LMFDB - Embedded newform 9075.2.a.bx.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9075,2,Mod(1,9075)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9075, 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("9075.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9075 = 3 \cdot 5^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9075.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:	$72.4642398343$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1815)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	9075.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.61803 q^{2} +1.00000 q^{3} +0.618034 q^{4} +1.61803 q^{6} -1.23607 q^{7} -2.23607 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.61803 q^{2} +1.00000 q^{3} +0.618034 q^{4} +1.61803 q^{6} -1.23607 q^{7} -2.23607 q^{8} +1.00000 q^{9} +0.618034 q^{12} +0.763932 q^{13} -2.00000 q^{14} -4.85410 q^{16} +3.47214 q^{17} +1.61803 q^{18} +2.47214 q^{19} -1.23607 q^{21} -8.70820 q^{23} -2.23607 q^{24} +1.23607 q^{26} +1.00000 q^{27} -0.763932 q^{28} -3.23607 q^{29} +6.70820 q^{31} -3.38197 q^{32} +5.61803 q^{34} +0.618034 q^{36} -5.23607 q^{37} +4.00000 q^{38} +0.763932 q^{39} -12.4721 q^{41} -2.00000 q^{42} -0.763932 q^{43} -14.0902 q^{46} +4.70820 q^{47} -4.85410 q^{48} -5.47214 q^{49} +3.47214 q^{51} +0.472136 q^{52} +7.94427 q^{53} +1.61803 q^{54} +2.76393 q^{56} +2.47214 q^{57} -5.23607 q^{58} -1.70820 q^{59} -7.47214 q^{61} +10.8541 q^{62} -1.23607 q^{63} +4.23607 q^{64} -6.76393 q^{67} +2.14590 q^{68} -8.70820 q^{69} -5.52786 q^{71} -2.23607 q^{72} -1.52786 q^{73} -8.47214 q^{74} +1.52786 q^{76} +1.23607 q^{78} +0.708204 q^{79} +1.00000 q^{81} -20.1803 q^{82} -4.00000 q^{83} -0.763932 q^{84} -1.23607 q^{86} -3.23607 q^{87} +9.23607 q^{89} -0.944272 q^{91} -5.38197 q^{92} +6.70820 q^{93} +7.61803 q^{94} -3.38197 q^{96} -17.2361 q^{97} -8.85410 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} + 2 q^{3} - q^{4} + q^{6} + 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + q^2 + 2 * q^3 - q^4 + q^6 + 2 * q^7 + 2 * q^9	$ 2 q + q^{2} + 2 q^{3} - q^{4} + q^{6} + 2 q^{7} + 2 q^{9} - q^{12} + 6 q^{13} - 4 q^{14} - 3 q^{16} - 2 q^{17} + q^{18} - 4 q^{19} + 2 q^{21} - 4 q^{23} - 2 q^{26} + 2 q^{27} - 6 q^{28} - 2 q^{29} - 9 q^{32} + 9 q^{34} - q^{36} - 6 q^{37} + 8 q^{38} + 6 q^{39} - 16 q^{41} - 4 q^{42} - 6 q^{43} - 17 q^{46} - 4 q^{47} - 3 q^{48} - 2 q^{49} - 2 q^{51} - 8 q^{52} - 2 q^{53} + q^{54} + 10 q^{56} - 4 q^{57} - 6 q^{58} + 10 q^{59} - 6 q^{61} + 15 q^{62} + 2 q^{63} + 4 q^{64} - 18 q^{67} + 11 q^{68} - 4 q^{69} - 20 q^{71} - 12 q^{73} - 8 q^{74} + 12 q^{76} - 2 q^{78} - 12 q^{79} + 2 q^{81} - 18 q^{82} - 8 q^{83} - 6 q^{84} + 2 q^{86} - 2 q^{87} + 14 q^{89} + 16 q^{91} - 13 q^{92} + 13 q^{94} - 9 q^{96} - 30 q^{97} - 11 q^{98}+O(q^{100}) $ 2 * q + q^2 + 2 * q^3 - q^4 + q^6 + 2 * q^7 + 2 * q^9 - q^12 + 6 * q^13 - 4 * q^14 - 3 * q^16 - 2 * q^17 + q^18 - 4 * q^19 + 2 * q^21 - 4 * q^23 - 2 * q^26 + 2 * q^27 - 6 * q^28 - 2 * q^29 - 9 * q^32 + 9 * q^34 - q^36 - 6 * q^37 + 8 * q^38 + 6 * q^39 - 16 * q^41 - 4 * q^42 - 6 * q^43 - 17 * q^46 - 4 * q^47 - 3 * q^48 - 2 * q^49 - 2 * q^51 - 8 * q^52 - 2 * q^53 + q^54 + 10 * q^56 - 4 * q^57 - 6 * q^58 + 10 * q^59 - 6 * q^61 + 15 * q^62 + 2 * q^63 + 4 * q^64 - 18 * q^67 + 11 * q^68 - 4 * q^69 - 20 * q^71 - 12 * q^73 - 8 * q^74 + 12 * q^76 - 2 * q^78 - 12 * q^79 + 2 * q^81 - 18 * q^82 - 8 * q^83 - 6 * q^84 + 2 * q^86 - 2 * q^87 + 14 * q^89 + 16 * q^91 - 13 * q^92 + 13 * q^94 - 9 * q^96 - 30 * q^97 - 11 * 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 $

$2$	1.61803	1.14412	0.572061	−	0.820211i	$-0.306144\pi$
$2$	1.61803	1.14412	0.572061	+	0.820211i	$0.306144\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0.618034	0.309017
$5$	0	0
$5$	0	0
$6$	1.61803	0.660560
$7$	−1.23607	−0.467190	−0.233595	−	0.972334i	$-0.575049\pi$
$7$	−1.23607	−0.467190	−0.233595	+	0.972334i	$0.575049\pi$
$8$	−2.23607	−0.790569
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0.618034	0.178411
$13$	0.763932	0.211877	0.105938	−	0.994373i	$-0.466215\pi$
$13$	0.763932	0.211877	0.105938	+	0.994373i	$0.466215\pi$
$14$	−2.00000	−0.534522
$15$	0	0
$16$	−4.85410	−1.21353
$17$	3.47214	0.842117	0.421058	−	0.907034i	$-0.361659\pi$
$17$	3.47214	0.842117	0.421058	+	0.907034i	$0.361659\pi$
$18$	1.61803	0.381374
$19$	2.47214	0.567147	0.283573	−	0.958951i	$-0.408480\pi$
$19$	2.47214	0.567147	0.283573	+	0.958951i	$0.408480\pi$
$20$	0	0
$21$	−1.23607	−0.269732
$22$	0	0
$23$	−8.70820	−1.81579	−0.907893	−	0.419202i	$-0.862310\pi$
$23$	−8.70820	−1.81579	−0.907893	+	0.419202i	$0.862310\pi$
$24$	−2.23607	−0.456435
$25$	0	0
$26$	1.23607	0.242413
$27$	1.00000	0.192450
$28$	−0.763932	−0.144370
$29$	−3.23607	−0.600923	−0.300461	−	0.953794i	$-0.597141\pi$
$29$	−3.23607	−0.600923	−0.300461	+	0.953794i	$0.597141\pi$
$30$	0	0
$31$	6.70820	1.20483	0.602414	−	0.798183i	$-0.294205\pi$
$31$	6.70820	1.20483	0.602414	+	0.798183i	$0.294205\pi$
$32$	−3.38197	−0.597853
$33$	0	0
$34$	5.61803	0.963485
$35$	0	0
$36$	0.618034	0.103006
$37$	−5.23607	−0.860804	−0.430402	−	0.902637i	$-0.641628\pi$
$37$	−5.23607	−0.860804	−0.430402	+	0.902637i	$0.641628\pi$
$38$	4.00000	0.648886
$39$	0.763932	0.122327
$40$	0	0
$41$	−12.4721	−1.94782	−0.973910	−	0.226934i	$-0.927130\pi$
$41$	−12.4721	−1.94782	−0.973910	+	0.226934i	$0.927130\pi$
$42$	−2.00000	−0.308607
$43$	−0.763932	−0.116499	−0.0582493	−	0.998302i	$-0.518552\pi$
$43$	−0.763932	−0.116499	−0.0582493	+	0.998302i	$0.518552\pi$
$44$	0	0
$45$	0	0
$46$	−14.0902	−2.07748
$47$	4.70820	0.686762	0.343381	−	0.939196i	$-0.388428\pi$
$47$	4.70820	0.686762	0.343381	+	0.939196i	$0.388428\pi$
$48$	−4.85410	−0.700629
$49$	−5.47214	−0.781734
$50$	0	0
$51$	3.47214	0.486196
$52$	0.472136	0.0654735
$53$	7.94427	1.09123	0.545615	−	0.838036i	$-0.316296\pi$
$53$	7.94427	1.09123	0.545615	+	0.838036i	$0.316296\pi$
$54$	1.61803	0.220187
$55$	0	0
$56$	2.76393	0.369346
$57$	2.47214	0.327442
$58$	−5.23607	−0.687529
$59$	−1.70820	−0.222389	−0.111195	−	0.993799i	$-0.535468\pi$
$59$	−1.70820	−0.222389	−0.111195	+	0.993799i	$0.535468\pi$
$60$	0	0
$61$	−7.47214	−0.956709	−0.478354	−	0.878167i	$-0.658767\pi$
$61$	−7.47214	−0.956709	−0.478354	+	0.878167i	$0.658767\pi$
$62$	10.8541	1.37847
$63$	−1.23607	−0.155730
$64$	4.23607	0.529508
$65$	0	0
$66$	0	0
$67$	−6.76393	−0.826346	−0.413173	−	0.910653i	$-0.635579\pi$
$67$	−6.76393	−0.826346	−0.413173	+	0.910653i	$0.635579\pi$
$68$	2.14590	0.260228
$69$	−8.70820	−1.04834
$70$	0	0
$71$	−5.52786	−0.656037	−0.328018	−	0.944671i	$-0.606381\pi$
$71$	−5.52786	−0.656037	−0.328018	+	0.944671i	$0.606381\pi$
$72$	−2.23607	−0.263523
$73$	−1.52786	−0.178823	−0.0894115	−	0.995995i	$-0.528499\pi$
$73$	−1.52786	−0.178823	−0.0894115	+	0.995995i	$0.528499\pi$
$74$	−8.47214	−0.984866
$75$	0	0
$76$	1.52786	0.175258
$77$	0	0
$78$	1.23607	0.139957
$79$	0.708204	0.0796792	0.0398396	−	0.999206i	$-0.487315\pi$
$79$	0.708204	0.0796792	0.0398396	+	0.999206i	$0.487315\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−20.1803	−2.22855
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	−0.763932	−0.0833518
$85$	0	0
$86$	−1.23607	−0.133289
$87$	−3.23607	−0.346943
$88$	0	0
$89$	9.23607	0.979021	0.489511	−	0.871997i	$-0.337175\pi$
$89$	9.23607	0.979021	0.489511	+	0.871997i	$0.337175\pi$
$90$	0	0
$91$	−0.944272	−0.0989866
$92$	−5.38197	−0.561109
$93$	6.70820	0.695608
$94$	7.61803	0.785740
$95$	0	0
$96$	−3.38197	−0.345170
$97$	−17.2361	−1.75006	−0.875029	−	0.484071i	$-0.839158\pi$
$97$	−17.2361	−1.75006	−0.875029	+	0.484071i	$0.839158\pi$
$98$	−8.85410	−0.894399
$99$	0	0
$100$	0	0
$101$	14.4721	1.44003	0.720016	−	0.693958i	$-0.244135\pi$
$101$	14.4721	1.44003	0.720016	+	0.693958i	$0.244135\pi$
$102$	5.61803	0.556268
$103$	−16.9443	−1.66957	−0.834784	−	0.550577i	$-0.814408\pi$
$103$	−16.9443	−1.66957	−0.834784	+	0.550577i	$0.814408\pi$
$104$	−1.70820	−0.167503
$105$	0	0
$106$	12.8541	1.24850
$107$	9.76393	0.943915	0.471957	−	0.881621i	$-0.343548\pi$
$107$	9.76393	0.943915	0.471957	+	0.881621i	$0.343548\pi$
$108$	0.618034	0.0594703
$109$	2.94427	0.282010	0.141005	−	0.990009i	$-0.454967\pi$
$109$	2.94427	0.282010	0.141005	+	0.990009i	$0.454967\pi$
$110$	0	0
$111$	−5.23607	−0.496986
$112$	6.00000	0.566947
$113$	−14.4164	−1.35618	−0.678091	−	0.734978i	$-0.737192\pi$
$113$	−14.4164	−1.35618	−0.678091	+	0.734978i	$0.737192\pi$
$114$	4.00000	0.374634
$115$	0	0
$116$	−2.00000	−0.185695
$117$	0.763932	0.0706255
$118$	−2.76393	−0.254441
$119$	−4.29180	−0.393428
$120$	0	0
$121$	0	0
$122$	−12.0902	−1.09459
$123$	−12.4721	−1.12457
$124$	4.14590	0.372313
$125$	0	0
$126$	−2.00000	−0.178174
$127$	−9.70820	−0.861464	−0.430732	−	0.902480i	$-0.641745\pi$
$127$	−9.70820	−0.861464	−0.430732	+	0.902480i	$0.641745\pi$
$128$	13.6180	1.20368
$129$	−0.763932	−0.0672605
$130$	0	0
$131$	−8.47214	−0.740214	−0.370107	−	0.928989i	$-0.620679\pi$
$131$	−8.47214	−0.740214	−0.370107	+	0.928989i	$0.620679\pi$
$132$	0	0
$133$	−3.05573	−0.264965
$134$	−10.9443	−0.945441
$135$	0	0
$136$	−7.76393	−0.665752
$137$	−8.52786	−0.728585	−0.364292	−	0.931285i	$-0.618689\pi$
$137$	−8.52786	−0.728585	−0.364292	+	0.931285i	$0.618689\pi$
$138$	−14.0902	−1.19943
$139$	20.7082	1.75645	0.878223	−	0.478251i	$-0.158729\pi$
$139$	20.7082	1.75645	0.878223	+	0.478251i	$0.158729\pi$
$140$	0	0
$141$	4.70820	0.396502
$142$	−8.94427	−0.750587
$143$	0	0
$144$	−4.85410	−0.404508
$145$	0	0
$146$	−2.47214	−0.204595
$147$	−5.47214	−0.451334
$148$	−3.23607	−0.266003
$149$	17.7082	1.45071	0.725356	−	0.688374i	$-0.241675\pi$
$149$	17.7082	1.45071	0.725356	+	0.688374i	$0.241675\pi$
$150$	0	0
$151$	−20.7082	−1.68521	−0.842605	−	0.538532i	$-0.818979\pi$
$151$	−20.7082	−1.68521	−0.842605	+	0.538532i	$0.818979\pi$
$152$	−5.52786	−0.448369
$153$	3.47214	0.280706
$154$	0	0
$155$	0	0
$156$	0.472136	0.0378011
$157$	−15.7082	−1.25365	−0.626826	−	0.779160i	$-0.715646\pi$
$157$	−15.7082	−1.25365	−0.626826	+	0.779160i	$0.715646\pi$
$158$	1.14590	0.0911628
$159$	7.94427	0.630022
$160$	0	0
$161$	10.7639	0.848317
$162$	1.61803	0.127125
$163$	16.1803	1.26734	0.633671	−	0.773603i	$-0.281547\pi$
$163$	16.1803	1.26734	0.633671	+	0.773603i	$0.281547\pi$
$164$	−7.70820	−0.601910
$165$	0	0
$166$	−6.47214	−0.502335
$167$	−19.1803	−1.48422	−0.742110	−	0.670279i	$-0.766175\pi$
$167$	−19.1803	−1.48422	−0.742110	+	0.670279i	$0.766175\pi$
$168$	2.76393	0.213242
$169$	−12.4164	−0.955108
$170$	0	0
$171$	2.47214	0.189049
$172$	−0.472136	−0.0360000
$173$	−14.9443	−1.13619	−0.568096	−	0.822962i	$-0.692320\pi$
$173$	−14.9443	−1.13619	−0.568096	+	0.822962i	$0.692320\pi$
$174$	−5.23607	−0.396945
$175$	0	0
$176$	0	0
$177$	−1.70820	−0.128396
$178$	14.9443	1.12012
$179$	2.18034	0.162966	0.0814831	−	0.996675i	$-0.474034\pi$
$179$	2.18034	0.162966	0.0814831	+	0.996675i	$0.474034\pi$
$180$	0	0
$181$	5.41641	0.402598	0.201299	−	0.979530i	$-0.435484\pi$
$181$	5.41641	0.402598	0.201299	+	0.979530i	$0.435484\pi$
$182$	−1.52786	−0.113253
$183$	−7.47214	−0.552356
$184$	19.4721	1.43550
$185$	0	0
$186$	10.8541	0.795861
$187$	0	0
$188$	2.90983	0.212221
$189$	−1.23607	−0.0899107
$190$	0	0
$191$	−21.5967	−1.56269	−0.781343	−	0.624102i	$-0.785465\pi$
$191$	−21.5967	−1.56269	−0.781343	+	0.624102i	$0.785465\pi$
$192$	4.23607	0.305712
$193$	−6.18034	−0.444871	−0.222435	−	0.974947i	$-0.571401\pi$
$193$	−6.18034	−0.444871	−0.222435	+	0.974947i	$0.571401\pi$
$194$	−27.8885	−2.00228
$195$	0	0
$196$	−3.38197	−0.241569
$197$	25.4164	1.81084	0.905422	−	0.424513i	$-0.139555\pi$
$197$	25.4164	1.81084	0.905422	+	0.424513i	$0.139555\pi$
$198$	0	0
$199$	11.7639	0.833923	0.416962	−	0.908924i	$-0.363095\pi$
$199$	11.7639	0.833923	0.416962	+	0.908924i	$0.363095\pi$
$200$	0	0
$201$	−6.76393	−0.477091
$202$	23.4164	1.64757
$203$	4.00000	0.280745
$204$	2.14590	0.150243
$205$	0	0
$206$	−27.4164	−1.91019
$207$	−8.70820	−0.605262
$208$	−3.70820	−0.257118
$209$	0	0
$210$	0	0
$211$	−6.23607	−0.429309	−0.214654	−	0.976690i	$-0.568862\pi$
$211$	−6.23607	−0.429309	−0.214654	+	0.976690i	$0.568862\pi$
$212$	4.90983	0.337209
$213$	−5.52786	−0.378763
$214$	15.7984	1.07995
$215$	0	0
$216$	−2.23607	−0.152145
$217$	−8.29180	−0.562884
$218$	4.76393	0.322654
$219$	−1.52786	−0.103243
$220$	0	0
$221$	2.65248	0.178425
$222$	−8.47214	−0.568613
$223$	−26.9443	−1.80432	−0.902161	−	0.431400i	$-0.858020\pi$
$223$	−26.9443	−1.80432	−0.902161	+	0.431400i	$0.858020\pi$
$224$	4.18034	0.279311
$225$	0	0
$226$	−23.3262	−1.55164
$227$	3.29180	0.218484	0.109242	−	0.994015i	$-0.465158\pi$
$227$	3.29180	0.218484	0.109242	+	0.994015i	$0.465158\pi$
$228$	1.52786	0.101185
$229$	7.00000	0.462573	0.231287	−	0.972886i	$-0.425707\pi$
$229$	7.00000	0.462573	0.231287	+	0.972886i	$0.425707\pi$
$230$	0	0
$231$	0	0
$232$	7.23607	0.475071
$233$	−24.8885	−1.63050	−0.815251	−	0.579107i	$-0.803401\pi$
$233$	−24.8885	−1.63050	−0.815251	+	0.579107i	$0.803401\pi$
$234$	1.23607	0.0808043
$235$	0	0
$236$	−1.05573	−0.0687220
$237$	0.708204	0.0460028
$238$	−6.94427	−0.450130
$239$	−6.65248	−0.430313	−0.215156	−	0.976580i	$-0.569026\pi$
$239$	−6.65248	−0.430313	−0.215156	+	0.976580i	$0.569026\pi$
$240$	0	0
$241$	0.0557281	0.00358976	0.00179488	−	0.999998i	$-0.499429\pi$
$241$	0.0557281	0.00358976	0.00179488	+	0.999998i	$0.499429\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	−4.61803	−0.295639
$245$	0	0
$246$	−20.1803	−1.28665
$247$	1.88854	0.120165
$248$	−15.0000	−0.952501
$249$	−4.00000	−0.253490
$250$	0	0
$251$	−10.2918	−0.649612	−0.324806	−	0.945781i	$-0.605299\pi$
$251$	−10.2918	−0.649612	−0.324806	+	0.945781i	$0.605299\pi$
$252$	−0.763932	−0.0481232
$253$	0	0
$254$	−15.7082	−0.985620
$255$	0	0
$256$	13.5623	0.847644
$257$	6.88854	0.429696	0.214848	−	0.976648i	$-0.431074\pi$
$257$	6.88854	0.429696	0.214848	+	0.976648i	$0.431074\pi$
$258$	−1.23607	−0.0769542
$259$	6.47214	0.402159
$260$	0	0
$261$	−3.23607	−0.200308
$262$	−13.7082	−0.846896
$263$	−20.1246	−1.24094	−0.620468	−	0.784231i	$-0.713057\pi$
$263$	−20.1246	−1.24094	−0.620468	+	0.784231i	$0.713057\pi$
$264$	0	0
$265$	0	0
$266$	−4.94427	−0.303153
$267$	9.23607	0.565238
$268$	−4.18034	−0.255355
$269$	24.9443	1.52088	0.760440	−	0.649409i	$-0.224984\pi$
$269$	24.9443	1.52088	0.760440	+	0.649409i	$0.224984\pi$
$270$	0	0
$271$	19.7639	1.20057	0.600287	−	0.799785i	$-0.295053\pi$
$271$	19.7639	1.20057	0.600287	+	0.799785i	$0.295053\pi$
$272$	−16.8541	−1.02193
$273$	−0.944272	−0.0571499
$274$	−13.7984	−0.833590
$275$	0	0
$276$	−5.38197	−0.323956
$277$	12.9443	0.777746	0.388873	−	0.921291i	$-0.372865\pi$
$277$	12.9443	0.777746	0.388873	+	0.921291i	$0.372865\pi$
$278$	33.5066	2.00959
$279$	6.70820	0.401610
$280$	0	0
$281$	−10.7639	−0.642122	−0.321061	−	0.947058i	$-0.604040\pi$
$281$	−10.7639	−0.642122	−0.321061	+	0.947058i	$0.604040\pi$
$282$	7.61803	0.453647
$283$	−23.8885	−1.42003	−0.710013	−	0.704188i	$-0.751311\pi$
$283$	−23.8885	−1.42003	−0.710013	+	0.704188i	$0.751311\pi$
$284$	−3.41641	−0.202727
$285$	0	0
$286$	0	0
$287$	15.4164	0.910002
$288$	−3.38197	−0.199284
$289$	−4.94427	−0.290840
$290$	0	0
$291$	−17.2361	−1.01040
$292$	−0.944272	−0.0552593
$293$	12.5279	0.731886	0.365943	−	0.930637i	$-0.380747\pi$
$293$	12.5279	0.731886	0.365943	+	0.930637i	$0.380747\pi$
$294$	−8.85410	−0.516382
$295$	0	0
$296$	11.7082	0.680526
$297$	0	0
$298$	28.6525	1.65979
$299$	−6.65248	−0.384723
$300$	0	0
$301$	0.944272	0.0544269
$302$	−33.5066	−1.92809
$303$	14.4721	0.831402
$304$	−12.0000	−0.688247
$305$	0	0
$306$	5.61803	0.321162
$307$	23.8885	1.36339	0.681696	−	0.731636i	$-0.261243\pi$
$307$	23.8885	1.36339	0.681696	+	0.731636i	$0.261243\pi$
$308$	0	0
$309$	−16.9443	−0.963926
$310$	0	0
$311$	1.23607	0.0700910	0.0350455	−	0.999386i	$-0.488842\pi$
$311$	1.23607	0.0700910	0.0350455	+	0.999386i	$0.488842\pi$
$312$	−1.70820	−0.0967080
$313$	−27.2361	−1.53947	−0.769737	−	0.638361i	$-0.779613\pi$
$313$	−27.2361	−1.53947	−0.769737	+	0.638361i	$0.779613\pi$
$314$	−25.4164	−1.43433
$315$	0	0
$316$	0.437694	0.0246222
$317$	−13.9443	−0.783188	−0.391594	−	0.920138i	$-0.628076\pi$
$317$	−13.9443	−0.783188	−0.391594	+	0.920138i	$0.628076\pi$
$318$	12.8541	0.720822
$319$	0	0
$320$	0	0
$321$	9.76393	0.544970
$322$	17.4164	0.970578
$323$	8.58359	0.477604
$324$	0.618034	0.0343352
$325$	0	0
$326$	26.1803	1.44999
$327$	2.94427	0.162819
$328$	27.8885	1.53989
$329$	−5.81966	−0.320848
$330$	0	0
$331$	5.18034	0.284737	0.142369	−	0.989814i	$-0.454528\pi$
$331$	5.18034	0.284737	0.142369	+	0.989814i	$0.454528\pi$
$332$	−2.47214	−0.135676
$333$	−5.23607	−0.286935
$334$	−31.0344	−1.69813
$335$	0	0
$336$	6.00000	0.327327
$337$	27.8885	1.51919	0.759593	−	0.650399i	$-0.225398\pi$
$337$	27.8885	1.51919	0.759593	+	0.650399i	$0.225398\pi$
$338$	−20.0902	−1.09276
$339$	−14.4164	−0.782992
$340$	0	0
$341$	0	0
$342$	4.00000	0.216295
$343$	15.4164	0.832408
$344$	1.70820	0.0921002
$345$	0	0
$346$	−24.1803	−1.29994
$347$	−9.76393	−0.524155	−0.262078	−	0.965047i	$-0.584408\pi$
$347$	−9.76393	−0.524155	−0.262078	+	0.965047i	$0.584408\pi$
$348$	−2.00000	−0.107211
$349$	15.0000	0.802932	0.401466	−	0.915874i	$-0.368501\pi$
$349$	15.0000	0.802932	0.401466	+	0.915874i	$0.368501\pi$
$350$	0	0
$351$	0.763932	0.0407757
$352$	0	0
$353$	7.00000	0.372572	0.186286	−	0.982496i	$-0.440355\pi$
$353$	7.00000	0.372572	0.186286	+	0.982496i	$0.440355\pi$
$354$	−2.76393	−0.146901
$355$	0	0
$356$	5.70820	0.302534
$357$	−4.29180	−0.227146
$358$	3.52786	0.186453
$359$	19.4164	1.02476	0.512379	−	0.858759i	$-0.328764\pi$
$359$	19.4164	1.02476	0.512379	+	0.858759i	$0.328764\pi$
$360$	0	0
$361$	−12.8885	−0.678344
$362$	8.76393	0.460622
$363$	0	0
$364$	−0.583592	−0.0305885
$365$	0	0
$366$	−12.0902	−0.631963
$367$	−9.23607	−0.482119	−0.241059	−	0.970510i	$-0.577495\pi$
$367$	−9.23607	−0.482119	−0.241059	+	0.970510i	$0.577495\pi$
$368$	42.2705	2.20350
$369$	−12.4721	−0.649273
$370$	0	0
$371$	−9.81966	−0.509811
$372$	4.14590	0.214955
$373$	−5.88854	−0.304897	−0.152449	−	0.988311i	$-0.548716\pi$
$373$	−5.88854	−0.304897	−0.152449	+	0.988311i	$0.548716\pi$
$374$	0	0
$375$	0	0
$376$	−10.5279	−0.542933
$377$	−2.47214	−0.127321
$378$	−2.00000	−0.102869
$379$	−18.5967	−0.955251	−0.477625	−	0.878564i	$-0.658502\pi$
$379$	−18.5967	−0.955251	−0.477625	+	0.878564i	$0.658502\pi$
$380$	0	0
$381$	−9.70820	−0.497366
$382$	−34.9443	−1.78790
$383$	−16.0000	−0.817562	−0.408781	−	0.912633i	$-0.634046\pi$
$383$	−16.0000	−0.817562	−0.408781	+	0.912633i	$0.634046\pi$
$384$	13.6180	0.694942
$385$	0	0
$386$	−10.0000	−0.508987
$387$	−0.763932	−0.0388328
$388$	−10.6525	−0.540798
$389$	27.5967	1.39921	0.699605	−	0.714529i	$-0.253359\pi$
$389$	27.5967	1.39921	0.699605	+	0.714529i	$0.253359\pi$
$390$	0	0
$391$	−30.2361	−1.52910
$392$	12.2361	0.618015
$393$	−8.47214	−0.427363
$394$	41.1246	2.07183
$395$	0	0
$396$	0	0
$397$	−9.41641	−0.472596	−0.236298	−	0.971681i	$-0.575934\pi$
$397$	−9.41641	−0.472596	−0.236298	+	0.971681i	$0.575934\pi$
$398$	19.0344	0.954110
$399$	−3.05573	−0.152978
$400$	0	0
$401$	3.05573	0.152596	0.0762979	−	0.997085i	$-0.475690\pi$
$401$	3.05573	0.152596	0.0762979	+	0.997085i	$0.475690\pi$
$402$	−10.9443	−0.545851
$403$	5.12461	0.255275
$404$	8.94427	0.444994
$405$	0	0
$406$	6.47214	0.321207
$407$	0	0
$408$	−7.76393	−0.384372
$409$	34.7771	1.71962	0.859808	−	0.510617i	$-0.170583\pi$
$409$	34.7771	1.71962	0.859808	+	0.510617i	$0.170583\pi$
$410$	0	0
$411$	−8.52786	−0.420649
$412$	−10.4721	−0.515925
$413$	2.11146	0.103898
$414$	−14.0902	−0.692494
$415$	0	0
$416$	−2.58359	−0.126671
$417$	20.7082	1.01409
$418$	0	0
$419$	−14.0000	−0.683945	−0.341972	−	0.939710i	$-0.611095\pi$
$419$	−14.0000	−0.683945	−0.341972	+	0.939710i	$0.611095\pi$
$420$	0	0
$421$	−23.8328	−1.16154	−0.580770	−	0.814068i	$-0.697249\pi$
$421$	−23.8328	−1.16154	−0.580770	+	0.814068i	$0.697249\pi$
$422$	−10.0902	−0.491182
$423$	4.70820	0.228921
$424$	−17.7639	−0.862693
$425$	0	0
$426$	−8.94427	−0.433351
$427$	9.23607	0.446965
$428$	6.03444	0.291686
$429$	0	0
$430$	0	0
$431$	28.1803	1.35740	0.678700	−	0.734416i	$-0.262544\pi$
$431$	28.1803	1.35740	0.678700	+	0.734416i	$0.262544\pi$
$432$	−4.85410	−0.233543
$433$	−20.7639	−0.997851	−0.498925	−	0.866645i	$-0.666272\pi$
$433$	−20.7639	−0.997851	−0.498925	+	0.866645i	$0.666272\pi$
$434$	−13.4164	−0.644008
$435$	0	0
$436$	1.81966	0.0871459
$437$	−21.5279	−1.02982
$438$	−2.47214	−0.118123
$439$	−32.1246	−1.53322	−0.766612	−	0.642111i	$-0.778059\pi$
$439$	−32.1246	−1.53322	−0.766612	+	0.642111i	$0.778059\pi$
$440$	0	0
$441$	−5.47214	−0.260578
$442$	4.29180	0.204140
$443$	19.4164	0.922501	0.461251	−	0.887270i	$-0.347401\pi$
$443$	19.4164	0.922501	0.461251	+	0.887270i	$0.347401\pi$
$444$	−3.23607	−0.153577
$445$	0	0
$446$	−43.5967	−2.06437
$447$	17.7082	0.837569
$448$	−5.23607	−0.247381
$449$	−10.0000	−0.471929	−0.235965	−	0.971762i	$-0.575825\pi$
$449$	−10.0000	−0.471929	−0.235965	+	0.971762i	$0.575825\pi$
$450$	0	0
$451$	0	0
$452$	−8.90983	−0.419083
$453$	−20.7082	−0.972956
$454$	5.32624	0.249973
$455$	0	0
$456$	−5.52786	−0.258866
$457$	10.3607	0.484652	0.242326	−	0.970195i	$-0.422090\pi$
$457$	10.3607	0.484652	0.242326	+	0.970195i	$0.422090\pi$
$458$	11.3262	0.529240
$459$	3.47214	0.162065
$460$	0	0
$461$	−7.81966	−0.364198	−0.182099	−	0.983280i	$-0.558289\pi$
$461$	−7.81966	−0.364198	−0.182099	+	0.983280i	$0.558289\pi$
$462$	0	0
$463$	−6.00000	−0.278844	−0.139422	−	0.990233i	$-0.544524\pi$
$463$	−6.00000	−0.278844	−0.139422	+	0.990233i	$0.544524\pi$
$464$	15.7082	0.729235
$465$	0	0
$466$	−40.2705	−1.86550
$467$	8.23607	0.381120	0.190560	−	0.981676i	$-0.438970\pi$
$467$	8.23607	0.381120	0.190560	+	0.981676i	$0.438970\pi$
$468$	0.472136	0.0218245
$469$	8.36068	0.386060
$470$	0	0
$471$	−15.7082	−0.723796
$472$	3.81966	0.175814
$473$	0	0
$474$	1.14590	0.0526328
$475$	0	0
$476$	−2.65248	−0.121576
$477$	7.94427	0.363743
$478$	−10.7639	−0.492331
$479$	13.7082	0.626344	0.313172	−	0.949696i	$-0.398608\pi$
$479$	13.7082	0.626344	0.313172	+	0.949696i	$0.398608\pi$
$480$	0	0
$481$	−4.00000	−0.182384
$482$	0.0901699	0.00410713
$483$	10.7639	0.489776
$484$	0	0
$485$	0	0
$486$	1.61803	0.0733955
$487$	−1.70820	−0.0774061	−0.0387031	−	0.999251i	$-0.512323\pi$
$487$	−1.70820	−0.0774061	−0.0387031	+	0.999251i	$0.512323\pi$
$488$	16.7082	0.756345
$489$	16.1803	0.731700
$490$	0	0
$491$	29.5279	1.33257	0.666287	−	0.745695i	$-0.267883\pi$
$491$	29.5279	1.33257	0.666287	+	0.745695i	$0.267883\pi$
$492$	−7.70820	−0.347513
$493$	−11.2361	−0.506047
$494$	3.05573	0.137484
$495$	0	0
$496$	−32.5623	−1.46209
$497$	6.83282	0.306494
$498$	−6.47214	−0.290023
$499$	16.9443	0.758530	0.379265	−	0.925288i	$-0.376177\pi$
$499$	16.9443	0.758530	0.379265	+	0.925288i	$0.376177\pi$
$500$	0	0
$501$	−19.1803	−0.856914
$502$	−16.6525	−0.743236
$503$	−0.819660	−0.0365468	−0.0182734	−	0.999833i	$-0.505817\pi$
$503$	−0.819660	−0.0365468	−0.0182734	+	0.999833i	$0.505817\pi$
$504$	2.76393	0.123115
$505$	0	0
$506$	0	0
$507$	−12.4164	−0.551432
$508$	−6.00000	−0.266207
$509$	−33.5967	−1.48915	−0.744575	−	0.667539i	$-0.767348\pi$
$509$	−33.5967	−1.48915	−0.744575	+	0.667539i	$0.767348\pi$
$510$	0	0
$511$	1.88854	0.0835443
$512$	−5.29180	−0.233867
$513$	2.47214	0.109147
$514$	11.1459	0.491624
$515$	0	0
$516$	−0.472136	−0.0207846
$517$	0	0
$518$	10.4721	0.460119
$519$	−14.9443	−0.655981
$520$	0	0
$521$	40.1803	1.76033	0.880166	−	0.474665i	$-0.157431\pi$
$521$	40.1803	1.76033	0.880166	+	0.474665i	$0.157431\pi$
$522$	−5.23607	−0.229176
$523$	42.5410	1.86019	0.930094	−	0.367320i	$-0.119725\pi$
$523$	42.5410	1.86019	0.930094	+	0.367320i	$0.119725\pi$
$524$	−5.23607	−0.228739
$525$	0	0
$526$	−32.5623	−1.41978
$527$	23.2918	1.01461
$528$	0	0
$529$	52.8328	2.29708
$530$	0	0
$531$	−1.70820	−0.0741297
$532$	−1.88854	−0.0818788
$533$	−9.52786	−0.412698
$534$	14.9443	0.646702
$535$	0	0
$536$	15.1246	0.653284
$537$	2.18034	0.0940886
$538$	40.3607	1.74007
$539$	0	0
$540$	0	0
$541$	−19.3050	−0.829985	−0.414992	−	0.909825i	$-0.636216\pi$
$541$	−19.3050	−0.829985	−0.414992	+	0.909825i	$0.636216\pi$
$542$	31.9787	1.37360
$543$	5.41641	0.232440
$544$	−11.7426	−0.503462
$545$	0	0
$546$	−1.52786	−0.0653865
$547$	39.5967	1.69303	0.846517	−	0.532361i	$-0.178695\pi$
$547$	39.5967	1.69303	0.846517	+	0.532361i	$0.178695\pi$
$548$	−5.27051	−0.225145
$549$	−7.47214	−0.318903
$550$	0	0
$551$	−8.00000	−0.340811
$552$	19.4721	0.828789
$553$	−0.875388	−0.0372253
$554$	20.9443	0.889837
$555$	0	0
$556$	12.7984	0.542772
$557$	−17.4721	−0.740318	−0.370159	−	0.928968i	$-0.620697\pi$
$557$	−17.4721	−0.740318	−0.370159	+	0.928968i	$0.620697\pi$
$558$	10.8541	0.459491
$559$	−0.583592	−0.0246833
$560$	0	0
$561$	0	0
$562$	−17.4164	−0.734667
$563$	16.9443	0.714116	0.357058	−	0.934082i	$-0.383780\pi$
$563$	16.9443	0.714116	0.357058	+	0.934082i	$0.383780\pi$
$564$	2.90983	0.122526
$565$	0	0
$566$	−38.6525	−1.62468
$567$	−1.23607	−0.0519100
$568$	12.3607	0.518643
$569$	6.29180	0.263766	0.131883	−	0.991265i	$-0.457898\pi$
$569$	6.29180	0.263766	0.131883	+	0.991265i	$0.457898\pi$
$570$	0	0
$571$	40.1246	1.67916	0.839581	−	0.543234i	$-0.182800\pi$
$571$	40.1246	1.67916	0.839581	+	0.543234i	$0.182800\pi$
$572$	0	0
$573$	−21.5967	−0.902217
$574$	24.9443	1.04115
$575$	0	0
$576$	4.23607	0.176503
$577$	−20.0000	−0.832611	−0.416305	−	0.909225i	$-0.636675\pi$
$577$	−20.0000	−0.832611	−0.416305	+	0.909225i	$0.636675\pi$
$578$	−8.00000	−0.332756
$579$	−6.18034	−0.256846
$580$	0	0
$581$	4.94427	0.205123
$582$	−27.8885	−1.15602
$583$	0	0
$584$	3.41641	0.141372
$585$	0	0
$586$	20.2705	0.837367
$587$	−19.6525	−0.811144	−0.405572	−	0.914063i	$-0.632928\pi$
$587$	−19.6525	−0.811144	−0.405572	+	0.914063i	$0.632928\pi$
$588$	−3.38197	−0.139470
$589$	16.5836	0.683315
$590$	0	0
$591$	25.4164	1.04549
$592$	25.4164	1.04461
$593$	−17.0557	−0.700395	−0.350197	−	0.936676i	$-0.613885\pi$
$593$	−17.0557	−0.700395	−0.350197	+	0.936676i	$0.613885\pi$
$594$	0	0
$595$	0	0
$596$	10.9443	0.448295
$597$	11.7639	0.481466
$598$	−10.7639	−0.440170
$599$	45.0132	1.83919	0.919594	−	0.392870i	$-0.128518\pi$
$599$	45.0132	1.83919	0.919594	+	0.392870i	$0.128518\pi$
$600$	0	0
$601$	−19.5279	−0.796558	−0.398279	−	0.917264i	$-0.630392\pi$
$601$	−19.5279	−0.796558	−0.398279	+	0.917264i	$0.630392\pi$
$602$	1.52786	0.0622711
$603$	−6.76393	−0.275449
$604$	−12.7984	−0.520758
$605$	0	0
$606$	23.4164	0.951227
$607$	19.7082	0.799931	0.399966	−	0.916530i	$-0.369022\pi$
$607$	19.7082	0.799931	0.399966	+	0.916530i	$0.369022\pi$
$608$	−8.36068	−0.339070
$609$	4.00000	0.162088
$610$	0	0
$611$	3.59675	0.145509
$612$	2.14590	0.0867428
$613$	38.8328	1.56844	0.784221	−	0.620481i	$-0.213063\pi$
$613$	38.8328	1.56844	0.784221	+	0.620481i	$0.213063\pi$
$614$	38.6525	1.55989
$615$	0	0
$616$	0	0
$617$	29.7771	1.19878	0.599390	−	0.800457i	$-0.295410\pi$
$617$	29.7771	1.19878	0.599390	+	0.800457i	$0.295410\pi$
$618$	−27.4164	−1.10285
$619$	−18.4721	−0.742458	−0.371229	−	0.928541i	$-0.621063\pi$
$619$	−18.4721	−0.742458	−0.371229	+	0.928541i	$0.621063\pi$
$620$	0	0
$621$	−8.70820	−0.349448
$622$	2.00000	0.0801927
$623$	−11.4164	−0.457389
$624$	−3.70820	−0.148447
$625$	0	0
$626$	−44.0689	−1.76135
$627$	0	0
$628$	−9.70820	−0.387400
$629$	−18.1803	−0.724898
$630$	0	0
$631$	−13.1803	−0.524701	−0.262351	−	0.964973i	$-0.584498\pi$
$631$	−13.1803	−0.524701	−0.262351	+	0.964973i	$0.584498\pi$
$632$	−1.58359	−0.0629919
$633$	−6.23607	−0.247861
$634$	−22.5623	−0.896064
$635$	0	0
$636$	4.90983	0.194687
$637$	−4.18034	−0.165631
$638$	0	0
$639$	−5.52786	−0.218679
$640$	0	0
$641$	31.1246	1.22935	0.614674	−	0.788781i	$-0.289288\pi$
$641$	31.1246	1.22935	0.614674	+	0.788781i	$0.289288\pi$
$642$	15.7984	0.623512
$643$	−24.8328	−0.979311	−0.489655	−	0.871916i	$-0.662877\pi$
$643$	−24.8328	−0.979311	−0.489655	+	0.871916i	$0.662877\pi$
$644$	6.65248	0.262144
$645$	0	0
$646$	13.8885	0.546437
$647$	11.7639	0.462488	0.231244	−	0.972896i	$-0.425720\pi$
$647$	11.7639	0.462488	0.231244	+	0.972896i	$0.425720\pi$
$648$	−2.23607	−0.0878410
$649$	0	0
$650$	0	0
$651$	−8.29180	−0.324981
$652$	10.0000	0.391630
$653$	36.8328	1.44138	0.720690	−	0.693258i	$-0.243825\pi$
$653$	36.8328	1.44138	0.720690	+	0.693258i	$0.243825\pi$
$654$	4.76393	0.186284
$655$	0	0
$656$	60.5410	2.36373
$657$	−1.52786	−0.0596077
$658$	−9.41641	−0.367090
$659$	13.5967	0.529654	0.264827	−	0.964296i	$-0.414685\pi$
$659$	13.5967	0.529654	0.264827	+	0.964296i	$0.414685\pi$
$660$	0	0
$661$	−25.7771	−1.00261	−0.501306	−	0.865270i	$-0.667147\pi$
$661$	−25.7771	−1.00261	−0.501306	+	0.865270i	$0.667147\pi$
$662$	8.38197	0.325774
$663$	2.65248	0.103014
$664$	8.94427	0.347105
$665$	0	0
$666$	−8.47214	−0.328289
$667$	28.1803	1.09115
$668$	−11.8541	−0.458649
$669$	−26.9443	−1.04173
$670$	0	0
$671$	0	0
$672$	4.18034	0.161260
$673$	21.5967	0.832493	0.416247	−	0.909252i	$-0.363345\pi$
$673$	21.5967	0.832493	0.416247	+	0.909252i	$0.363345\pi$
$674$	45.1246	1.73814
$675$	0	0
$676$	−7.67376	−0.295145
$677$	−21.4164	−0.823099	−0.411550	−	0.911387i	$-0.635012\pi$
$677$	−21.4164	−0.823099	−0.411550	+	0.911387i	$0.635012\pi$
$678$	−23.3262	−0.895839
$679$	21.3050	0.817609
$680$	0	0
$681$	3.29180	0.126142
$682$	0	0
$683$	24.0000	0.918334	0.459167	−	0.888350i	$-0.348148\pi$
$683$	24.0000	0.918334	0.459167	+	0.888350i	$0.348148\pi$
$684$	1.52786	0.0584193
$685$	0	0
$686$	24.9443	0.952377
$687$	7.00000	0.267067
$688$	3.70820	0.141374
$689$	6.06888	0.231206
$690$	0	0
$691$	29.5410	1.12379	0.561897	−	0.827207i	$-0.310072\pi$
$691$	29.5410	1.12379	0.561897	+	0.827207i	$0.310072\pi$
$692$	−9.23607	−0.351103
$693$	0	0
$694$	−15.7984	−0.599698
$695$	0	0
$696$	7.23607	0.274282
$697$	−43.3050	−1.64029
$698$	24.2705	0.918652
$699$	−24.8885	−0.941371
$700$	0	0
$701$	−13.8197	−0.521961	−0.260981	−	0.965344i	$-0.584046\pi$
$701$	−13.8197	−0.521961	−0.260981	+	0.965344i	$0.584046\pi$
$702$	1.23607	0.0466524
$703$	−12.9443	−0.488202
$704$	0	0
$705$	0	0
$706$	11.3262	0.426269
$707$	−17.8885	−0.672768
$708$	−1.05573	−0.0396767
$709$	−0.0557281	−0.00209291	−0.00104646	−	0.999999i	$-0.500333\pi$
$709$	−0.0557281	−0.00209291	−0.00104646	+	0.999999i	$0.500333\pi$
$710$	0	0
$711$	0.708204	0.0265597
$712$	−20.6525	−0.773984
$713$	−58.4164	−2.18771
$714$	−6.94427	−0.259883
$715$	0	0
$716$	1.34752	0.0503593
$717$	−6.65248	−0.248441
$718$	31.4164	1.17245
$719$	−18.3607	−0.684738	−0.342369	−	0.939566i	$-0.611229\pi$
$719$	−18.3607	−0.684738	−0.342369	+	0.939566i	$0.611229\pi$
$720$	0	0
$721$	20.9443	0.780005
$722$	−20.8541	−0.776109
$723$	0.0557281	0.00207255
$724$	3.34752	0.124410
$725$	0	0
$726$	0	0
$727$	8.83282	0.327591	0.163796	−	0.986494i	$-0.447626\pi$
$727$	8.83282	0.327591	0.163796	+	0.986494i	$0.447626\pi$
$728$	2.11146	0.0782558
$729$	1.00000	0.0370370
$730$	0	0
$731$	−2.65248	−0.0981054
$732$	−4.61803	−0.170687
$733$	12.1115	0.447347	0.223673	−	0.974664i	$-0.428195\pi$
$733$	12.1115	0.447347	0.223673	+	0.974664i	$0.428195\pi$
$734$	−14.9443	−0.551603
$735$	0	0
$736$	29.4508	1.08557
$737$	0	0
$738$	−20.1803	−0.742849
$739$	29.6525	1.09078	0.545392	−	0.838181i	$-0.316381\pi$
$739$	29.6525	1.09078	0.545392	+	0.838181i	$0.316381\pi$
$740$	0	0
$741$	1.88854	0.0693774
$742$	−15.8885	−0.583287
$743$	−2.23607	−0.0820334	−0.0410167	−	0.999158i	$-0.513060\pi$
$743$	−2.23607	−0.0820334	−0.0410167	+	0.999158i	$0.513060\pi$
$744$	−15.0000	−0.549927
$745$	0	0
$746$	−9.52786	−0.348840
$747$	−4.00000	−0.146352
$748$	0	0
$749$	−12.0689	−0.440987
$750$	0	0
$751$	42.0132	1.53308	0.766541	−	0.642195i	$-0.221976\pi$
$751$	42.0132	1.53308	0.766541	+	0.642195i	$0.221976\pi$
$752$	−22.8541	−0.833403
$753$	−10.2918	−0.375054
$754$	−4.00000	−0.145671
$755$	0	0
$756$	−0.763932	−0.0277839
$757$	−1.05573	−0.0383711	−0.0191855	−	0.999816i	$-0.506107\pi$
$757$	−1.05573	−0.0383711	−0.0191855	+	0.999816i	$0.506107\pi$
$758$	−30.0902	−1.09292
$759$	0	0
$760$	0	0
$761$	13.3475	0.483847	0.241924	−	0.970295i	$-0.422222\pi$
$761$	13.3475	0.483847	0.241924	+	0.970295i	$0.422222\pi$
$762$	−15.7082	−0.569048
$763$	−3.63932	−0.131752
$764$	−13.3475	−0.482896
$765$	0	0
$766$	−25.8885	−0.935391
$767$	−1.30495	−0.0471191
$768$	13.5623	0.489388
$769$	33.4721	1.20704	0.603518	−	0.797349i	$-0.293765\pi$
$769$	33.4721	1.20704	0.603518	+	0.797349i	$0.293765\pi$
$770$	0	0
$771$	6.88854	0.248085
$772$	−3.81966	−0.137473
$773$	−9.94427	−0.357671	−0.178835	−	0.983879i	$-0.557233\pi$
$773$	−9.94427	−0.357671	−0.178835	+	0.983879i	$0.557233\pi$
$774$	−1.23607	−0.0444295
$775$	0	0
$776$	38.5410	1.38354
$777$	6.47214	0.232187
$778$	44.6525	1.60087
$779$	−30.8328	−1.10470
$780$	0	0
$781$	0	0
$782$	−48.9230	−1.74948
$783$	−3.23607	−0.115648
$784$	26.5623	0.948654
$785$	0	0
$786$	−13.7082	−0.488955
$787$	−44.9443	−1.60209	−0.801045	−	0.598604i	$-0.795722\pi$
$787$	−44.9443	−1.60209	−0.801045	+	0.598604i	$0.795722\pi$
$788$	15.7082	0.559582
$789$	−20.1246	−0.716455
$790$	0	0
$791$	17.8197	0.633594
$792$	0	0
$793$	−5.70820	−0.202704
$794$	−15.2361	−0.540708
$795$	0	0
$796$	7.27051	0.257696
$797$	−30.0000	−1.06265	−0.531327	−	0.847167i	$-0.678307\pi$
$797$	−30.0000	−1.06265	−0.531327	+	0.847167i	$0.678307\pi$
$798$	−4.94427	−0.175025
$799$	16.3475	0.578334
$800$	0	0
$801$	9.23607	0.326340
$802$	4.94427	0.174588
$803$	0	0
$804$	−4.18034	−0.147429
$805$	0	0
$806$	8.29180	0.292066
$807$	24.9443	0.878080
$808$	−32.3607	−1.13844
$809$	8.00000	0.281265	0.140633	−	0.990062i	$-0.455086\pi$
$809$	8.00000	0.281265	0.140633	+	0.990062i	$0.455086\pi$
$810$	0	0
$811$	−48.1246	−1.68988	−0.844942	−	0.534858i	$-0.820365\pi$
$811$	−48.1246	−1.68988	−0.844942	+	0.534858i	$0.820365\pi$
$812$	2.47214	0.0867550
$813$	19.7639	0.693151
$814$	0	0
$815$	0	0
$816$	−16.8541	−0.590012
$817$	−1.88854	−0.0660718
$818$	56.2705	1.96745
$819$	−0.944272	−0.0329955
$820$	0	0
$821$	−37.7771	−1.31843	−0.659215	−	0.751955i	$-0.729111\pi$
$821$	−37.7771	−1.31843	−0.659215	+	0.751955i	$0.729111\pi$
$822$	−13.7984	−0.481274
$823$	42.2492	1.47272	0.736358	−	0.676592i	$-0.236544\pi$
$823$	42.2492	1.47272	0.736358	+	0.676592i	$0.236544\pi$
$824$	37.8885	1.31991
$825$	0	0
$826$	3.41641	0.118872
$827$	−8.94427	−0.311023	−0.155511	−	0.987834i	$-0.549703\pi$
$827$	−8.94427	−0.311023	−0.155511	+	0.987834i	$0.549703\pi$
$828$	−5.38197	−0.187036
$829$	−11.0000	−0.382046	−0.191023	−	0.981586i	$-0.561180\pi$
$829$	−11.0000	−0.382046	−0.191023	+	0.981586i	$0.561180\pi$
$830$	0	0
$831$	12.9443	0.449032
$832$	3.23607	0.112190
$833$	−19.0000	−0.658311
$834$	33.5066	1.16024
$835$	0	0
$836$	0	0
$837$	6.70820	0.231869
$838$	−22.6525	−0.782517
$839$	−16.3607	−0.564833	−0.282417	−	0.959292i	$-0.591136\pi$
$839$	−16.3607	−0.564833	−0.282417	+	0.959292i	$0.591136\pi$
$840$	0	0
$841$	−18.5279	−0.638892
$842$	−38.5623	−1.32894
$843$	−10.7639	−0.370730
$844$	−3.85410	−0.132664
$845$	0	0
$846$	7.61803	0.261913
$847$	0	0
$848$	−38.5623	−1.32424
$849$	−23.8885	−0.819853
$850$	0	0
$851$	45.5967	1.56304
$852$	−3.41641	−0.117044
$853$	3.88854	0.133141	0.0665706	−	0.997782i	$-0.478794\pi$
$853$	3.88854	0.133141	0.0665706	+	0.997782i	$0.478794\pi$
$854$	14.9443	0.511382
$855$	0	0
$856$	−21.8328	−0.746230
$857$	1.00000	0.0341593	0.0170797	−	0.999854i	$-0.494563\pi$
$857$	1.00000	0.0341593	0.0170797	+	0.999854i	$0.494563\pi$
$858$	0	0
$859$	26.8328	0.915524	0.457762	−	0.889075i	$-0.348651\pi$
$859$	26.8328	0.915524	0.457762	+	0.889075i	$0.348651\pi$
$860$	0	0
$861$	15.4164	0.525390
$862$	45.5967	1.55303
$863$	2.11146	0.0718748	0.0359374	−	0.999354i	$-0.488558\pi$
$863$	2.11146	0.0718748	0.0359374	+	0.999354i	$0.488558\pi$
$864$	−3.38197	−0.115057
$865$	0	0
$866$	−33.5967	−1.14166
$867$	−4.94427	−0.167916
$868$	−5.12461	−0.173941
$869$	0	0
$870$	0	0
$871$	−5.16718	−0.175083
$872$	−6.58359	−0.222949
$873$	−17.2361	−0.583353
$874$	−34.8328	−1.17824
$875$	0	0
$876$	−0.944272	−0.0319040
$877$	−51.3050	−1.73245	−0.866223	−	0.499658i	$-0.833459\pi$
$877$	−51.3050	−1.73245	−0.866223	+	0.499658i	$0.833459\pi$
$878$	−51.9787	−1.75420
$879$	12.5279	0.422554
$880$	0	0
$881$	−14.8328	−0.499730	−0.249865	−	0.968281i	$-0.580386\pi$
$881$	−14.8328	−0.499730	−0.249865	+	0.968281i	$0.580386\pi$
$882$	−8.85410	−0.298133
$883$	12.1803	0.409901	0.204951	−	0.978772i	$-0.434297\pi$
$883$	12.1803	0.409901	0.204951	+	0.978772i	$0.434297\pi$
$884$	1.63932	0.0551363
$885$	0	0
$886$	31.4164	1.05545
$887$	−0.944272	−0.0317055	−0.0158528	−	0.999874i	$-0.505046\pi$
$887$	−0.944272	−0.0317055	−0.0158528	+	0.999874i	$0.505046\pi$
$888$	11.7082	0.392902
$889$	12.0000	0.402467
$890$	0	0
$891$	0	0
$892$	−16.6525	−0.557566
$893$	11.6393	0.389495
$894$	28.6525	0.958282
$895$	0	0
$896$	−16.8328	−0.562345
$897$	−6.65248	−0.222120
$898$	−16.1803	−0.539945
$899$	−21.7082	−0.724009
$900$	0	0
$901$	27.5836	0.918943
$902$	0	0
$903$	0.944272	0.0314234
$904$	32.2361	1.07216
$905$	0	0
$906$	−33.5066	−1.11318
$907$	42.0000	1.39459	0.697294	−	0.716786i	$-0.254387\pi$
$907$	42.0000	1.39459	0.697294	+	0.716786i	$0.254387\pi$
$908$	2.03444	0.0675153
$909$	14.4721	0.480010
$910$	0	0
$911$	−45.9574	−1.52264	−0.761319	−	0.648378i	$-0.775448\pi$
$911$	−45.9574	−1.52264	−0.761319	+	0.648378i	$0.775448\pi$
$912$	−12.0000	−0.397360
$913$	0	0
$914$	16.7639	0.554502
$915$	0	0
$916$	4.32624	0.142943
$917$	10.4721	0.345820
$918$	5.61803	0.185423
$919$	23.4164	0.772436	0.386218	−	0.922408i	$-0.373781\pi$
$919$	23.4164	0.772436	0.386218	+	0.922408i	$0.373781\pi$
$920$	0	0
$921$	23.8885	0.787154
$922$	−12.6525	−0.416687
$923$	−4.22291	−0.138999
$924$	0	0
$925$	0	0
$926$	−9.70820	−0.319031
$927$	−16.9443	−0.556523
$928$	10.9443	0.359263
$929$	33.0557	1.08452	0.542262	−	0.840210i	$-0.317568\pi$
$929$	33.0557	1.08452	0.542262	+	0.840210i	$0.317568\pi$
$930$	0	0
$931$	−13.5279	−0.443358
$932$	−15.3820	−0.503853
$933$	1.23607	0.0404670
$934$	13.3262	0.436048
$935$	0	0
$936$	−1.70820	−0.0558344
$937$	−7.12461	−0.232751	−0.116375	−	0.993205i	$-0.537128\pi$
$937$	−7.12461	−0.232751	−0.116375	+	0.993205i	$0.537128\pi$
$938$	13.5279	0.441700
$939$	−27.2361	−0.888815
$940$	0	0
$941$	40.7639	1.32887	0.664433	−	0.747348i	$-0.268673\pi$
$941$	40.7639	1.32887	0.664433	+	0.747348i	$0.268673\pi$
$942$	−25.4164	−0.828111
$943$	108.610	3.53683
$944$	8.29180	0.269875
$945$	0	0
$946$	0	0
$947$	13.7639	0.447268	0.223634	−	0.974673i	$-0.428208\pi$
$947$	13.7639	0.447268	0.223634	+	0.974673i	$0.428208\pi$
$948$	0.437694	0.0142156
$949$	−1.16718	−0.0378884
$950$	0	0
$951$	−13.9443	−0.452174
$952$	9.59675	0.311032
$953$	−3.88854	−0.125962	−0.0629811	−	0.998015i	$-0.520061\pi$
$953$	−3.88854	−0.125962	−0.0629811	+	0.998015i	$0.520061\pi$
$954$	12.8541	0.416167
$955$	0	0
$956$	−4.11146	−0.132974
$957$	0	0
$958$	22.1803	0.716614
$959$	10.5410	0.340387
$960$	0	0
$961$	14.0000	0.451613
$962$	−6.47214	−0.208670
$963$	9.76393	0.314638
$964$	0.0344419	0.00110930
$965$	0	0
$966$	17.4164	0.560364
$967$	−32.0000	−1.02905	−0.514525	−	0.857475i	$-0.672032\pi$
$967$	−32.0000	−1.02905	−0.514525	+	0.857475i	$0.672032\pi$
$968$	0	0
$969$	8.58359	0.275745
$970$	0	0
$971$	24.7639	0.794712	0.397356	−	0.917664i	$-0.369928\pi$
$971$	24.7639	0.794712	0.397356	+	0.917664i	$0.369928\pi$
$972$	0.618034	0.0198234
$973$	−25.5967	−0.820594
$974$	−2.76393	−0.0885621
$975$	0	0
$976$	36.2705	1.16099
$977$	−1.94427	−0.0622028	−0.0311014	−	0.999516i	$-0.509901\pi$
$977$	−1.94427	−0.0622028	−0.0311014	+	0.999516i	$0.509901\pi$
$978$	26.1803	0.837155
$979$	0	0
$980$	0	0
$981$	2.94427	0.0940034
$982$	47.7771	1.52463
$983$	32.8197	1.04678	0.523392	−	0.852092i	$-0.324666\pi$
$983$	32.8197	1.04678	0.523392	+	0.852092i	$0.324666\pi$
$984$	27.8885	0.889054
$985$	0	0
$986$	−18.1803	−0.578980
$987$	−5.81966	−0.185242
$988$	1.16718	0.0371331
$989$	6.65248	0.211536
$990$	0	0
$991$	1.76393	0.0560331	0.0280166	−	0.999607i	$-0.491081\pi$
$991$	1.76393	0.0560331	0.0280166	+	0.999607i	$0.491081\pi$
$992$	−22.6869	−0.720310
$993$	5.18034	0.164393
$994$	11.0557	0.350666
$995$	0	0
$996$	−2.47214	−0.0783326
$997$	43.2361	1.36930	0.684650	−	0.728872i	$-0.259955\pi$
$997$	43.2361	1.36930	0.684650	+	0.728872i	$0.259955\pi$
$998$	27.4164	0.867851
$999$	−5.23607	−0.165662

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9075.2.a.bx.1.2		2
5.4	even	2		1815.2.a.f.1.1	✓	2
11.10	odd	2		9075.2.a.bd.1.1		2
15.14	odd	2		5445.2.a.x.1.2		2
55.54	odd	2		1815.2.a.j.1.2	yes	2
165.164	even	2		5445.2.a.o.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1815.2.a.f.1.1	✓	2	5.4	even	2
1815.2.a.j.1.2	yes	2	55.54	odd	2
5445.2.a.o.1.1		2	165.164	even	2
5445.2.a.x.1.2		2	15.14	odd	2
9075.2.a.bd.1.1		2	11.10	odd	2
9075.2.a.bx.1.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 9075 = 3 \cdot 5^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9075.a (trivial)

\(f(q)\)	\(=\)	\(q+1.61803 q^{2} +1.00000 q^{3} +0.618034 q^{4} +1.61803 q^{6} -1.23607 q^{7} -2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.61803 q^{2} +1.00000 q^{3} +0.618034 q^{4} +1.61803 q^{6} -1.23607 q^{7} -2.23607 q^{8} +1.00000 q^{9} +0.618034 q^{12} +0.763932 q^{13} -2.00000 q^{14} -4.85410 q^{16} +3.47214 q^{17} +1.61803 q^{18} +2.47214 q^{19} -1.23607 q^{21} -8.70820 q^{23} -2.23607 q^{24} +1.23607 q^{26} +1.00000 q^{27} -0.763932 q^{28} -3.23607 q^{29} +6.70820 q^{31} -3.38197 q^{32} +5.61803 q^{34} +0.618034 q^{36} -5.23607 q^{37} +4.00000 q^{38} +0.763932 q^{39} -12.4721 q^{41} -2.00000 q^{42} -0.763932 q^{43} -14.0902 q^{46} +4.70820 q^{47} -4.85410 q^{48} -5.47214 q^{49} +3.47214 q^{51} +0.472136 q^{52} +7.94427 q^{53} +1.61803 q^{54} +2.76393 q^{56} +2.47214 q^{57} -5.23607 q^{58} -1.70820 q^{59} -7.47214 q^{61} +10.8541 q^{62} -1.23607 q^{63} +4.23607 q^{64} -6.76393 q^{67} +2.14590 q^{68} -8.70820 q^{69} -5.52786 q^{71} -2.23607 q^{72} -1.52786 q^{73} -8.47214 q^{74} +1.52786 q^{76} +1.23607 q^{78} +0.708204 q^{79} +1.00000 q^{81} -20.1803 q^{82} -4.00000 q^{83} -0.763932 q^{84} -1.23607 q^{86} -3.23607 q^{87} +9.23607 q^{89} -0.944272 q^{91} -5.38197 q^{92} +6.70820 q^{93} +7.61803 q^{94} -3.38197 q^{96} -17.2361 q^{97} -8.85410 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} + 2 q^{3} - q^{4} + q^{6} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + q^2 + 2 * q^3 - q^4 + q^6 + 2 * q^7 + 2 * q^9	\( 2 q + q^{2} + 2 q^{3} - q^{4} + q^{6} + 2 q^{7} + 2 q^{9} - q^{12} + 6 q^{13} - 4 q^{14} - 3 q^{16} - 2 q^{17} + q^{18} - 4 q^{19} + 2 q^{21} - 4 q^{23} - 2 q^{26} + 2 q^{27} - 6 q^{28} - 2 q^{29} - 9 q^{32} + 9 q^{34} - q^{36} - 6 q^{37} + 8 q^{38} + 6 q^{39} - 16 q^{41} - 4 q^{42} - 6 q^{43} - 17 q^{46} - 4 q^{47} - 3 q^{48} - 2 q^{49} - 2 q^{51} - 8 q^{52} - 2 q^{53} + q^{54} + 10 q^{56} - 4 q^{57} - 6 q^{58} + 10 q^{59} - 6 q^{61} + 15 q^{62} + 2 q^{63} + 4 q^{64} - 18 q^{67} + 11 q^{68} - 4 q^{69} - 20 q^{71} - 12 q^{73} - 8 q^{74} + 12 q^{76} - 2 q^{78} - 12 q^{79} + 2 q^{81} - 18 q^{82} - 8 q^{83} - 6 q^{84} + 2 q^{86} - 2 q^{87} + 14 q^{89} + 16 q^{91} - 13 q^{92} + 13 q^{94} - 9 q^{96} - 30 q^{97} - 11 q^{98}+O(q^{100}) \) 2 * q + q^2 + 2 * q^3 - q^4 + q^6 + 2 * q^7 + 2 * q^9 - q^12 + 6 * q^13 - 4 * q^14 - 3 * q^16 - 2 * q^17 + q^18 - 4 * q^19 + 2 * q^21 - 4 * q^23 - 2 * q^26 + 2 * q^27 - 6 * q^28 - 2 * q^29 - 9 * q^32 + 9 * q^34 - q^36 - 6 * q^37 + 8 * q^38 + 6 * q^39 - 16 * q^41 - 4 * q^42 - 6 * q^43 - 17 * q^46 - 4 * q^47 - 3 * q^48 - 2 * q^49 - 2 * q^51 - 8 * q^52 - 2 * q^53 + q^54 + 10 * q^56 - 4 * q^57 - 6 * q^58 + 10 * q^59 - 6 * q^61 + 15 * q^62 + 2 * q^63 + 4 * q^64 - 18 * q^67 + 11 * q^68 - 4 * q^69 - 20 * q^71 - 12 * q^73 - 8 * q^74 + 12 * q^76 - 2 * q^78 - 12 * q^79 + 2 * q^81 - 18 * q^82 - 8 * q^83 - 6 * q^84 + 2 * q^86 - 2 * q^87 + 14 * q^89 + 16 * q^91 - 13 * q^92 + 13 * q^94 - 9 * q^96 - 30 * q^97 - 11 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more