LMFDB - Embedded newform 2550.2.a.bh.1.2

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			1.2
Root			$-0.618034$ 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)

$f(q)$	$=$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.23607 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{11} +1.00000 q^{12} -5.23607 q^{13} -1.23607 q^{14} +1.00000 q^{16} -1.00000 q^{17} -1.00000 q^{18} -6.47214 q^{19} +1.23607 q^{21} -2.00000 q^{22} -4.00000 q^{23} -1.00000 q^{24} +5.23607 q^{26} +1.00000 q^{27} +1.23607 q^{28} -4.00000 q^{29} +2.47214 q^{31} -1.00000 q^{32} +2.00000 q^{33} +1.00000 q^{34} +1.00000 q^{36} -2.00000 q^{37} +6.47214 q^{38} -5.23607 q^{39} -7.70820 q^{41} -1.23607 q^{42} -12.1803 q^{43} +2.00000 q^{44} +4.00000 q^{46} -10.4721 q^{47} +1.00000 q^{48} -5.47214 q^{49} -1.00000 q^{51} -5.23607 q^{52} +10.9443 q^{53} -1.00000 q^{54} -1.23607 q^{56} -6.47214 q^{57} +4.00000 q^{58} +11.7082 q^{59} +4.47214 q^{61} -2.47214 q^{62} +1.23607 q^{63} +1.00000 q^{64} -2.00000 q^{66} +1.70820 q^{67} -1.00000 q^{68} -4.00000 q^{69} +11.2361 q^{71} -1.00000 q^{72} -8.76393 q^{73} +2.00000 q^{74} -6.47214 q^{76} +2.47214 q^{77} +5.23607 q^{78} +0.944272 q^{79} +1.00000 q^{81} +7.70820 q^{82} -10.4721 q^{83} +1.23607 q^{84} +12.1803 q^{86} -4.00000 q^{87} -2.00000 q^{88} +12.4721 q^{89} -6.47214 q^{91} -4.00000 q^{92} +2.47214 q^{93} +10.4721 q^{94} -1.00000 q^{96} +1.70820 q^{97} +5.47214 q^{98} +2.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	1.23607	0.467190	0.233595	−	0.972334i	$-0.424951\pi$
$7$	1.23607	0.467190	0.233595	+	0.972334i	$0.424951\pi$
$8$	−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
$13$	−5.23607	−1.45222	−0.726112	−	0.687576i	$-0.758675\pi$
$13$	−5.23607	−1.45222	−0.726112	+	0.687576i	$0.758675\pi$
$14$	−1.23607	−0.330353
$15$	0	0
$16$	1.00000	0.250000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	−1.00000	−0.235702
$19$	−6.47214	−1.48481	−0.742405	−	0.669951i	$-0.766315\pi$
$19$	−6.47214	−1.48481	−0.742405	+	0.669951i	$0.766315\pi$
$20$	0	0
$21$	1.23607	0.269732
$22$	−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
$26$	5.23607	1.02688
$27$	1.00000	0.192450
$28$	1.23607	0.233595
$29$	−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
$31$	2.47214	0.444009	0.222004	−	0.975046i	$-0.428740\pi$
$31$	2.47214	0.444009	0.222004	+	0.975046i	$0.428740\pi$
$32$	−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	$0.552568\pi$
$38$	6.47214	1.04992
$39$	−5.23607	−0.838442
$40$	0	0
$41$	−7.70820	−1.20382	−0.601910	−	0.798564i	$-0.705593\pi$
$41$	−7.70820	−1.20382	−0.601910	+	0.798564i	$0.705593\pi$
$42$	−1.23607	−0.190729
$43$	−12.1803	−1.85748	−0.928742	−	0.370726i	$-0.879109\pi$
$43$	−12.1803	−1.85748	−0.928742	+	0.370726i	$0.879109\pi$
$44$	2.00000	0.301511
$45$	0	0
$46$	4.00000	0.589768
$47$	−10.4721	−1.52752	−0.763759	−	0.645501i	$-0.776648\pi$
$47$	−10.4721	−1.52752	−0.763759	+	0.645501i	$0.776648\pi$
$48$	1.00000	0.144338
$49$	−5.47214	−0.781734
$50$	0	0
$51$	−1.00000	−0.140028
$52$	−5.23607	−0.726112
$53$	10.9443	1.50331	0.751656	−	0.659556i	$-0.229256\pi$
$53$	10.9443	1.50331	0.751656	+	0.659556i	$0.229256\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	−1.23607	−0.165177
$57$	−6.47214	−0.857255
$58$	4.00000	0.525226
$59$	11.7082	1.52428	0.762139	−	0.647413i	$-0.224149\pi$
$59$	11.7082	1.52428	0.762139	+	0.647413i	$0.224149\pi$
$60$	0	0
$61$	4.47214	0.572598	0.286299	−	0.958140i	$-0.407575\pi$
$61$	4.47214	0.572598	0.286299	+	0.958140i	$0.407575\pi$
$62$	−2.47214	−0.313962
$63$	1.23607	0.155730
$64$	1.00000	0.125000
$65$	0	0
$66$	−2.00000	−0.246183
$67$	1.70820	0.208690	0.104345	−	0.994541i	$-0.466725\pi$
$67$	1.70820	0.208690	0.104345	+	0.994541i	$0.466725\pi$
$68$	−1.00000	−0.121268
$69$	−4.00000	−0.481543
$70$	0	0
$71$	11.2361	1.33348	0.666738	−	0.745292i	$-0.267690\pi$
$71$	11.2361	1.33348	0.666738	+	0.745292i	$0.267690\pi$
$72$	−1.00000	−0.117851
$73$	−8.76393	−1.02574	−0.512870	−	0.858466i	$-0.671418\pi$
$73$	−8.76393	−1.02574	−0.512870	+	0.858466i	$0.671418\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	−6.47214	−0.742405
$77$	2.47214	0.281726
$78$	5.23607	0.592868
$79$	0.944272	0.106239	0.0531194	−	0.998588i	$-0.483084\pi$
$79$	0.944272	0.106239	0.0531194	+	0.998588i	$0.483084\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	7.70820	0.851229
$83$	−10.4721	−1.14947	−0.574733	−	0.818341i	$-0.694894\pi$
$83$	−10.4721	−1.14947	−0.574733	+	0.818341i	$0.694894\pi$
$84$	1.23607	0.134866
$85$	0	0
$86$	12.1803	1.31344
$87$	−4.00000	−0.428845
$88$	−2.00000	−0.213201
$89$	12.4721	1.32204	0.661022	−	0.750367i	$-0.270123\pi$
$89$	12.4721	1.32204	0.661022	+	0.750367i	$0.270123\pi$
$90$	0	0
$91$	−6.47214	−0.678464
$92$	−4.00000	−0.417029
$93$	2.47214	0.256349
$94$	10.4721	1.08012
$95$	0	0
$96$	−1.00000	−0.102062
$97$	1.70820	0.173442	0.0867209	−	0.996233i	$-0.472361\pi$
$97$	1.70820	0.173442	0.0867209	+	0.996233i	$0.472361\pi$
$98$	5.47214	0.552769
$99$	2.00000	0.201008
$100$	0	0
$101$	−1.70820	−0.169973	−0.0849863	−	0.996382i	$-0.527085\pi$
$101$	−1.70820	−0.169973	−0.0849863	+	0.996382i	$0.527085\pi$
$102$	1.00000	0.0990148
$103$	−10.9443	−1.07837	−0.539186	−	0.842187i	$-0.681268\pi$
$103$	−10.9443	−1.07837	−0.539186	+	0.842187i	$0.681268\pi$
$104$	5.23607	0.513439
$105$	0	0
$106$	−10.9443	−1.06300
$107$	−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
$109$	−7.52786	−0.721039	−0.360519	−	0.932752i	$-0.617401\pi$
$109$	−7.52786	−0.721039	−0.360519	+	0.932752i	$0.617401\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	1.23607	0.116797
$113$	10.0000	0.940721	0.470360	−	0.882474i	$-0.344124\pi$
$113$	10.0000	0.940721	0.470360	+	0.882474i	$0.344124\pi$
$114$	6.47214	0.606171
$115$	0	0
$116$	−4.00000	−0.371391
$117$	−5.23607	−0.484075
$118$	−11.7082	−1.07783
$119$	−1.23607	−0.113310
$120$	0	0
$121$	−7.00000	−0.636364
$122$	−4.47214	−0.404888
$123$	−7.70820	−0.695025
$124$	2.47214	0.222004
$125$	0	0
$126$	−1.23607	−0.110118
$127$	−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
$129$	−12.1803	−1.07242
$130$	0	0
$131$	12.4721	1.08970	0.544848	−	0.838535i	$-0.316587\pi$
$131$	12.4721	1.08970	0.544848	+	0.838535i	$0.316587\pi$
$132$	2.00000	0.174078
$133$	−8.00000	−0.693688
$134$	−1.70820	−0.147566
$135$	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
$139$	1.52786	0.129592	0.0647959	−	0.997899i	$-0.479360\pi$
$139$	1.52786	0.129592	0.0647959	+	0.997899i	$0.479360\pi$
$140$	0	0
$141$	−10.4721	−0.881913
$142$	−11.2361	−0.942910
$143$	−10.4721	−0.875724
$144$	1.00000	0.0833333
$145$	0	0
$146$	8.76393	0.725308
$147$	−5.47214	−0.451334
$148$	−2.00000	−0.164399
$149$	12.1803	0.997852	0.498926	−	0.866644i	$-0.333728\pi$
$149$	12.1803	0.997852	0.498926	+	0.866644i	$0.333728\pi$
$150$	0	0
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	6.47214	0.524960
$153$	−1.00000	−0.0808452
$154$	−2.47214	−0.199210
$155$	0	0
$156$	−5.23607	−0.419221
$157$	−10.1803	−0.812480	−0.406240	−	0.913767i	$-0.633160\pi$
$157$	−10.1803	−0.812480	−0.406240	+	0.913767i	$0.633160\pi$
$158$	−0.944272	−0.0751222
$159$	10.9443	0.867937
$160$	0	0
$161$	−4.94427	−0.389663
$162$	−1.00000	−0.0785674
$163$	22.4721	1.76015	0.880077	−	0.474831i	$-0.157491\pi$
$163$	22.4721	1.76015	0.880077	+	0.474831i	$0.157491\pi$
$164$	−7.70820	−0.601910
$165$	0	0
$166$	10.4721	0.812795
$167$	13.8885	1.07473	0.537364	−	0.843350i	$-0.319420\pi$
$167$	13.8885	1.07473	0.537364	+	0.843350i	$0.319420\pi$
$168$	−1.23607	−0.0953647
$169$	14.4164	1.10895
$170$	0	0
$171$	−6.47214	−0.494937
$172$	−12.1803	−0.928742
$173$	15.8885	1.20798	0.603992	−	0.796991i	$-0.293576\pi$
$173$	15.8885	1.20798	0.603992	+	0.796991i	$0.293576\pi$
$174$	4.00000	0.303239
$175$	0	0
$176$	2.00000	0.150756
$177$	11.7082	0.880042
$178$	−12.4721	−0.934826
$179$	−7.70820	−0.576138	−0.288069	−	0.957610i	$-0.593013\pi$
$179$	−7.70820	−0.576138	−0.288069	+	0.957610i	$0.593013\pi$
$180$	0	0
$181$	0.472136	0.0350936	0.0175468	−	0.999846i	$-0.494414\pi$
$181$	0.472136	0.0350936	0.0175468	+	0.999846i	$0.494414\pi$
$182$	6.47214	0.479747
$183$	4.47214	0.330590
$184$	4.00000	0.294884
$185$	0	0
$186$	−2.47214	−0.181266
$187$	−2.00000	−0.146254
$188$	−10.4721	−0.763759
$189$	1.23607	0.0899107
$190$	0	0
$191$	5.52786	0.399982	0.199991	−	0.979798i	$-0.435909\pi$
$191$	5.52786	0.399982	0.199991	+	0.979798i	$0.435909\pi$
$192$	1.00000	0.0721688
$193$	23.5967	1.69853	0.849266	−	0.527966i	$-0.177045\pi$
$193$	23.5967	1.69853	0.849266	+	0.527966i	$0.177045\pi$
$194$	−1.70820	−0.122642
$195$	0	0
$196$	−5.47214	−0.390867
$197$	−15.8885	−1.13201	−0.566006	−	0.824401i	$-0.691512\pi$
$197$	−15.8885	−1.13201	−0.566006	+	0.824401i	$0.691512\pi$
$198$	−2.00000	−0.142134
$199$	−8.94427	−0.634043	−0.317021	−	0.948418i	$-0.602683\pi$
$199$	−8.94427	−0.634043	−0.317021	+	0.948418i	$0.602683\pi$
$200$	0	0
$201$	1.70820	0.120487
$202$	1.70820	0.120189
$203$	−4.94427	−0.347020
$204$	−1.00000	−0.0700140
$205$	0	0
$206$	10.9443	0.762524
$207$	−4.00000	−0.278019
$208$	−5.23607	−0.363056
$209$	−12.9443	−0.895374
$210$	0	0
$211$	−24.9443	−1.71723	−0.858617	−	0.512617i	$-0.828676\pi$
$211$	−24.9443	−1.71723	−0.858617	+	0.512617i	$0.828676\pi$
$212$	10.9443	0.751656
$213$	11.2361	0.769883
$214$	8.00000	0.546869
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	3.05573	0.207436
$218$	7.52786	0.509851
$219$	−8.76393	−0.592212
$220$	0	0
$221$	5.23607	0.352216
$222$	2.00000	0.134231
$223$	−9.05573	−0.606416	−0.303208	−	0.952924i	$-0.598058\pi$
$223$	−9.05573	−0.606416	−0.303208	+	0.952924i	$0.598058\pi$
$224$	−1.23607	−0.0825883
$225$	0	0
$226$	−10.0000	−0.665190
$227$	−9.88854	−0.656326	−0.328163	−	0.944621i	$-0.606429\pi$
$227$	−9.88854	−0.656326	−0.328163	+	0.944621i	$0.606429\pi$
$228$	−6.47214	−0.428628
$229$	−21.4164	−1.41524	−0.707618	−	0.706595i	$-0.750230\pi$
$229$	−21.4164	−1.41524	−0.707618	+	0.706595i	$0.750230\pi$
$230$	0	0
$231$	2.47214	0.162655
$232$	4.00000	0.262613
$233$	−5.41641	−0.354841	−0.177420	−	0.984135i	$-0.556775\pi$
$233$	−5.41641	−0.354841	−0.177420	+	0.984135i	$0.556775\pi$
$234$	5.23607	0.342292
$235$	0	0
$236$	11.7082	0.762139
$237$	0.944272	0.0613371
$238$	1.23607	0.0801224
$239$	−21.8885	−1.41585	−0.707926	−	0.706287i	$-0.750369\pi$
$239$	−21.8885	−1.41585	−0.707926	+	0.706287i	$0.750369\pi$
$240$	0	0
$241$	9.41641	0.606564	0.303282	−	0.952901i	$-0.401918\pi$
$241$	9.41641	0.606564	0.303282	+	0.952901i	$0.401918\pi$
$242$	7.00000	0.449977
$243$	1.00000	0.0641500
$244$	4.47214	0.286299
$245$	0	0
$246$	7.70820	0.491457
$247$	33.8885	2.15628
$248$	−2.47214	−0.156981
$249$	−10.4721	−0.663645
$250$	0	0
$251$	−10.1803	−0.642577	−0.321289	−	0.946981i	$-0.604116\pi$
$251$	−10.1803	−0.642577	−0.321289	+	0.946981i	$0.604116\pi$
$252$	1.23607	0.0778650
$253$	−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	$0.689739\pi$
$258$	12.1803	0.758315
$259$	−2.47214	−0.153611
$260$	0	0
$261$	−4.00000	−0.247594
$262$	−12.4721	−0.770531
$263$	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
$267$	12.4721	0.763282
$268$	1.70820	0.104345
$269$	−7.05573	−0.430195	−0.215098	−	0.976593i	$-0.569007\pi$
$269$	−7.05573	−0.430195	−0.215098	+	0.976593i	$0.569007\pi$
$270$	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
$273$	−6.47214	−0.391711
$274$	−2.00000	−0.120824
$275$	0	0
$276$	−4.00000	−0.240772
$277$	−26.3607	−1.58386	−0.791930	−	0.610612i	$-0.790923\pi$
$277$	−26.3607	−1.58386	−0.791930	+	0.610612i	$0.790923\pi$
$278$	−1.52786	−0.0916352
$279$	2.47214	0.148003
$280$	0	0
$281$	1.41641	0.0844958	0.0422479	−	0.999107i	$-0.486548\pi$
$281$	1.41641	0.0844958	0.0422479	+	0.999107i	$0.486548\pi$
$282$	10.4721	0.623607
$283$	25.8885	1.53891	0.769457	−	0.638699i	$-0.220527\pi$
$283$	25.8885	1.53891	0.769457	+	0.638699i	$0.220527\pi$
$284$	11.2361	0.666738
$285$	0	0
$286$	10.4721	0.619230
$287$	−9.52786	−0.562412
$288$	−1.00000	−0.0589256
$289$	1.00000	0.0588235
$290$	0	0
$291$	1.70820	0.100137
$292$	−8.76393	−0.512870
$293$	−29.4164	−1.71852	−0.859262	−	0.511535i	$-0.829077\pi$
$293$	−29.4164	−1.71852	−0.859262	+	0.511535i	$0.829077\pi$
$294$	5.47214	0.319141
$295$	0	0
$296$	2.00000	0.116248
$297$	2.00000	0.116052
$298$	−12.1803	−0.705588
$299$	20.9443	1.21124
$300$	0	0
$301$	−15.0557	−0.867798
$302$	4.00000	0.230174
$303$	−1.70820	−0.0981338
$304$	−6.47214	−0.371202
$305$	0	0
$306$	1.00000	0.0571662
$307$	−4.18034	−0.238585	−0.119292	−	0.992859i	$-0.538063\pi$
$307$	−4.18034	−0.238585	−0.119292	+	0.992859i	$0.538063\pi$
$308$	2.47214	0.140863
$309$	−10.9443	−0.622598
$310$	0	0
$311$	19.5967	1.11123	0.555615	−	0.831440i	$-0.312483\pi$
$311$	19.5967	1.11123	0.555615	+	0.831440i	$0.312483\pi$
$312$	5.23607	0.296434
$313$	21.7082	1.22702	0.613510	−	0.789687i	$-0.289757\pi$
$313$	21.7082	1.22702	0.613510	+	0.789687i	$0.289757\pi$
$314$	10.1803	0.574510
$315$	0	0
$316$	0.944272	0.0531194
$317$	−11.8885	−0.667727	−0.333864	−	0.942621i	$-0.608352\pi$
$317$	−11.8885	−0.667727	−0.333864	+	0.942621i	$0.608352\pi$
$318$	−10.9443	−0.613724
$319$	−8.00000	−0.447914
$320$	0	0
$321$	−8.00000	−0.446516
$322$	4.94427	0.275534
$323$	6.47214	0.360119
$324$	1.00000	0.0555556
$325$	0	0
$326$	−22.4721	−1.24462
$327$	−7.52786	−0.416292
$328$	7.70820	0.425614
$329$	−12.9443	−0.713641
$330$	0	0
$331$	−3.41641	−0.187783	−0.0938914	−	0.995582i	$-0.529931\pi$
$331$	−3.41641	−0.187783	−0.0938914	+	0.995582i	$0.529931\pi$
$332$	−10.4721	−0.574733
$333$	−2.00000	−0.109599
$334$	−13.8885	−0.759947
$335$	0	0
$336$	1.23607	0.0674330
$337$	20.1803	1.09929	0.549647	−	0.835397i	$-0.314762\pi$
$337$	20.1803	1.09929	0.549647	+	0.835397i	$0.314762\pi$
$338$	−14.4164	−0.784149
$339$	10.0000	0.543125
$340$	0	0
$341$	4.94427	0.267747
$342$	6.47214	0.349973
$343$	−15.4164	−0.832408
$344$	12.1803	0.656720
$345$	0	0
$346$	−15.8885	−0.854173
$347$	22.8328	1.22573	0.612865	−	0.790188i	$-0.290017\pi$
$347$	22.8328	1.22573	0.612865	+	0.790188i	$0.290017\pi$
$348$	−4.00000	−0.214423
$349$	−28.4721	−1.52408	−0.762039	−	0.647531i	$-0.775802\pi$
$349$	−28.4721	−1.52408	−0.762039	+	0.647531i	$0.775802\pi$
$350$	0	0
$351$	−5.23607	−0.279481
$352$	−2.00000	−0.106600
$353$	14.9443	0.795403	0.397702	−	0.917515i	$-0.369808\pi$
$353$	14.9443	0.795403	0.397702	+	0.917515i	$0.369808\pi$
$354$	−11.7082	−0.622284
$355$	0	0
$356$	12.4721	0.661022
$357$	−1.23607	−0.0654197
$358$	7.70820	0.407391
$359$	−9.52786	−0.502861	−0.251431	−	0.967875i	$-0.580901\pi$
$359$	−9.52786	−0.502861	−0.251431	+	0.967875i	$0.580901\pi$
$360$	0	0
$361$	22.8885	1.20466
$362$	−0.472136	−0.0248149
$363$	−7.00000	−0.367405
$364$	−6.47214	−0.339232
$365$	0	0
$366$	−4.47214	−0.233762
$367$	5.81966	0.303784	0.151892	−	0.988397i	$-0.451463\pi$
$367$	5.81966	0.303784	0.151892	+	0.988397i	$0.451463\pi$
$368$	−4.00000	−0.208514
$369$	−7.70820	−0.401273
$370$	0	0
$371$	13.5279	0.702332
$372$	2.47214	0.128174
$373$	1.23607	0.0640012	0.0320006	−	0.999488i	$-0.489812\pi$
$373$	1.23607	0.0640012	0.0320006	+	0.999488i	$0.489812\pi$
$374$	2.00000	0.103418
$375$	0	0
$376$	10.4721	0.540059
$377$	20.9443	1.07868
$378$	−1.23607	−0.0635765
$379$	−1.52786	−0.0784811	−0.0392406	−	0.999230i	$-0.512494\pi$
$379$	−1.52786	−0.0784811	−0.0392406	+	0.999230i	$0.512494\pi$
$380$	0	0
$381$	−14.0000	−0.717242
$382$	−5.52786	−0.282830
$383$	−17.8885	−0.914062	−0.457031	−	0.889451i	$-0.651087\pi$
$383$	−17.8885	−0.914062	−0.457031	+	0.889451i	$0.651087\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−23.5967	−1.20104
$387$	−12.1803	−0.619161
$388$	1.70820	0.0867209
$389$	20.1803	1.02318	0.511592	−	0.859229i	$-0.329056\pi$
$389$	20.1803	1.02318	0.511592	+	0.859229i	$0.329056\pi$
$390$	0	0
$391$	4.00000	0.202289
$392$	5.47214	0.276385
$393$	12.4721	0.629136
$394$	15.8885	0.800453
$395$	0	0
$396$	2.00000	0.100504
$397$	−3.88854	−0.195160	−0.0975802	−	0.995228i	$-0.531110\pi$
$397$	−3.88854	−0.195160	−0.0975802	+	0.995228i	$0.531110\pi$
$398$	8.94427	0.448336
$399$	−8.00000	−0.400501
$400$	0	0
$401$	−36.0689	−1.80119	−0.900597	−	0.434655i	$-0.856870\pi$
$401$	−36.0689	−1.80119	−0.900597	+	0.434655i	$0.856870\pi$
$402$	−1.70820	−0.0851975
$403$	−12.9443	−0.644800
$404$	−1.70820	−0.0849863
$405$	0	0
$406$	4.94427	0.245380
$407$	−4.00000	−0.198273
$408$	1.00000	0.0495074
$409$	26.3607	1.30345	0.651726	−	0.758455i	$-0.274045\pi$
$409$	26.3607	1.30345	0.651726	+	0.758455i	$0.274045\pi$
$410$	0	0
$411$	2.00000	0.0986527
$412$	−10.9443	−0.539186
$413$	14.4721	0.712127
$414$	4.00000	0.196589
$415$	0	0
$416$	5.23607	0.256719
$417$	1.52786	0.0748198
$418$	12.9443	0.633125
$419$	6.36068	0.310740	0.155370	−	0.987856i	$-0.450343\pi$
$419$	6.36068	0.310740	0.155370	+	0.987856i	$0.450343\pi$
$420$	0	0
$421$	−37.4164	−1.82356	−0.911782	−	0.410674i	$-0.865293\pi$
$421$	−37.4164	−1.82356	−0.911782	+	0.410674i	$0.865293\pi$
$422$	24.9443	1.21427
$423$	−10.4721	−0.509173
$424$	−10.9443	−0.531501
$425$	0	0
$426$	−11.2361	−0.544389
$427$	5.52786	0.267512
$428$	−8.00000	−0.386695
$429$	−10.4721	−0.505599
$430$	0	0
$431$	31.5967	1.52196	0.760981	−	0.648774i	$-0.224718\pi$
$431$	31.5967	1.52196	0.760981	+	0.648774i	$0.224718\pi$
$432$	1.00000	0.0481125
$433$	−28.3607	−1.36293	−0.681464	−	0.731852i	$-0.738656\pi$
$433$	−28.3607	−1.36293	−0.681464	+	0.731852i	$0.738656\pi$
$434$	−3.05573	−0.146680
$435$	0	0
$436$	−7.52786	−0.360519
$437$	25.8885	1.23842
$438$	8.76393	0.418757
$439$	13.5279	0.645650	0.322825	−	0.946459i	$-0.395368\pi$
$439$	13.5279	0.645650	0.322825	+	0.946459i	$0.395368\pi$
$440$	0	0
$441$	−5.47214	−0.260578
$442$	−5.23607	−0.249054
$443$	−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
$446$	9.05573	0.428801
$447$	12.1803	0.576110
$448$	1.23607	0.0583987
$449$	17.5967	0.830442	0.415221	−	0.909721i	$-0.363704\pi$
$449$	17.5967	0.830442	0.415221	+	0.909721i	$0.363704\pi$
$450$	0	0
$451$	−15.4164	−0.725930
$452$	10.0000	0.470360
$453$	−4.00000	−0.187936
$454$	9.88854	0.464092
$455$	0	0
$456$	6.47214	0.303086
$457$	16.3607	0.765320	0.382660	−	0.923889i	$-0.375008\pi$
$457$	16.3607	0.765320	0.382660	+	0.923889i	$0.375008\pi$
$458$	21.4164	1.00072
$459$	−1.00000	−0.0466760
$460$	0	0
$461$	−30.0689	−1.40045	−0.700224	−	0.713923i	$-0.746916\pi$
$461$	−30.0689	−1.40045	−0.700224	+	0.713923i	$0.746916\pi$
$462$	−2.47214	−0.115014
$463$	18.9443	0.880415	0.440207	−	0.897896i	$-0.354905\pi$
$463$	18.9443	0.880415	0.440207	+	0.897896i	$0.354905\pi$
$464$	−4.00000	−0.185695
$465$	0	0
$466$	5.41641	0.250910
$467$	−0.944272	−0.0436957	−0.0218478	−	0.999761i	$-0.506955\pi$
$467$	−0.944272	−0.0436957	−0.0218478	+	0.999761i	$0.506955\pi$
$468$	−5.23607	−0.242037
$469$	2.11146	0.0974980
$470$	0	0
$471$	−10.1803	−0.469085
$472$	−11.7082	−0.538914
$473$	−24.3607	−1.12011
$474$	−0.944272	−0.0433718
$475$	0	0
$476$	−1.23607	−0.0566551
$477$	10.9443	0.501104
$478$	21.8885	1.00116
$479$	−35.5967	−1.62646	−0.813228	−	0.581945i	$-0.802292\pi$
$479$	−35.5967	−1.62646	−0.813228	+	0.581945i	$0.802292\pi$
$480$	0	0
$481$	10.4721	0.477488
$482$	−9.41641	−0.428906
$483$	−4.94427	−0.224972
$484$	−7.00000	−0.318182
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	16.2918	0.738252	0.369126	−	0.929379i	$-0.379657\pi$
$487$	16.2918	0.738252	0.369126	+	0.929379i	$0.379657\pi$
$488$	−4.47214	−0.202444
$489$	22.4721	1.01623
$490$	0	0
$491$	12.2918	0.554721	0.277360	−	0.960766i	$-0.410540\pi$
$491$	12.2918	0.554721	0.277360	+	0.960766i	$0.410540\pi$
$492$	−7.70820	−0.347513
$493$	4.00000	0.180151
$494$	−33.8885	−1.52472
$495$	0	0
$496$	2.47214	0.111002
$497$	13.8885	0.622986
$498$	10.4721	0.469268
$499$	−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
$501$	13.8885	0.620494
$502$	10.1803	0.454371
$503$	27.4164	1.22244	0.611219	−	0.791462i	$-0.290680\pi$
$503$	27.4164	1.22244	0.611219	+	0.791462i	$0.290680\pi$
$504$	−1.23607	−0.0550588
$505$	0	0
$506$	8.00000	0.355643
$507$	14.4164	0.640255
$508$	−14.0000	−0.621150
$509$	1.34752	0.0597280	0.0298640	−	0.999554i	$-0.490493\pi$
$509$	1.34752	0.0597280	0.0298640	+	0.999554i	$0.490493\pi$
$510$	0	0
$511$	−10.8328	−0.479216
$512$	−1.00000	−0.0441942
$513$	−6.47214	−0.285752
$514$	18.0000	0.793946
$515$	0	0
$516$	−12.1803	−0.536210
$517$	−20.9443	−0.921128
$518$	2.47214	0.108619
$519$	15.8885	0.697430
$520$	0	0
$521$	5.81966	0.254964	0.127482	−	0.991841i	$-0.459311\pi$
$521$	5.81966	0.254964	0.127482	+	0.991841i	$0.459311\pi$
$522$	4.00000	0.175075
$523$	14.2918	0.624937	0.312468	−	0.949928i	$-0.398844\pi$
$523$	14.2918	0.624937	0.312468	+	0.949928i	$0.398844\pi$
$524$	12.4721	0.544848
$525$	0	0
$526$	−4.00000	−0.174408
$527$	−2.47214	−0.107688
$528$	2.00000	0.0870388
$529$	−7.00000	−0.304348
$530$	0	0
$531$	11.7082	0.508093
$532$	−8.00000	−0.346844
$533$	40.3607	1.74822
$534$	−12.4721	−0.539722
$535$	0	0
$536$	−1.70820	−0.0737832
$537$	−7.70820	−0.332634
$538$	7.05573	0.304194
$539$	−10.9443	−0.471403
$540$	0	0
$541$	−31.3050	−1.34590	−0.672952	−	0.739686i	$-0.734974\pi$
$541$	−31.3050	−1.34590	−0.672952	+	0.739686i	$0.734974\pi$
$542$	−4.00000	−0.171815
$543$	0.472136	0.0202613
$544$	1.00000	0.0428746
$545$	0	0
$546$	6.47214	0.276982
$547$	−0.944272	−0.0403742	−0.0201871	−	0.999796i	$-0.506426\pi$
$547$	−0.944272	−0.0403742	−0.0201871	+	0.999796i	$0.506426\pi$
$548$	2.00000	0.0854358
$549$	4.47214	0.190866
$550$	0	0
$551$	25.8885	1.10289
$552$	4.00000	0.170251
$553$	1.16718	0.0496337
$554$	26.3607	1.11996
$555$	0	0
$556$	1.52786	0.0647959
$557$	25.0557	1.06165	0.530823	−	0.847483i	$-0.321883\pi$
$557$	25.0557	1.06165	0.530823	+	0.847483i	$0.321883\pi$
$558$	−2.47214	−0.104654
$559$	63.7771	2.69748
$560$	0	0
$561$	−2.00000	−0.0844401
$562$	−1.41641	−0.0597476
$563$	−2.11146	−0.0889873	−0.0444936	−	0.999010i	$-0.514167\pi$
$563$	−2.11146	−0.0889873	−0.0444936	+	0.999010i	$0.514167\pi$
$564$	−10.4721	−0.440956
$565$	0	0
$566$	−25.8885	−1.08818
$567$	1.23607	0.0519100
$568$	−11.2361	−0.471455
$569$	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
$571$	36.3607	1.52165	0.760824	−	0.648959i	$-0.224795\pi$
$571$	36.3607	1.52165	0.760824	+	0.648959i	$0.224795\pi$
$572$	−10.4721	−0.437862
$573$	5.52786	0.230930
$574$	9.52786	0.397685
$575$	0	0
$576$	1.00000	0.0416667
$577$	35.4164	1.47440	0.737202	−	0.675672i	$-0.236147\pi$
$577$	35.4164	1.47440	0.737202	+	0.675672i	$0.236147\pi$
$578$	−1.00000	−0.0415945
$579$	23.5967	0.980647
$580$	0	0
$581$	−12.9443	−0.537019
$582$	−1.70820	−0.0708073
$583$	21.8885	0.906531
$584$	8.76393	0.362654
$585$	0	0
$586$	29.4164	1.21518
$587$	5.52786	0.228159	0.114080	−	0.993472i	$-0.463608\pi$
$587$	5.52786	0.228159	0.114080	+	0.993472i	$0.463608\pi$
$588$	−5.47214	−0.225667
$589$	−16.0000	−0.659269
$590$	0	0
$591$	−15.8885	−0.653567
$592$	−2.00000	−0.0821995
$593$	−30.9443	−1.27073	−0.635364	−	0.772212i	$-0.719150\pi$
$593$	−30.9443	−1.27073	−0.635364	+	0.772212i	$0.719150\pi$
$594$	−2.00000	−0.0820610
$595$	0	0
$596$	12.1803	0.498926
$597$	−8.94427	−0.366065
$598$	−20.9443	−0.856475
$599$	25.5279	1.04304	0.521520	−	0.853239i	$-0.325365\pi$
$599$	25.5279	1.04304	0.521520	+	0.853239i	$0.325365\pi$
$600$	0	0
$601$	−28.4721	−1.16140	−0.580701	−	0.814117i	$-0.697222\pi$
$601$	−28.4721	−1.16140	−0.580701	+	0.814117i	$0.697222\pi$
$602$	15.0557	0.613626
$603$	1.70820	0.0695634
$604$	−4.00000	−0.162758
$605$	0	0
$606$	1.70820	0.0693910
$607$	31.1246	1.26331	0.631655	−	0.775250i	$-0.282376\pi$
$607$	31.1246	1.26331	0.631655	+	0.775250i	$0.282376\pi$
$608$	6.47214	0.262480
$609$	−4.94427	−0.200352
$610$	0	0
$611$	54.8328	2.21830
$612$	−1.00000	−0.0404226
$613$	42.1803	1.70365	0.851824	−	0.523828i	$-0.175497\pi$
$613$	42.1803	1.70365	0.851824	+	0.523828i	$0.175497\pi$
$614$	4.18034	0.168705
$615$	0	0
$616$	−2.47214	−0.0996052
$617$	20.4721	0.824177	0.412089	−	0.911144i	$-0.364799\pi$
$617$	20.4721	0.824177	0.412089	+	0.911144i	$0.364799\pi$
$618$	10.9443	0.440243
$619$	−48.9443	−1.96724	−0.983618	−	0.180264i	$-0.942305\pi$
$619$	−48.9443	−1.96724	−0.983618	+	0.180264i	$0.942305\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	−19.5967	−0.785758
$623$	15.4164	0.617645
$624$	−5.23607	−0.209610
$625$	0	0
$626$	−21.7082	−0.867634
$627$	−12.9443	−0.516944
$628$	−10.1803	−0.406240
$629$	2.00000	0.0797452
$630$	0	0
$631$	−9.88854	−0.393657	−0.196828	−	0.980438i	$-0.563064\pi$
$631$	−9.88854	−0.393657	−0.196828	+	0.980438i	$0.563064\pi$
$632$	−0.944272	−0.0375611
$633$	−24.9443	−0.991446
$634$	11.8885	0.472154
$635$	0	0
$636$	10.9443	0.433969
$637$	28.6525	1.13525
$638$	8.00000	0.316723
$639$	11.2361	0.444492
$640$	0	0
$641$	11.7082	0.462446	0.231223	−	0.972901i	$-0.425727\pi$
$641$	11.7082	0.462446	0.231223	+	0.972901i	$0.425727\pi$
$642$	8.00000	0.315735
$643$	27.4164	1.08120	0.540599	−	0.841281i	$-0.318198\pi$
$643$	27.4164	1.08120	0.540599	+	0.841281i	$0.318198\pi$
$644$	−4.94427	−0.194832
$645$	0	0
$646$	−6.47214	−0.254643
$647$	20.9443	0.823404	0.411702	−	0.911318i	$-0.364934\pi$
$647$	20.9443	0.823404	0.411702	+	0.911318i	$0.364934\pi$
$648$	−1.00000	−0.0392837
$649$	23.4164	0.919174
$650$	0	0
$651$	3.05573	0.119763
$652$	22.4721	0.880077
$653$	3.52786	0.138056	0.0690280	−	0.997615i	$-0.478010\pi$
$653$	3.52786	0.138056	0.0690280	+	0.997615i	$0.478010\pi$
$654$	7.52786	0.294363
$655$	0	0
$656$	−7.70820	−0.300955
$657$	−8.76393	−0.341914
$658$	12.9443	0.504620
$659$	28.2918	1.10209	0.551046	−	0.834475i	$-0.314229\pi$
$659$	28.2918	1.10209	0.551046	+	0.834475i	$0.314229\pi$
$660$	0	0
$661$	13.4164	0.521838	0.260919	−	0.965361i	$-0.415974\pi$
$661$	13.4164	0.521838	0.260919	+	0.965361i	$0.415974\pi$
$662$	3.41641	0.132782
$663$	5.23607	0.203352
$664$	10.4721	0.406398
$665$	0	0
$666$	2.00000	0.0774984
$667$	16.0000	0.619522
$668$	13.8885	0.537364
$669$	−9.05573	−0.350115
$670$	0	0
$671$	8.94427	0.345290
$672$	−1.23607	−0.0476824
$673$	−4.18034	−0.161140	−0.0805701	−	0.996749i	$-0.525674\pi$
$673$	−4.18034	−0.161140	−0.0805701	+	0.996749i	$0.525674\pi$
$674$	−20.1803	−0.777318
$675$	0	0
$676$	14.4164	0.554477
$677$	9.05573	0.348040	0.174020	−	0.984742i	$-0.444324\pi$
$677$	9.05573	0.348040	0.174020	+	0.984742i	$0.444324\pi$
$678$	−10.0000	−0.384048
$679$	2.11146	0.0810303
$680$	0	0
$681$	−9.88854	−0.378930
$682$	−4.94427	−0.189326
$683$	32.9443	1.26058	0.630289	−	0.776361i	$-0.282936\pi$
$683$	32.9443	1.26058	0.630289	+	0.776361i	$0.282936\pi$
$684$	−6.47214	−0.247468
$685$	0	0
$686$	15.4164	0.588601
$687$	−21.4164	−0.817087
$688$	−12.1803	−0.464371
$689$	−57.3050	−2.18314
$690$	0	0
$691$	−4.94427	−0.188089	−0.0940445	−	0.995568i	$-0.529980\pi$
$691$	−4.94427	−0.188089	−0.0940445	+	0.995568i	$0.529980\pi$
$692$	15.8885	0.603992
$693$	2.47214	0.0939087
$694$	−22.8328	−0.866722
$695$	0	0
$696$	4.00000	0.151620
$697$	7.70820	0.291969
$698$	28.4721	1.07769
$699$	−5.41641	−0.204867
$700$	0	0
$701$	11.5967	0.438003	0.219002	−	0.975725i	$-0.429720\pi$
$701$	11.5967	0.438003	0.219002	+	0.975725i	$0.429720\pi$
$702$	5.23607	0.197623
$703$	12.9443	0.488202
$704$	2.00000	0.0753778
$705$	0	0
$706$	−14.9443	−0.562435
$707$	−2.11146	−0.0794095
$708$	11.7082	0.440021
$709$	−39.5279	−1.48450	−0.742250	−	0.670123i	$-0.766241\pi$
$709$	−39.5279	−1.48450	−0.742250	+	0.670123i	$0.766241\pi$
$710$	0	0
$711$	0.944272	0.0354130
$712$	−12.4721	−0.467413
$713$	−9.88854	−0.370329
$714$	1.23607	0.0462587
$715$	0	0
$716$	−7.70820	−0.288069
$717$	−21.8885	−0.817443
$718$	9.52786	0.355577
$719$	−36.5410	−1.36275	−0.681375	−	0.731934i	$-0.738618\pi$
$719$	−36.5410	−1.36275	−0.681375	+	0.731934i	$0.738618\pi$
$720$	0	0
$721$	−13.5279	−0.503804
$722$	−22.8885	−0.851823
$723$	9.41641	0.350200
$724$	0.472136	0.0175468
$725$	0	0
$726$	7.00000	0.259794
$727$	23.8885	0.885977	0.442989	−	0.896527i	$-0.353918\pi$
$727$	23.8885	0.885977	0.442989	+	0.896527i	$0.353918\pi$
$728$	6.47214	0.239873
$729$	1.00000	0.0370370
$730$	0	0
$731$	12.1803	0.450506
$732$	4.47214	0.165295
$733$	−38.5410	−1.42355	−0.711773	−	0.702410i	$-0.752107\pi$
$733$	−38.5410	−1.42355	−0.711773	+	0.702410i	$0.752107\pi$
$734$	−5.81966	−0.214808
$735$	0	0
$736$	4.00000	0.147442
$737$	3.41641	0.125845
$738$	7.70820	0.283743
$739$	33.5279	1.23334	0.616671	−	0.787221i	$-0.288481\pi$
$739$	33.5279	1.23334	0.616671	+	0.787221i	$0.288481\pi$
$740$	0	0
$741$	33.8885	1.24493
$742$	−13.5279	−0.496624
$743$	−33.5279	−1.23002	−0.615009	−	0.788520i	$-0.710848\pi$
$743$	−33.5279	−1.23002	−0.615009	+	0.788520i	$0.710848\pi$
$744$	−2.47214	−0.0906329
$745$	0	0
$746$	−1.23607	−0.0452557
$747$	−10.4721	−0.383155
$748$	−2.00000	−0.0731272
$749$	−9.88854	−0.361320
$750$	0	0
$751$	29.8885	1.09065	0.545324	−	0.838225i	$-0.316407\pi$
$751$	29.8885	1.09065	0.545324	+	0.838225i	$0.316407\pi$
$752$	−10.4721	−0.381880
$753$	−10.1803	−0.370992
$754$	−20.9443	−0.762745
$755$	0	0
$756$	1.23607	0.0449554
$757$	37.5967	1.36648	0.683239	−	0.730195i	$-0.260571\pi$
$757$	37.5967	1.36648	0.683239	+	0.730195i	$0.260571\pi$
$758$	1.52786	0.0554945
$759$	−8.00000	−0.290382
$760$	0	0
$761$	52.8328	1.91519	0.957594	−	0.288121i	$-0.0930305\pi$
$761$	52.8328	1.91519	0.957594	+	0.288121i	$0.0930305\pi$
$762$	14.0000	0.507166
$763$	−9.30495	−0.336862
$764$	5.52786	0.199991
$765$	0	0
$766$	17.8885	0.646339
$767$	−61.3050	−2.21359
$768$	1.00000	0.0360844
$769$	−34.3607	−1.23908	−0.619539	−	0.784966i	$-0.712680\pi$
$769$	−34.3607	−1.23908	−0.619539	+	0.784966i	$0.712680\pi$
$770$	0	0
$771$	−18.0000	−0.648254
$772$	23.5967	0.849266
$773$	−33.4164	−1.20190	−0.600952	−	0.799285i	$-0.705212\pi$
$773$	−33.4164	−1.20190	−0.600952	+	0.799285i	$0.705212\pi$
$774$	12.1803	0.437813
$775$	0	0
$776$	−1.70820	−0.0613209
$777$	−2.47214	−0.0886874
$778$	−20.1803	−0.723500
$779$	49.8885	1.78744
$780$	0	0
$781$	22.4721	0.804116
$782$	−4.00000	−0.143040
$783$	−4.00000	−0.142948
$784$	−5.47214	−0.195433
$785$	0	0
$786$	−12.4721	−0.444866
$787$	−46.4721	−1.65655	−0.828276	−	0.560320i	$-0.810678\pi$
$787$	−46.4721	−1.65655	−0.828276	+	0.560320i	$0.810678\pi$
$788$	−15.8885	−0.566006
$789$	4.00000	0.142404
$790$	0	0
$791$	12.3607	0.439495
$792$	−2.00000	−0.0710669
$793$	−23.4164	−0.831541
$794$	3.88854	0.137999
$795$	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
$799$	10.4721	0.370478
$800$	0	0
$801$	12.4721	0.440681
$802$	36.0689	1.27364
$803$	−17.5279	−0.618545
$804$	1.70820	0.0602437
$805$	0	0
$806$	12.9443	0.455943
$807$	−7.05573	−0.248373
$808$	1.70820	0.0600944
$809$	47.1246	1.65681	0.828407	−	0.560127i	$-0.189248\pi$
$809$	47.1246	1.65681	0.828407	+	0.560127i	$0.189248\pi$
$810$	0	0
$811$	42.2492	1.48357	0.741785	−	0.670637i	$-0.233979\pi$
$811$	42.2492	1.48357	0.741785	+	0.670637i	$0.233979\pi$
$812$	−4.94427	−0.173510
$813$	4.00000	0.140286
$814$	4.00000	0.140200
$815$	0	0
$816$	−1.00000	−0.0350070
$817$	78.8328	2.75801
$818$	−26.3607	−0.921680
$819$	−6.47214	−0.226155
$820$	0	0
$821$	15.4164	0.538036	0.269018	−	0.963135i	$-0.413301\pi$
$821$	15.4164	0.538036	0.269018	+	0.963135i	$0.413301\pi$
$822$	−2.00000	−0.0697580
$823$	−38.1803	−1.33088	−0.665441	−	0.746450i	$-0.731757\pi$
$823$	−38.1803	−1.33088	−0.665441	+	0.746450i	$0.731757\pi$
$824$	10.9443	0.381262
$825$	0	0
$826$	−14.4721	−0.503550
$827$	23.0557	0.801726	0.400863	−	0.916138i	$-0.368710\pi$
$827$	23.0557	0.801726	0.400863	+	0.916138i	$0.368710\pi$
$828$	−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
$831$	−26.3607	−0.914442
$832$	−5.23607	−0.181528
$833$	5.47214	0.189598
$834$	−1.52786	−0.0529056
$835$	0	0
$836$	−12.9443	−0.447687
$837$	2.47214	0.0854495
$838$	−6.36068	−0.219726
$839$	−22.0689	−0.761902	−0.380951	−	0.924595i	$-0.624403\pi$
$839$	−22.0689	−0.761902	−0.380951	+	0.924595i	$0.624403\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	37.4164	1.28945
$843$	1.41641	0.0487837
$844$	−24.9443	−0.858617
$845$	0	0
$846$	10.4721	0.360039
$847$	−8.65248	−0.297303
$848$	10.9443	0.375828
$849$	25.8885	0.888493
$850$	0	0
$851$	8.00000	0.274236
$852$	11.2361	0.384941
$853$	6.36068	0.217786	0.108893	−	0.994054i	$-0.465269\pi$
$853$	6.36068	0.217786	0.108893	+	0.994054i	$0.465269\pi$
$854$	−5.52786	−0.189160
$855$	0	0
$856$	8.00000	0.273434
$857$	−50.3607	−1.72029	−0.860144	−	0.510051i	$-0.829626\pi$
$857$	−50.3607	−1.72029	−0.860144	+	0.510051i	$0.829626\pi$
$858$	10.4721	0.357513
$859$	7.05573	0.240738	0.120369	−	0.992729i	$-0.461592\pi$
$859$	7.05573	0.240738	0.120369	+	0.992729i	$0.461592\pi$
$860$	0	0
$861$	−9.52786	−0.324709
$862$	−31.5967	−1.07619
$863$	21.3050	0.725229	0.362614	−	0.931939i	$-0.381884\pi$
$863$	21.3050	0.725229	0.362614	+	0.931939i	$0.381884\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	28.3607	0.963735
$867$	1.00000	0.0339618
$868$	3.05573	0.103718
$869$	1.88854	0.0640645
$870$	0	0
$871$	−8.94427	−0.303065
$872$	7.52786	0.254926
$873$	1.70820	0.0578139
$874$	−25.8885	−0.875693
$875$	0	0
$876$	−8.76393	−0.296106
$877$	40.4721	1.36665	0.683323	−	0.730116i	$-0.260534\pi$
$877$	40.4721	1.36665	0.683323	+	0.730116i	$0.260534\pi$
$878$	−13.5279	−0.456543
$879$	−29.4164	−0.992191
$880$	0	0
$881$	−7.12461	−0.240034	−0.120017	−	0.992772i	$-0.538295\pi$
$881$	−7.12461	−0.240034	−0.120017	+	0.992772i	$0.538295\pi$
$882$	5.47214	0.184256
$883$	27.2361	0.916567	0.458283	−	0.888806i	$-0.348465\pi$
$883$	27.2361	0.916567	0.458283	+	0.888806i	$0.348465\pi$
$884$	5.23607	0.176108
$885$	0	0
$886$	4.00000	0.134383
$887$	−8.94427	−0.300319	−0.150160	−	0.988662i	$-0.547979\pi$
$887$	−8.94427	−0.300319	−0.150160	+	0.988662i	$0.547979\pi$
$888$	2.00000	0.0671156
$889$	−17.3050	−0.580389
$890$	0	0
$891$	2.00000	0.0670025
$892$	−9.05573	−0.303208
$893$	67.7771	2.26807
$894$	−12.1803	−0.407372
$895$	0	0
$896$	−1.23607	−0.0412941
$897$	20.9443	0.699309
$898$	−17.5967	−0.587211
$899$	−9.88854	−0.329801
$900$	0	0
$901$	−10.9443	−0.364607
$902$	15.4164	0.513310
$903$	−15.0557	−0.501023
$904$	−10.0000	−0.332595
$905$	0	0
$906$	4.00000	0.132891
$907$	−4.36068	−0.144794	−0.0723970	−	0.997376i	$-0.523065\pi$
$907$	−4.36068	−0.144794	−0.0723970	+	0.997376i	$0.523065\pi$
$908$	−9.88854	−0.328163
$909$	−1.70820	−0.0566575
$910$	0	0
$911$	−12.5410	−0.415503	−0.207751	−	0.978182i	$-0.566614\pi$
$911$	−12.5410	−0.415503	−0.207751	+	0.978182i	$0.566614\pi$
$912$	−6.47214	−0.214314
$913$	−20.9443	−0.693154
$914$	−16.3607	−0.541163
$915$	0	0
$916$	−21.4164	−0.707618
$917$	15.4164	0.509095
$918$	1.00000	0.0330049
$919$	52.7214	1.73912	0.869559	−	0.493830i	$-0.164403\pi$
$919$	52.7214	1.73912	0.869559	+	0.493830i	$0.164403\pi$
$920$	0	0
$921$	−4.18034	−0.137747
$922$	30.0689	0.990266
$923$	−58.8328	−1.93651
$924$	2.47214	0.0813273
$925$	0	0
$926$	−18.9443	−0.622547
$927$	−10.9443	−0.359457
$928$	4.00000	0.131306
$929$	24.0689	0.789674	0.394837	−	0.918751i	$-0.370801\pi$
$929$	24.0689	0.789674	0.394837	+	0.918751i	$0.370801\pi$
$930$	0	0
$931$	35.4164	1.16073
$932$	−5.41641	−0.177420
$933$	19.5967	0.641569
$934$	0.944272	0.0308975
$935$	0	0
$936$	5.23607	0.171146
$937$	−17.5279	−0.572610	−0.286305	−	0.958138i	$-0.592427\pi$
$937$	−17.5279	−0.572610	−0.286305	+	0.958138i	$0.592427\pi$
$938$	−2.11146	−0.0689415
$939$	21.7082	0.708420
$940$	0	0
$941$	13.3050	0.433729	0.216865	−	0.976202i	$-0.430417\pi$
$941$	13.3050	0.433729	0.216865	+	0.976202i	$0.430417\pi$
$942$	10.1803	0.331693
$943$	30.8328	1.00405
$944$	11.7082	0.381070
$945$	0	0
$946$	24.3607	0.792034
$947$	−48.7214	−1.58323	−0.791616	−	0.611019i	$-0.790760\pi$
$947$	−48.7214	−1.58323	−0.791616	+	0.611019i	$0.790760\pi$
$948$	0.944272	0.0306685
$949$	45.8885	1.48961
$950$	0	0
$951$	−11.8885	−0.385512
$952$	1.23607	0.0400612
$953$	−26.9443	−0.872811	−0.436405	−	0.899750i	$-0.643749\pi$
$953$	−26.9443	−0.872811	−0.436405	+	0.899750i	$0.643749\pi$
$954$	−10.9443	−0.354334
$955$	0	0
$956$	−21.8885	−0.707926
$957$	−8.00000	−0.258603
$958$	35.5967	1.15008
$959$	2.47214	0.0798294
$960$	0	0
$961$	−24.8885	−0.802856
$962$	−10.4721	−0.337635
$963$	−8.00000	−0.257796
$964$	9.41641	0.303282
$965$	0	0
$966$	4.94427	0.159079
$967$	0.472136	0.0151829	0.00759143	−	0.999971i	$-0.497584\pi$
$967$	0.472136	0.0151829	0.00759143	+	0.999971i	$0.497584\pi$
$968$	7.00000	0.224989
$969$	6.47214	0.207915
$970$	0	0
$971$	−22.7639	−0.730529	−0.365265	−	0.930904i	$-0.619022\pi$
$971$	−22.7639	−0.730529	−0.365265	+	0.930904i	$0.619022\pi$
$972$	1.00000	0.0320750
$973$	1.88854	0.0605439
$974$	−16.2918	−0.522023
$975$	0	0
$976$	4.47214	0.143150
$977$	−26.9443	−0.862024	−0.431012	−	0.902346i	$-0.641843\pi$
$977$	−26.9443	−0.862024	−0.431012	+	0.902346i	$0.641843\pi$
$978$	−22.4721	−0.718580
$979$	24.9443	0.797222
$980$	0	0
$981$	−7.52786	−0.240346
$982$	−12.2918	−0.392247
$983$	−11.4164	−0.364127	−0.182063	−	0.983287i	$-0.558278\pi$
$983$	−11.4164	−0.364127	−0.182063	+	0.983287i	$0.558278\pi$
$984$	7.70820	0.245729
$985$	0	0
$986$	−4.00000	−0.127386
$987$	−12.9443	−0.412021
$988$	33.8885	1.07814
$989$	48.7214	1.54925
$990$	0	0
$991$	−10.8328	−0.344116	−0.172058	−	0.985087i	$-0.555042\pi$
$991$	−10.8328	−0.344116	−0.172058	+	0.985087i	$0.555042\pi$
$992$	−2.47214	−0.0784904
$993$	−3.41641	−0.108416
$994$	−13.8885	−0.440518
$995$	0	0
$996$	−10.4721	−0.331822
$997$	−41.0557	−1.30025	−0.650124	−	0.759828i	$-0.725283\pi$
$997$	−41.0557	−1.30025	−0.650124	+	0.759828i	$0.725283\pi$
$998$	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.1.2		2
3.2	odd	2		7650.2.a.da.1.2		2
5.2	odd	4		510.2.d.b.409.2	✓	4
5.3	odd	4		510.2.d.b.409.3	yes	4
5.4	even	2		2550.2.a.bk.1.1		2
15.2	even	4		1530.2.d.f.919.3		4
15.8	even	4		1530.2.d.f.919.2		4
15.14	odd	2		7650.2.a.cx.1.1		2
20.3	even	4		4080.2.m.m.2449.1		4
20.7	even	4		4080.2.m.m.2449.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
510.2.d.b.409.2	✓	4	5.2	odd	4
510.2.d.b.409.3	yes	4	5.3	odd	4
1530.2.d.f.919.2		4	15.8	even	4
1530.2.d.f.919.3		4	15.2	even	4
2550.2.a.bh.1.2		2	1.1	even	1	trivial
2550.2.a.bk.1.1		2	5.4	even	2
4080.2.m.m.2449.1		4	20.3	even	4
4080.2.m.m.2449.4		4	20.7	even	4
7650.2.a.cx.1.1		2	15.14	odd	2
7650.2.a.da.1.2		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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.23607 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{11} +1.00000 q^{12} -5.23607 q^{13} -1.23607 q^{14} +1.00000 q^{16} -1.00000 q^{17} -1.00000 q^{18} -6.47214 q^{19} +1.23607 q^{21} -2.00000 q^{22} -4.00000 q^{23} -1.00000 q^{24} +5.23607 q^{26} +1.00000 q^{27} +1.23607 q^{28} -4.00000 q^{29} +2.47214 q^{31} -1.00000 q^{32} +2.00000 q^{33} +1.00000 q^{34} +1.00000 q^{36} -2.00000 q^{37} +6.47214 q^{38} -5.23607 q^{39} -7.70820 q^{41} -1.23607 q^{42} -12.1803 q^{43} +2.00000 q^{44} +4.00000 q^{46} -10.4721 q^{47} +1.00000 q^{48} -5.47214 q^{49} -1.00000 q^{51} -5.23607 q^{52} +10.9443 q^{53} -1.00000 q^{54} -1.23607 q^{56} -6.47214 q^{57} +4.00000 q^{58} +11.7082 q^{59} +4.47214 q^{61} -2.47214 q^{62} +1.23607 q^{63} +1.00000 q^{64} -2.00000 q^{66} +1.70820 q^{67} -1.00000 q^{68} -4.00000 q^{69} +11.2361 q^{71} -1.00000 q^{72} -8.76393 q^{73} +2.00000 q^{74} -6.47214 q^{76} +2.47214 q^{77} +5.23607 q^{78} +0.944272 q^{79} +1.00000 q^{81} +7.70820 q^{82} -10.4721 q^{83} +1.23607 q^{84} +12.1803 q^{86} -4.00000 q^{87} -2.00000 q^{88} +12.4721 q^{89} -6.47214 q^{91} -4.00000 q^{92} +2.47214 q^{93} +10.4721 q^{94} -1.00000 q^{96} +1.70820 q^{97} +5.47214 q^{98} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 2 q^{7} - 2 q^{8} + 2 q^{9} + 4 q^{11} + 2 q^{12} - 6 q^{13} + 2 q^{14} + 2 q^{16} - 2 q^{17} - 2 q^{18} - 4 q^{19} - 2 q^{21} - 4 q^{22} - 8 q^{23} - 2 q^{24}+ \cdots + 4 q^{99}+O(q^{100}) \) 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

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Database

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