LMFDB - Embedded newform 8550.2.a.bn.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8550,2,Mod(1,8550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8550, 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("8550.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8550.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:	$68.2720937282$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 190)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	8550.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -4.41421 q^{7} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} -4.41421 q^{7} -1.00000 q^{8} +1.41421 q^{11} -5.82843 q^{13} +4.41421 q^{14} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{19} -1.41421 q^{22} +0.757359 q^{23} +5.82843 q^{26} -4.41421 q^{28} +0.171573 q^{29} +6.24264 q^{31} -1.00000 q^{32} -1.00000 q^{34} +8.48528 q^{37} +1.00000 q^{38} +4.24264 q^{41} -1.75736 q^{43} +1.41421 q^{44} -0.757359 q^{46} +12.4853 q^{49} -5.82843 q^{52} +5.48528 q^{53} +4.41421 q^{56} -0.171573 q^{58} -6.89949 q^{59} +14.2426 q^{61} -6.24264 q^{62} +1.00000 q^{64} -4.75736 q^{67} +1.00000 q^{68} +13.4142 q^{71} -11.4853 q^{73} -8.48528 q^{74} -1.00000 q^{76} -6.24264 q^{77} -6.48528 q^{79} -4.24264 q^{82} +14.4853 q^{83} +1.75736 q^{86} -1.41421 q^{88} -7.07107 q^{89} +25.7279 q^{91} +0.757359 q^{92} +0.343146 q^{97} -12.4853 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 6 q^{7} - 2 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 - 6 * q^7 - 2 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} - 6 q^{7} - 2 q^{8} - 6 q^{13} + 6 q^{14} + 2 q^{16} + 2 q^{17} - 2 q^{19} + 10 q^{23} + 6 q^{26} - 6 q^{28} + 6 q^{29} + 4 q^{31} - 2 q^{32} - 2 q^{34} + 2 q^{38} - 12 q^{43} - 10 q^{46} + 8 q^{49} - 6 q^{52} - 6 q^{53} + 6 q^{56} - 6 q^{58} + 6 q^{59} + 20 q^{61} - 4 q^{62} + 2 q^{64} - 18 q^{67} + 2 q^{68} + 24 q^{71} - 6 q^{73} - 2 q^{76} - 4 q^{77} + 4 q^{79} + 12 q^{83} + 12 q^{86} + 26 q^{91} + 10 q^{92} + 12 q^{97} - 8 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 6 * q^7 - 2 * q^8 - 6 * q^13 + 6 * q^14 + 2 * q^16 + 2 * q^17 - 2 * q^19 + 10 * q^23 + 6 * q^26 - 6 * q^28 + 6 * q^29 + 4 * q^31 - 2 * q^32 - 2 * q^34 + 2 * q^38 - 12 * q^43 - 10 * q^46 + 8 * q^49 - 6 * q^52 - 6 * q^53 + 6 * q^56 - 6 * q^58 + 6 * q^59 + 20 * q^61 - 4 * q^62 + 2 * q^64 - 18 * q^67 + 2 * q^68 + 24 * q^71 - 6 * q^73 - 2 * q^76 - 4 * q^77 + 4 * q^79 + 12 * q^83 + 12 * q^86 + 26 * q^91 + 10 * q^92 + 12 * q^97 - 8 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.41421	−1.66842	−0.834208	−	0.551450i	$-0.814075\pi$
$7$	−4.41421	−1.66842	−0.834208	+	0.551450i	$0.814075\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	0	0
$11$	1.41421	0.426401	0.213201	−	0.977008i	$-0.431611\pi$
$11$	1.41421	0.426401	0.213201	+	0.977008i	$0.431611\pi$
$12$	0	0
$13$	−5.82843	−1.61651	−0.808257	−	0.588829i	$-0.799589\pi$
$13$	−5.82843	−1.61651	−0.808257	+	0.588829i	$0.799589\pi$
$14$	4.41421	1.17975
$15$	0	0
$16$	1.00000	0.250000
$17$	1.00000	0.242536	0.121268	−	0.992620i	$-0.461304\pi$
$17$	1.00000	0.242536	0.121268	+	0.992620i	$0.461304\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	−1.41421	−0.301511
$23$	0.757359	0.157920	0.0789602	−	0.996878i	$-0.474840\pi$
$23$	0.757359	0.157920	0.0789602	+	0.996878i	$0.474840\pi$
$24$	0	0
$25$	0	0
$26$	5.82843	1.14305
$27$	0	0
$28$	−4.41421	−0.834208
$29$	0.171573	0.0318603	0.0159301	−	0.999873i	$-0.494929\pi$
$29$	0.171573	0.0318603	0.0159301	+	0.999873i	$0.494929\pi$
$30$	0	0
$31$	6.24264	1.12121	0.560606	−	0.828083i	$-0.310568\pi$
$31$	6.24264	1.12121	0.560606	+	0.828083i	$0.310568\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−1.00000	−0.171499
$35$	0	0
$36$	0	0
$37$	8.48528	1.39497	0.697486	−	0.716599i	$-0.254302\pi$
$37$	8.48528	1.39497	0.697486	+	0.716599i	$0.254302\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	0	0
$41$	4.24264	0.662589	0.331295	−	0.943527i	$-0.392515\pi$
$41$	4.24264	0.662589	0.331295	+	0.943527i	$0.392515\pi$
$42$	0	0
$43$	−1.75736	−0.267995	−0.133997	−	0.990982i	$-0.542781\pi$
$43$	−1.75736	−0.267995	−0.133997	+	0.990982i	$0.542781\pi$
$44$	1.41421	0.213201
$45$	0	0
$46$	−0.757359	−0.111667
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	12.4853	1.78361
$50$	0	0
$51$	0	0
$52$	−5.82843	−0.808257
$53$	5.48528	0.753461	0.376731	−	0.926323i	$-0.377048\pi$
$53$	5.48528	0.753461	0.376731	+	0.926323i	$0.377048\pi$
$54$	0	0
$55$	0	0
$56$	4.41421	0.589874
$57$	0	0
$58$	−0.171573	−0.0225286
$59$	−6.89949	−0.898238	−0.449119	−	0.893472i	$-0.648262\pi$
$59$	−6.89949	−0.898238	−0.449119	+	0.893472i	$0.648262\pi$
$60$	0	0
$61$	14.2426	1.82358	0.911792	−	0.410653i	$-0.134699\pi$
$61$	14.2426	1.82358	0.911792	+	0.410653i	$0.134699\pi$
$62$	−6.24264	−0.792816
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−4.75736	−0.581204	−0.290602	−	0.956844i	$-0.593856\pi$
$67$	−4.75736	−0.581204	−0.290602	+	0.956844i	$0.593856\pi$
$68$	1.00000	0.121268
$69$	0	0
$70$	0	0
$71$	13.4142	1.59197	0.795987	−	0.605314i	$-0.206952\pi$
$71$	13.4142	1.59197	0.795987	+	0.605314i	$0.206952\pi$
$72$	0	0
$73$	−11.4853	−1.34425	−0.672125	−	0.740437i	$-0.734618\pi$
$73$	−11.4853	−1.34425	−0.672125	+	0.740437i	$0.734618\pi$
$74$	−8.48528	−0.986394
$75$	0	0
$76$	−1.00000	−0.114708
$77$	−6.24264	−0.711415
$78$	0	0
$79$	−6.48528	−0.729651	−0.364826	−	0.931076i	$-0.618871\pi$
$79$	−6.48528	−0.729651	−0.364826	+	0.931076i	$0.618871\pi$
$80$	0	0
$81$	0	0
$82$	−4.24264	−0.468521
$83$	14.4853	1.58997	0.794983	−	0.606632i	$-0.207480\pi$
$83$	14.4853	1.58997	0.794983	+	0.606632i	$0.207480\pi$
$84$	0	0
$85$	0	0
$86$	1.75736	0.189501
$87$	0	0
$88$	−1.41421	−0.150756
$89$	−7.07107	−0.749532	−0.374766	−	0.927119i	$-0.622277\pi$
$89$	−7.07107	−0.749532	−0.374766	+	0.927119i	$0.622277\pi$
$90$	0	0
$91$	25.7279	2.69702
$92$	0.757359	0.0789602
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0.343146	0.0348412	0.0174206	−	0.999848i	$-0.494455\pi$
$97$	0.343146	0.0348412	0.0174206	+	0.999848i	$0.494455\pi$
$98$	−12.4853	−1.26120
$99$	0	0
$100$	0	0
$101$	−13.0711	−1.30062	−0.650310	−	0.759669i	$-0.725361\pi$
$101$	−13.0711	−1.30062	−0.650310	+	0.759669i	$0.725361\pi$
$102$	0	0
$103$	−4.24264	−0.418040	−0.209020	−	0.977911i	$-0.567027\pi$
$103$	−4.24264	−0.418040	−0.209020	+	0.977911i	$0.567027\pi$
$104$	5.82843	0.571524
$105$	0	0
$106$	−5.48528	−0.532778
$107$	19.7279	1.90717	0.953585	−	0.301124i	$-0.0973617\pi$
$107$	19.7279	1.90717	0.953585	+	0.301124i	$0.0973617\pi$
$108$	0	0
$109$	−17.9706	−1.72127	−0.860634	−	0.509224i	$-0.829932\pi$
$109$	−17.9706	−1.72127	−0.860634	+	0.509224i	$0.829932\pi$
$110$	0	0
$111$	0	0
$112$	−4.41421	−0.417104
$113$	−10.2426	−0.963547	−0.481773	−	0.876296i	$-0.660007\pi$
$113$	−10.2426	−0.963547	−0.481773	+	0.876296i	$0.660007\pi$
$114$	0	0
$115$	0	0
$116$	0.171573	0.0159301
$117$	0	0
$118$	6.89949	0.635150
$119$	−4.41421	−0.404650
$120$	0	0
$121$	−9.00000	−0.818182
$122$	−14.2426	−1.28947
$123$	0	0
$124$	6.24264	0.560606
$125$	0	0
$126$	0	0
$127$	2.48528	0.220533	0.110267	−	0.993902i	$-0.464830\pi$
$127$	2.48528	0.220533	0.110267	+	0.993902i	$0.464830\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0	0
$131$	16.9706	1.48272	0.741362	−	0.671105i	$-0.234180\pi$
$131$	16.9706	1.48272	0.741362	+	0.671105i	$0.234180\pi$
$132$	0	0
$133$	4.41421	0.382761
$134$	4.75736	0.410973
$135$	0	0
$136$	−1.00000	−0.0857493
$137$	13.0000	1.11066	0.555332	−	0.831628i	$-0.312591\pi$
$137$	13.0000	1.11066	0.555332	+	0.831628i	$0.312591\pi$
$138$	0	0
$139$	−12.0000	−1.01783	−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000	−1.01783	−0.508913	+	0.860818i	$0.669953\pi$
$140$	0	0
$141$	0	0
$142$	−13.4142	−1.12570
$143$	−8.24264	−0.689284
$144$	0	0
$145$	0	0
$146$	11.4853	0.950529
$147$	0	0
$148$	8.48528	0.697486
$149$	−17.6569	−1.44651	−0.723253	−	0.690583i	$-0.757354\pi$
$149$	−17.6569	−1.44651	−0.723253	+	0.690583i	$0.757354\pi$
$150$	0	0
$151$	−10.4853	−0.853280	−0.426640	−	0.904422i	$-0.640303\pi$
$151$	−10.4853	−0.853280	−0.426640	+	0.904422i	$0.640303\pi$
$152$	1.00000	0.0811107
$153$	0	0
$154$	6.24264	0.503046
$155$	0	0
$156$	0	0
$157$	−11.6569	−0.930318	−0.465159	−	0.885227i	$-0.654003\pi$
$157$	−11.6569	−0.930318	−0.465159	+	0.885227i	$0.654003\pi$
$158$	6.48528	0.515941
$159$	0	0
$160$	0	0
$161$	−3.34315	−0.263477
$162$	0	0
$163$	10.2426	0.802266	0.401133	−	0.916020i	$-0.368617\pi$
$163$	10.2426	0.802266	0.401133	+	0.916020i	$0.368617\pi$
$164$	4.24264	0.331295
$165$	0	0
$166$	−14.4853	−1.12428
$167$	−18.2426	−1.41166	−0.705829	−	0.708382i	$-0.749425\pi$
$167$	−18.2426	−1.41166	−0.705829	+	0.708382i	$0.749425\pi$
$168$	0	0
$169$	20.9706	1.61312
$170$	0	0
$171$	0	0
$172$	−1.75736	−0.133997
$173$	0.485281	0.0368953	0.0184476	−	0.999830i	$-0.494128\pi$
$173$	0.485281	0.0368953	0.0184476	+	0.999830i	$0.494128\pi$
$174$	0	0
$175$	0	0
$176$	1.41421	0.106600
$177$	0	0
$178$	7.07107	0.529999
$179$	11.6569	0.871274	0.435637	−	0.900122i	$-0.356523\pi$
$179$	11.6569	0.871274	0.435637	+	0.900122i	$0.356523\pi$
$180$	0	0
$181$	−8.48528	−0.630706	−0.315353	−	0.948974i	$-0.602123\pi$
$181$	−8.48528	−0.630706	−0.315353	+	0.948974i	$0.602123\pi$
$182$	−25.7279	−1.90708
$183$	0	0
$184$	−0.757359	−0.0558333
$185$	0	0
$186$	0	0
$187$	1.41421	0.103418
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−12.5563	−0.908546	−0.454273	−	0.890863i	$-0.650101\pi$
$191$	−12.5563	−0.908546	−0.454273	+	0.890863i	$0.650101\pi$
$192$	0	0
$193$	0.343146	0.0247002	0.0123501	−	0.999924i	$-0.496069\pi$
$193$	0.343146	0.0247002	0.0123501	+	0.999924i	$0.496069\pi$
$194$	−0.343146	−0.0246364
$195$	0	0
$196$	12.4853	0.891806
$197$	−11.7574	−0.837677	−0.418839	−	0.908061i	$-0.637563\pi$
$197$	−11.7574	−0.837677	−0.418839	+	0.908061i	$0.637563\pi$
$198$	0	0
$199$	9.24264	0.655193	0.327597	−	0.944818i	$-0.393761\pi$
$199$	9.24264	0.655193	0.327597	+	0.944818i	$0.393761\pi$
$200$	0	0
$201$	0	0
$202$	13.0711	0.919677
$203$	−0.757359	−0.0531562
$204$	0	0
$205$	0	0
$206$	4.24264	0.295599
$207$	0	0
$208$	−5.82843	−0.404129
$209$	−1.41421	−0.0978232
$210$	0	0
$211$	−19.7279	−1.35813	−0.679063	−	0.734080i	$-0.737614\pi$
$211$	−19.7279	−1.35813	−0.679063	+	0.734080i	$0.737614\pi$
$212$	5.48528	0.376731
$213$	0	0
$214$	−19.7279	−1.34857
$215$	0	0
$216$	0	0
$217$	−27.5563	−1.87065
$218$	17.9706	1.21712
$219$	0	0
$220$	0	0
$221$	−5.82843	−0.392062
$222$	0	0
$223$	15.1716	1.01596	0.507982	−	0.861368i	$-0.330392\pi$
$223$	15.1716	1.01596	0.507982	+	0.861368i	$0.330392\pi$
$224$	4.41421	0.294937
$225$	0	0
$226$	10.2426	0.681330
$227$	−16.7574	−1.11223	−0.556113	−	0.831107i	$-0.687708\pi$
$227$	−16.7574	−1.11223	−0.556113	+	0.831107i	$0.687708\pi$
$228$	0	0
$229$	14.9706	0.989283	0.494641	−	0.869097i	$-0.335299\pi$
$229$	14.9706	0.989283	0.494641	+	0.869097i	$0.335299\pi$
$230$	0	0
$231$	0	0
$232$	−0.171573	−0.0112643
$233$	−24.9706	−1.63588	−0.817938	−	0.575306i	$-0.804883\pi$
$233$	−24.9706	−1.63588	−0.817938	+	0.575306i	$0.804883\pi$
$234$	0	0
$235$	0	0
$236$	−6.89949	−0.449119
$237$	0	0
$238$	4.41421	0.286131
$239$	6.89949	0.446291	0.223146	−	0.974785i	$-0.428367\pi$
$239$	6.89949	0.446291	0.223146	+	0.974785i	$0.428367\pi$
$240$	0	0
$241$	8.97056	0.577845	0.288922	−	0.957353i	$-0.406703\pi$
$241$	8.97056	0.577845	0.288922	+	0.957353i	$0.406703\pi$
$242$	9.00000	0.578542
$243$	0	0
$244$	14.2426	0.911792
$245$	0	0
$246$	0	0
$247$	5.82843	0.370854
$248$	−6.24264	−0.396408
$249$	0	0
$250$	0	0
$251$	−3.55635	−0.224475	−0.112237	−	0.993681i	$-0.535802\pi$
$251$	−3.55635	−0.224475	−0.112237	+	0.993681i	$0.535802\pi$
$252$	0	0
$253$	1.07107	0.0673375
$254$	−2.48528	−0.155940
$255$	0	0
$256$	1.00000	0.0625000
$257$	20.7279	1.29297	0.646486	−	0.762926i	$-0.276238\pi$
$257$	20.7279	1.29297	0.646486	+	0.762926i	$0.276238\pi$
$258$	0	0
$259$	−37.4558	−2.32739
$260$	0	0
$261$	0	0
$262$	−16.9706	−1.04844
$263$	−26.9706	−1.66308	−0.831538	−	0.555468i	$-0.812539\pi$
$263$	−26.9706	−1.66308	−0.831538	+	0.555468i	$0.812539\pi$
$264$	0	0
$265$	0	0
$266$	−4.41421	−0.270653
$267$	0	0
$268$	−4.75736	−0.290602
$269$	16.6274	1.01379	0.506896	−	0.862007i	$-0.330793\pi$
$269$	16.6274	1.01379	0.506896	+	0.862007i	$0.330793\pi$
$270$	0	0
$271$	−27.2426	−1.65487	−0.827436	−	0.561560i	$-0.810202\pi$
$271$	−27.2426	−1.65487	−0.827436	+	0.561560i	$0.810202\pi$
$272$	1.00000	0.0606339
$273$	0	0
$274$	−13.0000	−0.785359
$275$	0	0
$276$	0	0
$277$	0	0		−	1.00000i	$-0.5\pi$
$277$	0	0			1.00000i	$0.5\pi$
$278$	12.0000	0.719712
$279$	0	0
$280$	0	0
$281$	−4.24264	−0.253095	−0.126547	−	0.991961i	$-0.540390\pi$
$281$	−4.24264	−0.253095	−0.126547	+	0.991961i	$0.540390\pi$
$282$	0	0
$283$	−32.1421	−1.91065	−0.955326	−	0.295555i	$-0.904496\pi$
$283$	−32.1421	−1.91065	−0.955326	+	0.295555i	$0.904496\pi$
$284$	13.4142	0.795987
$285$	0	0
$286$	8.24264	0.487398
$287$	−18.7279	−1.10547
$288$	0	0
$289$	−16.0000	−0.941176
$290$	0	0
$291$	0	0
$292$	−11.4853	−0.672125
$293$	−5.48528	−0.320454	−0.160227	−	0.987080i	$-0.551223\pi$
$293$	−5.48528	−0.320454	−0.160227	+	0.987080i	$0.551223\pi$
$294$	0	0
$295$	0	0
$296$	−8.48528	−0.493197
$297$	0	0
$298$	17.6569	1.02283
$299$	−4.41421	−0.255281
$300$	0	0
$301$	7.75736	0.447127
$302$	10.4853	0.603360
$303$	0	0
$304$	−1.00000	−0.0573539
$305$	0	0
$306$	0	0
$307$	17.6569	1.00773	0.503865	−	0.863782i	$-0.331911\pi$
$307$	17.6569	1.00773	0.503865	+	0.863782i	$0.331911\pi$
$308$	−6.24264	−0.355707
$309$	0	0
$310$	0	0
$311$	4.75736	0.269765	0.134883	−	0.990862i	$-0.456934\pi$
$311$	4.75736	0.269765	0.134883	+	0.990862i	$0.456934\pi$
$312$	0	0
$313$	7.97056	0.450523	0.225261	−	0.974298i	$-0.427676\pi$
$313$	7.97056	0.450523	0.225261	+	0.974298i	$0.427676\pi$
$314$	11.6569	0.657834
$315$	0	0
$316$	−6.48528	−0.364826
$317$	−9.48528	−0.532746	−0.266373	−	0.963870i	$-0.585825\pi$
$317$	−9.48528	−0.532746	−0.266373	+	0.963870i	$0.585825\pi$
$318$	0	0
$319$	0.242641	0.0135853
$320$	0	0
$321$	0	0
$322$	3.34315	0.186306
$323$	−1.00000	−0.0556415
$324$	0	0
$325$	0	0
$326$	−10.2426	−0.567287
$327$	0	0
$328$	−4.24264	−0.234261
$329$	0	0
$330$	0	0
$331$	−19.2426	−1.05767	−0.528836	−	0.848724i	$-0.677371\pi$
$331$	−19.2426	−1.05767	−0.528836	+	0.848724i	$0.677371\pi$
$332$	14.4853	0.794983
$333$	0	0
$334$	18.2426	0.998193
$335$	0	0
$336$	0	0
$337$	−14.1005	−0.768103	−0.384052	−	0.923312i	$-0.625472\pi$
$337$	−14.1005	−0.768103	−0.384052	+	0.923312i	$0.625472\pi$
$338$	−20.9706	−1.14065
$339$	0	0
$340$	0	0
$341$	8.82843	0.478086
$342$	0	0
$343$	−24.2132	−1.30739
$344$	1.75736	0.0947505
$345$	0	0
$346$	−0.485281	−0.0260889
$347$	−5.51472	−0.296046	−0.148023	−	0.988984i	$-0.547291\pi$
$347$	−5.51472	−0.296046	−0.148023	+	0.988984i	$0.547291\pi$
$348$	0	0
$349$	6.00000	0.321173	0.160586	−	0.987022i	$-0.448662\pi$
$349$	6.00000	0.321173	0.160586	+	0.987022i	$0.448662\pi$
$350$	0	0
$351$	0	0
$352$	−1.41421	−0.0753778
$353$	−2.51472	−0.133845	−0.0669225	−	0.997758i	$-0.521318\pi$
$353$	−2.51472	−0.133845	−0.0669225	+	0.997758i	$0.521318\pi$
$354$	0	0
$355$	0	0
$356$	−7.07107	−0.374766
$357$	0	0
$358$	−11.6569	−0.616084
$359$	22.7574	1.20109	0.600544	−	0.799592i	$-0.294951\pi$
$359$	22.7574	1.20109	0.600544	+	0.799592i	$0.294951\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	8.48528	0.445976
$363$	0	0
$364$	25.7279	1.34851
$365$	0	0
$366$	0	0
$367$	25.4558	1.32878	0.664392	−	0.747384i	$-0.268691\pi$
$367$	25.4558	1.32878	0.664392	+	0.747384i	$0.268691\pi$
$368$	0.757359	0.0394801
$369$	0	0
$370$	0	0
$371$	−24.2132	−1.25709
$372$	0	0
$373$	−9.00000	−0.466002	−0.233001	−	0.972476i	$-0.574855\pi$
$373$	−9.00000	−0.466002	−0.233001	+	0.972476i	$0.574855\pi$
$374$	−1.41421	−0.0731272
$375$	0	0
$376$	0	0
$377$	−1.00000	−0.0515026
$378$	0	0
$379$	11.2426	0.577496	0.288748	−	0.957405i	$-0.406761\pi$
$379$	11.2426	0.577496	0.288748	+	0.957405i	$0.406761\pi$
$380$	0	0
$381$	0	0
$382$	12.5563	0.642439
$383$	12.2426	0.625570	0.312785	−	0.949824i	$-0.398738\pi$
$383$	12.2426	0.625570	0.312785	+	0.949824i	$0.398738\pi$
$384$	0	0
$385$	0	0
$386$	−0.343146	−0.0174657
$387$	0	0
$388$	0.343146	0.0174206
$389$	−22.9289	−1.16254	−0.581272	−	0.813710i	$-0.697445\pi$
$389$	−22.9289	−1.16254	−0.581272	+	0.813710i	$0.697445\pi$
$390$	0	0
$391$	0.757359	0.0383013
$392$	−12.4853	−0.630602
$393$	0	0
$394$	11.7574	0.592327
$395$	0	0
$396$	0	0
$397$	−24.0000	−1.20453	−0.602263	−	0.798298i	$-0.705734\pi$
$397$	−24.0000	−1.20453	−0.602263	+	0.798298i	$0.705734\pi$
$398$	−9.24264	−0.463292
$399$	0	0
$400$	0	0
$401$	25.4142	1.26913	0.634563	−	0.772871i	$-0.281180\pi$
$401$	25.4142	1.26913	0.634563	+	0.772871i	$0.281180\pi$
$402$	0	0
$403$	−36.3848	−1.81245
$404$	−13.0711	−0.650310
$405$	0	0
$406$	0.757359	0.0375871
$407$	12.0000	0.594818
$408$	0	0
$409$	25.2132	1.24671	0.623356	−	0.781938i	$-0.285769\pi$
$409$	25.2132	1.24671	0.623356	+	0.781938i	$0.285769\pi$
$410$	0	0
$411$	0	0
$412$	−4.24264	−0.209020
$413$	30.4558	1.49863
$414$	0	0
$415$	0	0
$416$	5.82843	0.285762
$417$	0	0
$418$	1.41421	0.0691714
$419$	−19.4142	−0.948446	−0.474223	−	0.880405i	$-0.657271\pi$
$419$	−19.4142	−0.948446	−0.474223	+	0.880405i	$0.657271\pi$
$420$	0	0
$421$	19.4853	0.949655	0.474827	−	0.880079i	$-0.342511\pi$
$421$	19.4853	0.949655	0.474827	+	0.880079i	$0.342511\pi$
$422$	19.7279	0.960340
$423$	0	0
$424$	−5.48528	−0.266389
$425$	0	0
$426$	0	0
$427$	−62.8701	−3.04250
$428$	19.7279	0.953585
$429$	0	0
$430$	0	0
$431$	6.38478	0.307544	0.153772	−	0.988106i	$-0.450858\pi$
$431$	6.38478	0.307544	0.153772	+	0.988106i	$0.450858\pi$
$432$	0	0
$433$	9.55635	0.459249	0.229624	−	0.973279i	$-0.426250\pi$
$433$	9.55635	0.459249	0.229624	+	0.973279i	$0.426250\pi$
$434$	27.5563	1.32275
$435$	0	0
$436$	−17.9706	−0.860634
$437$	−0.757359	−0.0362294
$438$	0	0
$439$	−5.75736	−0.274784	−0.137392	−	0.990517i	$-0.543872\pi$
$439$	−5.75736	−0.274784	−0.137392	+	0.990517i	$0.543872\pi$
$440$	0	0
$441$	0	0
$442$	5.82843	0.277230
$443$	4.24264	0.201574	0.100787	−	0.994908i	$-0.467864\pi$
$443$	4.24264	0.201574	0.100787	+	0.994908i	$0.467864\pi$
$444$	0	0
$445$	0	0
$446$	−15.1716	−0.718395
$447$	0	0
$448$	−4.41421	−0.208552
$449$	−17.3137	−0.817084	−0.408542	−	0.912739i	$-0.633963\pi$
$449$	−17.3137	−0.817084	−0.408542	+	0.912739i	$0.633963\pi$
$450$	0	0
$451$	6.00000	0.282529
$452$	−10.2426	−0.481773
$453$	0	0
$454$	16.7574	0.786462
$455$	0	0
$456$	0	0
$457$	3.00000	0.140334	0.0701670	−	0.997535i	$-0.477647\pi$
$457$	3.00000	0.140334	0.0701670	+	0.997535i	$0.477647\pi$
$458$	−14.9706	−0.699528
$459$	0	0
$460$	0	0
$461$	3.55635	0.165636	0.0828178	−	0.996565i	$-0.473608\pi$
$461$	3.55635	0.165636	0.0828178	+	0.996565i	$0.473608\pi$
$462$	0	0
$463$	−14.1421	−0.657241	−0.328620	−	0.944462i	$-0.606584\pi$
$463$	−14.1421	−0.657241	−0.328620	+	0.944462i	$0.606584\pi$
$464$	0.171573	0.00796507
$465$	0	0
$466$	24.9706	1.15674
$467$	0.727922	0.0336842	0.0168421	−	0.999858i	$-0.494639\pi$
$467$	0.727922	0.0336842	0.0168421	+	0.999858i	$0.494639\pi$
$468$	0	0
$469$	21.0000	0.969690
$470$	0	0
$471$	0	0
$472$	6.89949	0.317575
$473$	−2.48528	−0.114273
$474$	0	0
$475$	0	0
$476$	−4.41421	−0.202325
$477$	0	0
$478$	−6.89949	−0.315576
$479$	31.1127	1.42158	0.710788	−	0.703407i	$-0.248339\pi$
$479$	31.1127	1.42158	0.710788	+	0.703407i	$0.248339\pi$
$480$	0	0
$481$	−49.4558	−2.25499
$482$	−8.97056	−0.408598
$483$	0	0
$484$	−9.00000	−0.409091
$485$	0	0
$486$	0	0
$487$	−37.7990	−1.71284	−0.856418	−	0.516283i	$-0.827315\pi$
$487$	−37.7990	−1.71284	−0.856418	+	0.516283i	$0.827315\pi$
$488$	−14.2426	−0.644734
$489$	0	0
$490$	0	0
$491$	33.5563	1.51438	0.757188	−	0.653197i	$-0.226572\pi$
$491$	33.5563	1.51438	0.757188	+	0.653197i	$0.226572\pi$
$492$	0	0
$493$	0.171573	0.00772725
$494$	−5.82843	−0.262233
$495$	0	0
$496$	6.24264	0.280303
$497$	−59.2132	−2.65608
$498$	0	0
$499$	−25.7574	−1.15306	−0.576529	−	0.817077i	$-0.695593\pi$
$499$	−25.7574	−1.15306	−0.576529	+	0.817077i	$0.695593\pi$
$500$	0	0
$501$	0	0
$502$	3.55635	0.158728
$503$	14.2721	0.636361	0.318180	−	0.948030i	$-0.396928\pi$
$503$	14.2721	0.636361	0.318180	+	0.948030i	$0.396928\pi$
$504$	0	0
$505$	0	0
$506$	−1.07107	−0.0476148
$507$	0	0
$508$	2.48528	0.110267
$509$	−28.9706	−1.28410	−0.642049	−	0.766664i	$-0.721915\pi$
$509$	−28.9706	−1.28410	−0.642049	+	0.766664i	$0.721915\pi$
$510$	0	0
$511$	50.6985	2.24277
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−20.7279	−0.914269
$515$	0	0
$516$	0	0
$517$	0	0
$518$	37.4558	1.64572
$519$	0	0
$520$	0	0
$521$	−23.3137	−1.02139	−0.510696	−	0.859761i	$-0.670612\pi$
$521$	−23.3137	−1.02139	−0.510696	+	0.859761i	$0.670612\pi$
$522$	0	0
$523$	2.27208	0.0993510	0.0496755	−	0.998765i	$-0.484181\pi$
$523$	2.27208	0.0993510	0.0496755	+	0.998765i	$0.484181\pi$
$524$	16.9706	0.741362
$525$	0	0
$526$	26.9706	1.17597
$527$	6.24264	0.271934
$528$	0	0
$529$	−22.4264	−0.975061
$530$	0	0
$531$	0	0
$532$	4.41421	0.191380
$533$	−24.7279	−1.07109
$534$	0	0
$535$	0	0
$536$	4.75736	0.205487
$537$	0	0
$538$	−16.6274	−0.716859
$539$	17.6569	0.760535
$540$	0	0
$541$	9.75736	0.419502	0.209751	−	0.977755i	$-0.432735\pi$
$541$	9.75736	0.419502	0.209751	+	0.977755i	$0.432735\pi$
$542$	27.2426	1.17017
$543$	0	0
$544$	−1.00000	−0.0428746
$545$	0	0
$546$	0	0
$547$	17.3137	0.740281	0.370140	−	0.928976i	$-0.379310\pi$
$547$	17.3137	0.740281	0.370140	+	0.928976i	$0.379310\pi$
$548$	13.0000	0.555332
$549$	0	0
$550$	0	0
$551$	−0.171573	−0.00730925
$552$	0	0
$553$	28.6274	1.21736
$554$	0	0
$555$	0	0
$556$	−12.0000	−0.508913
$557$	16.0000	0.677942	0.338971	−	0.940797i	$-0.389921\pi$
$557$	16.0000	0.677942	0.338971	+	0.940797i	$0.389921\pi$
$558$	0	0
$559$	10.2426	0.433218
$560$	0	0
$561$	0	0
$562$	4.24264	0.178965
$563$	4.97056	0.209484	0.104742	−	0.994499i	$-0.466598\pi$
$563$	4.97056	0.209484	0.104742	+	0.994499i	$0.466598\pi$
$564$	0	0
$565$	0	0
$566$	32.1421	1.35103
$567$	0	0
$568$	−13.4142	−0.562848
$569$	28.2843	1.18574	0.592869	−	0.805299i	$-0.297995\pi$
$569$	28.2843	1.18574	0.592869	+	0.805299i	$0.297995\pi$
$570$	0	0
$571$	−2.24264	−0.0938516	−0.0469258	−	0.998898i	$-0.514942\pi$
$571$	−2.24264	−0.0938516	−0.0469258	+	0.998898i	$0.514942\pi$
$572$	−8.24264	−0.344642
$573$	0	0
$574$	18.7279	0.781688
$575$	0	0
$576$	0	0
$577$	20.3137	0.845671	0.422835	−	0.906207i	$-0.361035\pi$
$577$	20.3137	0.845671	0.422835	+	0.906207i	$0.361035\pi$
$578$	16.0000	0.665512
$579$	0	0
$580$	0	0
$581$	−63.9411	−2.65272
$582$	0	0
$583$	7.75736	0.321277
$584$	11.4853	0.475264
$585$	0	0
$586$	5.48528	0.226595
$587$	30.2426	1.24825	0.624124	−	0.781326i	$-0.285456\pi$
$587$	30.2426	1.24825	0.624124	+	0.781326i	$0.285456\pi$
$588$	0	0
$589$	−6.24264	−0.257224
$590$	0	0
$591$	0	0
$592$	8.48528	0.348743
$593$	22.0000	0.903432	0.451716	−	0.892162i	$-0.350812\pi$
$593$	22.0000	0.903432	0.451716	+	0.892162i	$0.350812\pi$
$594$	0	0
$595$	0	0
$596$	−17.6569	−0.723253
$597$	0	0
$598$	4.41421	0.180511
$599$	15.2132	0.621595	0.310797	−	0.950476i	$-0.399404\pi$
$599$	15.2132	0.621595	0.310797	+	0.950476i	$0.399404\pi$
$600$	0	0
$601$	−20.2426	−0.825715	−0.412857	−	0.910796i	$-0.635469\pi$
$601$	−20.2426	−0.825715	−0.412857	+	0.910796i	$0.635469\pi$
$602$	−7.75736	−0.316166
$603$	0	0
$604$	−10.4853	−0.426640
$605$	0	0
$606$	0	0
$607$	−3.17157	−0.128730	−0.0643651	−	0.997926i	$-0.520502\pi$
$607$	−3.17157	−0.128730	−0.0643651	+	0.997926i	$0.520502\pi$
$608$	1.00000	0.0405554
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	11.6985	0.472497	0.236249	−	0.971693i	$-0.424082\pi$
$613$	11.6985	0.472497	0.236249	+	0.971693i	$0.424082\pi$
$614$	−17.6569	−0.712573
$615$	0	0
$616$	6.24264	0.251523
$617$	−4.48528	−0.180571	−0.0902853	−	0.995916i	$-0.528778\pi$
$617$	−4.48528	−0.180571	−0.0902853	+	0.995916i	$0.528778\pi$
$618$	0	0
$619$	7.75736	0.311795	0.155897	−	0.987773i	$-0.450173\pi$
$619$	7.75736	0.311795	0.155897	+	0.987773i	$0.450173\pi$
$620$	0	0
$621$	0	0
$622$	−4.75736	−0.190753
$623$	31.2132	1.25053
$624$	0	0
$625$	0	0
$626$	−7.97056	−0.318568
$627$	0	0
$628$	−11.6569	−0.465159
$629$	8.48528	0.338330
$630$	0	0
$631$	−38.9706	−1.55139	−0.775697	−	0.631106i	$-0.782601\pi$
$631$	−38.9706	−1.55139	−0.775697	+	0.631106i	$0.782601\pi$
$632$	6.48528	0.257971
$633$	0	0
$634$	9.48528	0.376709
$635$	0	0
$636$	0	0
$637$	−72.7696	−2.88323
$638$	−0.242641	−0.00960624
$639$	0	0
$640$	0	0
$641$	18.0416	0.712602	0.356301	−	0.934371i	$-0.384038\pi$
$641$	18.0416	0.712602	0.356301	+	0.934371i	$0.384038\pi$
$642$	0	0
$643$	14.4853	0.571244	0.285622	−	0.958342i	$-0.407800\pi$
$643$	14.4853	0.571244	0.285622	+	0.958342i	$0.407800\pi$
$644$	−3.34315	−0.131738
$645$	0	0
$646$	1.00000	0.0393445
$647$	−18.7574	−0.737428	−0.368714	−	0.929543i	$-0.620202\pi$
$647$	−18.7574	−0.737428	−0.368714	+	0.929543i	$0.620202\pi$
$648$	0	0
$649$	−9.75736	−0.383010
$650$	0	0
$651$	0	0
$652$	10.2426	0.401133
$653$	−28.9706	−1.13371	−0.566853	−	0.823819i	$-0.691839\pi$
$653$	−28.9706	−1.13371	−0.566853	+	0.823819i	$0.691839\pi$
$654$	0	0
$655$	0	0
$656$	4.24264	0.165647
$657$	0	0
$658$	0	0
$659$	−18.8995	−0.736220	−0.368110	−	0.929782i	$-0.619995\pi$
$659$	−18.8995	−0.736220	−0.368110	+	0.929782i	$0.619995\pi$
$660$	0	0
$661$	−18.4558	−0.717849	−0.358925	−	0.933367i	$-0.616856\pi$
$661$	−18.4558	−0.717849	−0.358925	+	0.933367i	$0.616856\pi$
$662$	19.2426	0.747886
$663$	0	0
$664$	−14.4853	−0.562138
$665$	0	0
$666$	0	0
$667$	0.129942	0.00503139
$668$	−18.2426	−0.705829
$669$	0	0
$670$	0	0
$671$	20.1421	0.777579
$672$	0	0
$673$	12.0000	0.462566	0.231283	−	0.972887i	$-0.425708\pi$
$673$	12.0000	0.462566	0.231283	+	0.972887i	$0.425708\pi$
$674$	14.1005	0.543131
$675$	0	0
$676$	20.9706	0.806560
$677$	−40.9411	−1.57350	−0.786748	−	0.617275i	$-0.788237\pi$
$677$	−40.9411	−1.57350	−0.786748	+	0.617275i	$0.788237\pi$
$678$	0	0
$679$	−1.51472	−0.0581296
$680$	0	0
$681$	0	0
$682$	−8.82843	−0.338058
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	0	0
$685$	0	0
$686$	24.2132	0.924464
$687$	0	0
$688$	−1.75736	−0.0669987
$689$	−31.9706	−1.21798
$690$	0	0
$691$	22.4853	0.855380	0.427690	−	0.903925i	$-0.359327\pi$
$691$	22.4853	0.855380	0.427690	+	0.903925i	$0.359327\pi$
$692$	0.485281	0.0184476
$693$	0	0
$694$	5.51472	0.209336
$695$	0	0
$696$	0	0
$697$	4.24264	0.160701
$698$	−6.00000	−0.227103
$699$	0	0
$700$	0	0
$701$	−16.9706	−0.640969	−0.320485	−	0.947254i	$-0.603846\pi$
$701$	−16.9706	−0.640969	−0.320485	+	0.947254i	$0.603846\pi$
$702$	0	0
$703$	−8.48528	−0.320028
$704$	1.41421	0.0533002
$705$	0	0
$706$	2.51472	0.0946427
$707$	57.6985	2.16997
$708$	0	0
$709$	−34.0000	−1.27690	−0.638448	−	0.769665i	$-0.720423\pi$
$709$	−34.0000	−1.27690	−0.638448	+	0.769665i	$0.720423\pi$
$710$	0	0
$711$	0	0
$712$	7.07107	0.264999
$713$	4.72792	0.177062
$714$	0	0
$715$	0	0
$716$	11.6569	0.435637
$717$	0	0
$718$	−22.7574	−0.849297
$719$	5.10051	0.190217	0.0951084	−	0.995467i	$-0.469680\pi$
$719$	5.10051	0.190217	0.0951084	+	0.995467i	$0.469680\pi$
$720$	0	0
$721$	18.7279	0.697464
$722$	−1.00000	−0.0372161
$723$	0	0
$724$	−8.48528	−0.315353
$725$	0	0
$726$	0	0
$727$	−9.72792	−0.360789	−0.180394	−	0.983594i	$-0.557737\pi$
$727$	−9.72792	−0.360789	−0.180394	+	0.983594i	$0.557737\pi$
$728$	−25.7279	−0.953540
$729$	0	0
$730$	0	0
$731$	−1.75736	−0.0649983
$732$	0	0
$733$	−12.0000	−0.443230	−0.221615	−	0.975134i	$-0.571133\pi$
$733$	−12.0000	−0.443230	−0.221615	+	0.975134i	$0.571133\pi$
$734$	−25.4558	−0.939592
$735$	0	0
$736$	−0.757359	−0.0279166
$737$	−6.72792	−0.247826
$738$	0	0
$739$	1.27208	0.0467941	0.0233971	−	0.999726i	$-0.492552\pi$
$739$	1.27208	0.0467941	0.0233971	+	0.999726i	$0.492552\pi$
$740$	0	0
$741$	0	0
$742$	24.2132	0.888895
$743$	6.72792	0.246824	0.123412	−	0.992356i	$-0.460616\pi$
$743$	6.72792	0.246824	0.123412	+	0.992356i	$0.460616\pi$
$744$	0	0
$745$	0	0
$746$	9.00000	0.329513
$747$	0	0
$748$	1.41421	0.0517088
$749$	−87.0833	−3.18195
$750$	0	0
$751$	−26.7279	−0.975316	−0.487658	−	0.873035i	$-0.662149\pi$
$751$	−26.7279	−0.975316	−0.487658	+	0.873035i	$0.662149\pi$
$752$	0	0
$753$	0	0
$754$	1.00000	0.0364179
$755$	0	0
$756$	0	0
$757$	−12.3431	−0.448619	−0.224310	−	0.974518i	$-0.572013\pi$
$757$	−12.3431	−0.448619	−0.224310	+	0.974518i	$0.572013\pi$
$758$	−11.2426	−0.408351
$759$	0	0
$760$	0	0
$761$	−43.9706	−1.59393	−0.796966	−	0.604024i	$-0.793563\pi$
$761$	−43.9706	−1.59393	−0.796966	+	0.604024i	$0.793563\pi$
$762$	0	0
$763$	79.3259	2.87179
$764$	−12.5563	−0.454273
$765$	0	0
$766$	−12.2426	−0.442345
$767$	40.2132	1.45201
$768$	0	0
$769$	−36.4558	−1.31463	−0.657316	−	0.753615i	$-0.728308\pi$
$769$	−36.4558	−1.31463	−0.657316	+	0.753615i	$0.728308\pi$
$770$	0	0
$771$	0	0
$772$	0.343146	0.0123501
$773$	13.9706	0.502486	0.251243	−	0.967924i	$-0.419161\pi$
$773$	13.9706	0.502486	0.251243	+	0.967924i	$0.419161\pi$
$774$	0	0
$775$	0	0
$776$	−0.343146	−0.0123182
$777$	0	0
$778$	22.9289	0.822042
$779$	−4.24264	−0.152008
$780$	0	0
$781$	18.9706	0.678820
$782$	−0.757359	−0.0270831
$783$	0	0
$784$	12.4853	0.445903
$785$	0	0
$786$	0	0
$787$	41.1838	1.46804	0.734021	−	0.679126i	$-0.237641\pi$
$787$	41.1838	1.46804	0.734021	+	0.679126i	$0.237641\pi$
$788$	−11.7574	−0.418839
$789$	0	0
$790$	0	0
$791$	45.2132	1.60760
$792$	0	0
$793$	−83.0122	−2.94785
$794$	24.0000	0.851728
$795$	0	0
$796$	9.24264	0.327597
$797$	−47.4853	−1.68201	−0.841007	−	0.541023i	$-0.818037\pi$
$797$	−47.4853	−1.68201	−0.841007	+	0.541023i	$0.818037\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	−25.4142	−0.897407
$803$	−16.2426	−0.573190
$804$	0	0
$805$	0	0
$806$	36.3848	1.28160
$807$	0	0
$808$	13.0711	0.459839
$809$	28.7990	1.01252	0.506259	−	0.862381i	$-0.331028\pi$
$809$	28.7990	1.01252	0.506259	+	0.862381i	$0.331028\pi$
$810$	0	0
$811$	52.6985	1.85049	0.925247	−	0.379365i	$-0.123858\pi$
$811$	52.6985	1.85049	0.925247	+	0.379365i	$0.123858\pi$
$812$	−0.757359	−0.0265781
$813$	0	0
$814$	−12.0000	−0.420600
$815$	0	0
$816$	0	0
$817$	1.75736	0.0614822
$818$	−25.2132	−0.881559
$819$	0	0
$820$	0	0
$821$	−6.34315	−0.221377	−0.110689	−	0.993855i	$-0.535306\pi$
$821$	−6.34315	−0.221377	−0.110689	+	0.993855i	$0.535306\pi$
$822$	0	0
$823$	15.7279	0.548241	0.274120	−	0.961695i	$-0.411613\pi$
$823$	15.7279	0.548241	0.274120	+	0.961695i	$0.411613\pi$
$824$	4.24264	0.147799
$825$	0	0
$826$	−30.4558	−1.05969
$827$	−20.6985	−0.719757	−0.359878	−	0.932999i	$-0.617182\pi$
$827$	−20.6985	−0.719757	−0.359878	+	0.932999i	$0.617182\pi$
$828$	0	0
$829$	10.4558	0.363146	0.181573	−	0.983377i	$-0.441881\pi$
$829$	10.4558	0.363146	0.181573	+	0.983377i	$0.441881\pi$
$830$	0	0
$831$	0	0
$832$	−5.82843	−0.202064
$833$	12.4853	0.432589
$834$	0	0
$835$	0	0
$836$	−1.41421	−0.0489116
$837$	0	0
$838$	19.4142	0.670653
$839$	−22.6274	−0.781185	−0.390593	−	0.920564i	$-0.627730\pi$
$839$	−22.6274	−0.781185	−0.390593	+	0.920564i	$0.627730\pi$
$840$	0	0
$841$	−28.9706	−0.998985
$842$	−19.4853	−0.671507
$843$	0	0
$844$	−19.7279	−0.679063
$845$	0	0
$846$	0	0
$847$	39.7279	1.36507
$848$	5.48528	0.188365
$849$	0	0
$850$	0	0
$851$	6.42641	0.220294
$852$	0	0
$853$	27.9411	0.956686	0.478343	−	0.878173i	$-0.341238\pi$
$853$	27.9411	0.956686	0.478343	+	0.878173i	$0.341238\pi$
$854$	62.8701	2.15137
$855$	0	0
$856$	−19.7279	−0.674286
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	4.72792	0.161315	0.0806573	−	0.996742i	$-0.474298\pi$
$859$	4.72792	0.161315	0.0806573	+	0.996742i	$0.474298\pi$
$860$	0	0
$861$	0	0
$862$	−6.38478	−0.217466
$863$	0.727922	0.0247788	0.0123894	−	0.999923i	$-0.496056\pi$
$863$	0.727922	0.0247788	0.0123894	+	0.999923i	$0.496056\pi$
$864$	0	0
$865$	0	0
$866$	−9.55635	−0.324738
$867$	0	0
$868$	−27.5563	−0.935323
$869$	−9.17157	−0.311124
$870$	0	0
$871$	27.7279	0.939525
$872$	17.9706	0.608560
$873$	0	0
$874$	0.757359	0.0256181
$875$	0	0
$876$	0	0
$877$	18.8579	0.636785	0.318392	−	0.947959i	$-0.396857\pi$
$877$	18.8579	0.636785	0.318392	+	0.947959i	$0.396857\pi$
$878$	5.75736	0.194301
$879$	0	0
$880$	0	0
$881$	39.5980	1.33409	0.667045	−	0.745018i	$-0.267559\pi$
$881$	39.5980	1.33409	0.667045	+	0.745018i	$0.267559\pi$
$882$	0	0
$883$	−14.4853	−0.487469	−0.243734	−	0.969842i	$-0.578372\pi$
$883$	−14.4853	−0.487469	−0.243734	+	0.969842i	$0.578372\pi$
$884$	−5.82843	−0.196031
$885$	0	0
$886$	−4.24264	−0.142534
$887$	−45.2132	−1.51811	−0.759055	−	0.651026i	$-0.774339\pi$
$887$	−45.2132	−1.51811	−0.759055	+	0.651026i	$0.774339\pi$
$888$	0	0
$889$	−10.9706	−0.367941
$890$	0	0
$891$	0	0
$892$	15.1716	0.507982
$893$	0	0
$894$	0	0
$895$	0	0
$896$	4.41421	0.147469
$897$	0	0
$898$	17.3137	0.577766
$899$	1.07107	0.0357221
$900$	0	0
$901$	5.48528	0.182741
$902$	−6.00000	−0.199778
$903$	0	0
$904$	10.2426	0.340665
$905$	0	0
$906$	0	0
$907$	−38.6985	−1.28496	−0.642481	−	0.766302i	$-0.722095\pi$
$907$	−38.6985	−1.28496	−0.642481	+	0.766302i	$0.722095\pi$
$908$	−16.7574	−0.556113
$909$	0	0
$910$	0	0
$911$	−40.2843	−1.33468	−0.667339	−	0.744754i	$-0.732567\pi$
$911$	−40.2843	−1.33468	−0.667339	+	0.744754i	$0.732567\pi$
$912$	0	0
$913$	20.4853	0.677964
$914$	−3.00000	−0.0992312
$915$	0	0
$916$	14.9706	0.494641
$917$	−74.9117	−2.47380
$918$	0	0
$919$	−6.21320	−0.204955	−0.102477	−	0.994735i	$-0.532677\pi$
$919$	−6.21320	−0.204955	−0.102477	+	0.994735i	$0.532677\pi$
$920$	0	0
$921$	0	0
$922$	−3.55635	−0.117122
$923$	−78.1838	−2.57345
$924$	0	0
$925$	0	0
$926$	14.1421	0.464739
$927$	0	0
$928$	−0.171573	−0.00563216
$929$	12.1716	0.399336	0.199668	−	0.979864i	$-0.436014\pi$
$929$	12.1716	0.399336	0.199668	+	0.979864i	$0.436014\pi$
$930$	0	0
$931$	−12.4853	−0.409189
$932$	−24.9706	−0.817938
$933$	0	0
$934$	−0.727922	−0.0238183
$935$	0	0
$936$	0	0
$937$	−27.0000	−0.882052	−0.441026	−	0.897494i	$-0.645385\pi$
$937$	−27.0000	−0.882052	−0.441026	+	0.897494i	$0.645385\pi$
$938$	−21.0000	−0.685674
$939$	0	0
$940$	0	0
$941$	−53.1421	−1.73238	−0.866192	−	0.499711i	$-0.833439\pi$
$941$	−53.1421	−1.73238	−0.866192	+	0.499711i	$0.833439\pi$
$942$	0	0
$943$	3.21320	0.104636
$944$	−6.89949	−0.224559
$945$	0	0
$946$	2.48528	0.0808035
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	0	0
$949$	66.9411	2.17300
$950$	0	0
$951$	0	0
$952$	4.41421	0.143065
$953$	18.7279	0.606657	0.303328	−	0.952886i	$-0.401902\pi$
$953$	18.7279	0.606657	0.303328	+	0.952886i	$0.401902\pi$
$954$	0	0
$955$	0	0
$956$	6.89949	0.223146
$957$	0	0
$958$	−31.1127	−1.00521
$959$	−57.3848	−1.85305
$960$	0	0
$961$	7.97056	0.257115
$962$	49.4558	1.59452
$963$	0	0
$964$	8.97056	0.288922
$965$	0	0
$966$	0	0
$967$	−43.1127	−1.38641	−0.693205	−	0.720740i	$-0.743802\pi$
$967$	−43.1127	−1.38641	−0.693205	+	0.720740i	$0.743802\pi$
$968$	9.00000	0.289271
$969$	0	0
$970$	0	0
$971$	−41.6569	−1.33683	−0.668416	−	0.743788i	$-0.733027\pi$
$971$	−41.6569	−1.33683	−0.668416	+	0.743788i	$0.733027\pi$
$972$	0	0
$973$	52.9706	1.69816
$974$	37.7990	1.21116
$975$	0	0
$976$	14.2426	0.455896
$977$	0.242641	0.00776276	0.00388138	−	0.999992i	$-0.498765\pi$
$977$	0.242641	0.00776276	0.00388138	+	0.999992i	$0.498765\pi$
$978$	0	0
$979$	−10.0000	−0.319601
$980$	0	0
$981$	0	0
$982$	−33.5563	−1.07083
$983$	−51.4558	−1.64119	−0.820593	−	0.571513i	$-0.806357\pi$
$983$	−51.4558	−1.64119	−0.820593	+	0.571513i	$0.806357\pi$
$984$	0	0
$985$	0	0
$986$	−0.171573	−0.00546399
$987$	0	0
$988$	5.82843	0.185427
$989$	−1.33095	−0.0423218
$990$	0	0
$991$	13.7574	0.437017	0.218508	−	0.975835i	$-0.429881\pi$
$991$	13.7574	0.437017	0.218508	+	0.975835i	$0.429881\pi$
$992$	−6.24264	−0.198204
$993$	0	0
$994$	59.2132	1.87813
$995$	0	0
$996$	0	0
$997$	−48.7279	−1.54323	−0.771614	−	0.636091i	$-0.780550\pi$
$997$	−48.7279	−1.54323	−0.771614	+	0.636091i	$0.780550\pi$
$998$	25.7574	0.815335
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8550.2.a.bn.1.1		2
3.2	odd	2		950.2.a.g.1.2		2
5.2	odd	4		1710.2.d.c.1369.1		4
5.3	odd	4		1710.2.d.c.1369.3		4
5.4	even	2		8550.2.a.cb.1.2		2
12.11	even	2		7600.2.a.bg.1.1		2
15.2	even	4		190.2.b.a.39.3	yes	4
15.8	even	4		190.2.b.a.39.2	✓	4
15.14	odd	2		950.2.a.f.1.1		2
60.23	odd	4		1520.2.d.e.609.2		4
60.47	odd	4		1520.2.d.e.609.3		4
60.59	even	2		7600.2.a.v.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
190.2.b.a.39.2	✓	4	15.8	even	4
190.2.b.a.39.3	yes	4	15.2	even	4
950.2.a.f.1.1		2	15.14	odd	2
950.2.a.g.1.2		2	3.2	odd	2
1520.2.d.e.609.2		4	60.23	odd	4
1520.2.d.e.609.3		4	60.47	odd	4
1710.2.d.c.1369.1		4	5.2	odd	4
1710.2.d.c.1369.3		4	5.3	odd	4
7600.2.a.v.1.2		2	60.59	even	2
7600.2.a.bg.1.1		2	12.11	even	2
8550.2.a.bn.1.1		2	1.1	even	1	trivial
8550.2.a.cb.1.2		2	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8550.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -4.41421 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -4.41421 q^{7} -1.00000 q^{8} +1.41421 q^{11} -5.82843 q^{13} +4.41421 q^{14} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{19} -1.41421 q^{22} +0.757359 q^{23} +5.82843 q^{26} -4.41421 q^{28} +0.171573 q^{29} +6.24264 q^{31} -1.00000 q^{32} -1.00000 q^{34} +8.48528 q^{37} +1.00000 q^{38} +4.24264 q^{41} -1.75736 q^{43} +1.41421 q^{44} -0.757359 q^{46} +12.4853 q^{49} -5.82843 q^{52} +5.48528 q^{53} +4.41421 q^{56} -0.171573 q^{58} -6.89949 q^{59} +14.2426 q^{61} -6.24264 q^{62} +1.00000 q^{64} -4.75736 q^{67} +1.00000 q^{68} +13.4142 q^{71} -11.4853 q^{73} -8.48528 q^{74} -1.00000 q^{76} -6.24264 q^{77} -6.48528 q^{79} -4.24264 q^{82} +14.4853 q^{83} +1.75736 q^{86} -1.41421 q^{88} -7.07107 q^{89} +25.7279 q^{91} +0.757359 q^{92} +0.343146 q^{97} -12.4853 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 6 q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 6 * q^7 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} - 6 q^{7} - 2 q^{8} - 6 q^{13} + 6 q^{14} + 2 q^{16} + 2 q^{17} - 2 q^{19} + 10 q^{23} + 6 q^{26} - 6 q^{28} + 6 q^{29} + 4 q^{31} - 2 q^{32} - 2 q^{34} + 2 q^{38} - 12 q^{43} - 10 q^{46} + 8 q^{49} - 6 q^{52} - 6 q^{53} + 6 q^{56} - 6 q^{58} + 6 q^{59} + 20 q^{61} - 4 q^{62} + 2 q^{64} - 18 q^{67} + 2 q^{68} + 24 q^{71} - 6 q^{73} - 2 q^{76} - 4 q^{77} + 4 q^{79} + 12 q^{83} + 12 q^{86} + 26 q^{91} + 10 q^{92} + 12 q^{97} - 8 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 6 * q^7 - 2 * q^8 - 6 * q^13 + 6 * q^14 + 2 * q^16 + 2 * q^17 - 2 * q^19 + 10 * q^23 + 6 * q^26 - 6 * q^28 + 6 * q^29 + 4 * q^31 - 2 * q^32 - 2 * q^34 + 2 * q^38 - 12 * q^43 - 10 * q^46 + 8 * q^49 - 6 * q^52 - 6 * q^53 + 6 * q^56 - 6 * q^58 + 6 * q^59 + 20 * q^61 - 4 * q^62 + 2 * q^64 - 18 * q^67 + 2 * q^68 + 24 * q^71 - 6 * q^73 - 2 * q^76 - 4 * q^77 + 4 * q^79 + 12 * q^83 + 12 * q^86 + 26 * q^91 + 10 * q^92 + 12 * q^97 - 8 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more