LMFDB - Embedded newform 976.2.a.j.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [976,2,Mod(1,976)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(976, 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("976.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 976 = 2^{4} \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	976.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:	$7.79339923728$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.20308.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 8x^{2} + 4x + 12 $ x^4 - x^3 - 8x^2 + 4x + 12
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 244)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.80027$ of defining polynomial
Character	$\chi$	$=$	976.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.04125 q^{3} +3.80027 q^{5} +0.342376 q^{7} -1.91579 q^{9} +O(q^{10})$	$q+1.04125 q^{3} +3.80027 q^{5} +0.342376 q^{7} -1.91579 q^{9} -0.698877 q^{11} +4.84153 q^{13} +3.95705 q^{15} +2.35650 q^{17} -4.64180 q^{19} +0.356500 q^{21} +4.29942 q^{23} +9.44207 q^{25} -5.11858 q^{27} -1.95705 q^{29} +3.87454 q^{31} -0.727707 q^{33} +1.30112 q^{35} -8.72430 q^{37} +5.04125 q^{39} +1.92573 q^{41} +12.8728 q^{43} -7.28053 q^{45} -2.91579 q^{47} -6.88278 q^{49} +2.45371 q^{51} -4.91579 q^{53} -2.65592 q^{55} -4.83329 q^{57} -3.94292 q^{59} -1.00000 q^{61} -0.655922 q^{63} +18.3991 q^{65} +2.25987 q^{67} +4.47679 q^{69} -13.4091 q^{71} -8.48503 q^{73} +9.83158 q^{75} -0.239279 q^{77} +0.134409 q^{79} +0.417636 q^{81} +5.83158 q^{83} +8.95534 q^{85} -2.03778 q^{87} -11.9683 q^{89} +1.65762 q^{91} +4.03438 q^{93} -17.6401 q^{95} +13.2011 q^{97} +1.33890 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 5 q^{5} + q^{7} + 12 q^{9}+O(q^{10}) $ 4 * q + 5 * q^5 + q^7 + 12 * q^9	$ 4 q + 5 q^{5} + q^{7} + 12 q^{9} + q^{11} + 5 q^{13} - 8 q^{15} + 6 q^{17} + 6 q^{19} - 2 q^{21} - 7 q^{23} + 3 q^{25} + 12 q^{27} + 16 q^{29} - 26 q^{33} + 9 q^{35} - 2 q^{37} + 16 q^{39} + 13 q^{41} + 8 q^{43} - 3 q^{45} + 8 q^{47} - 9 q^{49} + 28 q^{51} + 17 q^{55} - 8 q^{57} + 5 q^{59} - 4 q^{61} + 25 q^{63} + 15 q^{65} + 17 q^{67} + 10 q^{69} - 20 q^{71} - 23 q^{73} + 21 q^{77} - 7 q^{79} + 44 q^{81} - 16 q^{83} - 16 q^{85} - 12 q^{87} + 2 q^{89} + 7 q^{91} - 36 q^{93} - 18 q^{95} + 12 q^{97} + 13 q^{99}+O(q^{100}) $ 4 * q + 5 * q^5 + q^7 + 12 * q^9 + q^11 + 5 * q^13 - 8 * q^15 + 6 * q^17 + 6 * q^19 - 2 * q^21 - 7 * q^23 + 3 * q^25 + 12 * q^27 + 16 * q^29 - 26 * q^33 + 9 * q^35 - 2 * q^37 + 16 * q^39 + 13 * q^41 + 8 * q^43 - 3 * q^45 + 8 * q^47 - 9 * q^49 + 28 * q^51 + 17 * q^55 - 8 * q^57 + 5 * q^59 - 4 * q^61 + 25 * q^63 + 15 * q^65 + 17 * q^67 + 10 * q^69 - 20 * q^71 - 23 * q^73 + 21 * q^77 - 7 * q^79 + 44 * q^81 - 16 * q^83 - 16 * q^85 - 12 * q^87 + 2 * q^89 + 7 * q^91 - 36 * q^93 - 18 * q^95 + 12 * q^97 + 13 * 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
$3$	1.04125	0.601168	0.300584	−	0.953755i	$-0.402818\pi$
$3$	1.04125	0.601168	0.300584	+	0.953755i	$0.402818\pi$
$4$	0	0
$5$	3.80027	1.69953	0.849767	−	0.527159i	$-0.176743\pi$
$5$	3.80027	1.69953	0.849767	+	0.527159i	$0.176743\pi$
$6$	0	0
$7$	0.342376	0.129406	0.0647030	−	0.997905i	$-0.479390\pi$
$7$	0.342376	0.129406	0.0647030	+	0.997905i	$0.479390\pi$
$8$	0	0
$9$	−1.91579	−0.638597
$10$	0	0
$11$	−0.698877	−0.210719	−0.105360	−	0.994434i	$-0.533599\pi$
$11$	−0.698877	−0.210719	−0.105360	+	0.994434i	$0.533599\pi$
$12$	0	0
$13$	4.84153	1.34280	0.671399	−	0.741096i	$-0.265694\pi$
$13$	4.84153	1.34280	0.671399	+	0.741096i	$0.265694\pi$
$14$	0	0
$15$	3.95705	1.02170
$16$	0	0
$17$	2.35650	0.571535	0.285768	−	0.958299i	$-0.407751\pi$
$17$	2.35650	0.571535	0.285768	+	0.958299i	$0.407751\pi$
$18$	0	0
$19$	−4.64180	−1.06490	−0.532451	−	0.846461i	$-0.678729\pi$
$19$	−4.64180	−1.06490	−0.532451	+	0.846461i	$0.678729\pi$
$20$	0	0
$21$	0.356500	0.0777948
$22$	0	0
$23$	4.29942	0.896491	0.448246	−	0.893910i	$-0.352049\pi$
$23$	4.29942	0.896491	0.448246	+	0.893910i	$0.352049\pi$
$24$	0	0
$25$	9.44207	1.88841
$26$	0	0
$27$	−5.11858	−0.985072
$28$	0	0
$29$	−1.95705	−0.363414	−0.181707	−	0.983353i	$-0.558162\pi$
$29$	−1.95705	−0.363414	−0.181707	+	0.983353i	$0.558162\pi$
$30$	0	0
$31$	3.87454	0.695888	0.347944	−	0.937515i	$-0.386880\pi$
$31$	3.87454	0.695888	0.347944	+	0.937515i	$0.386880\pi$
$32$	0	0
$33$	−0.727707	−0.126678
$34$	0	0
$35$	1.30112	0.219930
$36$	0	0
$37$	−8.72430	−1.43427	−0.717133	−	0.696936i	$-0.754546\pi$
$37$	−8.72430	−1.43427	−0.717133	+	0.696936i	$0.754546\pi$
$38$	0	0
$39$	5.04125	0.807247
$40$	0	0
$41$	1.92573	0.300749	0.150375	−	0.988629i	$-0.451952\pi$
$41$	1.92573	0.300749	0.150375	+	0.988629i	$0.451952\pi$
$42$	0	0
$43$	12.8728	1.96309	0.981545	−	0.191233i	$-0.0612487\pi$
$43$	12.8728	1.96309	0.981545	+	0.191233i	$0.0612487\pi$
$44$	0	0
$45$	−7.28053	−1.08532
$46$	0	0
$47$	−2.91579	−0.425312	−0.212656	−	0.977127i	$-0.568211\pi$
$47$	−2.91579	−0.425312	−0.212656	+	0.977127i	$0.568211\pi$
$48$	0	0
$49$	−6.88278	−0.983254
$50$	0	0
$51$	2.45371	0.343589
$52$	0	0
$53$	−4.91579	−0.675236	−0.337618	−	0.941283i	$-0.609621\pi$
$53$	−4.91579	−0.675236	−0.337618	+	0.941283i	$0.609621\pi$
$54$	0	0
$55$	−2.65592	−0.358124
$56$	0	0
$57$	−4.83329	−0.640184
$58$	0	0
$59$	−3.94292	−0.513325	−0.256662	−	0.966501i	$-0.582623\pi$
$59$	−3.94292	−0.513325	−0.256662	+	0.966501i	$0.582623\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	0	0
$63$	−0.655922	−0.0826384
$64$	0	0
$65$	18.3991	2.28213
$66$	0	0
$67$	2.25987	0.276087	0.138044	−	0.990426i	$-0.455919\pi$
$67$	2.25987	0.276087	0.138044	+	0.990426i	$0.455919\pi$
$68$	0	0
$69$	4.47679	0.538942
$70$	0	0
$71$	−13.4091	−1.59136	−0.795681	−	0.605716i	$-0.792887\pi$
$71$	−13.4091	−1.59136	−0.795681	+	0.605716i	$0.792887\pi$
$72$	0	0
$73$	−8.48503	−0.993097	−0.496549	−	0.868009i	$-0.665400\pi$
$73$	−8.48503	−0.993097	−0.496549	+	0.868009i	$0.665400\pi$
$74$	0	0
$75$	9.83158	1.13525
$76$	0	0
$77$	−0.239279	−0.0272684
$78$	0	0
$79$	0.134409	0.0151222	0.00756112	−	0.999971i	$-0.497593\pi$
$79$	0.134409	0.0151222	0.00756112	+	0.999971i	$0.497593\pi$
$80$	0	0
$81$	0.417636	0.0464040
$82$	0	0
$83$	5.83158	0.640100	0.320050	−	0.947401i	$-0.396300\pi$
$83$	5.83158	0.640100	0.320050	+	0.947401i	$0.396300\pi$
$84$	0	0
$85$	8.95534	0.971343
$86$	0	0
$87$	−2.03778	−0.218473
$88$	0	0
$89$	−11.9683	−1.26864	−0.634321	−	0.773070i	$-0.718720\pi$
$89$	−11.9683	−1.26864	−0.634321	+	0.773070i	$0.718720\pi$
$90$	0	0
$91$	1.65762	0.173766
$92$	0	0
$93$	4.03438	0.418345
$94$	0	0
$95$	−17.6401	−1.80984
$96$	0	0
$97$	13.2011	1.34037	0.670184	−	0.742195i	$-0.266215\pi$
$97$	13.2011	1.34037	0.670184	+	0.742195i	$0.266215\pi$
$98$	0	0
$99$	1.33890	0.134565
$100$	0	0
$101$	15.4751	1.53983	0.769914	−	0.638147i	$-0.220299\pi$
$101$	15.4751	1.53983	0.769914	+	0.638147i	$0.220299\pi$
$102$	0	0
$103$	16.0938	1.58577	0.792885	−	0.609371i	$-0.208578\pi$
$103$	16.0938	1.58577	0.792885	+	0.609371i	$0.208578\pi$
$104$	0	0
$105$	1.35480	0.132215
$106$	0	0
$107$	18.9666	1.83357	0.916787	−	0.399375i	$-0.130773\pi$
$107$	18.9666	1.83357	0.916787	+	0.399375i	$0.130773\pi$
$108$	0	0
$109$	−9.12682	−0.874191	−0.437096	−	0.899415i	$-0.643993\pi$
$109$	−9.12682	−0.874191	−0.437096	+	0.899415i	$0.643993\pi$
$110$	0	0
$111$	−9.08421	−0.862235
$112$	0	0
$113$	13.6748	1.28642	0.643209	−	0.765691i	$-0.277603\pi$
$113$	13.6748	1.28642	0.643209	+	0.765691i	$0.277603\pi$
$114$	0	0
$115$	16.3390	1.52362
$116$	0	0
$117$	−9.27536	−0.857507
$118$	0	0
$119$	0.806810	0.0739601
$120$	0	0
$121$	−10.5116	−0.955597
$122$	0	0
$123$	2.00518	0.180801
$124$	0	0
$125$	16.8811	1.50989
$126$	0	0
$127$	4.24234	0.376447	0.188224	−	0.982126i	$-0.439727\pi$
$127$	4.24234	0.376447	0.188224	+	0.982126i	$0.439727\pi$
$128$	0	0
$129$	13.4039	1.18015
$130$	0	0
$131$	−20.0079	−1.74810	−0.874049	−	0.485838i	$-0.838515\pi$
$131$	−20.0079	−1.74810	−0.874049	+	0.485838i	$0.838515\pi$
$132$	0	0
$133$	−1.58924	−0.137805
$134$	0	0
$135$	−19.4520	−1.67416
$136$	0	0
$137$	−3.28053	−0.280275	−0.140137	−	0.990132i	$-0.544754\pi$
$137$	−3.28053	−0.280275	−0.140137	+	0.990132i	$0.544754\pi$
$138$	0	0
$139$	−17.7745	−1.50761	−0.753807	−	0.657096i	$-0.771785\pi$
$139$	−17.7745	−1.50761	−0.753807	+	0.657096i	$0.771785\pi$
$140$	0	0
$141$	−3.03608	−0.255684
$142$	0	0
$143$	−3.38363	−0.282953
$144$	0	0
$145$	−7.43730	−0.617635
$146$	0	0
$147$	−7.16671	−0.591101
$148$	0	0
$149$	−6.37604	−0.522346	−0.261173	−	0.965292i	$-0.584109\pi$
$149$	−6.37604	−0.522346	−0.261173	+	0.965292i	$0.584109\pi$
$150$	0	0
$151$	−23.1723	−1.88573	−0.942866	−	0.333173i	$-0.891881\pi$
$151$	−23.1723	−1.88573	−0.942866	+	0.333173i	$0.891881\pi$
$152$	0	0
$153$	−4.51456	−0.364981
$154$	0	0
$155$	14.7243	1.18268
$156$	0	0
$157$	−6.64180	−0.530073	−0.265037	−	0.964238i	$-0.585384\pi$
$157$	−6.64180	−0.530073	−0.265037	+	0.964238i	$0.585384\pi$
$158$	0	0
$159$	−5.11858	−0.405930
$160$	0	0
$161$	1.47202	0.116011
$162$	0	0
$163$	−10.3248	−0.808705	−0.404352	−	0.914603i	$-0.632503\pi$
$163$	−10.3248	−0.808705	−0.404352	+	0.914603i	$0.632503\pi$
$164$	0	0
$165$	−2.76549	−0.215293
$166$	0	0
$167$	−12.1056	−0.936758	−0.468379	−	0.883528i	$-0.655162\pi$
$167$	−12.1056	−0.936758	−0.468379	+	0.883528i	$0.655162\pi$
$168$	0	0
$169$	10.4404	0.803105
$170$	0	0
$171$	8.89272	0.680043
$172$	0	0
$173$	−8.37957	−0.637087	−0.318544	−	0.947908i	$-0.603194\pi$
$173$	−8.37957	−0.637087	−0.318544	+	0.947908i	$0.603194\pi$
$174$	0	0
$175$	3.23274	0.244372
$176$	0	0
$177$	−4.10558	−0.308594
$178$	0	0
$179$	11.6005	0.867065	0.433533	−	0.901138i	$-0.357267\pi$
$179$	11.6005	0.867065	0.433533	+	0.901138i	$0.357267\pi$
$180$	0	0
$181$	−0.887544	−0.0659706	−0.0329853	−	0.999456i	$-0.510501\pi$
$181$	−0.887544	−0.0659706	−0.0329853	+	0.999456i	$0.510501\pi$
$182$	0	0
$183$	−1.04125	−0.0769716
$184$	0	0
$185$	−33.1547	−2.43758
$186$	0	0
$187$	−1.64690	−0.120433
$188$	0	0
$189$	−1.75248	−0.127474
$190$	0	0
$191$	−24.2565	−1.75514	−0.877568	−	0.479452i	$-0.840835\pi$
$191$	−24.2565	−1.75514	−0.877568	+	0.479452i	$0.840835\pi$
$192$	0	0
$193$	−8.31695	−0.598667	−0.299334	−	0.954149i	$-0.596764\pi$
$193$	−8.31695	−0.598667	−0.299334	+	0.954149i	$0.596764\pi$
$194$	0	0
$195$	19.1581	1.37194
$196$	0	0
$197$	6.88108	0.490256	0.245128	−	0.969491i	$-0.421170\pi$
$197$	6.88108	0.490256	0.245128	+	0.969491i	$0.421170\pi$
$198$	0	0
$199$	11.7656	0.834038	0.417019	−	0.908898i	$-0.363075\pi$
$199$	11.7656	0.834038	0.417019	+	0.908898i	$0.363075\pi$
$200$	0	0
$201$	2.35310	0.165975
$202$	0	0
$203$	−0.670046	−0.0470280
$204$	0	0
$205$	7.31831	0.511133
$206$	0	0
$207$	−8.23680	−0.572497
$208$	0	0
$209$	3.24404	0.224395
$210$	0	0
$211$	10.1221	0.696831	0.348416	−	0.937340i	$-0.386720\pi$
$211$	10.1221	0.696831	0.348416	+	0.937340i	$0.386720\pi$
$212$	0	0
$213$	−13.9622	−0.956676
$214$	0	0
$215$	48.9203	3.33634
$216$	0	0
$217$	1.32655	0.0900521
$218$	0	0
$219$	−8.83506	−0.597018
$220$	0	0
$221$	11.4091	0.767456
$222$	0	0
$223$	−11.9859	−0.802634	−0.401317	−	0.915939i	$-0.631447\pi$
$223$	−11.9859	−0.802634	−0.401317	+	0.915939i	$0.631447\pi$
$224$	0	0
$225$	−18.0890	−1.20594
$226$	0	0
$227$	0.368852	0.0244816	0.0122408	−	0.999925i	$-0.496104\pi$
$227$	0.368852	0.0244816	0.0122408	+	0.999925i	$0.496104\pi$
$228$	0	0
$229$	−3.91443	−0.258673	−0.129336	−	0.991601i	$-0.541285\pi$
$229$	−3.91443	−0.258673	−0.129336	+	0.991601i	$0.541285\pi$
$230$	0	0
$231$	−0.249150	−0.0163929
$232$	0	0
$233$	−15.7769	−1.03358	−0.516788	−	0.856113i	$-0.672873\pi$
$233$	−15.7769	−1.03358	−0.516788	+	0.856113i	$0.672873\pi$
$234$	0	0
$235$	−11.0808	−0.722832
$236$	0	0
$237$	0.139954	0.00909100
$238$	0	0
$239$	−25.5259	−1.65114	−0.825568	−	0.564303i	$-0.809145\pi$
$239$	−25.5259	−1.65114	−0.825568	+	0.564303i	$0.809145\pi$
$240$	0	0
$241$	5.06126	0.326025	0.163012	−	0.986624i	$-0.447879\pi$
$241$	5.06126	0.326025	0.163012	+	0.986624i	$0.447879\pi$
$242$	0	0
$243$	15.7906	1.01297
$244$	0	0
$245$	−26.1564	−1.67107
$246$	0	0
$247$	−22.4734	−1.42995
$248$	0	0
$249$	6.07215	0.384807
$250$	0	0
$251$	18.9553	1.19645	0.598225	−	0.801328i	$-0.295873\pi$
$251$	18.9553	1.19645	0.598225	+	0.801328i	$0.295873\pi$
$252$	0	0
$253$	−3.00477	−0.188908
$254$	0	0
$255$	9.32478	0.583940
$256$	0	0
$257$	26.0791	1.62677	0.813385	−	0.581726i	$-0.197622\pi$
$257$	26.0791	1.62677	0.813385	+	0.581726i	$0.197622\pi$
$258$	0	0
$259$	−2.98700	−0.185603
$260$	0	0
$261$	3.74929	0.232075
$262$	0	0
$263$	−3.72260	−0.229545	−0.114773	−	0.993392i	$-0.536614\pi$
$263$	−3.72260	−0.229545	−0.114773	+	0.993392i	$0.536614\pi$
$264$	0	0
$265$	−18.6813	−1.14759
$266$	0	0
$267$	−12.4621	−0.762667
$268$	0	0
$269$	12.2028	0.744017	0.372009	−	0.928229i	$-0.378669\pi$
$269$	12.2028	0.744017	0.372009	+	0.928229i	$0.378669\pi$
$270$	0	0
$271$	−2.53104	−0.153750	−0.0768750	−	0.997041i	$-0.524494\pi$
$271$	−2.53104	−0.153750	−0.0768750	+	0.997041i	$0.524494\pi$
$272$	0	0
$273$	1.72601	0.104463
$274$	0	0
$275$	−6.59884	−0.397925
$276$	0	0
$277$	−22.7044	−1.36418	−0.682088	−	0.731270i	$-0.738928\pi$
$277$	−22.7044	−1.36418	−0.682088	+	0.731270i	$0.738928\pi$
$278$	0	0
$279$	−7.42281	−0.444392
$280$	0	0
$281$	−11.0148	−0.657087	−0.328543	−	0.944489i	$-0.606558\pi$
$281$	−11.0148	−0.657087	−0.328543	+	0.944489i	$0.606558\pi$
$282$	0	0
$283$	27.7061	1.64696	0.823479	−	0.567347i	$-0.192030\pi$
$283$	27.7061	1.64696	0.823479	+	0.567347i	$0.192030\pi$
$284$	0	0
$285$	−18.3678	−1.08801
$286$	0	0
$287$	0.659325	0.0389188
$288$	0	0
$289$	−11.4469	−0.673347
$290$	0	0
$291$	13.7457	0.805786
$292$	0	0
$293$	2.53622	0.148168	0.0740838	−	0.997252i	$-0.476397\pi$
$293$	2.53622	0.148168	0.0740838	+	0.997252i	$0.476397\pi$
$294$	0	0
$295$	−14.9842	−0.872412
$296$	0	0
$297$	3.57726	0.207574
$298$	0	0
$299$	20.8158	1.20381
$300$	0	0
$301$	4.40735	0.254036
$302$	0	0
$303$	16.1135	0.925695
$304$	0	0
$305$	−3.80027	−0.217603
$306$	0	0
$307$	−30.8322	−1.75969	−0.879844	−	0.475262i	$-0.842353\pi$
$307$	−30.8322	−1.75969	−0.879844	+	0.475262i	$0.842353\pi$
$308$	0	0
$309$	16.7577	0.953314
$310$	0	0
$311$	13.8966	0.788002	0.394001	−	0.919110i	$-0.371091\pi$
$311$	13.8966	0.788002	0.394001	+	0.919110i	$0.371091\pi$
$312$	0	0
$313$	−3.11858	−0.176273	−0.0881364	−	0.996108i	$-0.528091\pi$
$313$	−3.11858	−0.176273	−0.0881364	+	0.996108i	$0.528091\pi$
$314$	0	0
$315$	−2.49268	−0.140447
$316$	0	0
$317$	30.2785	1.70061	0.850305	−	0.526291i	$-0.176418\pi$
$317$	30.2785	1.70061	0.850305	+	0.526291i	$0.176418\pi$
$318$	0	0
$319$	1.36773	0.0765784
$320$	0	0
$321$	19.7491	1.10229
$322$	0	0
$323$	−10.9384	−0.608629
$324$	0	0
$325$	45.7140	2.53576
$326$	0	0
$327$	−9.50333	−0.525535
$328$	0	0
$329$	−0.998298	−0.0550380
$330$	0	0
$331$	−6.43665	−0.353790	−0.176895	−	0.984230i	$-0.556605\pi$
$331$	−6.43665	−0.353790	−0.176895	+	0.984230i	$0.556605\pi$
$332$	0	0
$333$	16.7140	0.915919
$334$	0	0
$335$	8.58812	0.469219
$336$	0	0
$337$	−20.1169	−1.09584	−0.547918	−	0.836532i	$-0.684579\pi$
$337$	−20.1169	−1.09584	−0.547918	+	0.836532i	$0.684579\pi$
$338$	0	0
$339$	14.2389	0.773353
$340$	0	0
$341$	−2.70783	−0.146637
$342$	0	0
$343$	−4.75313	−0.256645
$344$	0	0
$345$	17.0130	0.915949
$346$	0	0
$347$	−1.29830	−0.0696965	−0.0348483	−	0.999393i	$-0.511095\pi$
$347$	−1.29830	−0.0696965	−0.0348483	+	0.999393i	$0.511095\pi$
$348$	0	0
$349$	−18.3050	−0.979842	−0.489921	−	0.871767i	$-0.662974\pi$
$349$	−18.3050	−0.979842	−0.489921	+	0.871767i	$0.662974\pi$
$350$	0	0
$351$	−24.7818	−1.32275
$352$	0	0
$353$	17.2324	0.917188	0.458594	−	0.888646i	$-0.348353\pi$
$353$	17.2324	0.917188	0.458594	+	0.888646i	$0.348353\pi$
$354$	0	0
$355$	−50.9581	−2.70457
$356$	0	0
$357$	0.840093	0.0444625
$358$	0	0
$359$	4.58754	0.242121	0.121061	−	0.992645i	$-0.461370\pi$
$359$	4.58754	0.242121	0.121061	+	0.992645i	$0.461370\pi$
$360$	0	0
$361$	2.54629	0.134015
$362$	0	0
$363$	−10.9452	−0.574474
$364$	0	0
$365$	−32.2454	−1.68780
$366$	0	0
$367$	−2.77903	−0.145064	−0.0725320	−	0.997366i	$-0.523108\pi$
$367$	−2.77903	−0.145064	−0.0725320	+	0.997366i	$0.523108\pi$
$368$	0	0
$369$	−3.68930	−0.192058
$370$	0	0
$371$	−1.68305	−0.0873796
$372$	0	0
$373$	−10.5276	−0.545101	−0.272550	−	0.962142i	$-0.587867\pi$
$373$	−10.5276	−0.545101	−0.272550	+	0.962142i	$0.587867\pi$
$374$	0	0
$375$	17.5775	0.907697
$376$	0	0
$377$	−9.47508	−0.487992
$378$	0	0
$379$	−17.8745	−0.918153	−0.459077	−	0.888397i	$-0.651820\pi$
$379$	−17.8745	−0.918153	−0.459077	+	0.888397i	$0.651820\pi$
$380$	0	0
$381$	4.41735	0.226308
$382$	0	0
$383$	7.72318	0.394636	0.197318	−	0.980340i	$-0.436777\pi$
$383$	7.72318	0.394636	0.197318	+	0.980340i	$0.436777\pi$
$384$	0	0
$385$	−0.909325	−0.0463435
$386$	0	0
$387$	−24.6617	−1.25362
$388$	0	0
$389$	23.5463	1.19384	0.596922	−	0.802299i	$-0.296390\pi$
$389$	23.5463	1.19384	0.596922	+	0.802299i	$0.296390\pi$
$390$	0	0
$391$	10.1316	0.512376
$392$	0	0
$393$	−20.8333	−1.05090
$394$	0	0
$395$	0.510792	0.0257007
$396$	0	0
$397$	14.7209	0.738821	0.369410	−	0.929266i	$-0.379560\pi$
$397$	14.7209	0.738821	0.369410	+	0.929266i	$0.379560\pi$
$398$	0	0
$399$	−1.65480	−0.0828438
$400$	0	0
$401$	30.1829	1.50726	0.753631	−	0.657297i	$-0.228300\pi$
$401$	30.1829	1.50726	0.753631	+	0.657297i	$0.228300\pi$
$402$	0	0
$403$	18.7587	0.934436
$404$	0	0
$405$	1.58713	0.0788651
$406$	0	0
$407$	6.09721	0.302228
$408$	0	0
$409$	0.202791	0.0100274	0.00501369	−	0.999987i	$-0.498404\pi$
$409$	0.202791	0.0100274	0.00501369	+	0.999987i	$0.498404\pi$
$410$	0	0
$411$	−3.41586	−0.168492
$412$	0	0
$413$	−1.34996	−0.0664273
$414$	0	0
$415$	22.1616	1.08787
$416$	0	0
$417$	−18.5078	−0.906329
$418$	0	0
$419$	36.0396	1.76065	0.880324	−	0.474374i	$-0.157325\pi$
$419$	36.0396	1.76065	0.880324	+	0.474374i	$0.157325\pi$
$420$	0	0
$421$	39.0062	1.90105	0.950523	−	0.310655i	$-0.100548\pi$
$421$	39.0062	1.90105	0.950523	+	0.310655i	$0.100548\pi$
$422$	0	0
$423$	5.58605	0.271603
$424$	0	0
$425$	22.2502	1.07930
$426$	0	0
$427$	−0.342376	−0.0165688
$428$	0	0
$429$	−3.52321	−0.170102
$430$	0	0
$431$	0.564467	0.0271894	0.0135947	−	0.999908i	$-0.495673\pi$
$431$	0.564467	0.0271894	0.0135947	+	0.999908i	$0.495673\pi$
$432$	0	0
$433$	−3.63832	−0.174847	−0.0874233	−	0.996171i	$-0.527863\pi$
$433$	−3.63832	−0.174847	−0.0874233	+	0.996171i	$0.527863\pi$
$434$	0	0
$435$	−7.74412	−0.371302
$436$	0	0
$437$	−19.9570	−0.954675
$438$	0	0
$439$	10.8564	0.518146	0.259073	−	0.965858i	$-0.416583\pi$
$439$	10.8564	0.518146	0.259073	+	0.965858i	$0.416583\pi$
$440$	0	0
$441$	13.1860	0.627903
$442$	0	0
$443$	11.7373	0.557656	0.278828	−	0.960341i	$-0.410054\pi$
$443$	11.7373	0.557656	0.278828	+	0.960341i	$0.410054\pi$
$444$	0	0
$445$	−45.4830	−2.15610
$446$	0	0
$447$	−6.63907	−0.314017
$448$	0	0
$449$	32.4274	1.53034	0.765171	−	0.643827i	$-0.222654\pi$
$449$	32.4274	1.53034	0.765171	+	0.643827i	$0.222654\pi$
$450$	0	0
$451$	−1.34585	−0.0633736
$452$	0	0
$453$	−24.1282	−1.13364
$454$	0	0
$455$	6.29942	0.295321
$456$	0	0
$457$	−29.1746	−1.36473	−0.682365	−	0.731011i	$-0.739049\pi$
$457$	−29.1746	−1.36473	−0.682365	+	0.731011i	$0.739049\pi$
$458$	0	0
$459$	−12.0619	−0.563003
$460$	0	0
$461$	−1.03608	−0.0482549	−0.0241275	−	0.999709i	$-0.507681\pi$
$461$	−1.03608	−0.0482549	−0.0241275	+	0.999709i	$0.507681\pi$
$462$	0	0
$463$	23.2576	1.08087	0.540436	−	0.841385i	$-0.318259\pi$
$463$	23.2576	1.08087	0.540436	+	0.841385i	$0.318259\pi$
$464$	0	0
$465$	15.3317	0.710992
$466$	0	0
$467$	23.6802	1.09579	0.547895	−	0.836547i	$-0.315429\pi$
$467$	23.6802	1.09579	0.547895	+	0.836547i	$0.315429\pi$
$468$	0	0
$469$	0.773726	0.0357274
$470$	0	0
$471$	−6.91579	−0.318663
$472$	0	0
$473$	−8.99653	−0.413661
$474$	0	0
$475$	−43.8282	−2.01097
$476$	0	0
$477$	9.41764	0.431204
$478$	0	0
$479$	19.5264	0.892184	0.446092	−	0.894987i	$-0.352815\pi$
$479$	19.5264	0.892184	0.446092	+	0.894987i	$0.352815\pi$
$480$	0	0
$481$	−42.2389	−1.92593
$482$	0	0
$483$	1.53275	0.0697423
$484$	0	0
$485$	50.1677	2.27800
$486$	0	0
$487$	−5.04738	−0.228719	−0.114359	−	0.993439i	$-0.536482\pi$
$487$	−5.04738	−0.228719	−0.114359	+	0.993439i	$0.536482\pi$
$488$	0	0
$489$	−10.7508	−0.486167
$490$	0	0
$491$	25.7192	1.16069	0.580346	−	0.814370i	$-0.302917\pi$
$491$	25.7192	1.16069	0.580346	+	0.814370i	$0.302917\pi$
$492$	0	0
$493$	−4.61178	−0.207704
$494$	0	0
$495$	5.08819	0.228697
$496$	0	0
$497$	−4.59094	−0.205932
$498$	0	0
$499$	−36.3417	−1.62688	−0.813439	−	0.581650i	$-0.802407\pi$
$499$	−36.3417	−1.62688	−0.813439	+	0.581650i	$0.802407\pi$
$500$	0	0
$501$	−12.6050	−0.563149
$502$	0	0
$503$	−25.8112	−1.15087	−0.575433	−	0.817849i	$-0.695166\pi$
$503$	−25.8112	−1.15087	−0.575433	+	0.817849i	$0.695166\pi$
$504$	0	0
$505$	58.8095	2.61699
$506$	0	0
$507$	10.8711	0.482801
$508$	0	0
$509$	29.5061	1.30783	0.653916	−	0.756567i	$-0.273125\pi$
$509$	29.5061	1.30783	0.653916	+	0.756567i	$0.273125\pi$
$510$	0	0
$511$	−2.90507	−0.128513
$512$	0	0
$513$	23.7594	1.04900
$514$	0	0
$515$	61.1609	2.69507
$516$	0	0
$517$	2.03778	0.0896214
$518$	0	0
$519$	−8.72526	−0.382996
$520$	0	0
$521$	−0.536686	−0.0235126	−0.0117563	−	0.999931i	$-0.503742\pi$
$521$	−0.536686	−0.0235126	−0.0117563	+	0.999931i	$0.503742\pi$
$522$	0	0
$523$	−17.9377	−0.784363	−0.392181	−	0.919888i	$-0.628279\pi$
$523$	−17.9377	−0.784363	−0.392181	+	0.919888i	$0.628279\pi$
$524$	0	0
$525$	3.36610	0.146909
$526$	0	0
$527$	9.13035	0.397724
$528$	0	0
$529$	−4.51497	−0.196303
$530$	0	0
$531$	7.55382	0.327808
$532$	0	0
$533$	9.32349	0.403845
$534$	0	0
$535$	72.0784	3.11622
$536$	0	0
$537$	12.0791	0.521252
$538$	0	0
$539$	4.81021	0.207191
$540$	0	0
$541$	−36.5507	−1.57144	−0.785719	−	0.618584i	$-0.787707\pi$
$541$	−36.5507	−1.57144	−0.785719	+	0.618584i	$0.787707\pi$
$542$	0	0
$543$	−0.924158	−0.0396594
$544$	0	0
$545$	−34.6844	−1.48572
$546$	0	0
$547$	−33.6113	−1.43711	−0.718557	−	0.695468i	$-0.755197\pi$
$547$	−33.6113	−1.43711	−0.718557	+	0.695468i	$0.755197\pi$
$548$	0	0
$549$	1.91579	0.0817640
$550$	0	0
$551$	9.08421	0.387000
$552$	0	0
$553$	0.0460186	0.00195691
$554$	0	0
$555$	−34.5225	−1.46540
$556$	0	0
$557$	−25.7882	−1.09268	−0.546340	−	0.837564i	$-0.683979\pi$
$557$	−25.7882	−1.09268	−0.546340	+	0.837564i	$0.683979\pi$
$558$	0	0
$559$	62.3242	2.63603
$560$	0	0
$561$	−1.71484	−0.0724007
$562$	0	0
$563$	25.0413	1.05536	0.527681	−	0.849442i	$-0.323061\pi$
$563$	25.0413	1.05536	0.527681	+	0.849442i	$0.323061\pi$
$564$	0	0
$565$	51.9680	2.18631
$566$	0	0
$567$	0.142989	0.00600496
$568$	0	0
$569$	29.6319	1.24223	0.621116	−	0.783719i	$-0.286680\pi$
$569$	29.6319	1.24223	0.621116	+	0.783719i	$0.286680\pi$
$570$	0	0
$571$	24.9185	1.04281	0.521404	−	0.853310i	$-0.325409\pi$
$571$	24.9185	1.04281	0.521404	+	0.853310i	$0.325409\pi$
$572$	0	0
$573$	−25.2571	−1.05513
$574$	0	0
$575$	40.5954	1.69295
$576$	0	0
$577$	45.3905	1.88963	0.944815	−	0.327604i	$-0.106241\pi$
$577$	45.3905	1.88963	0.944815	+	0.327604i	$0.106241\pi$
$578$	0	0
$579$	−8.66005	−0.359899
$580$	0	0
$581$	1.99660	0.0828328
$582$	0	0
$583$	3.43553	0.142285
$584$	0	0
$585$	−35.2489	−1.45736
$586$	0	0
$587$	36.7752	1.51787	0.758937	−	0.651165i	$-0.225719\pi$
$587$	36.7752	1.51787	0.758937	+	0.651165i	$0.225719\pi$
$588$	0	0
$589$	−17.9848	−0.741052
$590$	0	0
$591$	7.16494	0.294726
$592$	0	0
$593$	−6.93274	−0.284693	−0.142347	−	0.989817i	$-0.545465\pi$
$593$	−6.93274	−0.284693	−0.142347	+	0.989817i	$0.545465\pi$
$594$	0	0
$595$	3.06610	0.125698
$596$	0	0
$597$	12.2509	0.501397
$598$	0	0
$599$	−38.3451	−1.56674	−0.783369	−	0.621557i	$-0.786501\pi$
$599$	−38.3451	−1.56674	−0.783369	+	0.621557i	$0.786501\pi$
$600$	0	0
$601$	31.3861	1.28027	0.640133	−	0.768264i	$-0.278879\pi$
$601$	31.3861	1.28027	0.640133	+	0.768264i	$0.278879\pi$
$602$	0	0
$603$	−4.32944	−0.176308
$604$	0	0
$605$	−39.9468	−1.62407
$606$	0	0
$607$	0.0164787	0.000668851 0	0.000334426	−	1.00000i	$-0.499894\pi$
$607$	0.0164787	0.000668851 0	0.000334426		1.00000i	$0.499894\pi$
$608$	0	0
$609$	−0.697687	−0.0282717
$610$	0	0
$611$	−14.1169	−0.571108
$612$	0	0
$613$	14.4621	0.584118	0.292059	−	0.956400i	$-0.405660\pi$
$613$	14.4621	0.584118	0.292059	+	0.956400i	$0.405660\pi$
$614$	0	0
$615$	7.62021	0.307277
$616$	0	0
$617$	−23.6254	−0.951122	−0.475561	−	0.879683i	$-0.657755\pi$
$617$	−23.6254	−0.951122	−0.475561	+	0.879683i	$0.657755\pi$
$618$	0	0
$619$	19.9932	0.803594	0.401797	−	0.915729i	$-0.368386\pi$
$619$	19.9932	0.803594	0.401797	+	0.915729i	$0.368386\pi$
$620$	0	0
$621$	−22.0069	−0.883108
$622$	0	0
$623$	−4.09768	−0.164170
$624$	0	0
$625$	16.9423	0.677694
$626$	0	0
$627$	3.37787	0.134899
$628$	0	0
$629$	−20.5588	−0.819734
$630$	0	0
$631$	−3.64292	−0.145022	−0.0725111	−	0.997368i	$-0.523101\pi$
$631$	−3.64292	−0.145022	−0.0725111	+	0.997368i	$0.523101\pi$
$632$	0	0
$633$	10.5396	0.418912
$634$	0	0
$635$	16.1221	0.639784
$636$	0	0
$637$	−33.3231	−1.32031
$638$	0	0
$639$	25.6890	1.01624
$640$	0	0
$641$	−17.4796	−0.690402	−0.345201	−	0.938529i	$-0.612189\pi$
$641$	−17.4796	−0.690402	−0.345201	+	0.938529i	$0.612189\pi$
$642$	0	0
$643$	−14.1682	−0.558739	−0.279370	−	0.960184i	$-0.590125\pi$
$643$	−14.1682	−0.558739	−0.279370	+	0.960184i	$0.590125\pi$
$644$	0	0
$645$	50.9384	2.00570
$646$	0	0
$647$	−24.8271	−0.976053	−0.488026	−	0.872829i	$-0.662283\pi$
$647$	−24.8271	−0.976053	−0.488026	+	0.872829i	$0.662283\pi$
$648$	0	0
$649$	2.75562	0.108167
$650$	0	0
$651$	1.38127	0.0541364
$652$	0	0
$653$	35.8989	1.40483	0.702417	−	0.711766i	$-0.252104\pi$
$653$	35.8989	1.40483	0.702417	+	0.711766i	$0.252104\pi$
$654$	0	0
$655$	−76.0355	−2.97095
$656$	0	0
$657$	16.2555	0.634189
$658$	0	0
$659$	14.2905	0.556678	0.278339	−	0.960483i	$-0.410216\pi$
$659$	14.2905	0.556678	0.278339	+	0.960483i	$0.410216\pi$
$660$	0	0
$661$	32.4383	1.26170	0.630852	−	0.775903i	$-0.282705\pi$
$661$	32.4383	1.26170	0.630852	+	0.775903i	$0.282705\pi$
$662$	0	0
$663$	11.8797	0.461370
$664$	0	0
$665$	−6.03955	−0.234204
$666$	0	0
$667$	−8.41416	−0.325798
$668$	0	0
$669$	−12.4803	−0.482517
$670$	0	0
$671$	0.698877	0.0269798
$672$	0	0
$673$	25.4834	0.982315	0.491157	−	0.871071i	$-0.336574\pi$
$673$	25.4834	0.982315	0.491157	+	0.871071i	$0.336574\pi$
$674$	0	0
$675$	−48.3300	−1.86022
$676$	0	0
$677$	30.9976	1.19134	0.595668	−	0.803231i	$-0.296888\pi$
$677$	30.9976	1.19134	0.595668	+	0.803231i	$0.296888\pi$
$678$	0	0
$679$	4.51974	0.173452
$680$	0	0
$681$	0.384069	0.0147175
$682$	0	0
$683$	−13.2406	−0.506639	−0.253320	−	0.967383i	$-0.581522\pi$
$683$	−13.2406	−0.506639	−0.253320	+	0.967383i	$0.581522\pi$
$684$	0	0
$685$	−12.4669	−0.476336
$686$	0	0
$687$	−4.07591	−0.155506
$688$	0	0
$689$	−23.7999	−0.906705
$690$	0	0
$691$	−28.7026	−1.09190	−0.545950	−	0.837818i	$-0.683831\pi$
$691$	−28.7026	−1.09190	−0.545950	+	0.837818i	$0.683831\pi$
$692$	0	0
$693$	0.458409	0.0174135
$694$	0	0
$695$	−67.5480	−2.56224
$696$	0	0
$697$	4.53799	0.171889
$698$	0	0
$699$	−16.4277	−0.621353
$700$	0	0
$701$	−28.1616	−1.06365	−0.531825	−	0.846855i	$-0.678493\pi$
$701$	−28.1616	−1.06365	−0.531825	+	0.846855i	$0.678493\pi$
$702$	0	0
$703$	40.4965	1.52735
$704$	0	0
$705$	−11.5379	−0.434543
$706$	0	0
$707$	5.29830	0.199263
$708$	0	0
$709$	34.6597	1.30167	0.650836	−	0.759218i	$-0.274419\pi$
$709$	34.6597	1.30167	0.650836	+	0.759218i	$0.274419\pi$
$710$	0	0
$711$	−0.257500	−0.00965702
$712$	0	0
$713$	16.6583	0.623857
$714$	0	0
$715$	−12.8587	−0.480889
$716$	0	0
$717$	−26.5790	−0.992609
$718$	0	0
$719$	−30.3813	−1.13303	−0.566516	−	0.824050i	$-0.691709\pi$
$719$	−30.3813	−1.13303	−0.566516	+	0.824050i	$0.691709\pi$
$720$	0	0
$721$	5.51014	0.205208
$722$	0	0
$723$	5.27005	0.195995
$724$	0	0
$725$	−18.4786	−0.686276
$726$	0	0
$727$	−34.7734	−1.28967	−0.644837	−	0.764320i	$-0.723075\pi$
$727$	−34.7734	−1.28967	−0.644837	+	0.764320i	$0.723075\pi$
$728$	0	0
$729$	15.1891	0.562560
$730$	0	0
$731$	30.3348	1.12197
$732$	0	0
$733$	13.0291	0.481242	0.240621	−	0.970619i	$-0.422649\pi$
$733$	13.0291	0.481242	0.240621	+	0.970619i	$0.422649\pi$
$734$	0	0
$735$	−27.2355	−1.00460
$736$	0	0
$737$	−1.57937	−0.0581769
$738$	0	0
$739$	45.6456	1.67910	0.839551	−	0.543281i	$-0.182818\pi$
$739$	45.6456	1.67910	0.839551	+	0.543281i	$0.182818\pi$
$740$	0	0
$741$	−23.4005	−0.859638
$742$	0	0
$743$	34.8072	1.27695	0.638476	−	0.769642i	$-0.279565\pi$
$743$	34.8072	1.27695	0.638476	+	0.769642i	$0.279565\pi$
$744$	0	0
$745$	−24.2307	−0.887744
$746$	0	0
$747$	−11.1721	−0.408766
$748$	0	0
$749$	6.49373	0.237276
$750$	0	0
$751$	−40.1196	−1.46399	−0.731993	−	0.681313i	$-0.761409\pi$
$751$	−40.1196	−1.46399	−0.731993	+	0.681313i	$0.761409\pi$
$752$	0	0
$753$	19.7373	0.719268
$754$	0	0
$755$	−88.0609	−3.20486
$756$	0	0
$757$	6.78550	0.246623	0.123312	−	0.992368i	$-0.460649\pi$
$757$	6.78550	0.246623	0.123312	+	0.992368i	$0.460649\pi$
$758$	0	0
$759$	−3.12872	−0.113565
$760$	0	0
$761$	−29.9914	−1.08719	−0.543594	−	0.839348i	$-0.682937\pi$
$761$	−29.9914	−1.08719	−0.543594	+	0.839348i	$0.682937\pi$
$762$	0	0
$763$	−3.12481	−0.113126
$764$	0	0
$765$	−17.1566	−0.620297
$766$	0	0
$767$	−19.0898	−0.689291
$768$	0	0
$769$	3.21239	0.115842	0.0579209	−	0.998321i	$-0.481553\pi$
$769$	3.21239	0.115842	0.0579209	+	0.998321i	$0.481553\pi$
$770$	0	0
$771$	27.1549	0.977961
$772$	0	0
$773$	−11.5689	−0.416104	−0.208052	−	0.978118i	$-0.566712\pi$
$773$	−11.5689	−0.416104	−0.208052	+	0.978118i	$0.566712\pi$
$774$	0	0
$775$	36.5837	1.31412
$776$	0	0
$777$	−3.11022	−0.111578
$778$	0	0
$779$	−8.93886	−0.320268
$780$	0	0
$781$	9.37128	0.335331
$782$	0	0
$783$	10.0173	0.357989
$784$	0	0
$785$	−25.2406	−0.900877
$786$	0	0
$787$	23.4174	0.834741	0.417370	−	0.908736i	$-0.362952\pi$
$787$	23.4174	0.834741	0.417370	+	0.908736i	$0.362952\pi$
$788$	0	0
$789$	−3.87617	−0.137995
$790$	0	0
$791$	4.68193	0.166470
$792$	0	0
$793$	−4.84153	−0.171928
$794$	0	0
$795$	−19.4520	−0.689892
$796$	0	0
$797$	−55.1657	−1.95407	−0.977035	−	0.213081i	$-0.931650\pi$
$797$	−55.1657	−1.95407	−0.977035	+	0.213081i	$0.931650\pi$
$798$	0	0
$799$	−6.87107	−0.243081
$800$	0	0
$801$	22.9289	0.810152
$802$	0	0
$803$	5.92999	0.209265
$804$	0	0
$805$	5.59408	0.197165
$806$	0	0
$807$	12.7062	0.447279
$808$	0	0
$809$	19.9206	0.700370	0.350185	−	0.936681i	$-0.386119\pi$
$809$	19.9206	0.700370	0.350185	+	0.936681i	$0.386119\pi$
$810$	0	0
$811$	28.2282	0.991227	0.495613	−	0.868543i	$-0.334943\pi$
$811$	28.2282	0.991227	0.495613	+	0.868543i	$0.334943\pi$
$812$	0	0
$813$	−2.63546	−0.0924295
$814$	0	0
$815$	−39.2372	−1.37442
$816$	0	0
$817$	−59.7531	−2.09050
$818$	0	0
$819$	−3.17566	−0.110967
$820$	0	0
$821$	10.4786	0.365704	0.182852	−	0.983140i	$-0.441467\pi$
$821$	10.4786	0.365704	0.182852	+	0.983140i	$0.441467\pi$
$822$	0	0
$823$	9.10115	0.317246	0.158623	−	0.987339i	$-0.449295\pi$
$823$	9.10115	0.317246	0.158623	+	0.987339i	$0.449295\pi$
$824$	0	0
$825$	−6.87107	−0.239220
$826$	0	0
$827$	−15.8547	−0.551320	−0.275660	−	0.961255i	$-0.588896\pi$
$827$	−15.8547	−0.551320	−0.275660	+	0.961255i	$0.588896\pi$
$828$	0	0
$829$	−20.5377	−0.713302	−0.356651	−	0.934238i	$-0.616081\pi$
$829$	−20.5377	−0.713302	−0.356651	+	0.934238i	$0.616081\pi$
$830$	0	0
$831$	−23.6410	−0.820099
$832$	0	0
$833$	−16.2193	−0.561964
$834$	0	0
$835$	−46.0045	−1.59205
$836$	0	0
$837$	−19.8322	−0.685499
$838$	0	0
$839$	−33.2038	−1.14632	−0.573161	−	0.819442i	$-0.694283\pi$
$839$	−33.2038	−1.14632	−0.573161	+	0.819442i	$0.694283\pi$
$840$	0	0
$841$	−25.1700	−0.867930
$842$	0	0
$843$	−11.4692	−0.395019
$844$	0	0
$845$	39.6762	1.36490
$846$	0	0
$847$	−3.59891	−0.123660
$848$	0	0
$849$	28.8491	0.990098
$850$	0	0
$851$	−37.5095	−1.28581
$852$	0	0
$853$	33.2371	1.13802	0.569008	−	0.822332i	$-0.307327\pi$
$853$	33.2371	1.13802	0.569008	+	0.822332i	$0.307327\pi$
$854$	0	0
$855$	33.7948	1.15576
$856$	0	0
$857$	24.5391	0.838239	0.419119	−	0.907931i	$-0.362339\pi$
$857$	24.5391	0.838239	0.419119	+	0.907931i	$0.362339\pi$
$858$	0	0
$859$	30.2457	1.03197	0.515986	−	0.856597i	$-0.327426\pi$
$859$	30.2457	1.03197	0.515986	+	0.856597i	$0.327426\pi$
$860$	0	0
$861$	0.686525	0.0233967
$862$	0	0
$863$	44.9811	1.53118	0.765588	−	0.643331i	$-0.222448\pi$
$863$	44.9811	1.53118	0.765588	+	0.643331i	$0.222448\pi$
$864$	0	0
$865$	−31.8447	−1.08275
$866$	0	0
$867$	−11.9191	−0.404795
$868$	0	0
$869$	−0.0939356	−0.00318655
$870$	0	0
$871$	10.9412	0.370729
$872$	0	0
$873$	−25.2905	−0.855955
$874$	0	0
$875$	5.77968	0.195389
$876$	0	0
$877$	12.7605	0.430890	0.215445	−	0.976516i	$-0.430880\pi$
$877$	12.7605	0.430890	0.215445	+	0.976516i	$0.430880\pi$
$878$	0	0
$879$	2.64085	0.0890735
$880$	0	0
$881$	40.3805	1.36045	0.680227	−	0.733001i	$-0.261881\pi$
$881$	40.3805	1.36045	0.680227	+	0.733001i	$0.261881\pi$
$882$	0	0
$883$	−19.3276	−0.650426	−0.325213	−	0.945641i	$-0.605436\pi$
$883$	−19.3276	−0.650426	−0.325213	+	0.945641i	$0.605436\pi$
$884$	0	0
$885$	−15.6023	−0.524466
$886$	0	0
$887$	23.6050	0.792580	0.396290	−	0.918125i	$-0.370298\pi$
$887$	23.6050	0.792580	0.396290	+	0.918125i	$0.370298\pi$
$888$	0	0
$889$	1.45248	0.0487145
$890$	0	0
$891$	−0.291876	−0.00977821
$892$	0	0
$893$	13.5345	0.452915
$894$	0	0
$895$	44.0852	1.47361
$896$	0	0
$897$	21.6745	0.723690
$898$	0	0
$899$	−7.58265	−0.252895
$900$	0	0
$901$	−11.5841	−0.385921
$902$	0	0
$903$	4.58917	0.152718
$904$	0	0
$905$	−3.37291	−0.112119
$906$	0	0
$907$	39.9433	1.32630	0.663148	−	0.748489i	$-0.269220\pi$
$907$	39.9433	1.32630	0.663148	+	0.748489i	$0.269220\pi$
$908$	0	0
$909$	−29.6470	−0.983330
$910$	0	0
$911$	23.2029	0.768745	0.384373	−	0.923178i	$-0.374418\pi$
$911$	23.2029	0.768745	0.384373	+	0.923178i	$0.374418\pi$
$912$	0	0
$913$	−4.07556	−0.134881
$914$	0	0
$915$	−3.95705	−0.130816
$916$	0	0
$917$	−6.85023	−0.226215
$918$	0	0
$919$	14.8923	0.491250	0.245625	−	0.969365i	$-0.421007\pi$
$919$	14.8923	0.491250	0.245625	+	0.969365i	$0.421007\pi$
$920$	0	0
$921$	−32.1042	−1.05787
$922$	0	0
$923$	−64.9203	−2.13688
$924$	0	0
$925$	−82.3755	−2.70849
$926$	0	0
$927$	−30.8324	−1.01267
$928$	0	0
$929$	−23.1860	−0.760709	−0.380355	−	0.924841i	$-0.624198\pi$
$929$	−23.1860	−0.760709	−0.380355	+	0.924841i	$0.624198\pi$
$930$	0	0
$931$	31.9485	1.04707
$932$	0	0
$933$	14.4698	0.473721
$934$	0	0
$935$	−6.25868	−0.204681
$936$	0	0
$937$	27.1743	0.887745	0.443872	−	0.896090i	$-0.353604\pi$
$937$	27.1743	0.887745	0.443872	+	0.896090i	$0.353604\pi$
$938$	0	0
$939$	−3.24723	−0.105969
$940$	0	0
$941$	42.5538	1.38721	0.693607	−	0.720354i	$-0.256020\pi$
$941$	42.5538	1.38721	0.693607	+	0.720354i	$0.256020\pi$
$942$	0	0
$943$	8.27954	0.269619
$944$	0	0
$945$	−6.65991	−0.216647
$946$	0	0
$947$	52.0564	1.69161	0.845803	−	0.533496i	$-0.179122\pi$
$947$	52.0564	1.69161	0.845803	+	0.533496i	$0.179122\pi$
$948$	0	0
$949$	−41.0805	−1.33353
$950$	0	0
$951$	31.5276	1.02235
$952$	0	0
$953$	31.5377	1.02161	0.510803	−	0.859698i	$-0.329348\pi$
$953$	31.5377	1.02161	0.510803	+	0.859698i	$0.329348\pi$
$954$	0	0
$955$	−92.1812	−2.98291
$956$	0	0
$957$	1.42416	0.0460364
$958$	0	0
$959$	−1.12318	−0.0362693
$960$	0	0
$961$	−15.9879	−0.515740
$962$	0	0
$963$	−36.3362	−1.17092
$964$	0	0
$965$	−31.6067	−1.01745
$966$	0	0
$967$	19.8146	0.637196	0.318598	−	0.947890i	$-0.396788\pi$
$967$	19.8146	0.637196	0.318598	+	0.947890i	$0.396788\pi$
$968$	0	0
$969$	−11.3896	−0.365888
$970$	0	0
$971$	21.3688	0.685758	0.342879	−	0.939380i	$-0.388598\pi$
$971$	21.3688	0.685758	0.342879	+	0.939380i	$0.388598\pi$
$972$	0	0
$973$	−6.08557	−0.195094
$974$	0	0
$975$	47.5999	1.52442
$976$	0	0
$977$	0.0309720	0.000990881 0	0.000495440	−	1.00000i	$-0.499842\pi$
$977$	0.0309720	0.000990881 0	0.000495440		1.00000i	$0.499842\pi$
$978$	0	0
$979$	8.36440	0.267327
$980$	0	0
$981$	17.4851	0.558256
$982$	0	0
$983$	−9.32315	−0.297362	−0.148681	−	0.988885i	$-0.547503\pi$
$983$	−9.32315	−0.297362	−0.148681	+	0.988885i	$0.547503\pi$
$984$	0	0
$985$	26.1500	0.833207
$986$	0	0
$987$	−1.03948	−0.0330870
$988$	0	0
$989$	55.3458	1.75989
$990$	0	0
$991$	32.0344	1.01761	0.508803	−	0.860883i	$-0.330088\pi$
$991$	32.0344	1.01761	0.508803	+	0.860883i	$0.330088\pi$
$992$	0	0
$993$	−6.70218	−0.212687
$994$	0	0
$995$	44.7123	1.41748
$996$	0	0
$997$	−34.7897	−1.10180	−0.550900	−	0.834571i	$-0.685715\pi$
$997$	−34.7897	−1.10180	−0.550900	+	0.834571i	$0.685715\pi$
$998$	0	0
$999$	44.6561	1.41286

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	976.2.a.j.1.3		4
3.2	odd	2		8784.2.a.bt.1.1		4
4.3	odd	2		244.2.a.b.1.2	✓	4
8.3	odd	2		3904.2.a.z.1.3		4
8.5	even	2		3904.2.a.ba.1.2		4
12.11	even	2		2196.2.a.j.1.1		4
20.3	even	4		6100.2.c.g.4149.4		8
20.7	even	4		6100.2.c.g.4149.5		8
20.19	odd	2		6100.2.a.k.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
244.2.a.b.1.2	✓	4	4.3	odd	2
976.2.a.j.1.3		4	1.1	even	1	trivial
2196.2.a.j.1.1		4	12.11	even	2
3904.2.a.z.1.3		4	8.3	odd	2
3904.2.a.ba.1.2		4	8.5	even	2
6100.2.a.k.1.3		4	20.19	odd	2
6100.2.c.g.4149.4		8	20.3	even	4
6100.2.c.g.4149.5		8	20.7	even	4
8784.2.a.bt.1.1		4	3.2	odd	2

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:	\( N \)	\(=\)	\( 976 = 2^{4} \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	976.a (trivial)

\(f(q)\)	\(=\)	\(q+1.04125 q^{3} +3.80027 q^{5} +0.342376 q^{7} -1.91579 q^{9} +O(q^{10})\)	\(q+1.04125 q^{3} +3.80027 q^{5} +0.342376 q^{7} -1.91579 q^{9} -0.698877 q^{11} +4.84153 q^{13} +3.95705 q^{15} +2.35650 q^{17} -4.64180 q^{19} +0.356500 q^{21} +4.29942 q^{23} +9.44207 q^{25} -5.11858 q^{27} -1.95705 q^{29} +3.87454 q^{31} -0.727707 q^{33} +1.30112 q^{35} -8.72430 q^{37} +5.04125 q^{39} +1.92573 q^{41} +12.8728 q^{43} -7.28053 q^{45} -2.91579 q^{47} -6.88278 q^{49} +2.45371 q^{51} -4.91579 q^{53} -2.65592 q^{55} -4.83329 q^{57} -3.94292 q^{59} -1.00000 q^{61} -0.655922 q^{63} +18.3991 q^{65} +2.25987 q^{67} +4.47679 q^{69} -13.4091 q^{71} -8.48503 q^{73} +9.83158 q^{75} -0.239279 q^{77} +0.134409 q^{79} +0.417636 q^{81} +5.83158 q^{83} +8.95534 q^{85} -2.03778 q^{87} -11.9683 q^{89} +1.65762 q^{91} +4.03438 q^{93} -17.6401 q^{95} +13.2011 q^{97} +1.33890 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 5 q^{5} + q^{7} + 12 q^{9}+O(q^{10}) \) 4 * q + 5 * q^5 + q^7 + 12 * q^9	\( 4 q + 5 q^{5} + q^{7} + 12 q^{9} + q^{11} + 5 q^{13} - 8 q^{15} + 6 q^{17} + 6 q^{19} - 2 q^{21} - 7 q^{23} + 3 q^{25} + 12 q^{27} + 16 q^{29} - 26 q^{33} + 9 q^{35} - 2 q^{37} + 16 q^{39} + 13 q^{41} + 8 q^{43} - 3 q^{45} + 8 q^{47} - 9 q^{49} + 28 q^{51} + 17 q^{55} - 8 q^{57} + 5 q^{59} - 4 q^{61} + 25 q^{63} + 15 q^{65} + 17 q^{67} + 10 q^{69} - 20 q^{71} - 23 q^{73} + 21 q^{77} - 7 q^{79} + 44 q^{81} - 16 q^{83} - 16 q^{85} - 12 q^{87} + 2 q^{89} + 7 q^{91} - 36 q^{93} - 18 q^{95} + 12 q^{97} + 13 q^{99}+O(q^{100}) \) 4 * q + 5 * q^5 + q^7 + 12 * q^9 + q^11 + 5 * q^13 - 8 * q^15 + 6 * q^17 + 6 * q^19 - 2 * q^21 - 7 * q^23 + 3 * q^25 + 12 * q^27 + 16 * q^29 - 26 * q^33 + 9 * q^35 - 2 * q^37 + 16 * q^39 + 13 * q^41 + 8 * q^43 - 3 * q^45 + 8 * q^47 - 9 * q^49 + 28 * q^51 + 17 * q^55 - 8 * q^57 + 5 * q^59 - 4 * q^61 + 25 * q^63 + 15 * q^65 + 17 * q^67 + 10 * q^69 - 20 * q^71 - 23 * q^73 + 21 * q^77 - 7 * q^79 + 44 * q^81 - 16 * q^83 - 16 * q^85 - 12 * q^87 + 2 * q^89 + 7 * q^91 - 36 * q^93 - 18 * q^95 + 12 * q^97 + 13 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more