LMFDB - Embedded newform 3330.2.a.bg.1.2

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:	$1$
Dimension:	$3$
Coefficient field:	3.3.892.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 8x + 10 $ x^3 - x^2 - 8*x + 10
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 370)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.31955$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.31955 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.31955 q^{7} -1.00000 q^{8} -1.00000 q^{10} +0.258786 q^{11} -5.87847 q^{13} +1.31955 q^{14} +1.00000 q^{16} -4.25879 q^{17} +2.93923 q^{19} +1.00000 q^{20} -0.258786 q^{22} +8.51757 q^{23} +1.00000 q^{25} +5.87847 q^{26} -1.31955 q^{28} +3.61968 q^{29} +3.95865 q^{31} -1.00000 q^{32} +4.25879 q^{34} -1.31955 q^{35} -1.00000 q^{37} -2.93923 q^{38} -1.00000 q^{40} +8.89789 q^{41} -5.61968 q^{43} +0.258786 q^{44} -8.51757 q^{46} -5.57834 q^{47} -5.25879 q^{49} -1.00000 q^{50} -5.87847 q^{52} -12.8979 q^{53} +0.258786 q^{55} +1.31955 q^{56} -3.61968 q^{58} -9.57834 q^{59} +0.380316 q^{61} -3.95865 q^{62} +1.00000 q^{64} -5.87847 q^{65} +5.57834 q^{67} -4.25879 q^{68} +1.31955 q^{70} -11.2394 q^{71} +2.51757 q^{73} +1.00000 q^{74} +2.93923 q^{76} -0.341481 q^{77} -7.69987 q^{79} +1.00000 q^{80} -8.89789 q^{82} +6.17860 q^{83} -4.25879 q^{85} +5.61968 q^{86} -0.258786 q^{88} -6.00000 q^{89} +7.75694 q^{91} +8.51757 q^{92} +5.57834 q^{94} +2.93923 q^{95} -12.8979 q^{97} +5.25879 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} - q^{7} - 3 q^{8}+O(q^{10}) $ 3 * q - 3 * q^2 + 3 * q^4 + 3 * q^5 - q^7 - 3 * q^8	$ 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} - q^{7} - 3 q^{8} - 3 q^{10} - 11 q^{11} + q^{14} + 3 q^{16} - q^{17} + 3 q^{20} + 11 q^{22} + 2 q^{23} + 3 q^{25} - q^{28} + 5 q^{29} + 3 q^{31} - 3 q^{32} + q^{34} - q^{35} - 3 q^{37} - 3 q^{40} + 9 q^{41} - 11 q^{43} - 11 q^{44} - 2 q^{46} - 2 q^{47} - 4 q^{49} - 3 q^{50} - 21 q^{53} - 11 q^{55} + q^{56} - 5 q^{58} - 14 q^{59} + 7 q^{61} - 3 q^{62} + 3 q^{64} + 2 q^{67} - q^{68} + q^{70} - 22 q^{71} - 16 q^{73} + 3 q^{74} - 7 q^{77} - 26 q^{79} + 3 q^{80} - 9 q^{82} - 2 q^{83} - q^{85} + 11 q^{86} + 11 q^{88} - 18 q^{89} - 12 q^{91} + 2 q^{92} + 2 q^{94} - 21 q^{97} + 4 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^4 + 3 * q^5 - q^7 - 3 * q^8 - 3 * q^10 - 11 * q^11 + q^14 + 3 * q^16 - q^17 + 3 * q^20 + 11 * q^22 + 2 * q^23 + 3 * q^25 - q^28 + 5 * q^29 + 3 * q^31 - 3 * q^32 + q^34 - q^35 - 3 * q^37 - 3 * q^40 + 9 * q^41 - 11 * q^43 - 11 * q^44 - 2 * q^46 - 2 * q^47 - 4 * q^49 - 3 * q^50 - 21 * q^53 - 11 * q^55 + q^56 - 5 * q^58 - 14 * q^59 + 7 * q^61 - 3 * q^62 + 3 * q^64 + 2 * q^67 - q^68 + q^70 - 22 * q^71 - 16 * q^73 + 3 * q^74 - 7 * q^77 - 26 * q^79 + 3 * q^80 - 9 * q^82 - 2 * q^83 - q^85 + 11 * q^86 + 11 * q^88 - 18 * q^89 - 12 * q^91 + 2 * q^92 + 2 * q^94 - 21 * q^97 + 4 * 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
$7$	−1.31955	−0.498743	−0.249372	−	0.968408i	$-0.580224\pi$
$7$	−1.31955	−0.498743	−0.249372	+	0.968408i	$0.580224\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−1.00000	−0.316228
$11$	0.258786	0.0780268	0.0390134	−	0.999239i	$-0.487578\pi$
$11$	0.258786	0.0780268	0.0390134	+	0.999239i	$0.487578\pi$
$12$	0	0
$13$	−5.87847	−1.63039	−0.815197	−	0.579184i	$-0.803371\pi$
$13$	−5.87847	−1.63039	−0.815197	+	0.579184i	$0.803371\pi$
$14$	1.31955	0.352665
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.25879	−1.03291	−0.516454	−	0.856315i	$-0.672748\pi$
$17$	−4.25879	−1.03291	−0.516454	+	0.856315i	$0.672748\pi$
$18$	0	0
$19$	2.93923	0.674307	0.337153	−	0.941450i	$-0.390536\pi$
$19$	2.93923	0.674307	0.337153	+	0.941450i	$0.390536\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	−0.258786	−0.0551733
$23$	8.51757	1.77604	0.888018	−	0.459808i	$-0.152082\pi$
$23$	8.51757	1.77604	0.888018	+	0.459808i	$0.152082\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	5.87847	1.15286
$27$	0	0
$28$	−1.31955	−0.249372
$29$	3.61968	0.672158	0.336079	−	0.941834i	$-0.390899\pi$
$29$	3.61968	0.672158	0.336079	+	0.941834i	$0.390899\pi$
$30$	0	0
$31$	3.95865	0.710995	0.355497	−	0.934677i	$-0.384311\pi$
$31$	3.95865	0.710995	0.355497	+	0.934677i	$0.384311\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	4.25879	0.730376
$35$	−1.31955	−0.223045
$36$	0	0
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	−2.93923	−0.476807
$39$	0	0
$40$	−1.00000	−0.158114
$41$	8.89789	1.38962	0.694808	−	0.719195i	$-0.255489\pi$
$41$	8.89789	1.38962	0.694808	+	0.719195i	$0.255489\pi$
$42$	0	0
$43$	−5.61968	−0.856994	−0.428497	−	0.903543i	$-0.640957\pi$
$43$	−5.61968	−0.856994	−0.428497	+	0.903543i	$0.640957\pi$
$44$	0.258786	0.0390134
$45$	0	0
$46$	−8.51757	−1.25585
$47$	−5.57834	−0.813684	−0.406842	−	0.913499i	$-0.633370\pi$
$47$	−5.57834	−0.813684	−0.406842	+	0.913499i	$0.633370\pi$
$48$	0	0
$49$	−5.25879	−0.751255
$50$	−1.00000	−0.141421
$51$	0	0
$52$	−5.87847	−0.815197
$53$	−12.8979	−1.77166	−0.885831	−	0.464009i	$-0.846411\pi$
$53$	−12.8979	−1.77166	−0.885831	+	0.464009i	$0.846411\pi$
$54$	0	0
$55$	0.258786	0.0348947
$56$	1.31955	0.176332
$57$	0	0
$58$	−3.61968	−0.475288
$59$	−9.57834	−1.24699	−0.623497	−	0.781826i	$-0.714288\pi$
$59$	−9.57834	−1.24699	−0.623497	+	0.781826i	$0.714288\pi$
$60$	0	0
$61$	0.380316	0.0486945	0.0243472	−	0.999704i	$-0.492249\pi$
$61$	0.380316	0.0486945	0.0243472	+	0.999704i	$0.492249\pi$
$62$	−3.95865	−0.502749
$63$	0	0
$64$	1.00000	0.125000
$65$	−5.87847	−0.729134
$66$	0	0
$67$	5.57834	0.681502	0.340751	−	0.940154i	$-0.389319\pi$
$67$	5.57834	0.681502	0.340751	+	0.940154i	$0.389319\pi$
$68$	−4.25879	−0.516454
$69$	0	0
$70$	1.31955	0.157716
$71$	−11.2394	−1.33387	−0.666934	−	0.745117i	$-0.732394\pi$
$71$	−11.2394	−1.33387	−0.666934	+	0.745117i	$0.732394\pi$
$72$	0	0
$73$	2.51757	0.294659	0.147330	−	0.989087i	$-0.452932\pi$
$73$	2.51757	0.294659	0.147330	+	0.989087i	$0.452932\pi$
$74$	1.00000	0.116248
$75$	0	0
$76$	2.93923	0.337153
$77$	−0.341481	−0.0389154
$78$	0	0
$79$	−7.69987	−0.866303	−0.433151	−	0.901321i	$-0.642598\pi$
$79$	−7.69987	−0.866303	−0.433151	+	0.901321i	$0.642598\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	−8.89789	−0.982607
$83$	6.17860	0.678190	0.339095	−	0.940752i	$-0.389879\pi$
$83$	6.17860	0.678190	0.339095	+	0.940752i	$0.389879\pi$
$84$	0	0
$85$	−4.25879	−0.461930
$86$	5.61968	0.605986
$87$	0	0
$88$	−0.258786	−0.0275866
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	7.75694	0.813148
$92$	8.51757	0.888018
$93$	0	0
$94$	5.57834	0.575361
$95$	2.93923	0.301559
$96$	0	0
$97$	−12.8979	−1.30958	−0.654791	−	0.755810i	$-0.727243\pi$
$97$	−12.8979	−1.30958	−0.654791	+	0.755810i	$0.727243\pi$
$98$	5.25879	0.531218
$99$	0	0
$100$	1.00000	0.100000
$101$	−10.3960	−1.03444	−0.517222	−	0.855851i	$-0.673034\pi$
$101$	−10.3960	−1.03444	−0.517222	+	0.855851i	$0.673034\pi$
$102$	0	0
$103$	−13.8785	−1.36749	−0.683743	−	0.729723i	$-0.739649\pi$
$103$	−13.8785	−1.36749	−0.683743	+	0.729723i	$0.739649\pi$
$104$	5.87847	0.576431
$105$	0	0
$106$	12.8979	1.25275
$107$	4.21744	0.407715	0.203858	−	0.979001i	$-0.434652\pi$
$107$	4.21744	0.407715	0.203858	+	0.979001i	$0.434652\pi$
$108$	0	0
$109$	16.1373	1.54567	0.772834	−	0.634608i	$-0.218838\pi$
$109$	16.1373	1.54567	0.772834	+	0.634608i	$0.218838\pi$
$110$	−0.258786	−0.0246742
$111$	0	0
$112$	−1.31955	−0.124686
$113$	−1.01942	−0.0958987	−0.0479494	−	0.998850i	$-0.515269\pi$
$113$	−1.01942	−0.0958987	−0.0479494	+	0.998850i	$0.515269\pi$
$114$	0	0
$115$	8.51757	0.794268
$116$	3.61968	0.336079
$117$	0	0
$118$	9.57834	0.881757
$119$	5.61968	0.515156
$120$	0	0
$121$	−10.9330	−0.993912
$122$	−0.380316	−0.0344322
$123$	0	0
$124$	3.95865	0.355497
$125$	1.00000	0.0894427
$126$	0	0
$127$	7.45681	0.661685	0.330842	−	0.943686i	$-0.392667\pi$
$127$	7.45681	0.661685	0.330842	+	0.943686i	$0.392667\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	5.87847	0.515576
$131$	−2.42166	−0.211582	−0.105791	−	0.994388i	$-0.533737\pi$
$131$	−2.42166	−0.211582	−0.105791	+	0.994388i	$0.533737\pi$
$132$	0	0
$133$	−3.87847	−0.336306
$134$	−5.57834	−0.481895
$135$	0	0
$136$	4.25879	0.365188
$137$	−15.7958	−1.34952	−0.674762	−	0.738035i	$-0.735754\pi$
$137$	−15.7958	−1.34952	−0.674762	+	0.738035i	$0.735754\pi$
$138$	0	0
$139$	−11.4155	−0.968247	−0.484123	−	0.875000i	$-0.660861\pi$
$139$	−11.4155	−0.968247	−0.484123	+	0.875000i	$0.660861\pi$
$140$	−1.31955	−0.111522
$141$	0	0
$142$	11.2394	0.943187
$143$	−1.52126	−0.127214
$144$	0	0
$145$	3.61968	0.300598
$146$	−2.51757	−0.208356
$147$	0	0
$148$	−1.00000	−0.0821995
$149$	−18.3960	−1.50706	−0.753531	−	0.657412i	$-0.771651\pi$
$149$	−18.3960	−1.50706	−0.753531	+	0.657412i	$0.771651\pi$
$150$	0	0
$151$	5.87847	0.478383	0.239192	−	0.970972i	$-0.423118\pi$
$151$	5.87847	0.478383	0.239192	+	0.970972i	$0.423118\pi$
$152$	−2.93923	−0.238403
$153$	0	0
$154$	0.341481	0.0275173
$155$	3.95865	0.317967
$156$	0	0
$157$	3.61968	0.288882	0.144441	−	0.989513i	$-0.453862\pi$
$157$	3.61968	0.288882	0.144441	+	0.989513i	$0.453862\pi$
$158$	7.69987	0.612569
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	−11.2394	−0.885786
$162$	0	0
$163$	−22.0546	−1.72745	−0.863723	−	0.503966i	$-0.831874\pi$
$163$	−22.0546	−1.72745	−0.863723	+	0.503966i	$0.831874\pi$
$164$	8.89789	0.694808
$165$	0	0
$166$	−6.17860	−0.479553
$167$	0.600267	0.0464500	0.0232250	−	0.999730i	$-0.492607\pi$
$167$	0.600267	0.0464500	0.0232250	+	0.999730i	$0.492607\pi$
$168$	0	0
$169$	21.5564	1.65819
$170$	4.25879	0.326634
$171$	0	0
$172$	−5.61968	−0.428497
$173$	22.6937	1.72537	0.862684	−	0.505744i	$-0.168782\pi$
$173$	22.6937	1.72537	0.862684	+	0.505744i	$0.168782\pi$
$174$	0	0
$175$	−1.31955	−0.0997487
$176$	0.258786	0.0195067
$177$	0	0
$178$	6.00000	0.449719
$179$	8.73501	0.652885	0.326443	−	0.945217i	$-0.394150\pi$
$179$	8.73501	0.652885	0.326443	+	0.945217i	$0.394150\pi$
$180$	0	0
$181$	10.6391	0.790798	0.395399	−	0.918509i	$-0.370606\pi$
$181$	10.6391	0.790798	0.395399	+	0.918509i	$0.370606\pi$
$182$	−7.75694	−0.574983
$183$	0	0
$184$	−8.51757	−0.627924
$185$	−1.00000	−0.0735215
$186$	0	0
$187$	−1.10211	−0.0805945
$188$	−5.57834	−0.406842
$189$	0	0
$190$	−2.93923	−0.213235
$191$	−22.1116	−1.59994	−0.799971	−	0.600039i	$-0.795152\pi$
$191$	−22.1116	−1.59994	−0.799971	+	0.600039i	$0.795152\pi$
$192$	0	0
$193$	7.03514	0.506401	0.253200	−	0.967414i	$-0.418517\pi$
$193$	7.03514	0.506401	0.253200	+	0.967414i	$0.418517\pi$
$194$	12.8979	0.926014
$195$	0	0
$196$	−5.25879	−0.375628
$197$	−25.7569	−1.83511	−0.917553	−	0.397614i	$-0.869838\pi$
$197$	−25.7569	−1.83511	−0.917553	+	0.397614i	$0.869838\pi$
$198$	0	0
$199$	−12.7350	−0.902761	−0.451380	−	0.892332i	$-0.649068\pi$
$199$	−12.7350	−0.902761	−0.451380	+	0.892332i	$0.649068\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	10.3960	0.731463
$203$	−4.77636	−0.335235
$204$	0	0
$205$	8.89789	0.621455
$206$	13.8785	0.966959
$207$	0	0
$208$	−5.87847	−0.407599
$209$	0.760632	0.0526140
$210$	0	0
$211$	−6.65483	−0.458137	−0.229069	−	0.973410i	$-0.573568\pi$
$211$	−6.65483	−0.458137	−0.229069	+	0.973410i	$0.573568\pi$
$212$	−12.8979	−0.885831
$213$	0	0
$214$	−4.21744	−0.288298
$215$	−5.61968	−0.383259
$216$	0	0
$217$	−5.22364	−0.354604
$218$	−16.1373	−1.09295
$219$	0	0
$220$	0.258786	0.0174473
$221$	25.0351	1.68405
$222$	0	0
$223$	−16.5589	−1.10887	−0.554434	−	0.832228i	$-0.687065\pi$
$223$	−16.5589	−1.10887	−0.554434	+	0.832228i	$0.687065\pi$
$224$	1.31955	0.0881662
$225$	0	0
$226$	1.01942	0.0678106
$227$	4.01573	0.266533	0.133267	−	0.991080i	$-0.457453\pi$
$227$	4.01573	0.266533	0.133267	+	0.991080i	$0.457453\pi$
$228$	0	0
$229$	25.5139	1.68600	0.843002	−	0.537910i	$-0.180786\pi$
$229$	25.5139	1.68600	0.843002	+	0.537910i	$0.180786\pi$
$230$	−8.51757	−0.561632
$231$	0	0
$232$	−3.61968	−0.237644
$233$	−7.96116	−0.521553	−0.260777	−	0.965399i	$-0.583979\pi$
$233$	−7.96116	−0.521553	−0.260777	+	0.965399i	$0.583979\pi$
$234$	0	0
$235$	−5.57834	−0.363891
$236$	−9.57834	−0.623497
$237$	0	0
$238$	−5.61968	−0.364270
$239$	−11.7983	−0.763168	−0.381584	−	0.924334i	$-0.624621\pi$
$239$	−11.7983	−0.763168	−0.381584	+	0.924334i	$0.624621\pi$
$240$	0	0
$241$	24.9963	1.61015	0.805077	−	0.593171i	$-0.202124\pi$
$241$	24.9963	1.61015	0.805077	+	0.593171i	$0.202124\pi$
$242$	10.9330	0.702802
$243$	0	0
$244$	0.380316	0.0243472
$245$	−5.25879	−0.335971
$246$	0	0
$247$	−17.2782	−1.09939
$248$	−3.95865	−0.251375
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	8.81770	0.556569	0.278284	−	0.960499i	$-0.410234\pi$
$251$	8.81770	0.556569	0.278284	+	0.960499i	$0.410234\pi$
$252$	0	0
$253$	2.20423	0.138578
$254$	−7.45681	−0.467882
$255$	0	0
$256$	1.00000	0.0625000
$257$	−11.8396	−0.738536	−0.369268	−	0.929323i	$-0.620392\pi$
$257$	−11.8396	−0.738536	−0.369268	+	0.929323i	$0.620392\pi$
$258$	0	0
$259$	1.31955	0.0819929
$260$	−5.87847	−0.364567
$261$	0	0
$262$	2.42166	0.149611
$263$	9.15919	0.564780	0.282390	−	0.959300i	$-0.408873\pi$
$263$	9.15919	0.564780	0.282390	+	0.959300i	$0.408873\pi$
$264$	0	0
$265$	−12.8979	−0.792311
$266$	3.87847	0.237804
$267$	0	0
$268$	5.57834	0.340751
$269$	−18.3960	−1.12163	−0.560813	−	0.827942i	$-0.689511\pi$
$269$	−18.3960	−1.12163	−0.560813	+	0.827942i	$0.689511\pi$
$270$	0	0
$271$	−27.2394	−1.65467	−0.827337	−	0.561706i	$-0.810145\pi$
$271$	−27.2394	−1.65467	−0.827337	+	0.561706i	$0.810145\pi$
$272$	−4.25879	−0.258227
$273$	0	0
$274$	15.7958	0.954258
$275$	0.258786	0.0156054
$276$	0	0
$277$	−21.8785	−1.31455	−0.657275	−	0.753651i	$-0.728291\pi$
$277$	−21.8785	−1.31455	−0.657275	+	0.753651i	$0.728291\pi$
$278$	11.4155	0.684654
$279$	0	0
$280$	1.31955	0.0788582
$281$	20.9963	1.25253	0.626267	−	0.779608i	$-0.284582\pi$
$281$	20.9963	1.25253	0.626267	+	0.779608i	$0.284582\pi$
$282$	0	0
$283$	18.9136	1.12430	0.562149	−	0.827036i	$-0.309975\pi$
$283$	18.9136	1.12430	0.562149	+	0.827036i	$0.309975\pi$
$284$	−11.2394	−0.666934
$285$	0	0
$286$	1.52126	0.0899542
$287$	−11.7412	−0.693062
$288$	0	0
$289$	1.13726	0.0668974
$290$	−3.61968	−0.212555
$291$	0	0
$292$	2.51757	0.147330
$293$	12.8979	0.753503	0.376751	−	0.926314i	$-0.377041\pi$
$293$	12.8979	0.753503	0.376751	+	0.926314i	$0.377041\pi$
$294$	0	0
$295$	−9.57834	−0.557672
$296$	1.00000	0.0581238
$297$	0	0
$298$	18.3960	1.06565
$299$	−50.0703	−2.89564
$300$	0	0
$301$	7.41546	0.427420
$302$	−5.87847	−0.338268
$303$	0	0
$304$	2.93923	0.168577
$305$	0.380316	0.0217768
$306$	0	0
$307$	−3.78256	−0.215882	−0.107941	−	0.994157i	$-0.534426\pi$
$307$	−3.78256	−0.215882	−0.107941	+	0.994157i	$0.534426\pi$
$308$	−0.341481	−0.0194577
$309$	0	0
$310$	−3.95865	−0.224836
$311$	15.2807	0.866490	0.433245	−	0.901276i	$-0.357369\pi$
$311$	15.2807	0.866490	0.433245	+	0.901276i	$0.357369\pi$
$312$	0	0
$313$	19.0351	1.07593	0.537965	−	0.842967i	$-0.319193\pi$
$313$	19.0351	1.07593	0.537965	+	0.842967i	$0.319193\pi$
$314$	−3.61968	−0.204271
$315$	0	0
$316$	−7.69987	−0.433151
$317$	24.1373	1.35568	0.677842	−	0.735208i	$-0.262915\pi$
$317$	24.1373	1.35568	0.677842	+	0.735208i	$0.262915\pi$
$318$	0	0
$319$	0.936722	0.0524464
$320$	1.00000	0.0559017
$321$	0	0
$322$	11.2394	0.626345
$323$	−12.5176	−0.696496
$324$	0	0
$325$	−5.87847	−0.326079
$326$	22.0546	1.22149
$327$	0	0
$328$	−8.89789	−0.491304
$329$	7.36090	0.405819
$330$	0	0
$331$	0.817705	0.0449451	0.0224726	−	0.999747i	$-0.492846\pi$
$331$	0.817705	0.0449451	0.0224726	+	0.999747i	$0.492846\pi$
$332$	6.17860	0.339095
$333$	0	0
$334$	−0.600267	−0.0328451
$335$	5.57834	0.304777
$336$	0	0
$337$	−19.7958	−1.07834	−0.539172	−	0.842195i	$-0.681263\pi$
$337$	−19.7958	−1.07834	−0.539172	+	0.842195i	$0.681263\pi$
$338$	−21.5564	−1.17251
$339$	0	0
$340$	−4.25879	−0.230965
$341$	1.02444	0.0554767
$342$	0	0
$343$	16.1761	0.873427
$344$	5.61968	0.302993
$345$	0	0
$346$	−22.6937	−1.22002
$347$	−15.7569	−0.845877	−0.422938	−	0.906158i	$-0.639001\pi$
$347$	−15.7569	−0.845877	−0.422938	+	0.906158i	$0.639001\pi$
$348$	0	0
$349$	−0.160365	−0.00858416	−0.00429208	−	0.999991i	$-0.501366\pi$
$349$	−0.160365	−0.00858416	−0.00429208	+	0.999991i	$0.501366\pi$
$350$	1.31955	0.0705330
$351$	0	0
$352$	−0.258786	−0.0137933
$353$	30.2075	1.60779	0.803893	−	0.594775i	$-0.202759\pi$
$353$	30.2075	1.60779	0.803893	+	0.594775i	$0.202759\pi$
$354$	0	0
$355$	−11.2394	−0.596524
$356$	−6.00000	−0.317999
$357$	0	0
$358$	−8.73501	−0.461660
$359$	7.39973	0.390543	0.195271	−	0.980749i	$-0.437441\pi$
$359$	7.39973	0.390543	0.195271	+	0.980749i	$0.437441\pi$
$360$	0	0
$361$	−10.3609	−0.545310
$362$	−10.6391	−0.559179
$363$	0	0
$364$	7.75694	0.406574
$365$	2.51757	0.131776
$366$	0	0
$367$	−13.9198	−0.726609	−0.363304	−	0.931671i	$-0.618351\pi$
$367$	−13.9198	−0.726609	−0.363304	+	0.931671i	$0.618351\pi$
$368$	8.51757	0.444009
$369$	0	0
$370$	1.00000	0.0519875
$371$	17.0194	0.883604
$372$	0	0
$373$	20.2357	1.04776	0.523882	−	0.851791i	$-0.324483\pi$
$373$	20.2357	1.04776	0.523882	+	0.851791i	$0.324483\pi$
$374$	1.10211	0.0569889
$375$	0	0
$376$	5.57834	0.287681
$377$	−21.2782	−1.09588
$378$	0	0
$379$	−27.9488	−1.43563	−0.717816	−	0.696233i	$-0.754858\pi$
$379$	−27.9488	−1.43563	−0.717816	+	0.696233i	$0.754858\pi$
$380$	2.93923	0.150780
$381$	0	0
$382$	22.1116	1.13133
$383$	−11.7569	−0.600752	−0.300376	−	0.953821i	$-0.597112\pi$
$383$	−11.7569	−0.600752	−0.300376	+	0.953821i	$0.597112\pi$
$384$	0	0
$385$	−0.341481	−0.0174035
$386$	−7.03514	−0.358079
$387$	0	0
$388$	−12.8979	−0.654791
$389$	9.58085	0.485768	0.242884	−	0.970055i	$-0.421907\pi$
$389$	9.58085	0.485768	0.242884	+	0.970055i	$0.421907\pi$
$390$	0	0
$391$	−36.2745	−1.83448
$392$	5.25879	0.265609
$393$	0	0
$394$	25.7569	1.29762
$395$	−7.69987	−0.387422
$396$	0	0
$397$	−32.4787	−1.63006	−0.815031	−	0.579418i	$-0.803280\pi$
$397$	−32.4787	−1.63006	−0.815031	+	0.579418i	$0.803280\pi$
$398$	12.7350	0.638348
$399$	0	0
$400$	1.00000	0.0500000
$401$	−7.27820	−0.363456	−0.181728	−	0.983349i	$-0.558169\pi$
$401$	−7.27820	−0.363456	−0.181728	+	0.983349i	$0.558169\pi$
$402$	0	0
$403$	−23.2708	−1.15920
$404$	−10.3960	−0.517222
$405$	0	0
$406$	4.77636	0.237047
$407$	−0.258786	−0.0128275
$408$	0	0
$409$	−32.9963	−1.63156	−0.815781	−	0.578361i	$-0.803693\pi$
$409$	−32.9963	−1.63156	−0.815781	+	0.578361i	$0.803693\pi$
$410$	−8.89789	−0.439435
$411$	0	0
$412$	−13.8785	−0.683743
$413$	12.6391	0.621930
$414$	0	0
$415$	6.17860	0.303296
$416$	5.87847	0.288216
$417$	0	0
$418$	−0.760632	−0.0372037
$419$	−25.7131	−1.25617	−0.628083	−	0.778146i	$-0.716160\pi$
$419$	−25.7131	−1.25617	−0.628083	+	0.778146i	$0.716160\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	6.65483	0.323952
$423$	0	0
$424$	12.8979	0.626377
$425$	−4.25879	−0.206581
$426$	0	0
$427$	−0.501846	−0.0242860
$428$	4.21744	0.203858
$429$	0	0
$430$	5.61968	0.271005
$431$	−17.9198	−0.863167	−0.431584	−	0.902073i	$-0.642045\pi$
$431$	−17.9198	−0.863167	−0.431584	+	0.902073i	$0.642045\pi$
$432$	0	0
$433$	4.96486	0.238596	0.119298	−	0.992859i	$-0.461936\pi$
$433$	4.96486	0.238596	0.119298	+	0.992859i	$0.461936\pi$
$434$	5.22364	0.250743
$435$	0	0
$436$	16.1373	0.772834
$437$	25.0351	1.19759
$438$	0	0
$439$	−13.8371	−0.660410	−0.330205	−	0.943909i	$-0.607118\pi$
$439$	−13.8371	−0.660410	−0.330205	+	0.943909i	$0.607118\pi$
$440$	−0.258786	−0.0123371
$441$	0	0
$442$	−25.0351	−1.19080
$443$	−14.3390	−0.681265	−0.340632	−	0.940197i	$-0.610641\pi$
$443$	−14.3390	−0.681265	−0.340632	+	0.940197i	$0.610641\pi$
$444$	0	0
$445$	−6.00000	−0.284427
$446$	16.5589	0.784088
$447$	0	0
$448$	−1.31955	−0.0623429
$449$	−12.4787	−0.588908	−0.294454	−	0.955666i	$-0.595138\pi$
$449$	−12.4787	−0.588908	−0.294454	+	0.955666i	$0.595138\pi$
$450$	0	0
$451$	2.30265	0.108427
$452$	−1.01942	−0.0479494
$453$	0	0
$454$	−4.01573	−0.188467
$455$	7.75694	0.363651
$456$	0	0
$457$	21.6585	1.01314	0.506571	−	0.862198i	$-0.330913\pi$
$457$	21.6585	1.01314	0.506571	+	0.862198i	$0.330913\pi$
$458$	−25.5139	−1.19219
$459$	0	0
$460$	8.51757	0.397134
$461$	29.4155	1.37001	0.685007	−	0.728536i	$-0.259799\pi$
$461$	29.4155	1.37001	0.685007	+	0.728536i	$0.259799\pi$
$462$	0	0
$463$	−9.03514	−0.419899	−0.209949	−	0.977712i	$-0.567330\pi$
$463$	−9.03514	−0.419899	−0.209949	+	0.977712i	$0.567330\pi$
$464$	3.61968	0.168040
$465$	0	0
$466$	7.96116	0.368794
$467$	−7.05825	−0.326617	−0.163308	−	0.986575i	$-0.552217\pi$
$467$	−7.05825	−0.326617	−0.163308	+	0.986575i	$0.552217\pi$
$468$	0	0
$469$	−7.36090	−0.339895
$470$	5.57834	0.257309
$471$	0	0
$472$	9.57834	0.440879
$473$	−1.45429	−0.0668685
$474$	0	0
$475$	2.93923	0.134861
$476$	5.61968	0.257578
$477$	0	0
$478$	11.7983	0.539641
$479$	29.3353	1.34036	0.670181	−	0.742197i	$-0.266216\pi$
$479$	29.3353	1.34036	0.670181	+	0.742197i	$0.266216\pi$
$480$	0	0
$481$	5.87847	0.268035
$482$	−24.9963	−1.13855
$483$	0	0
$484$	−10.9330	−0.496956
$485$	−12.8979	−0.585663
$486$	0	0
$487$	−37.9099	−1.71786	−0.858931	−	0.512091i	$-0.828871\pi$
$487$	−37.9099	−1.71786	−0.858931	+	0.512091i	$0.828871\pi$
$488$	−0.380316	−0.0172161
$489$	0	0
$490$	5.25879	0.237568
$491$	14.5564	0.656921	0.328461	−	0.944518i	$-0.393470\pi$
$491$	14.5564	0.656921	0.328461	+	0.944518i	$0.393470\pi$
$492$	0	0
$493$	−15.4155	−0.694277
$494$	17.2782	0.777383
$495$	0	0
$496$	3.95865	0.177749
$497$	14.8309	0.665258
$498$	0	0
$499$	23.9744	1.07324	0.536620	−	0.843824i	$-0.319701\pi$
$499$	23.9744	1.07324	0.536620	+	0.843824i	$0.319701\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	−8.81770	−0.393553
$503$	43.7569	1.95103	0.975513	−	0.219943i	$-0.0705871\pi$
$503$	43.7569	1.95103	0.975513	+	0.219943i	$0.0705871\pi$
$504$	0	0
$505$	−10.3960	−0.462618
$506$	−2.20423	−0.0979898
$507$	0	0
$508$	7.45681	0.330842
$509$	−11.7958	−0.522839	−0.261419	−	0.965225i	$-0.584191\pi$
$509$	−11.7958	−0.522839	−0.261419	+	0.965225i	$0.584191\pi$
$510$	0	0
$511$	−3.32206	−0.146959
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	11.8396	0.522224
$515$	−13.8785	−0.611558
$516$	0	0
$517$	−1.44359	−0.0634892
$518$	−1.31955	−0.0579777
$519$	0	0
$520$	5.87847	0.257788
$521$	−14.9806	−0.656311	−0.328156	−	0.944624i	$-0.606427\pi$
$521$	−14.9806	−0.656311	−0.328156	+	0.944624i	$0.606427\pi$
$522$	0	0
$523$	−27.5139	−1.20310	−0.601549	−	0.798836i	$-0.705450\pi$
$523$	−27.5139	−1.20310	−0.601549	+	0.798836i	$0.705450\pi$
$524$	−2.42166	−0.105791
$525$	0	0
$526$	−9.15919	−0.399359
$527$	−16.8591	−0.734392
$528$	0	0
$529$	49.5490	2.15431
$530$	12.8979	0.560248
$531$	0	0
$532$	−3.87847	−0.168153
$533$	−52.3060	−2.26562
$534$	0	0
$535$	4.21744	0.182336
$536$	−5.57834	−0.240947
$537$	0	0
$538$	18.3960	0.793110
$539$	−1.36090	−0.0586180
$540$	0	0
$541$	−11.0351	−0.474438	−0.237219	−	0.971456i	$-0.576236\pi$
$541$	−11.0351	−0.474438	−0.237219	+	0.971456i	$0.576236\pi$
$542$	27.2394	1.17003
$543$	0	0
$544$	4.25879	0.182594
$545$	16.1373	0.691244
$546$	0	0
$547$	−16.1761	−0.691640	−0.345820	−	0.938301i	$-0.612399\pi$
$547$	−16.1761	−0.691640	−0.345820	+	0.938301i	$0.612399\pi$
$548$	−15.7958	−0.674762
$549$	0	0
$550$	−0.258786	−0.0110347
$551$	10.6391	0.453241
$552$	0	0
$553$	10.1604	0.432063
$554$	21.8785	0.929527
$555$	0	0
$556$	−11.4155	−0.484123
$557$	−11.1567	−0.472723	−0.236362	−	0.971665i	$-0.575955\pi$
$557$	−11.1567	−0.472723	−0.236362	+	0.971665i	$0.575955\pi$
$558$	0	0
$559$	33.0351	1.39724
$560$	−1.31955	−0.0557612
$561$	0	0
$562$	−20.9963	−0.885676
$563$	41.7288	1.75866	0.879330	−	0.476213i	$-0.157991\pi$
$563$	41.7288	1.75866	0.879330	+	0.476213i	$0.157991\pi$
$564$	0	0
$565$	−1.01942	−0.0428872
$566$	−18.9136	−0.794998
$567$	0	0
$568$	11.2394	0.471593
$569$	24.0703	1.00908	0.504539	−	0.863389i	$-0.331662\pi$
$569$	24.0703	1.00908	0.504539	+	0.863389i	$0.331662\pi$
$570$	0	0
$571$	37.6511	1.57565	0.787825	−	0.615899i	$-0.211207\pi$
$571$	37.6511	1.57565	0.787825	+	0.615899i	$0.211207\pi$
$572$	−1.52126	−0.0636072
$573$	0	0
$574$	11.7412	0.490069
$575$	8.51757	0.355207
$576$	0	0
$577$	23.4312	0.975453	0.487726	−	0.872996i	$-0.337826\pi$
$577$	23.4312	0.975453	0.487726	+	0.872996i	$0.337826\pi$
$578$	−1.13726	−0.0473036
$579$	0	0
$580$	3.61968	0.150299
$581$	−8.15298	−0.338243
$582$	0	0
$583$	−3.33779	−0.138237
$584$	−2.51757	−0.104178
$585$	0	0
$586$	−12.8979	−0.532807
$587$	−6.30265	−0.260138	−0.130069	−	0.991505i	$-0.541520\pi$
$587$	−6.30265	−0.260138	−0.130069	+	0.991505i	$0.541520\pi$
$588$	0	0
$589$	11.6354	0.479429
$590$	9.57834	0.394334
$591$	0	0
$592$	−1.00000	−0.0410997
$593$	34.0315	1.39750	0.698752	−	0.715364i	$-0.253739\pi$
$593$	34.0315	1.39750	0.698752	+	0.715364i	$0.253739\pi$
$594$	0	0
$595$	5.61968	0.230385
$596$	−18.3960	−0.753531
$597$	0	0
$598$	50.0703	2.04753
$599$	−34.8359	−1.42336	−0.711679	−	0.702505i	$-0.752065\pi$
$599$	−34.8359	−1.42336	−0.711679	+	0.702505i	$0.752065\pi$
$600$	0	0
$601$	15.2236	0.620985	0.310493	−	0.950576i	$-0.399506\pi$
$601$	15.2236	0.620985	0.310493	+	0.950576i	$0.399506\pi$
$602$	−7.41546	−0.302232
$603$	0	0
$604$	5.87847	0.239192
$605$	−10.9330	−0.444491
$606$	0	0
$607$	−30.8309	−1.25139	−0.625694	−	0.780068i	$-0.715184\pi$
$607$	−30.8309	−1.25139	−0.625694	+	0.780068i	$0.715184\pi$
$608$	−2.93923	−0.119202
$609$	0	0
$610$	−0.380316	−0.0153985
$611$	32.7921	1.32663
$612$	0	0
$613$	−11.2112	−0.452817	−0.226409	−	0.974032i	$-0.572698\pi$
$613$	−11.2112	−0.452817	−0.226409	+	0.974032i	$0.572698\pi$
$614$	3.78256	0.152652
$615$	0	0
$616$	0.341481	0.0137587
$617$	−26.5176	−1.06756	−0.533779	−	0.845624i	$-0.679228\pi$
$617$	−26.5176	−1.06756	−0.533779	+	0.845624i	$0.679228\pi$
$618$	0	0
$619$	−3.41546	−0.137279	−0.0686394	−	0.997642i	$-0.521866\pi$
$619$	−3.41546	−0.137279	−0.0686394	+	0.997642i	$0.521866\pi$
$620$	3.95865	0.158983
$621$	0	0
$622$	−15.2807	−0.612701
$623$	7.91730	0.317200
$624$	0	0
$625$	1.00000	0.0400000
$626$	−19.0351	−0.760797
$627$	0	0
$628$	3.61968	0.144441
$629$	4.25879	0.169809
$630$	0	0
$631$	3.11533	0.124019	0.0620096	−	0.998076i	$-0.480249\pi$
$631$	3.11533	0.124019	0.0620096	+	0.998076i	$0.480249\pi$
$632$	7.69987	0.306284
$633$	0	0
$634$	−24.1373	−0.958613
$635$	7.45681	0.295914
$636$	0	0
$637$	30.9136	1.22484
$638$	−0.936722	−0.0370852
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	20.5018	0.809774	0.404887	−	0.914367i	$-0.367311\pi$
$641$	20.5018	0.809774	0.404887	+	0.914367i	$0.367311\pi$
$642$	0	0
$643$	5.70238	0.224880	0.112440	−	0.993659i	$-0.464133\pi$
$643$	5.70238	0.224880	0.112440	+	0.993659i	$0.464133\pi$
$644$	−11.2394	−0.442893
$645$	0	0
$646$	12.5176	0.492497
$647$	41.0351	1.61326	0.806629	−	0.591058i	$-0.201290\pi$
$647$	41.0351	1.61326	0.806629	+	0.591058i	$0.201290\pi$
$648$	0	0
$649$	−2.47874	−0.0972989
$650$	5.87847	0.230573
$651$	0	0
$652$	−22.0546	−0.863723
$653$	−14.9575	−0.585331	−0.292666	−	0.956215i	$-0.594542\pi$
$653$	−14.9575	−0.585331	−0.292666	+	0.956215i	$0.594542\pi$
$654$	0	0
$655$	−2.42166	−0.0946222
$656$	8.89789	0.347404
$657$	0	0
$658$	−7.36090	−0.286958
$659$	15.7569	0.613803	0.306902	−	0.951741i	$-0.400708\pi$
$659$	15.7569	0.613803	0.306902	+	0.951741i	$0.400708\pi$
$660$	0	0
$661$	11.6197	0.451953	0.225977	−	0.974133i	$-0.427443\pi$
$661$	11.6197	0.451953	0.225977	+	0.974133i	$0.427443\pi$
$662$	−0.817705	−0.0317810
$663$	0	0
$664$	−6.17860	−0.239776
$665$	−3.87847	−0.150401
$666$	0	0
$667$	30.8309	1.19378
$668$	0.600267	0.0232250
$669$	0	0
$670$	−5.57834	−0.215510
$671$	0.0984203	0.00379947
$672$	0	0
$673$	12.7218	0.490389	0.245195	−	0.969474i	$-0.421148\pi$
$673$	12.7218	0.490389	0.245195	+	0.969474i	$0.421148\pi$
$674$	19.7958	0.762505
$675$	0	0
$676$	21.5564	0.829093
$677$	21.5139	0.826846	0.413423	−	0.910539i	$-0.364333\pi$
$677$	21.5139	0.826846	0.413423	+	0.910539i	$0.364333\pi$
$678$	0	0
$679$	17.0194	0.653145
$680$	4.25879	0.163317
$681$	0	0
$682$	−1.02444	−0.0392279
$683$	−47.6900	−1.82481	−0.912403	−	0.409293i	$-0.865775\pi$
$683$	−47.6900	−1.82481	−0.912403	+	0.409293i	$0.865775\pi$
$684$	0	0
$685$	−15.7958	−0.603526
$686$	−16.1761	−0.617606
$687$	0	0
$688$	−5.61968	−0.214248
$689$	75.8198	2.88851
$690$	0	0
$691$	30.5721	1.16302	0.581509	−	0.813540i	$-0.302462\pi$
$691$	30.5721	1.16302	0.581509	+	0.813540i	$0.302462\pi$
$692$	22.6937	0.862684
$693$	0	0
$694$	15.7569	0.598125
$695$	−11.4155	−0.433013
$696$	0	0
$697$	−37.8942	−1.43534
$698$	0.160365	0.00606992
$699$	0	0
$700$	−1.31955	−0.0498743
$701$	−32.0703	−1.21128	−0.605639	−	0.795740i	$-0.707082\pi$
$701$	−32.0703	−1.21128	−0.605639	+	0.795740i	$0.707082\pi$
$702$	0	0
$703$	−2.93923	−0.110855
$704$	0.258786	0.00975335
$705$	0	0
$706$	−30.2075	−1.13688
$707$	13.7181	0.515922
$708$	0	0
$709$	10.6160	0.398692	0.199346	−	0.979929i	$-0.436118\pi$
$709$	10.6160	0.398692	0.199346	+	0.979929i	$0.436118\pi$
$710$	11.2394	0.421806
$711$	0	0
$712$	6.00000	0.224860
$713$	33.7181	1.26275
$714$	0	0
$715$	−1.52126	−0.0568920
$716$	8.73501	0.326443
$717$	0	0
$718$	−7.39973	−0.276156
$719$	32.4663	1.21079	0.605395	−	0.795925i	$-0.293015\pi$
$719$	32.4663	1.21079	0.605395	+	0.795925i	$0.293015\pi$
$720$	0	0
$721$	18.3133	0.682025
$722$	10.3609	0.385593
$723$	0	0
$724$	10.6391	0.395399
$725$	3.61968	0.134432
$726$	0	0
$727$	31.5139	1.16879	0.584393	−	0.811471i	$-0.301333\pi$
$727$	31.5139	1.16879	0.584393	+	0.811471i	$0.301333\pi$
$728$	−7.75694	−0.287491
$729$	0	0
$730$	−2.51757	−0.0931795
$731$	23.9330	0.885195
$732$	0	0
$733$	−16.8202	−0.621269	−0.310634	−	0.950529i	$-0.600541\pi$
$733$	−16.8202	−0.621269	−0.310634	+	0.950529i	$0.600541\pi$
$734$	13.9198	0.513790
$735$	0	0
$736$	−8.51757	−0.313962
$737$	1.44359	0.0531755
$738$	0	0
$739$	9.69735	0.356723	0.178361	−	0.983965i	$-0.442920\pi$
$739$	9.69735	0.356723	0.178361	+	0.983965i	$0.442920\pi$
$740$	−1.00000	−0.0367607
$741$	0	0
$742$	−17.0194	−0.624802
$743$	39.1153	1.43500	0.717501	−	0.696557i	$-0.245286\pi$
$743$	39.1153	1.43500	0.717501	+	0.696557i	$0.245286\pi$
$744$	0	0
$745$	−18.3960	−0.673979
$746$	−20.2357	−0.740881
$747$	0	0
$748$	−1.10211	−0.0402972
$749$	−5.56512	−0.203345
$750$	0	0
$751$	−26.1530	−0.954336	−0.477168	−	0.878812i	$-0.658337\pi$
$751$	−26.1530	−0.954336	−0.477168	+	0.878812i	$0.658337\pi$
$752$	−5.57834	−0.203421
$753$	0	0
$754$	21.2782	0.774906
$755$	5.87847	0.213939
$756$	0	0
$757$	−33.5915	−1.22091	−0.610453	−	0.792053i	$-0.709013\pi$
$757$	−33.5915	−1.22091	−0.610453	+	0.792053i	$0.709013\pi$
$758$	27.9488	1.01514
$759$	0	0
$760$	−2.93923	−0.106617
$761$	51.3766	1.86240	0.931201	−	0.364507i	$-0.118763\pi$
$761$	51.3766	1.86240	0.931201	+	0.364507i	$0.118763\pi$
$762$	0	0
$763$	−21.2939	−0.770892
$764$	−22.1116	−0.799971
$765$	0	0
$766$	11.7569	0.424795
$767$	56.3060	2.03309
$768$	0	0
$769$	27.5527	0.993576	0.496788	−	0.867872i	$-0.334513\pi$
$769$	27.5527	0.993576	0.496788	+	0.867872i	$0.334513\pi$
$770$	0.341481	0.0123061
$771$	0	0
$772$	7.03514	0.253200
$773$	−13.4155	−0.482521	−0.241260	−	0.970460i	$-0.577561\pi$
$773$	−13.4155	−0.482521	−0.241260	+	0.970460i	$0.577561\pi$
$774$	0	0
$775$	3.95865	0.142199
$776$	12.8979	0.463007
$777$	0	0
$778$	−9.58085	−0.343490
$779$	26.1530	0.937028
$780$	0	0
$781$	−2.90859	−0.104077
$782$	36.2745	1.29717
$783$	0	0
$784$	−5.25879	−0.187814
$785$	3.61968	0.129192
$786$	0	0
$787$	−25.4956	−0.908821	−0.454411	−	0.890792i	$-0.650150\pi$
$787$	−25.4956	−0.908821	−0.454411	+	0.890792i	$0.650150\pi$
$788$	−25.7569	−0.917553
$789$	0	0
$790$	7.69987	0.273949
$791$	1.34517	0.0478289
$792$	0	0
$793$	−2.23568	−0.0793912
$794$	32.4787	1.15263
$795$	0	0
$796$	−12.7350	−0.451380
$797$	−48.4663	−1.71677	−0.858383	−	0.513010i	$-0.828530\pi$
$797$	−48.4663	−1.71677	−0.858383	+	0.513010i	$0.828530\pi$
$798$	0	0
$799$	23.7569	0.840460
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	7.27820	0.257002
$803$	0.651511	0.0229913
$804$	0	0
$805$	−11.2394	−0.396136
$806$	23.2708	0.819680
$807$	0	0
$808$	10.3960	0.365731
$809$	12.9649	0.455820	0.227910	−	0.973682i	$-0.426811\pi$
$809$	12.9649	0.455820	0.227910	+	0.973682i	$0.426811\pi$
$810$	0	0
$811$	0.0776702	0.00272737	0.00136368	−	0.999999i	$-0.499566\pi$
$811$	0.0776702	0.00272737	0.00136368	+	0.999999i	$0.499566\pi$
$812$	−4.77636	−0.167617
$813$	0	0
$814$	0.258786	0.00907043
$815$	−22.0546	−0.772538
$816$	0	0
$817$	−16.5176	−0.577877
$818$	32.9963	1.15369
$819$	0	0
$820$	8.89789	0.310728
$821$	−36.3448	−1.26844	−0.634221	−	0.773152i	$-0.718679\pi$
$821$	−36.3448	−1.26844	−0.634221	+	0.773152i	$0.718679\pi$
$822$	0	0
$823$	−23.0996	−0.805201	−0.402601	−	0.915376i	$-0.631894\pi$
$823$	−23.0996	−0.805201	−0.402601	+	0.915376i	$0.631894\pi$
$824$	13.8785	0.483479
$825$	0	0
$826$	−12.6391	−0.439771
$827$	−15.2551	−0.530472	−0.265236	−	0.964184i	$-0.585450\pi$
$827$	−15.2551	−0.530472	−0.265236	+	0.964184i	$0.585450\pi$
$828$	0	0
$829$	28.1373	0.977247	0.488624	−	0.872495i	$-0.337499\pi$
$829$	28.1373	0.977247	0.488624	+	0.872495i	$0.337499\pi$
$830$	−6.17860	−0.214462
$831$	0	0
$832$	−5.87847	−0.203799
$833$	22.3960	0.775977
$834$	0	0
$835$	0.600267	0.0207731
$836$	0.760632	0.0263070
$837$	0	0
$838$	25.7131	0.888244
$839$	−4.24306	−0.146487	−0.0732434	−	0.997314i	$-0.523335\pi$
$839$	−4.24306	−0.146487	−0.0732434	+	0.997314i	$0.523335\pi$
$840$	0	0
$841$	−15.8979	−0.548203
$842$	−22.0000	−0.758170
$843$	0	0
$844$	−6.65483	−0.229069
$845$	21.5564	0.741563
$846$	0	0
$847$	14.4267	0.495707
$848$	−12.8979	−0.442915
$849$	0	0
$850$	4.25879	0.146075
$851$	−8.51757	−0.291979
$852$	0	0
$853$	36.4787	1.24901	0.624504	−	0.781022i	$-0.285301\pi$
$853$	36.4787	1.24901	0.624504	+	0.781022i	$0.285301\pi$
$854$	0.501846	0.0171728
$855$	0	0
$856$	−4.21744	−0.144149
$857$	−23.7288	−0.810561	−0.405280	−	0.914192i	$-0.632826\pi$
$857$	−23.7288	−0.810561	−0.405280	+	0.914192i	$0.632826\pi$
$858$	0	0
$859$	2.77887	0.0948138	0.0474069	−	0.998876i	$-0.484904\pi$
$859$	2.77887	0.0948138	0.0474069	+	0.998876i	$0.484904\pi$
$860$	−5.61968	−0.191630
$861$	0	0
$862$	17.9198	0.610351
$863$	−10.5151	−0.357937	−0.178968	−	0.983855i	$-0.557276\pi$
$863$	−10.5151	−0.357937	−0.178968	+	0.983855i	$0.557276\pi$
$864$	0	0
$865$	22.6937	0.771608
$866$	−4.96486	−0.168713
$867$	0	0
$868$	−5.22364	−0.177302
$869$	−1.99262	−0.0675948
$870$	0	0
$871$	−32.7921	−1.11112
$872$	−16.1373	−0.546476
$873$	0	0
$874$	−25.0351	−0.846826
$875$	−1.31955	−0.0446090
$876$	0	0
$877$	27.6197	0.932650	0.466325	−	0.884613i	$-0.345578\pi$
$877$	27.6197	0.932650	0.466325	+	0.884613i	$0.345578\pi$
$878$	13.8371	0.466980
$879$	0	0
$880$	0.258786	0.00872366
$881$	−10.2200	−0.344319	−0.172159	−	0.985069i	$-0.555074\pi$
$881$	−10.2200	−0.344319	−0.172159	+	0.985069i	$0.555074\pi$
$882$	0	0
$883$	53.5684	1.80272	0.901361	−	0.433069i	$-0.142569\pi$
$883$	53.5684	1.80272	0.901361	+	0.433069i	$0.142569\pi$
$884$	25.0351	0.842023
$885$	0	0
$886$	14.3390	0.481727
$887$	−36.5904	−1.22858	−0.614292	−	0.789079i	$-0.710558\pi$
$887$	−36.5904	−1.22858	−0.614292	+	0.789079i	$0.710558\pi$
$888$	0	0
$889$	−9.83963	−0.330011
$890$	6.00000	0.201120
$891$	0	0
$892$	−16.5589	−0.554434
$893$	−16.3960	−0.548673
$894$	0	0
$895$	8.73501	0.291979
$896$	1.31955	0.0440831
$897$	0	0
$898$	12.4787	0.416421
$899$	14.3291	0.477901
$900$	0	0
$901$	54.9293	1.82996
$902$	−2.30265	−0.0766697
$903$	0	0
$904$	1.01942	0.0339053
$905$	10.6391	0.353656
$906$	0	0
$907$	7.64279	0.253775	0.126887	−	0.991917i	$-0.459501\pi$
$907$	7.64279	0.253775	0.126887	+	0.991917i	$0.459501\pi$
$908$	4.01573	0.133267
$909$	0	0
$910$	−7.75694	−0.257140
$911$	−22.0132	−0.729330	−0.364665	−	0.931139i	$-0.618817\pi$
$911$	−22.0132	−0.729330	−0.364665	+	0.931139i	$0.618817\pi$
$912$	0	0
$913$	1.59893	0.0529170
$914$	−21.6585	−0.716400
$915$	0	0
$916$	25.5139	0.843002
$917$	3.19551	0.105525
$918$	0	0
$919$	23.6999	0.781786	0.390893	−	0.920436i	$-0.372166\pi$
$919$	23.6999	0.781786	0.390893	+	0.920436i	$0.372166\pi$
$920$	−8.51757	−0.280816
$921$	0	0
$922$	−29.4155	−0.968747
$923$	66.0703	2.17473
$924$	0	0
$925$	−1.00000	−0.0328798
$926$	9.03514	0.296913
$927$	0	0
$928$	−3.61968	−0.118822
$929$	−8.82022	−0.289382	−0.144691	−	0.989477i	$-0.546219\pi$
$929$	−8.82022	−0.289382	−0.144691	+	0.989477i	$0.546219\pi$
$930$	0	0
$931$	−15.4568	−0.506576
$932$	−7.96116	−0.260777
$933$	0	0
$934$	7.05825	0.230953
$935$	−1.10211	−0.0360429
$936$	0	0
$937$	40.0703	1.30904	0.654520	−	0.756045i	$-0.272871\pi$
$937$	40.0703	1.30904	0.654520	+	0.756045i	$0.272871\pi$
$938$	7.36090	0.240342
$939$	0	0
$940$	−5.57834	−0.181945
$941$	31.4312	1.02463	0.512314	−	0.858798i	$-0.328789\pi$
$941$	31.4312	1.02463	0.512314	+	0.858798i	$0.328789\pi$
$942$	0	0
$943$	75.7884	2.46801
$944$	−9.57834	−0.311748
$945$	0	0
$946$	1.45429	0.0472832
$947$	53.5370	1.73972	0.869859	−	0.493300i	$-0.164210\pi$
$947$	53.5370	1.73972	0.869859	+	0.493300i	$0.164210\pi$
$948$	0	0
$949$	−14.7995	−0.480411
$950$	−2.93923	−0.0953614
$951$	0	0
$952$	−5.61968	−0.182135
$953$	−5.23937	−0.169720	−0.0848599	−	0.996393i	$-0.527044\pi$
$953$	−5.23937	−0.169720	−0.0848599	+	0.996393i	$0.527044\pi$
$954$	0	0
$955$	−22.1116	−0.715516
$956$	−11.7983	−0.381584
$957$	0	0
$958$	−29.3353	−0.947780
$959$	20.8433	0.673066
$960$	0	0
$961$	−15.3291	−0.494486
$962$	−5.87847	−0.189529
$963$	0	0
$964$	24.9963	0.805077
$965$	7.03514	0.226469
$966$	0	0
$967$	−46.5929	−1.49833	−0.749163	−	0.662386i	$-0.769544\pi$
$967$	−46.5929	−1.49833	−0.749163	+	0.662386i	$0.769544\pi$
$968$	10.9330	0.351401
$969$	0	0
$970$	12.8979	0.414126
$971$	−38.1373	−1.22388	−0.611941	−	0.790903i	$-0.709611\pi$
$971$	−38.1373	−1.22388	−0.611941	+	0.790903i	$0.709611\pi$
$972$	0	0
$973$	15.0633	0.482907
$974$	37.9099	1.21471
$975$	0	0
$976$	0.380316	0.0121736
$977$	−40.8979	−1.30844	−0.654220	−	0.756305i	$-0.727003\pi$
$977$	−40.8979	−1.30844	−0.654220	+	0.756305i	$0.727003\pi$
$978$	0	0
$979$	−1.55271	−0.0496250
$980$	−5.25879	−0.167986
$981$	0	0
$982$	−14.5564	−0.464514
$983$	−50.4638	−1.60955	−0.804773	−	0.593583i	$-0.797713\pi$
$983$	−50.4638	−1.60955	−0.804773	+	0.593583i	$0.797713\pi$
$984$	0	0
$985$	−25.7569	−0.820684
$986$	15.4155	0.490928
$987$	0	0
$988$	−17.2782	−0.549693
$989$	−47.8661	−1.52205
$990$	0	0
$991$	−15.9587	−0.506943	−0.253472	−	0.967343i	$-0.581572\pi$
$991$	−15.9587	−0.506943	−0.253472	+	0.967343i	$0.581572\pi$
$992$	−3.95865	−0.125687
$993$	0	0
$994$	−14.8309	−0.470408
$995$	−12.7350	−0.403727
$996$	0	0
$997$	−9.23937	−0.292614	−0.146307	−	0.989239i	$-0.546739\pi$
$997$	−9.23937	−0.292614	−0.146307	+	0.989239i	$0.546739\pi$
$998$	−23.9744	−0.758896
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3330.2.a.bg.1.2		3
3.2	odd	2		370.2.a.g.1.3	✓	3
12.11	even	2		2960.2.a.u.1.1		3
15.2	even	4		1850.2.b.o.149.4		6
15.8	even	4		1850.2.b.o.149.3		6
15.14	odd	2		1850.2.a.z.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
370.2.a.g.1.3	✓	3	3.2	odd	2
1850.2.a.z.1.1		3	15.14	odd	2
1850.2.b.o.149.3		6	15.8	even	4
1850.2.b.o.149.4		6	15.2	even	4
2960.2.a.u.1.1		3	12.11	even	2
3330.2.a.bg.1.2		3	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:	\( N \)	\(=\)	\( 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3330.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.31955 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.31955 q^{7} -1.00000 q^{8} -1.00000 q^{10} +0.258786 q^{11} -5.87847 q^{13} +1.31955 q^{14} +1.00000 q^{16} -4.25879 q^{17} +2.93923 q^{19} +1.00000 q^{20} -0.258786 q^{22} +8.51757 q^{23} +1.00000 q^{25} +5.87847 q^{26} -1.31955 q^{28} +3.61968 q^{29} +3.95865 q^{31} -1.00000 q^{32} +4.25879 q^{34} -1.31955 q^{35} -1.00000 q^{37} -2.93923 q^{38} -1.00000 q^{40} +8.89789 q^{41} -5.61968 q^{43} +0.258786 q^{44} -8.51757 q^{46} -5.57834 q^{47} -5.25879 q^{49} -1.00000 q^{50} -5.87847 q^{52} -12.8979 q^{53} +0.258786 q^{55} +1.31955 q^{56} -3.61968 q^{58} -9.57834 q^{59} +0.380316 q^{61} -3.95865 q^{62} +1.00000 q^{64} -5.87847 q^{65} +5.57834 q^{67} -4.25879 q^{68} +1.31955 q^{70} -11.2394 q^{71} +2.51757 q^{73} +1.00000 q^{74} +2.93923 q^{76} -0.341481 q^{77} -7.69987 q^{79} +1.00000 q^{80} -8.89789 q^{82} +6.17860 q^{83} -4.25879 q^{85} +5.61968 q^{86} -0.258786 q^{88} -6.00000 q^{89} +7.75694 q^{91} +8.51757 q^{92} +5.57834 q^{94} +2.93923 q^{95} -12.8979 q^{97} +5.25879 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} - q^{7} - 3 q^{8}+O(q^{10}) \) 3 * q - 3 * q^2 + 3 * q^4 + 3 * q^5 - q^7 - 3 * q^8	\( 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} - q^{7} - 3 q^{8} - 3 q^{10} - 11 q^{11} + q^{14} + 3 q^{16} - q^{17} + 3 q^{20} + 11 q^{22} + 2 q^{23} + 3 q^{25} - q^{28} + 5 q^{29} + 3 q^{31} - 3 q^{32} + q^{34} - q^{35} - 3 q^{37} - 3 q^{40} + 9 q^{41} - 11 q^{43} - 11 q^{44} - 2 q^{46} - 2 q^{47} - 4 q^{49} - 3 q^{50} - 21 q^{53} - 11 q^{55} + q^{56} - 5 q^{58} - 14 q^{59} + 7 q^{61} - 3 q^{62} + 3 q^{64} + 2 q^{67} - q^{68} + q^{70} - 22 q^{71} - 16 q^{73} + 3 q^{74} - 7 q^{77} - 26 q^{79} + 3 q^{80} - 9 q^{82} - 2 q^{83} - q^{85} + 11 q^{86} + 11 q^{88} - 18 q^{89} - 12 q^{91} + 2 q^{92} + 2 q^{94} - 21 q^{97} + 4 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^4 + 3 * q^5 - q^7 - 3 * q^8 - 3 * q^10 - 11 * q^11 + q^14 + 3 * q^16 - q^17 + 3 * q^20 + 11 * q^22 + 2 * q^23 + 3 * q^25 - q^28 + 5 * q^29 + 3 * q^31 - 3 * q^32 + q^34 - q^35 - 3 * q^37 - 3 * q^40 + 9 * q^41 - 11 * q^43 - 11 * q^44 - 2 * q^46 - 2 * q^47 - 4 * q^49 - 3 * q^50 - 21 * q^53 - 11 * q^55 + q^56 - 5 * q^58 - 14 * q^59 + 7 * q^61 - 3 * q^62 + 3 * q^64 + 2 * q^67 - q^68 + q^70 - 22 * q^71 - 16 * q^73 + 3 * q^74 - 7 * q^77 - 26 * q^79 + 3 * q^80 - 9 * q^82 - 2 * q^83 - q^85 + 11 * q^86 + 11 * q^88 - 18 * q^89 - 12 * q^91 + 2 * q^92 + 2 * q^94 - 21 * q^97 + 4 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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