LMFDB - Embedded newform 2150.2.a.v.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2150,2,Mod(1,2150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2150, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2150.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2150 = 2 \cdot 5^{2} \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2150.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:	$17.1678364346$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 430)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	2150.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.41421 q^{3} +1.00000 q^{4} +1.41421 q^{6} -1.00000 q^{7} -1.00000 q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.41421 q^{3} +1.00000 q^{4} +1.41421 q^{6} -1.00000 q^{7} -1.00000 q^{8} -1.00000 q^{9} +0.585786 q^{11} -1.41421 q^{12} +1.82843 q^{13} +1.00000 q^{14} +1.00000 q^{16} -1.41421 q^{17} +1.00000 q^{18} -1.00000 q^{19} +1.41421 q^{21} -0.585786 q^{22} +5.07107 q^{23} +1.41421 q^{24} -1.82843 q^{26} +5.65685 q^{27} -1.00000 q^{28} +1.24264 q^{29} -0.414214 q^{31} -1.00000 q^{32} -0.828427 q^{33} +1.41421 q^{34} -1.00000 q^{36} -2.24264 q^{37} +1.00000 q^{38} -2.58579 q^{39} -1.82843 q^{41} -1.41421 q^{42} -1.00000 q^{43} +0.585786 q^{44} -5.07107 q^{46} -7.07107 q^{47} -1.41421 q^{48} -6.00000 q^{49} +2.00000 q^{51} +1.82843 q^{52} +5.65685 q^{53} -5.65685 q^{54} +1.00000 q^{56} +1.41421 q^{57} -1.24264 q^{58} -1.17157 q^{59} +0.0710678 q^{61} +0.414214 q^{62} +1.00000 q^{63} +1.00000 q^{64} +0.828427 q^{66} +1.24264 q^{67} -1.41421 q^{68} -7.17157 q^{69} +11.8995 q^{71} +1.00000 q^{72} +9.24264 q^{73} +2.24264 q^{74} -1.00000 q^{76} -0.585786 q^{77} +2.58579 q^{78} -8.41421 q^{79} -5.00000 q^{81} +1.82843 q^{82} -3.65685 q^{83} +1.41421 q^{84} +1.00000 q^{86} -1.75736 q^{87} -0.585786 q^{88} -9.07107 q^{89} -1.82843 q^{91} +5.07107 q^{92} +0.585786 q^{93} +7.07107 q^{94} +1.41421 q^{96} -13.5563 q^{97} +6.00000 q^{98} -0.585786 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{7} - 2 q^{8} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^7 - 2 * q^8 - 2 * q^9	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{7} - 2 q^{8} - 2 q^{9} + 4 q^{11} - 2 q^{13} + 2 q^{14} + 2 q^{16} + 2 q^{18} - 2 q^{19} - 4 q^{22} - 4 q^{23} + 2 q^{26} - 2 q^{28} - 6 q^{29} + 2 q^{31} - 2 q^{32} + 4 q^{33} - 2 q^{36} + 4 q^{37} + 2 q^{38} - 8 q^{39} + 2 q^{41} - 2 q^{43} + 4 q^{44} + 4 q^{46} - 12 q^{49} + 4 q^{51} - 2 q^{52} + 2 q^{56} + 6 q^{58} - 8 q^{59} - 14 q^{61} - 2 q^{62} + 2 q^{63} + 2 q^{64} - 4 q^{66} - 6 q^{67} - 20 q^{69} + 4 q^{71} + 2 q^{72} + 10 q^{73} - 4 q^{74} - 2 q^{76} - 4 q^{77} + 8 q^{78} - 14 q^{79} - 10 q^{81} - 2 q^{82} + 4 q^{83} + 2 q^{86} - 12 q^{87} - 4 q^{88} - 4 q^{89} + 2 q^{91} - 4 q^{92} + 4 q^{93} + 4 q^{97} + 12 q^{98} - 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^7 - 2 * q^8 - 2 * q^9 + 4 * q^11 - 2 * q^13 + 2 * q^14 + 2 * q^16 + 2 * q^18 - 2 * q^19 - 4 * q^22 - 4 * q^23 + 2 * q^26 - 2 * q^28 - 6 * q^29 + 2 * q^31 - 2 * q^32 + 4 * q^33 - 2 * q^36 + 4 * q^37 + 2 * q^38 - 8 * q^39 + 2 * q^41 - 2 * q^43 + 4 * q^44 + 4 * q^46 - 12 * q^49 + 4 * q^51 - 2 * q^52 + 2 * q^56 + 6 * q^58 - 8 * q^59 - 14 * q^61 - 2 * q^62 + 2 * q^63 + 2 * q^64 - 4 * q^66 - 6 * q^67 - 20 * q^69 + 4 * q^71 + 2 * q^72 + 10 * q^73 - 4 * q^74 - 2 * q^76 - 4 * q^77 + 8 * q^78 - 14 * q^79 - 10 * q^81 - 2 * q^82 + 4 * q^83 + 2 * q^86 - 12 * q^87 - 4 * q^88 - 4 * q^89 + 2 * q^91 - 4 * q^92 + 4 * q^93 + 4 * q^97 + 12 * q^98 - 4 * q^99

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
$3$	−1.41421	−0.816497	−0.408248	−	0.912871i	$-0.633860\pi$
$3$	−1.41421	−0.816497	−0.408248	+	0.912871i	$0.633860\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	1.41421	0.577350
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	−1.00000	−0.353553
$9$	−1.00000	−0.333333
$10$	0	0
$11$	0.585786	0.176621	0.0883106	−	0.996093i	$-0.471853\pi$
$11$	0.585786	0.176621	0.0883106	+	0.996093i	$0.471853\pi$
$12$	−1.41421	−0.408248
$13$	1.82843	0.507114	0.253557	−	0.967320i	$-0.418399\pi$
$13$	1.82843	0.507114	0.253557	+	0.967320i	$0.418399\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−1.41421	−0.342997	−0.171499	−	0.985184i	$-0.554861\pi$
$17$	−1.41421	−0.342997	−0.171499	+	0.985184i	$0.554861\pi$
$18$	1.00000	0.235702
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	0	0
$21$	1.41421	0.308607
$22$	−0.585786	−0.124890
$23$	5.07107	1.05739	0.528695	−	0.848812i	$-0.322681\pi$
$23$	5.07107	1.05739	0.528695	+	0.848812i	$0.322681\pi$
$24$	1.41421	0.288675
$25$	0	0
$26$	−1.82843	−0.358584
$27$	5.65685	1.08866
$28$	−1.00000	−0.188982
$29$	1.24264	0.230753	0.115376	−	0.993322i	$-0.463193\pi$
$29$	1.24264	0.230753	0.115376	+	0.993322i	$0.463193\pi$
$30$	0	0
$31$	−0.414214	−0.0743950	−0.0371975	−	0.999308i	$-0.511843\pi$
$31$	−0.414214	−0.0743950	−0.0371975	+	0.999308i	$0.511843\pi$
$32$	−1.00000	−0.176777
$33$	−0.828427	−0.144211
$34$	1.41421	0.242536
$35$	0	0
$36$	−1.00000	−0.166667
$37$	−2.24264	−0.368688	−0.184344	−	0.982862i	$-0.559016\pi$
$37$	−2.24264	−0.368688	−0.184344	+	0.982862i	$0.559016\pi$
$38$	1.00000	0.162221
$39$	−2.58579	−0.414057
$40$	0	0
$41$	−1.82843	−0.285552	−0.142776	−	0.989755i	$-0.545603\pi$
$41$	−1.82843	−0.285552	−0.142776	+	0.989755i	$0.545603\pi$
$42$	−1.41421	−0.218218
$43$	−1.00000	−0.152499
$43$	−1.00000	−0.152499
$44$	0.585786	0.0883106
$45$	0	0
$46$	−5.07107	−0.747688
$47$	−7.07107	−1.03142	−0.515711	−	0.856763i	$-0.672472\pi$
$47$	−7.07107	−1.03142	−0.515711	+	0.856763i	$0.672472\pi$
$48$	−1.41421	−0.204124
$49$	−6.00000	−0.857143
$50$	0	0
$51$	2.00000	0.280056
$52$	1.82843	0.253557
$53$	5.65685	0.777029	0.388514	−	0.921443i	$-0.372988\pi$
$53$	5.65685	0.777029	0.388514	+	0.921443i	$0.372988\pi$
$54$	−5.65685	−0.769800
$55$	0	0
$56$	1.00000	0.133631
$57$	1.41421	0.187317
$58$	−1.24264	−0.163167
$59$	−1.17157	−0.152526	−0.0762629	−	0.997088i	$-0.524299\pi$
$59$	−1.17157	−0.152526	−0.0762629	+	0.997088i	$0.524299\pi$
$60$	0	0
$61$	0.0710678	0.00909930	0.00454965	−	0.999990i	$-0.498552\pi$
$61$	0.0710678	0.00909930	0.00454965	+	0.999990i	$0.498552\pi$
$62$	0.414214	0.0526052
$63$	1.00000	0.125988
$64$	1.00000	0.125000
$65$	0	0
$66$	0.828427	0.101972
$67$	1.24264	0.151813	0.0759064	−	0.997115i	$-0.475815\pi$
$67$	1.24264	0.151813	0.0759064	+	0.997115i	$0.475815\pi$
$68$	−1.41421	−0.171499
$69$	−7.17157	−0.863356
$70$	0	0
$71$	11.8995	1.41221	0.706105	−	0.708107i	$-0.250451\pi$
$71$	11.8995	1.41221	0.706105	+	0.708107i	$0.250451\pi$
$72$	1.00000	0.117851
$73$	9.24264	1.08177	0.540885	−	0.841097i	$-0.318090\pi$
$73$	9.24264	1.08177	0.540885	+	0.841097i	$0.318090\pi$
$74$	2.24264	0.260702
$75$	0	0
$76$	−1.00000	−0.114708
$77$	−0.585786	−0.0667566
$78$	2.58579	0.292783
$79$	−8.41421	−0.946673	−0.473336	−	0.880882i	$-0.656951\pi$
$79$	−8.41421	−0.946673	−0.473336	+	0.880882i	$0.656951\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	1.82843	0.201916
$83$	−3.65685	−0.401392	−0.200696	−	0.979654i	$-0.564320\pi$
$83$	−3.65685	−0.401392	−0.200696	+	0.979654i	$0.564320\pi$
$84$	1.41421	0.154303
$85$	0	0
$86$	1.00000	0.107833
$87$	−1.75736	−0.188409
$88$	−0.585786	−0.0624450
$89$	−9.07107	−0.961531	−0.480766	−	0.876849i	$-0.659641\pi$
$89$	−9.07107	−0.961531	−0.480766	+	0.876849i	$0.659641\pi$
$90$	0	0
$91$	−1.82843	−0.191671
$92$	5.07107	0.528695
$93$	0.585786	0.0607432
$94$	7.07107	0.729325
$95$	0	0
$96$	1.41421	0.144338
$97$	−13.5563	−1.37644	−0.688219	−	0.725503i	$-0.741607\pi$
$97$	−13.5563	−1.37644	−0.688219	+	0.725503i	$0.741607\pi$
$98$	6.00000	0.606092
$99$	−0.585786	−0.0588738
$100$	0	0
$101$	1.41421	0.140720	0.0703598	−	0.997522i	$-0.477585\pi$
$101$	1.41421	0.140720	0.0703598	+	0.997522i	$0.477585\pi$
$102$	−2.00000	−0.198030
$103$	−0.828427	−0.0816274	−0.0408137	−	0.999167i	$-0.512995\pi$
$103$	−0.828427	−0.0816274	−0.0408137	+	0.999167i	$0.512995\pi$
$104$	−1.82843	−0.179292
$105$	0	0
$106$	−5.65685	−0.549442
$107$	−1.92893	−0.186477	−0.0932385	−	0.995644i	$-0.529722\pi$
$107$	−1.92893	−0.186477	−0.0932385	+	0.995644i	$0.529722\pi$
$108$	5.65685	0.544331
$109$	−10.8284	−1.03718	−0.518588	−	0.855024i	$-0.673542\pi$
$109$	−10.8284	−1.03718	−0.518588	+	0.855024i	$0.673542\pi$
$110$	0	0
$111$	3.17157	0.301032
$112$	−1.00000	−0.0944911
$113$	−4.89949	−0.460906	−0.230453	−	0.973083i	$-0.574021\pi$
$113$	−4.89949	−0.460906	−0.230453	+	0.973083i	$0.574021\pi$
$114$	−1.41421	−0.132453
$115$	0	0
$116$	1.24264	0.115376
$117$	−1.82843	−0.169038
$118$	1.17157	0.107852
$119$	1.41421	0.129641
$120$	0	0
$121$	−10.6569	−0.968805
$122$	−0.0710678	−0.00643418
$123$	2.58579	0.233153
$124$	−0.414214	−0.0371975
$125$	0	0
$126$	−1.00000	−0.0890871
$127$	−3.51472	−0.311881	−0.155940	−	0.987766i	$-0.549841\pi$
$127$	−3.51472	−0.311881	−0.155940	+	0.987766i	$0.549841\pi$
$128$	−1.00000	−0.0883883
$129$	1.41421	0.124515
$130$	0	0
$131$	6.00000	0.524222	0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000	0.524222	0.262111	+	0.965038i	$0.415581\pi$
$132$	−0.828427	−0.0721053
$133$	1.00000	0.0867110
$134$	−1.24264	−0.107348
$135$	0	0
$136$	1.41421	0.121268
$137$	−3.92893	−0.335671	−0.167836	−	0.985815i	$-0.553678\pi$
$137$	−3.92893	−0.335671	−0.167836	+	0.985815i	$0.553678\pi$
$138$	7.17157	0.610485
$139$	−6.24264	−0.529494	−0.264747	−	0.964318i	$-0.585288\pi$
$139$	−6.24264	−0.529494	−0.264747	+	0.964318i	$0.585288\pi$
$140$	0	0
$141$	10.0000	0.842152
$142$	−11.8995	−0.998583
$143$	1.07107	0.0895672
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−9.24264	−0.764926
$147$	8.48528	0.699854
$148$	−2.24264	−0.184344
$149$	−10.5563	−0.864810	−0.432405	−	0.901680i	$-0.642335\pi$
$149$	−10.5563	−0.864810	−0.432405	+	0.901680i	$0.642335\pi$
$150$	0	0
$151$	10.4853	0.853280	0.426640	−	0.904422i	$-0.359697\pi$
$151$	10.4853	0.853280	0.426640	+	0.904422i	$0.359697\pi$
$152$	1.00000	0.0811107
$153$	1.41421	0.114332
$154$	0.585786	0.0472040
$155$	0	0
$156$	−2.58579	−0.207029
$157$	−10.2426	−0.817452	−0.408726	−	0.912657i	$-0.634027\pi$
$157$	−10.2426	−0.817452	−0.408726	+	0.912657i	$0.634027\pi$
$158$	8.41421	0.669399
$159$	−8.00000	−0.634441
$160$	0	0
$161$	−5.07107	−0.399656
$162$	5.00000	0.392837
$163$	−13.6569	−1.06969	−0.534844	−	0.844951i	$-0.679630\pi$
$163$	−13.6569	−1.06969	−0.534844	+	0.844951i	$0.679630\pi$
$164$	−1.82843	−0.142776
$165$	0	0
$166$	3.65685	0.283827
$167$	−13.5563	−1.04902	−0.524511	−	0.851404i	$-0.675752\pi$
$167$	−13.5563	−1.04902	−0.524511	+	0.851404i	$0.675752\pi$
$168$	−1.41421	−0.109109
$169$	−9.65685	−0.742835
$170$	0	0
$171$	1.00000	0.0764719
$172$	−1.00000	−0.0762493
$173$	−0.514719	−0.0391333	−0.0195667	−	0.999809i	$-0.506229\pi$
$173$	−0.514719	−0.0391333	−0.0195667	+	0.999809i	$0.506229\pi$
$174$	1.75736	0.133225
$175$	0	0
$176$	0.585786	0.0441553
$177$	1.65685	0.124537
$178$	9.07107	0.679905
$179$	17.4853	1.30691	0.653456	−	0.756965i	$-0.273319\pi$
$179$	17.4853	1.30691	0.653456	+	0.756965i	$0.273319\pi$
$180$	0	0
$181$	−15.0711	−1.12022	−0.560112	−	0.828417i	$-0.689242\pi$
$181$	−15.0711	−1.12022	−0.560112	+	0.828417i	$0.689242\pi$
$182$	1.82843	0.135532
$183$	−0.100505	−0.00742955
$184$	−5.07107	−0.373844
$185$	0	0
$186$	−0.585786	−0.0429519
$187$	−0.828427	−0.0605806
$188$	−7.07107	−0.515711
$189$	−5.65685	−0.411476
$190$	0	0
$191$	19.8995	1.43988	0.719938	−	0.694038i	$-0.244170\pi$
$191$	19.8995	1.43988	0.719938	+	0.694038i	$0.244170\pi$
$192$	−1.41421	−0.102062
$193$	−6.82843	−0.491521	−0.245760	−	0.969331i	$-0.579038\pi$
$193$	−6.82843	−0.491521	−0.245760	+	0.969331i	$0.579038\pi$
$194$	13.5563	0.973289
$195$	0	0
$196$	−6.00000	−0.428571
$197$	−25.1421	−1.79130	−0.895651	−	0.444757i	$-0.853290\pi$
$197$	−25.1421	−1.79130	−0.895651	+	0.444757i	$0.853290\pi$
$198$	0.585786	0.0416300
$199$	−0.100505	−0.00712462	−0.00356231	−	0.999994i	$-0.501134\pi$
$199$	−0.100505	−0.00712462	−0.00356231	+	0.999994i	$0.501134\pi$
$200$	0	0
$201$	−1.75736	−0.123955
$202$	−1.41421	−0.0995037
$203$	−1.24264	−0.0872163
$204$	2.00000	0.140028
$205$	0	0
$206$	0.828427	0.0577193
$207$	−5.07107	−0.352464
$208$	1.82843	0.126779
$209$	−0.585786	−0.0405197
$210$	0	0
$211$	−1.65685	−0.114063	−0.0570313	−	0.998372i	$-0.518163\pi$
$211$	−1.65685	−0.114063	−0.0570313	+	0.998372i	$0.518163\pi$
$212$	5.65685	0.388514
$213$	−16.8284	−1.15306
$214$	1.92893	0.131859
$215$	0	0
$216$	−5.65685	−0.384900
$217$	0.414214	0.0281186
$218$	10.8284	0.733394
$219$	−13.0711	−0.883261
$220$	0	0
$221$	−2.58579	−0.173939
$222$	−3.17157	−0.212862
$223$	−16.1421	−1.08096	−0.540479	−	0.841358i	$-0.681757\pi$
$223$	−16.1421	−1.08096	−0.540479	+	0.841358i	$0.681757\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	4.89949	0.325910
$227$	−9.17157	−0.608739	−0.304369	−	0.952554i	$-0.598446\pi$
$227$	−9.17157	−0.608739	−0.304369	+	0.952554i	$0.598446\pi$
$228$	1.41421	0.0936586
$229$	0.828427	0.0547440	0.0273720	−	0.999625i	$-0.491286\pi$
$229$	0.828427	0.0547440	0.0273720	+	0.999625i	$0.491286\pi$
$230$	0	0
$231$	0.828427	0.0545065
$232$	−1.24264	−0.0815834
$233$	−8.48528	−0.555889	−0.277945	−	0.960597i	$-0.589653\pi$
$233$	−8.48528	−0.555889	−0.277945	+	0.960597i	$0.589653\pi$
$234$	1.82843	0.119528
$235$	0	0
$236$	−1.17157	−0.0762629
$237$	11.8995	0.772955
$238$	−1.41421	−0.0916698
$239$	11.5858	0.749422	0.374711	−	0.927142i	$-0.377742\pi$
$239$	11.5858	0.749422	0.374711	+	0.927142i	$0.377742\pi$
$240$	0	0
$241$	−21.4142	−1.37941	−0.689705	−	0.724090i	$-0.742260\pi$
$241$	−21.4142	−1.37941	−0.689705	+	0.724090i	$0.742260\pi$
$242$	10.6569	0.685049
$243$	−9.89949	−0.635053
$244$	0.0710678	0.00454965
$245$	0	0
$246$	−2.58579	−0.164864
$247$	−1.82843	−0.116340
$248$	0.414214	0.0263026
$249$	5.17157	0.327735
$250$	0	0
$251$	−10.9706	−0.692456	−0.346228	−	0.938150i	$-0.612538\pi$
$251$	−10.9706	−0.692456	−0.346228	+	0.938150i	$0.612538\pi$
$252$	1.00000	0.0629941
$253$	2.97056	0.186758
$254$	3.51472	0.220533
$255$	0	0
$256$	1.00000	0.0625000
$257$	−8.89949	−0.555135	−0.277568	−	0.960706i	$-0.589528\pi$
$257$	−8.89949	−0.555135	−0.277568	+	0.960706i	$0.589528\pi$
$258$	−1.41421	−0.0880451
$259$	2.24264	0.139351
$260$	0	0
$261$	−1.24264	−0.0769175
$262$	−6.00000	−0.370681
$263$	−17.4853	−1.07819	−0.539094	−	0.842245i	$-0.681233\pi$
$263$	−17.4853	−1.07819	−0.539094	+	0.842245i	$0.681233\pi$
$264$	0.828427	0.0509862
$265$	0	0
$266$	−1.00000	−0.0613139
$267$	12.8284	0.785087
$268$	1.24264	0.0759064
$269$	9.17157	0.559201	0.279600	−	0.960116i	$-0.409798\pi$
$269$	9.17157	0.559201	0.279600	+	0.960116i	$0.409798\pi$
$270$	0	0
$271$	8.89949	0.540606	0.270303	−	0.962775i	$-0.412876\pi$
$271$	8.89949	0.540606	0.270303	+	0.962775i	$0.412876\pi$
$272$	−1.41421	−0.0857493
$273$	2.58579	0.156499
$274$	3.92893	0.237355
$275$	0	0
$276$	−7.17157	−0.431678
$277$	0.485281	0.0291577	0.0145789	−	0.999894i	$-0.495359\pi$
$277$	0.485281	0.0291577	0.0145789	+	0.999894i	$0.495359\pi$
$278$	6.24264	0.374409
$279$	0.414214	0.0247983
$280$	0	0
$281$	−1.34315	−0.0801254	−0.0400627	−	0.999197i	$-0.512756\pi$
$281$	−1.34315	−0.0801254	−0.0400627	+	0.999197i	$0.512756\pi$
$282$	−10.0000	−0.595491
$283$	−18.7574	−1.11501	−0.557505	−	0.830174i	$-0.688241\pi$
$283$	−18.7574	−1.11501	−0.557505	+	0.830174i	$0.688241\pi$
$284$	11.8995	0.706105
$285$	0	0
$286$	−1.07107	−0.0633336
$287$	1.82843	0.107929
$288$	1.00000	0.0589256
$289$	−15.0000	−0.882353
$290$	0	0
$291$	19.1716	1.12386
$292$	9.24264	0.540885
$293$	−10.9706	−0.640907	−0.320454	−	0.947264i	$-0.603835\pi$
$293$	−10.9706	−0.640907	−0.320454	+	0.947264i	$0.603835\pi$
$294$	−8.48528	−0.494872
$295$	0	0
$296$	2.24264	0.130351
$297$	3.31371	0.192281
$298$	10.5563	0.611513
$299$	9.27208	0.536218
$300$	0	0
$301$	1.00000	0.0576390
$302$	−10.4853	−0.603360
$303$	−2.00000	−0.114897
$304$	−1.00000	−0.0573539
$305$	0	0
$306$	−1.41421	−0.0808452
$307$	5.92893	0.338382	0.169191	−	0.985583i	$-0.445885\pi$
$307$	5.92893	0.338382	0.169191	+	0.985583i	$0.445885\pi$
$308$	−0.585786	−0.0333783
$309$	1.17157	0.0666485
$310$	0	0
$311$	−4.07107	−0.230849	−0.115425	−	0.993316i	$-0.536823\pi$
$311$	−4.07107	−0.230849	−0.115425	+	0.993316i	$0.536823\pi$
$312$	2.58579	0.146391
$313$	1.31371	0.0742552	0.0371276	−	0.999311i	$-0.488179\pi$
$313$	1.31371	0.0742552	0.0371276	+	0.999311i	$0.488179\pi$
$314$	10.2426	0.578026
$315$	0	0
$316$	−8.41421	−0.473336
$317$	−0.857864	−0.0481825	−0.0240912	−	0.999710i	$-0.507669\pi$
$317$	−0.857864	−0.0481825	−0.0240912	+	0.999710i	$0.507669\pi$
$318$	8.00000	0.448618
$319$	0.727922	0.0407558
$320$	0	0
$321$	2.72792	0.152258
$322$	5.07107	0.282600
$323$	1.41421	0.0786889
$324$	−5.00000	−0.277778
$325$	0	0
$326$	13.6569	0.756383
$327$	15.3137	0.846850
$328$	1.82843	0.100958
$329$	7.07107	0.389841
$330$	0	0
$331$	−25.3137	−1.39137	−0.695684	−	0.718348i	$-0.744898\pi$
$331$	−25.3137	−1.39137	−0.695684	+	0.718348i	$0.744898\pi$
$332$	−3.65685	−0.200696
$333$	2.24264	0.122896
$334$	13.5563	0.741770
$335$	0	0
$336$	1.41421	0.0771517
$337$	8.34315	0.454480	0.227240	−	0.973839i	$-0.427030\pi$
$337$	8.34315	0.454480	0.227240	+	0.973839i	$0.427030\pi$
$338$	9.65685	0.525264
$339$	6.92893	0.376328
$340$	0	0
$341$	−0.242641	−0.0131397
$342$	−1.00000	−0.0540738
$343$	13.0000	0.701934
$344$	1.00000	0.0539164
$345$	0	0
$346$	0.514719	0.0276714
$347$	22.2426	1.19405	0.597024	−	0.802224i	$-0.296350\pi$
$347$	22.2426	1.19405	0.597024	+	0.802224i	$0.296350\pi$
$348$	−1.75736	−0.0942043
$349$	23.4558	1.25556	0.627781	−	0.778390i	$-0.283963\pi$
$349$	23.4558	1.25556	0.627781	+	0.778390i	$0.283963\pi$
$350$	0	0
$351$	10.3431	0.552076
$352$	−0.585786	−0.0312225
$353$	32.3848	1.72367	0.861834	−	0.507191i	$-0.169316\pi$
$353$	32.3848	1.72367	0.861834	+	0.507191i	$0.169316\pi$
$354$	−1.65685	−0.0880608
$355$	0	0
$356$	−9.07107	−0.480766
$357$	−2.00000	−0.105851
$358$	−17.4853	−0.924126
$359$	27.5269	1.45281	0.726407	−	0.687264i	$-0.241189\pi$
$359$	27.5269	1.45281	0.726407	+	0.687264i	$0.241189\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	15.0711	0.792118
$363$	15.0711	0.791026
$364$	−1.82843	−0.0958356
$365$	0	0
$366$	0.100505	0.00525348
$367$	31.3137	1.63456	0.817281	−	0.576239i	$-0.195480\pi$
$367$	31.3137	1.63456	0.817281	+	0.576239i	$0.195480\pi$
$368$	5.07107	0.264348
$369$	1.82843	0.0951841
$370$	0	0
$371$	−5.65685	−0.293689
$372$	0.585786	0.0303716
$373$	−14.7279	−0.762583	−0.381291	−	0.924455i	$-0.624521\pi$
$373$	−14.7279	−0.762583	−0.381291	+	0.924455i	$0.624521\pi$
$374$	0.828427	0.0428369
$375$	0	0
$376$	7.07107	0.364662
$377$	2.27208	0.117018
$378$	5.65685	0.290957
$379$	4.14214	0.212767	0.106384	−	0.994325i	$-0.466073\pi$
$379$	4.14214	0.212767	0.106384	+	0.994325i	$0.466073\pi$
$380$	0	0
$381$	4.97056	0.254650
$382$	−19.8995	−1.01815
$383$	8.31371	0.424811	0.212405	−	0.977182i	$-0.431870\pi$
$383$	8.31371	0.424811	0.212405	+	0.977182i	$0.431870\pi$
$384$	1.41421	0.0721688
$385$	0	0
$386$	6.82843	0.347558
$387$	1.00000	0.0508329
$388$	−13.5563	−0.688219
$389$	20.8284	1.05604	0.528022	−	0.849231i	$-0.322934\pi$
$389$	20.8284	1.05604	0.528022	+	0.849231i	$0.322934\pi$
$390$	0	0
$391$	−7.17157	−0.362682
$392$	6.00000	0.303046
$393$	−8.48528	−0.428026
$394$	25.1421	1.26664
$395$	0	0
$396$	−0.585786	−0.0294369
$397$	20.4853	1.02813	0.514063	−	0.857752i	$-0.328140\pi$
$397$	20.4853	1.02813	0.514063	+	0.857752i	$0.328140\pi$
$398$	0.100505	0.00503786
$399$	−1.41421	−0.0707992
$400$	0	0
$401$	−24.1127	−1.20413	−0.602065	−	0.798447i	$-0.705655\pi$
$401$	−24.1127	−1.20413	−0.602065	+	0.798447i	$0.705655\pi$
$402$	1.75736	0.0876491
$403$	−0.757359	−0.0377268
$404$	1.41421	0.0703598
$405$	0	0
$406$	1.24264	0.0616712
$407$	−1.31371	−0.0651181
$408$	−2.00000	−0.0990148
$409$	−27.2132	−1.34561	−0.672803	−	0.739822i	$-0.734910\pi$
$409$	−27.2132	−1.34561	−0.672803	+	0.739822i	$0.734910\pi$
$410$	0	0
$411$	5.55635	0.274074
$412$	−0.828427	−0.0408137
$413$	1.17157	0.0576493
$414$	5.07107	0.249229
$415$	0	0
$416$	−1.82843	−0.0896460
$417$	8.82843	0.432330
$418$	0.585786	0.0286518
$419$	−12.7990	−0.625272	−0.312636	−	0.949873i	$-0.601212\pi$
$419$	−12.7990	−0.625272	−0.312636	+	0.949873i	$0.601212\pi$
$420$	0	0
$421$	1.24264	0.0605626	0.0302813	−	0.999541i	$-0.490360\pi$
$421$	1.24264	0.0605626	0.0302813	+	0.999541i	$0.490360\pi$
$422$	1.65685	0.0806544
$423$	7.07107	0.343807
$424$	−5.65685	−0.274721
$425$	0	0
$426$	16.8284	0.815340
$427$	−0.0710678	−0.00343921
$428$	−1.92893	−0.0932385
$429$	−1.51472	−0.0731313
$430$	0	0
$431$	1.02944	0.0495862	0.0247931	−	0.999693i	$-0.492107\pi$
$431$	1.02944	0.0495862	0.0247931	+	0.999693i	$0.492107\pi$
$432$	5.65685	0.272166
$433$	36.0711	1.73346	0.866732	−	0.498773i	$-0.166216\pi$
$433$	36.0711	1.73346	0.866732	+	0.498773i	$0.166216\pi$
$434$	−0.414214	−0.0198829
$435$	0	0
$436$	−10.8284	−0.518588
$437$	−5.07107	−0.242582
$438$	13.0711	0.624560
$439$	−15.3137	−0.730883	−0.365442	−	0.930834i	$-0.619082\pi$
$439$	−15.3137	−0.730883	−0.365442	+	0.930834i	$0.619082\pi$
$440$	0	0
$441$	6.00000	0.285714
$442$	2.58579	0.122993
$443$	9.92893	0.471738	0.235869	−	0.971785i	$-0.424206\pi$
$443$	9.92893	0.471738	0.235869	+	0.971785i	$0.424206\pi$
$444$	3.17157	0.150516
$445$	0	0
$446$	16.1421	0.764352
$447$	14.9289	0.706114
$448$	−1.00000	−0.0472456
$449$	22.2426	1.04970	0.524848	−	0.851196i	$-0.324122\pi$
$449$	22.2426	1.04970	0.524848	+	0.851196i	$0.324122\pi$
$450$	0	0
$451$	−1.07107	−0.0504346
$452$	−4.89949	−0.230453
$453$	−14.8284	−0.696700
$454$	9.17157	0.430443
$455$	0	0
$456$	−1.41421	−0.0662266
$457$	30.4853	1.42604	0.713021	−	0.701143i	$-0.247327\pi$
$457$	30.4853	1.42604	0.713021	+	0.701143i	$0.247327\pi$
$458$	−0.828427	−0.0387099
$459$	−8.00000	−0.373408
$460$	0	0
$461$	−15.0711	−0.701930	−0.350965	−	0.936389i	$-0.614146\pi$
$461$	−15.0711	−0.701930	−0.350965	+	0.936389i	$0.614146\pi$
$462$	−0.828427	−0.0385419
$463$	−30.3137	−1.40880	−0.704399	−	0.709804i	$-0.748783\pi$
$463$	−30.3137	−1.40880	−0.704399	+	0.709804i	$0.748783\pi$
$464$	1.24264	0.0576881
$465$	0	0
$466$	8.48528	0.393073
$467$	20.9289	0.968475	0.484238	−	0.874936i	$-0.339097\pi$
$467$	20.9289	0.968475	0.484238	+	0.874936i	$0.339097\pi$
$468$	−1.82843	−0.0845191
$469$	−1.24264	−0.0573798
$470$	0	0
$471$	14.4853	0.667447
$472$	1.17157	0.0539260
$473$	−0.585786	−0.0269345
$474$	−11.8995	−0.546562
$475$	0	0
$476$	1.41421	0.0648204
$477$	−5.65685	−0.259010
$478$	−11.5858	−0.529922
$479$	9.17157	0.419060	0.209530	−	0.977802i	$-0.432807\pi$
$479$	9.17157	0.419060	0.209530	+	0.977802i	$0.432807\pi$
$480$	0	0
$481$	−4.10051	−0.186967
$482$	21.4142	0.975391
$483$	7.17157	0.326318
$484$	−10.6569	−0.484402
$485$	0	0
$486$	9.89949	0.449050
$487$	33.2132	1.50503	0.752517	−	0.658573i	$-0.228840\pi$
$487$	33.2132	1.50503	0.752517	+	0.658573i	$0.228840\pi$
$488$	−0.0710678	−0.00321709
$489$	19.3137	0.873396
$490$	0	0
$491$	16.6274	0.750385	0.375192	−	0.926947i	$-0.377577\pi$
$491$	16.6274	0.750385	0.375192	+	0.926947i	$0.377577\pi$
$492$	2.58579	0.116576
$493$	−1.75736	−0.0791475
$494$	1.82843	0.0822648
$495$	0	0
$496$	−0.414214	−0.0185987
$497$	−11.8995	−0.533765
$498$	−5.17157	−0.231744
$499$	−34.4558	−1.54246	−0.771228	−	0.636559i	$-0.780357\pi$
$499$	−34.4558	−1.54246	−0.771228	+	0.636559i	$0.780357\pi$
$500$	0	0
$501$	19.1716	0.856523
$502$	10.9706	0.489640
$503$	31.6569	1.41151	0.705755	−	0.708456i	$-0.250608\pi$
$503$	31.6569	1.41151	0.705755	+	0.708456i	$0.250608\pi$
$504$	−1.00000	−0.0445435
$505$	0	0
$506$	−2.97056	−0.132058
$507$	13.6569	0.606522
$508$	−3.51472	−0.155940
$509$	−20.4853	−0.907994	−0.453997	−	0.891003i	$-0.650002\pi$
$509$	−20.4853	−0.907994	−0.453997	+	0.891003i	$0.650002\pi$
$510$	0	0
$511$	−9.24264	−0.408870
$512$	−1.00000	−0.0441942
$513$	−5.65685	−0.249756
$514$	8.89949	0.392540
$515$	0	0
$516$	1.41421	0.0622573
$517$	−4.14214	−0.182171
$518$	−2.24264	−0.0985360
$519$	0.727922	0.0319522
$520$	0	0
$521$	−11.0711	−0.485032	−0.242516	−	0.970147i	$-0.577973\pi$
$521$	−11.0711	−0.485032	−0.242516	+	0.970147i	$0.577973\pi$
$522$	1.24264	0.0543889
$523$	−10.7279	−0.469099	−0.234550	−	0.972104i	$-0.575362\pi$
$523$	−10.7279	−0.469099	−0.234550	+	0.972104i	$0.575362\pi$
$524$	6.00000	0.262111
$525$	0	0
$526$	17.4853	0.762394
$527$	0.585786	0.0255173
$528$	−0.828427	−0.0360527
$529$	2.71573	0.118075
$530$	0	0
$531$	1.17157	0.0508419
$532$	1.00000	0.0433555
$533$	−3.34315	−0.144808
$534$	−12.8284	−0.555140
$535$	0	0
$536$	−1.24264	−0.0536739
$537$	−24.7279	−1.06709
$538$	−9.17157	−0.395415
$539$	−3.51472	−0.151390
$540$	0	0
$541$	7.85786	0.337836	0.168918	−	0.985630i	$-0.445973\pi$
$541$	7.85786	0.337836	0.168918	+	0.985630i	$0.445973\pi$
$542$	−8.89949	−0.382266
$543$	21.3137	0.914659
$544$	1.41421	0.0606339
$545$	0	0
$546$	−2.58579	−0.110661
$547$	12.0000	0.513083	0.256541	−	0.966533i	$-0.417417\pi$
$547$	12.0000	0.513083	0.256541	+	0.966533i	$0.417417\pi$
$548$	−3.92893	−0.167836
$549$	−0.0710678	−0.00303310
$550$	0	0
$551$	−1.24264	−0.0529383
$552$	7.17157	0.305242
$553$	8.41421	0.357809
$554$	−0.485281	−0.0206176
$555$	0	0
$556$	−6.24264	−0.264747
$557$	−41.8284	−1.77233	−0.886164	−	0.463372i	$-0.846639\pi$
$557$	−41.8284	−1.77233	−0.886164	+	0.463372i	$0.846639\pi$
$558$	−0.414214	−0.0175351
$559$	−1.82843	−0.0773342
$560$	0	0
$561$	1.17157	0.0494638
$562$	1.34315	0.0566572
$563$	1.92893	0.0812948	0.0406474	−	0.999174i	$-0.487058\pi$
$563$	1.92893	0.0812948	0.0406474	+	0.999174i	$0.487058\pi$
$564$	10.0000	0.421076
$565$	0	0
$566$	18.7574	0.788431
$567$	5.00000	0.209980
$568$	−11.8995	−0.499292
$569$	22.7990	0.955783	0.477892	−	0.878419i	$-0.341401\pi$
$569$	22.7990	0.955783	0.477892	+	0.878419i	$0.341401\pi$
$570$	0	0
$571$	33.0000	1.38101	0.690504	−	0.723329i	$-0.257389\pi$
$571$	33.0000	1.38101	0.690504	+	0.723329i	$0.257389\pi$
$572$	1.07107	0.0447836
$573$	−28.1421	−1.17565
$574$	−1.82843	−0.0763171
$575$	0	0
$576$	−1.00000	−0.0416667
$577$	5.58579	0.232539	0.116270	−	0.993218i	$-0.462906\pi$
$577$	5.58579	0.232539	0.116270	+	0.993218i	$0.462906\pi$
$578$	15.0000	0.623918
$579$	9.65685	0.401325
$580$	0	0
$581$	3.65685	0.151712
$582$	−19.1716	−0.794687
$583$	3.31371	0.137240
$584$	−9.24264	−0.382463
$585$	0	0
$586$	10.9706	0.453190
$587$	27.5563	1.13737	0.568686	−	0.822555i	$-0.307452\pi$
$587$	27.5563	1.13737	0.568686	+	0.822555i	$0.307452\pi$
$588$	8.48528	0.349927
$589$	0.414214	0.0170674
$590$	0	0
$591$	35.5563	1.46259
$592$	−2.24264	−0.0921720
$593$	7.24264	0.297420	0.148710	−	0.988881i	$-0.452488\pi$
$593$	7.24264	0.297420	0.148710	+	0.988881i	$0.452488\pi$
$594$	−3.31371	−0.135963
$595$	0	0
$596$	−10.5563	−0.432405
$597$	0.142136	0.00581722
$598$	−9.27208	−0.379163
$599$	9.85786	0.402781	0.201391	−	0.979511i	$-0.435454\pi$
$599$	9.85786	0.402781	0.201391	+	0.979511i	$0.435454\pi$
$600$	0	0
$601$	−28.0000	−1.14214	−0.571072	−	0.820900i	$-0.693472\pi$
$601$	−28.0000	−1.14214	−0.571072	+	0.820900i	$0.693472\pi$
$602$	−1.00000	−0.0407570
$603$	−1.24264	−0.0506042
$604$	10.4853	0.426640
$605$	0	0
$606$	2.00000	0.0812444
$607$	−6.00000	−0.243532	−0.121766	−	0.992559i	$-0.538856\pi$
$607$	−6.00000	−0.243532	−0.121766	+	0.992559i	$0.538856\pi$
$608$	1.00000	0.0405554
$609$	1.75736	0.0712118
$610$	0	0
$611$	−12.9289	−0.523049
$612$	1.41421	0.0571662
$613$	−38.1716	−1.54174	−0.770868	−	0.636995i	$-0.780177\pi$
$613$	−38.1716	−1.54174	−0.770868	+	0.636995i	$0.780177\pi$
$614$	−5.92893	−0.239272
$615$	0	0
$616$	0.585786	0.0236020
$617$	16.2843	0.655580	0.327790	−	0.944751i	$-0.393696\pi$
$617$	16.2843	0.655580	0.327790	+	0.944751i	$0.393696\pi$
$618$	−1.17157	−0.0471276
$619$	−0.100505	−0.00403964	−0.00201982	−	0.999998i	$-0.500643\pi$
$619$	−0.100505	−0.00403964	−0.00201982	+	0.999998i	$0.500643\pi$
$620$	0	0
$621$	28.6863	1.15114
$622$	4.07107	0.163235
$623$	9.07107	0.363425
$624$	−2.58579	−0.103514
$625$	0	0
$626$	−1.31371	−0.0525064
$627$	0.828427	0.0330842
$628$	−10.2426	−0.408726
$629$	3.17157	0.126459
$630$	0	0
$631$	−13.6985	−0.545328	−0.272664	−	0.962109i	$-0.587905\pi$
$631$	−13.6985	−0.545328	−0.272664	+	0.962109i	$0.587905\pi$
$632$	8.41421	0.334699
$633$	2.34315	0.0931317
$634$	0.857864	0.0340701
$635$	0	0
$636$	−8.00000	−0.317221
$637$	−10.9706	−0.434670
$638$	−0.727922	−0.0288187
$639$	−11.8995	−0.470737
$640$	0	0
$641$	42.7696	1.68930	0.844648	−	0.535322i	$-0.179810\pi$
$641$	42.7696	1.68930	0.844648	+	0.535322i	$0.179810\pi$
$642$	−2.72792	−0.107662
$643$	−39.7279	−1.56672	−0.783358	−	0.621571i	$-0.786495\pi$
$643$	−39.7279	−1.56672	−0.783358	+	0.621571i	$0.786495\pi$
$644$	−5.07107	−0.199828
$645$	0	0
$646$	−1.41421	−0.0556415
$647$	−5.20101	−0.204473	−0.102236	−	0.994760i	$-0.532600\pi$
$647$	−5.20101	−0.204473	−0.102236	+	0.994760i	$0.532600\pi$
$648$	5.00000	0.196419
$649$	−0.686292	−0.0269393
$650$	0	0
$651$	−0.585786	−0.0229588
$652$	−13.6569	−0.534844
$653$	6.58579	0.257722	0.128861	−	0.991663i	$-0.458868\pi$
$653$	6.58579	0.257722	0.128861	+	0.991663i	$0.458868\pi$
$654$	−15.3137	−0.598813
$655$	0	0
$656$	−1.82843	−0.0713881
$657$	−9.24264	−0.360590
$658$	−7.07107	−0.275659
$659$	24.3848	0.949896	0.474948	−	0.880014i	$-0.342467\pi$
$659$	24.3848	0.949896	0.474948	+	0.880014i	$0.342467\pi$
$660$	0	0
$661$	24.8701	0.967333	0.483667	−	0.875252i	$-0.339305\pi$
$661$	24.8701	0.967333	0.483667	+	0.875252i	$0.339305\pi$
$662$	25.3137	0.983845
$663$	3.65685	0.142020
$664$	3.65685	0.141913
$665$	0	0
$666$	−2.24264	−0.0869006
$667$	6.30152	0.243996
$668$	−13.5563	−0.524511
$669$	22.8284	0.882598
$670$	0	0
$671$	0.0416306	0.00160713
$672$	−1.41421	−0.0545545
$673$	−20.4142	−0.786910	−0.393455	−	0.919344i	$-0.628720\pi$
$673$	−20.4142	−0.786910	−0.393455	+	0.919344i	$0.628720\pi$
$674$	−8.34315	−0.321366
$675$	0	0
$676$	−9.65685	−0.371417
$677$	−43.4558	−1.67014	−0.835072	−	0.550141i	$-0.814574\pi$
$677$	−43.4558	−1.67014	−0.835072	+	0.550141i	$0.814574\pi$
$678$	−6.92893	−0.266104
$679$	13.5563	0.520245
$680$	0	0
$681$	12.9706	0.497033
$682$	0.242641	0.00929119
$683$	3.51472	0.134487	0.0672435	−	0.997737i	$-0.478580\pi$
$683$	3.51472	0.134487	0.0672435	+	0.997737i	$0.478580\pi$
$684$	1.00000	0.0382360
$685$	0	0
$686$	−13.0000	−0.496342
$687$	−1.17157	−0.0446983
$688$	−1.00000	−0.0381246
$689$	10.3431	0.394042
$690$	0	0
$691$	18.4853	0.703213	0.351607	−	0.936148i	$-0.385635\pi$
$691$	18.4853	0.703213	0.351607	+	0.936148i	$0.385635\pi$
$692$	−0.514719	−0.0195667
$693$	0.585786	0.0222522
$694$	−22.2426	−0.844319
$695$	0	0
$696$	1.75736	0.0666125
$697$	2.58579	0.0979436
$698$	−23.4558	−0.887817
$699$	12.0000	0.453882
$700$	0	0
$701$	11.2721	0.425741	0.212870	−	0.977080i	$-0.431719\pi$
$701$	11.2721	0.425741	0.212870	+	0.977080i	$0.431719\pi$
$702$	−10.3431	−0.390377
$703$	2.24264	0.0845828
$704$	0.585786	0.0220777
$705$	0	0
$706$	−32.3848	−1.21882
$707$	−1.41421	−0.0531870
$708$	1.65685	0.0622684
$709$	−7.21320	−0.270898	−0.135449	−	0.990784i	$-0.543248\pi$
$709$	−7.21320	−0.270898	−0.135449	+	0.990784i	$0.543248\pi$
$710$	0	0
$711$	8.41421	0.315558
$712$	9.07107	0.339953
$713$	−2.10051	−0.0786645
$714$	2.00000	0.0748481
$715$	0	0
$716$	17.4853	0.653456
$717$	−16.3848	−0.611901
$718$	−27.5269	−1.02730
$719$	−36.6274	−1.36597	−0.682986	−	0.730431i	$-0.739319\pi$
$719$	−36.6274	−1.36597	−0.682986	+	0.730431i	$0.739319\pi$
$720$	0	0
$721$	0.828427	0.0308522
$722$	18.0000	0.669891
$723$	30.2843	1.12628
$724$	−15.0711	−0.560112
$725$	0	0
$726$	−15.0711	−0.559340
$727$	14.0000	0.519231	0.259616	−	0.965712i	$-0.416404\pi$
$727$	14.0000	0.519231	0.259616	+	0.965712i	$0.416404\pi$
$728$	1.82843	0.0677660
$729$	29.0000	1.07407
$730$	0	0
$731$	1.41421	0.0523066
$732$	−0.100505	−0.00371477
$733$	−27.7574	−1.02524	−0.512621	−	0.858615i	$-0.671325\pi$
$733$	−27.7574	−1.02524	−0.512621	+	0.858615i	$0.671325\pi$
$734$	−31.3137	−1.15581
$735$	0	0
$736$	−5.07107	−0.186922
$737$	0.727922	0.0268134
$738$	−1.82843	−0.0673053
$739$	8.02944	0.295368	0.147684	−	0.989035i	$-0.452818\pi$
$739$	8.02944	0.295368	0.147684	+	0.989035i	$0.452818\pi$
$740$	0	0
$741$	2.58579	0.0949912
$742$	5.65685	0.207670
$743$	−2.31371	−0.0848817	−0.0424409	−	0.999099i	$-0.513513\pi$
$743$	−2.31371	−0.0848817	−0.0424409	+	0.999099i	$0.513513\pi$
$744$	−0.585786	−0.0214760
$745$	0	0
$746$	14.7279	0.539228
$747$	3.65685	0.133797
$748$	−0.828427	−0.0302903
$749$	1.92893	0.0704816
$750$	0	0
$751$	−8.72792	−0.318486	−0.159243	−	0.987239i	$-0.550905\pi$
$751$	−8.72792	−0.318486	−0.159243	+	0.987239i	$0.550905\pi$
$752$	−7.07107	−0.257855
$753$	15.5147	0.565388
$754$	−2.27208	−0.0827442
$755$	0	0
$756$	−5.65685	−0.205738
$757$	−36.4264	−1.32394	−0.661970	−	0.749530i	$-0.730279\pi$
$757$	−36.4264	−1.32394	−0.661970	+	0.749530i	$0.730279\pi$
$758$	−4.14214	−0.150449
$759$	−4.20101	−0.152487
$760$	0	0
$761$	−44.8284	−1.62503	−0.812515	−	0.582941i	$-0.801902\pi$
$761$	−44.8284	−1.62503	−0.812515	+	0.582941i	$0.801902\pi$
$762$	−4.97056	−0.180064
$763$	10.8284	0.392015
$764$	19.8995	0.719938
$765$	0	0
$766$	−8.31371	−0.300386
$767$	−2.14214	−0.0773480
$768$	−1.41421	−0.0510310
$769$	21.0000	0.757279	0.378640	−	0.925544i	$-0.376392\pi$
$769$	21.0000	0.757279	0.378640	+	0.925544i	$0.376392\pi$
$770$	0	0
$771$	12.5858	0.453266
$772$	−6.82843	−0.245760
$773$	25.0711	0.901744	0.450872	−	0.892589i	$-0.351113\pi$
$773$	25.0711	0.901744	0.450872	+	0.892589i	$0.351113\pi$
$774$	−1.00000	−0.0359443
$775$	0	0
$776$	13.5563	0.486645
$777$	−3.17157	−0.113780
$778$	−20.8284	−0.746735
$779$	1.82843	0.0655102
$780$	0	0
$781$	6.97056	0.249426
$782$	7.17157	0.256455
$783$	7.02944	0.251212
$784$	−6.00000	−0.214286
$785$	0	0
$786$	8.48528	0.302660
$787$	−31.2426	−1.11368	−0.556840	−	0.830620i	$-0.687986\pi$
$787$	−31.2426	−1.11368	−0.556840	+	0.830620i	$0.687986\pi$
$788$	−25.1421	−0.895651
$789$	24.7279	0.880337
$790$	0	0
$791$	4.89949	0.174206
$792$	0.585786	0.0208150
$793$	0.129942	0.00461439
$794$	−20.4853	−0.726995
$795$	0	0
$796$	−0.100505	−0.00356231
$797$	42.9411	1.52105	0.760526	−	0.649307i	$-0.224941\pi$
$797$	42.9411	1.52105	0.760526	+	0.649307i	$0.224941\pi$
$798$	1.41421	0.0500626
$799$	10.0000	0.353775
$800$	0	0
$801$	9.07107	0.320510
$802$	24.1127	0.851449
$803$	5.41421	0.191063
$804$	−1.75736	−0.0619773
$805$	0	0
$806$	0.757359	0.0266768
$807$	−12.9706	−0.456585
$808$	−1.41421	−0.0497519
$809$	−28.7990	−1.01252	−0.506259	−	0.862381i	$-0.668972\pi$
$809$	−28.7990	−1.01252	−0.506259	+	0.862381i	$0.668972\pi$
$810$	0	0
$811$	−48.1127	−1.68947	−0.844733	−	0.535188i	$-0.820241\pi$
$811$	−48.1127	−1.68947	−0.844733	+	0.535188i	$0.820241\pi$
$812$	−1.24264	−0.0436081
$813$	−12.5858	−0.441403
$814$	1.31371	0.0460455
$815$	0	0
$816$	2.00000	0.0700140
$817$	1.00000	0.0349856
$818$	27.2132	0.951487
$819$	1.82843	0.0638904
$820$	0	0
$821$	−20.1421	−0.702965	−0.351483	−	0.936194i	$-0.614322\pi$
$821$	−20.1421	−0.702965	−0.351483	+	0.936194i	$0.614322\pi$
$822$	−5.55635	−0.193800
$823$	−13.5147	−0.471093	−0.235547	−	0.971863i	$-0.575688\pi$
$823$	−13.5147	−0.471093	−0.235547	+	0.971863i	$0.575688\pi$
$824$	0.828427	0.0288596
$825$	0	0
$826$	−1.17157	−0.0407642
$827$	43.5269	1.51358	0.756790	−	0.653659i	$-0.226767\pi$
$827$	43.5269	1.51358	0.756790	+	0.653659i	$0.226767\pi$
$828$	−5.07107	−0.176232
$829$	3.24264	0.112622	0.0563108	−	0.998413i	$-0.482066\pi$
$829$	3.24264	0.112622	0.0563108	+	0.998413i	$0.482066\pi$
$830$	0	0
$831$	−0.686292	−0.0238072
$832$	1.82843	0.0633893
$833$	8.48528	0.293998
$834$	−8.82843	−0.305703
$835$	0	0
$836$	−0.585786	−0.0202598
$837$	−2.34315	−0.0809910
$838$	12.7990	0.442134
$839$	−51.0122	−1.76114	−0.880568	−	0.473919i	$-0.842839\pi$
$839$	−51.0122	−1.76114	−0.880568	+	0.473919i	$0.842839\pi$
$840$	0	0
$841$	−27.4558	−0.946753
$842$	−1.24264	−0.0428242
$843$	1.89949	0.0654221
$844$	−1.65685	−0.0570313
$845$	0	0
$846$	−7.07107	−0.243108
$847$	10.6569	0.366174
$848$	5.65685	0.194257
$849$	26.5269	0.910401
$850$	0	0
$851$	−11.3726	−0.389847
$852$	−16.8284	−0.576532
$853$	35.9411	1.23060	0.615300	−	0.788293i	$-0.289035\pi$
$853$	35.9411	1.23060	0.615300	+	0.788293i	$0.289035\pi$
$854$	0.0710678	0.00243189
$855$	0	0
$856$	1.92893	0.0659295
$857$	−47.0711	−1.60792	−0.803959	−	0.594685i	$-0.797277\pi$
$857$	−47.0711	−1.60792	−0.803959	+	0.594685i	$0.797277\pi$
$858$	1.51472	0.0517116
$859$	−16.5147	−0.563475	−0.281737	−	0.959492i	$-0.590911\pi$
$859$	−16.5147	−0.563475	−0.281737	+	0.959492i	$0.590911\pi$
$860$	0	0
$861$	−2.58579	−0.0881234
$862$	−1.02944	−0.0350628
$863$	40.8284	1.38982	0.694908	−	0.719099i	$-0.255445\pi$
$863$	40.8284	1.38982	0.694908	+	0.719099i	$0.255445\pi$
$864$	−5.65685	−0.192450
$865$	0	0
$866$	−36.0711	−1.22574
$867$	21.2132	0.720438
$868$	0.414214	0.0140593
$869$	−4.92893	−0.167203
$870$	0	0
$871$	2.27208	0.0769864
$872$	10.8284	0.366697
$873$	13.5563	0.458813
$874$	5.07107	0.171531
$875$	0	0
$876$	−13.0711	−0.441630
$877$	18.6274	0.629003	0.314502	−	0.949257i	$-0.398163\pi$
$877$	18.6274	0.629003	0.314502	+	0.949257i	$0.398163\pi$
$878$	15.3137	0.516813
$879$	15.5147	0.523298
$880$	0	0
$881$	−1.20101	−0.0404631	−0.0202315	−	0.999795i	$-0.506440\pi$
$881$	−1.20101	−0.0404631	−0.0202315	+	0.999795i	$0.506440\pi$
$882$	−6.00000	−0.202031
$883$	−26.2132	−0.882145	−0.441072	−	0.897472i	$-0.645402\pi$
$883$	−26.2132	−0.882145	−0.441072	+	0.897472i	$0.645402\pi$
$884$	−2.58579	−0.0869694
$885$	0	0
$886$	−9.92893	−0.333569
$887$	56.1127	1.88408	0.942040	−	0.335501i	$-0.108905\pi$
$887$	56.1127	1.88408	0.942040	+	0.335501i	$0.108905\pi$
$888$	−3.17157	−0.106431
$889$	3.51472	0.117880
$890$	0	0
$891$	−2.92893	−0.0981229
$892$	−16.1421	−0.540479
$893$	7.07107	0.236624
$894$	−14.9289	−0.499298
$895$	0	0
$896$	1.00000	0.0334077
$897$	−13.1127	−0.437820
$898$	−22.2426	−0.742247
$899$	−0.514719	−0.0171668
$900$	0	0
$901$	−8.00000	−0.266519
$902$	1.07107	0.0356627
$903$	−1.41421	−0.0470621
$904$	4.89949	0.162955
$905$	0	0
$906$	14.8284	0.492641
$907$	−14.8995	−0.494730	−0.247365	−	0.968922i	$-0.579565\pi$
$907$	−14.8995	−0.494730	−0.247365	+	0.968922i	$0.579565\pi$
$908$	−9.17157	−0.304369
$909$	−1.41421	−0.0469065
$910$	0	0
$911$	−20.3848	−0.675378	−0.337689	−	0.941258i	$-0.609645\pi$
$911$	−20.3848	−0.675378	−0.337689	+	0.941258i	$0.609645\pi$
$912$	1.41421	0.0468293
$913$	−2.14214	−0.0708943
$914$	−30.4853	−1.00836
$915$	0	0
$916$	0.828427	0.0273720
$917$	−6.00000	−0.198137
$918$	8.00000	0.264039
$919$	−34.8995	−1.15123	−0.575614	−	0.817722i	$-0.695237\pi$
$919$	−34.8995	−1.15123	−0.575614	+	0.817722i	$0.695237\pi$
$920$	0	0
$921$	−8.38478	−0.276288
$922$	15.0711	0.496339
$923$	21.7574	0.716152
$924$	0.828427	0.0272533
$925$	0	0
$926$	30.3137	0.996170
$927$	0.828427	0.0272091
$928$	−1.24264	−0.0407917
$929$	−13.7990	−0.452730	−0.226365	−	0.974043i	$-0.572684\pi$
$929$	−13.7990	−0.452730	−0.226365	+	0.974043i	$0.572684\pi$
$930$	0	0
$931$	6.00000	0.196642
$932$	−8.48528	−0.277945
$933$	5.75736	0.188487
$934$	−20.9289	−0.684816
$935$	0	0
$936$	1.82843	0.0597640
$937$	34.4264	1.12466	0.562331	−	0.826912i	$-0.309905\pi$
$937$	34.4264	1.12466	0.562331	+	0.826912i	$0.309905\pi$
$938$	1.24264	0.0405737
$939$	−1.85786	−0.0606291
$940$	0	0
$941$	−53.0711	−1.73007	−0.865034	−	0.501714i	$-0.832703\pi$
$941$	−53.0711	−1.73007	−0.865034	+	0.501714i	$0.832703\pi$
$942$	−14.4853	−0.471956
$943$	−9.27208	−0.301940
$944$	−1.17157	−0.0381314
$945$	0	0
$946$	0.585786	0.0190456
$947$	−59.1838	−1.92321	−0.961607	−	0.274430i	$-0.911511\pi$
$947$	−59.1838	−1.92321	−0.961607	+	0.274430i	$0.911511\pi$
$948$	11.8995	0.386478
$949$	16.8995	0.548581
$950$	0	0
$951$	1.21320	0.0393408
$952$	−1.41421	−0.0458349
$953$	9.10051	0.294794	0.147397	−	0.989077i	$-0.452910\pi$
$953$	9.10051	0.294794	0.147397	+	0.989077i	$0.452910\pi$
$954$	5.65685	0.183147
$955$	0	0
$956$	11.5858	0.374711
$957$	−1.02944	−0.0332770
$958$	−9.17157	−0.296320
$959$	3.92893	0.126872
$960$	0	0
$961$	−30.8284	−0.994465
$962$	4.10051	0.132206
$963$	1.92893	0.0621590
$964$	−21.4142	−0.689705
$965$	0	0
$966$	−7.17157	−0.230742
$967$	−36.2426	−1.16548	−0.582742	−	0.812657i	$-0.698020\pi$
$967$	−36.2426	−1.16548	−0.582742	+	0.812657i	$0.698020\pi$
$968$	10.6569	0.342524
$969$	−2.00000	−0.0642493
$970$	0	0
$971$	21.0711	0.676203	0.338101	−	0.941110i	$-0.390215\pi$
$971$	21.0711	0.676203	0.338101	+	0.941110i	$0.390215\pi$
$972$	−9.89949	−0.317526
$973$	6.24264	0.200130
$974$	−33.2132	−1.06422
$975$	0	0
$976$	0.0710678	0.00227483
$977$	17.0711	0.546152	0.273076	−	0.961992i	$-0.411959\pi$
$977$	17.0711	0.546152	0.273076	+	0.961992i	$0.411959\pi$
$978$	−19.3137	−0.617584
$979$	−5.31371	−0.169827
$980$	0	0
$981$	10.8284	0.345725
$982$	−16.6274	−0.530602
$983$	27.3431	0.872111	0.436055	−	0.899920i	$-0.356375\pi$
$983$	27.3431	0.872111	0.436055	+	0.899920i	$0.356375\pi$
$984$	−2.58579	−0.0824319
$985$	0	0
$986$	1.75736	0.0559657
$987$	−10.0000	−0.318304
$988$	−1.82843	−0.0581700
$989$	−5.07107	−0.161251
$990$	0	0
$991$	41.2548	1.31050	0.655251	−	0.755411i	$-0.272563\pi$
$991$	41.2548	1.31050	0.655251	+	0.755411i	$0.272563\pi$
$992$	0.414214	0.0131513
$993$	35.7990	1.13605
$994$	11.8995	0.377429
$995$	0	0
$996$	5.17157	0.163868
$997$	−37.1716	−1.17724	−0.588618	−	0.808411i	$-0.700328\pi$
$997$	−37.1716	−1.17724	−0.588618	+	0.808411i	$0.700328\pi$
$998$	34.4558	1.09068
$999$	−12.6863	−0.401377

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2150.2.a.v.1.1		2
5.2	odd	4		2150.2.b.o.1549.2		4
5.3	odd	4		2150.2.b.o.1549.3		4
5.4	even	2		430.2.a.g.1.2	✓	2
15.14	odd	2		3870.2.a.bc.1.2		2
20.19	odd	2		3440.2.a.j.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
430.2.a.g.1.2	✓	2	5.4	even	2
2150.2.a.v.1.1		2	1.1	even	1	trivial
2150.2.b.o.1549.2		4	5.2	odd	4
2150.2.b.o.1549.3		4	5.3	odd	4
3440.2.a.j.1.1		2	20.19	odd	2
3870.2.a.bc.1.2		2	15.14	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:	\( N \)	\(=\)	\( 2150 = 2 \cdot 5^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2150.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.41421 q^{3} +1.00000 q^{4} +1.41421 q^{6} -1.00000 q^{7} -1.00000 q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.41421 q^{3} +1.00000 q^{4} +1.41421 q^{6} -1.00000 q^{7} -1.00000 q^{8} -1.00000 q^{9} +0.585786 q^{11} -1.41421 q^{12} +1.82843 q^{13} +1.00000 q^{14} +1.00000 q^{16} -1.41421 q^{17} +1.00000 q^{18} -1.00000 q^{19} +1.41421 q^{21} -0.585786 q^{22} +5.07107 q^{23} +1.41421 q^{24} -1.82843 q^{26} +5.65685 q^{27} -1.00000 q^{28} +1.24264 q^{29} -0.414214 q^{31} -1.00000 q^{32} -0.828427 q^{33} +1.41421 q^{34} -1.00000 q^{36} -2.24264 q^{37} +1.00000 q^{38} -2.58579 q^{39} -1.82843 q^{41} -1.41421 q^{42} -1.00000 q^{43} +0.585786 q^{44} -5.07107 q^{46} -7.07107 q^{47} -1.41421 q^{48} -6.00000 q^{49} +2.00000 q^{51} +1.82843 q^{52} +5.65685 q^{53} -5.65685 q^{54} +1.00000 q^{56} +1.41421 q^{57} -1.24264 q^{58} -1.17157 q^{59} +0.0710678 q^{61} +0.414214 q^{62} +1.00000 q^{63} +1.00000 q^{64} +0.828427 q^{66} +1.24264 q^{67} -1.41421 q^{68} -7.17157 q^{69} +11.8995 q^{71} +1.00000 q^{72} +9.24264 q^{73} +2.24264 q^{74} -1.00000 q^{76} -0.585786 q^{77} +2.58579 q^{78} -8.41421 q^{79} -5.00000 q^{81} +1.82843 q^{82} -3.65685 q^{83} +1.41421 q^{84} +1.00000 q^{86} -1.75736 q^{87} -0.585786 q^{88} -9.07107 q^{89} -1.82843 q^{91} +5.07107 q^{92} +0.585786 q^{93} +7.07107 q^{94} +1.41421 q^{96} -13.5563 q^{97} +6.00000 q^{98} -0.585786 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{7} - 2 q^{8} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^7 - 2 * q^8 - 2 * q^9	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{7} - 2 q^{8} - 2 q^{9} + 4 q^{11} - 2 q^{13} + 2 q^{14} + 2 q^{16} + 2 q^{18} - 2 q^{19} - 4 q^{22} - 4 q^{23} + 2 q^{26} - 2 q^{28} - 6 q^{29} + 2 q^{31} - 2 q^{32} + 4 q^{33} - 2 q^{36} + 4 q^{37} + 2 q^{38} - 8 q^{39} + 2 q^{41} - 2 q^{43} + 4 q^{44} + 4 q^{46} - 12 q^{49} + 4 q^{51} - 2 q^{52} + 2 q^{56} + 6 q^{58} - 8 q^{59} - 14 q^{61} - 2 q^{62} + 2 q^{63} + 2 q^{64} - 4 q^{66} - 6 q^{67} - 20 q^{69} + 4 q^{71} + 2 q^{72} + 10 q^{73} - 4 q^{74} - 2 q^{76} - 4 q^{77} + 8 q^{78} - 14 q^{79} - 10 q^{81} - 2 q^{82} + 4 q^{83} + 2 q^{86} - 12 q^{87} - 4 q^{88} - 4 q^{89} + 2 q^{91} - 4 q^{92} + 4 q^{93} + 4 q^{97} + 12 q^{98} - 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^7 - 2 * q^8 - 2 * q^9 + 4 * q^11 - 2 * q^13 + 2 * q^14 + 2 * q^16 + 2 * q^18 - 2 * q^19 - 4 * q^22 - 4 * q^23 + 2 * q^26 - 2 * q^28 - 6 * q^29 + 2 * q^31 - 2 * q^32 + 4 * q^33 - 2 * q^36 + 4 * q^37 + 2 * q^38 - 8 * q^39 + 2 * q^41 - 2 * q^43 + 4 * q^44 + 4 * q^46 - 12 * q^49 + 4 * q^51 - 2 * q^52 + 2 * q^56 + 6 * q^58 - 8 * q^59 - 14 * q^61 - 2 * q^62 + 2 * q^63 + 2 * q^64 - 4 * q^66 - 6 * q^67 - 20 * q^69 + 4 * q^71 + 2 * q^72 + 10 * q^73 - 4 * q^74 - 2 * q^76 - 4 * q^77 + 8 * q^78 - 14 * q^79 - 10 * q^81 - 2 * q^82 + 4 * q^83 + 2 * q^86 - 12 * q^87 - 4 * q^88 - 4 * q^89 + 2 * q^91 - 4 * q^92 + 4 * q^93 + 4 * q^97 + 12 * q^98 - 4 * q^99

Introduction

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