LMFDB - Embedded newform 2550.2.a.bh.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2550,2,Mod(1,2550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2550, 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("2550.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2550.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:	$20.3618525154$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 510)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	2550.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -3.23607 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{11} +1.00000 q^{12} -0.763932 q^{13} +3.23607 q^{14} +1.00000 q^{16} -1.00000 q^{17} -1.00000 q^{18} +2.47214 q^{19} -3.23607 q^{21} -2.00000 q^{22} -4.00000 q^{23} -1.00000 q^{24} +0.763932 q^{26} +1.00000 q^{27} -3.23607 q^{28} -4.00000 q^{29} -6.47214 q^{31} -1.00000 q^{32} +2.00000 q^{33} +1.00000 q^{34} +1.00000 q^{36} -2.00000 q^{37} -2.47214 q^{38} -0.763932 q^{39} +5.70820 q^{41} +3.23607 q^{42} +10.1803 q^{43} +2.00000 q^{44} +4.00000 q^{46} -1.52786 q^{47} +1.00000 q^{48} +3.47214 q^{49} -1.00000 q^{51} -0.763932 q^{52} -6.94427 q^{53} -1.00000 q^{54} +3.23607 q^{56} +2.47214 q^{57} +4.00000 q^{58} -1.70820 q^{59} -4.47214 q^{61} +6.47214 q^{62} -3.23607 q^{63} +1.00000 q^{64} -2.00000 q^{66} -11.7082 q^{67} -1.00000 q^{68} -4.00000 q^{69} +6.76393 q^{71} -1.00000 q^{72} -13.2361 q^{73} +2.00000 q^{74} +2.47214 q^{76} -6.47214 q^{77} +0.763932 q^{78} -16.9443 q^{79} +1.00000 q^{81} -5.70820 q^{82} -1.52786 q^{83} -3.23607 q^{84} -10.1803 q^{86} -4.00000 q^{87} -2.00000 q^{88} +3.52786 q^{89} +2.47214 q^{91} -4.00000 q^{92} -6.47214 q^{93} +1.52786 q^{94} -1.00000 q^{96} -11.7082 q^{97} -3.47214 q^{98} +2.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 2 q^{7} - 2 q^{8} + 2 q^{9} + 4 q^{11} + 2 q^{12} - 6 q^{13} + 2 q^{14} + 2 q^{16} - 2 q^{17} - 2 q^{18} - 4 q^{19} - 2 q^{21} - 4 q^{22} - 8 q^{23} - 2 q^{24}+ \cdots + 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 - 2 * q^7 - 2 * q^8 + 2 * q^9 + 4 * q^11 + 2 * q^12 - 6 * q^13 + 2 * q^14 + 2 * q^16 - 2 * q^17 - 2 * q^18 - 4 * q^19 - 2 * q^21 - 4 * q^22 - 8 * q^23 - 2 * q^24 + 6 * q^26 + 2 * q^27 - 2 * q^28 - 8 * q^29 - 4 * q^31 - 2 * q^32 + 4 * q^33 + 2 * q^34 + 2 * q^36 - 4 * q^37 + 4 * q^38 - 6 * q^39 - 2 * q^41 + 2 * q^42 - 2 * q^43 + 4 * q^44 + 8 * q^46 - 12 * q^47 + 2 * q^48 - 2 * q^49 - 2 * q^51 - 6 * q^52 + 4 * q^53 - 2 * q^54 + 2 * q^56 - 4 * q^57 + 8 * q^58 + 10 * q^59 + 4 * q^62 - 2 * q^63 + 2 * q^64 - 4 * q^66 - 10 * q^67 - 2 * q^68 - 8 * q^69 + 18 * q^71 - 2 * q^72 - 22 * q^73 + 4 * q^74 - 4 * q^76 - 4 * q^77 + 6 * q^78 - 16 * q^79 + 2 * q^81 + 2 * q^82 - 12 * q^83 - 2 * q^84 + 2 * q^86 - 8 * q^87 - 4 * q^88 + 16 * q^89 - 4 * q^91 - 8 * q^92 - 4 * q^93 + 12 * q^94 - 2 * q^96 - 10 * q^97 + 2 * q^98 + 4 * q^99

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	−3.23607	−1.22312	−0.611559	−	0.791199i	$-0.709457\pi$
$7$	−3.23607	−1.22312	−0.611559	+	0.791199i	$0.709457\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	1.00000	0.288675
$13$	−0.763932	−0.211877	−0.105938	−	0.994373i	$-0.533785\pi$
$13$	−0.763932	−0.211877	−0.105938	+	0.994373i	$0.533785\pi$
$14$	3.23607	0.864876
$15$	0	0
$16$	1.00000	0.250000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	−1.00000	−0.235702
$19$	2.47214	0.567147	0.283573	−	0.958951i	$-0.408480\pi$
$19$	2.47214	0.567147	0.283573	+	0.958951i	$0.408480\pi$
$20$	0	0
$21$	−3.23607	−0.706168
$22$	−2.00000	−0.426401
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	0.763932	0.149819
$27$	1.00000	0.192450
$28$	−3.23607	−0.611559
$29$	−4.00000	−0.742781	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000	−0.742781	−0.371391	+	0.928477i	$0.621119\pi$
$30$	0	0
$31$	−6.47214	−1.16243	−0.581215	−	0.813750i	$-0.697422\pi$
$31$	−6.47214	−1.16243	−0.581215	+	0.813750i	$0.697422\pi$
$32$	−1.00000	−0.176777
$33$	2.00000	0.348155
$34$	1.00000	0.171499
$35$	0	0
$36$	1.00000	0.166667
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	−2.47214	−0.401033
$39$	−0.763932	−0.122327
$40$	0	0
$41$	5.70820	0.891472	0.445736	−	0.895165i	$-0.352942\pi$
$41$	5.70820	0.891472	0.445736	+	0.895165i	$0.352942\pi$
$42$	3.23607	0.499336
$43$	10.1803	1.55249	0.776244	−	0.630433i	$-0.217123\pi$
$43$	10.1803	1.55249	0.776244	+	0.630433i	$0.217123\pi$
$44$	2.00000	0.301511
$45$	0	0
$46$	4.00000	0.589768
$47$	−1.52786	−0.222862	−0.111431	−	0.993772i	$-0.535543\pi$
$47$	−1.52786	−0.222862	−0.111431	+	0.993772i	$0.535543\pi$
$48$	1.00000	0.144338
$49$	3.47214	0.496019
$50$	0	0
$51$	−1.00000	−0.140028
$52$	−0.763932	−0.105938
$53$	−6.94427	−0.953869	−0.476935	−	0.878939i	$-0.658252\pi$
$53$	−6.94427	−0.953869	−0.476935	+	0.878939i	$0.658252\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	3.23607	0.432438
$57$	2.47214	0.327442
$58$	4.00000	0.525226
$59$	−1.70820	−0.222389	−0.111195	−	0.993799i	$-0.535468\pi$
$59$	−1.70820	−0.222389	−0.111195	+	0.993799i	$0.535468\pi$
$60$	0	0
$61$	−4.47214	−0.572598	−0.286299	−	0.958140i	$-0.592425\pi$
$61$	−4.47214	−0.572598	−0.286299	+	0.958140i	$0.592425\pi$
$62$	6.47214	0.821962
$63$	−3.23607	−0.407706
$64$	1.00000	0.125000
$65$	0	0
$66$	−2.00000	−0.246183
$67$	−11.7082	−1.43038	−0.715192	−	0.698928i	$-0.753661\pi$
$67$	−11.7082	−1.43038	−0.715192	+	0.698928i	$0.753661\pi$
$68$	−1.00000	−0.121268
$69$	−4.00000	−0.481543
$70$	0	0
$71$	6.76393	0.802731	0.401366	−	0.915918i	$-0.368536\pi$
$71$	6.76393	0.802731	0.401366	+	0.915918i	$0.368536\pi$
$72$	−1.00000	−0.117851
$73$	−13.2361	−1.54916	−0.774582	−	0.632473i	$-0.782040\pi$
$73$	−13.2361	−1.54916	−0.774582	+	0.632473i	$0.782040\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	2.47214	0.283573
$77$	−6.47214	−0.737568
$78$	0.763932	0.0864983
$79$	−16.9443	−1.90638	−0.953190	−	0.302373i	$-0.902221\pi$
$79$	−16.9443	−1.90638	−0.953190	+	0.302373i	$0.902221\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−5.70820	−0.630366
$83$	−1.52786	−0.167705	−0.0838524	−	0.996478i	$-0.526722\pi$
$83$	−1.52786	−0.167705	−0.0838524	+	0.996478i	$0.526722\pi$
$84$	−3.23607	−0.353084
$85$	0	0
$86$	−10.1803	−1.09777
$87$	−4.00000	−0.428845
$88$	−2.00000	−0.213201
$89$	3.52786	0.373953	0.186976	−	0.982364i	$-0.440131\pi$
$89$	3.52786	0.373953	0.186976	+	0.982364i	$0.440131\pi$
$90$	0	0
$91$	2.47214	0.259150
$92$	−4.00000	−0.417029
$93$	−6.47214	−0.671129
$94$	1.52786	0.157587
$95$	0	0
$96$	−1.00000	−0.102062
$97$	−11.7082	−1.18879	−0.594394	−	0.804174i	$-0.702608\pi$
$97$	−11.7082	−1.18879	−0.594394	+	0.804174i	$0.702608\pi$
$98$	−3.47214	−0.350739
$99$	2.00000	0.201008
$100$	0	0
$101$	11.7082	1.16501	0.582505	−	0.812827i	$-0.302073\pi$
$101$	11.7082	1.16501	0.582505	+	0.812827i	$0.302073\pi$
$102$	1.00000	0.0990148
$103$	6.94427	0.684239	0.342120	−	0.939656i	$-0.388855\pi$
$103$	6.94427	0.684239	0.342120	+	0.939656i	$0.388855\pi$
$104$	0.763932	0.0749097
$105$	0	0
$106$	6.94427	0.674487
$107$	−8.00000	−0.773389	−0.386695	−	0.922208i	$-0.626383\pi$
$107$	−8.00000	−0.773389	−0.386695	+	0.922208i	$0.626383\pi$
$108$	1.00000	0.0962250
$109$	−16.4721	−1.57774	−0.788872	−	0.614557i	$-0.789335\pi$
$109$	−16.4721	−1.57774	−0.788872	+	0.614557i	$0.789335\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	−3.23607	−0.305780
$113$	10.0000	0.940721	0.470360	−	0.882474i	$-0.344124\pi$
$113$	10.0000	0.940721	0.470360	+	0.882474i	$0.344124\pi$
$114$	−2.47214	−0.231537
$115$	0	0
$116$	−4.00000	−0.371391
$117$	−0.763932	−0.0706255
$118$	1.70820	0.157253
$119$	3.23607	0.296650
$120$	0	0
$121$	−7.00000	−0.636364
$122$	4.47214	0.404888
$123$	5.70820	0.514691
$124$	−6.47214	−0.581215
$125$	0	0
$126$	3.23607	0.288292
$127$	−14.0000	−1.24230	−0.621150	−	0.783692i	$-0.713334\pi$
$127$	−14.0000	−1.24230	−0.621150	+	0.783692i	$0.713334\pi$
$128$	−1.00000	−0.0883883
$129$	10.1803	0.896329
$130$	0	0
$131$	3.52786	0.308231	0.154115	−	0.988053i	$-0.450747\pi$
$131$	3.52786	0.308231	0.154115	+	0.988053i	$0.450747\pi$
$132$	2.00000	0.174078
$133$	−8.00000	−0.693688
$134$	11.7082	1.01143
$135$	0	0
$136$	1.00000	0.0857493
$137$	2.00000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	4.00000	0.340503
$139$	10.4721	0.888235	0.444117	−	0.895969i	$-0.353517\pi$
$139$	10.4721	0.888235	0.444117	+	0.895969i	$0.353517\pi$
$140$	0	0
$141$	−1.52786	−0.128669
$142$	−6.76393	−0.567617
$143$	−1.52786	−0.127766
$144$	1.00000	0.0833333
$145$	0	0
$146$	13.2361	1.09542
$147$	3.47214	0.286377
$148$	−2.00000	−0.164399
$149$	−10.1803	−0.834006	−0.417003	−	0.908905i	$-0.636920\pi$
$149$	−10.1803	−0.834006	−0.417003	+	0.908905i	$0.636920\pi$
$150$	0	0
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	−2.47214	−0.200517
$153$	−1.00000	−0.0808452
$154$	6.47214	0.521540
$155$	0	0
$156$	−0.763932	−0.0611635
$157$	12.1803	0.972097	0.486048	−	0.873932i	$-0.338438\pi$
$157$	12.1803	0.972097	0.486048	+	0.873932i	$0.338438\pi$
$158$	16.9443	1.34801
$159$	−6.94427	−0.550717
$160$	0	0
$161$	12.9443	1.02015
$162$	−1.00000	−0.0785674
$163$	13.5279	1.05958	0.529792	−	0.848128i	$-0.322270\pi$
$163$	13.5279	1.05958	0.529792	+	0.848128i	$0.322270\pi$
$164$	5.70820	0.445736
$165$	0	0
$166$	1.52786	0.118585
$167$	−21.8885	−1.69379	−0.846893	−	0.531763i	$-0.821530\pi$
$167$	−21.8885	−1.69379	−0.846893	+	0.531763i	$0.821530\pi$
$168$	3.23607	0.249668
$169$	−12.4164	−0.955108
$170$	0	0
$171$	2.47214	0.189049
$172$	10.1803	0.776244
$173$	−19.8885	−1.51210	−0.756049	−	0.654515i	$-0.772873\pi$
$173$	−19.8885	−1.51210	−0.756049	+	0.654515i	$0.772873\pi$
$174$	4.00000	0.303239
$175$	0	0
$176$	2.00000	0.150756
$177$	−1.70820	−0.128396
$178$	−3.52786	−0.264425
$179$	5.70820	0.426651	0.213326	−	0.976981i	$-0.431570\pi$
$179$	5.70820	0.426651	0.213326	+	0.976981i	$0.431570\pi$
$180$	0	0
$181$	−8.47214	−0.629729	−0.314864	−	0.949137i	$-0.601959\pi$
$181$	−8.47214	−0.629729	−0.314864	+	0.949137i	$0.601959\pi$
$182$	−2.47214	−0.183247
$183$	−4.47214	−0.330590
$184$	4.00000	0.294884
$185$	0	0
$186$	6.47214	0.474560
$187$	−2.00000	−0.146254
$188$	−1.52786	−0.111431
$189$	−3.23607	−0.235389
$190$	0	0
$191$	14.4721	1.04717	0.523584	−	0.851974i	$-0.324595\pi$
$191$	14.4721	1.04717	0.523584	+	0.851974i	$0.324595\pi$
$192$	1.00000	0.0721688
$193$	−25.5967	−1.84249	−0.921247	−	0.388978i	$-0.872828\pi$
$193$	−25.5967	−1.84249	−0.921247	+	0.388978i	$0.872828\pi$
$194$	11.7082	0.840600
$195$	0	0
$196$	3.47214	0.248010
$197$	19.8885	1.41700	0.708500	−	0.705711i	$-0.249372\pi$
$197$	19.8885	1.41700	0.708500	+	0.705711i	$0.249372\pi$
$198$	−2.00000	−0.142134
$199$	8.94427	0.634043	0.317021	−	0.948418i	$-0.397317\pi$
$199$	8.94427	0.634043	0.317021	+	0.948418i	$0.397317\pi$
$200$	0	0
$201$	−11.7082	−0.825833
$202$	−11.7082	−0.823786
$203$	12.9443	0.908510
$204$	−1.00000	−0.0700140
$205$	0	0
$206$	−6.94427	−0.483830
$207$	−4.00000	−0.278019
$208$	−0.763932	−0.0529692
$209$	4.94427	0.342002
$210$	0	0
$211$	−7.05573	−0.485736	−0.242868	−	0.970059i	$-0.578088\pi$
$211$	−7.05573	−0.485736	−0.242868	+	0.970059i	$0.578088\pi$
$212$	−6.94427	−0.476935
$213$	6.76393	0.463457
$214$	8.00000	0.546869
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	20.9443	1.42179
$218$	16.4721	1.11563
$219$	−13.2361	−0.894411
$220$	0	0
$221$	0.763932	0.0513876
$222$	2.00000	0.134231
$223$	−26.9443	−1.80432	−0.902161	−	0.431400i	$-0.858020\pi$
$223$	−26.9443	−1.80432	−0.902161	+	0.431400i	$0.858020\pi$
$224$	3.23607	0.216219
$225$	0	0
$226$	−10.0000	−0.665190
$227$	25.8885	1.71828	0.859142	−	0.511738i	$-0.170998\pi$
$227$	25.8885	1.71828	0.859142	+	0.511738i	$0.170998\pi$
$228$	2.47214	0.163721
$229$	5.41641	0.357926	0.178963	−	0.983856i	$-0.442726\pi$
$229$	5.41641	0.357926	0.178963	+	0.983856i	$0.442726\pi$
$230$	0	0
$231$	−6.47214	−0.425835
$232$	4.00000	0.262613
$233$	21.4164	1.40304	0.701518	−	0.712652i	$-0.252506\pi$
$233$	21.4164	1.40304	0.701518	+	0.712652i	$0.252506\pi$
$234$	0.763932	0.0499398
$235$	0	0
$236$	−1.70820	−0.111195
$237$	−16.9443	−1.10065
$238$	−3.23607	−0.209763
$239$	13.8885	0.898375	0.449188	−	0.893437i	$-0.351713\pi$
$239$	13.8885	0.898375	0.449188	+	0.893437i	$0.351713\pi$
$240$	0	0
$241$	−17.4164	−1.12189	−0.560945	−	0.827853i	$-0.689562\pi$
$241$	−17.4164	−1.12189	−0.560945	+	0.827853i	$0.689562\pi$
$242$	7.00000	0.449977
$243$	1.00000	0.0641500
$244$	−4.47214	−0.286299
$245$	0	0
$246$	−5.70820	−0.363942
$247$	−1.88854	−0.120165
$248$	6.47214	0.410981
$249$	−1.52786	−0.0968244
$250$	0	0
$251$	12.1803	0.768816	0.384408	−	0.923163i	$-0.374406\pi$
$251$	12.1803	0.768816	0.384408	+	0.923163i	$0.374406\pi$
$252$	−3.23607	−0.203853
$253$	−8.00000	−0.502956
$254$	14.0000	0.878438
$255$	0	0
$256$	1.00000	0.0625000
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	−10.1803	−0.633800
$259$	6.47214	0.402159
$260$	0	0
$261$	−4.00000	−0.247594
$262$	−3.52786	−0.217952
$263$	4.00000	0.246651	0.123325	−	0.992366i	$-0.460644\pi$
$263$	4.00000	0.246651	0.123325	+	0.992366i	$0.460644\pi$
$264$	−2.00000	−0.123091
$265$	0	0
$266$	8.00000	0.490511
$267$	3.52786	0.215902
$268$	−11.7082	−0.715192
$269$	−24.9443	−1.52088	−0.760440	−	0.649409i	$-0.775016\pi$
$269$	−24.9443	−1.52088	−0.760440	+	0.649409i	$0.775016\pi$
$270$	0	0
$271$	4.00000	0.242983	0.121491	−	0.992592i	$-0.461232\pi$
$271$	4.00000	0.242983	0.121491	+	0.992592i	$0.461232\pi$
$272$	−1.00000	−0.0606339
$273$	2.47214	0.149620
$274$	−2.00000	−0.120824
$275$	0	0
$276$	−4.00000	−0.240772
$277$	18.3607	1.10319	0.551593	−	0.834113i	$-0.314020\pi$
$277$	18.3607	1.10319	0.551593	+	0.834113i	$0.314020\pi$
$278$	−10.4721	−0.628077
$279$	−6.47214	−0.387477
$280$	0	0
$281$	−25.4164	−1.51622	−0.758108	−	0.652129i	$-0.773876\pi$
$281$	−25.4164	−1.51622	−0.758108	+	0.652129i	$0.773876\pi$
$282$	1.52786	0.0909830
$283$	−9.88854	−0.587813	−0.293906	−	0.955834i	$-0.594955\pi$
$283$	−9.88854	−0.587813	−0.293906	+	0.955834i	$0.594955\pi$
$284$	6.76393	0.401366
$285$	0	0
$286$	1.52786	0.0903445
$287$	−18.4721	−1.09038
$288$	−1.00000	−0.0589256
$289$	1.00000	0.0588235
$290$	0	0
$291$	−11.7082	−0.686347
$292$	−13.2361	−0.774582
$293$	−2.58359	−0.150935	−0.0754675	−	0.997148i	$-0.524045\pi$
$293$	−2.58359	−0.150935	−0.0754675	+	0.997148i	$0.524045\pi$
$294$	−3.47214	−0.202499
$295$	0	0
$296$	2.00000	0.116248
$297$	2.00000	0.116052
$298$	10.1803	0.589731
$299$	3.05573	0.176717
$300$	0	0
$301$	−32.9443	−1.89888
$302$	4.00000	0.230174
$303$	11.7082	0.672619
$304$	2.47214	0.141787
$305$	0	0
$306$	1.00000	0.0571662
$307$	18.1803	1.03761	0.518803	−	0.854894i	$-0.326378\pi$
$307$	18.1803	1.03761	0.518803	+	0.854894i	$0.326378\pi$
$308$	−6.47214	−0.368784
$309$	6.94427	0.395046
$310$	0	0
$311$	−29.5967	−1.67828	−0.839139	−	0.543917i	$-0.816941\pi$
$311$	−29.5967	−1.67828	−0.839139	+	0.543917i	$0.816941\pi$
$312$	0.763932	0.0432491
$313$	8.29180	0.468680	0.234340	−	0.972155i	$-0.424707\pi$
$313$	8.29180	0.468680	0.234340	+	0.972155i	$0.424707\pi$
$314$	−12.1803	−0.687376
$315$	0	0
$316$	−16.9443	−0.953190
$317$	23.8885	1.34171	0.670857	−	0.741587i	$-0.265926\pi$
$317$	23.8885	1.34171	0.670857	+	0.741587i	$0.265926\pi$
$318$	6.94427	0.389415
$319$	−8.00000	−0.447914
$320$	0	0
$321$	−8.00000	−0.446516
$322$	−12.9443	−0.721356
$323$	−2.47214	−0.137553
$324$	1.00000	0.0555556
$325$	0	0
$326$	−13.5279	−0.749239
$327$	−16.4721	−0.910911
$328$	−5.70820	−0.315183
$329$	4.94427	0.272587
$330$	0	0
$331$	23.4164	1.28708	0.643541	−	0.765412i	$-0.277465\pi$
$331$	23.4164	1.28708	0.643541	+	0.765412i	$0.277465\pi$
$332$	−1.52786	−0.0838524
$333$	−2.00000	−0.109599
$334$	21.8885	1.19769
$335$	0	0
$336$	−3.23607	−0.176542
$337$	−2.18034	−0.118771	−0.0593853	−	0.998235i	$-0.518914\pi$
$337$	−2.18034	−0.118771	−0.0593853	+	0.998235i	$0.518914\pi$
$338$	12.4164	0.675364
$339$	10.0000	0.543125
$340$	0	0
$341$	−12.9443	−0.700972
$342$	−2.47214	−0.133678
$343$	11.4164	0.616428
$344$	−10.1803	−0.548887
$345$	0	0
$346$	19.8885	1.06921
$347$	−30.8328	−1.65519	−0.827596	−	0.561324i	$-0.810292\pi$
$347$	−30.8328	−1.65519	−0.827596	+	0.561324i	$0.810292\pi$
$348$	−4.00000	−0.214423
$349$	−19.5279	−1.04530	−0.522651	−	0.852547i	$-0.675057\pi$
$349$	−19.5279	−1.04530	−0.522651	+	0.852547i	$0.675057\pi$
$350$	0	0
$351$	−0.763932	−0.0407757
$352$	−2.00000	−0.106600
$353$	−2.94427	−0.156708	−0.0783539	−	0.996926i	$-0.524966\pi$
$353$	−2.94427	−0.156708	−0.0783539	+	0.996926i	$0.524966\pi$
$354$	1.70820	0.0907900
$355$	0	0
$356$	3.52786	0.186976
$357$	3.23607	0.171271
$358$	−5.70820	−0.301688
$359$	−18.4721	−0.974922	−0.487461	−	0.873145i	$-0.662077\pi$
$359$	−18.4721	−0.974922	−0.487461	+	0.873145i	$0.662077\pi$
$360$	0	0
$361$	−12.8885	−0.678344
$362$	8.47214	0.445286
$363$	−7.00000	−0.367405
$364$	2.47214	0.129575
$365$	0	0
$366$	4.47214	0.233762
$367$	28.1803	1.47100	0.735501	−	0.677524i	$-0.236947\pi$
$367$	28.1803	1.47100	0.735501	+	0.677524i	$0.236947\pi$
$368$	−4.00000	−0.208514
$369$	5.70820	0.297157
$370$	0	0
$371$	22.4721	1.16670
$372$	−6.47214	−0.335565
$373$	−3.23607	−0.167557	−0.0837786	−	0.996484i	$-0.526699\pi$
$373$	−3.23607	−0.167557	−0.0837786	+	0.996484i	$0.526699\pi$
$374$	2.00000	0.103418
$375$	0	0
$376$	1.52786	0.0787936
$377$	3.05573	0.157378
$378$	3.23607	0.166445
$379$	−10.4721	−0.537917	−0.268959	−	0.963152i	$-0.586680\pi$
$379$	−10.4721	−0.537917	−0.268959	+	0.963152i	$0.586680\pi$
$380$	0	0
$381$	−14.0000	−0.717242
$382$	−14.4721	−0.740459
$383$	17.8885	0.914062	0.457031	−	0.889451i	$-0.348913\pi$
$383$	17.8885	0.914062	0.457031	+	0.889451i	$0.348913\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	25.5967	1.30284
$387$	10.1803	0.517496
$388$	−11.7082	−0.594394
$389$	−2.18034	−0.110548	−0.0552738	−	0.998471i	$-0.517603\pi$
$389$	−2.18034	−0.110548	−0.0552738	+	0.998471i	$0.517603\pi$
$390$	0	0
$391$	4.00000	0.202289
$392$	−3.47214	−0.175369
$393$	3.52786	0.177957
$394$	−19.8885	−1.00197
$395$	0	0
$396$	2.00000	0.100504
$397$	31.8885	1.60044	0.800220	−	0.599706i	$-0.204716\pi$
$397$	31.8885	1.60044	0.800220	+	0.599706i	$0.204716\pi$
$398$	−8.94427	−0.448336
$399$	−8.00000	−0.400501
$400$	0	0
$401$	22.0689	1.10207	0.551034	−	0.834483i	$-0.314234\pi$
$401$	22.0689	1.10207	0.551034	+	0.834483i	$0.314234\pi$
$402$	11.7082	0.583952
$403$	4.94427	0.246292
$404$	11.7082	0.582505
$405$	0	0
$406$	−12.9443	−0.642413
$407$	−4.00000	−0.198273
$408$	1.00000	0.0495074
$409$	−18.3607	−0.907877	−0.453939	−	0.891033i	$-0.649981\pi$
$409$	−18.3607	−0.907877	−0.453939	+	0.891033i	$0.649981\pi$
$410$	0	0
$411$	2.00000	0.0986527
$412$	6.94427	0.342120
$413$	5.52786	0.272008
$414$	4.00000	0.196589
$415$	0	0
$416$	0.763932	0.0374548
$417$	10.4721	0.512823
$418$	−4.94427	−0.241832
$419$	−38.3607	−1.87404	−0.937021	−	0.349273i	$-0.886429\pi$
$419$	−38.3607	−1.87404	−0.937021	+	0.349273i	$0.886429\pi$
$420$	0	0
$421$	−10.5836	−0.515813	−0.257906	−	0.966170i	$-0.583033\pi$
$421$	−10.5836	−0.515813	−0.257906	+	0.966170i	$0.583033\pi$
$422$	7.05573	0.343467
$423$	−1.52786	−0.0742873
$424$	6.94427	0.337244
$425$	0	0
$426$	−6.76393	−0.327714
$427$	14.4721	0.700356
$428$	−8.00000	−0.386695
$429$	−1.52786	−0.0737660
$430$	0	0
$431$	−17.5967	−0.847606	−0.423803	−	0.905755i	$-0.639305\pi$
$431$	−17.5967	−0.847606	−0.423803	+	0.905755i	$0.639305\pi$
$432$	1.00000	0.0481125
$433$	16.3607	0.786244	0.393122	−	0.919486i	$-0.371395\pi$
$433$	16.3607	0.786244	0.393122	+	0.919486i	$0.371395\pi$
$434$	−20.9443	−1.00536
$435$	0	0
$436$	−16.4721	−0.788872
$437$	−9.88854	−0.473033
$438$	13.2361	0.632444
$439$	22.4721	1.07254	0.536268	−	0.844048i	$-0.319834\pi$
$439$	22.4721	1.07254	0.536268	+	0.844048i	$0.319834\pi$
$440$	0	0
$441$	3.47214	0.165340
$442$	−0.763932	−0.0363365
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	−2.00000	−0.0949158
$445$	0	0
$446$	26.9443	1.27585
$447$	−10.1803	−0.481514
$448$	−3.23607	−0.152890
$449$	−31.5967	−1.49114	−0.745571	−	0.666426i	$-0.767823\pi$
$449$	−31.5967	−1.49114	−0.745571	+	0.666426i	$0.767823\pi$
$450$	0	0
$451$	11.4164	0.537578
$452$	10.0000	0.470360
$453$	−4.00000	−0.187936
$454$	−25.8885	−1.21501
$455$	0	0
$456$	−2.47214	−0.115768
$457$	−28.3607	−1.32666	−0.663328	−	0.748328i	$-0.730857\pi$
$457$	−28.3607	−1.32666	−0.663328	+	0.748328i	$0.730857\pi$
$458$	−5.41641	−0.253092
$459$	−1.00000	−0.0466760
$460$	0	0
$461$	28.0689	1.30730	0.653649	−	0.756798i	$-0.273237\pi$
$461$	28.0689	1.30730	0.653649	+	0.756798i	$0.273237\pi$
$462$	6.47214	0.301111
$463$	1.05573	0.0490638	0.0245319	−	0.999699i	$-0.492190\pi$
$463$	1.05573	0.0490638	0.0245319	+	0.999699i	$0.492190\pi$
$464$	−4.00000	−0.185695
$465$	0	0
$466$	−21.4164	−0.992096
$467$	16.9443	0.784087	0.392044	−	0.919947i	$-0.371768\pi$
$467$	16.9443	0.784087	0.392044	+	0.919947i	$0.371768\pi$
$468$	−0.763932	−0.0353128
$469$	37.8885	1.74953
$470$	0	0
$471$	12.1803	0.561240
$472$	1.70820	0.0786265
$473$	20.3607	0.936185
$474$	16.9443	0.778276
$475$	0	0
$476$	3.23607	0.148325
$477$	−6.94427	−0.317956
$478$	−13.8885	−0.635247
$479$	13.5967	0.621251	0.310626	−	0.950532i	$-0.399461\pi$
$479$	13.5967	0.621251	0.310626	+	0.950532i	$0.399461\pi$
$480$	0	0
$481$	1.52786	0.0696646
$482$	17.4164	0.793296
$483$	12.9443	0.588985
$484$	−7.00000	−0.318182
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	29.7082	1.34621	0.673104	−	0.739548i	$-0.264961\pi$
$487$	29.7082	1.34621	0.673104	+	0.739548i	$0.264961\pi$
$488$	4.47214	0.202444
$489$	13.5279	0.611751
$490$	0	0
$491$	25.7082	1.16020	0.580098	−	0.814547i	$-0.303014\pi$
$491$	25.7082	1.16020	0.580098	+	0.814547i	$0.303014\pi$
$492$	5.70820	0.257346
$493$	4.00000	0.180151
$494$	1.88854	0.0849696
$495$	0	0
$496$	−6.47214	−0.290607
$497$	−21.8885	−0.981835
$498$	1.52786	0.0684652
$499$	−16.0000	−0.716258	−0.358129	−	0.933672i	$-0.616585\pi$
$499$	−16.0000	−0.716258	−0.358129	+	0.933672i	$0.616585\pi$
$500$	0	0
$501$	−21.8885	−0.977908
$502$	−12.1803	−0.543635
$503$	0.583592	0.0260211	0.0130105	−	0.999915i	$-0.495858\pi$
$503$	0.583592	0.0260211	0.0130105	+	0.999915i	$0.495858\pi$
$504$	3.23607	0.144146
$505$	0	0
$506$	8.00000	0.355643
$507$	−12.4164	−0.551432
$508$	−14.0000	−0.621150
$509$	32.6525	1.44730	0.723648	−	0.690169i	$-0.242464\pi$
$509$	32.6525	1.44730	0.723648	+	0.690169i	$0.242464\pi$
$510$	0	0
$511$	42.8328	1.89481
$512$	−1.00000	−0.0441942
$513$	2.47214	0.109147
$514$	18.0000	0.793946
$515$	0	0
$516$	10.1803	0.448164
$517$	−3.05573	−0.134391
$518$	−6.47214	−0.284369
$519$	−19.8885	−0.873010
$520$	0	0
$521$	28.1803	1.23460	0.617302	−	0.786727i	$-0.288226\pi$
$521$	28.1803	1.23460	0.617302	+	0.786727i	$0.288226\pi$
$522$	4.00000	0.175075
$523$	27.7082	1.21160	0.605798	−	0.795619i	$-0.292854\pi$
$523$	27.7082	1.21160	0.605798	+	0.795619i	$0.292854\pi$
$524$	3.52786	0.154115
$525$	0	0
$526$	−4.00000	−0.174408
$527$	6.47214	0.281931
$528$	2.00000	0.0870388
$529$	−7.00000	−0.304348
$530$	0	0
$531$	−1.70820	−0.0741297
$532$	−8.00000	−0.346844
$533$	−4.36068	−0.188882
$534$	−3.52786	−0.152666
$535$	0	0
$536$	11.7082	0.505717
$537$	5.70820	0.246327
$538$	24.9443	1.07542
$539$	6.94427	0.299111
$540$	0	0
$541$	31.3050	1.34590	0.672952	−	0.739686i	$-0.265026\pi$
$541$	31.3050	1.34590	0.672952	+	0.739686i	$0.265026\pi$
$542$	−4.00000	−0.171815
$543$	−8.47214	−0.363574
$544$	1.00000	0.0428746
$545$	0	0
$546$	−2.47214	−0.105798
$547$	16.9443	0.724485	0.362242	−	0.932084i	$-0.382011\pi$
$547$	16.9443	0.724485	0.362242	+	0.932084i	$0.382011\pi$
$548$	2.00000	0.0854358
$549$	−4.47214	−0.190866
$550$	0	0
$551$	−9.88854	−0.421266
$552$	4.00000	0.170251
$553$	54.8328	2.33173
$554$	−18.3607	−0.780071
$555$	0	0
$556$	10.4721	0.444117
$557$	42.9443	1.81961	0.909804	−	0.415039i	$-0.136232\pi$
$557$	42.9443	1.81961	0.909804	+	0.415039i	$0.136232\pi$
$558$	6.47214	0.273987
$559$	−7.77709	−0.328936
$560$	0	0
$561$	−2.00000	−0.0844401
$562$	25.4164	1.07213
$563$	−37.8885	−1.59681	−0.798406	−	0.602120i	$-0.794323\pi$
$563$	−37.8885	−1.59681	−0.798406	+	0.602120i	$0.794323\pi$
$564$	−1.52786	−0.0643347
$565$	0	0
$566$	9.88854	0.415646
$567$	−3.23607	−0.135902
$568$	−6.76393	−0.283808
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	−8.36068	−0.349884	−0.174942	−	0.984579i	$-0.555974\pi$
$571$	−8.36068	−0.349884	−0.174942	+	0.984579i	$0.555974\pi$
$572$	−1.52786	−0.0638832
$573$	14.4721	0.604582
$574$	18.4721	0.771012
$575$	0	0
$576$	1.00000	0.0416667
$577$	8.58359	0.357340	0.178670	−	0.983909i	$-0.442821\pi$
$577$	8.58359	0.357340	0.178670	+	0.983909i	$0.442821\pi$
$578$	−1.00000	−0.0415945
$579$	−25.5967	−1.06376
$580$	0	0
$581$	4.94427	0.205123
$582$	11.7082	0.485321
$583$	−13.8885	−0.575205
$584$	13.2361	0.547712
$585$	0	0
$586$	2.58359	0.106727
$587$	14.4721	0.597329	0.298664	−	0.954358i	$-0.403459\pi$
$587$	14.4721	0.597329	0.298664	+	0.954358i	$0.403459\pi$
$588$	3.47214	0.143188
$589$	−16.0000	−0.659269
$590$	0	0
$591$	19.8885	0.818105
$592$	−2.00000	−0.0821995
$593$	−13.0557	−0.536134	−0.268067	−	0.963400i	$-0.586385\pi$
$593$	−13.0557	−0.536134	−0.268067	+	0.963400i	$0.586385\pi$
$594$	−2.00000	−0.0820610
$595$	0	0
$596$	−10.1803	−0.417003
$597$	8.94427	0.366065
$598$	−3.05573	−0.124958
$599$	34.4721	1.40849	0.704247	−	0.709955i	$-0.251285\pi$
$599$	34.4721	1.40849	0.704247	+	0.709955i	$0.251285\pi$
$600$	0	0
$601$	−19.5279	−0.796558	−0.398279	−	0.917264i	$-0.630392\pi$
$601$	−19.5279	−0.796558	−0.398279	+	0.917264i	$0.630392\pi$
$602$	32.9443	1.34271
$603$	−11.7082	−0.476795
$604$	−4.00000	−0.162758
$605$	0	0
$606$	−11.7082	−0.475613
$607$	−9.12461	−0.370357	−0.185178	−	0.982705i	$-0.559286\pi$
$607$	−9.12461	−0.370357	−0.185178	+	0.982705i	$0.559286\pi$
$608$	−2.47214	−0.100258
$609$	12.9443	0.524528
$610$	0	0
$611$	1.16718	0.0472192
$612$	−1.00000	−0.0404226
$613$	19.8197	0.800509	0.400254	−	0.916404i	$-0.368922\pi$
$613$	19.8197	0.800509	0.400254	+	0.916404i	$0.368922\pi$
$614$	−18.1803	−0.733699
$615$	0	0
$616$	6.47214	0.260770
$617$	11.5279	0.464094	0.232047	−	0.972705i	$-0.425458\pi$
$617$	11.5279	0.464094	0.232047	+	0.972705i	$0.425458\pi$
$618$	−6.94427	−0.279340
$619$	−31.0557	−1.24824	−0.624118	−	0.781330i	$-0.714541\pi$
$619$	−31.0557	−1.24824	−0.624118	+	0.781330i	$0.714541\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	29.5967	1.18672
$623$	−11.4164	−0.457389
$624$	−0.763932	−0.0305818
$625$	0	0
$626$	−8.29180	−0.331407
$627$	4.94427	0.197455
$628$	12.1803	0.486048
$629$	2.00000	0.0797452
$630$	0	0
$631$	25.8885	1.03061	0.515303	−	0.857008i	$-0.327679\pi$
$631$	25.8885	1.03061	0.515303	+	0.857008i	$0.327679\pi$
$632$	16.9443	0.674007
$633$	−7.05573	−0.280440
$634$	−23.8885	−0.948735
$635$	0	0
$636$	−6.94427	−0.275358
$637$	−2.65248	−0.105095
$638$	8.00000	0.316723
$639$	6.76393	0.267577
$640$	0	0
$641$	−1.70820	−0.0674700	−0.0337350	−	0.999431i	$-0.510740\pi$
$641$	−1.70820	−0.0674700	−0.0337350	+	0.999431i	$0.510740\pi$
$642$	8.00000	0.315735
$643$	0.583592	0.0230146	0.0115073	−	0.999934i	$-0.496337\pi$
$643$	0.583592	0.0230146	0.0115073	+	0.999934i	$0.496337\pi$
$644$	12.9443	0.510076
$645$	0	0
$646$	2.47214	0.0972649
$647$	3.05573	0.120133	0.0600665	−	0.998194i	$-0.480869\pi$
$647$	3.05573	0.120133	0.0600665	+	0.998194i	$0.480869\pi$
$648$	−1.00000	−0.0392837
$649$	−3.41641	−0.134106
$650$	0	0
$651$	20.9443	0.820871
$652$	13.5279	0.529792
$653$	12.4721	0.488072	0.244036	−	0.969766i	$-0.421528\pi$
$653$	12.4721	0.488072	0.244036	+	0.969766i	$0.421528\pi$
$654$	16.4721	0.644111
$655$	0	0
$656$	5.70820	0.222868
$657$	−13.2361	−0.516388
$658$	−4.94427	−0.192748
$659$	41.7082	1.62472	0.812360	−	0.583156i	$-0.198182\pi$
$659$	41.7082	1.62472	0.812360	+	0.583156i	$0.198182\pi$
$660$	0	0
$661$	−13.4164	−0.521838	−0.260919	−	0.965361i	$-0.584026\pi$
$661$	−13.4164	−0.521838	−0.260919	+	0.965361i	$0.584026\pi$
$662$	−23.4164	−0.910105
$663$	0.763932	0.0296687
$664$	1.52786	0.0592926
$665$	0	0
$666$	2.00000	0.0774984
$667$	16.0000	0.619522
$668$	−21.8885	−0.846893
$669$	−26.9443	−1.04173
$670$	0	0
$671$	−8.94427	−0.345290
$672$	3.23607	0.124834
$673$	18.1803	0.700801	0.350400	−	0.936600i	$-0.386046\pi$
$673$	18.1803	0.700801	0.350400	+	0.936600i	$0.386046\pi$
$674$	2.18034	0.0839836
$675$	0	0
$676$	−12.4164	−0.477554
$677$	26.9443	1.03555	0.517776	−	0.855516i	$-0.326760\pi$
$677$	26.9443	1.03555	0.517776	+	0.855516i	$0.326760\pi$
$678$	−10.0000	−0.384048
$679$	37.8885	1.45403
$680$	0	0
$681$	25.8885	0.992051
$682$	12.9443	0.495662
$683$	15.0557	0.576091	0.288046	−	0.957617i	$-0.406994\pi$
$683$	15.0557	0.576091	0.288046	+	0.957617i	$0.406994\pi$
$684$	2.47214	0.0945245
$685$	0	0
$686$	−11.4164	−0.435880
$687$	5.41641	0.206649
$688$	10.1803	0.388122
$689$	5.30495	0.202103
$690$	0	0
$691$	12.9443	0.492423	0.246212	−	0.969216i	$-0.420814\pi$
$691$	12.9443	0.492423	0.246212	+	0.969216i	$0.420814\pi$
$692$	−19.8885	−0.756049
$693$	−6.47214	−0.245856
$694$	30.8328	1.17040
$695$	0	0
$696$	4.00000	0.151620
$697$	−5.70820	−0.216214
$698$	19.5279	0.739141
$699$	21.4164	0.810043
$700$	0	0
$701$	−37.5967	−1.42001	−0.710005	−	0.704197i	$-0.751307\pi$
$701$	−37.5967	−1.42001	−0.710005	+	0.704197i	$0.751307\pi$
$702$	0.763932	0.0288328
$703$	−4.94427	−0.186477
$704$	2.00000	0.0753778
$705$	0	0
$706$	2.94427	0.110809
$707$	−37.8885	−1.42495
$708$	−1.70820	−0.0641982
$709$	−48.4721	−1.82041	−0.910205	−	0.414159i	$-0.864076\pi$
$709$	−48.4721	−1.82041	−0.910205	+	0.414159i	$0.864076\pi$
$710$	0	0
$711$	−16.9443	−0.635460
$712$	−3.52786	−0.132212
$713$	25.8885	0.969534
$714$	−3.23607	−0.121107
$715$	0	0
$716$	5.70820	0.213326
$717$	13.8885	0.518677
$718$	18.4721	0.689374
$719$	30.5410	1.13899	0.569494	−	0.821996i	$-0.307139\pi$
$719$	30.5410	1.13899	0.569494	+	0.821996i	$0.307139\pi$
$720$	0	0
$721$	−22.4721	−0.836906
$722$	12.8885	0.479662
$723$	−17.4164	−0.647723
$724$	−8.47214	−0.314864
$725$	0	0
$726$	7.00000	0.259794
$727$	−11.8885	−0.440922	−0.220461	−	0.975396i	$-0.570756\pi$
$727$	−11.8885	−0.440922	−0.220461	+	0.975396i	$0.570756\pi$
$728$	−2.47214	−0.0916235
$729$	1.00000	0.0370370
$730$	0	0
$731$	−10.1803	−0.376533
$732$	−4.47214	−0.165295
$733$	28.5410	1.05419	0.527093	−	0.849807i	$-0.323282\pi$
$733$	28.5410	1.05419	0.527093	+	0.849807i	$0.323282\pi$
$734$	−28.1803	−1.04016
$735$	0	0
$736$	4.00000	0.147442
$737$	−23.4164	−0.862554
$738$	−5.70820	−0.210122
$739$	42.4721	1.56236	0.781181	−	0.624304i	$-0.214617\pi$
$739$	42.4721	1.56236	0.781181	+	0.624304i	$0.214617\pi$
$740$	0	0
$741$	−1.88854	−0.0693774
$742$	−22.4721	−0.824978
$743$	−42.4721	−1.55815	−0.779076	−	0.626930i	$-0.784311\pi$
$743$	−42.4721	−1.55815	−0.779076	+	0.626930i	$0.784311\pi$
$744$	6.47214	0.237280
$745$	0	0
$746$	3.23607	0.118481
$747$	−1.52786	−0.0559016
$748$	−2.00000	−0.0731272
$749$	25.8885	0.945947
$750$	0	0
$751$	−5.88854	−0.214876	−0.107438	−	0.994212i	$-0.534265\pi$
$751$	−5.88854	−0.214876	−0.107438	+	0.994212i	$0.534265\pi$
$752$	−1.52786	−0.0557155
$753$	12.1803	0.443876
$754$	−3.05573	−0.111283
$755$	0	0
$756$	−3.23607	−0.117695
$757$	−11.5967	−0.421491	−0.210745	−	0.977541i	$-0.567589\pi$
$757$	−11.5967	−0.421491	−0.210745	+	0.977541i	$0.567589\pi$
$758$	10.4721	0.380365
$759$	−8.00000	−0.290382
$760$	0	0
$761$	−0.832816	−0.0301895	−0.0150948	−	0.999886i	$-0.504805\pi$
$761$	−0.832816	−0.0301895	−0.0150948	+	0.999886i	$0.504805\pi$
$762$	14.0000	0.507166
$763$	53.3050	1.92977
$764$	14.4721	0.523584
$765$	0	0
$766$	−17.8885	−0.646339
$767$	1.30495	0.0471191
$768$	1.00000	0.0360844
$769$	10.3607	0.373616	0.186808	−	0.982396i	$-0.440186\pi$
$769$	10.3607	0.373616	0.186808	+	0.982396i	$0.440186\pi$
$770$	0	0
$771$	−18.0000	−0.648254
$772$	−25.5967	−0.921247
$773$	−6.58359	−0.236795	−0.118398	−	0.992966i	$-0.537776\pi$
$773$	−6.58359	−0.236795	−0.118398	+	0.992966i	$0.537776\pi$
$774$	−10.1803	−0.365925
$775$	0	0
$776$	11.7082	0.420300
$777$	6.47214	0.232187
$778$	2.18034	0.0781690
$779$	14.1115	0.505595
$780$	0	0
$781$	13.5279	0.484065
$782$	−4.00000	−0.143040
$783$	−4.00000	−0.142948
$784$	3.47214	0.124005
$785$	0	0
$786$	−3.52786	−0.125835
$787$	−37.5279	−1.33772	−0.668862	−	0.743387i	$-0.733218\pi$
$787$	−37.5279	−1.33772	−0.668862	+	0.743387i	$0.733218\pi$
$788$	19.8885	0.708500
$789$	4.00000	0.142404
$790$	0	0
$791$	−32.3607	−1.15061
$792$	−2.00000	−0.0710669
$793$	3.41641	0.121320
$794$	−31.8885	−1.13168
$795$	0	0
$796$	8.94427	0.317021
$797$	−46.0000	−1.62940	−0.814702	−	0.579880i	$-0.803099\pi$
$797$	−46.0000	−1.62940	−0.814702	+	0.579880i	$0.803099\pi$
$798$	8.00000	0.283197
$799$	1.52786	0.0540519
$800$	0	0
$801$	3.52786	0.124651
$802$	−22.0689	−0.779279
$803$	−26.4721	−0.934181
$804$	−11.7082	−0.412917
$805$	0	0
$806$	−4.94427	−0.174155
$807$	−24.9443	−0.878080
$808$	−11.7082	−0.411893
$809$	6.87539	0.241726	0.120863	−	0.992669i	$-0.461434\pi$
$809$	6.87539	0.241726	0.120863	+	0.992669i	$0.461434\pi$
$810$	0	0
$811$	−38.2492	−1.34311	−0.671556	−	0.740954i	$-0.734374\pi$
$811$	−38.2492	−1.34311	−0.671556	+	0.740954i	$0.734374\pi$
$812$	12.9443	0.454255
$813$	4.00000	0.140286
$814$	4.00000	0.140200
$815$	0	0
$816$	−1.00000	−0.0350070
$817$	25.1672	0.880488
$818$	18.3607	0.641966
$819$	2.47214	0.0863834
$820$	0	0
$821$	−11.4164	−0.398435	−0.199218	−	0.979955i	$-0.563840\pi$
$821$	−11.4164	−0.398435	−0.199218	+	0.979955i	$0.563840\pi$
$822$	−2.00000	−0.0697580
$823$	−15.8197	−0.551439	−0.275719	−	0.961238i	$-0.588916\pi$
$823$	−15.8197	−0.551439	−0.275719	+	0.961238i	$0.588916\pi$
$824$	−6.94427	−0.241915
$825$	0	0
$826$	−5.52786	−0.192339
$827$	40.9443	1.42377	0.711886	−	0.702295i	$-0.247841\pi$
$827$	40.9443	1.42377	0.711886	+	0.702295i	$0.247841\pi$
$828$	−4.00000	−0.139010
$829$	34.0000	1.18087	0.590434	−	0.807086i	$-0.298956\pi$
$829$	34.0000	1.18087	0.590434	+	0.807086i	$0.298956\pi$
$830$	0	0
$831$	18.3607	0.636925
$832$	−0.763932	−0.0264846
$833$	−3.47214	−0.120302
$834$	−10.4721	−0.362620
$835$	0	0
$836$	4.94427	0.171001
$837$	−6.47214	−0.223710
$838$	38.3607	1.32515
$839$	36.0689	1.24524	0.622618	−	0.782526i	$-0.286069\pi$
$839$	36.0689	1.24524	0.622618	+	0.782526i	$0.286069\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	10.5836	0.364735
$843$	−25.4164	−0.875388
$844$	−7.05573	−0.242868
$845$	0	0
$846$	1.52786	0.0525290
$847$	22.6525	0.778348
$848$	−6.94427	−0.238467
$849$	−9.88854	−0.339374
$850$	0	0
$851$	8.00000	0.274236
$852$	6.76393	0.231728
$853$	−38.3607	−1.31344	−0.656722	−	0.754132i	$-0.728058\pi$
$853$	−38.3607	−1.31344	−0.656722	+	0.754132i	$0.728058\pi$
$854$	−14.4721	−0.495226
$855$	0	0
$856$	8.00000	0.273434
$857$	−5.63932	−0.192636	−0.0963178	−	0.995351i	$-0.530706\pi$
$857$	−5.63932	−0.192636	−0.0963178	+	0.995351i	$0.530706\pi$
$858$	1.52786	0.0521604
$859$	24.9443	0.851088	0.425544	−	0.904938i	$-0.360083\pi$
$859$	24.9443	0.851088	0.425544	+	0.904938i	$0.360083\pi$
$860$	0	0
$861$	−18.4721	−0.629529
$862$	17.5967	0.599348
$863$	−41.3050	−1.40604	−0.703018	−	0.711172i	$-0.748165\pi$
$863$	−41.3050	−1.40604	−0.703018	+	0.711172i	$0.748165\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	−16.3607	−0.555959
$867$	1.00000	0.0339618
$868$	20.9443	0.710895
$869$	−33.8885	−1.14959
$870$	0	0
$871$	8.94427	0.303065
$872$	16.4721	0.557817
$873$	−11.7082	−0.396263
$874$	9.88854	0.334485
$875$	0	0
$876$	−13.2361	−0.447205
$877$	31.5279	1.06462	0.532310	−	0.846549i	$-0.321324\pi$
$877$	31.5279	1.06462	0.532310	+	0.846549i	$0.321324\pi$
$878$	−22.4721	−0.758398
$879$	−2.58359	−0.0871424
$880$	0	0
$881$	33.1246	1.11600	0.557998	−	0.829842i	$-0.311570\pi$
$881$	33.1246	1.11600	0.557998	+	0.829842i	$0.311570\pi$
$882$	−3.47214	−0.116913
$883$	22.7639	0.766067	0.383034	−	0.923734i	$-0.374879\pi$
$883$	22.7639	0.766067	0.383034	+	0.923734i	$0.374879\pi$
$884$	0.763932	0.0256938
$885$	0	0
$886$	4.00000	0.134383
$887$	8.94427	0.300319	0.150160	−	0.988662i	$-0.452021\pi$
$887$	8.94427	0.300319	0.150160	+	0.988662i	$0.452021\pi$
$888$	2.00000	0.0671156
$889$	45.3050	1.51948
$890$	0	0
$891$	2.00000	0.0670025
$892$	−26.9443	−0.902161
$893$	−3.77709	−0.126395
$894$	10.1803	0.340481
$895$	0	0
$896$	3.23607	0.108109
$897$	3.05573	0.102028
$898$	31.5967	1.05440
$899$	25.8885	0.863431
$900$	0	0
$901$	6.94427	0.231347
$902$	−11.4164	−0.380125
$903$	−32.9443	−1.09632
$904$	−10.0000	−0.332595
$905$	0	0
$906$	4.00000	0.132891
$907$	40.3607	1.34015	0.670077	−	0.742291i	$-0.266261\pi$
$907$	40.3607	1.34015	0.670077	+	0.742291i	$0.266261\pi$
$908$	25.8885	0.859142
$909$	11.7082	0.388337
$910$	0	0
$911$	54.5410	1.80702	0.903512	−	0.428562i	$-0.140980\pi$
$911$	54.5410	1.80702	0.903512	+	0.428562i	$0.140980\pi$
$912$	2.47214	0.0818606
$913$	−3.05573	−0.101130
$914$	28.3607	0.938088
$915$	0	0
$916$	5.41641	0.178963
$917$	−11.4164	−0.377003
$918$	1.00000	0.0330049
$919$	−36.7214	−1.21133	−0.605663	−	0.795721i	$-0.707092\pi$
$919$	−36.7214	−1.21133	−0.605663	+	0.795721i	$0.707092\pi$
$920$	0	0
$921$	18.1803	0.599063
$922$	−28.0689	−0.924399
$923$	−5.16718	−0.170080
$924$	−6.47214	−0.212918
$925$	0	0
$926$	−1.05573	−0.0346934
$927$	6.94427	0.228080
$928$	4.00000	0.131306
$929$	−34.0689	−1.11776	−0.558882	−	0.829247i	$-0.688769\pi$
$929$	−34.0689	−1.11776	−0.558882	+	0.829247i	$0.688769\pi$
$930$	0	0
$931$	8.58359	0.281316
$932$	21.4164	0.701518
$933$	−29.5967	−0.968954
$934$	−16.9443	−0.554434
$935$	0	0
$936$	0.763932	0.0249699
$937$	−26.4721	−0.864807	−0.432403	−	0.901680i	$-0.642334\pi$
$937$	−26.4721	−0.864807	−0.432403	+	0.901680i	$0.642334\pi$
$938$	−37.8885	−1.23710
$939$	8.29180	0.270593
$940$	0	0
$941$	−49.3050	−1.60730	−0.803648	−	0.595105i	$-0.797110\pi$
$941$	−49.3050	−1.60730	−0.803648	+	0.595105i	$0.797110\pi$
$942$	−12.1803	−0.396857
$943$	−22.8328	−0.743539
$944$	−1.70820	−0.0555973
$945$	0	0
$946$	−20.3607	−0.661983
$947$	40.7214	1.32327	0.661633	−	0.749828i	$-0.269864\pi$
$947$	40.7214	1.32327	0.661633	+	0.749828i	$0.269864\pi$
$948$	−16.9443	−0.550324
$949$	10.1115	0.328232
$950$	0	0
$951$	23.8885	0.774639
$952$	−3.23607	−0.104882
$953$	−9.05573	−0.293344	−0.146672	−	0.989185i	$-0.546856\pi$
$953$	−9.05573	−0.293344	−0.146672	+	0.989185i	$0.546856\pi$
$954$	6.94427	0.224829
$955$	0	0
$956$	13.8885	0.449188
$957$	−8.00000	−0.258603
$958$	−13.5967	−0.439291
$959$	−6.47214	−0.208996
$960$	0	0
$961$	10.8885	0.351243
$962$	−1.52786	−0.0492603
$963$	−8.00000	−0.257796
$964$	−17.4164	−0.560945
$965$	0	0
$966$	−12.9443	−0.416475
$967$	−8.47214	−0.272446	−0.136223	−	0.990678i	$-0.543496\pi$
$967$	−8.47214	−0.272446	−0.136223	+	0.990678i	$0.543496\pi$
$968$	7.00000	0.224989
$969$	−2.47214	−0.0794164
$970$	0	0
$971$	−27.2361	−0.874047	−0.437024	−	0.899450i	$-0.643967\pi$
$971$	−27.2361	−0.874047	−0.437024	+	0.899450i	$0.643967\pi$
$972$	1.00000	0.0320750
$973$	−33.8885	−1.08642
$974$	−29.7082	−0.951912
$975$	0	0
$976$	−4.47214	−0.143150
$977$	−9.05573	−0.289718	−0.144859	−	0.989452i	$-0.546273\pi$
$977$	−9.05573	−0.289718	−0.144859	+	0.989452i	$0.546273\pi$
$978$	−13.5279	−0.432573
$979$	7.05573	0.225502
$980$	0	0
$981$	−16.4721	−0.525915
$982$	−25.7082	−0.820382
$983$	15.4164	0.491707	0.245854	−	0.969307i	$-0.420932\pi$
$983$	15.4164	0.491707	0.245854	+	0.969307i	$0.420932\pi$
$984$	−5.70820	−0.181971
$985$	0	0
$986$	−4.00000	−0.127386
$987$	4.94427	0.157378
$988$	−1.88854	−0.0600826
$989$	−40.7214	−1.29486
$990$	0	0
$991$	42.8328	1.36063	0.680315	−	0.732920i	$-0.261843\pi$
$991$	42.8328	1.36063	0.680315	+	0.732920i	$0.261843\pi$
$992$	6.47214	0.205491
$993$	23.4164	0.743097
$994$	21.8885	0.694262
$995$	0	0
$996$	−1.52786	−0.0484122
$997$	−58.9443	−1.86678	−0.933392	−	0.358859i	$-0.883166\pi$
$997$	−58.9443	−1.86678	−0.933392	+	0.358859i	$0.883166\pi$
$998$	16.0000	0.506471
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2550.2.a.bh.1.1		2
3.2	odd	2		7650.2.a.da.1.1		2
5.2	odd	4		510.2.d.b.409.1	✓	4
5.3	odd	4		510.2.d.b.409.4	yes	4
5.4	even	2		2550.2.a.bk.1.2		2
15.2	even	4		1530.2.d.f.919.4		4
15.8	even	4		1530.2.d.f.919.1		4
15.14	odd	2		7650.2.a.cx.1.2		2
20.3	even	4		4080.2.m.m.2449.2		4
20.7	even	4		4080.2.m.m.2449.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
510.2.d.b.409.1	✓	4	5.2	odd	4
510.2.d.b.409.4	yes	4	5.3	odd	4
1530.2.d.f.919.1		4	15.8	even	4
1530.2.d.f.919.4		4	15.2	even	4
2550.2.a.bh.1.1		2	1.1	even	1	trivial
2550.2.a.bk.1.2		2	5.4	even	2
4080.2.m.m.2449.2		4	20.3	even	4
4080.2.m.m.2449.3		4	20.7	even	4
7650.2.a.cx.1.2		2	15.14	odd	2
7650.2.a.da.1.1		2	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2550.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -3.23607 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{11} +1.00000 q^{12} -0.763932 q^{13} +3.23607 q^{14} +1.00000 q^{16} -1.00000 q^{17} -1.00000 q^{18} +2.47214 q^{19} -3.23607 q^{21} -2.00000 q^{22} -4.00000 q^{23} -1.00000 q^{24} +0.763932 q^{26} +1.00000 q^{27} -3.23607 q^{28} -4.00000 q^{29} -6.47214 q^{31} -1.00000 q^{32} +2.00000 q^{33} +1.00000 q^{34} +1.00000 q^{36} -2.00000 q^{37} -2.47214 q^{38} -0.763932 q^{39} +5.70820 q^{41} +3.23607 q^{42} +10.1803 q^{43} +2.00000 q^{44} +4.00000 q^{46} -1.52786 q^{47} +1.00000 q^{48} +3.47214 q^{49} -1.00000 q^{51} -0.763932 q^{52} -6.94427 q^{53} -1.00000 q^{54} +3.23607 q^{56} +2.47214 q^{57} +4.00000 q^{58} -1.70820 q^{59} -4.47214 q^{61} +6.47214 q^{62} -3.23607 q^{63} +1.00000 q^{64} -2.00000 q^{66} -11.7082 q^{67} -1.00000 q^{68} -4.00000 q^{69} +6.76393 q^{71} -1.00000 q^{72} -13.2361 q^{73} +2.00000 q^{74} +2.47214 q^{76} -6.47214 q^{77} +0.763932 q^{78} -16.9443 q^{79} +1.00000 q^{81} -5.70820 q^{82} -1.52786 q^{83} -3.23607 q^{84} -10.1803 q^{86} -4.00000 q^{87} -2.00000 q^{88} +3.52786 q^{89} +2.47214 q^{91} -4.00000 q^{92} -6.47214 q^{93} +1.52786 q^{94} -1.00000 q^{96} -11.7082 q^{97} -3.47214 q^{98} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 2 q^{7} - 2 q^{8} + 2 q^{9} + 4 q^{11} + 2 q^{12} - 6 q^{13} + 2 q^{14} + 2 q^{16} - 2 q^{17} - 2 q^{18} - 4 q^{19} - 2 q^{21} - 4 q^{22} - 8 q^{23} - 2 q^{24}+ \cdots + 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 - 2 * q^7 - 2 * q^8 + 2 * q^9 + 4 * q^11 + 2 * q^12 - 6 * q^13 + 2 * q^14 + 2 * q^16 - 2 * q^17 - 2 * q^18 - 4 * q^19 - 2 * q^21 - 4 * q^22 - 8 * q^23 - 2 * q^24 + 6 * q^26 + 2 * q^27 - 2 * q^28 - 8 * q^29 - 4 * q^31 - 2 * q^32 + 4 * q^33 + 2 * q^34 + 2 * q^36 - 4 * q^37 + 4 * q^38 - 6 * q^39 - 2 * q^41 + 2 * q^42 - 2 * q^43 + 4 * q^44 + 8 * q^46 - 12 * q^47 + 2 * q^48 - 2 * q^49 - 2 * q^51 - 6 * q^52 + 4 * q^53 - 2 * q^54 + 2 * q^56 - 4 * q^57 + 8 * q^58 + 10 * q^59 + 4 * q^62 - 2 * q^63 + 2 * q^64 - 4 * q^66 - 10 * q^67 - 2 * q^68 - 8 * q^69 + 18 * q^71 - 2 * q^72 - 22 * q^73 + 4 * q^74 - 4 * q^76 - 4 * q^77 + 6 * q^78 - 16 * q^79 + 2 * q^81 + 2 * q^82 - 12 * q^83 - 2 * q^84 + 2 * q^86 - 8 * q^87 - 4 * q^88 + 16 * q^89 - 4 * q^91 - 8 * q^92 - 4 * q^93 + 12 * q^94 - 2 * q^96 - 10 * q^97 + 2 * q^98 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more