LMFDB - Embedded newform 4012.2.a.j.1.20

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:	$0$
Dimension:	$21$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
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)

$f(q)$	$=$	$q+3.31779 q^{3} -0.252330 q^{5} +4.50552 q^{7} +8.00775 q^{9} +O(q^{10})$	$q+3.31779 q^{3} -0.252330 q^{5} +4.50552 q^{7} +8.00775 q^{9} -3.72793 q^{11} +1.68271 q^{13} -0.837180 q^{15} +1.00000 q^{17} -7.83723 q^{19} +14.9484 q^{21} +3.99701 q^{23} -4.93633 q^{25} +16.6147 q^{27} +6.14458 q^{29} +8.20425 q^{31} -12.3685 q^{33} -1.13688 q^{35} -4.08058 q^{37} +5.58287 q^{39} +2.05483 q^{41} -9.83147 q^{43} -2.02060 q^{45} +6.19779 q^{47} +13.2997 q^{49} +3.31779 q^{51} -2.82986 q^{53} +0.940669 q^{55} -26.0023 q^{57} +1.00000 q^{59} +0.798137 q^{61} +36.0791 q^{63} -0.424598 q^{65} +2.51151 q^{67} +13.2613 q^{69} +2.18932 q^{71} +11.6656 q^{73} -16.3777 q^{75} -16.7962 q^{77} -8.17539 q^{79} +31.1008 q^{81} +6.98848 q^{83} -0.252330 q^{85} +20.3864 q^{87} -13.6916 q^{89} +7.58146 q^{91} +27.2200 q^{93} +1.97757 q^{95} -7.98239 q^{97} -29.8523 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 21 q + 9 q^{3} + q^{5} + 11 q^{7} + 26 q^{9}+O(q^{10}) $ 21 * q + 9 * q^3 + q^5 + 11 * q^7 + 26 * q^9	$ 21 q + 9 q^{3} + q^{5} + 11 q^{7} + 26 q^{9} + 12 q^{11} - 4 q^{13} + 9 q^{15} + 21 q^{17} + 4 q^{19} + 8 q^{21} + 19 q^{23} + 26 q^{25} + 33 q^{27} - q^{29} + 13 q^{31} + 11 q^{33} + 15 q^{35} - 4 q^{37} + 18 q^{39} + 9 q^{41} + 7 q^{43} + 7 q^{45} + 33 q^{47} + 36 q^{49} + 9 q^{51} + 7 q^{53} + 12 q^{55} + 26 q^{57} + 21 q^{59} + 3 q^{61} + 51 q^{63} + q^{65} + 8 q^{69} + 55 q^{71} + 22 q^{73} + 14 q^{75} - 15 q^{77} + 28 q^{79} + 25 q^{81} + 54 q^{83} + q^{85} + 34 q^{87} + 30 q^{89} + 35 q^{91} - 5 q^{93} + 42 q^{95} + 9 q^{97} + 20 q^{99}+O(q^{100}) $ 21 * q + 9 * q^3 + q^5 + 11 * q^7 + 26 * q^9 + 12 * q^11 - 4 * q^13 + 9 * q^15 + 21 * q^17 + 4 * q^19 + 8 * q^21 + 19 * q^23 + 26 * q^25 + 33 * q^27 - q^29 + 13 * q^31 + 11 * q^33 + 15 * q^35 - 4 * q^37 + 18 * q^39 + 9 * q^41 + 7 * q^43 + 7 * q^45 + 33 * q^47 + 36 * q^49 + 9 * q^51 + 7 * q^53 + 12 * q^55 + 26 * q^57 + 21 * q^59 + 3 * q^61 + 51 * q^63 + q^65 + 8 * q^69 + 55 * q^71 + 22 * q^73 + 14 * q^75 - 15 * q^77 + 28 * q^79 + 25 * q^81 + 54 * q^83 + q^85 + 34 * q^87 + 30 * q^89 + 35 * q^91 - 5 * q^93 + 42 * q^95 + 9 * q^97 + 20 * 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
$3$	3.31779	1.91553	0.957764	−	0.287554i	$-0.0928421\pi$
$3$	3.31779	1.91553	0.957764	+	0.287554i	$0.0928421\pi$
$4$	0	0
$5$	−0.252330	−0.112846	−0.0564228	−	0.998407i	$-0.517969\pi$
$5$	−0.252330	−0.112846	−0.0564228	+	0.998407i	$0.517969\pi$
$6$	0	0
$7$	4.50552	1.70292	0.851462	−	0.524416i	$-0.175716\pi$
$7$	4.50552	1.70292	0.851462	+	0.524416i	$0.175716\pi$
$8$	0	0
$9$	8.00775	2.66925
$10$	0	0
$11$	−3.72793	−1.12401	−0.562006	−	0.827133i	$-0.689970\pi$
$11$	−3.72793	−1.12401	−0.562006	+	0.827133i	$0.689970\pi$
$12$	0	0
$13$	1.68271	0.466699	0.233349	−	0.972393i	$-0.425031\pi$
$13$	1.68271	0.466699	0.233349	+	0.972393i	$0.425031\pi$
$14$	0	0
$15$	−0.837180	−0.216159
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−7.83723	−1.79798	−0.898992	−	0.437966i	$-0.855699\pi$
$19$	−7.83723	−1.79798	−0.898992	+	0.437966i	$0.855699\pi$
$20$	0	0
$21$	14.9484	3.26200
$22$	0	0
$23$	3.99701	0.833435	0.416717	−	0.909036i	$-0.363180\pi$
$23$	3.99701	0.833435	0.416717	+	0.909036i	$0.363180\pi$
$24$	0	0
$25$	−4.93633	−0.987266
$26$	0	0
$27$	16.6147	3.19750
$28$	0	0
$29$	6.14458	1.14102	0.570510	−	0.821291i	$-0.306746\pi$
$29$	6.14458	1.14102	0.570510	+	0.821291i	$0.306746\pi$
$30$	0	0
$31$	8.20425	1.47353	0.736764	−	0.676150i	$-0.236353\pi$
$31$	8.20425	1.47353	0.736764	+	0.676150i	$0.236353\pi$
$32$	0	0
$33$	−12.3685	−2.15308
$34$	0	0
$35$	−1.13688	−0.192167
$36$	0	0
$37$	−4.08058	−0.670843	−0.335421	−	0.942068i	$-0.608879\pi$
$37$	−4.08058	−0.670843	−0.335421	+	0.942068i	$0.608879\pi$
$38$	0	0
$39$	5.58287	0.893975
$40$	0	0
$41$	2.05483	0.320911	0.160456	−	0.987043i	$-0.448704\pi$
$41$	2.05483	0.320911	0.160456	+	0.987043i	$0.448704\pi$
$42$	0	0
$43$	−9.83147	−1.49929	−0.749643	−	0.661843i	$-0.769775\pi$
$43$	−9.83147	−1.49929	−0.749643	+	0.661843i	$0.769775\pi$
$44$	0	0
$45$	−2.02060	−0.301213
$46$	0	0
$47$	6.19779	0.904041	0.452020	−	0.892008i	$-0.350703\pi$
$47$	6.19779	0.904041	0.452020	+	0.892008i	$0.350703\pi$
$48$	0	0
$49$	13.2997	1.89995
$50$	0	0
$51$	3.31779	0.464584
$52$	0	0
$53$	−2.82986	−0.388711	−0.194355	−	0.980931i	$-0.562262\pi$
$53$	−2.82986	−0.388711	−0.194355	+	0.980931i	$0.562262\pi$
$54$	0	0
$55$	0.940669	0.126840
$56$	0	0
$57$	−26.0023	−3.44409
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	0.798137	0.102191	0.0510955	−	0.998694i	$-0.483729\pi$
$61$	0.798137	0.102191	0.0510955	+	0.998694i	$0.483729\pi$
$62$	0	0
$63$	36.0791	4.54553
$64$	0	0
$65$	−0.424598	−0.0526649
$66$	0	0
$67$	2.51151	0.306829	0.153415	−	0.988162i	$-0.450973\pi$
$67$	2.51151	0.306829	0.153415	+	0.988162i	$0.450973\pi$
$68$	0	0
$69$	13.2613	1.59647
$70$	0	0
$71$	2.18932	0.259824	0.129912	−	0.991526i	$-0.458531\pi$
$71$	2.18932	0.259824	0.129912	+	0.991526i	$0.458531\pi$
$72$	0	0
$73$	11.6656	1.36535	0.682677	−	0.730720i	$-0.260816\pi$
$73$	11.6656	1.36535	0.682677	+	0.730720i	$0.260816\pi$
$74$	0	0
$75$	−16.3777	−1.89114
$76$	0	0
$77$	−16.7962	−1.91411
$78$	0	0
$79$	−8.17539	−0.919803	−0.459902	−	0.887970i	$-0.652115\pi$
$79$	−8.17539	−0.919803	−0.459902	+	0.887970i	$0.652115\pi$
$80$	0	0
$81$	31.1008	3.45565
$82$	0	0
$83$	6.98848	0.767085	0.383543	−	0.923523i	$-0.374704\pi$
$83$	6.98848	0.767085	0.383543	+	0.923523i	$0.374704\pi$
$84$	0	0
$85$	−0.252330	−0.0273691
$86$	0	0
$87$	20.3864	2.18566
$88$	0	0
$89$	−13.6916	−1.45131	−0.725656	−	0.688058i	$-0.758463\pi$
$89$	−13.6916	−1.45131	−0.725656	+	0.688058i	$0.758463\pi$
$90$	0	0
$91$	7.58146	0.794753
$92$	0	0
$93$	27.2200	2.82258
$94$	0	0
$95$	1.97757	0.202894
$96$	0	0
$97$	−7.98239	−0.810489	−0.405244	−	0.914208i	$-0.632814\pi$
$97$	−7.98239	−0.810489	−0.405244	+	0.914208i	$0.632814\pi$
$98$	0	0
$99$	−29.8523	−3.00027
$100$	0	0
$101$	−4.51657	−0.449416	−0.224708	−	0.974426i	$-0.572143\pi$
$101$	−4.51657	−0.449416	−0.224708	+	0.974426i	$0.572143\pi$
$102$	0	0
$103$	15.3514	1.51261	0.756307	−	0.654217i	$-0.227002\pi$
$103$	15.3514	1.51261	0.756307	+	0.654217i	$0.227002\pi$
$104$	0	0
$105$	−3.77193	−0.368102
$106$	0	0
$107$	7.90135	0.763852	0.381926	−	0.924193i	$-0.375261\pi$
$107$	7.90135	0.763852	0.381926	+	0.924193i	$0.375261\pi$
$108$	0	0
$109$	−13.9016	−1.33153	−0.665767	−	0.746160i	$-0.731896\pi$
$109$	−13.9016	−1.33153	−0.665767	+	0.746160i	$0.731896\pi$
$110$	0	0
$111$	−13.5385	−1.28502
$112$	0	0
$113$	−15.3995	−1.44866	−0.724330	−	0.689453i	$-0.757851\pi$
$113$	−15.3995	−1.44866	−0.724330	+	0.689453i	$0.757851\pi$
$114$	0	0
$115$	−1.00857	−0.0940494
$116$	0	0
$117$	13.4747	1.24574
$118$	0	0
$119$	4.50552	0.413020
$120$	0	0
$121$	2.89745	0.263405
$122$	0	0
$123$	6.81752	0.614715
$124$	0	0
$125$	2.50724	0.224254
$126$	0	0
$127$	−8.07523	−0.716561	−0.358280	−	0.933614i	$-0.616637\pi$
$127$	−8.07523	−0.716561	−0.358280	+	0.933614i	$0.616637\pi$
$128$	0	0
$129$	−32.6188	−2.87192
$130$	0	0
$131$	−9.58784	−0.837694	−0.418847	−	0.908057i	$-0.637566\pi$
$131$	−9.58784	−0.837694	−0.418847	+	0.908057i	$0.637566\pi$
$132$	0	0
$133$	−35.3108	−3.06183
$134$	0	0
$135$	−4.19239	−0.360823
$136$	0	0
$137$	21.8427	1.86615	0.933076	−	0.359680i	$-0.117114\pi$
$137$	21.8427	1.86615	0.933076	+	0.359680i	$0.117114\pi$
$138$	0	0
$139$	−3.67763	−0.311933	−0.155966	−	0.987762i	$-0.549849\pi$
$139$	−3.67763	−0.311933	−0.155966	+	0.987762i	$0.549849\pi$
$140$	0	0
$141$	20.5630	1.73172
$142$	0	0
$143$	−6.27301	−0.524575
$144$	0	0
$145$	−1.55046	−0.128759
$146$	0	0
$147$	44.1256	3.63941
$148$	0	0
$149$	3.19293	0.261575	0.130787	−	0.991410i	$-0.458249\pi$
$149$	3.19293	0.261575	0.130787	+	0.991410i	$0.458249\pi$
$150$	0	0
$151$	−18.7348	−1.52461	−0.762306	−	0.647216i	$-0.775933\pi$
$151$	−18.7348	−1.52461	−0.762306	+	0.647216i	$0.775933\pi$
$152$	0	0
$153$	8.00775	0.647388
$154$	0	0
$155$	−2.07018	−0.166281
$156$	0	0
$157$	−2.95159	−0.235562	−0.117781	−	0.993040i	$-0.537578\pi$
$157$	−2.95159	−0.235562	−0.117781	+	0.993040i	$0.537578\pi$
$158$	0	0
$159$	−9.38888	−0.744587
$160$	0	0
$161$	18.0086	1.41928
$162$	0	0
$163$	11.5514	0.904776	0.452388	−	0.891821i	$-0.350572\pi$
$163$	11.5514	0.904776	0.452388	+	0.891821i	$0.350572\pi$
$164$	0	0
$165$	3.12095	0.242965
$166$	0	0
$167$	−21.7060	−1.67966	−0.839832	−	0.542847i	$-0.817346\pi$
$167$	−21.7060	−1.67966	−0.839832	+	0.542847i	$0.817346\pi$
$168$	0	0
$169$	−10.1685	−0.782192
$170$	0	0
$171$	−62.7586	−4.79927
$172$	0	0
$173$	−20.0547	−1.52473	−0.762364	−	0.647148i	$-0.775961\pi$
$173$	−20.0547	−1.52473	−0.762364	+	0.647148i	$0.775961\pi$
$174$	0	0
$175$	−22.2407	−1.68124
$176$	0	0
$177$	3.31779	0.249381
$178$	0	0
$179$	−7.28101	−0.544208	−0.272104	−	0.962268i	$-0.587720\pi$
$179$	−7.28101	−0.544208	−0.272104	+	0.962268i	$0.587720\pi$
$180$	0	0
$181$	−1.21620	−0.0903991	−0.0451995	−	0.998978i	$-0.514392\pi$
$181$	−1.21620	−0.0903991	−0.0451995	+	0.998978i	$0.514392\pi$
$182$	0	0
$183$	2.64805	0.195750
$184$	0	0
$185$	1.02965	0.0757016
$186$	0	0
$187$	−3.72793	−0.272613
$188$	0	0
$189$	74.8577	5.44510
$190$	0	0
$191$	−10.3691	−0.750279	−0.375139	−	0.926968i	$-0.622405\pi$
$191$	−10.3691	−0.750279	−0.375139	+	0.926968i	$0.622405\pi$
$192$	0	0
$193$	18.9046	1.36078	0.680392	−	0.732848i	$-0.261810\pi$
$193$	18.9046	1.36078	0.680392	+	0.732848i	$0.261810\pi$
$194$	0	0
$195$	−1.40873	−0.100881
$196$	0	0
$197$	4.65335	0.331537	0.165769	−	0.986165i	$-0.446989\pi$
$197$	4.65335	0.331537	0.165769	+	0.986165i	$0.446989\pi$
$198$	0	0
$199$	−2.66417	−0.188858	−0.0944291	−	0.995532i	$-0.530103\pi$
$199$	−2.66417	−0.188858	−0.0944291	+	0.995532i	$0.530103\pi$
$200$	0	0
$201$	8.33266	0.587741
$202$	0	0
$203$	27.6845	1.94307
$204$	0	0
$205$	−0.518497	−0.0362134
$206$	0	0
$207$	32.0071	2.22465
$208$	0	0
$209$	29.2166	2.02096
$210$	0	0
$211$	−13.4674	−0.927137	−0.463568	−	0.886061i	$-0.653431\pi$
$211$	−13.4674	−0.927137	−0.463568	+	0.886061i	$0.653431\pi$
$212$	0	0
$213$	7.26370	0.497700
$214$	0	0
$215$	2.48078	0.169188
$216$	0	0
$217$	36.9644	2.50931
$218$	0	0
$219$	38.7040	2.61538
$220$	0	0
$221$	1.68271	0.113191
$222$	0	0
$223$	−23.3225	−1.56179	−0.780896	−	0.624661i	$-0.785237\pi$
$223$	−23.3225	−1.56179	−0.780896	+	0.624661i	$0.785237\pi$
$224$	0	0
$225$	−39.5289	−2.63526
$226$	0	0
$227$	−3.79415	−0.251826	−0.125913	−	0.992041i	$-0.540186\pi$
$227$	−3.79415	−0.251826	−0.125913	+	0.992041i	$0.540186\pi$
$228$	0	0
$229$	11.1363	0.735911	0.367955	−	0.929843i	$-0.380058\pi$
$229$	11.1363	0.735911	0.367955	+	0.929843i	$0.380058\pi$
$230$	0	0
$231$	−55.7265	−3.66653
$232$	0	0
$233$	19.6741	1.28890	0.644448	−	0.764648i	$-0.277087\pi$
$233$	19.6741	1.28890	0.644448	+	0.764648i	$0.277087\pi$
$234$	0	0
$235$	−1.56389	−0.102017
$236$	0	0
$237$	−27.1243	−1.76191
$238$	0	0
$239$	8.58561	0.555357	0.277679	−	0.960674i	$-0.410435\pi$
$239$	8.58561	0.555357	0.277679	+	0.960674i	$0.410435\pi$
$240$	0	0
$241$	−20.7503	−1.33664	−0.668322	−	0.743872i	$-0.732987\pi$
$241$	−20.7503	−1.33664	−0.668322	+	0.743872i	$0.732987\pi$
$242$	0	0
$243$	53.3421	3.42190
$244$	0	0
$245$	−3.35591	−0.214401
$246$	0	0
$247$	−13.1878	−0.839117
$248$	0	0
$249$	23.1863	1.46937
$250$	0	0
$251$	11.1588	0.704335	0.352168	−	0.935937i	$-0.385445\pi$
$251$	11.1588	0.704335	0.352168	+	0.935937i	$0.385445\pi$
$252$	0	0
$253$	−14.9006	−0.936791
$254$	0	0
$255$	−0.837180	−0.0524262
$256$	0	0
$257$	28.3351	1.76750	0.883748	−	0.467962i	$-0.155012\pi$
$257$	28.3351	1.76750	0.883748	+	0.467962i	$0.155012\pi$
$258$	0	0
$259$	−18.3851	−1.14239
$260$	0	0
$261$	49.2042	3.04567
$262$	0	0
$263$	20.3897	1.25728	0.628641	−	0.777696i	$-0.283612\pi$
$263$	20.3897	1.25728	0.628641	+	0.777696i	$0.283612\pi$
$264$	0	0
$265$	0.714059	0.0438643
$266$	0	0
$267$	−45.4260	−2.78003
$268$	0	0
$269$	−15.9653	−0.973424	−0.486712	−	0.873563i	$-0.661804\pi$
$269$	−15.9653	−0.973424	−0.486712	+	0.873563i	$0.661804\pi$
$270$	0	0
$271$	−3.85501	−0.234175	−0.117088	−	0.993122i	$-0.537356\pi$
$271$	−3.85501	−0.234175	−0.117088	+	0.993122i	$0.537356\pi$
$272$	0	0
$273$	25.1537	1.52237
$274$	0	0
$275$	18.4023	1.10970
$276$	0	0
$277$	−2.71396	−0.163066	−0.0815330	−	0.996671i	$-0.525982\pi$
$277$	−2.71396	−0.163066	−0.0815330	+	0.996671i	$0.525982\pi$
$278$	0	0
$279$	65.6976	3.93321
$280$	0	0
$281$	27.9458	1.66711	0.833553	−	0.552440i	$-0.186303\pi$
$281$	27.9458	1.66711	0.833553	+	0.552440i	$0.186303\pi$
$282$	0	0
$283$	−3.83116	−0.227739	−0.113869	−	0.993496i	$-0.536325\pi$
$283$	−3.83116	−0.227739	−0.113869	+	0.993496i	$0.536325\pi$
$284$	0	0
$285$	6.56117	0.388650
$286$	0	0
$287$	9.25809	0.546488
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−26.4839	−1.55251
$292$	0	0
$293$	−1.78066	−0.104027	−0.0520137	−	0.998646i	$-0.516564\pi$
$293$	−1.78066	−0.104027	−0.0520137	+	0.998646i	$0.516564\pi$
$294$	0	0
$295$	−0.252330	−0.0146912
$296$	0	0
$297$	−61.9384	−3.59403
$298$	0	0
$299$	6.72580	0.388963
$300$	0	0
$301$	−44.2958	−2.55317
$302$	0	0
$303$	−14.9851	−0.860869
$304$	0	0
$305$	−0.201394	−0.0115318
$306$	0	0
$307$	10.5660	0.603035	0.301518	−	0.953461i	$-0.402507\pi$
$307$	10.5660	0.603035	0.301518	+	0.953461i	$0.402507\pi$
$308$	0	0
$309$	50.9326	2.89746
$310$	0	0
$311$	3.36226	0.190656	0.0953281	−	0.995446i	$-0.469610\pi$
$311$	3.36226	0.190656	0.0953281	+	0.995446i	$0.469610\pi$
$312$	0	0
$313$	−33.4238	−1.88922	−0.944612	−	0.328188i	$-0.893562\pi$
$313$	−33.4238	−1.88922	−0.944612	+	0.328188i	$0.893562\pi$
$314$	0	0
$315$	−9.10384	−0.512943
$316$	0	0
$317$	12.8288	0.720536	0.360268	−	0.932849i	$-0.382685\pi$
$317$	12.8288	0.720536	0.360268	+	0.932849i	$0.382685\pi$
$318$	0	0
$319$	−22.9065	−1.28252
$320$	0	0
$321$	26.2150	1.46318
$322$	0	0
$323$	−7.83723	−0.436075
$324$	0	0
$325$	−8.30639	−0.460756
$326$	0	0
$327$	−46.1227	−2.55059
$328$	0	0
$329$	27.9242	1.53951
$330$	0	0
$331$	−9.13335	−0.502014	−0.251007	−	0.967985i	$-0.580762\pi$
$331$	−9.13335	−0.502014	−0.251007	+	0.967985i	$0.580762\pi$
$332$	0	0
$333$	−32.6763	−1.79065
$334$	0	0
$335$	−0.633729	−0.0346243
$336$	0	0
$337$	−28.2144	−1.53694	−0.768468	−	0.639889i	$-0.778980\pi$
$337$	−28.2144	−1.53694	−0.768468	+	0.639889i	$0.778980\pi$
$338$	0	0
$339$	−51.0923	−2.77495
$340$	0	0
$341$	−30.5849	−1.65626
$342$	0	0
$343$	28.3833	1.53255
$344$	0	0
$345$	−3.34622	−0.180154
$346$	0	0
$347$	−28.9878	−1.55615	−0.778074	−	0.628172i	$-0.783803\pi$
$347$	−28.9878	−1.55615	−0.778074	+	0.628172i	$0.783803\pi$
$348$	0	0
$349$	22.7092	1.21559	0.607797	−	0.794092i	$-0.292053\pi$
$349$	22.7092	1.21559	0.607797	+	0.794092i	$0.292053\pi$
$350$	0	0
$351$	27.9576	1.49227
$352$	0	0
$353$	5.83208	0.310410	0.155205	−	0.987882i	$-0.450396\pi$
$353$	5.83208	0.310410	0.155205	+	0.987882i	$0.450396\pi$
$354$	0	0
$355$	−0.552431	−0.0293200
$356$	0	0
$357$	14.9484	0.791152
$358$	0	0
$359$	11.5363	0.608863	0.304432	−	0.952534i	$-0.401533\pi$
$359$	11.5363	0.608863	0.304432	+	0.952534i	$0.401533\pi$
$360$	0	0
$361$	42.4221	2.23274
$362$	0	0
$363$	9.61316	0.504560
$364$	0	0
$365$	−2.94358	−0.154074
$366$	0	0
$367$	4.39173	0.229246	0.114623	−	0.993409i	$-0.463434\pi$
$367$	4.39173	0.229246	0.114623	+	0.993409i	$0.463434\pi$
$368$	0	0
$369$	16.4546	0.856592
$370$	0	0
$371$	−12.7500	−0.661945
$372$	0	0
$373$	−1.17405	−0.0607901	−0.0303951	−	0.999538i	$-0.509677\pi$
$373$	−1.17405	−0.0607901	−0.0303951	+	0.999538i	$0.509677\pi$
$374$	0	0
$375$	8.31849	0.429565
$376$	0	0
$377$	10.3395	0.532512
$378$	0	0
$379$	22.9865	1.18073	0.590367	−	0.807135i	$-0.298983\pi$
$379$	22.9865	1.18073	0.590367	+	0.807135i	$0.298983\pi$
$380$	0	0
$381$	−26.7919	−1.37259
$382$	0	0
$383$	−33.1657	−1.69469	−0.847345	−	0.531043i	$-0.821800\pi$
$383$	−33.1657	−1.69469	−0.847345	+	0.531043i	$0.821800\pi$
$384$	0	0
$385$	4.23820	0.215999
$386$	0	0
$387$	−78.7280	−4.00197
$388$	0	0
$389$	−22.2981	−1.13056	−0.565279	−	0.824900i	$-0.691231\pi$
$389$	−22.2981	−1.13056	−0.565279	+	0.824900i	$0.691231\pi$
$390$	0	0
$391$	3.99701	0.202138
$392$	0	0
$393$	−31.8105	−1.60463
$394$	0	0
$395$	2.06290	0.103796
$396$	0	0
$397$	−11.4367	−0.573992	−0.286996	−	0.957932i	$-0.592657\pi$
$397$	−11.4367	−0.573992	−0.286996	+	0.957932i	$0.592657\pi$
$398$	0	0
$399$	−117.154	−5.86502
$400$	0	0
$401$	9.58152	0.478478	0.239239	−	0.970961i	$-0.423102\pi$
$401$	9.58152	0.478478	0.239239	+	0.970961i	$0.423102\pi$
$402$	0	0
$403$	13.8053	0.687693
$404$	0	0
$405$	−7.84768	−0.389955
$406$	0	0
$407$	15.2121	0.754036
$408$	0	0
$409$	−28.9107	−1.42954	−0.714772	−	0.699358i	$-0.753469\pi$
$409$	−28.9107	−1.42954	−0.714772	+	0.699358i	$0.753469\pi$
$410$	0	0
$411$	72.4697	3.57467
$412$	0	0
$413$	4.50552	0.221702
$414$	0	0
$415$	−1.76340	−0.0865621
$416$	0	0
$417$	−12.2016	−0.597516
$418$	0	0
$419$	−12.9475	−0.632529	−0.316265	−	0.948671i	$-0.602429\pi$
$419$	−12.9475	−0.632529	−0.316265	+	0.948671i	$0.602429\pi$
$420$	0	0
$421$	−13.2796	−0.647206	−0.323603	−	0.946193i	$-0.604894\pi$
$421$	−13.2796	−0.647206	−0.323603	+	0.946193i	$0.604894\pi$
$422$	0	0
$423$	49.6304	2.41311
$424$	0	0
$425$	−4.93633	−0.239447
$426$	0	0
$427$	3.59602	0.174023
$428$	0	0
$429$	−20.8125	−1.00484
$430$	0	0
$431$	29.4612	1.41910	0.709549	−	0.704656i	$-0.248899\pi$
$431$	29.4612	1.41910	0.709549	+	0.704656i	$0.248899\pi$
$432$	0	0
$433$	2.89900	0.139317	0.0696585	−	0.997571i	$-0.477809\pi$
$433$	2.89900	0.139317	0.0696585	+	0.997571i	$0.477809\pi$
$434$	0	0
$435$	−5.14412	−0.246641
$436$	0	0
$437$	−31.3255	−1.49850
$438$	0	0
$439$	2.79097	0.133206	0.0666028	−	0.997780i	$-0.478784\pi$
$439$	2.79097	0.133206	0.0666028	+	0.997780i	$0.478784\pi$
$440$	0	0
$441$	106.500	5.07145
$442$	0	0
$443$	−8.54668	−0.406065	−0.203033	−	0.979172i	$-0.565080\pi$
$443$	−8.54668	−0.406065	−0.203033	+	0.979172i	$0.565080\pi$
$444$	0	0
$445$	3.45482	0.163774
$446$	0	0
$447$	10.5935	0.501054
$448$	0	0
$449$	35.4861	1.67469	0.837346	−	0.546673i	$-0.184106\pi$
$449$	35.4861	1.67469	0.837346	+	0.546673i	$0.184106\pi$
$450$	0	0
$451$	−7.66028	−0.360708
$452$	0	0
$453$	−62.1580	−2.92044
$454$	0	0
$455$	−1.91303	−0.0896843
$456$	0	0
$457$	−21.0682	−0.985528	−0.492764	−	0.870163i	$-0.664013\pi$
$457$	−21.0682	−0.985528	−0.492764	+	0.870163i	$0.664013\pi$
$458$	0	0
$459$	16.6147	0.775507
$460$	0	0
$461$	−38.4136	−1.78910	−0.894550	−	0.446968i	$-0.852504\pi$
$461$	−38.4136	−1.78910	−0.894550	+	0.446968i	$0.852504\pi$
$462$	0	0
$463$	22.8778	1.06322	0.531610	−	0.846989i	$-0.321587\pi$
$463$	22.8778	1.06322	0.531610	+	0.846989i	$0.321587\pi$
$464$	0	0
$465$	−6.86844	−0.318516
$466$	0	0
$467$	2.03959	0.0943808	0.0471904	−	0.998886i	$-0.484973\pi$
$467$	2.03959	0.0943808	0.0471904	+	0.998886i	$0.484973\pi$
$468$	0	0
$469$	11.3156	0.522507
$470$	0	0
$471$	−9.79276	−0.451226
$472$	0	0
$473$	36.6510	1.68522
$474$	0	0
$475$	38.6871	1.77509
$476$	0	0
$477$	−22.6608	−1.03757
$478$	0	0
$479$	29.2224	1.33521	0.667603	−	0.744518i	$-0.267321\pi$
$479$	29.2224	1.33521	0.667603	+	0.744518i	$0.267321\pi$
$480$	0	0
$481$	−6.86641	−0.313081
$482$	0	0
$483$	59.7488	2.71867
$484$	0	0
$485$	2.01420	0.0914600
$486$	0	0
$487$	−32.8597	−1.48902	−0.744508	−	0.667614i	$-0.767316\pi$
$487$	−32.8597	−1.48902	−0.744508	+	0.667614i	$0.767316\pi$
$488$	0	0
$489$	38.3252	1.73312
$490$	0	0
$491$	12.5772	0.567600	0.283800	−	0.958883i	$-0.408405\pi$
$491$	12.5772	0.567600	0.283800	+	0.958883i	$0.408405\pi$
$492$	0	0
$493$	6.14458	0.276738
$494$	0	0
$495$	7.53265	0.338567
$496$	0	0
$497$	9.86400	0.442461
$498$	0	0
$499$	22.6019	1.01180	0.505899	−	0.862593i	$-0.331161\pi$
$499$	22.6019	1.01180	0.505899	+	0.862593i	$0.331161\pi$
$500$	0	0
$501$	−72.0161	−3.21744
$502$	0	0
$503$	31.3484	1.39776	0.698878	−	0.715240i	$-0.253683\pi$
$503$	31.3484	1.39776	0.698878	+	0.715240i	$0.253683\pi$
$504$	0	0
$505$	1.13967	0.0507146
$506$	0	0
$507$	−33.7370	−1.49831
$508$	0	0
$509$	12.9604	0.574460	0.287230	−	0.957862i	$-0.407266\pi$
$509$	12.9604	0.574460	0.287230	+	0.957862i	$0.407266\pi$
$510$	0	0
$511$	52.5595	2.32510
$512$	0	0
$513$	−130.213	−5.74905
$514$	0	0
$515$	−3.87361	−0.170692
$516$	0	0
$517$	−23.1049	−1.01615
$518$	0	0
$519$	−66.5372	−2.92066
$520$	0	0
$521$	14.7186	0.644832	0.322416	−	0.946598i	$-0.395505\pi$
$521$	14.7186	0.644832	0.322416	+	0.946598i	$0.395505\pi$
$522$	0	0
$523$	−36.2356	−1.58447	−0.792236	−	0.610215i	$-0.791083\pi$
$523$	−36.2356	−1.58447	−0.792236	+	0.610215i	$0.791083\pi$
$524$	0	0
$525$	−73.7901	−3.22046
$526$	0	0
$527$	8.20425	0.357383
$528$	0	0
$529$	−7.02389	−0.305387
$530$	0	0
$531$	8.00775	0.347507
$532$	0	0
$533$	3.45768	0.149769
$534$	0	0
$535$	−1.99375	−0.0861973
$536$	0	0
$537$	−24.1569	−1.04245
$538$	0	0
$539$	−49.5802	−2.13557
$540$	0	0
$541$	2.60029	0.111795	0.0558977	−	0.998437i	$-0.482198\pi$
$541$	2.60029	0.111795	0.0558977	+	0.998437i	$0.482198\pi$
$542$	0	0
$543$	−4.03508	−0.173162
$544$	0	0
$545$	3.50780	0.150258
$546$	0	0
$547$	−34.1808	−1.46146	−0.730732	−	0.682664i	$-0.760821\pi$
$547$	−34.1808	−1.46146	−0.730732	+	0.682664i	$0.760821\pi$
$548$	0	0
$549$	6.39128	0.272773
$550$	0	0
$551$	−48.1564	−2.05153
$552$	0	0
$553$	−36.8344	−1.56636
$554$	0	0
$555$	3.41618	0.145009
$556$	0	0
$557$	24.5674	1.04095	0.520477	−	0.853876i	$-0.325754\pi$
$557$	24.5674	1.04095	0.520477	+	0.853876i	$0.325754\pi$
$558$	0	0
$559$	−16.5435	−0.699715
$560$	0	0
$561$	−12.3685	−0.522198
$562$	0	0
$563$	−35.1232	−1.48027	−0.740133	−	0.672461i	$-0.765237\pi$
$563$	−35.1232	−1.48027	−0.740133	+	0.672461i	$0.765237\pi$
$564$	0	0
$565$	3.88575	0.163475
$566$	0	0
$567$	140.125	5.88471
$568$	0	0
$569$	−3.89498	−0.163286	−0.0816431	−	0.996662i	$-0.526017\pi$
$569$	−3.89498	−0.163286	−0.0816431	+	0.996662i	$0.526017\pi$
$570$	0	0
$571$	31.1828	1.30496	0.652479	−	0.757807i	$-0.273729\pi$
$571$	31.1828	1.30496	0.652479	+	0.757807i	$0.273729\pi$
$572$	0	0
$573$	−34.4024	−1.43718
$574$	0	0
$575$	−19.7306	−0.822822
$576$	0	0
$577$	16.5964	0.690916	0.345458	−	0.938434i	$-0.387724\pi$
$577$	16.5964	0.690916	0.345458	+	0.938434i	$0.387724\pi$
$578$	0	0
$579$	62.7216	2.60662
$580$	0	0
$581$	31.4867	1.30629
$582$	0	0
$583$	10.5495	0.436916
$584$	0	0
$585$	−3.40007	−0.140576
$586$	0	0
$587$	0.953338	0.0393485	0.0196742	−	0.999806i	$-0.493737\pi$
$587$	0.953338	0.0393485	0.0196742	+	0.999806i	$0.493737\pi$
$588$	0	0
$589$	−64.2986	−2.64938
$590$	0	0
$591$	15.4388	0.635069
$592$	0	0
$593$	−22.5414	−0.925665	−0.462832	−	0.886446i	$-0.653167\pi$
$593$	−22.5414	−0.925665	−0.462832	+	0.886446i	$0.653167\pi$
$594$	0	0
$595$	−1.13688	−0.0466075
$596$	0	0
$597$	−8.83917	−0.361763
$598$	0	0
$599$	43.5271	1.77847	0.889234	−	0.457453i	$-0.151238\pi$
$599$	43.5271	1.77847	0.889234	+	0.457453i	$0.151238\pi$
$600$	0	0
$601$	−16.9006	−0.689391	−0.344696	−	0.938714i	$-0.612018\pi$
$601$	−16.9006	−0.689391	−0.344696	+	0.938714i	$0.612018\pi$
$602$	0	0
$603$	20.1115	0.819005
$604$	0	0
$605$	−0.731116	−0.0297241
$606$	0	0
$607$	45.5613	1.84927	0.924637	−	0.380849i	$-0.124368\pi$
$607$	45.5613	1.84927	0.924637	+	0.380849i	$0.124368\pi$
$608$	0	0
$609$	91.8514	3.72201
$610$	0	0
$611$	10.4291	0.421915
$612$	0	0
$613$	−46.7531	−1.88834	−0.944170	−	0.329458i	$-0.893134\pi$
$613$	−46.7531	−1.88834	−0.944170	+	0.329458i	$0.893134\pi$
$614$	0	0
$615$	−1.72027	−0.0693678
$616$	0	0
$617$	−15.2052	−0.612138	−0.306069	−	0.952009i	$-0.599014\pi$
$617$	−15.2052	−0.612138	−0.306069	+	0.952009i	$0.599014\pi$
$618$	0	0
$619$	−6.27928	−0.252385	−0.126193	−	0.992006i	$-0.540276\pi$
$619$	−6.27928	−0.252385	−0.126193	+	0.992006i	$0.540276\pi$
$620$	0	0
$621$	66.4091	2.66491
$622$	0	0
$623$	−61.6879	−2.47147
$624$	0	0
$625$	24.0490	0.961960
$626$	0	0
$627$	96.9347	3.87120
$628$	0	0
$629$	−4.08058	−0.162703
$630$	0	0
$631$	5.85407	0.233047	0.116523	−	0.993188i	$-0.462825\pi$
$631$	5.85407	0.233047	0.116523	+	0.993188i	$0.462825\pi$
$632$	0	0
$633$	−44.6822	−1.77596
$634$	0	0
$635$	2.03762	0.0808607
$636$	0	0
$637$	22.3794	0.886705
$638$	0	0
$639$	17.5315	0.693536
$640$	0	0
$641$	−13.3942	−0.529039	−0.264519	−	0.964380i	$-0.585213\pi$
$641$	−13.3942	−0.529039	−0.264519	+	0.964380i	$0.585213\pi$
$642$	0	0
$643$	−22.9250	−0.904073	−0.452037	−	0.891999i	$-0.649302\pi$
$643$	−22.9250	−0.904073	−0.452037	+	0.891999i	$0.649302\pi$
$644$	0	0
$645$	8.23071	0.324084
$646$	0	0
$647$	29.0804	1.14327	0.571634	−	0.820509i	$-0.306310\pi$
$647$	29.0804	1.14327	0.571634	+	0.820509i	$0.306310\pi$
$648$	0	0
$649$	−3.72793	−0.146334
$650$	0	0
$651$	122.640	4.80665
$652$	0	0
$653$	−20.0228	−0.783553	−0.391776	−	0.920060i	$-0.628139\pi$
$653$	−20.0228	−0.783553	−0.391776	+	0.920060i	$0.628139\pi$
$654$	0	0
$655$	2.41930	0.0945300
$656$	0	0
$657$	93.4152	3.64447
$658$	0	0
$659$	−15.8953	−0.619193	−0.309596	−	0.950868i	$-0.600194\pi$
$659$	−15.8953	−0.619193	−0.309596	+	0.950868i	$0.600194\pi$
$660$	0	0
$661$	18.3607	0.714149	0.357075	−	0.934076i	$-0.383774\pi$
$661$	18.3607	0.714149	0.357075	+	0.934076i	$0.383774\pi$
$662$	0	0
$663$	5.58287	0.216821
$664$	0	0
$665$	8.90997	0.345514
$666$	0	0
$667$	24.5600	0.950965
$668$	0	0
$669$	−77.3793	−2.99166
$670$	0	0
$671$	−2.97540	−0.114864
$672$	0	0
$673$	1.26851	0.0488975	0.0244488	−	0.999701i	$-0.492217\pi$
$673$	1.26851	0.0488975	0.0244488	+	0.999701i	$0.492217\pi$
$674$	0	0
$675$	−82.0156	−3.15678
$676$	0	0
$677$	17.0549	0.655475	0.327737	−	0.944769i	$-0.393714\pi$
$677$	17.0549	0.655475	0.327737	+	0.944769i	$0.393714\pi$
$678$	0	0
$679$	−35.9648	−1.38020
$680$	0	0
$681$	−12.5882	−0.482380
$682$	0	0
$683$	5.86710	0.224498	0.112249	−	0.993680i	$-0.464195\pi$
$683$	5.86710	0.224498	0.112249	+	0.993680i	$0.464195\pi$
$684$	0	0
$685$	−5.51159	−0.210587
$686$	0	0
$687$	36.9481	1.40966
$688$	0	0
$689$	−4.76182	−0.181411
$690$	0	0
$691$	14.6055	0.555620	0.277810	−	0.960636i	$-0.410391\pi$
$691$	14.6055	0.555620	0.277810	+	0.960636i	$0.410391\pi$
$692$	0	0
$693$	−134.500	−5.10924
$694$	0	0
$695$	0.927979	0.0352002
$696$	0	0
$697$	2.05483	0.0778324
$698$	0	0
$699$	65.2747	2.46892
$700$	0	0
$701$	21.2715	0.803414	0.401707	−	0.915768i	$-0.368417\pi$
$701$	21.2715	0.803414	0.401707	+	0.915768i	$0.368417\pi$
$702$	0	0
$703$	31.9804	1.20616
$704$	0	0
$705$	−5.18867	−0.195416
$706$	0	0
$707$	−20.3495	−0.765321
$708$	0	0
$709$	−14.7727	−0.554799	−0.277399	−	0.960755i	$-0.589473\pi$
$709$	−14.7727	−0.554799	−0.277399	+	0.960755i	$0.589473\pi$
$710$	0	0
$711$	−65.4665	−2.45519
$712$	0	0
$713$	32.7925	1.22809
$714$	0	0
$715$	1.58287	0.0591960
$716$	0	0
$717$	28.4853	1.06380
$718$	0	0
$719$	8.35960	0.311761	0.155880	−	0.987776i	$-0.450179\pi$
$719$	8.35960	0.311761	0.155880	+	0.987776i	$0.450179\pi$
$720$	0	0
$721$	69.1658	2.57587
$722$	0	0
$723$	−68.8452	−2.56038
$724$	0	0
$725$	−30.3317	−1.12649
$726$	0	0
$727$	−32.5707	−1.20798	−0.603991	−	0.796991i	$-0.706424\pi$
$727$	−32.5707	−1.20798	−0.603991	+	0.796991i	$0.706424\pi$
$728$	0	0
$729$	83.6755	3.09909
$730$	0	0
$731$	−9.83147	−0.363630
$732$	0	0
$733$	44.2634	1.63491	0.817453	−	0.575996i	$-0.195385\pi$
$733$	44.2634	1.63491	0.817453	+	0.575996i	$0.195385\pi$
$734$	0	0
$735$	−11.1342	−0.410692
$736$	0	0
$737$	−9.36272	−0.344880
$738$	0	0
$739$	12.1337	0.446346	0.223173	−	0.974779i	$-0.428358\pi$
$739$	12.1337	0.446346	0.223173	+	0.974779i	$0.428358\pi$
$740$	0	0
$741$	−43.7542	−1.60735
$742$	0	0
$743$	−24.6129	−0.902960	−0.451480	−	0.892281i	$-0.649104\pi$
$743$	−24.6129	−0.902960	−0.451480	+	0.892281i	$0.649104\pi$
$744$	0	0
$745$	−0.805672	−0.0295175
$746$	0	0
$747$	55.9620	2.04754
$748$	0	0
$749$	35.5996	1.30078
$750$	0	0
$751$	−9.44438	−0.344630	−0.172315	−	0.985042i	$-0.555125\pi$
$751$	−9.44438	−0.344630	−0.172315	+	0.985042i	$0.555125\pi$
$752$	0	0
$753$	37.0225	1.34917
$754$	0	0
$755$	4.72735	0.172046
$756$	0	0
$757$	4.95551	0.180111	0.0900555	−	0.995937i	$-0.471296\pi$
$757$	4.95551	0.180111	0.0900555	+	0.995937i	$0.471296\pi$
$758$	0	0
$759$	−49.4370	−1.79445
$760$	0	0
$761$	13.3977	0.485667	0.242834	−	0.970068i	$-0.421923\pi$
$761$	13.3977	0.485667	0.242834	+	0.970068i	$0.421923\pi$
$762$	0	0
$763$	−62.6340	−2.26750
$764$	0	0
$765$	−2.02060	−0.0730549
$766$	0	0
$767$	1.68271	0.0607590
$768$	0	0
$769$	23.6230	0.851867	0.425933	−	0.904755i	$-0.359946\pi$
$769$	23.6230	0.851867	0.425933	+	0.904755i	$0.359946\pi$
$770$	0	0
$771$	94.0101	3.38569
$772$	0	0
$773$	11.7537	0.422751	0.211376	−	0.977405i	$-0.432206\pi$
$773$	11.7537	0.422751	0.211376	+	0.977405i	$0.432206\pi$
$774$	0	0
$775$	−40.4989	−1.45476
$776$	0	0
$777$	−60.9980	−2.18829
$778$	0	0
$779$	−16.1042	−0.576993
$780$	0	0
$781$	−8.16162	−0.292046
$782$	0	0
$783$	102.090	3.64841
$784$	0	0
$785$	0.744775	0.0265822
$786$	0	0
$787$	−35.7544	−1.27451	−0.637253	−	0.770655i	$-0.719929\pi$
$787$	−35.7544	−1.27451	−0.637253	+	0.770655i	$0.719929\pi$
$788$	0	0
$789$	67.6487	2.40836
$790$	0	0
$791$	−69.3825	−2.46696
$792$	0	0
$793$	1.34303	0.0476924
$794$	0	0
$795$	2.36910	0.0840233
$796$	0	0
$797$	8.49163	0.300789	0.150394	−	0.988626i	$-0.451946\pi$
$797$	8.49163	0.300789	0.150394	+	0.988626i	$0.451946\pi$
$798$	0	0
$799$	6.19779	0.219262
$800$	0	0
$801$	−109.639	−3.87391
$802$	0	0
$803$	−43.4885	−1.53468
$804$	0	0
$805$	−4.54412	−0.160159
$806$	0	0
$807$	−52.9697	−1.86462
$808$	0	0
$809$	−29.5731	−1.03973	−0.519867	−	0.854247i	$-0.674019\pi$
$809$	−29.5731	−1.03973	−0.519867	+	0.854247i	$0.674019\pi$
$810$	0	0
$811$	50.8026	1.78392	0.891960	−	0.452114i	$-0.149330\pi$
$811$	50.8026	1.78392	0.891960	+	0.452114i	$0.149330\pi$
$812$	0	0
$813$	−12.7901	−0.448569
$814$	0	0
$815$	−2.91477	−0.102100
$816$	0	0
$817$	77.0515	2.69569
$818$	0	0
$819$	60.7104	2.12139
$820$	0	0
$821$	4.03332	0.140764	0.0703820	−	0.997520i	$-0.477578\pi$
$821$	4.03332	0.140764	0.0703820	+	0.997520i	$0.477578\pi$
$822$	0	0
$823$	−15.2462	−0.531448	−0.265724	−	0.964049i	$-0.585611\pi$
$823$	−15.2462	−0.531448	−0.265724	+	0.964049i	$0.585611\pi$
$824$	0	0
$825$	61.0550	2.12566
$826$	0	0
$827$	24.4634	0.850675	0.425337	−	0.905035i	$-0.360155\pi$
$827$	24.4634	0.850675	0.425337	+	0.905035i	$0.360155\pi$
$828$	0	0
$829$	−5.95625	−0.206869	−0.103435	−	0.994636i	$-0.532983\pi$
$829$	−5.95625	−0.206869	−0.103435	+	0.994636i	$0.532983\pi$
$830$	0	0
$831$	−9.00435	−0.312358
$832$	0	0
$833$	13.2997	0.460806
$834$	0	0
$835$	5.47709	0.189543
$836$	0	0
$837$	136.311	4.71160
$838$	0	0
$839$	−49.1517	−1.69690	−0.848452	−	0.529272i	$-0.822465\pi$
$839$	−49.1517	−1.69690	−0.848452	+	0.529272i	$0.822465\pi$
$840$	0	0
$841$	8.75582	0.301925
$842$	0	0
$843$	92.7183	3.19339
$844$	0	0
$845$	2.56582	0.0882669
$846$	0	0
$847$	13.0545	0.448559
$848$	0	0
$849$	−12.7110	−0.436240
$850$	0	0
$851$	−16.3101	−0.559104
$852$	0	0
$853$	5.62137	0.192472	0.0962360	−	0.995359i	$-0.469320\pi$
$853$	5.62137	0.192472	0.0962360	+	0.995359i	$0.469320\pi$
$854$	0	0
$855$	15.8359	0.541576
$856$	0	0
$857$	−40.1871	−1.37277	−0.686383	−	0.727241i	$-0.740802\pi$
$857$	−40.1871	−1.37277	−0.686383	+	0.727241i	$0.740802\pi$
$858$	0	0
$859$	47.4913	1.62038	0.810191	−	0.586166i	$-0.199363\pi$
$859$	47.4913	1.62038	0.810191	+	0.586166i	$0.199363\pi$
$860$	0	0
$861$	30.7164	1.04681
$862$	0	0
$863$	31.2595	1.06409	0.532043	−	0.846717i	$-0.321424\pi$
$863$	31.2595	1.06409	0.532043	+	0.846717i	$0.321424\pi$
$864$	0	0
$865$	5.06040	0.172059
$866$	0	0
$867$	3.31779	0.112678
$868$	0	0
$869$	30.4773	1.03387
$870$	0	0
$871$	4.22613	0.143197
$872$	0	0
$873$	−63.9210	−2.16340
$874$	0	0
$875$	11.2964	0.381888
$876$	0	0
$877$	−6.91850	−0.233621	−0.116811	−	0.993154i	$-0.537267\pi$
$877$	−6.91850	−0.233621	−0.116811	+	0.993154i	$0.537267\pi$
$878$	0	0
$879$	−5.90786	−0.199267
$880$	0	0
$881$	−15.5560	−0.524096	−0.262048	−	0.965055i	$-0.584398\pi$
$881$	−15.5560	−0.524096	−0.262048	+	0.965055i	$0.584398\pi$
$882$	0	0
$883$	7.42884	0.250000	0.125000	−	0.992157i	$-0.460107\pi$
$883$	7.42884	0.250000	0.125000	+	0.992157i	$0.460107\pi$
$884$	0	0
$885$	−0.837180	−0.0281415
$886$	0	0
$887$	31.4595	1.05631	0.528153	−	0.849149i	$-0.322885\pi$
$887$	31.4595	1.05631	0.528153	+	0.849149i	$0.322885\pi$
$888$	0	0
$889$	−36.3831	−1.22025
$890$	0	0
$891$	−115.942	−3.88419
$892$	0	0
$893$	−48.5735	−1.62545
$894$	0	0
$895$	1.83722	0.0614115
$896$	0	0
$897$	22.3148	0.745070
$898$	0	0
$899$	50.4117	1.68132
$900$	0	0
$901$	−2.82986	−0.0942762
$902$	0	0
$903$	−146.964	−4.89067
$904$	0	0
$905$	0.306883	0.0102011
$906$	0	0
$907$	−5.03835	−0.167296	−0.0836478	−	0.996495i	$-0.526657\pi$
$907$	−5.03835	−0.167296	−0.0836478	+	0.996495i	$0.526657\pi$
$908$	0	0
$909$	−36.1676	−1.19960
$910$	0	0
$911$	−45.9150	−1.52123	−0.760616	−	0.649202i	$-0.775103\pi$
$911$	−45.9150	−1.52123	−0.760616	+	0.649202i	$0.775103\pi$
$912$	0	0
$913$	−26.0525	−0.862214
$914$	0	0
$915$	−0.668184	−0.0220895
$916$	0	0
$917$	−43.1982	−1.42653
$918$	0	0
$919$	34.8711	1.15029	0.575146	−	0.818051i	$-0.304945\pi$
$919$	34.8711	1.15029	0.575146	+	0.818051i	$0.304945\pi$
$920$	0	0
$921$	35.0559	1.15513
$922$	0	0
$923$	3.68398	0.121260
$924$	0	0
$925$	20.1431	0.662300
$926$	0	0
$927$	122.930	4.03755
$928$	0	0
$929$	−39.3421	−1.29077	−0.645387	−	0.763856i	$-0.723304\pi$
$929$	−39.3421	−1.29077	−0.645387	+	0.763856i	$0.723304\pi$
$930$	0	0
$931$	−104.233	−3.41608
$932$	0	0
$933$	11.1553	0.365207
$934$	0	0
$935$	0.940669	0.0307632
$936$	0	0
$937$	20.6772	0.675496	0.337748	−	0.941237i	$-0.390335\pi$
$937$	20.6772	0.675496	0.337748	+	0.941237i	$0.390335\pi$
$938$	0	0
$939$	−110.893	−3.61886
$940$	0	0
$941$	−28.0563	−0.914611	−0.457305	−	0.889310i	$-0.651185\pi$
$941$	−28.0563	−0.914611	−0.457305	+	0.889310i	$0.651185\pi$
$942$	0	0
$943$	8.21320	0.267459
$944$	0	0
$945$	−18.8889	−0.614455
$946$	0	0
$947$	38.6471	1.25586	0.627931	−	0.778269i	$-0.283902\pi$
$947$	38.6471	1.25586	0.627931	+	0.778269i	$0.283902\pi$
$948$	0	0
$949$	19.6298	0.637209
$950$	0	0
$951$	42.5632	1.38021
$952$	0	0
$953$	−56.2865	−1.82330	−0.911649	−	0.410971i	$-0.865190\pi$
$953$	−56.2865	−1.82330	−0.911649	+	0.410971i	$0.865190\pi$
$954$	0	0
$955$	2.61643	0.0846656
$956$	0	0
$957$	−75.9992	−2.45670
$958$	0	0
$959$	98.4128	3.17792
$960$	0	0
$961$	36.3098	1.17128
$962$	0	0
$963$	63.2720	2.03891
$964$	0	0
$965$	−4.77021	−0.153558
$966$	0	0
$967$	−37.6734	−1.21150	−0.605748	−	0.795657i	$-0.707126\pi$
$967$	−37.6734	−1.21150	−0.605748	+	0.795657i	$0.707126\pi$
$968$	0	0
$969$	−26.0023	−0.835314
$970$	0	0
$971$	9.03215	0.289856	0.144928	−	0.989442i	$-0.453705\pi$
$971$	9.03215	0.289856	0.144928	+	0.989442i	$0.453705\pi$
$972$	0	0
$973$	−16.5696	−0.531198
$974$	0	0
$975$	−27.5589	−0.882591
$976$	0	0
$977$	27.0846	0.866513	0.433257	−	0.901271i	$-0.357364\pi$
$977$	27.0846	0.866513	0.433257	+	0.901271i	$0.357364\pi$
$978$	0	0
$979$	51.0415	1.63129
$980$	0	0
$981$	−111.321	−3.55420
$982$	0	0
$983$	0.547953	0.0174770	0.00873849	−	0.999962i	$-0.497218\pi$
$983$	0.547953	0.0174770	0.00873849	+	0.999962i	$0.497218\pi$
$984$	0	0
$985$	−1.17418	−0.0374125
$986$	0	0
$987$	92.6469	2.94898
$988$	0	0
$989$	−39.2965	−1.24956
$990$	0	0
$991$	−33.4061	−1.06118	−0.530590	−	0.847629i	$-0.678029\pi$
$991$	−33.4061	−1.06118	−0.530590	+	0.847629i	$0.678029\pi$
$992$	0	0
$993$	−30.3026	−0.961623
$994$	0	0
$995$	0.672251	0.0213118
$996$	0	0
$997$	−28.7151	−0.909416	−0.454708	−	0.890640i	$-0.650256\pi$
$997$	−28.7151	−0.909416	−0.454708	+	0.890640i	$0.650256\pi$
$998$	0	0
$999$	−67.7975	−2.14502

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.a.j.1.20	✓	21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.a.j.1.20	✓	21	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q+3.31779 q^{3} -0.252330 q^{5} +4.50552 q^{7} +8.00775 q^{9} +O(q^{10})\)	\(q+3.31779 q^{3} -0.252330 q^{5} +4.50552 q^{7} +8.00775 q^{9} -3.72793 q^{11} +1.68271 q^{13} -0.837180 q^{15} +1.00000 q^{17} -7.83723 q^{19} +14.9484 q^{21} +3.99701 q^{23} -4.93633 q^{25} +16.6147 q^{27} +6.14458 q^{29} +8.20425 q^{31} -12.3685 q^{33} -1.13688 q^{35} -4.08058 q^{37} +5.58287 q^{39} +2.05483 q^{41} -9.83147 q^{43} -2.02060 q^{45} +6.19779 q^{47} +13.2997 q^{49} +3.31779 q^{51} -2.82986 q^{53} +0.940669 q^{55} -26.0023 q^{57} +1.00000 q^{59} +0.798137 q^{61} +36.0791 q^{63} -0.424598 q^{65} +2.51151 q^{67} +13.2613 q^{69} +2.18932 q^{71} +11.6656 q^{73} -16.3777 q^{75} -16.7962 q^{77} -8.17539 q^{79} +31.1008 q^{81} +6.98848 q^{83} -0.252330 q^{85} +20.3864 q^{87} -13.6916 q^{89} +7.58146 q^{91} +27.2200 q^{93} +1.97757 q^{95} -7.98239 q^{97} -29.8523 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q + 9 q^{3} + q^{5} + 11 q^{7} + 26 q^{9}+O(q^{10}) \) 21 * q + 9 * q^3 + q^5 + 11 * q^7 + 26 * q^9	\( 21 q + 9 q^{3} + q^{5} + 11 q^{7} + 26 q^{9} + 12 q^{11} - 4 q^{13} + 9 q^{15} + 21 q^{17} + 4 q^{19} + 8 q^{21} + 19 q^{23} + 26 q^{25} + 33 q^{27} - q^{29} + 13 q^{31} + 11 q^{33} + 15 q^{35} - 4 q^{37} + 18 q^{39} + 9 q^{41} + 7 q^{43} + 7 q^{45} + 33 q^{47} + 36 q^{49} + 9 q^{51} + 7 q^{53} + 12 q^{55} + 26 q^{57} + 21 q^{59} + 3 q^{61} + 51 q^{63} + q^{65} + 8 q^{69} + 55 q^{71} + 22 q^{73} + 14 q^{75} - 15 q^{77} + 28 q^{79} + 25 q^{81} + 54 q^{83} + q^{85} + 34 q^{87} + 30 q^{89} + 35 q^{91} - 5 q^{93} + 42 q^{95} + 9 q^{97} + 20 q^{99}+O(q^{100}) \) 21 * q + 9 * q^3 + q^5 + 11 * q^7 + 26 * q^9 + 12 * q^11 - 4 * q^13 + 9 * q^15 + 21 * q^17 + 4 * q^19 + 8 * q^21 + 19 * q^23 + 26 * q^25 + 33 * q^27 - q^29 + 13 * q^31 + 11 * q^33 + 15 * q^35 - 4 * q^37 + 18 * q^39 + 9 * q^41 + 7 * q^43 + 7 * q^45 + 33 * q^47 + 36 * q^49 + 9 * q^51 + 7 * q^53 + 12 * q^55 + 26 * q^57 + 21 * q^59 + 3 * q^61 + 51 * q^63 + q^65 + 8 * q^69 + 55 * q^71 + 22 * q^73 + 14 * q^75 - 15 * q^77 + 28 * q^79 + 25 * q^81 + 54 * q^83 + q^85 + 34 * q^87 + 30 * q^89 + 35 * q^91 - 5 * q^93 + 42 * q^95 + 9 * q^97 + 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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