.3535 q^{53} -8.79851 q^{55} +6.39926 q^{57} -1.00000 q^{59} +1.39926 q^{61} -2.00000 q^{63} -14.9542 q^{65} -2.44506 q^{67} -0.222528 q^{69} +6.35346 q^{71} +2.00000 q^{73} -14.3535 q^{75} -2.00000 q^{77} -5.00000 q^{79} +1.00000 q^{81} -13.9752 q^{83} +4.39926 q^{85} -2.39926 q^{87} -4.57598 q^{89} -3.39926 q^{91} +6.57598 q^{93} -28.1520 q^{95} -15.7527 q^{97} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + q^{5} + 3 q^{7} - 6 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 + q^5 + 3 * q^7 - 6 * q^9	\( 3 q - 3 q^{3} + q^{5} + 3 q^{7} - 6 q^{9} - 6 q^{11} + 2 q^{13} - q^{15} + 3 q^{17} - 7 q^{19} - 3 q^{21} + 20 q^{25} + 15 q^{27} - 5 q^{29} + 4 q^{31} + 6 q^{33} + q^{35} + 6 q^{37} - 2 q^{39} - 7 q^{41} - 8 q^{43} - 2 q^{45} - 12 q^{47} - 18 q^{49} - 3 q^{51} + 17 q^{53} - 2 q^{55} + 7 q^{57} - 3 q^{59} - 8 q^{61} - 6 q^{63} - 34 q^{65} - 6 q^{67} - 4 q^{71} + 6 q^{73} - 20 q^{75} - 6 q^{77} - 15 q^{79} + 3 q^{81} - 6 q^{83} + q^{85} + 5 q^{87} + 10 q^{89} + 2 q^{91} - 4 q^{93} - 37 q^{95} - 12 q^{97} + 12 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + q^5 + 3 * q^7 - 6 * q^9 - 6 * q^11 + 2 * q^13 - q^15 + 3 * q^17 - 7 * q^19 - 3 * q^21 + 20 * q^25 + 15 * q^27 - 5 * q^29 + 4 * q^31 + 6 * q^33 + q^35 + 6 * q^37 - 2 * q^39 - 7 * q^41 - 8 * q^43 - 2 * q^45 - 12 * q^47 - 18 * q^49 - 3 * q^51 + 17 * q^53 - 2 * q^55 + 7 * q^57 - 3 * q^59 - 8 * q^61 - 6 * q^63 - 34 * q^65 - 6 * q^67 - 4 * q^71 + 6 * q^73 - 20 * q^75 - 6 * q^77 - 15 * q^79 + 3 * q^81 - 6 * q^83 + q^85 + 5 * q^87 + 10 * q^89 + 2 * q^91 - 4 * q^93 - 37 * q^95 - 12 * q^97 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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