LMFDB - Embedded newform 3330.2.a.bj.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3330,2,Mod(1,3330)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3330, 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("3330.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3330.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:	$26.5901838731$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.54764.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 9x^{2} + 3x + 2 $ x^4 - x^3 - 9x^2 + 3x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1110)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.339102$ of defining polynomial
Character	$\chi$	$=$	3330.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.84556 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.84556 q^{7} -1.00000 q^{8} +1.00000 q^{10} -0.832640 q^{11} +3.21973 q^{13} -1.84556 q^{14} +1.00000 q^{16} +7.95030 q^{17} +4.20681 q^{19} -1.00000 q^{20} +0.832640 q^{22} +6.57614 q^{23} +1.00000 q^{25} -3.21973 q^{26} +1.84556 q^{28} -3.03945 q^{29} -4.73057 q^{31} -1.00000 q^{32} -7.95030 q^{34} -1.84556 q^{35} -1.00000 q^{37} -4.20681 q^{38} +1.00000 q^{40} -2.73057 q^{41} -5.03945 q^{43} -0.832640 q^{44} -6.57614 q^{46} -9.11766 q^{47} -3.59390 q^{49} -1.00000 q^{50} +3.21973 q^{52} -3.95030 q^{53} +0.832640 q^{55} -1.84556 q^{56} +3.03945 q^{58} +3.69113 q^{59} +5.40878 q^{61} +4.73057 q^{62} +1.00000 q^{64} -3.21973 q^{65} +11.7959 q^{67} +7.95030 q^{68} +1.84556 q^{70} +13.7700 q^{71} -3.52860 q^{73} +1.00000 q^{74} +4.20681 q^{76} -1.53669 q^{77} +4.00000 q^{79} -1.00000 q^{80} +2.73057 q^{82} +3.55445 q^{83} -7.95030 q^{85} +5.03945 q^{86} +0.832640 q^{88} +6.55029 q^{89} +5.94222 q^{91} +6.57614 q^{92} +9.11766 q^{94} -4.20681 q^{95} -10.4918 q^{97} +3.59390 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 4 q^{4} - 4 q^{5} + 4 q^{7} - 4 q^{8}+O(q^{10}) $ 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^5 + 4 * q^7 - 4 * q^8	$ 4 q - 4 q^{2} + 4 q^{4} - 4 q^{5} + 4 q^{7} - 4 q^{8} + 4 q^{10} - 2 q^{11} - 3 q^{13} - 4 q^{14} + 4 q^{16} - 6 q^{17} + 3 q^{19} - 4 q^{20} + 2 q^{22} + q^{23} + 4 q^{25} + 3 q^{26} + 4 q^{28} + 3 q^{29} + 3 q^{31} - 4 q^{32} + 6 q^{34} - 4 q^{35} - 4 q^{37} - 3 q^{38} + 4 q^{40} + 11 q^{41} - 5 q^{43} - 2 q^{44} - q^{46} + 14 q^{49} - 4 q^{50} - 3 q^{52} + 22 q^{53} + 2 q^{55} - 4 q^{56} - 3 q^{58} + 8 q^{59} - 5 q^{61} - 3 q^{62} + 4 q^{64} + 3 q^{65} + 6 q^{67} - 6 q^{68} + 4 q^{70} + 18 q^{71} - 5 q^{73} + 4 q^{74} + 3 q^{76} + 4 q^{77} + 16 q^{79} - 4 q^{80} - 11 q^{82} + q^{83} + 6 q^{85} + 5 q^{86} + 2 q^{88} + 5 q^{89} - 13 q^{91} + q^{92} - 3 q^{95} + 7 q^{97} - 14 q^{98}+O(q^{100}) $ 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^5 + 4 * q^7 - 4 * q^8 + 4 * q^10 - 2 * q^11 - 3 * q^13 - 4 * q^14 + 4 * q^16 - 6 * q^17 + 3 * q^19 - 4 * q^20 + 2 * q^22 + q^23 + 4 * q^25 + 3 * q^26 + 4 * q^28 + 3 * q^29 + 3 * q^31 - 4 * q^32 + 6 * q^34 - 4 * q^35 - 4 * q^37 - 3 * q^38 + 4 * q^40 + 11 * q^41 - 5 * q^43 - 2 * q^44 - q^46 + 14 * q^49 - 4 * q^50 - 3 * q^52 + 22 * q^53 + 2 * q^55 - 4 * q^56 - 3 * q^58 + 8 * q^59 - 5 * q^61 - 3 * q^62 + 4 * q^64 + 3 * q^65 + 6 * q^67 - 6 * q^68 + 4 * q^70 + 18 * q^71 - 5 * q^73 + 4 * q^74 + 3 * q^76 + 4 * q^77 + 16 * q^79 - 4 * q^80 - 11 * q^82 + q^83 + 6 * q^85 + 5 * q^86 + 2 * q^88 + 5 * q^89 - 13 * q^91 + q^92 - 3 * q^95 + 7 * q^97 - 14 * q^98

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
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.84556	0.697557	0.348779	−	0.937205i	$-0.386596\pi$
$7$	1.84556	0.697557	0.348779	+	0.937205i	$0.386596\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	−0.832640	−0.251050	−0.125525	−	0.992090i	$-0.540062\pi$
$11$	−0.832640	−0.251050	−0.125525	+	0.992090i	$0.540062\pi$
$12$	0	0
$13$	3.21973	0.892992	0.446496	−	0.894786i	$-0.352672\pi$
$13$	3.21973	0.892992	0.446496	+	0.894786i	$0.352672\pi$
$14$	−1.84556	−0.493248
$15$	0	0
$16$	1.00000	0.250000
$17$	7.95030	1.92823	0.964116	−	0.265482i	$-0.0855311\pi$
$17$	7.95030	1.92823	0.964116	+	0.265482i	$0.0855311\pi$
$18$	0	0
$19$	4.20681	0.965108	0.482554	−	0.875866i	$-0.339709\pi$
$19$	4.20681	0.965108	0.482554	+	0.875866i	$0.339709\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	0.832640	0.177519
$23$	6.57614	1.37122	0.685610	−	0.727969i	$-0.259536\pi$
$23$	6.57614	1.37122	0.685610	+	0.727969i	$0.259536\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−3.21973	−0.631441
$27$	0	0
$28$	1.84556	0.348779
$29$	−3.03945	−0.564411	−0.282206	−	0.959354i	$-0.591066\pi$
$29$	−3.03945	−0.564411	−0.282206	+	0.959354i	$0.591066\pi$
$30$	0	0
$31$	−4.73057	−0.849636	−0.424818	−	0.905279i	$-0.639662\pi$
$31$	−4.73057	−0.849636	−0.424818	+	0.905279i	$0.639662\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−7.95030	−1.36347
$35$	−1.84556	−0.311957
$36$	0	0
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	−4.20681	−0.682434
$39$	0	0
$40$	1.00000	0.158114
$41$	−2.73057	−0.426444	−0.213222	−	0.977004i	$-0.568396\pi$
$41$	−2.73057	−0.426444	−0.213222	+	0.977004i	$0.568396\pi$
$42$	0	0
$43$	−5.03945	−0.768508	−0.384254	−	0.923227i	$-0.625541\pi$
$43$	−5.03945	−0.768508	−0.384254	+	0.923227i	$0.625541\pi$
$44$	−0.832640	−0.125525
$45$	0	0
$46$	−6.57614	−0.969598
$47$	−9.11766	−1.32995	−0.664974	−	0.746867i	$-0.731557\pi$
$47$	−9.11766	−1.32995	−0.664974	+	0.746867i	$0.731557\pi$
$48$	0	0
$49$	−3.59390	−0.513414
$50$	−1.00000	−0.141421
$51$	0	0
$52$	3.21973	0.446496
$53$	−3.95030	−0.542616	−0.271308	−	0.962493i	$-0.587456\pi$
$53$	−3.95030	−0.542616	−0.271308	+	0.962493i	$0.587456\pi$
$54$	0	0
$55$	0.832640	0.112273
$56$	−1.84556	−0.246624
$57$	0	0
$58$	3.03945	0.399099
$59$	3.69113	0.480544	0.240272	−	0.970706i	$-0.422763\pi$
$59$	3.69113	0.480544	0.240272	+	0.970706i	$0.422763\pi$
$60$	0	0
$61$	5.40878	0.692523	0.346261	−	0.938138i	$-0.387451\pi$
$61$	5.40878	0.692523	0.346261	+	0.938138i	$0.387451\pi$
$62$	4.73057	0.600783
$63$	0	0
$64$	1.00000	0.125000
$65$	−3.21973	−0.399358
$66$	0	0
$67$	11.7959	1.44109	0.720547	−	0.693406i	$-0.243891\pi$
$67$	11.7959	1.44109	0.720547	+	0.693406i	$0.243891\pi$
$68$	7.95030	0.964116
$69$	0	0
$70$	1.84556	0.220587
$71$	13.7700	1.63420	0.817100	−	0.576495i	$-0.195580\pi$
$71$	13.7700	1.63420	0.817100	+	0.576495i	$0.195580\pi$
$72$	0	0
$73$	−3.52860	−0.412992	−0.206496	−	0.978447i	$-0.566206\pi$
$73$	−3.52860	−0.412992	−0.206496	+	0.978447i	$0.566206\pi$
$74$	1.00000	0.116248
$75$	0	0
$76$	4.20681	0.482554
$77$	−1.53669	−0.175122
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	2.73057	0.301541
$83$	3.55445	0.390151	0.195076	−	0.980788i	$-0.437505\pi$
$83$	3.55445	0.390151	0.195076	+	0.980788i	$0.437505\pi$
$84$	0	0
$85$	−7.95030	−0.862331
$86$	5.03945	0.543417
$87$	0	0
$88$	0.832640	0.0887597
$89$	6.55029	0.694329	0.347165	−	0.937804i	$-0.387144\pi$
$89$	6.55029	0.694329	0.347165	+	0.937804i	$0.387144\pi$
$90$	0	0
$91$	5.94222	0.622913
$92$	6.57614	0.685610
$93$	0	0
$94$	9.11766	0.940415
$95$	−4.20681	−0.431609
$96$	0	0
$97$	−10.4918	−1.06528	−0.532642	−	0.846341i	$-0.678801\pi$
$97$	−10.4918	−1.06528	−0.532642	+	0.846341i	$0.678801\pi$
$98$	3.59390	0.363038
$99$	0	0
$100$	1.00000	0.100000
$101$	3.63067	0.361265	0.180633	−	0.983551i	$-0.442186\pi$
$101$	3.63067	0.361265	0.180633	+	0.983551i	$0.442186\pi$
$102$	0	0
$103$	−9.11766	−0.898390	−0.449195	−	0.893434i	$-0.648289\pi$
$103$	−9.11766	−0.898390	−0.449195	+	0.893434i	$0.648289\pi$
$104$	−3.21973	−0.315720
$105$	0	0
$106$	3.95030	0.383687
$107$	−8.91086	−0.861445	−0.430723	−	0.902484i	$-0.641741\pi$
$107$	−8.91086	−0.861445	−0.430723	+	0.902484i	$0.641741\pi$
$108$	0	0
$109$	11.2979	1.08215	0.541073	−	0.840975i	$-0.318018\pi$
$109$	11.2979	1.08215	0.541073	+	0.840975i	$0.318018\pi$
$110$	−0.832640	−0.0793891
$111$	0	0
$112$	1.84556	0.174389
$113$	11.0394	1.03850	0.519252	−	0.854621i	$-0.326211\pi$
$113$	11.0394	1.03850	0.519252	+	0.854621i	$0.326211\pi$
$114$	0	0
$115$	−6.57614	−0.613228
$116$	−3.03945	−0.282206
$117$	0	0
$118$	−3.69113	−0.339796
$119$	14.6728	1.34505
$120$	0	0
$121$	−10.3067	−0.936974
$122$	−5.40878	−0.489688
$123$	0	0
$124$	−4.73057	−0.424818
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−0.136678	−0.0121282	−0.00606410	−	0.999982i	$-0.501930\pi$
$127$	−0.136678	−0.0121282	−0.00606410	+	0.999982i	$0.501930\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	3.21973	0.282389
$131$	−2.07889	−0.181634	−0.0908169	−	0.995868i	$-0.528948\pi$
$131$	−2.07889	−0.181634	−0.0908169	+	0.995868i	$0.528948\pi$
$132$	0	0
$133$	7.76393	0.673218
$134$	−11.7959	−1.01901
$135$	0	0
$136$	−7.95030	−0.681733
$137$	−3.06597	−0.261943	−0.130972	−	0.991386i	$-0.541810\pi$
$137$	−3.06597	−0.261943	−0.130972	+	0.991386i	$0.541810\pi$
$138$	0	0
$139$	22.1917	1.88228	0.941139	−	0.338021i	$-0.109757\pi$
$139$	22.1917	1.88228	0.941139	+	0.338021i	$0.109757\pi$
$140$	−1.84556	−0.155979
$141$	0	0
$142$	−13.7700	−1.15555
$143$	−2.68088	−0.224186
$144$	0	0
$145$	3.03945	0.252412
$146$	3.52860	0.292029
$147$	0	0
$148$	−1.00000	−0.0821995
$149$	14.4999	1.18788	0.593940	−	0.804510i	$-0.297572\pi$
$149$	14.4999	1.18788	0.593940	+	0.804510i	$0.297572\pi$
$150$	0	0
$151$	8.24142	0.670677	0.335339	−	0.942098i	$-0.391149\pi$
$151$	8.24142	0.670677	0.335339	+	0.942098i	$0.391149\pi$
$152$	−4.20681	−0.341217
$153$	0	0
$154$	1.53669	0.123830
$155$	4.73057	0.379969
$156$	0	0
$157$	−6.11283	−0.487857	−0.243928	−	0.969793i	$-0.578436\pi$
$157$	−6.11283	−0.487857	−0.243928	+	0.969793i	$0.578436\pi$
$158$	−4.00000	−0.318223
$159$	0	0
$160$	1.00000	0.0790569
$161$	12.1367	0.956504
$162$	0	0
$163$	−21.4373	−1.67910	−0.839549	−	0.543283i	$-0.817181\pi$
$163$	−21.4373	−1.67910	−0.839549	+	0.543283i	$0.817181\pi$
$164$	−2.73057	−0.213222
$165$	0	0
$166$	−3.55445	−0.275879
$167$	22.5815	1.74741	0.873704	−	0.486458i	$-0.161711\pi$
$167$	22.5815	1.74741	0.873704	+	0.486458i	$0.161711\pi$
$168$	0	0
$169$	−2.63334	−0.202565
$170$	7.95030	0.609760
$171$	0	0
$172$	−5.03945	−0.384254
$173$	6.48916	0.493361	0.246681	−	0.969097i	$-0.420660\pi$
$173$	6.48916	0.493361	0.246681	+	0.969097i	$0.420660\pi$
$174$	0	0
$175$	1.84556	0.139511
$176$	−0.832640	−0.0627626
$177$	0	0
$178$	−6.55029	−0.490965
$179$	−16.2095	−1.21155	−0.605777	−	0.795635i	$-0.707138\pi$
$179$	−16.2095	−1.21155	−0.605777	+	0.795635i	$0.707138\pi$
$180$	0	0
$181$	5.38225	0.400060	0.200030	−	0.979790i	$-0.435896\pi$
$181$	5.38225	0.400060	0.200030	+	0.979790i	$0.435896\pi$
$182$	−5.94222	−0.440466
$183$	0	0
$184$	−6.57614	−0.484799
$185$	1.00000	0.0735215
$186$	0	0
$187$	−6.61974	−0.484083
$188$	−9.11766	−0.664974
$189$	0	0
$190$	4.20681	0.305194
$191$	10.5680	0.764677	0.382339	−	0.924022i	$-0.375119\pi$
$191$	10.5680	0.764677	0.382339	+	0.924022i	$0.375119\pi$
$192$	0	0
$193$	3.11766	0.224414	0.112207	−	0.993685i	$-0.464208\pi$
$193$	3.11766	0.224414	0.112207	+	0.993685i	$0.464208\pi$
$194$	10.4918	0.753269
$195$	0	0
$196$	−3.59390	−0.256707
$197$	−0.576137	−0.0410481	−0.0205240	−	0.999789i	$-0.506533\pi$
$197$	−0.576137	−0.0410481	−0.0205240	+	0.999789i	$0.506533\pi$
$198$	0	0
$199$	6.64359	0.470952	0.235476	−	0.971880i	$-0.424335\pi$
$199$	6.64359	0.470952	0.235476	+	0.971880i	$0.424335\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	−3.63067	−0.255453
$203$	−5.60949	−0.393709
$204$	0	0
$205$	2.73057	0.190712
$206$	9.11766	0.635258
$207$	0	0
$208$	3.21973	0.223248
$209$	−3.50276	−0.242291
$210$	0	0
$211$	−8.83531	−0.608248	−0.304124	−	0.952632i	$-0.598364\pi$
$211$	−8.83531	−0.608248	−0.304124	+	0.952632i	$0.598364\pi$
$212$	−3.95030	−0.271308
$213$	0	0
$214$	8.91086	0.609134
$215$	5.03945	0.343687
$216$	0	0
$217$	−8.73057	−0.592670
$218$	−11.2979	−0.765193
$219$	0	0
$220$	0.832640	0.0561366
$221$	25.5978	1.72190
$222$	0	0
$223$	17.8828	1.19752	0.598762	−	0.800927i	$-0.295660\pi$
$223$	17.8828	1.19752	0.598762	+	0.800927i	$0.295660\pi$
$224$	−1.84556	−0.123312
$225$	0	0
$226$	−11.0394	−0.734333
$227$	−25.1959	−1.67231	−0.836155	−	0.548494i	$-0.815201\pi$
$227$	−25.1959	−1.67231	−0.836155	+	0.548494i	$0.815201\pi$
$228$	0	0
$229$	−5.48699	−0.362591	−0.181295	−	0.983429i	$-0.558029\pi$
$229$	−5.48699	−0.362591	−0.181295	+	0.983429i	$0.558029\pi$
$230$	6.57614	0.433618
$231$	0	0
$232$	3.03945	0.199549
$233$	28.5788	1.87226	0.936130	−	0.351654i	$-0.114381\pi$
$233$	28.5788	1.87226	0.936130	+	0.351654i	$0.114381\pi$
$234$	0	0
$235$	9.11766	0.594771
$236$	3.69113	0.240272
$237$	0	0
$238$	−14.6728	−0.951096
$239$	−1.77811	−0.115016	−0.0575081	−	0.998345i	$-0.518316\pi$
$239$	−1.77811	−0.115016	−0.0575081	+	0.998345i	$0.518316\pi$
$240$	0	0
$241$	22.2612	1.43397	0.716984	−	0.697090i	$-0.245522\pi$
$241$	22.2612	1.43397	0.716984	+	0.697090i	$0.245522\pi$
$242$	10.3067	0.662540
$243$	0	0
$244$	5.40878	0.346261
$245$	3.59390	0.229606
$246$	0	0
$247$	13.5448	0.861834
$248$	4.73057	0.300392
$249$	0	0
$250$	1.00000	0.0632456
$251$	28.6748	1.80994	0.904968	−	0.425479i	$-0.139894\pi$
$251$	28.6748	1.80994	0.904968	+	0.425479i	$0.139894\pi$
$252$	0	0
$253$	−5.47556	−0.344245
$254$	0.136678	0.00857593
$255$	0	0
$256$	1.00000	0.0625000
$257$	−9.19388	−0.573499	−0.286749	−	0.958006i	$-0.592575\pi$
$257$	−9.19388	−0.573499	−0.286749	+	0.958006i	$0.592575\pi$
$258$	0	0
$259$	−1.84556	−0.114678
$260$	−3.21973	−0.199679
$261$	0	0
$262$	2.07889	0.128434
$263$	8.02652	0.494937	0.247468	−	0.968896i	$-0.420401\pi$
$263$	8.02652	0.494937	0.247468	+	0.968896i	$0.420401\pi$
$264$	0	0
$265$	3.95030	0.242665
$266$	−7.76393	−0.476037
$267$	0	0
$268$	11.7959	0.720547
$269$	−21.3849	−1.30386	−0.651931	−	0.758278i	$-0.726041\pi$
$269$	−21.3849	−1.30386	−0.651931	+	0.758278i	$0.726041\pi$
$270$	0	0
$271$	2.64359	0.160587	0.0802934	−	0.996771i	$-0.474414\pi$
$271$	2.64359	0.160587	0.0802934	+	0.996771i	$0.474414\pi$
$272$	7.95030	0.482058
$273$	0	0
$274$	3.06597	0.185222
$275$	−0.832640	−0.0502101
$276$	0	0
$277$	−14.4456	−0.867949	−0.433975	−	0.900925i	$-0.642889\pi$
$277$	−14.4456	−0.867949	−0.433975	+	0.900925i	$0.642889\pi$
$278$	−22.1917	−1.33097
$279$	0	0
$280$	1.84556	0.110294
$281$	11.2197	0.669313	0.334656	−	0.942340i	$-0.391380\pi$
$281$	11.2197	0.669313	0.334656	+	0.942340i	$0.391380\pi$
$282$	0	0
$283$	3.86332	0.229651	0.114825	−	0.993386i	$-0.463369\pi$
$283$	3.86332	0.229651	0.114825	+	0.993386i	$0.463369\pi$
$284$	13.7700	0.817100
$285$	0	0
$286$	2.68088	0.158524
$287$	−5.03945	−0.297469
$288$	0	0
$289$	46.2073	2.71808
$290$	−3.03945	−0.178482
$291$	0	0
$292$	−3.52860	−0.206496
$293$	16.1544	0.943752	0.471876	−	0.881665i	$-0.343577\pi$
$293$	16.1544	0.943752	0.471876	+	0.881665i	$0.343577\pi$
$294$	0	0
$295$	−3.69113	−0.214906
$296$	1.00000	0.0581238
$297$	0	0
$298$	−14.4999	−0.839958
$299$	21.1734	1.22449
$300$	0	0
$301$	−9.30062	−0.536079
$302$	−8.24142	−0.474240
$303$	0	0
$304$	4.20681	0.241277
$305$	−5.40878	−0.309706
$306$	0	0
$307$	6.28303	0.358591	0.179296	−	0.983795i	$-0.442618\pi$
$307$	6.28303	0.358591	0.179296	+	0.983795i	$0.442618\pi$
$308$	−1.53669	−0.0875611
$309$	0	0
$310$	−4.73057	−0.268679
$311$	−20.6312	−1.16989	−0.584943	−	0.811074i	$-0.698883\pi$
$311$	−20.6312	−1.16989	−0.584943	+	0.811074i	$0.698883\pi$
$312$	0	0
$313$	−7.32180	−0.413852	−0.206926	−	0.978357i	$-0.566346\pi$
$313$	−7.32180	−0.413852	−0.206926	+	0.978357i	$0.566346\pi$
$314$	6.11283	0.344967
$315$	0	0
$316$	4.00000	0.225018
$317$	−19.1442	−1.07524	−0.537622	−	0.843186i	$-0.680677\pi$
$317$	−19.1442	−1.07524	−0.537622	+	0.843186i	$0.680677\pi$
$318$	0	0
$319$	2.53077	0.141696
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	−12.1367	−0.676351
$323$	33.4454	1.86095
$324$	0	0
$325$	3.21973	0.178598
$326$	21.4373	1.18730
$327$	0	0
$328$	2.73057	0.150771
$329$	−16.8272	−0.927715
$330$	0	0
$331$	0.908184	0.0499183	0.0249591	−	0.999688i	$-0.492054\pi$
$331$	0.908184	0.0499183	0.0249591	+	0.999688i	$0.492054\pi$
$332$	3.55445	0.195076
$333$	0	0
$334$	−22.5815	−1.23560
$335$	−11.7959	−0.644477
$336$	0	0
$337$	−8.11083	−0.441825	−0.220913	−	0.975294i	$-0.570904\pi$
$337$	−8.11083	−0.441825	−0.220913	+	0.975294i	$0.570904\pi$
$338$	2.63334	0.143235
$339$	0	0
$340$	−7.95030	−0.431166
$341$	3.93887	0.213302
$342$	0	0
$343$	−19.5517	−1.05569
$344$	5.03945	0.271709
$345$	0	0
$346$	−6.48916	−0.348859
$347$	−9.35641	−0.502278	−0.251139	−	0.967951i	$-0.580805\pi$
$347$	−9.35641	−0.502278	−0.251139	+	0.967951i	$0.580805\pi$
$348$	0	0
$349$	−35.4665	−1.89848	−0.949239	−	0.314556i	$-0.898144\pi$
$349$	−35.4665	−1.89848	−0.949239	+	0.314556i	$0.898144\pi$
$350$	−1.84556	−0.0986495
$351$	0	0
$352$	0.832640	0.0443799
$353$	36.0923	1.92100	0.960500	−	0.278279i	$-0.0897639\pi$
$353$	36.0923	1.92100	0.960500	+	0.278279i	$0.0897639\pi$
$354$	0	0
$355$	−13.7700	−0.730837
$356$	6.55029	0.347165
$357$	0	0
$358$	16.2095	0.856698
$359$	−2.07889	−0.109720	−0.0548599	−	0.998494i	$-0.517471\pi$
$359$	−2.07889	−0.109720	−0.0548599	+	0.998494i	$0.517471\pi$
$360$	0	0
$361$	−1.30278	−0.0685674
$362$	−5.38225	−0.282885
$363$	0	0
$364$	5.94222	0.311457
$365$	3.52860	0.184696
$366$	0	0
$367$	34.2645	1.78859	0.894297	−	0.447474i	$-0.147676\pi$
$367$	34.2645	1.78859	0.894297	+	0.447474i	$0.147676\pi$
$368$	6.57614	0.342805
$369$	0	0
$370$	−1.00000	−0.0519875
$371$	−7.29053	−0.378506
$372$	0	0
$373$	−6.00000	−0.310668	−0.155334	−	0.987862i	$-0.549645\pi$
$373$	−6.00000	−0.310668	−0.155334	+	0.987862i	$0.549645\pi$
$374$	6.61974	0.342299
$375$	0	0
$376$	9.11766	0.470208
$377$	−9.78620	−0.504015
$378$	0	0
$379$	29.8217	1.53184	0.765919	−	0.642937i	$-0.222284\pi$
$379$	29.8217	1.53184	0.765919	+	0.642937i	$0.222284\pi$
$380$	−4.20681	−0.215805
$381$	0	0
$382$	−10.5680	−0.540708
$383$	6.78027	0.346456	0.173228	−	0.984882i	$-0.444580\pi$
$383$	6.78027	0.346456	0.173228	+	0.984882i	$0.444580\pi$
$384$	0	0
$385$	1.53669	0.0783170
$386$	−3.11766	−0.158685
$387$	0	0
$388$	−10.4918	−0.532642
$389$	−35.1959	−1.78450	−0.892251	−	0.451540i	$-0.850875\pi$
$389$	−35.1959	−1.78450	−0.892251	+	0.451540i	$0.850875\pi$
$390$	0	0
$391$	52.2823	2.64403
$392$	3.59390	0.181519
$393$	0	0
$394$	0.576137	0.0290254
$395$	−4.00000	−0.201262
$396$	0	0
$397$	−3.56054	−0.178698	−0.0893492	−	0.996000i	$-0.528479\pi$
$397$	−3.56054	−0.178698	−0.0893492	+	0.996000i	$0.528479\pi$
$398$	−6.64359	−0.333013
$399$	0	0
$400$	1.00000	0.0500000
$401$	−1.44971	−0.0723950	−0.0361975	−	0.999345i	$-0.511525\pi$
$401$	−1.44971	−0.0723950	−0.0361975	+	0.999345i	$0.511525\pi$
$402$	0	0
$403$	−15.2312	−0.758718
$404$	3.63067	0.180633
$405$	0	0
$406$	5.60949	0.278394
$407$	0.832640	0.0412724
$408$	0	0
$409$	25.1781	1.24498	0.622489	−	0.782629i	$-0.286122\pi$
$409$	25.1781	1.24498	0.622489	+	0.782629i	$0.286122\pi$
$410$	−2.73057	−0.134853
$411$	0	0
$412$	−9.11766	−0.449195
$413$	6.81221	0.335207
$414$	0	0
$415$	−3.55445	−0.174481
$416$	−3.21973	−0.157860
$417$	0	0
$418$	3.50276	0.171325
$419$	−31.6938	−1.54834	−0.774172	−	0.632976i	$-0.781833\pi$
$419$	−31.6938	−1.54834	−0.774172	+	0.632976i	$0.781833\pi$
$420$	0	0
$421$	−5.14351	−0.250679	−0.125340	−	0.992114i	$-0.540002\pi$
$421$	−5.14351	−0.250679	−0.125340	+	0.992114i	$0.540002\pi$
$422$	8.83531	0.430096
$423$	0	0
$424$	3.95030	0.191844
$425$	7.95030	0.385646
$426$	0	0
$427$	9.98224	0.483075
$428$	−8.91086	−0.430723
$429$	0	0
$430$	−5.03945	−0.243024
$431$	−2.72448	−0.131234	−0.0656168	−	0.997845i	$-0.520902\pi$
$431$	−2.72448	−0.131234	−0.0656168	+	0.997845i	$0.520902\pi$
$432$	0	0
$433$	−8.68088	−0.417176	−0.208588	−	0.978004i	$-0.566887\pi$
$433$	−8.68088	−0.417176	−0.208588	+	0.978004i	$0.566887\pi$
$434$	8.73057	0.419081
$435$	0	0
$436$	11.2979	0.541073
$437$	27.6645	1.32337
$438$	0	0
$439$	7.64752	0.364996	0.182498	−	0.983206i	$-0.441582\pi$
$439$	7.64752	0.364996	0.182498	+	0.983206i	$0.441582\pi$
$440$	−0.832640	−0.0396946
$441$	0	0
$442$	−25.5978	−1.21756
$443$	−16.7917	−0.797798	−0.398899	−	0.916995i	$-0.630608\pi$
$443$	−16.7917	−0.797798	−0.398899	+	0.916995i	$0.630608\pi$
$444$	0	0
$445$	−6.55029	−0.310514
$446$	−17.8828	−0.846777
$447$	0	0
$448$	1.84556	0.0871947
$449$	−26.1047	−1.23196	−0.615979	−	0.787762i	$-0.711240\pi$
$449$	−26.1047	−1.23196	−0.615979	+	0.787762i	$0.711240\pi$
$450$	0	0
$451$	2.27359	0.107059
$452$	11.0394	0.519252
$453$	0	0
$454$	25.1959	1.18250
$455$	−5.94222	−0.278575
$456$	0	0
$457$	−17.6646	−0.826315	−0.413158	−	0.910660i	$-0.635574\pi$
$457$	−17.6646	−0.826315	−0.413158	+	0.910660i	$0.635574\pi$
$458$	5.48699	0.256390
$459$	0	0
$460$	−6.57614	−0.306614
$461$	21.2231	0.988457	0.494229	−	0.869332i	$-0.335451\pi$
$461$	21.2231	0.988457	0.494229	+	0.869332i	$0.335451\pi$
$462$	0	0
$463$	32.5530	1.51286	0.756432	−	0.654072i	$-0.226941\pi$
$463$	32.5530	1.51286	0.756432	+	0.654072i	$0.226941\pi$
$464$	−3.03945	−0.141103
$465$	0	0
$466$	−28.5788	−1.32389
$467$	−15.1170	−0.699531	−0.349765	−	0.936837i	$-0.613739\pi$
$467$	−15.1170	−0.699531	−0.349765	+	0.936837i	$0.613739\pi$
$468$	0	0
$469$	21.7700	1.00525
$470$	−9.11766	−0.420566
$471$	0	0
$472$	−3.69113	−0.169898
$473$	4.19605	0.192934
$474$	0	0
$475$	4.20681	0.193022
$476$	14.6728	0.672526
$477$	0	0
$478$	1.77811	0.0813288
$479$	−3.44971	−0.157621	−0.0788106	−	0.996890i	$-0.525112\pi$
$479$	−3.44971	−0.157621	−0.0788106	+	0.996890i	$0.525112\pi$
$480$	0	0
$481$	−3.21973	−0.146807
$482$	−22.2612	−1.01397
$483$	0	0
$484$	−10.3067	−0.468487
$485$	10.4918	0.476409
$486$	0	0
$487$	−15.6619	−0.709707	−0.354853	−	0.934922i	$-0.615469\pi$
$487$	−15.6619	−0.709707	−0.354853	+	0.934922i	$0.615469\pi$
$488$	−5.40878	−0.244844
$489$	0	0
$490$	−3.59390	−0.162356
$491$	−7.89793	−0.356429	−0.178214	−	0.983992i	$-0.557032\pi$
$491$	−7.89793	−0.356429	−0.178214	+	0.983992i	$0.557032\pi$
$492$	0	0
$493$	−24.1645	−1.08832
$494$	−13.5448	−0.609408
$495$	0	0
$496$	−4.73057	−0.212409
$497$	25.4134	1.13995
$498$	0	0
$499$	5.87209	0.262871	0.131435	−	0.991325i	$-0.458041\pi$
$499$	5.87209	0.262871	0.131435	+	0.991325i	$0.458041\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	−28.6748	−1.27982
$503$	−3.33056	−0.148502	−0.0742512	−	0.997240i	$-0.523657\pi$
$503$	−3.33056	−0.148502	−0.0742512	+	0.997240i	$0.523657\pi$
$504$	0	0
$505$	−3.63067	−0.161563
$506$	5.47556	0.243418
$507$	0	0
$508$	−0.136678	−0.00606410
$509$	30.9508	1.37187	0.685935	−	0.727663i	$-0.259393\pi$
$509$	30.9508	1.37187	0.685935	+	0.727663i	$0.259393\pi$
$510$	0	0
$511$	−6.51226	−0.288085
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	9.19388	0.405525
$515$	9.11766	0.401772
$516$	0	0
$517$	7.59173	0.333884
$518$	1.84556	0.0810894
$519$	0	0
$520$	3.21973	0.141194
$521$	−7.45306	−0.326524	−0.163262	−	0.986583i	$-0.552202\pi$
$521$	−7.45306	−0.326524	−0.163262	+	0.986583i	$0.552202\pi$
$522$	0	0
$523$	31.5917	1.38141	0.690705	−	0.723137i	$-0.257300\pi$
$523$	31.5917	1.38141	0.690705	+	0.723137i	$0.257300\pi$
$524$	−2.07889	−0.0908169
$525$	0	0
$526$	−8.02652	−0.349973
$527$	−37.6095	−1.63830
$528$	0	0
$529$	20.2456	0.880242
$530$	−3.95030	−0.171590
$531$	0	0
$532$	7.76393	0.336609
$533$	−8.79171	−0.380811
$534$	0	0
$535$	8.91086	0.385250
$536$	−11.7959	−0.509504
$537$	0	0
$538$	21.3849	0.921970
$539$	2.99242	0.128893
$540$	0	0
$541$	36.0816	1.55127	0.775634	−	0.631183i	$-0.217430\pi$
$541$	36.0816	1.55127	0.775634	+	0.631183i	$0.217430\pi$
$542$	−2.64359	−0.113552
$543$	0	0
$544$	−7.95030	−0.340866
$545$	−11.2979	−0.483951
$546$	0	0
$547$	−38.7420	−1.65649	−0.828244	−	0.560367i	$-0.810660\pi$
$547$	−38.7420	−1.65649	−0.828244	+	0.560367i	$0.810660\pi$
$548$	−3.06597	−0.130972
$549$	0	0
$550$	0.832640	0.0355039
$551$	−12.7864	−0.544717
$552$	0	0
$553$	7.38225	0.313925
$554$	14.4456	0.613733
$555$	0	0
$556$	22.1917	0.941139
$557$	23.3618	0.989869	0.494935	−	0.868930i	$-0.335192\pi$
$557$	23.3618	0.989869	0.494935	+	0.868930i	$0.335192\pi$
$558$	0	0
$559$	−16.2257	−0.686272
$560$	−1.84556	−0.0779893
$561$	0	0
$562$	−11.2197	−0.473276
$563$	−39.0351	−1.64513	−0.822567	−	0.568668i	$-0.807459\pi$
$563$	−39.0351	−1.64513	−0.822567	+	0.568668i	$0.807459\pi$
$564$	0	0
$565$	−11.0394	−0.464433
$566$	−3.86332	−0.162388
$567$	0	0
$568$	−13.7700	−0.577777
$569$	−35.4809	−1.48744	−0.743718	−	0.668493i	$-0.766940\pi$
$569$	−35.4809	−1.48744	−0.743718	+	0.668493i	$0.766940\pi$
$570$	0	0
$571$	37.0093	1.54879	0.774395	−	0.632702i	$-0.218054\pi$
$571$	37.0093	1.54879	0.774395	+	0.632702i	$0.218054\pi$
$572$	−2.68088	−0.112093
$573$	0	0
$574$	5.03945	0.210342
$575$	6.57614	0.274244
$576$	0	0
$577$	−15.7354	−0.655074	−0.327537	−	0.944838i	$-0.606219\pi$
$577$	−15.7354	−0.655074	−0.327537	+	0.944838i	$0.606219\pi$
$578$	−46.2073	−1.92197
$579$	0	0
$580$	3.03945	0.126206
$581$	6.55996	0.272153
$582$	0	0
$583$	3.28918	0.136224
$584$	3.52860	0.146015
$585$	0	0
$586$	−16.1544	−0.667334
$587$	−37.3006	−1.53956	−0.769781	−	0.638309i	$-0.779634\pi$
$587$	−37.3006	−1.53956	−0.769781	+	0.638309i	$0.779634\pi$
$588$	0	0
$589$	−19.9006	−0.819990
$590$	3.69113	0.151961
$591$	0	0
$592$	−1.00000	−0.0410997
$593$	17.9829	0.738470	0.369235	−	0.929336i	$-0.379620\pi$
$593$	17.9829	0.738470	0.369235	+	0.929336i	$0.379620\pi$
$594$	0	0
$595$	−14.6728	−0.601526
$596$	14.4999	0.593940
$597$	0	0
$598$	−21.1734	−0.865844
$599$	32.9998	1.34834	0.674168	−	0.738578i	$-0.264502\pi$
$599$	32.9998	1.34834	0.674168	+	0.738578i	$0.264502\pi$
$600$	0	0
$601$	33.0334	1.34746	0.673729	−	0.738978i	$-0.264691\pi$
$601$	33.0334	1.34746	0.673729	+	0.738978i	$0.264691\pi$
$602$	9.30062	0.379065
$603$	0	0
$604$	8.24142	0.335339
$605$	10.3067	0.419027
$606$	0	0
$607$	27.6102	1.12066	0.560331	−	0.828269i	$-0.310674\pi$
$607$	27.6102	1.12066	0.560331	+	0.828269i	$0.310674\pi$
$608$	−4.20681	−0.170609
$609$	0	0
$610$	5.40878	0.218995
$611$	−29.3564	−1.18763
$612$	0	0
$613$	1.62999	0.0658348	0.0329174	−	0.999458i	$-0.489520\pi$
$613$	1.62999	0.0658348	0.0329174	+	0.999458i	$0.489520\pi$
$614$	−6.28303	−0.253562
$615$	0	0
$616$	1.53669	0.0619150
$617$	−28.6143	−1.15197	−0.575985	−	0.817460i	$-0.695381\pi$
$617$	−28.6143	−1.15197	−0.575985	+	0.817460i	$0.695381\pi$
$618$	0	0
$619$	19.6830	0.791128	0.395564	−	0.918438i	$-0.370549\pi$
$619$	19.6830	0.791128	0.395564	+	0.918438i	$0.370549\pi$
$620$	4.73057	0.189984
$621$	0	0
$622$	20.6312	0.827235
$623$	12.0890	0.484335
$624$	0	0
$625$	1.00000	0.0400000
$626$	7.32180	0.292638
$627$	0	0
$628$	−6.11283	−0.243928
$629$	−7.95030	−0.316999
$630$	0	0
$631$	−5.76193	−0.229379	−0.114689	−	0.993401i	$-0.536587\pi$
$631$	−5.76193	−0.229379	−0.114689	+	0.993401i	$0.536587\pi$
$632$	−4.00000	−0.159111
$633$	0	0
$634$	19.1442	0.760313
$635$	0.136678	0.00542390
$636$	0	0
$637$	−11.5714	−0.458474
$638$	−2.53077	−0.100194
$639$	0	0
$640$	1.00000	0.0395285
$641$	−20.8967	−0.825369	−0.412685	−	0.910874i	$-0.635409\pi$
$641$	−20.8967	−0.825369	−0.412685	+	0.910874i	$0.635409\pi$
$642$	0	0
$643$	30.6197	1.20752	0.603762	−	0.797164i	$-0.293668\pi$
$643$	30.6197	1.20752	0.603762	+	0.797164i	$0.293668\pi$
$644$	12.1367	0.478252
$645$	0	0
$646$	−33.4454	−1.31589
$647$	10.6278	0.417823	0.208912	−	0.977935i	$-0.433008\pi$
$647$	10.6278	0.417823	0.208912	+	0.977935i	$0.433008\pi$
$648$	0	0
$649$	−3.07338	−0.120641
$650$	−3.21973	−0.126288
$651$	0	0
$652$	−21.4373	−0.839549
$653$	5.90061	0.230909	0.115454	−	0.993313i	$-0.463168\pi$
$653$	5.90061	0.230909	0.115454	+	0.993313i	$0.463168\pi$
$654$	0	0
$655$	2.07889	0.0812291
$656$	−2.73057	−0.106611
$657$	0	0
$658$	16.8272	0.655994
$659$	−19.8305	−0.772486	−0.386243	−	0.922397i	$-0.626227\pi$
$659$	−19.8305	−0.772486	−0.386243	+	0.922397i	$0.626227\pi$
$660$	0	0
$661$	−9.32379	−0.362653	−0.181327	−	0.983423i	$-0.558039\pi$
$661$	−9.32379	−0.362653	−0.181327	+	0.983423i	$0.558039\pi$
$662$	−0.908184	−0.0352976
$663$	0	0
$664$	−3.55445	−0.137939
$665$	−7.76393	−0.301072
$666$	0	0
$667$	−19.9878	−0.773931
$668$	22.5815	0.873704
$669$	0	0
$670$	11.7959	0.455714
$671$	−4.50357	−0.173858
$672$	0	0
$673$	33.0113	1.27249	0.636245	−	0.771487i	$-0.280487\pi$
$673$	33.0113	1.27249	0.636245	+	0.771487i	$0.280487\pi$
$674$	8.11083	0.312418
$675$	0	0
$676$	−2.63334	−0.101282
$677$	−26.1245	−1.00405	−0.502023	−	0.864854i	$-0.667411\pi$
$677$	−26.1245	−1.00405	−0.502023	+	0.864854i	$0.667411\pi$
$678$	0	0
$679$	−19.3633	−0.743097
$680$	7.95030	0.304880
$681$	0	0
$682$	−3.93887	−0.150827
$683$	39.3957	1.50743	0.753717	−	0.657199i	$-0.228259\pi$
$683$	39.3957	1.50743	0.753717	+	0.657199i	$0.228259\pi$
$684$	0	0
$685$	3.06597	0.117145
$686$	19.5517	0.746488
$687$	0	0
$688$	−5.03945	−0.192127
$689$	−12.7189	−0.484552
$690$	0	0
$691$	−9.45306	−0.359611	−0.179806	−	0.983702i	$-0.557547\pi$
$691$	−9.45306	−0.359611	−0.179806	+	0.983702i	$0.557547\pi$
$692$	6.48916	0.246681
$693$	0	0
$694$	9.35641	0.355164
$695$	−22.1917	−0.841780
$696$	0	0
$697$	−21.7089	−0.822283
$698$	35.4665	1.34243
$699$	0	0
$700$	1.84556	0.0697557
$701$	−37.2842	−1.40821	−0.704103	−	0.710098i	$-0.748651\pi$
$701$	−37.2842	−1.40821	−0.704103	+	0.710098i	$0.748651\pi$
$702$	0	0
$703$	−4.20681	−0.158663
$704$	−0.832640	−0.0313813
$705$	0	0
$706$	−36.0923	−1.35835
$707$	6.70063	0.252003
$708$	0	0
$709$	−19.8110	−0.744016	−0.372008	−	0.928230i	$-0.621331\pi$
$709$	−19.8110	−0.744016	−0.372008	+	0.928230i	$0.621331\pi$
$710$	13.7700	0.516780
$711$	0	0
$712$	−6.55029	−0.245483
$713$	−31.1089	−1.16504
$714$	0	0
$715$	2.68088	0.100259
$716$	−16.2095	−0.605777
$717$	0	0
$718$	2.07889	0.0775836
$719$	19.2136	0.716548	0.358274	−	0.933617i	$-0.383365\pi$
$719$	19.2136	0.716548	0.358274	+	0.933617i	$0.383365\pi$
$720$	0	0
$721$	−16.8272	−0.626679
$722$	1.30278	0.0484845
$723$	0	0
$724$	5.38225	0.200030
$725$	−3.03945	−0.112882
$726$	0	0
$727$	−45.0183	−1.66964	−0.834818	−	0.550527i	$-0.814427\pi$
$727$	−45.0183	−1.66964	−0.834818	+	0.550527i	$0.814427\pi$
$728$	−5.94222	−0.220233
$729$	0	0
$730$	−3.52860	−0.130599
$731$	−40.0651	−1.48186
$732$	0	0
$733$	−20.7262	−0.765541	−0.382771	−	0.923843i	$-0.625030\pi$
$733$	−20.7262	−0.765541	−0.382771	+	0.923843i	$0.625030\pi$
$734$	−34.2645	−1.26473
$735$	0	0
$736$	−6.57614	−0.242400
$737$	−9.82171	−0.361787
$738$	0	0
$739$	24.2176	0.890858	0.445429	−	0.895317i	$-0.353051\pi$
$739$	24.2176	0.890858	0.445429	+	0.895317i	$0.353051\pi$
$740$	1.00000	0.0367607
$741$	0	0
$742$	7.29053	0.267644
$743$	−13.9530	−0.511885	−0.255943	−	0.966692i	$-0.582386\pi$
$743$	−13.9530	−0.511885	−0.255943	+	0.966692i	$0.582386\pi$
$744$	0	0
$745$	−14.4999	−0.531236
$746$	6.00000	0.219676
$747$	0	0
$748$	−6.61974	−0.242042
$749$	−16.4456	−0.600907
$750$	0	0
$751$	41.6176	1.51865	0.759324	−	0.650713i	$-0.225530\pi$
$751$	41.6176	1.51865	0.759324	+	0.650713i	$0.225530\pi$
$752$	−9.11766	−0.332487
$753$	0	0
$754$	9.78620	0.356392
$755$	−8.24142	−0.299936
$756$	0	0
$757$	−45.4034	−1.65021	−0.825107	−	0.564977i	$-0.808885\pi$
$757$	−45.4034	−1.65021	−0.825107	+	0.564977i	$0.808885\pi$
$758$	−29.8217	−1.08317
$759$	0	0
$760$	4.20681	0.152597
$761$	−18.9347	−0.686383	−0.343191	−	0.939266i	$-0.611508\pi$
$761$	−18.9347	−0.686383	−0.343191	+	0.939266i	$0.611508\pi$
$762$	0	0
$763$	20.8511	0.754860
$764$	10.5680	0.382339
$765$	0	0
$766$	−6.78027	−0.244981
$767$	11.8844	0.429122
$768$	0	0
$769$	31.0884	1.12108	0.560538	−	0.828129i	$-0.310594\pi$
$769$	31.0884	1.12108	0.560538	+	0.828129i	$0.310594\pi$
$770$	−1.53669	−0.0553785
$771$	0	0
$772$	3.11766	0.112207
$773$	−34.3542	−1.23564	−0.617818	−	0.786321i	$-0.711983\pi$
$773$	−34.3542	−1.23564	−0.617818	+	0.786321i	$0.711983\pi$
$774$	0	0
$775$	−4.73057	−0.169927
$776$	10.4918	0.376635
$777$	0	0
$778$	35.1959	1.26183
$779$	−11.4870	−0.411564
$780$	0	0
$781$	−11.4655	−0.410267
$782$	−52.2823	−1.86961
$783$	0	0
$784$	−3.59390	−0.128353
$785$	6.11283	0.218176
$786$	0	0
$787$	36.0053	1.28345	0.641726	−	0.766934i	$-0.278219\pi$
$787$	36.0053	1.28345	0.641726	+	0.766934i	$0.278219\pi$
$788$	−0.576137	−0.0205240
$789$	0	0
$790$	4.00000	0.142314
$791$	20.3740	0.724416
$792$	0	0
$793$	17.4148	0.618418
$794$	3.56054	0.126359
$795$	0	0
$796$	6.64359	0.235476
$797$	−43.2107	−1.53060	−0.765300	−	0.643674i	$-0.777409\pi$
$797$	−43.2107	−1.53060	−0.765300	+	0.643674i	$0.777409\pi$
$798$	0	0
$799$	−72.4882	−2.56445
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	1.44971	0.0511910
$803$	2.93806	0.103682
$804$	0	0
$805$	−12.1367	−0.427762
$806$	15.2312	0.536495
$807$	0	0
$808$	−3.63067	−0.127727
$809$	32.2209	1.13283	0.566414	−	0.824121i	$-0.308331\pi$
$809$	32.2209	1.13283	0.566414	+	0.824121i	$0.308331\pi$
$810$	0	0
$811$	8.87892	0.311781	0.155890	−	0.987774i	$-0.450175\pi$
$811$	8.87892	0.311781	0.155890	+	0.987774i	$0.450175\pi$
$812$	−5.60949	−0.196855
$813$	0	0
$814$	−0.832640	−0.0291840
$815$	21.4373	0.750916
$816$	0	0
$817$	−21.2000	−0.741693
$818$	−25.1781	−0.880332
$819$	0	0
$820$	2.73057	0.0953558
$821$	−2.31155	−0.0806735	−0.0403368	−	0.999186i	$-0.512843\pi$
$821$	−2.31155	−0.0806735	−0.0403368	+	0.999186i	$0.512843\pi$
$822$	0	0
$823$	13.6280	0.475042	0.237521	−	0.971382i	$-0.423665\pi$
$823$	13.6280	0.475042	0.237521	+	0.971382i	$0.423665\pi$
$824$	9.11766	0.317629
$825$	0	0
$826$	−6.81221	−0.237027
$827$	17.8298	0.620003	0.310001	−	0.950736i	$-0.399670\pi$
$827$	17.8298	0.620003	0.310001	+	0.950736i	$0.399670\pi$
$828$	0	0
$829$	−29.7388	−1.03287	−0.516435	−	0.856326i	$-0.672741\pi$
$829$	−29.7388	−1.03287	−0.516435	+	0.856326i	$0.672741\pi$
$830$	3.55445	0.123377
$831$	0	0
$832$	3.21973	0.111624
$833$	−28.5726	−0.989980
$834$	0	0
$835$	−22.5815	−0.781464
$836$	−3.50276	−0.121145
$837$	0	0
$838$	31.6938	1.09484
$839$	10.4556	0.360969	0.180484	−	0.983578i	$-0.442234\pi$
$839$	10.4556	0.360969	0.180484	+	0.983578i	$0.442234\pi$
$840$	0	0
$841$	−19.7618	−0.681440
$842$	5.14351	0.177257
$843$	0	0
$844$	−8.83531	−0.304124
$845$	2.63334	0.0905897
$846$	0	0
$847$	−19.0217	−0.653593
$848$	−3.95030	−0.135654
$849$	0	0
$850$	−7.95030	−0.272693
$851$	−6.57614	−0.225427
$852$	0	0
$853$	−15.6333	−0.535275	−0.267638	−	0.963520i	$-0.586243\pi$
$853$	−15.6333	−0.535275	−0.267638	+	0.963520i	$0.586243\pi$
$854$	−9.98224	−0.341585
$855$	0	0
$856$	8.91086	0.304567
$857$	14.4945	0.495123	0.247561	−	0.968872i	$-0.420371\pi$
$857$	14.4945	0.495123	0.247561	+	0.968872i	$0.420371\pi$
$858$	0	0
$859$	3.38493	0.115492	0.0577461	−	0.998331i	$-0.481609\pi$
$859$	3.38493	0.115492	0.0577461	+	0.998331i	$0.481609\pi$
$860$	5.03945	0.171844
$861$	0	0
$862$	2.72448	0.0927962
$863$	56.3924	1.91962	0.959810	−	0.280649i	$-0.0905498\pi$
$863$	56.3924	1.91962	0.959810	+	0.280649i	$0.0905498\pi$
$864$	0	0
$865$	−6.48916	−0.220638
$866$	8.68088	0.294988
$867$	0	0
$868$	−8.73057	−0.296335
$869$	−3.33056	−0.112982
$870$	0	0
$871$	37.9795	1.28689
$872$	−11.2979	−0.382597
$873$	0	0
$874$	−27.6645	−0.935767
$875$	−1.84556	−0.0623914
$876$	0	0
$877$	−32.1755	−1.08649	−0.543245	−	0.839574i	$-0.682805\pi$
$877$	−32.1755	−1.08649	−0.543245	+	0.839574i	$0.682805\pi$
$878$	−7.64752	−0.258091
$879$	0	0
$880$	0.832640	0.0280683
$881$	−14.6097	−0.492212	−0.246106	−	0.969243i	$-0.579151\pi$
$881$	−14.6097	−0.492212	−0.246106	+	0.969243i	$0.579151\pi$
$882$	0	0
$883$	18.5203	0.623259	0.311630	−	0.950204i	$-0.399125\pi$
$883$	18.5203	0.623259	0.311630	+	0.950204i	$0.399125\pi$
$884$	25.5978	0.860948
$885$	0	0
$886$	16.7917	0.564128
$887$	28.5611	0.958986	0.479493	−	0.877546i	$-0.340821\pi$
$887$	28.5611	0.958986	0.479493	+	0.877546i	$0.340821\pi$
$888$	0	0
$889$	−0.252248	−0.00846012
$890$	6.55029	0.219566
$891$	0	0
$892$	17.8828	0.598762
$893$	−38.3562	−1.28354
$894$	0	0
$895$	16.2095	0.541823
$896$	−1.84556	−0.0616559
$897$	0	0
$898$	26.1047	0.871126
$899$	14.3783	0.479544
$900$	0	0
$901$	−31.4061	−1.04629
$902$	−2.27359	−0.0757021
$903$	0	0
$904$	−11.0394	−0.367167
$905$	−5.38225	−0.178912
$906$	0	0
$907$	−25.8428	−0.858097	−0.429048	−	0.903282i	$-0.641151\pi$
$907$	−25.8428	−0.858097	−0.429048	+	0.903282i	$0.641151\pi$
$908$	−25.1959	−0.836155
$909$	0	0
$910$	5.94222	0.196983
$911$	−24.9998	−0.828281	−0.414141	−	0.910213i	$-0.635918\pi$
$911$	−24.9998	−0.828281	−0.414141	+	0.910213i	$0.635918\pi$
$912$	0	0
$913$	−2.95958	−0.0979477
$914$	17.6646	0.584293
$915$	0	0
$916$	−5.48699	−0.181295
$917$	−3.83673	−0.126700
$918$	0	0
$919$	−34.3931	−1.13452	−0.567262	−	0.823537i	$-0.691997\pi$
$919$	−34.3931	−1.13452	−0.567262	+	0.823537i	$0.691997\pi$
$920$	6.57614	0.216809
$921$	0	0
$922$	−21.2231	−0.698945
$923$	44.3357	1.45933
$924$	0	0
$925$	−1.00000	−0.0328798
$926$	−32.5530	−1.06976
$927$	0	0
$928$	3.03945	0.0997747
$929$	−55.6310	−1.82519	−0.912597	−	0.408860i	$-0.865926\pi$
$929$	−55.6310	−1.82519	−0.912597	+	0.408860i	$0.865926\pi$
$930$	0	0
$931$	−15.1188	−0.495499
$932$	28.5788	0.936130
$933$	0	0
$934$	15.1170	0.494643
$935$	6.61974	0.216489
$936$	0	0
$937$	−53.9576	−1.76272	−0.881360	−	0.472446i	$-0.843371\pi$
$937$	−53.9576	−1.76272	−0.881360	+	0.472446i	$0.843371\pi$
$938$	−21.7700	−0.710816
$939$	0	0
$940$	9.11766	0.297385
$941$	50.1092	1.63351	0.816756	−	0.576983i	$-0.195770\pi$
$941$	50.1092	1.63351	0.816756	+	0.576983i	$0.195770\pi$
$942$	0	0
$943$	−17.9566	−0.584748
$944$	3.69113	0.120136
$945$	0	0
$946$	−4.19605	−0.136425
$947$	−39.3795	−1.27966	−0.639831	−	0.768516i	$-0.720996\pi$
$947$	−39.3795	−1.27966	−0.639831	+	0.768516i	$0.720996\pi$
$948$	0	0
$949$	−11.3611	−0.368798
$950$	−4.20681	−0.136487
$951$	0	0
$952$	−14.6728	−0.475548
$953$	20.5096	0.664371	0.332185	−	0.943214i	$-0.392214\pi$
$953$	20.5096	0.664371	0.332185	+	0.943214i	$0.392214\pi$
$954$	0	0
$955$	−10.5680	−0.341974
$956$	−1.77811	−0.0575081
$957$	0	0
$958$	3.44971	0.111455
$959$	−5.65844	−0.182721
$960$	0	0
$961$	−8.62168	−0.278119
$962$	3.21973	0.103808
$963$	0	0
$964$	22.2612	0.716984
$965$	−3.11766	−0.100361
$966$	0	0
$967$	18.6265	0.598988	0.299494	−	0.954098i	$-0.403182\pi$
$967$	18.6265	0.598988	0.299494	+	0.954098i	$0.403182\pi$
$968$	10.3067	0.331270
$969$	0	0
$970$	−10.4918	−0.336872
$971$	3.56656	0.114456	0.0572282	−	0.998361i	$-0.481774\pi$
$971$	3.56656	0.114456	0.0572282	+	0.998361i	$0.481774\pi$
$972$	0	0
$973$	40.9562	1.31300
$974$	15.6619	0.501838
$975$	0	0
$976$	5.40878	0.173131
$977$	43.6998	1.39808	0.699041	−	0.715082i	$-0.253611\pi$
$977$	43.6998	1.39808	0.699041	+	0.715082i	$0.253611\pi$
$978$	0	0
$979$	−5.45404	−0.174312
$980$	3.59390	0.114803
$981$	0	0
$982$	7.89793	0.252033
$983$	−53.1096	−1.69393	−0.846966	−	0.531647i	$-0.821573\pi$
$983$	−53.1096	−1.69393	−0.846966	+	0.531647i	$0.821573\pi$
$984$	0	0
$985$	0.576137	0.0183572
$986$	24.1645	0.769555
$987$	0	0
$988$	13.5448	0.430917
$989$	−33.1401	−1.05379
$990$	0	0
$991$	−9.14283	−0.290432	−0.145216	−	0.989400i	$-0.546388\pi$
$991$	−9.14283	−0.290432	−0.145216	+	0.989400i	$0.546388\pi$
$992$	4.73057	0.150196
$993$	0	0
$994$	−25.4134	−0.806066
$995$	−6.64359	−0.210616
$996$	0	0
$997$	−15.6280	−0.494944	−0.247472	−	0.968895i	$-0.579600\pi$
$997$	−15.6280	−0.494944	−0.247472	+	0.968895i	$0.579600\pi$
$998$	−5.87209	−0.185878
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3330.2.a.bj.1.3		4
3.2	odd	2		1110.2.a.s.1.3	✓	4
12.11	even	2		8880.2.a.cg.1.2		4
15.14	odd	2		5550.2.a.cj.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1110.2.a.s.1.3	✓	4	3.2	odd	2
3330.2.a.bj.1.3		4	1.1	even	1	trivial
5550.2.a.cj.1.2		4	15.14	odd	2
8880.2.a.cg.1.2		4	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:	\( N \)	\(=\)	\( 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3330.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.84556 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.84556 q^{7} -1.00000 q^{8} +1.00000 q^{10} -0.832640 q^{11} +3.21973 q^{13} -1.84556 q^{14} +1.00000 q^{16} +7.95030 q^{17} +4.20681 q^{19} -1.00000 q^{20} +0.832640 q^{22} +6.57614 q^{23} +1.00000 q^{25} -3.21973 q^{26} +1.84556 q^{28} -3.03945 q^{29} -4.73057 q^{31} -1.00000 q^{32} -7.95030 q^{34} -1.84556 q^{35} -1.00000 q^{37} -4.20681 q^{38} +1.00000 q^{40} -2.73057 q^{41} -5.03945 q^{43} -0.832640 q^{44} -6.57614 q^{46} -9.11766 q^{47} -3.59390 q^{49} -1.00000 q^{50} +3.21973 q^{52} -3.95030 q^{53} +0.832640 q^{55} -1.84556 q^{56} +3.03945 q^{58} +3.69113 q^{59} +5.40878 q^{61} +4.73057 q^{62} +1.00000 q^{64} -3.21973 q^{65} +11.7959 q^{67} +7.95030 q^{68} +1.84556 q^{70} +13.7700 q^{71} -3.52860 q^{73} +1.00000 q^{74} +4.20681 q^{76} -1.53669 q^{77} +4.00000 q^{79} -1.00000 q^{80} +2.73057 q^{82} +3.55445 q^{83} -7.95030 q^{85} +5.03945 q^{86} +0.832640 q^{88} +6.55029 q^{89} +5.94222 q^{91} +6.57614 q^{92} +9.11766 q^{94} -4.20681 q^{95} -10.4918 q^{97} +3.59390 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{5} + 4 q^{7} - 4 q^{8}+O(q^{10}) \) 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^5 + 4 * q^7 - 4 * q^8	\( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{5} + 4 q^{7} - 4 q^{8} + 4 q^{10} - 2 q^{11} - 3 q^{13} - 4 q^{14} + 4 q^{16} - 6 q^{17} + 3 q^{19} - 4 q^{20} + 2 q^{22} + q^{23} + 4 q^{25} + 3 q^{26} + 4 q^{28} + 3 q^{29} + 3 q^{31} - 4 q^{32} + 6 q^{34} - 4 q^{35} - 4 q^{37} - 3 q^{38} + 4 q^{40} + 11 q^{41} - 5 q^{43} - 2 q^{44} - q^{46} + 14 q^{49} - 4 q^{50} - 3 q^{52} + 22 q^{53} + 2 q^{55} - 4 q^{56} - 3 q^{58} + 8 q^{59} - 5 q^{61} - 3 q^{62} + 4 q^{64} + 3 q^{65} + 6 q^{67} - 6 q^{68} + 4 q^{70} + 18 q^{71} - 5 q^{73} + 4 q^{74} + 3 q^{76} + 4 q^{77} + 16 q^{79} - 4 q^{80} - 11 q^{82} + q^{83} + 6 q^{85} + 5 q^{86} + 2 q^{88} + 5 q^{89} - 13 q^{91} + q^{92} - 3 q^{95} + 7 q^{97} - 14 q^{98}+O(q^{100}) \) 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^5 + 4 * q^7 - 4 * q^8 + 4 * q^10 - 2 * q^11 - 3 * q^13 - 4 * q^14 + 4 * q^16 - 6 * q^17 + 3 * q^19 - 4 * q^20 + 2 * q^22 + q^23 + 4 * q^25 + 3 * q^26 + 4 * q^28 + 3 * q^29 + 3 * q^31 - 4 * q^32 + 6 * q^34 - 4 * q^35 - 4 * q^37 - 3 * q^38 + 4 * q^40 + 11 * q^41 - 5 * q^43 - 2 * q^44 - q^46 + 14 * q^49 - 4 * q^50 - 3 * q^52 + 22 * q^53 + 2 * q^55 - 4 * q^56 - 3 * q^58 + 8 * q^59 - 5 * q^61 - 3 * q^62 + 4 * q^64 + 3 * q^65 + 6 * q^67 - 6 * q^68 + 4 * q^70 + 18 * q^71 - 5 * q^73 + 4 * q^74 + 3 * q^76 + 4 * q^77 + 16 * q^79 - 4 * q^80 - 11 * q^82 + q^83 + 6 * q^85 + 5 * q^86 + 2 * q^88 + 5 * q^89 - 13 * q^91 + q^92 - 3 * q^95 + 7 * q^97 - 14 * q^98

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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