LMFDB - Embedded newform 4014.2.a.u.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4014,2,Mod(1,4014)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4014, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4014.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4014 = 2 \cdot 3^{2} \cdot 223 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4014.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:	$32.0519513713$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.103354048.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 8x^{4} + 14x^{3} + 13x^{2} - 16x + 3 $ x^6 - 2x^5 - 8x^4 + 14x^3 + 13x^2 - 16*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$2.55499$ of defining polynomial
Character	$\chi$	$=$	4014.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +3.52797 q^{5} +0.783532 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +3.52797 q^{5} +0.783532 q^{7} +1.00000 q^{8} +3.52797 q^{10} +3.05111 q^{11} +1.18945 q^{13} +0.783532 q^{14} +1.00000 q^{16} -2.89159 q^{17} -4.10806 q^{19} +3.52797 q^{20} +3.05111 q^{22} +1.44501 q^{23} +7.44658 q^{25} +1.18945 q^{26} +0.783532 q^{28} -7.36289 q^{29} +5.50813 q^{31} +1.00000 q^{32} -2.89159 q^{34} +2.76428 q^{35} +8.23040 q^{37} -4.10806 q^{38} +3.52797 q^{40} +6.93097 q^{41} +7.31409 q^{43} +3.05111 q^{44} +1.44501 q^{46} +1.42058 q^{47} -6.38608 q^{49} +7.44658 q^{50} +1.18945 q^{52} +11.0409 q^{53} +10.7642 q^{55} +0.783532 q^{56} -7.36289 q^{58} -13.7805 q^{59} -8.23816 q^{61} +5.50813 q^{62} +1.00000 q^{64} +4.19635 q^{65} +5.12627 q^{67} -2.89159 q^{68} +2.76428 q^{70} +4.22298 q^{71} -9.93226 q^{73} +8.23040 q^{74} -4.10806 q^{76} +2.39064 q^{77} -1.51298 q^{79} +3.52797 q^{80} +6.93097 q^{82} +0.890303 q^{83} -10.2015 q^{85} +7.31409 q^{86} +3.05111 q^{88} -17.4504 q^{89} +0.931972 q^{91} +1.44501 q^{92} +1.42058 q^{94} -14.4931 q^{95} +8.73111 q^{97} -6.38608 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{2} + 6 q^{4} + 2 q^{5} + 8 q^{7} + 6 q^{8}+O(q^{10}) $ 6 * q + 6 * q^2 + 6 * q^4 + 2 * q^5 + 8 * q^7 + 6 * q^8	$ 6 q + 6 q^{2} + 6 q^{4} + 2 q^{5} + 8 q^{7} + 6 q^{8} + 2 q^{10} + 2 q^{11} - 2 q^{13} + 8 q^{14} + 6 q^{16} + 2 q^{17} - 2 q^{19} + 2 q^{20} + 2 q^{22} + 22 q^{23} + 12 q^{25} - 2 q^{26} + 8 q^{28} + 8 q^{29} - 12 q^{31} + 6 q^{32} + 2 q^{34} + 20 q^{35} - 2 q^{38} + 2 q^{40} + 28 q^{41} + 14 q^{43} + 2 q^{44} + 22 q^{46} + 2 q^{47} + 12 q^{50} - 2 q^{52} + 26 q^{53} + 6 q^{55} + 8 q^{56} + 8 q^{58} + 4 q^{59} - 6 q^{61} - 12 q^{62} + 6 q^{64} + 16 q^{65} + 18 q^{67} + 2 q^{68} + 20 q^{70} + 12 q^{71} - 20 q^{73} - 2 q^{76} + 20 q^{77} + 12 q^{79} + 2 q^{80} + 28 q^{82} + 12 q^{83} - 10 q^{85} + 14 q^{86} + 2 q^{88} - 2 q^{89} - 22 q^{91} + 22 q^{92} + 2 q^{94} + 6 q^{95} - 6 q^{97}+O(q^{100}) $ 6 * q + 6 * q^2 + 6 * q^4 + 2 * q^5 + 8 * q^7 + 6 * q^8 + 2 * q^10 + 2 * q^11 - 2 * q^13 + 8 * q^14 + 6 * q^16 + 2 * q^17 - 2 * q^19 + 2 * q^20 + 2 * q^22 + 22 * q^23 + 12 * q^25 - 2 * q^26 + 8 * q^28 + 8 * q^29 - 12 * q^31 + 6 * q^32 + 2 * q^34 + 20 * q^35 - 2 * q^38 + 2 * q^40 + 28 * q^41 + 14 * q^43 + 2 * q^44 + 22 * q^46 + 2 * q^47 + 12 * q^50 - 2 * q^52 + 26 * q^53 + 6 * q^55 + 8 * q^56 + 8 * q^58 + 4 * q^59 - 6 * q^61 - 12 * q^62 + 6 * q^64 + 16 * q^65 + 18 * q^67 + 2 * q^68 + 20 * q^70 + 12 * q^71 - 20 * q^73 - 2 * q^76 + 20 * q^77 + 12 * q^79 + 2 * q^80 + 28 * q^82 + 12 * q^83 - 10 * q^85 + 14 * q^86 + 2 * q^88 - 2 * q^89 - 22 * q^91 + 22 * q^92 + 2 * q^94 + 6 * q^95 - 6 * q^97

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	3.52797	1.57776	0.788878	−	0.614549i	$-0.210662\pi$
$5$	3.52797	1.57776	0.788878	+	0.614549i	$0.210662\pi$
$6$	0	0
$7$	0.783532	0.296147	0.148074	−	0.988976i	$-0.452693\pi$
$7$	0.783532	0.296147	0.148074	+	0.988976i	$0.452693\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	3.52797	1.11564
$11$	3.05111	0.919943	0.459972	−	0.887934i	$-0.347860\pi$
$11$	3.05111	0.919943	0.459972	+	0.887934i	$0.347860\pi$
$12$	0	0
$13$	1.18945	0.329894	0.164947	−	0.986302i	$-0.447255\pi$
$13$	1.18945	0.329894	0.164947	+	0.986302i	$0.447255\pi$
$14$	0.783532	0.209408
$15$	0	0
$16$	1.00000	0.250000
$17$	−2.89159	−0.701315	−0.350657	−	0.936504i	$-0.614042\pi$
$17$	−2.89159	−0.701315	−0.350657	+	0.936504i	$0.614042\pi$
$18$	0	0
$19$	−4.10806	−0.942454	−0.471227	−	0.882012i	$-0.656189\pi$
$19$	−4.10806	−0.942454	−0.471227	+	0.882012i	$0.656189\pi$
$20$	3.52797	0.788878
$21$	0	0
$22$	3.05111	0.650498
$23$	1.44501	0.301306	0.150653	−	0.988587i	$-0.451862\pi$
$23$	1.44501	0.301306	0.150653	+	0.988587i	$0.451862\pi$
$24$	0	0
$25$	7.44658	1.48932
$26$	1.18945	0.233270
$27$	0	0
$28$	0.783532	0.148074
$29$	−7.36289	−1.36725	−0.683627	−	0.729831i	$-0.739599\pi$
$29$	−7.36289	−1.36725	−0.683627	+	0.729831i	$0.739599\pi$
$30$	0	0
$31$	5.50813	0.989290	0.494645	−	0.869095i	$-0.335298\pi$
$31$	5.50813	0.989290	0.494645	+	0.869095i	$0.335298\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−2.89159	−0.495904
$35$	2.76428	0.467249
$36$	0	0
$37$	8.23040	1.35307	0.676535	−	0.736411i	$-0.263481\pi$
$37$	8.23040	1.35307	0.676535	+	0.736411i	$0.263481\pi$
$38$	−4.10806	−0.666416
$39$	0	0
$40$	3.52797	0.557821
$41$	6.93097	1.08244	0.541218	−	0.840882i	$-0.317963\pi$
$41$	6.93097	1.08244	0.541218	+	0.840882i	$0.317963\pi$
$42$	0	0
$43$	7.31409	1.11539	0.557694	−	0.830047i	$-0.311686\pi$
$43$	7.31409	1.11539	0.557694	+	0.830047i	$0.311686\pi$
$44$	3.05111	0.459972
$45$	0	0
$46$	1.44501	0.213055
$47$	1.42058	0.207213	0.103606	−	0.994618i	$-0.466962\pi$
$47$	1.42058	0.207213	0.103606	+	0.994618i	$0.466962\pi$
$48$	0	0
$49$	−6.38608	−0.912297
$50$	7.44658	1.05311
$51$	0	0
$52$	1.18945	0.164947
$53$	11.0409	1.51659	0.758295	−	0.651911i	$-0.226032\pi$
$53$	11.0409	1.51659	0.758295	+	0.651911i	$0.226032\pi$
$54$	0	0
$55$	10.7642	1.45145
$56$	0.783532	0.104704
$57$	0	0
$58$	−7.36289	−0.966795
$59$	−13.7805	−1.79407	−0.897037	−	0.441956i	$-0.854285\pi$
$59$	−13.7805	−1.79407	−0.897037	+	0.441956i	$0.854285\pi$
$60$	0	0
$61$	−8.23816	−1.05479	−0.527394	−	0.849621i	$-0.676831\pi$
$61$	−8.23816	−1.05479	−0.527394	+	0.849621i	$0.676831\pi$
$62$	5.50813	0.699533
$63$	0	0
$64$	1.00000	0.125000
$65$	4.19635	0.520493
$66$	0	0
$67$	5.12627	0.626273	0.313137	−	0.949708i	$-0.398620\pi$
$67$	5.12627	0.626273	0.313137	+	0.949708i	$0.398620\pi$
$68$	−2.89159	−0.350657
$69$	0	0
$70$	2.76428	0.330395
$71$	4.22298	0.501175	0.250588	−	0.968094i	$-0.419376\pi$
$71$	4.22298	0.501175	0.250588	+	0.968094i	$0.419376\pi$
$72$	0	0
$73$	−9.93226	−1.16248	−0.581242	−	0.813731i	$-0.697433\pi$
$73$	−9.93226	−1.16248	−0.581242	+	0.813731i	$0.697433\pi$
$74$	8.23040	0.956764
$75$	0	0
$76$	−4.10806	−0.471227
$77$	2.39064	0.272439
$78$	0	0
$79$	−1.51298	−0.170223	−0.0851116	−	0.996371i	$-0.527125\pi$
$79$	−1.51298	−0.170223	−0.0851116	+	0.996371i	$0.527125\pi$
$80$	3.52797	0.394439
$81$	0	0
$82$	6.93097	0.765398
$83$	0.890303	0.0977234	0.0488617	−	0.998806i	$-0.484441\pi$
$83$	0.890303	0.0977234	0.0488617	+	0.998806i	$0.484441\pi$
$84$	0	0
$85$	−10.2015	−1.10650
$86$	7.31409	0.788699
$87$	0	0
$88$	3.05111	0.325249
$89$	−17.4504	−1.84973	−0.924867	−	0.380291i	$-0.875824\pi$
$89$	−17.4504	−1.84973	−0.924867	+	0.380291i	$0.875824\pi$
$90$	0	0
$91$	0.931972	0.0976972
$92$	1.44501	0.150653
$93$	0	0
$94$	1.42058	0.146521
$95$	−14.4931	−1.48696
$96$	0	0
$97$	8.73111	0.886510	0.443255	−	0.896396i	$-0.353824\pi$
$97$	8.73111	0.886510	0.443255	+	0.896396i	$0.353824\pi$
$98$	−6.38608	−0.645091
$99$	0	0
$100$	7.44658	0.744658
$101$	−2.92077	−0.290628	−0.145314	−	0.989386i	$-0.546419\pi$
$101$	−2.92077	−0.290628	−0.145314	+	0.989386i	$0.546419\pi$
$102$	0	0
$103$	−15.1041	−1.48825	−0.744127	−	0.668039i	$-0.767134\pi$
$103$	−15.1041	−1.48825	−0.744127	+	0.668039i	$0.767134\pi$
$104$	1.18945	0.116635
$105$	0	0
$106$	11.0409	1.07239
$107$	6.13316	0.592915	0.296458	−	0.955046i	$-0.404195\pi$
$107$	6.13316	0.592915	0.296458	+	0.955046i	$0.404195\pi$
$108$	0	0
$109$	−2.17737	−0.208555	−0.104277	−	0.994548i	$-0.533253\pi$
$109$	−2.17737	−0.208555	−0.104277	+	0.994548i	$0.533253\pi$
$110$	10.7642	1.02633
$111$	0	0
$112$	0.783532	0.0740368
$113$	10.5809	0.995368	0.497684	−	0.867358i	$-0.334184\pi$
$113$	10.5809	0.995368	0.497684	+	0.867358i	$0.334184\pi$
$114$	0	0
$115$	5.09796	0.475387
$116$	−7.36289	−0.683627
$117$	0	0
$118$	−13.7805	−1.26860
$119$	−2.26566	−0.207693
$120$	0	0
$121$	−1.69075	−0.153704
$122$	−8.23816	−0.745848
$123$	0	0
$124$	5.50813	0.494645
$125$	8.63148	0.772023
$126$	0	0
$127$	−18.2359	−1.61818	−0.809089	−	0.587686i	$-0.800039\pi$
$127$	−18.2359	−1.61818	−0.809089	+	0.587686i	$0.800039\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	4.19635	0.368044
$131$	−0.238894	−0.0208723	−0.0104361	−	0.999946i	$-0.503322\pi$
$131$	−0.238894	−0.0208723	−0.0104361	+	0.999946i	$0.503322\pi$
$132$	0	0
$133$	−3.21880	−0.279105
$134$	5.12627	0.442842
$135$	0	0
$136$	−2.89159	−0.247952
$137$	19.1861	1.63918	0.819591	−	0.572949i	$-0.194201\pi$
$137$	19.1861	1.63918	0.819591	+	0.572949i	$0.194201\pi$
$138$	0	0
$139$	16.1731	1.37179	0.685894	−	0.727702i	$-0.259411\pi$
$139$	16.1731	1.37179	0.685894	+	0.727702i	$0.259411\pi$
$140$	2.76428	0.233624
$141$	0	0
$142$	4.22298	0.354384
$143$	3.62914	0.303484
$144$	0	0
$145$	−25.9761	−2.15720
$146$	−9.93226	−0.822000
$147$	0	0
$148$	8.23040	0.676535
$149$	−16.4459	−1.34730	−0.673650	−	0.739050i	$-0.735275\pi$
$149$	−16.4459	−1.34730	−0.673650	+	0.739050i	$0.735275\pi$
$150$	0	0
$151$	16.7307	1.36152	0.680762	−	0.732505i	$-0.261649\pi$
$151$	16.7307	1.36152	0.680762	+	0.732505i	$0.261649\pi$
$152$	−4.10806	−0.333208
$153$	0	0
$154$	2.39064	0.192643
$155$	19.4325	1.56086
$156$	0	0
$157$	−1.33176	−0.106286	−0.0531430	−	0.998587i	$-0.516924\pi$
$157$	−1.33176	−0.106286	−0.0531430	+	0.998587i	$0.516924\pi$
$158$	−1.51298	−0.120366
$159$	0	0
$160$	3.52797	0.278911
$161$	1.13221	0.0892308
$162$	0	0
$163$	2.82946	0.221620	0.110810	−	0.993842i	$-0.464655\pi$
$163$	2.82946	0.221620	0.110810	+	0.993842i	$0.464655\pi$
$164$	6.93097	0.541218
$165$	0	0
$166$	0.890303	0.0691009
$167$	8.37656	0.648198	0.324099	−	0.946023i	$-0.394939\pi$
$167$	8.37656	0.648198	0.324099	+	0.946023i	$0.394939\pi$
$168$	0	0
$169$	−11.5852	−0.891170
$170$	−10.2015	−0.782417
$171$	0	0
$172$	7.31409	0.557694
$173$	−13.4817	−1.02499	−0.512497	−	0.858689i	$-0.671280\pi$
$173$	−13.4817	−1.02499	−0.512497	+	0.858689i	$0.671280\pi$
$174$	0	0
$175$	5.83464	0.441057
$176$	3.05111	0.229986
$177$	0	0
$178$	−17.4504	−1.30796
$179$	−4.67837	−0.349678	−0.174839	−	0.984597i	$-0.555940\pi$
$179$	−4.67837	−0.349678	−0.174839	+	0.984597i	$0.555940\pi$
$180$	0	0
$181$	−15.1260	−1.12431	−0.562154	−	0.827033i	$-0.690027\pi$
$181$	−15.1260	−1.12431	−0.562154	+	0.827033i	$0.690027\pi$
$182$	0.931972	0.0690824
$183$	0	0
$184$	1.44501	0.106528
$185$	29.0366	2.13481
$186$	0	0
$187$	−8.82256	−0.645170
$188$	1.42058	0.103606
$189$	0	0
$190$	−14.4931	−1.05144
$191$	13.0409	0.943607	0.471804	−	0.881704i	$-0.343603\pi$
$191$	13.0409	0.943607	0.471804	+	0.881704i	$0.343603\pi$
$192$	0	0
$193$	−8.03075	−0.578066	−0.289033	−	0.957319i	$-0.593334\pi$
$193$	−8.03075	−0.578066	−0.289033	+	0.957319i	$0.593334\pi$
$194$	8.73111	0.626857
$195$	0	0
$196$	−6.38608	−0.456148
$197$	21.7171	1.54728	0.773640	−	0.633626i	$-0.218434\pi$
$197$	21.7171	1.54728	0.773640	+	0.633626i	$0.218434\pi$
$198$	0	0
$199$	−6.18321	−0.438316	−0.219158	−	0.975689i	$-0.570331\pi$
$199$	−6.18321	−0.438316	−0.219158	+	0.975689i	$0.570331\pi$
$200$	7.44658	0.526553
$201$	0	0
$202$	−2.92077	−0.205505
$203$	−5.76906	−0.404909
$204$	0	0
$205$	24.4523	1.70782
$206$	−15.1041	−1.05235
$207$	0	0
$208$	1.18945	0.0824735
$209$	−12.5341	−0.867004
$210$	0	0
$211$	−22.5418	−1.55184	−0.775921	−	0.630830i	$-0.782714\pi$
$211$	−22.5418	−1.55184	−0.775921	+	0.630830i	$0.782714\pi$
$212$	11.0409	0.758295
$213$	0	0
$214$	6.13316	0.419254
$215$	25.8039	1.75981
$216$	0	0
$217$	4.31580	0.292976
$218$	−2.17737	−0.147470
$219$	0	0
$220$	10.7642	0.725723
$221$	−3.43941	−0.231360
$222$	0	0
$223$	−1.00000	−0.0669650
$223$	−1.00000	−0.0669650
$224$	0.783532	0.0523520
$225$	0	0
$226$	10.5809	0.703831
$227$	8.08139	0.536380	0.268190	−	0.963366i	$-0.413574\pi$
$227$	8.08139	0.536380	0.268190	+	0.963366i	$0.413574\pi$
$228$	0	0
$229$	−20.8687	−1.37904	−0.689520	−	0.724267i	$-0.742178\pi$
$229$	−20.8687	−1.37904	−0.689520	+	0.724267i	$0.742178\pi$
$230$	5.09796	0.336149
$231$	0	0
$232$	−7.36289	−0.483398
$233$	−5.85766	−0.383748	−0.191874	−	0.981420i	$-0.561456\pi$
$233$	−5.85766	−0.383748	−0.191874	+	0.981420i	$0.561456\pi$
$234$	0	0
$235$	5.01176	0.326931
$236$	−13.7805	−0.897037
$237$	0	0
$238$	−2.26566	−0.146861
$239$	3.61755	0.234000	0.117000	−	0.993132i	$-0.462672\pi$
$239$	3.61755	0.234000	0.117000	+	0.993132i	$0.462672\pi$
$240$	0	0
$241$	12.9608	0.834879	0.417440	−	0.908705i	$-0.362928\pi$
$241$	12.9608	0.834879	0.417440	+	0.908705i	$0.362928\pi$
$242$	−1.69075	−0.108685
$243$	0	0
$244$	−8.23816	−0.527394
$245$	−22.5299	−1.43938
$246$	0	0
$247$	−4.88633	−0.310910
$248$	5.50813	0.349767
$249$	0	0
$250$	8.63148	0.545903
$251$	25.8640	1.63252	0.816262	−	0.577682i	$-0.196043\pi$
$251$	25.8640	1.63252	0.816262	+	0.577682i	$0.196043\pi$
$252$	0	0
$253$	4.40888	0.277184
$254$	−18.2359	−1.14422
$255$	0	0
$256$	1.00000	0.0625000
$257$	5.78774	0.361030	0.180515	−	0.983572i	$-0.442224\pi$
$257$	5.78774	0.361030	0.180515	+	0.983572i	$0.442224\pi$
$258$	0	0
$259$	6.44878	0.400708
$260$	4.19635	0.260246
$261$	0	0
$262$	−0.238894	−0.0147589
$263$	8.18878	0.504942	0.252471	−	0.967604i	$-0.418757\pi$
$263$	8.18878	0.504942	0.252471	+	0.967604i	$0.418757\pi$
$264$	0	0
$265$	38.9522	2.39281
$266$	−3.21880	−0.197357
$267$	0	0
$268$	5.12627	0.313137
$269$	26.9445	1.64283	0.821417	−	0.570329i	$-0.193184\pi$
$269$	26.9445	1.64283	0.821417	+	0.570329i	$0.193184\pi$
$270$	0	0
$271$	−12.4355	−0.755404	−0.377702	−	0.925927i	$-0.623286\pi$
$271$	−12.4355	−0.755404	−0.377702	+	0.925927i	$0.623286\pi$
$272$	−2.89159	−0.175329
$273$	0	0
$274$	19.1861	1.15908
$275$	22.7203	1.37009
$276$	0	0
$277$	22.8323	1.37186	0.685931	−	0.727667i	$-0.259395\pi$
$277$	22.8323	1.37186	0.685931	+	0.727667i	$0.259395\pi$
$278$	16.1731	0.970000
$279$	0	0
$280$	2.76428	0.165197
$281$	−10.6943	−0.637967	−0.318983	−	0.947760i	$-0.603341\pi$
$281$	−10.6943	−0.637967	−0.318983	+	0.947760i	$0.603341\pi$
$282$	0	0
$283$	4.54628	0.270248	0.135124	−	0.990829i	$-0.456857\pi$
$283$	4.54628	0.270248	0.135124	+	0.990829i	$0.456857\pi$
$284$	4.22298	0.250588
$285$	0	0
$286$	3.62914	0.214595
$287$	5.43064	0.320560
$288$	0	0
$289$	−8.63868	−0.508158
$290$	−25.9761	−1.52537
$291$	0	0
$292$	−9.93226	−0.581242
$293$	−24.3970	−1.42529	−0.712644	−	0.701526i	$-0.752502\pi$
$293$	−24.3970	−1.42529	−0.712644	+	0.701526i	$0.752502\pi$
$294$	0	0
$295$	−48.6174	−2.83061
$296$	8.23040	0.478382
$297$	0	0
$298$	−16.4459	−0.952685
$299$	1.71877	0.0993989
$300$	0	0
$301$	5.73083	0.330319
$302$	16.7307	0.962742
$303$	0	0
$304$	−4.10806	−0.235614
$305$	−29.0640	−1.66420
$306$	0	0
$307$	−6.36759	−0.363418	−0.181709	−	0.983352i	$-0.558163\pi$
$307$	−6.36759	−0.363418	−0.181709	+	0.983352i	$0.558163\pi$
$308$	2.39064	0.136219
$309$	0	0
$310$	19.4325	1.10369
$311$	−10.3838	−0.588809	−0.294405	−	0.955681i	$-0.595121\pi$
$311$	−10.3838	−0.588809	−0.294405	+	0.955681i	$0.595121\pi$
$312$	0	0
$313$	0.659990	0.0373048	0.0186524	−	0.999826i	$-0.494062\pi$
$313$	0.659990	0.0373048	0.0186524	+	0.999826i	$0.494062\pi$
$314$	−1.33176	−0.0751555
$315$	0	0
$316$	−1.51298	−0.0851116
$317$	−17.3596	−0.975011	−0.487506	−	0.873120i	$-0.662093\pi$
$317$	−17.3596	−0.975011	−0.487506	+	0.873120i	$0.662093\pi$
$318$	0	0
$319$	−22.4650	−1.25780
$320$	3.52797	0.197220
$321$	0	0
$322$	1.13221	0.0630957
$323$	11.8789	0.660957
$324$	0	0
$325$	8.85734	0.491317
$326$	2.82946	0.156709
$327$	0	0
$328$	6.93097	0.382699
$329$	1.11307	0.0613655
$330$	0	0
$331$	6.75952	0.371537	0.185768	−	0.982594i	$-0.440523\pi$
$331$	6.75952	0.371537	0.185768	+	0.982594i	$0.440523\pi$
$332$	0.890303	0.0488617
$333$	0	0
$334$	8.37656	0.458345
$335$	18.0853	0.988107
$336$	0	0
$337$	−32.2196	−1.75512	−0.877558	−	0.479471i	$-0.840829\pi$
$337$	−32.2196	−1.75512	−0.877558	+	0.479471i	$0.840829\pi$
$338$	−11.5852	−0.630152
$339$	0	0
$340$	−10.2015	−0.553252
$341$	16.8059	0.910090
$342$	0	0
$343$	−10.4884	−0.566322
$344$	7.31409	0.394349
$345$	0	0
$346$	−13.4817	−0.724781
$347$	−27.4569	−1.47397	−0.736983	−	0.675911i	$-0.763750\pi$
$347$	−27.4569	−1.47397	−0.736983	+	0.675911i	$0.763750\pi$
$348$	0	0
$349$	−15.1318	−0.809985	−0.404992	−	0.914320i	$-0.632726\pi$
$349$	−15.1318	−0.809985	−0.404992	+	0.914320i	$0.632726\pi$
$350$	5.83464	0.311875
$351$	0	0
$352$	3.05111	0.162625
$353$	−6.66086	−0.354522	−0.177261	−	0.984164i	$-0.556724\pi$
$353$	−6.66086	−0.354522	−0.177261	+	0.984164i	$0.556724\pi$
$354$	0	0
$355$	14.8985	0.790732
$356$	−17.4504	−0.924867
$357$	0	0
$358$	−4.67837	−0.247260
$359$	20.9666	1.10658	0.553288	−	0.832990i	$-0.313373\pi$
$359$	20.9666	1.10658	0.553288	+	0.832990i	$0.313373\pi$
$360$	0	0
$361$	−2.12382	−0.111780
$362$	−15.1260	−0.795006
$363$	0	0
$364$	0.931972	0.0488486
$365$	−35.0407	−1.83412
$366$	0	0
$367$	−18.2504	−0.952661	−0.476330	−	0.879266i	$-0.658033\pi$
$367$	−18.2504	−0.952661	−0.476330	+	0.879266i	$0.658033\pi$
$368$	1.44501	0.0753264
$369$	0	0
$370$	29.0366	1.50954
$371$	8.65094	0.449134
$372$	0	0
$373$	−4.01623	−0.207953	−0.103976	−	0.994580i	$-0.533157\pi$
$373$	−4.01623	−0.207953	−0.103976	+	0.994580i	$0.533157\pi$
$374$	−8.82256	−0.456204
$375$	0	0
$376$	1.42058	0.0732607
$377$	−8.75779	−0.451049
$378$	0	0
$379$	3.42929	0.176151	0.0880755	−	0.996114i	$-0.471928\pi$
$379$	3.42929	0.176151	0.0880755	+	0.996114i	$0.471928\pi$
$380$	−14.4931	−0.743482
$381$	0	0
$382$	13.0409	0.667231
$383$	−16.6667	−0.851627	−0.425814	−	0.904811i	$-0.640012\pi$
$383$	−16.6667	−0.851627	−0.425814	+	0.904811i	$0.640012\pi$
$384$	0	0
$385$	8.43411	0.429842
$386$	−8.03075	−0.408755
$387$	0	0
$388$	8.73111	0.443255
$389$	15.0620	0.763675	0.381837	−	0.924230i	$-0.375291\pi$
$389$	15.0620	0.763675	0.381837	+	0.924230i	$0.375291\pi$
$390$	0	0
$391$	−4.17838	−0.211310
$392$	−6.38608	−0.322546
$393$	0	0
$394$	21.7171	1.09409
$395$	−5.33774	−0.268571
$396$	0	0
$397$	13.4446	0.674763	0.337381	−	0.941368i	$-0.390459\pi$
$397$	13.4446	0.674763	0.337381	+	0.941368i	$0.390459\pi$
$398$	−6.18321	−0.309937
$399$	0	0
$400$	7.44658	0.372329
$401$	25.2981	1.26333	0.631663	−	0.775243i	$-0.282373\pi$
$401$	25.2981	1.26333	0.631663	+	0.775243i	$0.282373\pi$
$402$	0	0
$403$	6.55165	0.326361
$404$	−2.92077	−0.145314
$405$	0	0
$406$	−5.76906	−0.286314
$407$	25.1118	1.24475
$408$	0	0
$409$	−10.7393	−0.531027	−0.265513	−	0.964107i	$-0.585541\pi$
$409$	−10.7393	−0.531027	−0.265513	+	0.964107i	$0.585541\pi$
$410$	24.4523	1.20761
$411$	0	0
$412$	−15.1041	−0.744127
$413$	−10.7975	−0.531310
$414$	0	0
$415$	3.14096	0.154184
$416$	1.18945	0.0583176
$417$	0	0
$418$	−12.5341	−0.613065
$419$	32.3250	1.57918	0.789589	−	0.613636i	$-0.210294\pi$
$419$	32.3250	1.57918	0.789589	+	0.613636i	$0.210294\pi$
$420$	0	0
$421$	−17.9165	−0.873195	−0.436598	−	0.899657i	$-0.643817\pi$
$421$	−17.9165	−0.873195	−0.436598	+	0.899657i	$0.643817\pi$
$422$	−22.5418	−1.09732
$423$	0	0
$424$	11.0409	0.536196
$425$	−21.5325	−1.04448
$426$	0	0
$427$	−6.45487	−0.312373
$428$	6.13316	0.296458
$429$	0	0
$430$	25.8039	1.24437
$431$	25.3476	1.22095	0.610475	−	0.792035i	$-0.290978\pi$
$431$	25.3476	1.22095	0.610475	+	0.792035i	$0.290978\pi$
$432$	0	0
$433$	−16.9687	−0.815464	−0.407732	−	0.913102i	$-0.633680\pi$
$433$	−16.9687	−0.815464	−0.407732	+	0.913102i	$0.633680\pi$
$434$	4.31580	0.207165
$435$	0	0
$436$	−2.17737	−0.104277
$437$	−5.93619	−0.283967
$438$	0	0
$439$	14.2557	0.680386	0.340193	−	0.940356i	$-0.389508\pi$
$439$	14.2557	0.680386	0.340193	+	0.940356i	$0.389508\pi$
$440$	10.7642	0.513164
$441$	0	0
$442$	−3.43941	−0.163596
$443$	−20.6507	−0.981143	−0.490571	−	0.871401i	$-0.663212\pi$
$443$	−20.6507	−0.981143	−0.490571	+	0.871401i	$0.663212\pi$
$444$	0	0
$445$	−61.5644	−2.91843
$446$	−1.00000	−0.0473514
$447$	0	0
$448$	0.783532	0.0370184
$449$	0.367555	0.0173460	0.00867299	−	0.999962i	$-0.497239\pi$
$449$	0.367555	0.0173460	0.00867299	+	0.999962i	$0.497239\pi$
$450$	0	0
$451$	21.1471	0.995779
$452$	10.5809	0.497684
$453$	0	0
$454$	8.08139	0.379278
$455$	3.28797	0.154143
$456$	0	0
$457$	−22.3013	−1.04321	−0.521604	−	0.853187i	$-0.674666\pi$
$457$	−22.3013	−1.04321	−0.521604	+	0.853187i	$0.674666\pi$
$458$	−20.8687	−0.975128
$459$	0	0
$460$	5.09796	0.237693
$461$	−2.75287	−0.128214	−0.0641071	−	0.997943i	$-0.520420\pi$
$461$	−2.75287	−0.128214	−0.0641071	+	0.997943i	$0.520420\pi$
$462$	0	0
$463$	19.8757	0.923701	0.461850	−	0.886958i	$-0.347186\pi$
$463$	19.8757	0.923701	0.461850	+	0.886958i	$0.347186\pi$
$464$	−7.36289	−0.341814
$465$	0	0
$466$	−5.85766	−0.271351
$467$	17.9849	0.832242	0.416121	−	0.909309i	$-0.363389\pi$
$467$	17.9849	0.832242	0.416121	+	0.909309i	$0.363389\pi$
$468$	0	0
$469$	4.01660	0.185469
$470$	5.01176	0.231175
$471$	0	0
$472$	−13.7805	−0.634301
$473$	22.3161	1.02609
$474$	0	0
$475$	−30.5910	−1.40361
$476$	−2.26566	−0.103846
$477$	0	0
$478$	3.61755	0.165463
$479$	−10.2988	−0.470565	−0.235282	−	0.971927i	$-0.575601\pi$
$479$	−10.2988	−0.470565	−0.235282	+	0.971927i	$0.575601\pi$
$480$	0	0
$481$	9.78965	0.446369
$482$	12.9608	0.590349
$483$	0	0
$484$	−1.69075	−0.0768521
$485$	30.8031	1.39870
$486$	0	0
$487$	36.4302	1.65081	0.825404	−	0.564542i	$-0.190947\pi$
$487$	36.4302	1.65081	0.825404	+	0.564542i	$0.190947\pi$
$488$	−8.23816	−0.372924
$489$	0	0
$490$	−22.5299	−1.01780
$491$	21.4204	0.966688	0.483344	−	0.875430i	$-0.339422\pi$
$491$	21.4204	0.966688	0.483344	+	0.875430i	$0.339422\pi$
$492$	0	0
$493$	21.2905	0.958876
$494$	−4.88633	−0.219847
$495$	0	0
$496$	5.50813	0.247322
$497$	3.30884	0.148422
$498$	0	0
$499$	33.7171	1.50938	0.754691	−	0.656080i	$-0.227787\pi$
$499$	33.7171	1.50938	0.754691	+	0.656080i	$0.227787\pi$
$500$	8.63148	0.386012
$501$	0	0
$502$	25.8640	1.15437
$503$	8.53594	0.380599	0.190299	−	0.981726i	$-0.439054\pi$
$503$	8.53594	0.380599	0.190299	+	0.981726i	$0.439054\pi$
$504$	0	0
$505$	−10.3044	−0.458540
$506$	4.40888	0.195999
$507$	0	0
$508$	−18.2359	−0.809089
$509$	6.19315	0.274507	0.137253	−	0.990536i	$-0.456173\pi$
$509$	6.19315	0.274507	0.137253	+	0.990536i	$0.456173\pi$
$510$	0	0
$511$	−7.78225	−0.344266
$512$	1.00000	0.0441942
$513$	0	0
$514$	5.78774	0.255286
$515$	−53.2869	−2.34810
$516$	0	0
$517$	4.33434	0.190624
$518$	6.44878	0.283343
$519$	0	0
$520$	4.19635	0.184022
$521$	−24.3357	−1.06616	−0.533082	−	0.846063i	$-0.678966\pi$
$521$	−24.3357	−1.06616	−0.533082	+	0.846063i	$0.678966\pi$
$522$	0	0
$523$	−10.9082	−0.476983	−0.238491	−	0.971145i	$-0.576653\pi$
$523$	−10.9082	−0.476983	−0.238491	+	0.971145i	$0.576653\pi$
$524$	−0.238894	−0.0104361
$525$	0	0
$526$	8.18878	0.357048
$527$	−15.9273	−0.693803
$528$	0	0
$529$	−20.9119	−0.909215
$530$	38.9522	1.69197
$531$	0	0
$532$	−3.21880	−0.139553
$533$	8.24404	0.357089
$534$	0	0
$535$	21.6376	0.935476
$536$	5.12627	0.221421
$537$	0	0
$538$	26.9445	1.16166
$539$	−19.4846	−0.839261
$540$	0	0
$541$	−26.6059	−1.14388	−0.571940	−	0.820296i	$-0.693809\pi$
$541$	−26.6059	−1.14388	−0.571940	+	0.820296i	$0.693809\pi$
$542$	−12.4355	−0.534151
$543$	0	0
$544$	−2.89159	−0.123976
$545$	−7.68172	−0.329049
$546$	0	0
$547$	26.0712	1.11473	0.557363	−	0.830269i	$-0.311813\pi$
$547$	26.0712	1.11473	0.557363	+	0.830269i	$0.311813\pi$
$548$	19.1861	0.819591
$549$	0	0
$550$	22.7203	0.968798
$551$	30.2472	1.28858
$552$	0	0
$553$	−1.18547	−0.0504112
$554$	22.8323	0.970052
$555$	0	0
$556$	16.1731	0.685894
$557$	−36.8080	−1.55960	−0.779802	−	0.626026i	$-0.784680\pi$
$557$	−36.8080	−1.55960	−0.779802	+	0.626026i	$0.784680\pi$
$558$	0	0
$559$	8.69974	0.367960
$560$	2.76428	0.116812
$561$	0	0
$562$	−10.6943	−0.451111
$563$	−24.1530	−1.01793	−0.508963	−	0.860788i	$-0.669971\pi$
$563$	−24.1530	−1.01793	−0.508963	+	0.860788i	$0.669971\pi$
$564$	0	0
$565$	37.3291	1.57045
$566$	4.54628	0.191094
$567$	0	0
$568$	4.22298	0.177192
$569$	16.2346	0.680588	0.340294	−	0.940319i	$-0.389473\pi$
$569$	16.2346	0.680588	0.340294	+	0.940319i	$0.389473\pi$
$570$	0	0
$571$	−32.7102	−1.36888	−0.684440	−	0.729069i	$-0.739953\pi$
$571$	−32.7102	−1.36888	−0.684440	+	0.729069i	$0.739953\pi$
$572$	3.62914	0.151742
$573$	0	0
$574$	5.43064	0.226670
$575$	10.7604	0.448739
$576$	0	0
$577$	11.4803	0.477931	0.238965	−	0.971028i	$-0.423192\pi$
$577$	11.4803	0.477931	0.238965	+	0.971028i	$0.423192\pi$
$578$	−8.63868	−0.359322
$579$	0	0
$580$	−25.9761	−1.07860
$581$	0.697581	0.0289405
$582$	0	0
$583$	33.6871	1.39518
$584$	−9.93226	−0.411000
$585$	0	0
$586$	−24.3970	−1.00783
$587$	−46.2462	−1.90879	−0.954393	−	0.298552i	$-0.903496\pi$
$587$	−46.2462	−1.90879	−0.954393	+	0.298552i	$0.903496\pi$
$588$	0	0
$589$	−22.6277	−0.932360
$590$	−48.6174	−2.00155
$591$	0	0
$592$	8.23040	0.338267
$593$	40.1654	1.64939	0.824697	−	0.565575i	$-0.191346\pi$
$593$	40.1654	1.64939	0.824697	+	0.565575i	$0.191346\pi$
$594$	0	0
$595$	−7.99318	−0.327688
$596$	−16.4459	−0.673650
$597$	0	0
$598$	1.71877	0.0702856
$599$	−11.8520	−0.484258	−0.242129	−	0.970244i	$-0.577846\pi$
$599$	−11.8520	−0.484258	−0.242129	+	0.970244i	$0.577846\pi$
$600$	0	0
$601$	10.0953	0.411796	0.205898	−	0.978573i	$-0.433988\pi$
$601$	10.0953	0.411796	0.205898	+	0.978573i	$0.433988\pi$
$602$	5.73083	0.233571
$603$	0	0
$604$	16.7307	0.680762
$605$	−5.96491	−0.242508
$606$	0	0
$607$	−16.1692	−0.656287	−0.328144	−	0.944628i	$-0.606423\pi$
$607$	−16.1692	−0.656287	−0.328144	+	0.944628i	$0.606423\pi$
$608$	−4.10806	−0.166604
$609$	0	0
$610$	−29.0640	−1.17677
$611$	1.68971	0.0683582
$612$	0	0
$613$	−23.7318	−0.958519	−0.479259	−	0.877673i	$-0.659095\pi$
$613$	−23.7318	−0.958519	−0.479259	+	0.877673i	$0.659095\pi$
$614$	−6.36759	−0.256975
$615$	0	0
$616$	2.39064	0.0963217
$617$	22.0191	0.886456	0.443228	−	0.896409i	$-0.353833\pi$
$617$	22.0191	0.886456	0.443228	+	0.896409i	$0.353833\pi$
$618$	0	0
$619$	−15.2932	−0.614687	−0.307344	−	0.951599i	$-0.599440\pi$
$619$	−15.2932	−0.614687	−0.307344	+	0.951599i	$0.599440\pi$
$620$	19.4325	0.780429
$621$	0	0
$622$	−10.3838	−0.416351
$623$	−13.6729	−0.547794
$624$	0	0
$625$	−6.78131	−0.271252
$626$	0.659990	0.0263785
$627$	0	0
$628$	−1.33176	−0.0531430
$629$	−23.7990	−0.948927
$630$	0	0
$631$	36.3162	1.44572	0.722862	−	0.690992i	$-0.242826\pi$
$631$	36.3162	1.44572	0.722862	+	0.690992i	$0.242826\pi$
$632$	−1.51298	−0.0601830
$633$	0	0
$634$	−17.3596	−0.689437
$635$	−64.3359	−2.55309
$636$	0	0
$637$	−7.59592	−0.300961
$638$	−22.4650	−0.889397
$639$	0	0
$640$	3.52797	0.139455
$641$	14.2498	0.562833	0.281416	−	0.959586i	$-0.409196\pi$
$641$	14.2498	0.562833	0.281416	+	0.959586i	$0.409196\pi$
$642$	0	0
$643$	−27.2028	−1.07277	−0.536387	−	0.843972i	$-0.680211\pi$
$643$	−27.2028	−1.07277	−0.536387	+	0.843972i	$0.680211\pi$
$644$	1.13221	0.0446154
$645$	0	0
$646$	11.8789	0.467367
$647$	−8.85985	−0.348317	−0.174158	−	0.984718i	$-0.555720\pi$
$647$	−8.85985	−0.348317	−0.174158	+	0.984718i	$0.555720\pi$
$648$	0	0
$649$	−42.0459	−1.65045
$650$	8.85734	0.347413
$651$	0	0
$652$	2.82946	0.110810
$653$	16.2155	0.634561	0.317280	−	0.948332i	$-0.397230\pi$
$653$	16.2155	0.634561	0.317280	+	0.948332i	$0.397230\pi$
$654$	0	0
$655$	−0.842811	−0.0329313
$656$	6.93097	0.270609
$657$	0	0
$658$	1.11307	0.0433919
$659$	35.5332	1.38418	0.692088	−	0.721813i	$-0.256691\pi$
$659$	35.5332	1.38418	0.692088	+	0.721813i	$0.256691\pi$
$660$	0	0
$661$	21.4795	0.835457	0.417729	−	0.908572i	$-0.362826\pi$
$661$	21.4795	0.835457	0.417729	+	0.908572i	$0.362826\pi$
$662$	6.75952	0.262716
$663$	0	0
$664$	0.890303	0.0345505
$665$	−11.3558	−0.440360
$666$	0	0
$667$	−10.6395	−0.411961
$668$	8.37656	0.324099
$669$	0	0
$670$	18.0853	0.698697
$671$	−25.1355	−0.970346
$672$	0	0
$673$	−21.0447	−0.811212	−0.405606	−	0.914048i	$-0.632940\pi$
$673$	−21.0447	−0.811212	−0.405606	+	0.914048i	$0.632940\pi$
$674$	−32.2196	−1.24105
$675$	0	0
$676$	−11.5852	−0.445585
$677$	−32.7242	−1.25769	−0.628846	−	0.777530i	$-0.716472\pi$
$677$	−32.7242	−1.25769	−0.628846	+	0.777530i	$0.716472\pi$
$678$	0	0
$679$	6.84110	0.262538
$680$	−10.2015	−0.391208
$681$	0	0
$682$	16.8059	0.643531
$683$	3.42427	0.131026	0.0655131	−	0.997852i	$-0.479132\pi$
$683$	3.42427	0.131026	0.0655131	+	0.997852i	$0.479132\pi$
$684$	0	0
$685$	67.6881	2.58623
$686$	−10.4884	−0.400450
$687$	0	0
$688$	7.31409	0.278847
$689$	13.1327	0.500314
$690$	0	0
$691$	−18.2775	−0.695309	−0.347655	−	0.937623i	$-0.613022\pi$
$691$	−18.2775	−0.695309	−0.347655	+	0.937623i	$0.613022\pi$
$692$	−13.4817	−0.512497
$693$	0	0
$694$	−27.4569	−1.04225
$695$	57.0584	2.16435
$696$	0	0
$697$	−20.0416	−0.759128
$698$	−15.1318	−0.572746
$699$	0	0
$700$	5.83464	0.220529
$701$	−35.3366	−1.33464	−0.667322	−	0.744770i	$-0.732559\pi$
$701$	−35.3366	−1.33464	−0.667322	+	0.744770i	$0.732559\pi$
$702$	0	0
$703$	−33.8110	−1.27521
$704$	3.05111	0.114993
$705$	0	0
$706$	−6.66086	−0.250685
$707$	−2.28852	−0.0860687
$708$	0	0
$709$	−29.3811	−1.10343	−0.551716	−	0.834032i	$-0.686027\pi$
$709$	−29.3811	−1.10343	−0.551716	+	0.834032i	$0.686027\pi$
$710$	14.8985	0.559132
$711$	0	0
$712$	−17.4504	−0.653980
$713$	7.95931	0.298078
$714$	0	0
$715$	12.8035	0.478824
$716$	−4.67837	−0.174839
$717$	0	0
$718$	20.9666	0.782467
$719$	−27.8880	−1.04005	−0.520023	−	0.854152i	$-0.674077\pi$
$719$	−27.8880	−1.04005	−0.520023	+	0.854152i	$0.674077\pi$
$720$	0	0
$721$	−11.8346	−0.440742
$722$	−2.12382	−0.0790406
$723$	0	0
$724$	−15.1260	−0.562154
$725$	−54.8284	−2.03628
$726$	0	0
$727$	−41.3594	−1.53393	−0.766967	−	0.641687i	$-0.778235\pi$
$727$	−41.3594	−1.53393	−0.766967	+	0.641687i	$0.778235\pi$
$728$	0.931972	0.0345412
$729$	0	0
$730$	−35.0407	−1.29692
$731$	−21.1494	−0.782238
$732$	0	0
$733$	−1.44182	−0.0532547	−0.0266273	−	0.999645i	$-0.508477\pi$
$733$	−1.44182	−0.0532547	−0.0266273	+	0.999645i	$0.508477\pi$
$734$	−18.2504	−0.673633
$735$	0	0
$736$	1.44501	0.0532638
$737$	15.6408	0.576136
$738$	0	0
$739$	−21.2968	−0.783415	−0.391708	−	0.920090i	$-0.628116\pi$
$739$	−21.2968	−0.783415	−0.391708	+	0.920090i	$0.628116\pi$
$740$	29.0366	1.06741
$741$	0	0
$742$	8.65094	0.317586
$743$	4.52558	0.166027	0.0830137	−	0.996548i	$-0.473545\pi$
$743$	4.52558	0.166027	0.0830137	+	0.996548i	$0.473545\pi$
$744$	0	0
$745$	−58.0207	−2.12571
$746$	−4.01623	−0.147045
$747$	0	0
$748$	−8.82256	−0.322585
$749$	4.80553	0.175590
$750$	0	0
$751$	8.90680	0.325014	0.162507	−	0.986707i	$-0.448042\pi$
$751$	8.90680	0.325014	0.162507	+	0.986707i	$0.448042\pi$
$752$	1.42058	0.0518032
$753$	0	0
$754$	−8.75779	−0.318940
$755$	59.0253	2.14815
$756$	0	0
$757$	−39.1924	−1.42447	−0.712236	−	0.701940i	$-0.752318\pi$
$757$	−39.1924	−1.42447	−0.712236	+	0.701940i	$0.752318\pi$
$758$	3.42929	0.124557
$759$	0	0
$760$	−14.4931	−0.525721
$761$	39.1352	1.41865	0.709324	−	0.704882i	$-0.249000\pi$
$761$	39.1352	1.41865	0.709324	+	0.704882i	$0.249000\pi$
$762$	0	0
$763$	−1.70604	−0.0617629
$764$	13.0409	0.471804
$765$	0	0
$766$	−16.6667	−0.602191
$767$	−16.3913	−0.591854
$768$	0	0
$769$	4.32876	0.156099	0.0780495	−	0.996949i	$-0.475131\pi$
$769$	4.32876	0.156099	0.0780495	+	0.996949i	$0.475131\pi$
$770$	8.43411	0.303944
$771$	0	0
$772$	−8.03075	−0.289033
$773$	−30.2094	−1.08656	−0.543279	−	0.839552i	$-0.682817\pi$
$773$	−30.2094	−1.08656	−0.543279	+	0.839552i	$0.682817\pi$
$774$	0	0
$775$	41.0168	1.47337
$776$	8.73111	0.313428
$777$	0	0
$778$	15.0620	0.540000
$779$	−28.4729	−1.02015
$780$	0	0
$781$	12.8848	0.461053
$782$	−4.17838	−0.149419
$783$	0	0
$784$	−6.38608	−0.228074
$785$	−4.69841	−0.167693
$786$	0	0
$787$	39.7104	1.41552	0.707761	−	0.706452i	$-0.249705\pi$
$787$	39.7104	1.41552	0.707761	+	0.706452i	$0.249705\pi$
$788$	21.7171	0.773640
$789$	0	0
$790$	−5.33774	−0.189908
$791$	8.29048	0.294776
$792$	0	0
$793$	−9.79888	−0.347969
$794$	13.4446	0.477129
$795$	0	0
$796$	−6.18321	−0.219158
$797$	−46.6363	−1.65194	−0.825972	−	0.563712i	$-0.809373\pi$
$797$	−46.6363	−1.65194	−0.825972	+	0.563712i	$0.809373\pi$
$798$	0	0
$799$	−4.10774	−0.145321
$800$	7.44658	0.263277
$801$	0	0
$802$	25.2981	0.893306
$803$	−30.3044	−1.06942
$804$	0	0
$805$	3.99441	0.140785
$806$	6.55165	0.230772
$807$	0	0
$808$	−2.92077	−0.102752
$809$	8.26934	0.290734	0.145367	−	0.989378i	$-0.453564\pi$
$809$	8.26934	0.290734	0.145367	+	0.989378i	$0.453564\pi$
$810$	0	0
$811$	26.8790	0.943848	0.471924	−	0.881639i	$-0.343560\pi$
$811$	26.8790	0.943848	0.471924	+	0.881639i	$0.343560\pi$
$812$	−5.76906	−0.202454
$813$	0	0
$814$	25.1118	0.880169
$815$	9.98225	0.349663
$816$	0	0
$817$	−30.0467	−1.05120
$818$	−10.7393	−0.375493
$819$	0	0
$820$	24.4523	0.853910
$821$	−24.3930	−0.851322	−0.425661	−	0.904883i	$-0.639958\pi$
$821$	−24.3930	−0.851322	−0.425661	+	0.904883i	$0.639958\pi$
$822$	0	0
$823$	47.5002	1.65575	0.827876	−	0.560911i	$-0.189549\pi$
$823$	47.5002	1.65575	0.827876	+	0.560911i	$0.189549\pi$
$824$	−15.1041	−0.526177
$825$	0	0
$826$	−10.7975	−0.375693
$827$	6.96077	0.242050	0.121025	−	0.992649i	$-0.461382\pi$
$827$	6.96077	0.242050	0.121025	+	0.992649i	$0.461382\pi$
$828$	0	0
$829$	23.6070	0.819905	0.409952	−	0.912107i	$-0.365545\pi$
$829$	23.6070	0.819905	0.409952	+	0.912107i	$0.365545\pi$
$830$	3.14096	0.109024
$831$	0	0
$832$	1.18945	0.0412368
$833$	18.4659	0.639807
$834$	0	0
$835$	29.5523	1.02270
$836$	−12.5341	−0.433502
$837$	0	0
$838$	32.3250	1.11665
$839$	24.5241	0.846665	0.423332	−	0.905974i	$-0.360860\pi$
$839$	24.5241	0.846665	0.423332	+	0.905974i	$0.360860\pi$
$840$	0	0
$841$	25.2122	0.869386
$842$	−17.9165	−0.617442
$843$	0	0
$844$	−22.5418	−0.775921
$845$	−40.8723	−1.40605
$846$	0	0
$847$	−1.32475	−0.0455191
$848$	11.0409	0.379148
$849$	0	0
$850$	−21.5325	−0.738559
$851$	11.8930	0.407687
$852$	0	0
$853$	−28.9238	−0.990332	−0.495166	−	0.868798i	$-0.664893\pi$
$853$	−28.9238	−0.990332	−0.495166	+	0.868798i	$0.664893\pi$
$854$	−6.45487	−0.220881
$855$	0	0
$856$	6.13316	0.209627
$857$	−22.6446	−0.773527	−0.386763	−	0.922179i	$-0.626407\pi$
$857$	−22.6446	−0.773527	−0.386763	+	0.922179i	$0.626407\pi$
$858$	0	0
$859$	49.3504	1.68381	0.841907	−	0.539623i	$-0.181433\pi$
$859$	49.3504	1.68381	0.841907	+	0.539623i	$0.181433\pi$
$860$	25.8039	0.879906
$861$	0	0
$862$	25.3476	0.863343
$863$	−15.3714	−0.523247	−0.261624	−	0.965170i	$-0.584258\pi$
$863$	−15.3714	−0.523247	−0.261624	+	0.965170i	$0.584258\pi$
$864$	0	0
$865$	−47.5631	−1.61719
$866$	−16.9687	−0.576620
$867$	0	0
$868$	4.31580	0.146488
$869$	−4.61625	−0.156596
$870$	0	0
$871$	6.09744	0.206604
$872$	−2.17737	−0.0737352
$873$	0	0
$874$	−5.93619	−0.200795
$875$	6.76304	0.228633
$876$	0	0
$877$	−7.76067	−0.262059	−0.131030	−	0.991378i	$-0.541828\pi$
$877$	−7.76067	−0.262059	−0.131030	+	0.991378i	$0.541828\pi$
$878$	14.2557	0.481106
$879$	0	0
$880$	10.7642	0.362862
$881$	−15.5210	−0.522915	−0.261457	−	0.965215i	$-0.584203\pi$
$881$	−15.5210	−0.522915	−0.261457	+	0.965215i	$0.584203\pi$
$882$	0	0
$883$	−55.3606	−1.86303	−0.931517	−	0.363699i	$-0.881514\pi$
$883$	−55.3606	−1.86303	−0.931517	+	0.363699i	$0.881514\pi$
$884$	−3.43941	−0.115680
$885$	0	0
$886$	−20.6507	−0.693773
$887$	19.9019	0.668240	0.334120	−	0.942531i	$-0.391561\pi$
$887$	19.9019	0.668240	0.334120	+	0.942531i	$0.391561\pi$
$888$	0	0
$889$	−14.2884	−0.479219
$890$	−61.5644	−2.06364
$891$	0	0
$892$	−1.00000	−0.0334825
$893$	−5.83582	−0.195288
$894$	0	0
$895$	−16.5052	−0.551707
$896$	0.783532	0.0261760
$897$	0	0
$898$	0.367555	0.0122655
$899$	−40.5558	−1.35261
$900$	0	0
$901$	−31.9259	−1.06361
$902$	21.1471	0.704122
$903$	0	0
$904$	10.5809	0.351916
$905$	−53.3641	−1.77388
$906$	0	0
$907$	37.4900	1.24484	0.622418	−	0.782685i	$-0.286151\pi$
$907$	37.4900	1.24484	0.622418	+	0.782685i	$0.286151\pi$
$908$	8.08139	0.268190
$909$	0	0
$910$	3.28797	0.108995
$911$	−48.4629	−1.60565	−0.802824	−	0.596217i	$-0.796670\pi$
$911$	−48.4629	−1.60565	−0.802824	+	0.596217i	$0.796670\pi$
$912$	0	0
$913$	2.71641	0.0899000
$914$	−22.3013	−0.737660
$915$	0	0
$916$	−20.8687	−0.689520
$917$	−0.187181	−0.00618126
$918$	0	0
$919$	−23.7153	−0.782296	−0.391148	−	0.920328i	$-0.627922\pi$
$919$	−23.7153	−0.782296	−0.391148	+	0.920328i	$0.627922\pi$
$920$	5.09796	0.168075
$921$	0	0
$922$	−2.75287	−0.0906611
$923$	5.02302	0.165335
$924$	0	0
$925$	61.2884	2.01515
$926$	19.8757	0.653155
$927$	0	0
$928$	−7.36289	−0.241699
$929$	29.5260	0.968715	0.484358	−	0.874870i	$-0.339053\pi$
$929$	29.5260	0.968715	0.484358	+	0.874870i	$0.339053\pi$
$930$	0	0
$931$	26.2344	0.859798
$932$	−5.85766	−0.191874
$933$	0	0
$934$	17.9849	0.588484
$935$	−31.1258	−1.01792
$936$	0	0
$937$	−24.7490	−0.808514	−0.404257	−	0.914645i	$-0.632470\pi$
$937$	−24.7490	−0.808514	−0.404257	+	0.914645i	$0.632470\pi$
$938$	4.01660	0.131147
$939$	0	0
$940$	5.01176	0.163466
$941$	−51.0695	−1.66482	−0.832410	−	0.554161i	$-0.813039\pi$
$941$	−51.0695	−1.66482	−0.832410	+	0.554161i	$0.813039\pi$
$942$	0	0
$943$	10.0153	0.326144
$944$	−13.7805	−0.448518
$945$	0	0
$946$	22.3161	0.725558
$947$	−12.7755	−0.415148	−0.207574	−	0.978219i	$-0.566557\pi$
$947$	−12.7755	−0.415148	−0.207574	+	0.978219i	$0.566557\pi$
$948$	0	0
$949$	−11.8139	−0.383496
$950$	−30.5910	−0.992504
$951$	0	0
$952$	−2.26566	−0.0734304
$953$	14.7868	0.478992	0.239496	−	0.970897i	$-0.423018\pi$
$953$	14.7868	0.478992	0.239496	+	0.970897i	$0.423018\pi$
$954$	0	0
$955$	46.0080	1.48878
$956$	3.61755	0.117000
$957$	0	0
$958$	−10.2988	−0.332740
$959$	15.0329	0.485439
$960$	0	0
$961$	−0.660490	−0.0213061
$962$	9.78965	0.315631
$963$	0	0
$964$	12.9608	0.417440
$965$	−28.3323	−0.912048
$966$	0	0
$967$	−21.3916	−0.687906	−0.343953	−	0.938987i	$-0.611766\pi$
$967$	−21.3916	−0.687906	−0.343953	+	0.938987i	$0.611766\pi$
$968$	−1.69075	−0.0543427
$969$	0	0
$970$	30.8031	0.989028
$971$	46.5631	1.49428	0.747140	−	0.664667i	$-0.231426\pi$
$971$	46.5631	1.49428	0.747140	+	0.664667i	$0.231426\pi$
$972$	0	0
$973$	12.6722	0.406251
$974$	36.4302	1.16730
$975$	0	0
$976$	−8.23816	−0.263697
$977$	−12.9709	−0.414976	−0.207488	−	0.978238i	$-0.566529\pi$
$977$	−12.9709	−0.414976	−0.207488	+	0.978238i	$0.566529\pi$
$978$	0	0
$979$	−53.2429	−1.70165
$980$	−22.5299	−0.719691
$981$	0	0
$982$	21.4204	0.683552
$983$	−57.8000	−1.84353	−0.921767	−	0.387745i	$-0.873254\pi$
$983$	−57.8000	−1.84353	−0.921767	+	0.387745i	$0.873254\pi$
$984$	0	0
$985$	76.6173	2.44123
$986$	21.2905	0.678028
$987$	0	0
$988$	−4.88633	−0.155455
$989$	10.5689	0.336073
$990$	0	0
$991$	−8.46606	−0.268933	−0.134467	−	0.990918i	$-0.542932\pi$
$991$	−8.46606	−0.268933	−0.134467	+	0.990918i	$0.542932\pi$
$992$	5.50813	0.174883
$993$	0	0
$994$	3.30884	0.104950
$995$	−21.8142	−0.691557
$996$	0	0
$997$	26.8774	0.851217	0.425609	−	0.904907i	$-0.360060\pi$
$997$	26.8774	0.851217	0.425609	+	0.904907i	$0.360060\pi$
$998$	33.7171	1.06729
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4014.2.a.u.1.6	yes	6
3.2	odd	2		4014.2.a.t.1.1	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4014.2.a.t.1.1	✓	6	3.2	odd	2
4014.2.a.u.1.6	yes	6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4014 = 2 \cdot 3^{2} \cdot 223 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4014.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +3.52797 q^{5} +0.783532 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +3.52797 q^{5} +0.783532 q^{7} +1.00000 q^{8} +3.52797 q^{10} +3.05111 q^{11} +1.18945 q^{13} +0.783532 q^{14} +1.00000 q^{16} -2.89159 q^{17} -4.10806 q^{19} +3.52797 q^{20} +3.05111 q^{22} +1.44501 q^{23} +7.44658 q^{25} +1.18945 q^{26} +0.783532 q^{28} -7.36289 q^{29} +5.50813 q^{31} +1.00000 q^{32} -2.89159 q^{34} +2.76428 q^{35} +8.23040 q^{37} -4.10806 q^{38} +3.52797 q^{40} +6.93097 q^{41} +7.31409 q^{43} +3.05111 q^{44} +1.44501 q^{46} +1.42058 q^{47} -6.38608 q^{49} +7.44658 q^{50} +1.18945 q^{52} +11.0409 q^{53} +10.7642 q^{55} +0.783532 q^{56} -7.36289 q^{58} -13.7805 q^{59} -8.23816 q^{61} +5.50813 q^{62} +1.00000 q^{64} +4.19635 q^{65} +5.12627 q^{67} -2.89159 q^{68} +2.76428 q^{70} +4.22298 q^{71} -9.93226 q^{73} +8.23040 q^{74} -4.10806 q^{76} +2.39064 q^{77} -1.51298 q^{79} +3.52797 q^{80} +6.93097 q^{82} +0.890303 q^{83} -10.2015 q^{85} +7.31409 q^{86} +3.05111 q^{88} -17.4504 q^{89} +0.931972 q^{91} +1.44501 q^{92} +1.42058 q^{94} -14.4931 q^{95} +8.73111 q^{97} -6.38608 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{2} + 6 q^{4} + 2 q^{5} + 8 q^{7} + 6 q^{8}+O(q^{10}) \) 6 * q + 6 * q^2 + 6 * q^4 + 2 * q^5 + 8 * q^7 + 6 * q^8	\( 6 q + 6 q^{2} + 6 q^{4} + 2 q^{5} + 8 q^{7} + 6 q^{8} + 2 q^{10} + 2 q^{11} - 2 q^{13} + 8 q^{14} + 6 q^{16} + 2 q^{17} - 2 q^{19} + 2 q^{20} + 2 q^{22} + 22 q^{23} + 12 q^{25} - 2 q^{26} + 8 q^{28} + 8 q^{29} - 12 q^{31} + 6 q^{32} + 2 q^{34} + 20 q^{35} - 2 q^{38} + 2 q^{40} + 28 q^{41} + 14 q^{43} + 2 q^{44} + 22 q^{46} + 2 q^{47} + 12 q^{50} - 2 q^{52} + 26 q^{53} + 6 q^{55} + 8 q^{56} + 8 q^{58} + 4 q^{59} - 6 q^{61} - 12 q^{62} + 6 q^{64} + 16 q^{65} + 18 q^{67} + 2 q^{68} + 20 q^{70} + 12 q^{71} - 20 q^{73} - 2 q^{76} + 20 q^{77} + 12 q^{79} + 2 q^{80} + 28 q^{82} + 12 q^{83} - 10 q^{85} + 14 q^{86} + 2 q^{88} - 2 q^{89} - 22 q^{91} + 22 q^{92} + 2 q^{94} + 6 q^{95} - 6 q^{97}+O(q^{100}) \) 6 * q + 6 * q^2 + 6 * q^4 + 2 * q^5 + 8 * q^7 + 6 * q^8 + 2 * q^10 + 2 * q^11 - 2 * q^13 + 8 * q^14 + 6 * q^16 + 2 * q^17 - 2 * q^19 + 2 * q^20 + 2 * q^22 + 22 * q^23 + 12 * q^25 - 2 * q^26 + 8 * q^28 + 8 * q^29 - 12 * q^31 + 6 * q^32 + 2 * q^34 + 20 * q^35 - 2 * q^38 + 2 * q^40 + 28 * q^41 + 14 * q^43 + 2 * q^44 + 22 * q^46 + 2 * q^47 + 12 * q^50 - 2 * q^52 + 26 * q^53 + 6 * q^55 + 8 * q^56 + 8 * q^58 + 4 * q^59 - 6 * q^61 - 12 * q^62 + 6 * q^64 + 16 * q^65 + 18 * q^67 + 2 * q^68 + 20 * q^70 + 12 * q^71 - 20 * q^73 - 2 * q^76 + 20 * q^77 + 12 * q^79 + 2 * q^80 + 28 * q^82 + 12 * q^83 - 10 * q^85 + 14 * q^86 + 2 * q^88 - 2 * q^89 - 22 * q^91 + 22 * q^92 + 2 * q^94 + 6 * q^95 - 6 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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