LMFDB - Embedded newform 4016.2.a.f.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4016,2,Mod(1,4016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4016, 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("4016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4016 = 2^{4} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4016.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:	$32.0679214517$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.60853001.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 8x^{4} + 15x^{3} + 20x^{2} - 12x - 5 $ x^6 - 3x^5 - 8x^4 + 15x^3 + 20x^2 - 12*x - 5
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 502)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.303212$ of defining polynomial
Character	$\chi$	$=$	4016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.63228 q^{3} +2.87032 q^{5} -4.63886 q^{7} -0.335646 q^{9} +O(q^{10})$	$q+1.63228 q^{3} +2.87032 q^{5} -4.63886 q^{7} -0.335646 q^{9} +2.30164 q^{11} -5.47674 q^{13} +4.68517 q^{15} +3.36682 q^{17} -6.90916 q^{19} -7.57194 q^{21} +0.413133 q^{23} +3.23871 q^{25} -5.44472 q^{27} +3.71261 q^{29} +4.81396 q^{31} +3.75693 q^{33} -13.3150 q^{35} -9.04384 q^{37} -8.93960 q^{39} -1.24070 q^{41} -5.17421 q^{43} -0.963410 q^{45} -3.72748 q^{47} +14.5190 q^{49} +5.49561 q^{51} -4.50833 q^{53} +6.60642 q^{55} -11.2777 q^{57} +10.3879 q^{59} -7.69836 q^{61} +1.55701 q^{63} -15.7200 q^{65} +12.5478 q^{67} +0.674351 q^{69} -16.5669 q^{71} -8.59665 q^{73} +5.28650 q^{75} -10.6770 q^{77} -0.0752223 q^{79} -7.88040 q^{81} -9.54676 q^{83} +9.66383 q^{85} +6.06003 q^{87} -17.0363 q^{89} +25.4058 q^{91} +7.85776 q^{93} -19.8315 q^{95} +16.0360 q^{97} -0.772535 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - q^{3} - q^{5} - 6 q^{7} + 15 q^{9}+O(q^{10}) $ 6 * q - q^3 - q^5 - 6 * q^7 + 15 * q^9	$ 6 q - q^{3} - q^{5} - 6 q^{7} + 15 q^{9} - q^{11} - 5 q^{13} - 2 q^{15} + 8 q^{17} - 3 q^{19} - 14 q^{21} - 18 q^{23} - q^{25} - 16 q^{27} + q^{29} - 6 q^{31} - 16 q^{33} - 6 q^{35} - 13 q^{37} + 6 q^{39} + 4 q^{41} + 5 q^{43} - 23 q^{45} - 8 q^{47} + 8 q^{49} + 16 q^{51} - 3 q^{53} + 30 q^{55} - 24 q^{57} + 5 q^{59} - 61 q^{61} + 27 q^{63} - q^{65} + 13 q^{67} - 21 q^{69} - 22 q^{71} + 6 q^{73} + 30 q^{75} + 4 q^{77} - 28 q^{79} + 2 q^{81} - 14 q^{83} - 16 q^{85} + 24 q^{87} + 18 q^{89} + 16 q^{91} + 27 q^{93} - 20 q^{95} + 16 q^{97} - 5 q^{99}+O(q^{100}) $ 6 * q - q^3 - q^5 - 6 * q^7 + 15 * q^9 - q^11 - 5 * q^13 - 2 * q^15 + 8 * q^17 - 3 * q^19 - 14 * q^21 - 18 * q^23 - q^25 - 16 * q^27 + q^29 - 6 * q^31 - 16 * q^33 - 6 * q^35 - 13 * q^37 + 6 * q^39 + 4 * q^41 + 5 * q^43 - 23 * q^45 - 8 * q^47 + 8 * q^49 + 16 * q^51 - 3 * q^53 + 30 * q^55 - 24 * q^57 + 5 * q^59 - 61 * q^61 + 27 * q^63 - q^65 + 13 * q^67 - 21 * q^69 - 22 * q^71 + 6 * q^73 + 30 * q^75 + 4 * q^77 - 28 * q^79 + 2 * q^81 - 14 * q^83 - 16 * q^85 + 24 * q^87 + 18 * q^89 + 16 * q^91 + 27 * q^93 - 20 * q^95 + 16 * q^97 - 5 * 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
$3$	1.63228	0.942400	0.471200	−	0.882026i	$-0.343821\pi$
$3$	1.63228	0.942400	0.471200	+	0.882026i	$0.343821\pi$
$4$	0	0
$5$	2.87032	1.28364	0.641822	−	0.766854i	$-0.278179\pi$
$5$	2.87032	1.28364	0.641822	+	0.766854i	$0.278179\pi$
$6$	0	0
$7$	−4.63886	−1.75332	−0.876662	−	0.481107i	$-0.840235\pi$
$7$	−4.63886	−1.75332	−0.876662	+	0.481107i	$0.840235\pi$
$8$	0	0
$9$	−0.335646	−0.111882
$10$	0	0
$11$	2.30164	0.693970	0.346985	−	0.937871i	$-0.387206\pi$
$11$	2.30164	0.693970	0.346985	+	0.937871i	$0.387206\pi$
$12$	0	0
$13$	−5.47674	−1.51897	−0.759487	−	0.650522i	$-0.774550\pi$
$13$	−5.47674	−1.51897	−0.759487	+	0.650522i	$0.774550\pi$
$14$	0	0
$15$	4.68517	1.20971
$16$	0	0
$17$	3.36682	0.816574	0.408287	−	0.912854i	$-0.366126\pi$
$17$	3.36682	0.816574	0.408287	+	0.912854i	$0.366126\pi$
$18$	0	0
$19$	−6.90916	−1.58507	−0.792535	−	0.609826i	$-0.791239\pi$
$19$	−6.90916	−1.58507	−0.792535	+	0.609826i	$0.791239\pi$
$20$	0	0
$21$	−7.57194	−1.65233
$22$	0	0
$23$	0.413133	0.0861442	0.0430721	−	0.999072i	$-0.486285\pi$
$23$	0.413133	0.0861442	0.0430721	+	0.999072i	$0.486285\pi$
$24$	0	0
$25$	3.23871	0.647742
$26$	0	0
$27$	−5.44472	−1.04784
$28$	0	0
$29$	3.71261	0.689414	0.344707	−	0.938710i	$-0.387978\pi$
$29$	3.71261	0.689414	0.344707	+	0.938710i	$0.387978\pi$
$30$	0	0
$31$	4.81396	0.864613	0.432306	−	0.901727i	$-0.357700\pi$
$31$	4.81396	0.864613	0.432306	+	0.901727i	$0.357700\pi$
$32$	0	0
$33$	3.75693	0.653997
$34$	0	0
$35$	−13.3150	−2.25064
$36$	0	0
$37$	−9.04384	−1.48680	−0.743399	−	0.668848i	$-0.766788\pi$
$37$	−9.04384	−1.48680	−0.743399	+	0.668848i	$0.766788\pi$
$38$	0	0
$39$	−8.93960	−1.43148
$40$	0	0
$41$	−1.24070	−0.193765	−0.0968826	−	0.995296i	$-0.530887\pi$
$41$	−1.24070	−0.193765	−0.0968826	+	0.995296i	$0.530887\pi$
$42$	0	0
$43$	−5.17421	−0.789059	−0.394530	−	0.918883i	$-0.629092\pi$
$43$	−5.17421	−0.789059	−0.394530	+	0.918883i	$0.629092\pi$
$44$	0	0
$45$	−0.963410	−0.143617
$46$	0	0
$47$	−3.72748	−0.543709	−0.271855	−	0.962338i	$-0.587637\pi$
$47$	−3.72748	−0.543709	−0.271855	+	0.962338i	$0.587637\pi$
$48$	0	0
$49$	14.5190	2.07414
$50$	0	0
$51$	5.49561	0.769539
$52$	0	0
$53$	−4.50833	−0.619267	−0.309633	−	0.950856i	$-0.600206\pi$
$53$	−4.50833	−0.619267	−0.309633	+	0.950856i	$0.600206\pi$
$54$	0	0
$55$	6.60642	0.890810
$56$	0	0
$57$	−11.2777	−1.49377
$58$	0	0
$59$	10.3879	1.35239	0.676194	−	0.736723i	$-0.263628\pi$
$59$	10.3879	1.35239	0.676194	+	0.736723i	$0.263628\pi$
$60$	0	0
$61$	−7.69836	−0.985674	−0.492837	−	0.870122i	$-0.664040\pi$
$61$	−7.69836	−0.985674	−0.492837	+	0.870122i	$0.664040\pi$
$62$	0	0
$63$	1.55701	0.196165
$64$	0	0
$65$	−15.7200	−1.94982
$66$	0	0
$67$	12.5478	1.53295	0.766477	−	0.642272i	$-0.222008\pi$
$67$	12.5478	1.53295	0.766477	+	0.642272i	$0.222008\pi$
$68$	0	0
$69$	0.674351	0.0811823
$70$	0	0
$71$	−16.5669	−1.96613	−0.983064	−	0.183260i	$-0.941335\pi$
$71$	−16.5669	−1.96613	−0.983064	+	0.183260i	$0.941335\pi$
$72$	0	0
$73$	−8.59665	−1.00616	−0.503081	−	0.864239i	$-0.667800\pi$
$73$	−8.59665	−1.00616	−0.503081	+	0.864239i	$0.667800\pi$
$74$	0	0
$75$	5.28650	0.610432
$76$	0	0
$77$	−10.6770	−1.21675
$78$	0	0
$79$	−0.0752223	−0.00846317	−0.00423159	−	0.999991i	$-0.501347\pi$
$79$	−0.0752223	−0.00846317	−0.00423159	+	0.999991i	$0.501347\pi$
$80$	0	0
$81$	−7.88040	−0.875600
$82$	0	0
$83$	−9.54676	−1.04789	−0.523946	−	0.851751i	$-0.675541\pi$
$83$	−9.54676	−1.04789	−0.523946	+	0.851751i	$0.675541\pi$
$84$	0	0
$85$	9.66383	1.04819
$86$	0	0
$87$	6.06003	0.649704
$88$	0	0
$89$	−17.0363	−1.80585	−0.902924	−	0.429801i	$-0.858584\pi$
$89$	−17.0363	−1.80585	−0.902924	+	0.429801i	$0.858584\pi$
$90$	0	0
$91$	25.4058	2.66325
$92$	0	0
$93$	7.85776	0.814811
$94$	0	0
$95$	−19.8315	−2.03467
$96$	0	0
$97$	16.0360	1.62821	0.814103	−	0.580721i	$-0.197229\pi$
$97$	16.0360	1.62821	0.814103	+	0.580721i	$0.197229\pi$
$98$	0	0
$99$	−0.772535	−0.0776427
$100$	0	0
$101$	−11.7563	−1.16980	−0.584900	−	0.811105i	$-0.698866\pi$
$101$	−11.7563	−1.16980	−0.584900	+	0.811105i	$0.698866\pi$
$102$	0	0
$103$	11.1363	1.09729	0.548647	−	0.836054i	$-0.315143\pi$
$103$	11.1363	1.09729	0.548647	+	0.836054i	$0.315143\pi$
$104$	0	0
$105$	−21.7339	−2.12101
$106$	0	0
$107$	3.44009	0.332566	0.166283	−	0.986078i	$-0.446823\pi$
$107$	3.44009	0.332566	0.166283	+	0.986078i	$0.446823\pi$
$108$	0	0
$109$	−9.11018	−0.872597	−0.436298	−	0.899802i	$-0.643711\pi$
$109$	−9.11018	−0.872597	−0.436298	+	0.899802i	$0.643711\pi$
$110$	0	0
$111$	−14.7621	−1.40116
$112$	0	0
$113$	−6.31389	−0.593961	−0.296980	−	0.954884i	$-0.595980\pi$
$113$	−6.31389	−0.593961	−0.296980	+	0.954884i	$0.595980\pi$
$114$	0	0
$115$	1.18582	0.110579
$116$	0	0
$117$	1.83825	0.169946
$118$	0	0
$119$	−15.6182	−1.43172
$120$	0	0
$121$	−5.70247	−0.518406
$122$	0	0
$123$	−2.02518	−0.182604
$124$	0	0
$125$	−5.05546	−0.452174
$126$	0	0
$127$	−0.0794803	−0.00705274	−0.00352637	−	0.999994i	$-0.501122\pi$
$127$	−0.0794803	−0.00705274	−0.00352637	+	0.999994i	$0.501122\pi$
$128$	0	0
$129$	−8.44578	−0.743609
$130$	0	0
$131$	−2.53326	−0.221332	−0.110666	−	0.993858i	$-0.535298\pi$
$131$	−2.53326	−0.221332	−0.110666	+	0.993858i	$0.535298\pi$
$132$	0	0
$133$	32.0506	2.77914
$134$	0	0
$135$	−15.6281	−1.34505
$136$	0	0
$137$	9.81927	0.838917	0.419459	−	0.907774i	$-0.362220\pi$
$137$	9.81927	0.838917	0.419459	+	0.907774i	$0.362220\pi$
$138$	0	0
$139$	−9.23340	−0.783167	−0.391584	−	0.920143i	$-0.628073\pi$
$139$	−9.23340	−0.783167	−0.391584	+	0.920143i	$0.628073\pi$
$140$	0	0
$141$	−6.08431	−0.512392
$142$	0	0
$143$	−12.6055	−1.05412
$144$	0	0
$145$	10.6564	0.884962
$146$	0	0
$147$	23.6992	1.95467
$148$	0	0
$149$	12.9714	1.06266	0.531329	−	0.847166i	$-0.321693\pi$
$149$	12.9714	1.06266	0.531329	+	0.847166i	$0.321693\pi$
$150$	0	0
$151$	−20.6481	−1.68032	−0.840158	−	0.542342i	$-0.817538\pi$
$151$	−20.6481	−1.68032	−0.840158	+	0.542342i	$0.817538\pi$
$152$	0	0
$153$	−1.13006	−0.0913599
$154$	0	0
$155$	13.8176	1.10986
$156$	0	0
$157$	−9.18457	−0.733008	−0.366504	−	0.930416i	$-0.619445\pi$
$157$	−9.18457	−0.733008	−0.366504	+	0.930416i	$0.619445\pi$
$158$	0	0
$159$	−7.35888	−0.583597
$160$	0	0
$161$	−1.91647	−0.151039
$162$	0	0
$163$	16.9288	1.32597	0.662984	−	0.748633i	$-0.269290\pi$
$163$	16.9288	1.32597	0.662984	+	0.748633i	$0.269290\pi$
$164$	0	0
$165$	10.7836	0.839500
$166$	0	0
$167$	1.88414	0.145799	0.0728996	−	0.997339i	$-0.476775\pi$
$167$	1.88414	0.145799	0.0728996	+	0.997339i	$0.476775\pi$
$168$	0	0
$169$	16.9947	1.30728
$170$	0	0
$171$	2.31903	0.177341
$172$	0	0
$173$	−0.363353	−0.0276252	−0.0138126	−	0.999905i	$-0.504397\pi$
$173$	−0.363353	−0.0276252	−0.0138126	+	0.999905i	$0.504397\pi$
$174$	0	0
$175$	−15.0239	−1.13570
$176$	0	0
$177$	16.9560	1.27449
$178$	0	0
$179$	0.946542	0.0707479	0.0353739	−	0.999374i	$-0.488738\pi$
$179$	0.946542	0.0707479	0.0353739	+	0.999374i	$0.488738\pi$
$180$	0	0
$181$	−15.6678	−1.16458	−0.582288	−	0.812983i	$-0.697842\pi$
$181$	−15.6678	−1.16458	−0.582288	+	0.812983i	$0.697842\pi$
$182$	0	0
$183$	−12.5659	−0.928900
$184$	0	0
$185$	−25.9587	−1.90852
$186$	0	0
$187$	7.74920	0.566677
$188$	0	0
$189$	25.2573	1.83720
$190$	0	0
$191$	13.2104	0.955873	0.477937	−	0.878394i	$-0.341385\pi$
$191$	13.2104	0.955873	0.477937	+	0.878394i	$0.341385\pi$
$192$	0	0
$193$	5.29266	0.380974	0.190487	−	0.981690i	$-0.438993\pi$
$193$	5.29266	0.380974	0.190487	+	0.981690i	$0.438993\pi$
$194$	0	0
$195$	−25.6595	−1.83751
$196$	0	0
$197$	23.2327	1.65526	0.827631	−	0.561272i	$-0.189688\pi$
$197$	23.2327	1.65526	0.827631	+	0.561272i	$0.189688\pi$
$198$	0	0
$199$	9.79983	0.694691	0.347346	−	0.937737i	$-0.387083\pi$
$199$	9.79983	0.694691	0.347346	+	0.937737i	$0.387083\pi$
$200$	0	0
$201$	20.4815	1.44466
$202$	0	0
$203$	−17.2223	−1.20877
$204$	0	0
$205$	−3.56121	−0.248726
$206$	0	0
$207$	−0.138667	−0.00963799
$208$	0	0
$209$	−15.9024	−1.09999
$210$	0	0
$211$	−9.74926	−0.671167	−0.335583	−	0.942011i	$-0.608933\pi$
$211$	−9.74926	−0.671167	−0.335583	+	0.942011i	$0.608933\pi$
$212$	0	0
$213$	−27.0419	−1.85288
$214$	0	0
$215$	−14.8516	−1.01287
$216$	0	0
$217$	−22.3313	−1.51595
$218$	0	0
$219$	−14.0322	−0.948207
$220$	0	0
$221$	−18.4392	−1.24035
$222$	0	0
$223$	4.06139	0.271971	0.135985	−	0.990711i	$-0.456580\pi$
$223$	4.06139	0.271971	0.135985	+	0.990711i	$0.456580\pi$
$224$	0	0
$225$	−1.08706	−0.0724707
$226$	0	0
$227$	3.73589	0.247960	0.123980	−	0.992285i	$-0.460434\pi$
$227$	3.73589	0.247960	0.123980	+	0.992285i	$0.460434\pi$
$228$	0	0
$229$	−15.0833	−0.996731	−0.498366	−	0.866967i	$-0.666066\pi$
$229$	−15.0833	−0.996731	−0.498366	+	0.866967i	$0.666066\pi$
$230$	0	0
$231$	−17.4279	−1.14667
$232$	0	0
$233$	16.6564	1.09119	0.545597	−	0.838047i	$-0.316303\pi$
$233$	16.6564	1.09119	0.545597	+	0.838047i	$0.316303\pi$
$234$	0	0
$235$	−10.6991	−0.697929
$236$	0	0
$237$	−0.122784	−0.00797569
$238$	0	0
$239$	8.31383	0.537777	0.268888	−	0.963171i	$-0.413344\pi$
$239$	8.31383	0.537777	0.268888	+	0.963171i	$0.413344\pi$
$240$	0	0
$241$	3.76545	0.242554	0.121277	−	0.992619i	$-0.461301\pi$
$241$	3.76545	0.242554	0.121277	+	0.992619i	$0.461301\pi$
$242$	0	0
$243$	3.47111	0.222672
$244$	0	0
$245$	41.6741	2.66246
$246$	0	0
$247$	37.8397	2.40768
$248$	0	0
$249$	−15.5830	−0.987534
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	0.950883	0.0597815
$254$	0	0
$255$	15.7741	0.987814
$256$	0	0
$257$	17.0430	1.06311	0.531555	−	0.847024i	$-0.321608\pi$
$257$	17.0430	1.06311	0.531555	+	0.847024i	$0.321608\pi$
$258$	0	0
$259$	41.9531	2.60684
$260$	0	0
$261$	−1.24612	−0.0771330
$262$	0	0
$263$	−8.78669	−0.541810	−0.270905	−	0.962606i	$-0.587323\pi$
$263$	−8.78669	−0.541810	−0.270905	+	0.962606i	$0.587323\pi$
$264$	0	0
$265$	−12.9403	−0.794918
$266$	0	0
$267$	−27.8081	−1.70183
$268$	0	0
$269$	15.4573	0.942448	0.471224	−	0.882013i	$-0.343812\pi$
$269$	15.4573	0.942448	0.471224	+	0.882013i	$0.343812\pi$
$270$	0	0
$271$	6.89090	0.418592	0.209296	−	0.977852i	$-0.432883\pi$
$271$	6.89090	0.418592	0.209296	+	0.977852i	$0.432883\pi$
$272$	0	0
$273$	41.4695	2.50985
$274$	0	0
$275$	7.45434	0.449513
$276$	0	0
$277$	−8.15544	−0.490013	−0.245006	−	0.969521i	$-0.578790\pi$
$277$	−8.15544	−0.490013	−0.245006	+	0.969521i	$0.578790\pi$
$278$	0	0
$279$	−1.61579	−0.0967346
$280$	0	0
$281$	4.16579	0.248510	0.124255	−	0.992250i	$-0.460346\pi$
$281$	4.16579	0.248510	0.124255	+	0.992250i	$0.460346\pi$
$282$	0	0
$283$	11.1550	0.663097	0.331548	−	0.943438i	$-0.392429\pi$
$283$	11.1550	0.663097	0.331548	+	0.943438i	$0.392429\pi$
$284$	0	0
$285$	−32.3706	−1.91747
$286$	0	0
$287$	5.75545	0.339733
$288$	0	0
$289$	−5.66453	−0.333207
$290$	0	0
$291$	26.1753	1.53442
$292$	0	0
$293$	18.4528	1.07802	0.539011	−	0.842299i	$-0.318798\pi$
$293$	18.4528	1.07802	0.539011	+	0.842299i	$0.318798\pi$
$294$	0	0
$295$	29.8165	1.73599
$296$	0	0
$297$	−12.5318	−0.727168
$298$	0	0
$299$	−2.26262	−0.130851
$300$	0	0
$301$	24.0024	1.38348
$302$	0	0
$303$	−19.1897	−1.10242
$304$	0	0
$305$	−22.0967	−1.26525
$306$	0	0
$307$	12.9615	0.739750	0.369875	−	0.929082i	$-0.379401\pi$
$307$	12.9615	0.739750	0.369875	+	0.929082i	$0.379401\pi$
$308$	0	0
$309$	18.1776	1.03409
$310$	0	0
$311$	−18.8328	−1.06791	−0.533955	−	0.845513i	$-0.679295\pi$
$311$	−18.8328	−1.06791	−0.533955	+	0.845513i	$0.679295\pi$
$312$	0	0
$313$	17.7906	1.00558	0.502791	−	0.864408i	$-0.332307\pi$
$313$	17.7906	1.00558	0.502791	+	0.864408i	$0.332307\pi$
$314$	0	0
$315$	4.46912	0.251807
$316$	0	0
$317$	33.0228	1.85474	0.927372	−	0.374141i	$-0.122063\pi$
$317$	33.0228	1.85474	0.927372	+	0.374141i	$0.122063\pi$
$318$	0	0
$319$	8.54508	0.478432
$320$	0	0
$321$	5.61521	0.313410
$322$	0	0
$323$	−23.2619	−1.29433
$324$	0	0
$325$	−17.7376	−0.983903
$326$	0	0
$327$	−14.8704	−0.822335
$328$	0	0
$329$	17.2913	0.953298
$330$	0	0
$331$	18.1365	0.996873	0.498436	−	0.866926i	$-0.333908\pi$
$331$	18.1365	0.996873	0.498436	+	0.866926i	$0.333908\pi$
$332$	0	0
$333$	3.03553	0.166346
$334$	0	0
$335$	36.0160	1.96777
$336$	0	0
$337$	25.0332	1.36365	0.681823	−	0.731517i	$-0.261187\pi$
$337$	25.0332	1.36365	0.681823	+	0.731517i	$0.261187\pi$
$338$	0	0
$339$	−10.3061	−0.559749
$340$	0	0
$341$	11.0800	0.600015
$342$	0	0
$343$	−34.8796	−1.88332
$344$	0	0
$345$	1.93560	0.104209
$346$	0	0
$347$	−34.9223	−1.87473	−0.937363	−	0.348353i	$-0.886741\pi$
$347$	−34.9223	−1.87473	−0.937363	+	0.348353i	$0.886741\pi$
$348$	0	0
$349$	−15.6814	−0.839408	−0.419704	−	0.907661i	$-0.637866\pi$
$349$	−15.6814	−0.839408	−0.419704	+	0.907661i	$0.637866\pi$
$350$	0	0
$351$	29.8193	1.59164
$352$	0	0
$353$	10.6859	0.568753	0.284377	−	0.958713i	$-0.408213\pi$
$353$	10.6859	0.568753	0.284377	+	0.958713i	$0.408213\pi$
$354$	0	0
$355$	−47.5522	−2.52381
$356$	0	0
$357$	−25.4933	−1.34925
$358$	0	0
$359$	−34.6653	−1.82956	−0.914782	−	0.403947i	$-0.867638\pi$
$359$	−34.6653	−1.82956	−0.914782	+	0.403947i	$0.867638\pi$
$360$	0	0
$361$	28.7365	1.51245
$362$	0	0
$363$	−9.30805	−0.488546
$364$	0	0
$365$	−24.6751	−1.29155
$366$	0	0
$367$	1.47086	0.0767783	0.0383891	−	0.999263i	$-0.487777\pi$
$367$	1.47086	0.0767783	0.0383891	+	0.999263i	$0.487777\pi$
$368$	0	0
$369$	0.416437	0.0216788
$370$	0	0
$371$	20.9135	1.08578
$372$	0	0
$373$	18.1346	0.938976	0.469488	−	0.882939i	$-0.344439\pi$
$373$	18.1346	0.938976	0.469488	+	0.882939i	$0.344439\pi$
$374$	0	0
$375$	−8.25195	−0.426129
$376$	0	0
$377$	−20.3330	−1.04720
$378$	0	0
$379$	−20.5720	−1.05671	−0.528355	−	0.849023i	$-0.677191\pi$
$379$	−20.5720	−1.05671	−0.528355	+	0.849023i	$0.677191\pi$
$380$	0	0
$381$	−0.129734	−0.00664650
$382$	0	0
$383$	23.3434	1.19279	0.596397	−	0.802690i	$-0.296598\pi$
$383$	23.3434	1.19279	0.596397	+	0.802690i	$0.296598\pi$
$384$	0	0
$385$	−30.6463	−1.56188
$386$	0	0
$387$	1.73670	0.0882815
$388$	0	0
$389$	−7.03637	−0.356758	−0.178379	−	0.983962i	$-0.557085\pi$
$389$	−7.03637	−0.356758	−0.178379	+	0.983962i	$0.557085\pi$
$390$	0	0
$391$	1.39094	0.0703431
$392$	0	0
$393$	−4.13500	−0.208583
$394$	0	0
$395$	−0.215912	−0.0108637
$396$	0	0
$397$	36.5747	1.83563	0.917817	−	0.397005i	$-0.129950\pi$
$397$	36.5747	1.83563	0.917817	+	0.397005i	$0.129950\pi$
$398$	0	0
$399$	52.3157	2.61906
$400$	0	0
$401$	−32.3014	−1.61306	−0.806528	−	0.591196i	$-0.798656\pi$
$401$	−32.3014	−1.61306	−0.806528	+	0.591196i	$0.798656\pi$
$402$	0	0
$403$	−26.3648	−1.31332
$404$	0	0
$405$	−22.6192	−1.12396
$406$	0	0
$407$	−20.8156	−1.03179
$408$	0	0
$409$	8.42789	0.416733	0.208366	−	0.978051i	$-0.433185\pi$
$409$	8.42789	0.416733	0.208366	+	0.978051i	$0.433185\pi$
$410$	0	0
$411$	16.0279	0.790596
$412$	0	0
$413$	−48.1880	−2.37117
$414$	0	0
$415$	−27.4022	−1.34512
$416$	0	0
$417$	−15.0715	−0.738057
$418$	0	0
$419$	4.41936	0.215900	0.107950	−	0.994156i	$-0.465571\pi$
$419$	4.41936	0.215900	0.107950	+	0.994156i	$0.465571\pi$
$420$	0	0
$421$	−30.4368	−1.48340	−0.741699	−	0.670733i	$-0.765980\pi$
$421$	−30.4368	−1.48340	−0.741699	+	0.670733i	$0.765980\pi$
$422$	0	0
$423$	1.25112	0.0608313
$424$	0	0
$425$	10.9042	0.528929
$426$	0	0
$427$	35.7116	1.72821
$428$	0	0
$429$	−20.5757	−0.993405
$430$	0	0
$431$	−34.9690	−1.68440	−0.842200	−	0.539165i	$-0.818740\pi$
$431$	−34.9690	−1.68440	−0.842200	+	0.539165i	$0.818740\pi$
$432$	0	0
$433$	−19.7229	−0.947821	−0.473910	−	0.880573i	$-0.657158\pi$
$433$	−19.7229	−0.947821	−0.473910	+	0.880573i	$0.657158\pi$
$434$	0	0
$435$	17.3942	0.833988
$436$	0	0
$437$	−2.85440	−0.136545
$438$	0	0
$439$	25.5127	1.21765	0.608827	−	0.793303i	$-0.291640\pi$
$439$	25.5127	1.21765	0.608827	+	0.793303i	$0.291640\pi$
$440$	0	0
$441$	−4.87325	−0.232059
$442$	0	0
$443$	−37.1938	−1.76713	−0.883566	−	0.468308i	$-0.844864\pi$
$443$	−37.1938	−1.76713	−0.883566	+	0.468308i	$0.844864\pi$
$444$	0	0
$445$	−48.8996	−2.31807
$446$	0	0
$447$	21.1730	1.00145
$448$	0	0
$449$	−3.32128	−0.156741	−0.0783705	−	0.996924i	$-0.524972\pi$
$449$	−3.32128	−0.156741	−0.0783705	+	0.996924i	$0.524972\pi$
$450$	0	0
$451$	−2.85565	−0.134467
$452$	0	0
$453$	−33.7035	−1.58353
$454$	0	0
$455$	72.9227	3.41867
$456$	0	0
$457$	9.90999	0.463570	0.231785	−	0.972767i	$-0.425543\pi$
$457$	9.90999	0.463570	0.231785	+	0.972767i	$0.425543\pi$
$458$	0	0
$459$	−18.3314	−0.855637
$460$	0	0
$461$	−11.4921	−0.535242	−0.267621	−	0.963524i	$-0.586238\pi$
$461$	−11.4921	−0.535242	−0.267621	+	0.963524i	$0.586238\pi$
$462$	0	0
$463$	−41.3658	−1.92243	−0.961216	−	0.275797i	$-0.911058\pi$
$463$	−41.3658	−1.92243	−0.961216	+	0.275797i	$0.911058\pi$
$464$	0	0
$465$	22.5542	1.04593
$466$	0	0
$467$	−12.7058	−0.587956	−0.293978	−	0.955812i	$-0.594979\pi$
$467$	−12.7058	−0.587956	−0.293978	+	0.955812i	$0.594979\pi$
$468$	0	0
$469$	−58.2073	−2.68776
$470$	0	0
$471$	−14.9918	−0.690787
$472$	0	0
$473$	−11.9091	−0.547583
$474$	0	0
$475$	−22.3768	−1.02672
$476$	0	0
$477$	1.51320	0.0692848
$478$	0	0
$479$	34.6466	1.58304	0.791522	−	0.611141i	$-0.209289\pi$
$479$	34.6466	1.58304	0.791522	+	0.611141i	$0.209289\pi$
$480$	0	0
$481$	49.5308	2.25841
$482$	0	0
$483$	−3.12822	−0.142339
$484$	0	0
$485$	46.0283	2.09004
$486$	0	0
$487$	−29.4549	−1.33473	−0.667364	−	0.744732i	$-0.732577\pi$
$487$	−29.4549	−1.33473	−0.667364	+	0.744732i	$0.732577\pi$
$488$	0	0
$489$	27.6327	1.24959
$490$	0	0
$491$	−10.6479	−0.480534	−0.240267	−	0.970707i	$-0.577235\pi$
$491$	−10.6479	−0.480534	−0.240267	+	0.970707i	$0.577235\pi$
$492$	0	0
$493$	12.4997	0.562957
$494$	0	0
$495$	−2.21742	−0.0996656
$496$	0	0
$497$	76.8515	3.44726
$498$	0	0
$499$	41.0716	1.83862	0.919310	−	0.393535i	$-0.128748\pi$
$499$	41.0716	1.83862	0.919310	+	0.393535i	$0.128748\pi$
$500$	0	0
$501$	3.07546	0.137401
$502$	0	0
$503$	−14.2097	−0.633581	−0.316791	−	0.948495i	$-0.602605\pi$
$503$	−14.2097	−0.633581	−0.316791	+	0.948495i	$0.602605\pi$
$504$	0	0
$505$	−33.7444	−1.50161
$506$	0	0
$507$	27.7402	1.23198
$508$	0	0
$509$	−12.2017	−0.540829	−0.270415	−	0.962744i	$-0.587161\pi$
$509$	−12.2017	−0.540829	−0.270415	+	0.962744i	$0.587161\pi$
$510$	0	0
$511$	39.8786	1.76413
$512$	0	0
$513$	37.6185	1.66090
$514$	0	0
$515$	31.9647	1.40853
$516$	0	0
$517$	−8.57931	−0.377318
$518$	0	0
$519$	−0.593095	−0.0260340
$520$	0	0
$521$	9.30863	0.407818	0.203909	−	0.978990i	$-0.434635\pi$
$521$	9.30863	0.407818	0.203909	+	0.978990i	$0.434635\pi$
$522$	0	0
$523$	−21.9797	−0.961107	−0.480554	−	0.876965i	$-0.659564\pi$
$523$	−21.9797	−0.961107	−0.480554	+	0.876965i	$0.659564\pi$
$524$	0	0
$525$	−24.5233	−1.07029
$526$	0	0
$527$	16.2077	0.706020
$528$	0	0
$529$	−22.8293	−0.992579
$530$	0	0
$531$	−3.48666	−0.151308
$532$	0	0
$533$	6.79501	0.294324
$534$	0	0
$535$	9.87414	0.426897
$536$	0	0
$537$	1.54503	0.0666728
$538$	0	0
$539$	33.4175	1.43939
$540$	0	0
$541$	−32.7094	−1.40629	−0.703144	−	0.711048i	$-0.748221\pi$
$541$	−32.7094	−1.40629	−0.703144	+	0.711048i	$0.748221\pi$
$542$	0	0
$543$	−25.5743	−1.09750
$544$	0	0
$545$	−26.1491	−1.12010
$546$	0	0
$547$	12.0570	0.515518	0.257759	−	0.966209i	$-0.417016\pi$
$547$	12.0570	0.515518	0.257759	+	0.966209i	$0.417016\pi$
$548$	0	0
$549$	2.58393	0.110279
$550$	0	0
$551$	−25.6510	−1.09277
$552$	0	0
$553$	0.348946	0.0148387
$554$	0	0
$555$	−42.3720	−1.79859
$556$	0	0
$557$	−36.8171	−1.55999	−0.779996	−	0.625785i	$-0.784779\pi$
$557$	−36.8171	−1.55999	−0.779996	+	0.625785i	$0.784779\pi$
$558$	0	0
$559$	28.3378	1.19856
$560$	0	0
$561$	12.6489	0.534037
$562$	0	0
$563$	−0.368628	−0.0155358	−0.00776791	−	0.999970i	$-0.502473\pi$
$563$	−0.368628	−0.0155358	−0.00776791	+	0.999970i	$0.502473\pi$
$564$	0	0
$565$	−18.1229	−0.762434
$566$	0	0
$567$	36.5561	1.53521
$568$	0	0
$569$	24.0359	1.00764	0.503819	−	0.863809i	$-0.331928\pi$
$569$	24.0359	1.00764	0.503819	+	0.863809i	$0.331928\pi$
$570$	0	0
$571$	22.5164	0.942282	0.471141	−	0.882058i	$-0.343842\pi$
$571$	22.5164	0.942282	0.471141	+	0.882058i	$0.343842\pi$
$572$	0	0
$573$	21.5632	0.900815
$574$	0	0
$575$	1.33802	0.0557992
$576$	0	0
$577$	2.72462	0.113427	0.0567137	−	0.998390i	$-0.481938\pi$
$577$	2.72462	0.113427	0.0567137	+	0.998390i	$0.481938\pi$
$578$	0	0
$579$	8.63912	0.359030
$580$	0	0
$581$	44.2861	1.83730
$582$	0	0
$583$	−10.3765	−0.429753
$584$	0	0
$585$	5.27635	0.218150
$586$	0	0
$587$	−18.2284	−0.752365	−0.376182	−	0.926546i	$-0.622763\pi$
$587$	−18.2284	−0.752365	−0.376182	+	0.926546i	$0.622763\pi$
$588$	0	0
$589$	−33.2604	−1.37047
$590$	0	0
$591$	37.9224	1.55992
$592$	0	0
$593$	11.1564	0.458140	0.229070	−	0.973410i	$-0.426431\pi$
$593$	11.1564	0.458140	0.229070	+	0.973410i	$0.426431\pi$
$594$	0	0
$595$	−44.8292	−1.83782
$596$	0	0
$597$	15.9961	0.654677
$598$	0	0
$599$	1.81180	0.0740281	0.0370140	−	0.999315i	$-0.488215\pi$
$599$	1.81180	0.0740281	0.0370140	+	0.999315i	$0.488215\pi$
$600$	0	0
$601$	−12.9149	−0.526810	−0.263405	−	0.964685i	$-0.584846\pi$
$601$	−12.9149	−0.526810	−0.263405	+	0.964685i	$0.584846\pi$
$602$	0	0
$603$	−4.21161	−0.171510
$604$	0	0
$605$	−16.3679	−0.665449
$606$	0	0
$607$	−30.1082	−1.22205	−0.611026	−	0.791610i	$-0.709243\pi$
$607$	−30.1082	−1.22205	−0.611026	+	0.791610i	$0.709243\pi$
$608$	0	0
$609$	−28.1116	−1.13914
$610$	0	0
$611$	20.4145	0.825881
$612$	0	0
$613$	23.1449	0.934814	0.467407	−	0.884042i	$-0.345188\pi$
$613$	23.1449	0.934814	0.467407	+	0.884042i	$0.345188\pi$
$614$	0	0
$615$	−5.81291	−0.234399
$616$	0	0
$617$	16.0919	0.647837	0.323919	−	0.946085i	$-0.395000\pi$
$617$	16.0919	0.647837	0.323919	+	0.946085i	$0.395000\pi$
$618$	0	0
$619$	−15.4108	−0.619413	−0.309706	−	0.950832i	$-0.600231\pi$
$619$	−15.4108	−0.619413	−0.309706	+	0.950832i	$0.600231\pi$
$620$	0	0
$621$	−2.24940	−0.0902652
$622$	0	0
$623$	79.0291	3.16624
$624$	0	0
$625$	−30.7043	−1.22817
$626$	0	0
$627$	−25.9572	−1.03663
$628$	0	0
$629$	−30.4490	−1.21408
$630$	0	0
$631$	14.0058	0.557561	0.278780	−	0.960355i	$-0.410070\pi$
$631$	14.0058	0.557561	0.278780	+	0.960355i	$0.410070\pi$
$632$	0	0
$633$	−15.9136	−0.632507
$634$	0	0
$635$	−0.228134	−0.00905320
$636$	0	0
$637$	−79.5168	−3.15057
$638$	0	0
$639$	5.56061	0.219974
$640$	0	0
$641$	−17.1361	−0.676836	−0.338418	−	0.940996i	$-0.609892\pi$
$641$	−17.1361	−0.676836	−0.338418	+	0.940996i	$0.609892\pi$
$642$	0	0
$643$	31.5872	1.24568	0.622839	−	0.782350i	$-0.285979\pi$
$643$	31.5872	1.24568	0.622839	+	0.782350i	$0.285979\pi$
$644$	0	0
$645$	−24.2421	−0.954530
$646$	0	0
$647$	28.6672	1.12702	0.563512	−	0.826108i	$-0.309450\pi$
$647$	28.6672	1.12702	0.563512	+	0.826108i	$0.309450\pi$
$648$	0	0
$649$	23.9092	0.938517
$650$	0	0
$651$	−36.4510	−1.42863
$652$	0	0
$653$	−11.6536	−0.456042	−0.228021	−	0.973656i	$-0.573226\pi$
$653$	−11.6536	−0.456042	−0.228021	+	0.973656i	$0.573226\pi$
$654$	0	0
$655$	−7.27125	−0.284111
$656$	0	0
$657$	2.88543	0.112571
$658$	0	0
$659$	−31.9217	−1.24349	−0.621747	−	0.783218i	$-0.713577\pi$
$659$	−31.9217	−1.24349	−0.621747	+	0.783218i	$0.713577\pi$
$660$	0	0
$661$	30.7329	1.19537	0.597685	−	0.801731i	$-0.296087\pi$
$661$	30.7329	1.19537	0.597685	+	0.801731i	$0.296087\pi$
$662$	0	0
$663$	−30.0980	−1.16891
$664$	0	0
$665$	91.9954	3.56743
$666$	0	0
$667$	1.53380	0.0593890
$668$	0	0
$669$	6.62935	0.256305
$670$	0	0
$671$	−17.7188	−0.684028
$672$	0	0
$673$	0.990387	0.0381766	0.0190883	−	0.999818i	$-0.493924\pi$
$673$	0.990387	0.0381766	0.0190883	+	0.999818i	$0.493924\pi$
$674$	0	0
$675$	−17.6339	−0.678728
$676$	0	0
$677$	−3.16119	−0.121494	−0.0607471	−	0.998153i	$-0.519348\pi$
$677$	−3.16119	−0.121494	−0.0607471	+	0.998153i	$0.519348\pi$
$678$	0	0
$679$	−74.3886	−2.85477
$680$	0	0
$681$	6.09803	0.233677
$682$	0	0
$683$	38.1167	1.45850	0.729248	−	0.684250i	$-0.239870\pi$
$683$	38.1167	1.45850	0.729248	+	0.684250i	$0.239870\pi$
$684$	0	0
$685$	28.1844	1.07687
$686$	0	0
$687$	−24.6202	−0.939320
$688$	0	0
$689$	24.6910	0.940651
$690$	0	0
$691$	−30.3637	−1.15509	−0.577546	−	0.816358i	$-0.695989\pi$
$691$	−30.3637	−1.15509	−0.577546	+	0.816358i	$0.695989\pi$
$692$	0	0
$693$	3.58368	0.136133
$694$	0	0
$695$	−26.5028	−1.00531
$696$	0	0
$697$	−4.17722	−0.158224
$698$	0	0
$699$	27.1879	1.02834
$700$	0	0
$701$	21.5013	0.812091	0.406046	−	0.913853i	$-0.366907\pi$
$701$	21.5013	0.812091	0.406046	+	0.913853i	$0.366907\pi$
$702$	0	0
$703$	62.4854	2.35668
$704$	0	0
$705$	−17.4639	−0.657729
$706$	0	0
$707$	54.5360	2.05104
$708$	0	0
$709$	22.1149	0.830542	0.415271	−	0.909698i	$-0.363687\pi$
$709$	22.1149	0.830542	0.415271	+	0.909698i	$0.363687\pi$
$710$	0	0
$711$	0.0252481	0.000946877 0
$712$	0	0
$713$	1.98881	0.0744814
$714$	0	0
$715$	−36.1817	−1.35312
$716$	0	0
$717$	13.5705	0.506801
$718$	0	0
$719$	−34.6275	−1.29139	−0.645694	−	0.763596i	$-0.723432\pi$
$719$	−34.6275	−1.29139	−0.645694	+	0.763596i	$0.723432\pi$
$720$	0	0
$721$	−51.6598	−1.92391
$722$	0	0
$723$	6.14629	0.228583
$724$	0	0
$725$	12.0241	0.446562
$726$	0	0
$727$	10.6479	0.394907	0.197454	−	0.980312i	$-0.436733\pi$
$727$	10.6479	0.394907	0.197454	+	0.980312i	$0.436733\pi$
$728$	0	0
$729$	29.3071	1.08545
$730$	0	0
$731$	−17.4206	−0.644325
$732$	0	0
$733$	−1.67308	−0.0617966	−0.0308983	−	0.999523i	$-0.509837\pi$
$733$	−1.67308	−0.0617966	−0.0308983	+	0.999523i	$0.509837\pi$
$734$	0	0
$735$	68.0241	2.50911
$736$	0	0
$737$	28.8804	1.06382
$738$	0	0
$739$	23.8919	0.878878	0.439439	−	0.898272i	$-0.355177\pi$
$739$	23.8919	0.878878	0.439439	+	0.898272i	$0.355177\pi$
$740$	0	0
$741$	61.7651	2.26900
$742$	0	0
$743$	−42.4663	−1.55794	−0.778968	−	0.627064i	$-0.784257\pi$
$743$	−42.4663	−1.55794	−0.778968	+	0.627064i	$0.784257\pi$
$744$	0	0
$745$	37.2320	1.36407
$746$	0	0
$747$	3.20433	0.117240
$748$	0	0
$749$	−15.9581	−0.583096
$750$	0	0
$751$	−23.8577	−0.870581	−0.435290	−	0.900290i	$-0.643354\pi$
$751$	−23.8577	−0.870581	−0.435290	+	0.900290i	$0.643354\pi$
$752$	0	0
$753$	1.63228	0.0594838
$754$	0	0
$755$	−59.2664	−2.15693
$756$	0	0
$757$	−1.16655	−0.0423989	−0.0211994	−	0.999775i	$-0.506748\pi$
$757$	−1.16655	−0.0423989	−0.0211994	+	0.999775i	$0.506748\pi$
$758$	0	0
$759$	1.55211	0.0563381
$760$	0	0
$761$	−2.91355	−0.105616	−0.0528080	−	0.998605i	$-0.516817\pi$
$761$	−2.91355	−0.105616	−0.0528080	+	0.998605i	$0.516817\pi$
$762$	0	0
$763$	42.2608	1.52994
$764$	0	0
$765$	−3.24363	−0.117274
$766$	0	0
$767$	−56.8918	−2.05424
$768$	0	0
$769$	8.81372	0.317831	0.158915	−	0.987292i	$-0.449200\pi$
$769$	8.81372	0.317831	0.158915	+	0.987292i	$0.449200\pi$
$770$	0	0
$771$	27.8190	1.00188
$772$	0	0
$773$	−11.8965	−0.427887	−0.213943	−	0.976846i	$-0.568631\pi$
$773$	−11.8965	−0.427887	−0.213943	+	0.976846i	$0.568631\pi$
$774$	0	0
$775$	15.5910	0.560046
$776$	0	0
$777$	68.4794	2.45669
$778$	0	0
$779$	8.57221	0.307131
$780$	0	0
$781$	−38.1310	−1.36443
$782$	0	0
$783$	−20.2141	−0.722394
$784$	0	0
$785$	−26.3626	−0.940921
$786$	0	0
$787$	33.8613	1.20703	0.603513	−	0.797353i	$-0.293767\pi$
$787$	33.8613	1.20703	0.603513	+	0.797353i	$0.293767\pi$
$788$	0	0
$789$	−14.3424	−0.510602
$790$	0	0
$791$	29.2892	1.04141
$792$	0	0
$793$	42.1619	1.49721
$794$	0	0
$795$	−21.1223	−0.749131
$796$	0	0
$797$	−13.0031	−0.460595	−0.230298	−	0.973120i	$-0.573970\pi$
$797$	−13.0031	−0.460595	−0.230298	+	0.973120i	$0.573970\pi$
$798$	0	0
$799$	−12.5498	−0.443979
$800$	0	0
$801$	5.71818	0.202042
$802$	0	0
$803$	−19.7864	−0.698246
$804$	0	0
$805$	−5.50086	−0.193880
$806$	0	0
$807$	25.2307	0.888163
$808$	0	0
$809$	−35.6551	−1.25357	−0.626784	−	0.779193i	$-0.715629\pi$
$809$	−35.6551	−1.25357	−0.626784	+	0.779193i	$0.715629\pi$
$810$	0	0
$811$	−46.9020	−1.64695	−0.823476	−	0.567350i	$-0.807969\pi$
$811$	−46.9020	−1.64695	−0.823476	+	0.567350i	$0.807969\pi$
$812$	0	0
$813$	11.2479	0.394482
$814$	0	0
$815$	48.5911	1.70207
$816$	0	0
$817$	35.7494	1.25071
$818$	0	0
$819$	−8.52736	−0.297970
$820$	0	0
$821$	19.4661	0.679373	0.339687	−	0.940539i	$-0.389679\pi$
$821$	19.4661	0.679373	0.339687	+	0.940539i	$0.389679\pi$
$822$	0	0
$823$	−25.7226	−0.896632	−0.448316	−	0.893875i	$-0.647976\pi$
$823$	−25.7226	−0.896632	−0.448316	+	0.893875i	$0.647976\pi$
$824$	0	0
$825$	12.1676	0.423621
$826$	0	0
$827$	46.3912	1.61318	0.806590	−	0.591111i	$-0.201311\pi$
$827$	46.3912	1.61318	0.806590	+	0.591111i	$0.201311\pi$
$828$	0	0
$829$	−31.5959	−1.09737	−0.548685	−	0.836029i	$-0.684871\pi$
$829$	−31.5959	−1.09737	−0.548685	+	0.836029i	$0.684871\pi$
$830$	0	0
$831$	−13.3120	−0.461788
$832$	0	0
$833$	48.8829	1.69369
$834$	0	0
$835$	5.40808	0.187154
$836$	0	0
$837$	−26.2107	−0.905974
$838$	0	0
$839$	40.4925	1.39796	0.698978	−	0.715144i	$-0.253639\pi$
$839$	40.4925	1.39796	0.698978	+	0.715144i	$0.253639\pi$
$840$	0	0
$841$	−15.2165	−0.524709
$842$	0	0
$843$	6.79975	0.234196
$844$	0	0
$845$	48.7801	1.67809
$846$	0	0
$847$	26.4529	0.908933
$848$	0	0
$849$	18.2082	0.624902
$850$	0	0
$851$	−3.73631	−0.128079
$852$	0	0
$853$	34.7367	1.18936	0.594681	−	0.803962i	$-0.297278\pi$
$853$	34.7367	1.18936	0.594681	+	0.803962i	$0.297278\pi$
$854$	0	0
$855$	6.65635	0.227642
$856$	0	0
$857$	−11.1365	−0.380417	−0.190208	−	0.981744i	$-0.560916\pi$
$857$	−11.1365	−0.380417	−0.190208	+	0.981744i	$0.560916\pi$
$858$	0	0
$859$	28.9208	0.986764	0.493382	−	0.869813i	$-0.335760\pi$
$859$	28.9208	0.986764	0.493382	+	0.869813i	$0.335760\pi$
$860$	0	0
$861$	9.39453	0.320165
$862$	0	0
$863$	−1.14400	−0.0389421	−0.0194711	−	0.999810i	$-0.506198\pi$
$863$	−1.14400	−0.0389421	−0.0194711	+	0.999810i	$0.506198\pi$
$864$	0	0
$865$	−1.04294	−0.0354609
$866$	0	0
$867$	−9.24612	−0.314015
$868$	0	0
$869$	−0.173134	−0.00587319
$870$	0	0
$871$	−68.7208	−2.32852
$872$	0	0
$873$	−5.38241	−0.182167
$874$	0	0
$875$	23.4516	0.792807
$876$	0	0
$877$	41.4237	1.39878	0.699390	−	0.714740i	$-0.253455\pi$
$877$	41.4237	1.39878	0.699390	+	0.714740i	$0.253455\pi$
$878$	0	0
$879$	30.1202	1.01593
$880$	0	0
$881$	−24.6217	−0.829525	−0.414762	−	0.909930i	$-0.636135\pi$
$881$	−24.6217	−0.829525	−0.414762	+	0.909930i	$0.636135\pi$
$882$	0	0
$883$	25.0682	0.843611	0.421805	−	0.906686i	$-0.361397\pi$
$883$	25.0682	0.843611	0.421805	+	0.906686i	$0.361397\pi$
$884$	0	0
$885$	48.6691	1.63599
$886$	0	0
$887$	4.77843	0.160444	0.0802221	−	0.996777i	$-0.474437\pi$
$887$	4.77843	0.160444	0.0802221	+	0.996777i	$0.474437\pi$
$888$	0	0
$889$	0.368698	0.0123657
$890$	0	0
$891$	−18.1378	−0.607640
$892$	0	0
$893$	25.7538	0.861817
$894$	0	0
$895$	2.71687	0.0908151
$896$	0	0
$897$	−3.69325	−0.123314
$898$	0	0
$899$	17.8723	0.596076
$900$	0	0
$901$	−15.1787	−0.505677
$902$	0	0
$903$	39.1788	1.30379
$904$	0	0
$905$	−44.9714	−1.49490
$906$	0	0
$907$	7.15637	0.237623	0.118812	−	0.992917i	$-0.462092\pi$
$907$	7.15637	0.237623	0.118812	+	0.992917i	$0.462092\pi$
$908$	0	0
$909$	3.94597	0.130880
$910$	0	0
$911$	31.4930	1.04341	0.521705	−	0.853126i	$-0.325296\pi$
$911$	31.4930	1.04341	0.521705	+	0.853126i	$0.325296\pi$
$912$	0	0
$913$	−21.9732	−0.727206
$914$	0	0
$915$	−36.0682	−1.19238
$916$	0	0
$917$	11.7514	0.388066
$918$	0	0
$919$	−5.78404	−0.190798	−0.0953989	−	0.995439i	$-0.530413\pi$
$919$	−5.78404	−0.190798	−0.0953989	+	0.995439i	$0.530413\pi$
$920$	0	0
$921$	21.1568	0.697140
$922$	0	0
$923$	90.7326	2.98650
$924$	0	0
$925$	−29.2904	−0.963062
$926$	0	0
$927$	−3.73786	−0.122767
$928$	0	0
$929$	−11.8961	−0.390299	−0.195149	−	0.980774i	$-0.562519\pi$
$929$	−11.8961	−0.390299	−0.195149	+	0.980774i	$0.562519\pi$
$930$	0	0
$931$	−100.314	−3.28766
$932$	0	0
$933$	−30.7405	−1.00640
$934$	0	0
$935$	22.2426	0.727412
$936$	0	0
$937$	11.2316	0.366922	0.183461	−	0.983027i	$-0.441270\pi$
$937$	11.2316	0.366922	0.183461	+	0.983027i	$0.441270\pi$
$938$	0	0
$939$	29.0393	0.947660
$940$	0	0
$941$	−32.7463	−1.06750	−0.533749	−	0.845643i	$-0.679217\pi$
$941$	−32.7463	−1.06750	−0.533749	+	0.845643i	$0.679217\pi$
$942$	0	0
$943$	−0.512576	−0.0166918
$944$	0	0
$945$	72.4964	2.35831
$946$	0	0
$947$	−2.88535	−0.0937613	−0.0468807	−	0.998900i	$-0.514928\pi$
$947$	−2.88535	−0.0937613	−0.0468807	+	0.998900i	$0.514928\pi$
$948$	0	0
$949$	47.0816	1.52833
$950$	0	0
$951$	53.9026	1.74791
$952$	0	0
$953$	−21.4667	−0.695375	−0.347688	−	0.937610i	$-0.613033\pi$
$953$	−21.4667	−0.695375	−0.347688	+	0.937610i	$0.613033\pi$
$954$	0	0
$955$	37.9181	1.22700
$956$	0	0
$957$	13.9480	0.450875
$958$	0	0
$959$	−45.5502	−1.47089
$960$	0	0
$961$	−7.82578	−0.252444
$962$	0	0
$963$	−1.15465	−0.0372082
$964$	0	0
$965$	15.1916	0.489035
$966$	0	0
$967$	0.0340542	0.00109511	0.000547555	−	1.00000i	$-0.499826\pi$
$967$	0.0340542	0.00109511	0.000547555		1.00000i	$0.499826\pi$
$968$	0	0
$969$	−37.9700	−1.21977
$970$	0	0
$971$	−4.96680	−0.159392	−0.0796962	−	0.996819i	$-0.525395\pi$
$971$	−4.96680	−0.159392	−0.0796962	+	0.996819i	$0.525395\pi$
$972$	0	0
$973$	42.8325	1.37315
$974$	0	0
$975$	−28.9528	−0.927231
$976$	0	0
$977$	26.6350	0.852129	0.426064	−	0.904693i	$-0.359900\pi$
$977$	26.6350	0.852129	0.426064	+	0.904693i	$0.359900\pi$
$978$	0	0
$979$	−39.2115	−1.25320
$980$	0	0
$981$	3.05779	0.0976279
$982$	0	0
$983$	−20.5273	−0.654718	−0.327359	−	0.944900i	$-0.606159\pi$
$983$	−20.5273	−0.654718	−0.327359	+	0.944900i	$0.606159\pi$
$984$	0	0
$985$	66.6852	2.12477
$986$	0	0
$987$	28.2243	0.898389
$988$	0	0
$989$	−2.13764	−0.0679729
$990$	0	0
$991$	−45.8505	−1.45649	−0.728245	−	0.685317i	$-0.759664\pi$
$991$	−45.8505	−1.45649	−0.728245	+	0.685317i	$0.759664\pi$
$992$	0	0
$993$	29.6040	0.939453
$994$	0	0
$995$	28.1286	0.891736
$996$	0	0
$997$	−40.6335	−1.28688	−0.643438	−	0.765498i	$-0.722493\pi$
$997$	−40.6335	−1.28688	−0.643438	+	0.765498i	$0.722493\pi$
$998$	0	0
$999$	49.2412	1.55792

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.f.1.4		6
4.3	odd	2		502.2.a.e.1.3	✓	6
12.11	even	2		4518.2.a.x.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
502.2.a.e.1.3	✓	6	4.3	odd	2
4016.2.a.f.1.4		6	1.1	even	1	trivial
4518.2.a.x.1.1		6	12.11	even	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:	\( N \)	\(=\)	\( 4016 = 2^{4} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4016.a (trivial)

\(f(q)\)	\(=\)	\(q+1.63228 q^{3} +2.87032 q^{5} -4.63886 q^{7} -0.335646 q^{9} +O(q^{10})\)	\(q+1.63228 q^{3} +2.87032 q^{5} -4.63886 q^{7} -0.335646 q^{9} +2.30164 q^{11} -5.47674 q^{13} +4.68517 q^{15} +3.36682 q^{17} -6.90916 q^{19} -7.57194 q^{21} +0.413133 q^{23} +3.23871 q^{25} -5.44472 q^{27} +3.71261 q^{29} +4.81396 q^{31} +3.75693 q^{33} -13.3150 q^{35} -9.04384 q^{37} -8.93960 q^{39} -1.24070 q^{41} -5.17421 q^{43} -0.963410 q^{45} -3.72748 q^{47} +14.5190 q^{49} +5.49561 q^{51} -4.50833 q^{53} +6.60642 q^{55} -11.2777 q^{57} +10.3879 q^{59} -7.69836 q^{61} +1.55701 q^{63} -15.7200 q^{65} +12.5478 q^{67} +0.674351 q^{69} -16.5669 q^{71} -8.59665 q^{73} +5.28650 q^{75} -10.6770 q^{77} -0.0752223 q^{79} -7.88040 q^{81} -9.54676 q^{83} +9.66383 q^{85} +6.06003 q^{87} -17.0363 q^{89} +25.4058 q^{91} +7.85776 q^{93} -19.8315 q^{95} +16.0360 q^{97} -0.772535 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{3} - q^{5} - 6 q^{7} + 15 q^{9}+O(q^{10}) \) 6 * q - q^3 - q^5 - 6 * q^7 + 15 * q^9	\( 6 q - q^{3} - q^{5} - 6 q^{7} + 15 q^{9} - q^{11} - 5 q^{13} - 2 q^{15} + 8 q^{17} - 3 q^{19} - 14 q^{21} - 18 q^{23} - q^{25} - 16 q^{27} + q^{29} - 6 q^{31} - 16 q^{33} - 6 q^{35} - 13 q^{37} + 6 q^{39} + 4 q^{41} + 5 q^{43} - 23 q^{45} - 8 q^{47} + 8 q^{49} + 16 q^{51} - 3 q^{53} + 30 q^{55} - 24 q^{57} + 5 q^{59} - 61 q^{61} + 27 q^{63} - q^{65} + 13 q^{67} - 21 q^{69} - 22 q^{71} + 6 q^{73} + 30 q^{75} + 4 q^{77} - 28 q^{79} + 2 q^{81} - 14 q^{83} - 16 q^{85} + 24 q^{87} + 18 q^{89} + 16 q^{91} + 27 q^{93} - 20 q^{95} + 16 q^{97} - 5 q^{99}+O(q^{100}) \) 6 * q - q^3 - q^5 - 6 * q^7 + 15 * q^9 - q^11 - 5 * q^13 - 2 * q^15 + 8 * q^17 - 3 * q^19 - 14 * q^21 - 18 * q^23 - q^25 - 16 * q^27 + q^29 - 6 * q^31 - 16 * q^33 - 6 * q^35 - 13 * q^37 + 6 * q^39 + 4 * q^41 + 5 * q^43 - 23 * q^45 - 8 * q^47 + 8 * q^49 + 16 * q^51 - 3 * q^53 + 30 * q^55 - 24 * q^57 + 5 * q^59 - 61 * q^61 + 27 * q^63 - q^65 + 13 * q^67 - 21 * q^69 - 22 * q^71 + 6 * q^73 + 30 * q^75 + 4 * q^77 - 28 * q^79 + 2 * q^81 - 14 * q^83 - 16 * q^85 + 24 * q^87 + 18 * q^89 + 16 * q^91 + 27 * q^93 - 20 * q^95 + 16 * q^97 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more