LMFDB - Embedded newform 4012.2.a.j.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:	$0$
Dimension:	$21$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
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)

$f(q)$	$=$	$q-2.50436 q^{3} +2.93974 q^{5} -0.515336 q^{7} +3.27181 q^{9} +O(q^{10})$	$q-2.50436 q^{3} +2.93974 q^{5} -0.515336 q^{7} +3.27181 q^{9} +4.74343 q^{11} +0.532089 q^{13} -7.36215 q^{15} +1.00000 q^{17} +4.25204 q^{19} +1.29058 q^{21} +8.33615 q^{23} +3.64206 q^{25} -0.680698 q^{27} -2.15941 q^{29} +0.646894 q^{31} -11.8792 q^{33} -1.51495 q^{35} -5.29614 q^{37} -1.33254 q^{39} -5.05695 q^{41} +7.82493 q^{43} +9.61825 q^{45} +7.28544 q^{47} -6.73443 q^{49} -2.50436 q^{51} +3.75243 q^{53} +13.9444 q^{55} -10.6486 q^{57} +1.00000 q^{59} +8.88245 q^{61} -1.68608 q^{63} +1.56420 q^{65} -4.98701 q^{67} -20.8767 q^{69} -12.4029 q^{71} -0.881251 q^{73} -9.12101 q^{75} -2.44446 q^{77} +10.4905 q^{79} -8.11071 q^{81} +1.63807 q^{83} +2.93974 q^{85} +5.40794 q^{87} -8.25085 q^{89} -0.274204 q^{91} -1.62005 q^{93} +12.4999 q^{95} +5.34584 q^{97} +15.5196 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.50436	−1.44589	−0.722946	−	0.690905i	$-0.757212\pi$
$3$	−2.50436	−1.44589	−0.722946	+	0.690905i	$0.757212\pi$
$4$	0	0
$5$	2.93974	1.31469	0.657345	−	0.753590i	$-0.271679\pi$
$5$	2.93974	1.31469	0.657345	+	0.753590i	$0.271679\pi$
$6$	0	0
$7$	−0.515336	−0.194779	−0.0973893	−	0.995246i	$-0.531049\pi$
$7$	−0.515336	−0.194779	−0.0973893	+	0.995246i	$0.531049\pi$
$8$	0	0
$9$	3.27181	1.09060
$10$	0	0
$11$	4.74343	1.43020	0.715098	−	0.699024i	$-0.246382\pi$
$11$	4.74343	1.43020	0.715098	+	0.699024i	$0.246382\pi$
$12$	0	0
$13$	0.532089	0.147575	0.0737874	−	0.997274i	$-0.476491\pi$
$13$	0.532089	0.147575	0.0737874	+	0.997274i	$0.476491\pi$
$14$	0	0
$15$	−7.36215	−1.90090
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	4.25204	0.975484	0.487742	−	0.872988i	$-0.337821\pi$
$19$	4.25204	0.975484	0.487742	+	0.872988i	$0.337821\pi$
$20$	0	0
$21$	1.29058	0.281629
$22$	0	0
$23$	8.33615	1.73821	0.869104	−	0.494630i	$-0.164696\pi$
$23$	8.33615	1.73821	0.869104	+	0.494630i	$0.164696\pi$
$24$	0	0
$25$	3.64206	0.728411
$26$	0	0
$27$	−0.680698	−0.131000
$28$	0	0
$29$	−2.15941	−0.400993	−0.200496	−	0.979694i	$-0.564255\pi$
$29$	−2.15941	−0.400993	−0.200496	+	0.979694i	$0.564255\pi$
$30$	0	0
$31$	0.646894	0.116186	0.0580928	−	0.998311i	$-0.481498\pi$
$31$	0.646894	0.116186	0.0580928	+	0.998311i	$0.481498\pi$
$32$	0	0
$33$	−11.8792	−2.06791
$34$	0	0
$35$	−1.51495	−0.256074
$36$	0	0
$37$	−5.29614	−0.870680	−0.435340	−	0.900266i	$-0.643372\pi$
$37$	−5.29614	−0.870680	−0.435340	+	0.900266i	$0.643372\pi$
$38$	0	0
$39$	−1.33254	−0.213377
$40$	0	0
$41$	−5.05695	−0.789762	−0.394881	−	0.918732i	$-0.629214\pi$
$41$	−5.05695	−0.789762	−0.394881	+	0.918732i	$0.629214\pi$
$42$	0	0
$43$	7.82493	1.19329	0.596646	−	0.802505i	$-0.296500\pi$
$43$	7.82493	1.19329	0.596646	+	0.802505i	$0.296500\pi$
$44$	0	0
$45$	9.61825	1.43380
$46$	0	0
$47$	7.28544	1.06269	0.531345	−	0.847155i	$-0.321687\pi$
$47$	7.28544	1.06269	0.531345	+	0.847155i	$0.321687\pi$
$48$	0	0
$49$	−6.73443	−0.962061
$50$	0	0
$51$	−2.50436	−0.350680
$52$	0	0
$53$	3.75243	0.515435	0.257718	−	0.966220i	$-0.417030\pi$
$53$	3.75243	0.515435	0.257718	+	0.966220i	$0.417030\pi$
$54$	0	0
$55$	13.9444	1.88027
$56$	0	0
$57$	−10.6486	−1.41044
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	8.88245	1.13728	0.568641	−	0.822586i	$-0.307470\pi$
$61$	8.88245	1.13728	0.568641	+	0.822586i	$0.307470\pi$
$62$	0	0
$63$	−1.68608	−0.212426
$64$	0	0
$65$	1.56420	0.194015
$66$	0	0
$67$	−4.98701	−0.609260	−0.304630	−	0.952471i	$-0.598533\pi$
$67$	−4.98701	−0.609260	−0.304630	+	0.952471i	$0.598533\pi$
$68$	0	0
$69$	−20.8767	−2.51326
$70$	0	0
$71$	−12.4029	−1.47196	−0.735979	−	0.677005i	$-0.763278\pi$
$71$	−12.4029	−1.47196	−0.735979	+	0.677005i	$0.763278\pi$
$72$	0	0
$73$	−0.881251	−0.103143	−0.0515713	−	0.998669i	$-0.516423\pi$
$73$	−0.881251	−0.103143	−0.0515713	+	0.998669i	$0.516423\pi$
$74$	0	0
$75$	−9.12101	−1.05320
$76$	0	0
$77$	−2.44446	−0.278572
$78$	0	0
$79$	10.4905	1.18027	0.590136	−	0.807304i	$-0.299074\pi$
$79$	10.4905	1.18027	0.590136	+	0.807304i	$0.299074\pi$
$80$	0	0
$81$	−8.11071	−0.901190
$82$	0	0
$83$	1.63807	0.179801	0.0899006	−	0.995951i	$-0.471345\pi$
$83$	1.63807	0.179801	0.0899006	+	0.995951i	$0.471345\pi$
$84$	0	0
$85$	2.93974	0.318859
$86$	0	0
$87$	5.40794	0.579792
$88$	0	0
$89$	−8.25085	−0.874589	−0.437294	−	0.899318i	$-0.644063\pi$
$89$	−8.25085	−0.874589	−0.437294	+	0.899318i	$0.644063\pi$
$90$	0	0
$91$	−0.274204	−0.0287444
$92$	0	0
$93$	−1.62005	−0.167992
$94$	0	0
$95$	12.4999	1.28246
$96$	0	0
$97$	5.34584	0.542788	0.271394	−	0.962468i	$-0.412515\pi$
$97$	5.34584	0.542788	0.271394	+	0.962468i	$0.412515\pi$
$98$	0	0
$99$	15.5196	1.55977
$100$	0	0
$101$	−3.91817	−0.389873	−0.194936	−	0.980816i	$-0.562450\pi$
$101$	−3.91817	−0.389873	−0.194936	+	0.980816i	$0.562450\pi$
$102$	0	0
$103$	−6.25870	−0.616688	−0.308344	−	0.951275i	$-0.599775\pi$
$103$	−6.25870	−0.616688	−0.308344	+	0.951275i	$0.599775\pi$
$104$	0	0
$105$	3.79398	0.370255
$106$	0	0
$107$	0.418948	0.0405012	0.0202506	−	0.999795i	$-0.493554\pi$
$107$	0.418948	0.0405012	0.0202506	+	0.999795i	$0.493554\pi$
$108$	0	0
$109$	−3.97392	−0.380633	−0.190316	−	0.981723i	$-0.560951\pi$
$109$	−3.97392	−0.380633	−0.190316	+	0.981723i	$0.560951\pi$
$110$	0	0
$111$	13.2634	1.25891
$112$	0	0
$113$	−14.4546	−1.35977	−0.679886	−	0.733318i	$-0.737971\pi$
$113$	−14.4546	−1.35977	−0.679886	+	0.733318i	$0.737971\pi$
$114$	0	0
$115$	24.5061	2.28520
$116$	0	0
$117$	1.74089	0.160945
$118$	0	0
$119$	−0.515336	−0.0472408
$120$	0	0
$121$	11.5001	1.04546
$122$	0	0
$123$	12.6644	1.14191
$124$	0	0
$125$	−3.99200	−0.357055
$126$	0	0
$127$	7.73048	0.685969	0.342985	−	0.939341i	$-0.388562\pi$
$127$	7.73048	0.685969	0.342985	+	0.939341i	$0.388562\pi$
$128$	0	0
$129$	−19.5964	−1.72537
$130$	0	0
$131$	−17.9091	−1.56472	−0.782362	−	0.622824i	$-0.785985\pi$
$131$	−17.9091	−1.56472	−0.782362	+	0.622824i	$0.785985\pi$
$132$	0	0
$133$	−2.19123	−0.190003
$134$	0	0
$135$	−2.00107	−0.172225
$136$	0	0
$137$	13.8235	1.18102	0.590510	−	0.807030i	$-0.298927\pi$
$137$	13.8235	1.18102	0.590510	+	0.807030i	$0.298927\pi$
$138$	0	0
$139$	14.3571	1.21775	0.608877	−	0.793264i	$-0.291620\pi$
$139$	14.3571	1.21775	0.608877	+	0.793264i	$0.291620\pi$
$140$	0	0
$141$	−18.2453	−1.53653
$142$	0	0
$143$	2.52392	0.211061
$144$	0	0
$145$	−6.34810	−0.527181
$146$	0	0
$147$	16.8654	1.39104
$148$	0	0
$149$	7.47932	0.612729	0.306365	−	0.951914i	$-0.400887\pi$
$149$	7.47932	0.612729	0.306365	+	0.951914i	$0.400887\pi$
$150$	0	0
$151$	−4.31109	−0.350831	−0.175416	−	0.984494i	$-0.556127\pi$
$151$	−4.31109	−0.350831	−0.175416	+	0.984494i	$0.556127\pi$
$152$	0	0
$153$	3.27181	0.264510
$154$	0	0
$155$	1.90170	0.152748
$156$	0	0
$157$	−4.42780	−0.353377	−0.176688	−	0.984267i	$-0.556538\pi$
$157$	−4.42780	−0.353377	−0.176688	+	0.984267i	$0.556538\pi$
$158$	0	0
$159$	−9.39742	−0.745264
$160$	0	0
$161$	−4.29592	−0.338566
$162$	0	0
$163$	−20.5506	−1.60964	−0.804822	−	0.593517i	$-0.797739\pi$
$163$	−20.5506	−1.60964	−0.804822	+	0.593517i	$0.797739\pi$
$164$	0	0
$165$	−34.9218	−2.71866
$166$	0	0
$167$	−19.6676	−1.52192	−0.760962	−	0.648796i	$-0.775273\pi$
$167$	−19.6676	−1.52192	−0.760962	+	0.648796i	$0.775273\pi$
$168$	0	0
$169$	−12.7169	−0.978222
$170$	0	0
$171$	13.9118	1.06386
$172$	0	0
$173$	0.192440	0.0146310	0.00731549	−	0.999973i	$-0.497671\pi$
$173$	0.192440	0.0146310	0.00731549	+	0.999973i	$0.497671\pi$
$174$	0	0
$175$	−1.87688	−0.141879
$176$	0	0
$177$	−2.50436	−0.188239
$178$	0	0
$179$	20.0557	1.49904	0.749518	−	0.661984i	$-0.230285\pi$
$179$	20.0557	1.49904	0.749518	+	0.661984i	$0.230285\pi$
$180$	0	0
$181$	23.4846	1.74560	0.872800	−	0.488078i	$-0.162302\pi$
$181$	23.4846	1.74560	0.872800	+	0.488078i	$0.162302\pi$
$182$	0	0
$183$	−22.2448	−1.64439
$184$	0	0
$185$	−15.5693	−1.14468
$186$	0	0
$187$	4.74343	0.346874
$188$	0	0
$189$	0.350788	0.0255161
$190$	0	0
$191$	13.3491	0.965910	0.482955	−	0.875645i	$-0.339563\pi$
$191$	13.3491	0.965910	0.482955	+	0.875645i	$0.339563\pi$
$192$	0	0
$193$	8.45438	0.608559	0.304280	−	0.952583i	$-0.401584\pi$
$193$	8.45438	0.608559	0.304280	+	0.952583i	$0.401584\pi$
$194$	0	0
$195$	−3.91732	−0.280525
$196$	0	0
$197$	16.0756	1.14534	0.572671	−	0.819786i	$-0.305907\pi$
$197$	16.0756	1.14534	0.572671	+	0.819786i	$0.305907\pi$
$198$	0	0
$199$	−22.2290	−1.57577	−0.787884	−	0.615823i	$-0.788824\pi$
$199$	−22.2290	−1.57577	−0.787884	+	0.615823i	$0.788824\pi$
$200$	0	0
$201$	12.4893	0.880924
$202$	0	0
$203$	1.11282	0.0781048
$204$	0	0
$205$	−14.8661	−1.03829
$206$	0	0
$207$	27.2743	1.89569
$208$	0	0
$209$	20.1692	1.39513
$210$	0	0
$211$	1.39545	0.0960669	0.0480334	−	0.998846i	$-0.484705\pi$
$211$	1.39545	0.0960669	0.0480334	+	0.998846i	$0.484705\pi$
$212$	0	0
$213$	31.0614	2.12829
$214$	0	0
$215$	23.0032	1.56881
$216$	0	0
$217$	−0.333367	−0.0226305
$218$	0	0
$219$	2.20697	0.149133
$220$	0	0
$221$	0.532089	0.0357922
$222$	0	0
$223$	19.7418	1.32201	0.661004	−	0.750382i	$-0.270131\pi$
$223$	19.7418	1.32201	0.661004	+	0.750382i	$0.270131\pi$
$224$	0	0
$225$	11.9161	0.794407
$226$	0	0
$227$	13.4458	0.892430	0.446215	−	0.894926i	$-0.352772\pi$
$227$	13.4458	0.892430	0.446215	+	0.894926i	$0.352772\pi$
$228$	0	0
$229$	−7.77569	−0.513832	−0.256916	−	0.966434i	$-0.582706\pi$
$229$	−7.77569	−0.513832	−0.256916	+	0.966434i	$0.582706\pi$
$230$	0	0
$231$	6.12179	0.402784
$232$	0	0
$233$	20.3580	1.33370	0.666850	−	0.745192i	$-0.267642\pi$
$233$	20.3580	1.33370	0.666850	+	0.745192i	$0.267642\pi$
$234$	0	0
$235$	21.4173	1.39711
$236$	0	0
$237$	−26.2719	−1.70654
$238$	0	0
$239$	−18.0127	−1.16514	−0.582572	−	0.812779i	$-0.697954\pi$
$239$	−18.0127	−1.16514	−0.582572	+	0.812779i	$0.697954\pi$
$240$	0	0
$241$	26.9894	1.73854	0.869269	−	0.494339i	$-0.164590\pi$
$241$	26.9894	1.73854	0.869269	+	0.494339i	$0.164590\pi$
$242$	0	0
$243$	22.3542	1.43402
$244$	0	0
$245$	−19.7975	−1.26481
$246$	0	0
$247$	2.26246	0.143957
$248$	0	0
$249$	−4.10231	−0.259973
$250$	0	0
$251$	1.04847	0.0661790	0.0330895	−	0.999452i	$-0.489465\pi$
$251$	1.04847	0.0661790	0.0330895	+	0.999452i	$0.489465\pi$
$252$	0	0
$253$	39.5419	2.48598
$254$	0	0
$255$	−7.36215	−0.461036
$256$	0	0
$257$	16.3723	1.02128	0.510638	−	0.859796i	$-0.329409\pi$
$257$	16.3723	1.02128	0.510638	+	0.859796i	$0.329409\pi$
$258$	0	0
$259$	2.72929	0.169590
$260$	0	0
$261$	−7.06517	−0.437323
$262$	0	0
$263$	2.28261	0.140751	0.0703757	−	0.997521i	$-0.477580\pi$
$263$	2.28261	0.140751	0.0703757	+	0.997521i	$0.477580\pi$
$264$	0	0
$265$	11.0311	0.677638
$266$	0	0
$267$	20.6631	1.26456
$268$	0	0
$269$	15.2818	0.931751	0.465875	−	0.884850i	$-0.345740\pi$
$269$	15.2818	0.931751	0.465875	+	0.884850i	$0.345740\pi$
$270$	0	0
$271$	19.8596	1.20639	0.603193	−	0.797595i	$-0.293895\pi$
$271$	19.8596	1.20639	0.603193	+	0.797595i	$0.293895\pi$
$272$	0	0
$273$	0.686706	0.0415613
$274$	0	0
$275$	17.2758	1.04177
$276$	0	0
$277$	29.7694	1.78867	0.894334	−	0.447399i	$-0.147650\pi$
$277$	29.7694	1.78867	0.894334	+	0.447399i	$0.147650\pi$
$278$	0	0
$279$	2.11651	0.126712
$280$	0	0
$281$	−27.8669	−1.66240	−0.831200	−	0.555973i	$-0.812346\pi$
$281$	−27.8669	−1.66240	−0.831200	+	0.555973i	$0.812346\pi$
$282$	0	0
$283$	14.0325	0.834143	0.417072	−	0.908874i	$-0.363056\pi$
$283$	14.0325	0.834143	0.417072	+	0.908874i	$0.363056\pi$
$284$	0	0
$285$	−31.3041	−1.85430
$286$	0	0
$287$	2.60602	0.153829
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−13.3879	−0.784812
$292$	0	0
$293$	−12.8205	−0.748982	−0.374491	−	0.927231i	$-0.622182\pi$
$293$	−12.8205	−0.748982	−0.374491	+	0.927231i	$0.622182\pi$
$294$	0	0
$295$	2.93974	0.171158
$296$	0	0
$297$	−3.22884	−0.187356
$298$	0	0
$299$	4.43557	0.256516
$300$	0	0
$301$	−4.03247	−0.232428
$302$	0	0
$303$	9.81250	0.563714
$304$	0	0
$305$	26.1121	1.49517
$306$	0	0
$307$	2.16820	0.123746	0.0618728	−	0.998084i	$-0.480293\pi$
$307$	2.16820	0.123746	0.0618728	+	0.998084i	$0.480293\pi$
$308$	0	0
$309$	15.6740	0.891664
$310$	0	0
$311$	−11.2107	−0.635703	−0.317852	−	0.948140i	$-0.602961\pi$
$311$	−11.2107	−0.635703	−0.317852	+	0.948140i	$0.602961\pi$
$312$	0	0
$313$	15.1392	0.855717	0.427858	−	0.903846i	$-0.359268\pi$
$313$	15.1392	0.855717	0.427858	+	0.903846i	$0.359268\pi$
$314$	0	0
$315$	−4.95663	−0.279274
$316$	0	0
$317$	−30.0384	−1.68713	−0.843563	−	0.537031i	$-0.819546\pi$
$317$	−30.0384	−1.68713	−0.843563	+	0.537031i	$0.819546\pi$
$318$	0	0
$319$	−10.2430	−0.573498
$320$	0	0
$321$	−1.04920	−0.0585604
$322$	0	0
$323$	4.25204	0.236590
$324$	0	0
$325$	1.93790	0.107495
$326$	0	0
$327$	9.95212	0.550353
$328$	0	0
$329$	−3.75445	−0.206989
$330$	0	0
$331$	−11.2267	−0.617077	−0.308539	−	0.951212i	$-0.599840\pi$
$331$	−11.2267	−0.617077	−0.308539	+	0.951212i	$0.599840\pi$
$332$	0	0
$333$	−17.3279	−0.949565
$334$	0	0
$335$	−14.6605	−0.800989
$336$	0	0
$337$	1.81306	0.0987634	0.0493817	−	0.998780i	$-0.484275\pi$
$337$	1.81306	0.0987634	0.0493817	+	0.998780i	$0.484275\pi$
$338$	0	0
$339$	36.1994	1.96608
$340$	0	0
$341$	3.06849	0.166168
$342$	0	0
$343$	7.07784	0.382168
$344$	0	0
$345$	−61.3720	−3.30416
$346$	0	0
$347$	28.2938	1.51889	0.759445	−	0.650571i	$-0.225470\pi$
$347$	28.2938	1.51889	0.759445	+	0.650571i	$0.225470\pi$
$348$	0	0
$349$	9.36168	0.501119	0.250560	−	0.968101i	$-0.419385\pi$
$349$	9.36168	0.501119	0.250560	+	0.968101i	$0.419385\pi$
$350$	0	0
$351$	−0.362192	−0.0193324
$352$	0	0
$353$	27.5966	1.46882	0.734410	−	0.678706i	$-0.237459\pi$
$353$	27.5966	1.46882	0.734410	+	0.678706i	$0.237459\pi$
$354$	0	0
$355$	−36.4614	−1.93517
$356$	0	0
$357$	1.29058	0.0683050
$358$	0	0
$359$	11.7291	0.619038	0.309519	−	0.950893i	$-0.399832\pi$
$359$	11.7291	0.619038	0.309519	+	0.950893i	$0.399832\pi$
$360$	0	0
$361$	−0.920193	−0.0484312
$362$	0	0
$363$	−28.8003	−1.51163
$364$	0	0
$365$	−2.59065	−0.135601
$366$	0	0
$367$	4.53707	0.236833	0.118417	−	0.992964i	$-0.462218\pi$
$367$	4.53707	0.236833	0.118417	+	0.992964i	$0.462218\pi$
$368$	0	0
$369$	−16.5453	−0.861316
$370$	0	0
$371$	−1.93376	−0.100396
$372$	0	0
$373$	−3.63462	−0.188193	−0.0940967	−	0.995563i	$-0.529996\pi$
$373$	−3.63462	−0.188193	−0.0940967	+	0.995563i	$0.529996\pi$
$374$	0	0
$375$	9.99739	0.516263
$376$	0	0
$377$	−1.14900	−0.0591764
$378$	0	0
$379$	−31.4307	−1.61449	−0.807243	−	0.590220i	$-0.799041\pi$
$379$	−31.4307	−1.61449	−0.807243	+	0.590220i	$0.799041\pi$
$380$	0	0
$381$	−19.3599	−0.991837
$382$	0	0
$383$	−6.32263	−0.323071	−0.161536	−	0.986867i	$-0.551645\pi$
$383$	−6.32263	−0.323071	−0.161536	+	0.986867i	$0.551645\pi$
$384$	0	0
$385$	−7.18606	−0.366236
$386$	0	0
$387$	25.6017	1.30141
$388$	0	0
$389$	−26.5618	−1.34674	−0.673369	−	0.739306i	$-0.735154\pi$
$389$	−26.5618	−1.34674	−0.673369	+	0.739306i	$0.735154\pi$
$390$	0	0
$391$	8.33615	0.421577
$392$	0	0
$393$	44.8507	2.26242
$394$	0	0
$395$	30.8393	1.55169
$396$	0	0
$397$	−18.1278	−0.909806	−0.454903	−	0.890541i	$-0.650326\pi$
$397$	−18.1278	−0.909806	−0.454903	+	0.890541i	$0.650326\pi$
$398$	0	0
$399$	5.48761	0.274724
$400$	0	0
$401$	20.7780	1.03760	0.518801	−	0.854895i	$-0.326378\pi$
$401$	20.7780	1.03760	0.518801	+	0.854895i	$0.326378\pi$
$402$	0	0
$403$	0.344205	0.0171461
$404$	0	0
$405$	−23.8433	−1.18479
$406$	0	0
$407$	−25.1219	−1.24524
$408$	0	0
$409$	−8.45681	−0.418162	−0.209081	−	0.977898i	$-0.567047\pi$
$409$	−8.45681	−0.418162	−0.209081	+	0.977898i	$0.567047\pi$
$410$	0	0
$411$	−34.6189	−1.70763
$412$	0	0
$413$	−0.515336	−0.0253580
$414$	0	0
$415$	4.81549	0.236383
$416$	0	0
$417$	−35.9553	−1.76074
$418$	0	0
$419$	−17.0027	−0.830637	−0.415319	−	0.909676i	$-0.636330\pi$
$419$	−17.0027	−0.830637	−0.415319	+	0.909676i	$0.636330\pi$
$420$	0	0
$421$	17.3992	0.847988	0.423994	−	0.905665i	$-0.360628\pi$
$421$	17.3992	0.847988	0.423994	+	0.905665i	$0.360628\pi$
$422$	0	0
$423$	23.8365	1.15897
$424$	0	0
$425$	3.64206	0.176666
$426$	0	0
$427$	−4.57744	−0.221518
$428$	0	0
$429$	−6.32081	−0.305171
$430$	0	0
$431$	−17.3092	−0.833757	−0.416878	−	0.908962i	$-0.636876\pi$
$431$	−17.3092	−0.833757	−0.416878	+	0.908962i	$0.636876\pi$
$432$	0	0
$433$	−1.95098	−0.0937583	−0.0468792	−	0.998901i	$-0.514928\pi$
$433$	−1.95098	−0.0937583	−0.0468792	+	0.998901i	$0.514928\pi$
$434$	0	0
$435$	15.8979	0.762247
$436$	0	0
$437$	35.4456	1.69559
$438$	0	0
$439$	12.1410	0.579456	0.289728	−	0.957109i	$-0.406435\pi$
$439$	12.1410	0.579456	0.289728	+	0.957109i	$0.406435\pi$
$440$	0	0
$441$	−22.0337	−1.04923
$442$	0	0
$443$	12.4807	0.592975	0.296487	−	0.955037i	$-0.404185\pi$
$443$	12.4807	0.592975	0.296487	+	0.955037i	$0.404185\pi$
$444$	0	0
$445$	−24.2553	−1.14981
$446$	0	0
$447$	−18.7309	−0.885940
$448$	0	0
$449$	1.98389	0.0936256	0.0468128	−	0.998904i	$-0.485094\pi$
$449$	1.98389	0.0936256	0.0468128	+	0.998904i	$0.485094\pi$
$450$	0	0
$451$	−23.9872	−1.12952
$452$	0	0
$453$	10.7965	0.507264
$454$	0	0
$455$	−0.806089	−0.0377900
$456$	0	0
$457$	−19.4632	−0.910449	−0.455224	−	0.890377i	$-0.650441\pi$
$457$	−19.4632	−0.910449	−0.455224	+	0.890377i	$0.650441\pi$
$458$	0	0
$459$	−0.680698	−0.0317722
$460$	0	0
$461$	−15.4024	−0.717363	−0.358682	−	0.933460i	$-0.616774\pi$
$461$	−15.4024	−0.717363	−0.358682	+	0.933460i	$0.616774\pi$
$462$	0	0
$463$	36.5204	1.69725	0.848623	−	0.528999i	$-0.177432\pi$
$463$	36.5204	1.69725	0.848623	+	0.528999i	$0.177432\pi$
$464$	0	0
$465$	−4.76253	−0.220857
$466$	0	0
$467$	8.85746	0.409874	0.204937	−	0.978775i	$-0.434301\pi$
$467$	8.85746	0.409874	0.204937	+	0.978775i	$0.434301\pi$
$468$	0	0
$469$	2.56998	0.118671
$470$	0	0
$471$	11.0888	0.510944
$472$	0	0
$473$	37.1170	1.70664
$474$	0	0
$475$	15.4862	0.710553
$476$	0	0
$477$	12.2772	0.562135
$478$	0	0
$479$	22.2458	1.01643	0.508217	−	0.861229i	$-0.330305\pi$
$479$	22.2458	1.01643	0.508217	+	0.861229i	$0.330305\pi$
$480$	0	0
$481$	−2.81802	−0.128491
$482$	0	0
$483$	10.7585	0.489529
$484$	0	0
$485$	15.7154	0.713598
$486$	0	0
$487$	−12.7856	−0.579371	−0.289686	−	0.957122i	$-0.593551\pi$
$487$	−12.7856	−0.579371	−0.289686	+	0.957122i	$0.593551\pi$
$488$	0	0
$489$	51.4659	2.32737
$490$	0	0
$491$	−34.0144	−1.53505	−0.767524	−	0.641020i	$-0.778511\pi$
$491$	−34.0144	−1.53505	−0.767524	+	0.641020i	$0.778511\pi$
$492$	0	0
$493$	−2.15941	−0.0972550
$494$	0	0
$495$	45.6234	2.05062
$496$	0	0
$497$	6.39167	0.286706
$498$	0	0
$499$	−0.549329	−0.0245913	−0.0122957	−	0.999924i	$-0.503914\pi$
$499$	−0.549329	−0.0245913	−0.0122957	+	0.999924i	$0.503914\pi$
$500$	0	0
$501$	49.2547	2.20054
$502$	0	0
$503$	17.4195	0.776695	0.388347	−	0.921513i	$-0.373046\pi$
$503$	17.4195	0.776695	0.388347	+	0.921513i	$0.373046\pi$
$504$	0	0
$505$	−11.5184	−0.512562
$506$	0	0
$507$	31.8476	1.41440
$508$	0	0
$509$	−33.7383	−1.49542	−0.747712	−	0.664024i	$-0.768847\pi$
$509$	−33.7383	−1.49542	−0.747712	+	0.664024i	$0.768847\pi$
$510$	0	0
$511$	0.454140	0.0200900
$512$	0	0
$513$	−2.89435	−0.127789
$514$	0	0
$515$	−18.3989	−0.810754
$516$	0	0
$517$	34.5579	1.51986
$518$	0	0
$519$	−0.481940	−0.0211548
$520$	0	0
$521$	−33.7945	−1.48056	−0.740282	−	0.672297i	$-0.765308\pi$
$521$	−33.7945	−1.48056	−0.740282	+	0.672297i	$0.765308\pi$
$522$	0	0
$523$	−33.6942	−1.47335	−0.736673	−	0.676249i	$-0.763604\pi$
$523$	−33.6942	−1.47335	−0.736673	+	0.676249i	$0.763604\pi$
$524$	0	0
$525$	4.70038	0.205142
$526$	0	0
$527$	0.646894	0.0281791
$528$	0	0
$529$	46.4914	2.02136
$530$	0	0
$531$	3.27181	0.141984
$532$	0	0
$533$	−2.69074	−0.116549
$534$	0	0
$535$	1.23160	0.0532466
$536$	0	0
$537$	−50.2267	−2.16744
$538$	0	0
$539$	−31.9443	−1.37594
$540$	0	0
$541$	−15.7237	−0.676016	−0.338008	−	0.941143i	$-0.609753\pi$
$541$	−15.7237	−0.676016	−0.338008	+	0.941143i	$0.609753\pi$
$542$	0	0
$543$	−58.8139	−2.52395
$544$	0	0
$545$	−11.6823	−0.500414
$546$	0	0
$547$	−8.48334	−0.362721	−0.181361	−	0.983417i	$-0.558050\pi$
$547$	−8.48334	−0.362721	−0.181361	+	0.983417i	$0.558050\pi$
$548$	0	0
$549$	29.0616	1.24032
$550$	0	0
$551$	−9.18190	−0.391162
$552$	0	0
$553$	−5.40612	−0.229892
$554$	0	0
$555$	38.9910	1.65508
$556$	0	0
$557$	−7.75178	−0.328453	−0.164227	−	0.986423i	$-0.552513\pi$
$557$	−7.75178	−0.328453	−0.164227	+	0.986423i	$0.552513\pi$
$558$	0	0
$559$	4.16356	0.176100
$560$	0	0
$561$	−11.8792	−0.501542
$562$	0	0
$563$	25.7280	1.08430	0.542152	−	0.840280i	$-0.317610\pi$
$563$	25.7280	1.08430	0.542152	+	0.840280i	$0.317610\pi$
$564$	0	0
$565$	−42.4927	−1.78768
$566$	0	0
$567$	4.17974	0.175532
$568$	0	0
$569$	−20.0156	−0.839097	−0.419548	−	0.907733i	$-0.637812\pi$
$569$	−20.0156	−0.839097	−0.419548	+	0.907733i	$0.637812\pi$
$570$	0	0
$571$	6.69660	0.280244	0.140122	−	0.990134i	$-0.455250\pi$
$571$	6.69660	0.280244	0.140122	+	0.990134i	$0.455250\pi$
$572$	0	0
$573$	−33.4310	−1.39660
$574$	0	0
$575$	30.3607	1.26613
$576$	0	0
$577$	−14.3457	−0.597220	−0.298610	−	0.954375i	$-0.596523\pi$
$577$	−14.3457	−0.597220	−0.298610	+	0.954375i	$0.596523\pi$
$578$	0	0
$579$	−21.1728	−0.879911
$580$	0	0
$581$	−0.844155	−0.0350214
$582$	0	0
$583$	17.7994	0.737174
$584$	0	0
$585$	5.11776	0.211593
$586$	0	0
$587$	−3.84159	−0.158559	−0.0792796	−	0.996852i	$-0.525262\pi$
$587$	−3.84159	−0.158559	−0.0792796	+	0.996852i	$0.525262\pi$
$588$	0	0
$589$	2.75061	0.113337
$590$	0	0
$591$	−40.2591	−1.65604
$592$	0	0
$593$	−4.85073	−0.199195	−0.0995977	−	0.995028i	$-0.531756\pi$
$593$	−4.85073	−0.199195	−0.0995977	+	0.995028i	$0.531756\pi$
$594$	0	0
$595$	−1.51495	−0.0621070
$596$	0	0
$597$	55.6692	2.27839
$598$	0	0
$599$	10.5869	0.432570	0.216285	−	0.976330i	$-0.430606\pi$
$599$	10.5869	0.432570	0.216285	+	0.976330i	$0.430606\pi$
$600$	0	0
$601$	2.34081	0.0954836	0.0477418	−	0.998860i	$-0.484798\pi$
$601$	2.34081	0.0954836	0.0477418	+	0.998860i	$0.484798\pi$
$602$	0	0
$603$	−16.3165	−0.664460
$604$	0	0
$605$	33.8072	1.37446
$606$	0	0
$607$	11.7198	0.475694	0.237847	−	0.971303i	$-0.423558\pi$
$607$	11.7198	0.475694	0.237847	+	0.971303i	$0.423558\pi$
$608$	0	0
$609$	−2.78690	−0.112931
$610$	0	0
$611$	3.87650	0.156826
$612$	0	0
$613$	9.54866	0.385667	0.192833	−	0.981232i	$-0.438232\pi$
$613$	9.54866	0.385667	0.192833	+	0.981232i	$0.438232\pi$
$614$	0	0
$615$	37.2300	1.50126
$616$	0	0
$617$	13.5368	0.544970	0.272485	−	0.962160i	$-0.412154\pi$
$617$	13.5368	0.544970	0.272485	+	0.962160i	$0.412154\pi$
$618$	0	0
$619$	7.08344	0.284707	0.142354	−	0.989816i	$-0.454533\pi$
$619$	7.08344	0.284707	0.142354	+	0.989816i	$0.454533\pi$
$620$	0	0
$621$	−5.67440	−0.227706
$622$	0	0
$623$	4.25196	0.170351
$624$	0	0
$625$	−29.9457	−1.19783
$626$	0	0
$627$	−50.5109	−2.01721
$628$	0	0
$629$	−5.29614	−0.211171
$630$	0	0
$631$	−33.5250	−1.33461	−0.667305	−	0.744784i	$-0.732552\pi$
$631$	−33.5250	−1.33461	−0.667305	+	0.744784i	$0.732552\pi$
$632$	0	0
$633$	−3.49471	−0.138902
$634$	0	0
$635$	22.7256	0.901837
$636$	0	0
$637$	−3.58331	−0.141976
$638$	0	0
$639$	−40.5800	−1.60532
$640$	0	0
$641$	−1.29128	−0.0510027	−0.0255013	−	0.999675i	$-0.508118\pi$
$641$	−1.29128	−0.0510027	−0.0255013	+	0.999675i	$0.508118\pi$
$642$	0	0
$643$	27.4360	1.08197	0.540985	−	0.841032i	$-0.318052\pi$
$643$	27.4360	1.08197	0.540985	+	0.841032i	$0.318052\pi$
$644$	0	0
$645$	−57.6084	−2.26833
$646$	0	0
$647$	1.61143	0.0633518	0.0316759	−	0.999498i	$-0.489916\pi$
$647$	1.61143	0.0633518	0.0316759	+	0.999498i	$0.489916\pi$
$648$	0	0
$649$	4.74343	0.186196
$650$	0	0
$651$	0.834871	0.0327212
$652$	0	0
$653$	−6.42149	−0.251292	−0.125646	−	0.992075i	$-0.540100\pi$
$653$	−6.42149	−0.251292	−0.125646	+	0.992075i	$0.540100\pi$
$654$	0	0
$655$	−52.6480	−2.05713
$656$	0	0
$657$	−2.88328	−0.112488
$658$	0	0
$659$	−43.0705	−1.67779	−0.838895	−	0.544293i	$-0.816798\pi$
$659$	−43.0705	−1.67779	−0.838895	+	0.544293i	$0.816798\pi$
$660$	0	0
$661$	30.6538	1.19229	0.596147	−	0.802876i	$-0.296698\pi$
$661$	30.6538	1.19229	0.596147	+	0.802876i	$0.296698\pi$
$662$	0	0
$663$	−1.33254	−0.0517516
$664$	0	0
$665$	−6.44163	−0.249796
$666$	0	0
$667$	−18.0012	−0.697008
$668$	0	0
$669$	−49.4405	−1.91148
$670$	0	0
$671$	42.1332	1.62654
$672$	0	0
$673$	−9.52866	−0.367303	−0.183651	−	0.982991i	$-0.558792\pi$
$673$	−9.52866	−0.367303	−0.183651	+	0.982991i	$0.558792\pi$
$674$	0	0
$675$	−2.47914	−0.0954221
$676$	0	0
$677$	−25.8650	−0.994074	−0.497037	−	0.867729i	$-0.665579\pi$
$677$	−25.8650	−0.994074	−0.497037	+	0.867729i	$0.665579\pi$
$678$	0	0
$679$	−2.75490	−0.105723
$680$	0	0
$681$	−33.6731	−1.29036
$682$	0	0
$683$	9.54105	0.365078	0.182539	−	0.983199i	$-0.441568\pi$
$683$	9.54105	0.365078	0.182539	+	0.983199i	$0.441568\pi$
$684$	0	0
$685$	40.6374	1.55268
$686$	0	0
$687$	19.4731	0.742945
$688$	0	0
$689$	1.99662	0.0760653
$690$	0	0
$691$	10.6723	0.405995	0.202997	−	0.979179i	$-0.434932\pi$
$691$	10.6723	0.405995	0.202997	+	0.979179i	$0.434932\pi$
$692$	0	0
$693$	−7.99779	−0.303811
$694$	0	0
$695$	42.2061	1.60097
$696$	0	0
$697$	−5.05695	−0.191545
$698$	0	0
$699$	−50.9838	−1.92838
$700$	0	0
$701$	35.5290	1.34191	0.670955	−	0.741498i	$-0.265884\pi$
$701$	35.5290	1.34191	0.670955	+	0.741498i	$0.265884\pi$
$702$	0	0
$703$	−22.5194	−0.849335
$704$	0	0
$705$	−53.6365	−2.02007
$706$	0	0
$707$	2.01917	0.0759389
$708$	0	0
$709$	14.2457	0.535009	0.267504	−	0.963557i	$-0.413801\pi$
$709$	14.2457	0.535009	0.267504	+	0.963557i	$0.413801\pi$
$710$	0	0
$711$	34.3228	1.28721
$712$	0	0
$713$	5.39260	0.201955
$714$	0	0
$715$	7.41967	0.277480
$716$	0	0
$717$	45.1102	1.68467
$718$	0	0
$719$	−39.4414	−1.47092	−0.735458	−	0.677571i	$-0.763033\pi$
$719$	−39.4414	−1.47092	−0.735458	+	0.677571i	$0.763033\pi$
$720$	0	0
$721$	3.22533	0.120118
$722$	0	0
$723$	−67.5910	−2.51374
$724$	0	0
$725$	−7.86470	−0.292088
$726$	0	0
$727$	−37.9954	−1.40917	−0.704587	−	0.709618i	$-0.748868\pi$
$727$	−37.9954	−1.40917	−0.704587	+	0.709618i	$0.748868\pi$
$728$	0	0
$729$	−31.6508	−1.17225
$730$	0	0
$731$	7.82493	0.289416
$732$	0	0
$733$	−37.5914	−1.38847	−0.694235	−	0.719748i	$-0.744257\pi$
$733$	−37.5914	−1.38847	−0.694235	+	0.719748i	$0.744257\pi$
$734$	0	0
$735$	49.5799	1.82878
$736$	0	0
$737$	−23.6555	−0.871362
$738$	0	0
$739$	−19.5082	−0.717621	−0.358810	−	0.933410i	$-0.616818\pi$
$739$	−19.5082	−0.717621	−0.358810	+	0.933410i	$0.616818\pi$
$740$	0	0
$741$	−5.66601	−0.208146
$742$	0	0
$743$	46.5745	1.70865	0.854327	−	0.519735i	$-0.173969\pi$
$743$	46.5745	1.70865	0.854327	+	0.519735i	$0.173969\pi$
$744$	0	0
$745$	21.9872	0.805550
$746$	0	0
$747$	5.35944	0.196092
$748$	0	0
$749$	−0.215899	−0.00788877
$750$	0	0
$751$	−5.40319	−0.197165	−0.0985826	−	0.995129i	$-0.531431\pi$
$751$	−5.40319	−0.197165	−0.0985826	+	0.995129i	$0.531431\pi$
$752$	0	0
$753$	−2.62575	−0.0956877
$754$	0	0
$755$	−12.6735	−0.461234
$756$	0	0
$757$	−35.6992	−1.29751	−0.648754	−	0.760998i	$-0.724710\pi$
$757$	−35.6992	−1.29751	−0.648754	+	0.760998i	$0.724710\pi$
$758$	0	0
$759$	−99.0271	−3.59445
$760$	0	0
$761$	37.3597	1.35429	0.677144	−	0.735850i	$-0.263217\pi$
$761$	37.3597	1.35429	0.677144	+	0.735850i	$0.263217\pi$
$762$	0	0
$763$	2.04790	0.0741391
$764$	0	0
$765$	9.61825	0.347749
$766$	0	0
$767$	0.532089	0.0192126
$768$	0	0
$769$	1.50523	0.0542798	0.0271399	−	0.999632i	$-0.491360\pi$
$769$	1.50523	0.0542798	0.0271399	+	0.999632i	$0.491360\pi$
$770$	0	0
$771$	−41.0021	−1.47665
$772$	0	0
$773$	50.4437	1.81433	0.907167	−	0.420772i	$-0.138241\pi$
$773$	50.4437	1.81433	0.907167	+	0.420772i	$0.138241\pi$
$774$	0	0
$775$	2.35602	0.0846308
$776$	0	0
$777$	−6.83512	−0.245209
$778$	0	0
$779$	−21.5023	−0.770400
$780$	0	0
$781$	−58.8324	−2.10519
$782$	0	0
$783$	1.46991	0.0525302
$784$	0	0
$785$	−13.0166	−0.464581
$786$	0	0
$787$	−50.6359	−1.80497	−0.902487	−	0.430717i	$-0.858261\pi$
$787$	−50.6359	−1.80497	−0.902487	+	0.430717i	$0.858261\pi$
$788$	0	0
$789$	−5.71646	−0.203511
$790$	0	0
$791$	7.44896	0.264855
$792$	0	0
$793$	4.72625	0.167834
$794$	0	0
$795$	−27.6259	−0.979791
$796$	0	0
$797$	−45.9667	−1.62822	−0.814112	−	0.580708i	$-0.802776\pi$
$797$	−45.9667	−1.62822	−0.814112	+	0.580708i	$0.802776\pi$
$798$	0	0
$799$	7.28544	0.257740
$800$	0	0
$801$	−26.9952	−0.953828
$802$	0	0
$803$	−4.18015	−0.147514
$804$	0	0
$805$	−12.6289	−0.445109
$806$	0	0
$807$	−38.2712	−1.34721
$808$	0	0
$809$	−5.82677	−0.204858	−0.102429	−	0.994740i	$-0.532661\pi$
$809$	−5.82677	−0.204858	−0.102429	+	0.994740i	$0.532661\pi$
$810$	0	0
$811$	6.88557	0.241785	0.120893	−	0.992666i	$-0.461424\pi$
$811$	6.88557	0.241785	0.120893	+	0.992666i	$0.461424\pi$
$812$	0	0
$813$	−49.7356	−1.74430
$814$	0	0
$815$	−60.4132	−2.11618
$816$	0	0
$817$	33.2719	1.16404
$818$	0	0
$819$	−0.897143	−0.0313487
$820$	0	0
$821$	−19.7555	−0.689472	−0.344736	−	0.938700i	$-0.612032\pi$
$821$	−19.7555	−0.689472	−0.344736	+	0.938700i	$0.612032\pi$
$822$	0	0
$823$	−8.74116	−0.304698	−0.152349	−	0.988327i	$-0.548684\pi$
$823$	−8.74116	−0.304698	−0.152349	+	0.988327i	$0.548684\pi$
$824$	0	0
$825$	−43.2648	−1.50629
$826$	0	0
$827$	8.87004	0.308441	0.154221	−	0.988036i	$-0.450713\pi$
$827$	8.87004	0.308441	0.154221	+	0.988036i	$0.450713\pi$
$828$	0	0
$829$	−19.9228	−0.691946	−0.345973	−	0.938244i	$-0.612451\pi$
$829$	−19.9228	−0.691946	−0.345973	+	0.938244i	$0.612451\pi$
$830$	0	0
$831$	−74.5532	−2.58622
$832$	0	0
$833$	−6.73443	−0.233334
$834$	0	0
$835$	−57.8176	−2.00086
$836$	0	0
$837$	−0.440339	−0.0152203
$838$	0	0
$839$	−9.38828	−0.324120	−0.162060	−	0.986781i	$-0.551814\pi$
$839$	−9.38828	−0.324120	−0.162060	+	0.986781i	$0.551814\pi$
$840$	0	0
$841$	−24.3369	−0.839205
$842$	0	0
$843$	69.7887	2.40365
$844$	0	0
$845$	−37.3843	−1.28606
$846$	0	0
$847$	−5.92641	−0.203634
$848$	0	0
$849$	−35.1423	−1.20608
$850$	0	0
$851$	−44.1494	−1.51342
$852$	0	0
$853$	9.52087	0.325988	0.162994	−	0.986627i	$-0.447885\pi$
$853$	9.52087	0.325988	0.162994	+	0.986627i	$0.447885\pi$
$854$	0	0
$855$	40.8971	1.39865
$856$	0	0
$857$	5.39941	0.184441	0.0922203	−	0.995739i	$-0.470604\pi$
$857$	5.39941	0.184441	0.0922203	+	0.995739i	$0.470604\pi$
$858$	0	0
$859$	44.1624	1.50680	0.753400	−	0.657562i	$-0.228412\pi$
$859$	44.1624	1.50680	0.753400	+	0.657562i	$0.228412\pi$
$860$	0	0
$861$	−6.52642	−0.222420
$862$	0	0
$863$	33.7808	1.14991	0.574956	−	0.818184i	$-0.305019\pi$
$863$	33.7808	1.14991	0.574956	+	0.818184i	$0.305019\pi$
$864$	0	0
$865$	0.565725	0.0192352
$866$	0	0
$867$	−2.50436	−0.0850524
$868$	0	0
$869$	49.7608	1.68802
$870$	0	0
$871$	−2.65353	−0.0899115
$872$	0	0
$873$	17.4905	0.591965
$874$	0	0
$875$	2.05722	0.0695467
$876$	0	0
$877$	54.0745	1.82597	0.912984	−	0.407996i	$-0.133772\pi$
$877$	54.0745	1.82597	0.912984	+	0.407996i	$0.133772\pi$
$878$	0	0
$879$	32.1071	1.08295
$880$	0	0
$881$	−22.9006	−0.771540	−0.385770	−	0.922595i	$-0.626064\pi$
$881$	−22.9006	−0.771540	−0.385770	+	0.922595i	$0.626064\pi$
$882$	0	0
$883$	−28.8748	−0.971715	−0.485858	−	0.874038i	$-0.661493\pi$
$883$	−28.8748	−0.971715	−0.485858	+	0.874038i	$0.661493\pi$
$884$	0	0
$885$	−7.36215	−0.247476
$886$	0	0
$887$	12.5859	0.422594	0.211297	−	0.977422i	$-0.432231\pi$
$887$	12.5859	0.422594	0.211297	+	0.977422i	$0.432231\pi$
$888$	0	0
$889$	−3.98379	−0.133612
$890$	0	0
$891$	−38.4725	−1.28888
$892$	0	0
$893$	30.9779	1.03664
$894$	0	0
$895$	58.9586	1.97077
$896$	0	0
$897$	−11.1083	−0.370894
$898$	0	0
$899$	−1.39691	−0.0465895
$900$	0	0
$901$	3.75243	0.125011
$902$	0	0
$903$	10.0987	0.336065
$904$	0	0
$905$	69.0387	2.29492
$906$	0	0
$907$	−23.7741	−0.789406	−0.394703	−	0.918809i	$-0.629152\pi$
$907$	−23.7741	−0.789406	−0.394703	+	0.918809i	$0.629152\pi$
$908$	0	0
$909$	−12.8195	−0.425196
$910$	0	0
$911$	55.6455	1.84362	0.921809	−	0.387645i	$-0.126712\pi$
$911$	55.6455	1.84362	0.921809	+	0.387645i	$0.126712\pi$
$912$	0	0
$913$	7.77005	0.257151
$914$	0	0
$915$	−65.3940	−2.16186
$916$	0	0
$917$	9.22919	0.304775
$918$	0	0
$919$	−8.23504	−0.271649	−0.135824	−	0.990733i	$-0.543368\pi$
$919$	−8.23504	−0.271649	−0.135824	+	0.990733i	$0.543368\pi$
$920$	0	0
$921$	−5.42995	−0.178923
$922$	0	0
$923$	−6.59946	−0.217224
$924$	0	0
$925$	−19.2888	−0.634213
$926$	0	0
$927$	−20.4773	−0.672561
$928$	0	0
$929$	36.8967	1.21054	0.605271	−	0.796019i	$-0.293065\pi$
$929$	36.8967	1.21054	0.605271	+	0.796019i	$0.293065\pi$
$930$	0	0
$931$	−28.6350	−0.938475
$932$	0	0
$933$	28.0757	0.919158
$934$	0	0
$935$	13.9444	0.456031
$936$	0	0
$937$	−41.1209	−1.34336	−0.671681	−	0.740840i	$-0.734428\pi$
$937$	−41.1209	−1.34336	−0.671681	+	0.740840i	$0.734428\pi$
$938$	0	0
$939$	−37.9139	−1.23727
$940$	0	0
$941$	−17.1421	−0.558816	−0.279408	−	0.960172i	$-0.590138\pi$
$941$	−17.1421	−0.558816	−0.279408	+	0.960172i	$0.590138\pi$
$942$	0	0
$943$	−42.1555	−1.37277
$944$	0	0
$945$	1.03122	0.0335457
$946$	0	0
$947$	31.8042	1.03350	0.516749	−	0.856137i	$-0.327142\pi$
$947$	31.8042	1.03350	0.516749	+	0.856137i	$0.327142\pi$
$948$	0	0
$949$	−0.468904	−0.0152213
$950$	0	0
$951$	75.2269	2.43940
$952$	0	0
$953$	−11.1999	−0.362801	−0.181401	−	0.983409i	$-0.558063\pi$
$953$	−11.1999	−0.362801	−0.181401	+	0.983409i	$0.558063\pi$
$954$	0	0
$955$	39.2430	1.26987
$956$	0	0
$957$	25.6522	0.829216
$958$	0	0
$959$	−7.12374	−0.230037
$960$	0	0
$961$	−30.5815	−0.986501
$962$	0	0
$963$	1.37072	0.0441707
$964$	0	0
$965$	24.8536	0.800067
$966$	0	0
$967$	−42.5894	−1.36958	−0.684792	−	0.728739i	$-0.740107\pi$
$967$	−42.5894	−1.36958	−0.684792	+	0.728739i	$0.740107\pi$
$968$	0	0
$969$	−10.6486	−0.342083
$970$	0	0
$971$	−10.7046	−0.343528	−0.171764	−	0.985138i	$-0.554947\pi$
$971$	−10.7046	−0.343528	−0.171764	+	0.985138i	$0.554947\pi$
$972$	0	0
$973$	−7.39873	−0.237193
$974$	0	0
$975$	−4.85319	−0.155426
$976$	0	0
$977$	28.1764	0.901444	0.450722	−	0.892664i	$-0.351167\pi$
$977$	28.1764	0.901444	0.450722	+	0.892664i	$0.351167\pi$
$978$	0	0
$979$	−39.1373	−1.25083
$980$	0	0
$981$	−13.0019	−0.415119
$982$	0	0
$983$	41.5356	1.32478	0.662389	−	0.749160i	$-0.269542\pi$
$983$	41.5356	1.32478	0.662389	+	0.749160i	$0.269542\pi$
$984$	0	0
$985$	47.2581	1.50577
$986$	0	0
$987$	9.40247	0.299284
$988$	0	0
$989$	65.2298	2.07419
$990$	0	0
$991$	6.17258	0.196079	0.0980393	−	0.995183i	$-0.468743\pi$
$991$	6.17258	0.196079	0.0980393	+	0.995183i	$0.468743\pi$
$992$	0	0
$993$	28.1158	0.892227
$994$	0	0
$995$	−65.3473	−2.07165
$996$	0	0
$997$	−9.23057	−0.292335	−0.146168	−	0.989260i	$-0.546694\pi$
$997$	−9.23057	−0.292335	−0.146168	+	0.989260i	$0.546694\pi$
$998$	0	0
$999$	3.60507	0.114059

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.a.j.1.3	✓	21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.a.j.1.3	✓	21	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q-2.50436 q^{3} +2.93974 q^{5} -0.515336 q^{7} +3.27181 q^{9} +O(q^{10})\)	\(q-2.50436 q^{3} +2.93974 q^{5} -0.515336 q^{7} +3.27181 q^{9} +4.74343 q^{11} +0.532089 q^{13} -7.36215 q^{15} +1.00000 q^{17} +4.25204 q^{19} +1.29058 q^{21} +8.33615 q^{23} +3.64206 q^{25} -0.680698 q^{27} -2.15941 q^{29} +0.646894 q^{31} -11.8792 q^{33} -1.51495 q^{35} -5.29614 q^{37} -1.33254 q^{39} -5.05695 q^{41} +7.82493 q^{43} +9.61825 q^{45} +7.28544 q^{47} -6.73443 q^{49} -2.50436 q^{51} +3.75243 q^{53} +13.9444 q^{55} -10.6486 q^{57} +1.00000 q^{59} +8.88245 q^{61} -1.68608 q^{63} +1.56420 q^{65} -4.98701 q^{67} -20.8767 q^{69} -12.4029 q^{71} -0.881251 q^{73} -9.12101 q^{75} -2.44446 q^{77} +10.4905 q^{79} -8.11071 q^{81} +1.63807 q^{83} +2.93974 q^{85} +5.40794 q^{87} -8.25085 q^{89} -0.274204 q^{91} -1.62005 q^{93} +12.4999 q^{95} +5.34584 q^{97} +15.5196 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q + 9 q^{3} + q^{5} + 11 q^{7} + 26 q^{9}+O(q^{10}) \) 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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