LMFDB - Embedded newform 8450.2.a.cc.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8450,2,Mod(1,8450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8450, 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("8450.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8450 = 2 \cdot 5^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8450.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:	$67.4735897080$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.940.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 7x - 4 $ x^3 - 7*x - 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 130)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.89511$ of defining polynomial
Character	$\chi$	$=$	8450.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +2.89511 q^{3} +1.00000 q^{4} +2.89511 q^{6} -4.38164 q^{7} +1.00000 q^{8} +5.38164 q^{9} -2.00000 q^{11} +2.89511 q^{12} -4.38164 q^{14} +1.00000 q^{16} -5.86818 q^{17} +5.38164 q^{18} +0.973070 q^{19} -12.6853 q^{21} -2.00000 q^{22} -7.79021 q^{23} +2.89511 q^{24} +6.89511 q^{27} -4.38164 q^{28} +0.973070 q^{29} +1.79021 q^{31} +1.00000 q^{32} -5.79021 q^{33} -5.86818 q^{34} +5.38164 q^{36} -0.591429 q^{37} +0.973070 q^{38} -4.81714 q^{41} -12.6853 q^{42} -4.68532 q^{43} -2.00000 q^{44} -7.79021 q^{46} +0.381642 q^{47} +2.89511 q^{48} +12.1988 q^{49} -16.9890 q^{51} -7.79021 q^{53} +6.89511 q^{54} -4.38164 q^{56} +2.81714 q^{57} +0.973070 q^{58} +0.973070 q^{59} -0.817143 q^{61} +1.79021 q^{62} -23.5804 q^{63} +1.00000 q^{64} -5.79021 q^{66} +1.79021 q^{67} -5.86818 q^{68} -22.5535 q^{69} +3.92204 q^{71} +5.38164 q^{72} -6.00000 q^{73} -0.591429 q^{74} +0.973070 q^{76} +8.76328 q^{77} +10.9731 q^{79} +3.81714 q^{81} -4.81714 q^{82} -6.97307 q^{83} -12.6853 q^{84} -4.68532 q^{86} +2.81714 q^{87} -2.00000 q^{88} -0.973070 q^{89} -7.79021 q^{92} +5.18286 q^{93} +0.381642 q^{94} +2.89511 q^{96} -18.6074 q^{97} +12.1988 q^{98} -10.7633 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} - 2 q^{7} + 3 q^{8} + 5 q^{9} - 6 q^{11} - 2 q^{14} + 3 q^{16} - 4 q^{17} + 5 q^{18} - 2 q^{19} - 12 q^{21} - 6 q^{22} - 6 q^{23} + 12 q^{27} - 2 q^{28} - 2 q^{29} - 12 q^{31} + 3 q^{32}+ \cdots - 10 q^{99}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 - 2 * q^7 + 3 * q^8 + 5 * q^9 - 6 * q^11 - 2 * q^14 + 3 * q^16 - 4 * q^17 + 5 * q^18 - 2 * q^19 - 12 * q^21 - 6 * q^22 - 6 * q^23 + 12 * q^27 - 2 * q^28 - 2 * q^29 - 12 * q^31 + 3 * q^32 - 4 * q^34 + 5 * q^36 - 8 * q^37 - 2 * q^38 - 2 * q^41 - 12 * q^42 + 12 * q^43 - 6 * q^44 - 6 * q^46 - 10 * q^47 + 13 * q^49 - 10 * q^51 - 6 * q^53 + 12 * q^54 - 2 * q^56 - 4 * q^57 - 2 * q^58 - 2 * q^59 + 10 * q^61 - 12 * q^62 - 36 * q^63 + 3 * q^64 - 12 * q^67 - 4 * q^68 - 28 * q^69 + 8 * q^71 + 5 * q^72 - 18 * q^73 - 8 * q^74 - 2 * q^76 + 4 * q^77 + 28 * q^79 - q^81 - 2 * q^82 - 16 * q^83 - 12 * q^84 + 12 * q^86 - 4 * q^87 - 6 * q^88 + 2 * q^89 - 6 * q^92 + 28 * q^93 - 10 * q^94 - 26 * q^97 + 13 * q^98 - 10 * q^99

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	2.89511	1.67149	0.835745	−	0.549117i	$-0.185036\pi$
$3$	2.89511	1.67149	0.835745	+	0.549117i	$0.185036\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	2.89511	1.18192
$7$	−4.38164	−1.65610	−0.828052	−	0.560651i	$-0.810551\pi$
$7$	−4.38164	−1.65610	−0.828052	+	0.560651i	$0.810551\pi$
$8$	1.00000	0.353553
$9$	5.38164	1.79388
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	2.89511	0.835745
$13$	0	0
$13$	0	0
$14$	−4.38164	−1.17104
$15$	0	0
$16$	1.00000	0.250000
$17$	−5.86818	−1.42324	−0.711621	−	0.702564i	$-0.752039\pi$
$17$	−5.86818	−1.42324	−0.711621	+	0.702564i	$0.752039\pi$
$18$	5.38164	1.26847
$19$	0.973070	0.223238	0.111619	−	0.993751i	$-0.464396\pi$
$19$	0.973070	0.223238	0.111619	+	0.993751i	$0.464396\pi$
$20$	0	0
$21$	−12.6853	−2.76816
$22$	−2.00000	−0.426401
$23$	−7.79021	−1.62437	−0.812186	−	0.583399i	$-0.801722\pi$
$23$	−7.79021	−1.62437	−0.812186	+	0.583399i	$0.801722\pi$
$24$	2.89511	0.590961
$25$	0	0
$26$	0	0
$27$	6.89511	1.32696
$28$	−4.38164	−0.828052
$29$	0.973070	0.180695	0.0903473	−	0.995910i	$-0.471202\pi$
$29$	0.973070	0.180695	0.0903473	+	0.995910i	$0.471202\pi$
$30$	0	0
$31$	1.79021	0.321532	0.160766	−	0.986993i	$-0.448604\pi$
$31$	1.79021	0.321532	0.160766	+	0.986993i	$0.448604\pi$
$32$	1.00000	0.176777
$33$	−5.79021	−1.00795
$34$	−5.86818	−1.00638
$35$	0	0
$36$	5.38164	0.896940
$37$	−0.591429	−0.0972303	−0.0486151	−	0.998818i	$-0.515481\pi$
$37$	−0.591429	−0.0972303	−0.0486151	+	0.998818i	$0.515481\pi$
$38$	0.973070	0.157853
$39$	0	0
$40$	0	0
$41$	−4.81714	−0.752311	−0.376156	−	0.926556i	$-0.622754\pi$
$41$	−4.81714	−0.752311	−0.376156	+	0.926556i	$0.622754\pi$
$42$	−12.6853	−1.95739
$43$	−4.68532	−0.714505	−0.357252	−	0.934008i	$-0.616286\pi$
$43$	−4.68532	−0.714505	−0.357252	+	0.934008i	$0.616286\pi$
$44$	−2.00000	−0.301511
$45$	0	0
$46$	−7.79021	−1.14860
$47$	0.381642	0.0556682	0.0278341	−	0.999613i	$-0.491139\pi$
$47$	0.381642	0.0556682	0.0278341	+	0.999613i	$0.491139\pi$
$48$	2.89511	0.417873
$49$	12.1988	1.74268
$50$	0	0
$51$	−16.9890	−2.37894
$52$	0	0
$53$	−7.79021	−1.07007	−0.535034	−	0.844831i	$-0.679701\pi$
$53$	−7.79021	−1.07007	−0.535034	+	0.844831i	$0.679701\pi$
$54$	6.89511	0.938305
$55$	0	0
$56$	−4.38164	−0.585522
$57$	2.81714	0.373140
$58$	0.973070	0.127770
$59$	0.973070	0.126683	0.0633415	−	0.997992i	$-0.479824\pi$
$59$	0.973070	0.126683	0.0633415	+	0.997992i	$0.479824\pi$
$60$	0	0
$61$	−0.817143	−0.104624	−0.0523122	−	0.998631i	$-0.516659\pi$
$61$	−0.817143	−0.104624	−0.0523122	+	0.998631i	$0.516659\pi$
$62$	1.79021	0.227357
$63$	−23.5804	−2.97085
$64$	1.00000	0.125000
$65$	0	0
$66$	−5.79021	−0.712726
$67$	1.79021	0.218709	0.109355	−	0.994003i	$-0.465122\pi$
$67$	1.79021	0.218709	0.109355	+	0.994003i	$0.465122\pi$
$68$	−5.86818	−0.711621
$69$	−22.5535	−2.71512
$70$	0	0
$71$	3.92204	0.465460	0.232730	−	0.972541i	$-0.425234\pi$
$71$	3.92204	0.465460	0.232730	+	0.972541i	$0.425234\pi$
$72$	5.38164	0.634233
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	−0.591429	−0.0687522
$75$	0	0
$76$	0.973070	0.111619
$77$	8.76328	0.998669
$78$	0	0
$79$	10.9731	1.23457	0.617283	−	0.786741i	$-0.288233\pi$
$79$	10.9731	1.23457	0.617283	+	0.786741i	$0.288233\pi$
$80$	0	0
$81$	3.81714	0.424127
$82$	−4.81714	−0.531964
$83$	−6.97307	−0.765394	−0.382697	−	0.923874i	$-0.625005\pi$
$83$	−6.97307	−0.765394	−0.382697	+	0.923874i	$0.625005\pi$
$84$	−12.6853	−1.38408
$85$	0	0
$86$	−4.68532	−0.505231
$87$	2.81714	0.302029
$88$	−2.00000	−0.213201
$89$	−0.973070	−0.103145	−0.0515726	−	0.998669i	$-0.516423\pi$
$89$	−0.973070	−0.103145	−0.0515726	+	0.998669i	$0.516423\pi$
$90$	0	0
$91$	0	0
$92$	−7.79021	−0.812186
$93$	5.18286	0.537437
$94$	0.381642	0.0393633
$95$	0	0
$96$	2.89511	0.295481
$97$	−18.6074	−1.88929	−0.944645	−	0.328093i	$-0.893594\pi$
$97$	−18.6074	−1.88929	−0.944645	+	0.328093i	$0.893594\pi$
$98$	12.1988	1.23226
$99$	−10.7633	−1.08175
$100$	0	0
$101$	14.3437	1.42725	0.713626	−	0.700527i	$-0.247052\pi$
$101$	14.3437	1.42725	0.713626	+	0.700527i	$0.247052\pi$
$102$	−16.9890	−1.68216
$103$	−9.73635	−0.959351	−0.479676	−	0.877446i	$-0.659246\pi$
$103$	−9.73635	−0.959351	−0.479676	+	0.877446i	$0.659246\pi$
$104$	0	0
$105$	0	0
$106$	−7.79021	−0.756652
$107$	5.79021	0.559761	0.279881	−	0.960035i	$-0.409705\pi$
$107$	5.79021	0.559761	0.279881	+	0.960035i	$0.409705\pi$
$108$	6.89511	0.663482
$109$	0.685320	0.0656417	0.0328209	−	0.999461i	$-0.489551\pi$
$109$	0.685320	0.0656417	0.0328209	+	0.999461i	$0.489551\pi$
$110$	0	0
$111$	−1.71225	−0.162519
$112$	−4.38164	−0.414026
$113$	−4.00000	−0.376288	−0.188144	−	0.982141i	$-0.560247\pi$
$113$	−4.00000	−0.376288	−0.188144	+	0.982141i	$0.560247\pi$
$114$	2.81714	0.263850
$115$	0	0
$116$	0.973070	0.0903473
$117$	0	0
$118$	0.973070	0.0895784
$119$	25.7122	2.35704
$120$	0	0
$121$	−7.00000	−0.636364
$122$	−0.817143	−0.0739806
$123$	−13.9461	−1.25748
$124$	1.79021	0.160766
$125$	0	0
$126$	−23.5804	−2.10071
$127$	−11.7902	−1.04621	−0.523106	−	0.852268i	$-0.675227\pi$
$127$	−11.7902	−1.04621	−0.523106	+	0.852268i	$0.675227\pi$
$128$	1.00000	0.0883883
$129$	−13.5645	−1.19429
$130$	0	0
$131$	−16.9890	−1.48434	−0.742168	−	0.670214i	$-0.766202\pi$
$131$	−16.9890	−1.48434	−0.742168	+	0.670214i	$0.766202\pi$
$132$	−5.79021	−0.503973
$133$	−4.26365	−0.369705
$134$	1.79021	0.154651
$135$	0	0
$136$	−5.86818	−0.503192
$137$	4.20979	0.359666	0.179833	−	0.983697i	$-0.442444\pi$
$137$	4.20979	0.359666	0.179833	+	0.983697i	$0.442444\pi$
$138$	−22.5535	−1.91988
$139$	−6.17185	−0.523490	−0.261745	−	0.965137i	$-0.584298\pi$
$139$	−6.17185	−0.523490	−0.261745	+	0.965137i	$0.584298\pi$
$140$	0	0
$141$	1.10489	0.0930488
$142$	3.92204	0.329130
$143$	0	0
$144$	5.38164	0.448470
$145$	0	0
$146$	−6.00000	−0.496564
$147$	35.3168	2.91288
$148$	−0.591429	−0.0486151
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	−15.5025	−1.26157	−0.630786	−	0.775957i	$-0.717268\pi$
$151$	−15.5025	−1.26157	−0.630786	+	0.775957i	$0.717268\pi$
$152$	0.973070	0.0789264
$153$	−31.5804	−2.55313
$154$	8.76328	0.706166
$155$	0	0
$156$	0	0
$157$	−8.97307	−0.716129	−0.358064	−	0.933697i	$-0.616563\pi$
$157$	−8.97307	−0.716129	−0.358064	+	0.933697i	$0.616563\pi$
$158$	10.9731	0.872971
$159$	−22.5535	−1.78861
$160$	0	0
$161$	34.1339	2.69013
$162$	3.81714	0.299903
$163$	−12.6074	−0.987484	−0.493742	−	0.869608i	$-0.664371\pi$
$163$	−12.6074	−0.987484	−0.493742	+	0.869608i	$0.664371\pi$
$164$	−4.81714	−0.376156
$165$	0	0
$166$	−6.97307	−0.541215
$167$	23.1609	1.79224	0.896120	−	0.443811i	$-0.146374\pi$
$167$	23.1609	1.79224	0.896120	+	0.443811i	$0.146374\pi$
$168$	−12.6853	−0.978694
$169$	0	0
$170$	0	0
$171$	5.23672	0.400462
$172$	−4.68532	−0.357252
$173$	19.7902	1.50462	0.752311	−	0.658808i	$-0.228939\pi$
$173$	19.7902	1.50462	0.752311	+	0.658808i	$0.228939\pi$
$174$	2.81714	0.213567
$175$	0	0
$176$	−2.00000	−0.150756
$177$	2.81714	0.211749
$178$	−0.973070	−0.0729347
$179$	6.17185	0.461306	0.230653	−	0.973036i	$-0.425914\pi$
$179$	6.17185	0.461306	0.230653	+	0.973036i	$0.425914\pi$
$180$	0	0
$181$	−19.3706	−1.43981	−0.719904	−	0.694074i	$-0.755814\pi$
$181$	−19.3706	−1.43981	−0.719904	+	0.694074i	$0.755814\pi$
$182$	0	0
$183$	−2.36571	−0.174879
$184$	−7.79021	−0.574302
$185$	0	0
$186$	5.18286	0.380026
$187$	11.7364	0.858247
$188$	0.381642	0.0278341
$189$	−30.2119	−2.19759
$190$	0	0
$191$	−3.73635	−0.270353	−0.135177	−	0.990822i	$-0.543160\pi$
$191$	−3.73635	−0.270353	−0.135177	+	0.990822i	$0.543160\pi$
$192$	2.89511	0.208936
$193$	15.3706	1.10640	0.553201	−	0.833048i	$-0.313406\pi$
$193$	15.3706	1.10640	0.553201	+	0.833048i	$0.313406\pi$
$194$	−18.6074	−1.33593
$195$	0	0
$196$	12.1988	0.871342
$197$	26.9890	1.92289	0.961443	−	0.275004i	$-0.0886790\pi$
$197$	26.9890	1.92289	0.961443	+	0.275004i	$0.0886790\pi$
$198$	−10.7633	−0.764913
$199$	23.3168	1.65288	0.826441	−	0.563023i	$-0.190362\pi$
$199$	23.3168	1.65288	0.826441	+	0.563023i	$0.190362\pi$
$200$	0	0
$201$	5.18286	0.365571
$202$	14.3437	1.00922
$203$	−4.26365	−0.299249
$204$	−16.9890	−1.18947
$205$	0	0
$206$	−9.73635	−0.678364
$207$	−41.9241	−2.91393
$208$	0	0
$209$	−1.94614	−0.134617
$210$	0	0
$211$	15.3547	1.05706	0.528531	−	0.848914i	$-0.322743\pi$
$211$	15.3547	1.05706	0.528531	+	0.848914i	$0.322743\pi$
$212$	−7.79021	−0.535034
$213$	11.3547	0.778012
$214$	5.79021	0.395811
$215$	0	0
$216$	6.89511	0.469153
$217$	−7.84407	−0.532490
$218$	0.685320	0.0464157
$219$	−17.3706	−1.17380
$220$	0	0
$221$	0	0
$222$	−1.71225	−0.114919
$223$	24.8331	1.66295	0.831473	−	0.555566i	$-0.187498\pi$
$223$	24.8331	1.66295	0.831473	+	0.555566i	$0.187498\pi$
$224$	−4.38164	−0.292761
$225$	0	0
$226$	−4.00000	−0.266076
$227$	−21.5266	−1.42877	−0.714384	−	0.699754i	$-0.753293\pi$
$227$	−21.5266	−1.42877	−0.714384	+	0.699754i	$0.753293\pi$
$228$	2.81714	0.186570
$229$	2.63146	0.173892	0.0869459	−	0.996213i	$-0.472289\pi$
$229$	2.63146	0.173892	0.0869459	+	0.996213i	$0.472289\pi$
$230$	0	0
$231$	25.3706	1.66927
$232$	0.973070	0.0638852
$233$	29.0290	1.90175	0.950877	−	0.309568i	$-0.100184\pi$
$233$	29.0290	1.90175	0.950877	+	0.309568i	$0.100184\pi$
$234$	0	0
$235$	0	0
$236$	0.973070	0.0633415
$237$	31.7682	2.06357
$238$	25.7122	1.66668
$239$	−2.13182	−0.137896	−0.0689481	−	0.997620i	$-0.521964\pi$
$239$	−2.13182	−0.137896	−0.0689481	+	0.997620i	$0.521964\pi$
$240$	0	0
$241$	−22.4996	−1.44933	−0.724665	−	0.689102i	$-0.758005\pi$
$241$	−22.4996	−1.44933	−0.724665	+	0.689102i	$0.758005\pi$
$242$	−7.00000	−0.449977
$243$	−9.63429	−0.618040
$244$	−0.817143	−0.0523122
$245$	0	0
$246$	−13.9461	−0.889173
$247$	0	0
$248$	1.79021	0.113679
$249$	−20.1878	−1.27935
$250$	0	0
$251$	−0.419574	−0.0264833	−0.0132416	−	0.999912i	$-0.504215\pi$
$251$	−0.419574	−0.0264833	−0.0132416	+	0.999912i	$0.504215\pi$
$252$	−23.5804	−1.48543
$253$	15.5804	0.979533
$254$	−11.7902	−0.739784
$255$	0	0
$256$	1.00000	0.0625000
$257$	−19.6584	−1.22626	−0.613128	−	0.789983i	$-0.710089\pi$
$257$	−19.6584	−1.22626	−0.613128	+	0.789983i	$0.710089\pi$
$258$	−13.5645	−0.844489
$259$	2.59143	0.161024
$260$	0	0
$261$	5.23672	0.324145
$262$	−16.9890	−1.04958
$263$	10.7633	0.663692	0.331846	−	0.943333i	$-0.392328\pi$
$263$	10.7633	0.663692	0.331846	+	0.943333i	$0.392328\pi$
$264$	−5.79021	−0.356363
$265$	0	0
$266$	−4.26365	−0.261421
$267$	−2.81714	−0.172406
$268$	1.79021	0.109355
$269$	−14.7633	−0.900133	−0.450067	−	0.892995i	$-0.648600\pi$
$269$	−14.7633	−0.900133	−0.450067	+	0.892995i	$0.648600\pi$
$270$	0	0
$271$	27.2388	1.65464	0.827320	−	0.561731i	$-0.189864\pi$
$271$	27.2388	1.65464	0.827320	+	0.561731i	$0.189864\pi$
$272$	−5.86818	−0.355810
$273$	0	0
$274$	4.20979	0.254323
$275$	0	0
$276$	−22.5535	−1.35756
$277$	2.87100	0.172502	0.0862509	−	0.996273i	$-0.472511\pi$
$277$	2.87100	0.172502	0.0862509	+	0.996273i	$0.472511\pi$
$278$	−6.17185	−0.370163
$279$	9.63429	0.576790
$280$	0	0
$281$	−5.39264	−0.321698	−0.160849	−	0.986979i	$-0.551423\pi$
$281$	−5.39264	−0.321698	−0.160849	+	0.986979i	$0.551423\pi$
$282$	1.10489	0.0657954
$283$	24.9511	1.48319	0.741593	−	0.670850i	$-0.234070\pi$
$283$	24.9511	1.48319	0.741593	+	0.670850i	$0.234070\pi$
$284$	3.92204	0.232730
$285$	0	0
$286$	0	0
$287$	21.1070	1.24591
$288$	5.38164	0.317116
$289$	17.4355	1.02562
$290$	0	0
$291$	−53.8703	−3.15793
$292$	−6.00000	−0.351123
$293$	24.9351	1.45673	0.728363	−	0.685191i	$-0.240281\pi$
$293$	24.9351	1.45673	0.728363	+	0.685191i	$0.240281\pi$
$294$	35.3168	2.05972
$295$	0	0
$296$	−0.591429	−0.0343761
$297$	−13.7902	−0.800189
$298$	0	0
$299$	0	0
$300$	0	0
$301$	20.5294	1.18329
$302$	−15.5025	−0.892066
$303$	41.5266	2.38564
$304$	0.973070	0.0558094
$305$	0	0
$306$	−31.5804	−1.80533
$307$	−14.8171	−0.845659	−0.422829	−	0.906209i	$-0.638963\pi$
$307$	−14.8171	−0.845659	−0.422829	+	0.906209i	$0.638963\pi$
$308$	8.76328	0.499334
$309$	−28.1878	−1.60355
$310$	0	0
$311$	12.8710	0.729848	0.364924	−	0.931037i	$-0.381095\pi$
$311$	12.8710	0.729848	0.364924	+	0.931037i	$0.381095\pi$
$312$	0	0
$313$	−19.9220	−1.12606	−0.563030	−	0.826436i	$-0.690364\pi$
$313$	−19.9220	−1.12606	−0.563030	+	0.826436i	$0.690364\pi$
$314$	−8.97307	−0.506380
$315$	0	0
$316$	10.9731	0.617283
$317$	−3.94614	−0.221637	−0.110819	−	0.993841i	$-0.535347\pi$
$317$	−3.94614	−0.221637	−0.110819	+	0.993841i	$0.535347\pi$
$318$	−22.5535	−1.26474
$319$	−1.94614	−0.108963
$320$	0	0
$321$	16.7633	0.935635
$322$	34.1339	1.90221
$323$	−5.71015	−0.317721
$324$	3.81714	0.212063
$325$	0	0
$326$	−12.6074	−0.698257
$327$	1.98407	0.109719
$328$	−4.81714	−0.265982
$329$	−1.67222	−0.0921923
$330$	0	0
$331$	−21.4245	−1.17760	−0.588798	−	0.808280i	$-0.700399\pi$
$331$	−21.4245	−1.17760	−0.588798	+	0.808280i	$0.700399\pi$
$332$	−6.97307	−0.382697
$333$	−3.18286	−0.174420
$334$	23.1609	1.26731
$335$	0	0
$336$	−12.6853	−0.692041
$337$	11.2388	0.612217	0.306109	−	0.951997i	$-0.400973\pi$
$337$	11.2388	0.612217	0.306109	+	0.951997i	$0.400973\pi$
$338$	0	0
$339$	−11.5804	−0.628962
$340$	0	0
$341$	−3.58043	−0.193891
$342$	5.23672	0.283169
$343$	−22.7792	−1.22996
$344$	−4.68532	−0.252616
$345$	0	0
$346$	19.7902	1.06393
$347$	16.9490	0.909868	0.454934	−	0.890525i	$-0.349663\pi$
$347$	16.9490	0.909868	0.454934	+	0.890525i	$0.349663\pi$
$348$	2.81714	0.151015
$349$	10.2119	0.546630	0.273315	−	0.961925i	$-0.411880\pi$
$349$	10.2119	0.546630	0.273315	+	0.961925i	$0.411880\pi$
$350$	0	0
$351$	0	0
$352$	−2.00000	−0.106600
$353$	−14.9511	−0.795765	−0.397882	−	0.917436i	$-0.630255\pi$
$353$	−14.9511	−0.795765	−0.397882	+	0.917436i	$0.630255\pi$
$354$	2.81714	0.149729
$355$	0	0
$356$	−0.973070	−0.0515726
$357$	74.4397	3.93977
$358$	6.17185	0.326193
$359$	−11.8441	−0.625106	−0.312553	−	0.949900i	$-0.601184\pi$
$359$	−11.8441	−0.625106	−0.312553	+	0.949900i	$0.601184\pi$
$360$	0	0
$361$	−18.0531	−0.950165
$362$	−19.3706	−1.01810
$363$	−20.2657	−1.06368
$364$	0	0
$365$	0	0
$366$	−2.36571	−0.123658
$367$	−16.0539	−0.838005	−0.419002	−	0.907985i	$-0.637620\pi$
$367$	−16.0539	−0.838005	−0.419002	+	0.907985i	$0.637620\pi$
$368$	−7.79021	−0.406093
$369$	−25.9241	−1.34956
$370$	0	0
$371$	34.1339	1.77214
$372$	5.18286	0.268719
$373$	6.87100	0.355767	0.177883	−	0.984052i	$-0.443075\pi$
$373$	6.87100	0.355767	0.177883	+	0.984052i	$0.443075\pi$
$374$	11.7364	0.606872
$375$	0	0
$376$	0.381642	0.0196817
$377$	0	0
$378$	−30.2119	−1.55393
$379$	−1.84407	−0.0947236	−0.0473618	−	0.998878i	$-0.515081\pi$
$379$	−1.84407	−0.0947236	−0.0473618	+	0.998878i	$0.515081\pi$
$380$	0	0
$381$	−34.1339	−1.74873
$382$	−3.73635	−0.191168
$383$	−9.25264	−0.472788	−0.236394	−	0.971657i	$-0.575966\pi$
$383$	−9.25264	−0.472788	−0.236394	+	0.971657i	$0.575966\pi$
$384$	2.89511	0.147740
$385$	0	0
$386$	15.3706	0.782345
$387$	−25.2147	−1.28174
$388$	−18.6074	−0.944645
$389$	−21.2686	−1.07836	−0.539180	−	0.842191i	$-0.681266\pi$
$389$	−21.2686	−1.07836	−0.539180	+	0.842191i	$0.681266\pi$
$390$	0	0
$391$	45.7143	2.31187
$392$	12.1988	0.616132
$393$	−49.1850	−2.48105
$394$	26.9890	1.35969
$395$	0	0
$396$	−10.7633	−0.540875
$397$	−33.4727	−1.67995	−0.839974	−	0.542627i	$-0.817430\pi$
$397$	−33.4727	−1.67995	−0.839974	+	0.542627i	$0.817430\pi$
$398$	23.3168	1.16876
$399$	−12.3437	−0.617958
$400$	0	0
$401$	−17.7364	−0.885711	−0.442856	−	0.896593i	$-0.646035\pi$
$401$	−17.7364	−0.885711	−0.442856	+	0.896593i	$0.646035\pi$
$402$	5.18286	0.258497
$403$	0	0
$404$	14.3437	0.713626
$405$	0	0
$406$	−4.26365	−0.211601
$407$	1.18286	0.0586321
$408$	−16.9890	−0.841081
$409$	−7.68249	−0.379875	−0.189937	−	0.981796i	$-0.560829\pi$
$409$	−7.68249	−0.379875	−0.189937	+	0.981796i	$0.560829\pi$
$410$	0	0
$411$	12.1878	0.601179
$412$	−9.73635	−0.479676
$413$	−4.26365	−0.209800
$414$	−41.9241	−2.06046
$415$	0	0
$416$	0	0
$417$	−17.8682	−0.875008
$418$	−1.94614	−0.0951889
$419$	11.8061	0.576768	0.288384	−	0.957515i	$-0.406882\pi$
$419$	11.8061	0.576768	0.288384	+	0.957515i	$0.406882\pi$
$420$	0	0
$421$	26.3678	1.28509	0.642544	−	0.766249i	$-0.277879\pi$
$421$	26.3678	1.28509	0.642544	+	0.766249i	$0.277879\pi$
$422$	15.3547	0.747456
$423$	2.05386	0.0998620
$424$	−7.79021	−0.378326
$425$	0	0
$426$	11.3547	0.550138
$427$	3.58043	0.173269
$428$	5.79021	0.279881
$429$	0	0
$430$	0	0
$431$	−1.71225	−0.0824761	−0.0412381	−	0.999149i	$-0.513130\pi$
$431$	−1.71225	−0.0824761	−0.0412381	+	0.999149i	$0.513130\pi$
$432$	6.89511	0.331741
$433$	11.9220	0.572936	0.286468	−	0.958090i	$-0.407519\pi$
$433$	11.9220	0.572936	0.286468	+	0.958090i	$0.407519\pi$
$434$	−7.84407	−0.376528
$435$	0	0
$436$	0.685320	0.0328209
$437$	−7.58043	−0.362621
$438$	−17.3706	−0.830001
$439$	−12.3437	−0.589133	−0.294567	−	0.955631i	$-0.595175\pi$
$439$	−12.3437	−0.589133	−0.294567	+	0.955631i	$0.595175\pi$
$440$	0	0
$441$	65.6495	3.12617
$442$	0	0
$443$	28.1580	1.33783	0.668914	−	0.743340i	$-0.266759\pi$
$443$	28.1580	1.33783	0.668914	+	0.743340i	$0.266759\pi$
$444$	−1.71225	−0.0812597
$445$	0	0
$446$	24.8331	1.17588
$447$	0	0
$448$	−4.38164	−0.207013
$449$	29.0531	1.37110	0.685551	−	0.728025i	$-0.259561\pi$
$449$	29.0531	1.37110	0.685551	+	0.728025i	$0.259561\pi$
$450$	0	0
$451$	9.63429	0.453661
$452$	−4.00000	−0.188144
$453$	−44.8813	−2.10871
$454$	−21.5266	−1.01029
$455$	0	0
$456$	2.81714	0.131925
$457$	−11.5266	−0.539190	−0.269595	−	0.962974i	$-0.586890\pi$
$457$	−11.5266	−0.539190	−0.269595	+	0.962974i	$0.586890\pi$
$458$	2.63146	0.122960
$459$	−40.4617	−1.88859
$460$	0	0
$461$	−28.4217	−1.32373	−0.661865	−	0.749623i	$-0.730235\pi$
$461$	−28.4217	−1.32373	−0.661865	+	0.749623i	$0.730235\pi$
$462$	25.3706	1.18035
$463$	−27.4727	−1.27677	−0.638383	−	0.769719i	$-0.720396\pi$
$463$	−27.4727	−1.27677	−0.638383	+	0.769719i	$0.720396\pi$
$464$	0.973070	0.0451737
$465$	0	0
$466$	29.0290	1.34474
$467$	18.2098	0.842648	0.421324	−	0.906910i	$-0.361565\pi$
$467$	18.2098	0.842648	0.421324	+	0.906910i	$0.361565\pi$
$468$	0	0
$469$	−7.84407	−0.362206
$470$	0	0
$471$	−25.9780	−1.19700
$472$	0.973070	0.0447892
$473$	9.37064	0.430862
$474$	31.7682	1.45916
$475$	0	0
$476$	25.7122	1.17852
$477$	−41.9241	−1.91957
$478$	−2.13182	−0.0975073
$479$	33.6045	1.53543	0.767715	−	0.640791i	$-0.221394\pi$
$479$	33.6045	1.53543	0.767715	+	0.640791i	$0.221394\pi$
$480$	0	0
$481$	0	0
$482$	−22.4996	−1.02483
$483$	98.8213	4.49653
$484$	−7.00000	−0.318182
$485$	0	0
$486$	−9.63429	−0.437020
$487$	8.00000	0.362515	0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000	0.362515	0.181257	+	0.983436i	$0.441983\pi$
$488$	−0.817143	−0.0369903
$489$	−36.4996	−1.65057
$490$	0	0
$491$	−19.8061	−0.893839	−0.446919	−	0.894574i	$-0.647479\pi$
$491$	−19.8061	−0.893839	−0.446919	+	0.894574i	$0.647479\pi$
$492$	−13.9461	−0.628741
$493$	−5.71015	−0.257172
$494$	0	0
$495$	0	0
$496$	1.79021	0.0803829
$497$	−17.1850	−0.770851
$498$	−20.1878	−0.904636
$499$	−9.12900	−0.408670	−0.204335	−	0.978901i	$-0.565503\pi$
$499$	−9.12900	−0.408670	−0.204335	+	0.978901i	$0.565503\pi$
$500$	0	0
$501$	67.0531	2.99571
$502$	−0.419574	−0.0187265
$503$	−10.6074	−0.472959	−0.236479	−	0.971637i	$-0.575994\pi$
$503$	−10.6074	−0.472959	−0.236479	+	0.971637i	$0.575994\pi$
$504$	−23.5804	−1.05036
$505$	0	0
$506$	15.5804	0.692634
$507$	0	0
$508$	−11.7902	−0.523106
$509$	5.63429	0.249735	0.124868	−	0.992173i	$-0.460149\pi$
$509$	5.63429	0.249735	0.124868	+	0.992173i	$0.460149\pi$
$510$	0	0
$511$	26.2899	1.16299
$512$	1.00000	0.0441942
$513$	6.70942	0.296228
$514$	−19.6584	−0.867094
$515$	0	0
$516$	−13.5645	−0.597144
$517$	−0.763283	−0.0335692
$518$	2.59143	0.113861
$519$	57.2948	2.51496
$520$	0	0
$521$	−22.8012	−0.998939	−0.499470	−	0.866331i	$-0.666472\pi$
$521$	−22.8012	−0.998939	−0.499470	+	0.866331i	$0.666472\pi$
$522$	5.23672	0.229205
$523$	−23.6286	−1.03321	−0.516604	−	0.856224i	$-0.672804\pi$
$523$	−23.6286	−1.03321	−0.516604	+	0.856224i	$0.672804\pi$
$524$	−16.9890	−0.742168
$525$	0	0
$526$	10.7633	0.469301
$527$	−10.5053	−0.457617
$528$	−5.79021	−0.251987
$529$	37.6874	1.63858
$530$	0	0
$531$	5.23672	0.227254
$532$	−4.26365	−0.184853
$533$	0	0
$534$	−2.81714	−0.121910
$535$	0	0
$536$	1.79021	0.0773254
$537$	17.8682	0.771069
$538$	−14.7633	−0.636490
$539$	−24.3976	−1.05088
$540$	0	0
$541$	−11.3147	−0.486456	−0.243228	−	0.969969i	$-0.578206\pi$
$541$	−11.3147	−0.486456	−0.243228	+	0.969969i	$0.578206\pi$
$542$	27.2388	1.17001
$543$	−56.0801	−2.40663
$544$	−5.86818	−0.251596
$545$	0	0
$546$	0	0
$547$	−16.8412	−0.720080	−0.360040	−	0.932937i	$-0.617237\pi$
$547$	−16.8412	−0.720080	−0.360040	+	0.932937i	$0.617237\pi$
$548$	4.20979	0.179833
$549$	−4.39757	−0.187684
$550$	0	0
$551$	0.946866	0.0403379
$552$	−22.5535	−0.959941
$553$	−48.0801	−2.04457
$554$	2.87100	0.121977
$555$	0	0
$556$	−6.17185	−0.261745
$557$	−35.7523	−1.51487	−0.757436	−	0.652909i	$-0.773548\pi$
$557$	−35.7523	−1.51487	−0.757436	+	0.652909i	$0.773548\pi$
$558$	9.63429	0.407852
$559$	0	0
$560$	0	0
$561$	33.9780	1.43455
$562$	−5.39264	−0.227475
$563$	13.3685	0.563417	0.281708	−	0.959500i	$-0.409099\pi$
$563$	13.3685	0.563417	0.281708	+	0.959500i	$0.409099\pi$
$564$	1.10489	0.0465244
$565$	0	0
$566$	24.9511	1.04877
$567$	−16.7254	−0.702399
$568$	3.92204	0.164565
$569$	−29.9621	−1.25608	−0.628038	−	0.778183i	$-0.716142\pi$
$569$	−29.9621	−1.25608	−0.628038	+	0.778183i	$0.716142\pi$
$570$	0	0
$571$	30.5156	1.27704	0.638518	−	0.769607i	$-0.279548\pi$
$571$	30.5156	1.27704	0.638518	+	0.769607i	$0.279548\pi$
$572$	0	0
$573$	−10.8171	−0.451893
$574$	21.1070	0.880989
$575$	0	0
$576$	5.38164	0.224235
$577$	33.1609	1.38050	0.690252	−	0.723569i	$-0.257500\pi$
$577$	33.1609	1.38050	0.690252	+	0.723569i	$0.257500\pi$
$578$	17.4355	0.725221
$579$	44.4996	1.84934
$580$	0	0
$581$	30.5535	1.26757
$582$	−53.8703	−2.23299
$583$	15.5804	0.645275
$584$	−6.00000	−0.248282
$585$	0	0
$586$	24.9351	1.03006
$587$	14.1339	0.583369	0.291685	−	0.956515i	$-0.405784\pi$
$587$	14.1339	0.583369	0.291685	+	0.956515i	$0.405784\pi$
$588$	35.3168	1.45644
$589$	1.74200	0.0717780
$590$	0	0
$591$	78.1360	3.21409
$592$	−0.591429	−0.0243076
$593$	−9.39264	−0.385710	−0.192855	−	0.981227i	$-0.561775\pi$
$593$	−9.39264	−0.385710	−0.192855	+	0.981227i	$0.561775\pi$
$594$	−13.7902	−0.565819
$595$	0	0
$596$	0	0
$597$	67.5046	2.76278
$598$	0	0
$599$	45.5584	1.86147	0.930733	−	0.365699i	$-0.119170\pi$
$599$	45.5584	1.86147	0.930733	+	0.365699i	$0.119170\pi$
$600$	0	0
$601$	26.7254	1.09015	0.545075	−	0.838387i	$-0.316501\pi$
$601$	26.7254	1.09015	0.545075	+	0.838387i	$0.316501\pi$
$602$	20.5294	0.836716
$603$	9.63429	0.392338
$604$	−15.5025	−0.630786
$605$	0	0
$606$	41.5266	1.68690
$607$	30.0801	1.22091	0.610456	−	0.792050i	$-0.290986\pi$
$607$	30.0801	1.22091	0.610456	+	0.792050i	$0.290986\pi$
$608$	0.973070	0.0394632
$609$	−12.3437	−0.500192
$610$	0	0
$611$	0	0
$612$	−31.5804	−1.27656
$613$	−17.2686	−0.697471	−0.348735	−	0.937221i	$-0.613389\pi$
$613$	−17.2686	−0.697471	−0.348735	+	0.937221i	$0.613389\pi$
$614$	−14.8171	−0.597971
$615$	0	0
$616$	8.76328	0.353083
$617$	−14.3437	−0.577456	−0.288728	−	0.957411i	$-0.593232\pi$
$617$	−14.3437	−0.577456	−0.288728	+	0.957411i	$0.593232\pi$
$618$	−28.1878	−1.13388
$619$	38.4514	1.54549	0.772747	−	0.634715i	$-0.218882\pi$
$619$	38.4514	1.54549	0.772747	+	0.634715i	$0.218882\pi$
$620$	0	0
$621$	−53.7143	−2.15548
$622$	12.8710	0.516080
$623$	4.26365	0.170819
$624$	0	0
$625$	0	0
$626$	−19.9220	−0.796245
$627$	−5.63429	−0.225012
$628$	−8.97307	−0.358064
$629$	3.47061	0.138382
$630$	0	0
$631$	8.18568	0.325867	0.162933	−	0.986637i	$-0.447904\pi$
$631$	8.18568	0.325867	0.162933	+	0.986637i	$0.447904\pi$
$632$	10.9731	0.436485
$633$	44.4535	1.76687
$634$	−3.94614	−0.156721
$635$	0	0
$636$	−22.5535	−0.894304
$637$	0	0
$638$	−1.94614	−0.0770485
$639$	21.1070	0.834980
$640$	0	0
$641$	−12.3657	−0.488416	−0.244208	−	0.969723i	$-0.578528\pi$
$641$	−12.3657	−0.488416	−0.244208	+	0.969723i	$0.578528\pi$
$642$	16.7633	0.661594
$643$	−46.8972	−1.84945	−0.924723	−	0.380642i	$-0.875703\pi$
$643$	−46.8972	−1.84945	−0.924723	+	0.380642i	$0.875703\pi$
$644$	34.1339	1.34506
$645$	0	0
$646$	−5.71015	−0.224663
$647$	26.1878	1.02955	0.514774	−	0.857326i	$-0.327876\pi$
$647$	26.1878	1.02955	0.514774	+	0.857326i	$0.327876\pi$
$648$	3.81714	0.149952
$649$	−1.94614	−0.0763927
$650$	0	0
$651$	−22.7094	−0.890053
$652$	−12.6074	−0.493742
$653$	10.7633	0.421200	0.210600	−	0.977572i	$-0.432458\pi$
$653$	10.7633	0.421200	0.210600	+	0.977572i	$0.432458\pi$
$654$	1.98407	0.0775834
$655$	0	0
$656$	−4.81714	−0.188078
$657$	−32.2899	−1.25975
$658$	−1.67222	−0.0651898
$659$	31.4727	1.22600	0.613001	−	0.790082i	$-0.289962\pi$
$659$	31.4727	1.22600	0.613001	+	0.790082i	$0.289962\pi$
$660$	0	0
$661$	−25.2147	−0.980739	−0.490369	−	0.871515i	$-0.663138\pi$
$661$	−25.2147	−0.980739	−0.490369	+	0.871515i	$0.663138\pi$
$662$	−21.4245	−0.832687
$663$	0	0
$664$	−6.97307	−0.270608
$665$	0	0
$666$	−3.18286	−0.123333
$667$	−7.58043	−0.293515
$668$	23.1609	0.896120
$669$	71.8944	2.77960
$670$	0	0
$671$	1.63429	0.0630909
$672$	−12.6853	−0.489347
$673$	−23.2388	−0.895791	−0.447895	−	0.894086i	$-0.647826\pi$
$673$	−23.2388	−0.895791	−0.447895	+	0.894086i	$0.647826\pi$
$674$	11.2388	0.432903
$675$	0	0
$676$	0	0
$677$	47.2629	1.81646	0.908231	−	0.418470i	$-0.137433\pi$
$677$	47.2629	1.81646	0.908231	+	0.418470i	$0.137433\pi$
$678$	−11.5804	−0.444744
$679$	81.5308	3.12886
$680$	0	0
$681$	−62.3217	−2.38817
$682$	−3.58043	−0.137102
$683$	−6.97307	−0.266817	−0.133409	−	0.991061i	$-0.542592\pi$
$683$	−6.97307	−0.266817	−0.133409	+	0.991061i	$0.542592\pi$
$684$	5.23672	0.200231
$685$	0	0
$686$	−22.7792	−0.869714
$687$	7.61836	0.290658
$688$	−4.68532	−0.178626
$689$	0	0
$690$	0	0
$691$	32.2899	1.22836	0.614182	−	0.789164i	$-0.289486\pi$
$691$	32.2899	1.22836	0.614182	+	0.789164i	$0.289486\pi$
$692$	19.7902	0.752311
$693$	47.1609	1.79149
$694$	16.9490	0.643374
$695$	0	0
$696$	2.81714	0.106784
$697$	28.2678	1.07072
$698$	10.2119	0.386526
$699$	84.0421	3.17877
$700$	0	0
$701$	−0.521643	−0.0197022	−0.00985109	−	0.999951i	$-0.503136\pi$
$701$	−0.521643	−0.0197022	−0.00985109	+	0.999951i	$0.503136\pi$
$702$	0	0
$703$	−0.575502	−0.0217055
$704$	−2.00000	−0.0753778
$705$	0	0
$706$	−14.9511	−0.562691
$707$	−62.8490	−2.36368
$708$	2.81714	0.105875
$709$	33.2147	1.24740	0.623702	−	0.781662i	$-0.285628\pi$
$709$	33.2147	1.24740	0.623702	+	0.781662i	$0.285628\pi$
$710$	0	0
$711$	59.0531	2.21467
$712$	−0.973070	−0.0364674
$713$	−13.9461	−0.522287
$714$	74.4397	2.78584
$715$	0	0
$716$	6.17185	0.230653
$717$	−6.17185	−0.230492
$718$	−11.8441	−0.442017
$719$	−4.26365	−0.159007	−0.0795036	−	0.996835i	$-0.525334\pi$
$719$	−4.26365	−0.159007	−0.0795036	+	0.996835i	$0.525334\pi$
$720$	0	0
$721$	42.6612	1.58879
$722$	−18.0531	−0.671868
$723$	−65.1388	−2.42254
$724$	−19.3706	−0.719904
$725$	0	0
$726$	−20.2657	−0.752132
$727$	−47.5266	−1.76266	−0.881331	−	0.472499i	$-0.843352\pi$
$727$	−47.5266	−1.76266	−0.881331	+	0.472499i	$0.843352\pi$
$728$	0	0
$729$	−39.3437	−1.45717
$730$	0	0
$731$	27.4943	1.01691
$732$	−2.36571	−0.0874393
$733$	44.8593	1.65692	0.828458	−	0.560052i	$-0.189219\pi$
$733$	44.8593	1.65692	0.828458	+	0.560052i	$0.189219\pi$
$734$	−16.0539	−0.592559
$735$	0	0
$736$	−7.79021	−0.287151
$737$	−3.58043	−0.131887
$738$	−25.9241	−0.954281
$739$	18.4514	0.678747	0.339373	−	0.940652i	$-0.389785\pi$
$739$	18.4514	0.678747	0.339373	+	0.940652i	$0.389785\pi$
$740$	0	0
$741$	0	0
$742$	34.1339	1.25310
$743$	−14.2201	−0.521684	−0.260842	−	0.965382i	$-0.584000\pi$
$743$	−14.2201	−0.521684	−0.260842	+	0.965382i	$0.584000\pi$
$744$	5.18286	0.190013
$745$	0	0
$746$	6.87100	0.251565
$747$	−37.5266	−1.37303
$748$	11.7364	0.429124
$749$	−25.3706	−0.927023
$750$	0	0
$751$	−28.7951	−1.05075	−0.525375	−	0.850871i	$-0.676075\pi$
$751$	−28.7951	−1.05075	−0.525375	+	0.850871i	$0.676075\pi$
$752$	0.381642	0.0139170
$753$	−1.21471	−0.0442665
$754$	0	0
$755$	0	0
$756$	−30.2119	−1.09880
$757$	−13.7364	−0.499256	−0.249628	−	0.968342i	$-0.580308\pi$
$757$	−13.7364	−0.499256	−0.249628	+	0.968342i	$0.580308\pi$
$758$	−1.84407	−0.0669797
$759$	45.1070	1.63728
$760$	0	0
$761$	−5.39264	−0.195483	−0.0977416	−	0.995212i	$-0.531162\pi$
$761$	−5.39264	−0.195483	−0.0977416	+	0.995212i	$0.531162\pi$
$762$	−34.1339	−1.23654
$763$	−3.00282	−0.108710
$764$	−3.73635	−0.135177
$765$	0	0
$766$	−9.25264	−0.334312
$767$	0	0
$768$	2.89511	0.104468
$769$	2.34371	0.0845163	0.0422582	−	0.999107i	$-0.486545\pi$
$769$	2.34371	0.0845163	0.0422582	+	0.999107i	$0.486545\pi$
$770$	0	0
$771$	−56.9131	−2.04968
$772$	15.3706	0.553201
$773$	−29.3547	−1.05582	−0.527908	−	0.849302i	$-0.677023\pi$
$773$	−29.3547	−1.05582	−0.527908	+	0.849302i	$0.677023\pi$
$774$	−25.2147	−0.906324
$775$	0	0
$776$	−18.6074	−0.667965
$777$	7.50246	0.269149
$778$	−21.2686	−0.762515
$779$	−4.68742	−0.167944
$780$	0	0
$781$	−7.84407	−0.280683
$782$	45.7143	1.63474
$783$	6.70942	0.239775
$784$	12.1988	0.435671
$785$	0	0
$786$	−49.1850	−1.75437
$787$	34.6336	1.23455	0.617277	−	0.786746i	$-0.288236\pi$
$787$	34.6336	1.23455	0.617277	+	0.786746i	$0.288236\pi$
$788$	26.9890	0.961443
$789$	31.1609	1.10936
$790$	0	0
$791$	17.5266	0.623173
$792$	−10.7633	−0.382457
$793$	0	0
$794$	−33.4727	−1.18790
$795$	0	0
$796$	23.3168	0.826441
$797$	−27.8221	−0.985508	−0.492754	−	0.870169i	$-0.664010\pi$
$797$	−27.8221	−0.985508	−0.492754	+	0.870169i	$0.664010\pi$
$798$	−12.3437	−0.436963
$799$	−2.23954	−0.0792293
$800$	0	0
$801$	−5.23672	−0.185030
$802$	−17.7364	−0.626292
$803$	12.0000	0.423471
$804$	5.18286	0.182785
$805$	0	0
$806$	0	0
$807$	−42.7413	−1.50456
$808$	14.3437	0.504610
$809$	−23.1768	−0.814852	−0.407426	−	0.913238i	$-0.633574\pi$
$809$	−23.1768	−0.814852	−0.407426	+	0.913238i	$0.633574\pi$
$810$	0	0
$811$	−36.0857	−1.26714	−0.633570	−	0.773685i	$-0.718411\pi$
$811$	−36.0857	−1.26714	−0.633570	+	0.773685i	$0.718411\pi$
$812$	−4.26365	−0.149625
$813$	78.8593	2.76572
$814$	1.18286	0.0414591
$815$	0	0
$816$	−16.9890	−0.594734
$817$	−4.55915	−0.159504
$818$	−7.68249	−0.268612
$819$	0	0
$820$	0	0
$821$	−38.3196	−1.33736	−0.668682	−	0.743549i	$-0.733141\pi$
$821$	−38.3196	−1.33736	−0.668682	+	0.743549i	$0.733141\pi$
$822$	12.1878	0.425098
$823$	−53.3486	−1.85962	−0.929808	−	0.368044i	$-0.880027\pi$
$823$	−53.3486	−1.85962	−0.929808	+	0.368044i	$0.880027\pi$
$824$	−9.73635	−0.339182
$825$	0	0
$826$	−4.26365	−0.148351
$827$	14.5053	0.504398	0.252199	−	0.967675i	$-0.418846\pi$
$827$	14.5053	0.504398	0.252199	+	0.967675i	$0.418846\pi$
$828$	−41.9241	−1.45696
$829$	−13.3926	−0.465146	−0.232573	−	0.972579i	$-0.574714\pi$
$829$	−13.3926	−0.465146	−0.232573	+	0.972579i	$0.574714\pi$
$830$	0	0
$831$	8.31186	0.288335
$832$	0	0
$833$	−71.5846	−2.48026
$834$	−17.8682	−0.618724
$835$	0	0
$836$	−1.94614	−0.0673087
$837$	12.3437	0.426661
$838$	11.8061	0.407836
$839$	−2.52164	−0.0870568	−0.0435284	−	0.999052i	$-0.513860\pi$
$839$	−2.52164	−0.0870568	−0.0435284	+	0.999052i	$0.513860\pi$
$840$	0	0
$841$	−28.0531	−0.967349
$842$	26.3678	0.908695
$843$	−15.6123	−0.537715
$844$	15.3547	0.528531
$845$	0	0
$846$	2.05386	0.0706131
$847$	30.6715	1.05388
$848$	−7.79021	−0.267517
$849$	72.2360	2.47913
$850$	0	0
$851$	4.60736	0.157938
$852$	11.3547	0.389006
$853$	36.4078	1.24658	0.623290	−	0.781990i	$-0.285795\pi$
$853$	36.4078	1.24658	0.623290	+	0.781990i	$0.285795\pi$
$854$	3.58043	0.122520
$855$	0	0
$856$	5.79021	0.197905
$857$	−21.4188	−0.731654	−0.365827	−	0.930683i	$-0.619214\pi$
$857$	−21.4188	−0.731654	−0.365827	+	0.930683i	$0.619214\pi$
$858$	0	0
$859$	−41.9461	−1.43118	−0.715592	−	0.698519i	$-0.753843\pi$
$859$	−41.9461	−1.43118	−0.715592	+	0.698519i	$0.753843\pi$
$860$	0	0
$861$	61.1070	2.08252
$862$	−1.71225	−0.0583194
$863$	38.3278	1.30469	0.652346	−	0.757921i	$-0.273785\pi$
$863$	38.3278	1.30469	0.652346	+	0.757921i	$0.273785\pi$
$864$	6.89511	0.234576
$865$	0	0
$866$	11.9220	0.405127
$867$	50.4776	1.71431
$868$	−7.84407	−0.266245
$869$	−21.9461	−0.744472
$870$	0	0
$871$	0	0
$872$	0.685320	0.0232078
$873$	−100.138	−3.38916
$874$	−7.58043	−0.256412
$875$	0	0
$876$	−17.3706	−0.586900
$877$	−43.3327	−1.46324	−0.731621	−	0.681712i	$-0.761236\pi$
$877$	−43.3327	−1.46324	−0.731621	+	0.681712i	$0.761236\pi$
$878$	−12.3437	−0.416580
$879$	72.1899	2.43490
$880$	0	0
$881$	1.09107	0.0367590	0.0183795	−	0.999831i	$-0.494149\pi$
$881$	1.09107	0.0367590	0.0183795	+	0.999831i	$0.494149\pi$
$882$	65.6495	2.21053
$883$	−28.3735	−0.954843	−0.477422	−	0.878674i	$-0.658429\pi$
$883$	−28.3735	−0.954843	−0.477422	+	0.878674i	$0.658429\pi$
$884$	0	0
$885$	0	0
$886$	28.1580	0.945987
$887$	−4.50529	−0.151273	−0.0756364	−	0.997135i	$-0.524099\pi$
$887$	−4.50529	−0.151273	−0.0756364	+	0.997135i	$0.524099\pi$
$888$	−1.71225	−0.0574593
$889$	51.6605	1.73264
$890$	0	0
$891$	−7.63429	−0.255758
$892$	24.8331	0.831473
$893$	0.371364	0.0124272
$894$	0	0
$895$	0	0
$896$	−4.38164	−0.146380
$897$	0	0
$898$	29.0531	0.969516
$899$	1.74200	0.0580991
$900$	0	0
$901$	45.7143	1.52297
$902$	9.63429	0.320787
$903$	59.4348	1.97787
$904$	−4.00000	−0.133038
$905$	0	0
$906$	−44.8813	−1.49108
$907$	48.0503	1.59548	0.797742	−	0.602999i	$-0.206028\pi$
$907$	48.0503	1.59548	0.797742	+	0.602999i	$0.206028\pi$
$908$	−21.5266	−0.714384
$909$	77.1927	2.56032
$910$	0	0
$911$	12.5755	0.416645	0.208322	−	0.978060i	$-0.433200\pi$
$911$	12.5755	0.416645	0.208322	+	0.978060i	$0.433200\pi$
$912$	2.81714	0.0932849
$913$	13.9461	0.461550
$914$	−11.5266	−0.381265
$915$	0	0
$916$	2.63146	0.0869459
$917$	74.4397	2.45822
$918$	−40.4617	−1.33544
$919$	23.6881	0.781400	0.390700	−	0.920518i	$-0.372233\pi$
$919$	23.6881	0.781400	0.390700	+	0.920518i	$0.372233\pi$
$920$	0	0
$921$	−42.8972	−1.41351
$922$	−28.4217	−0.936018
$923$	0	0
$924$	25.3706	0.834633
$925$	0	0
$926$	−27.4727	−0.902809
$927$	−52.3976	−1.72096
$928$	0.973070	0.0319426
$929$	−45.0850	−1.47919	−0.739595	−	0.673052i	$-0.764983\pi$
$929$	−45.0850	−1.47919	−0.739595	+	0.673052i	$0.764983\pi$
$930$	0	0
$931$	11.8703	0.389033
$932$	29.0290	0.950877
$933$	37.2629	1.21993
$934$	18.2098	0.595842
$935$	0	0
$936$	0	0
$937$	−28.7951	−0.940696	−0.470348	−	0.882481i	$-0.655872\pi$
$937$	−28.7951	−0.940696	−0.470348	+	0.882481i	$0.655872\pi$
$938$	−7.84407	−0.256118
$939$	−57.6764	−1.88220
$940$	0	0
$941$	2.84690	0.0928062	0.0464031	−	0.998923i	$-0.485224\pi$
$941$	2.84690	0.0928062	0.0464031	+	0.998923i	$0.485224\pi$
$942$	−25.9780	−0.846409
$943$	37.5266	1.22203
$944$	0.973070	0.0316707
$945$	0	0
$946$	9.37064	0.304666
$947$	11.3168	0.367746	0.183873	−	0.982950i	$-0.441136\pi$
$947$	11.3168	0.367746	0.183873	+	0.982950i	$0.441136\pi$
$948$	31.7682	1.03178
$949$	0	0
$950$	0	0
$951$	−11.4245	−0.370465
$952$	25.7122	0.833339
$953$	28.4535	0.921700	0.460850	−	0.887478i	$-0.347545\pi$
$953$	28.4535	0.921700	0.460850	+	0.887478i	$0.347545\pi$
$954$	−41.9241	−1.35734
$955$	0	0
$956$	−2.13182	−0.0689481
$957$	−5.63429	−0.182131
$958$	33.6045	1.08571
$959$	−18.4458	−0.595645
$960$	0	0
$961$	−27.7951	−0.896617
$962$	0	0
$963$	31.1609	1.00414
$964$	−22.4996	−0.724665
$965$	0	0
$966$	98.8213	3.17952
$967$	−1.48863	−0.0478713	−0.0239356	−	0.999714i	$-0.507620\pi$
$967$	−1.48863	−0.0478713	−0.0239356	+	0.999714i	$0.507620\pi$
$968$	−7.00000	−0.224989
$969$	−16.5315	−0.531068
$970$	0	0
$971$	49.6446	1.59317	0.796585	−	0.604527i	$-0.206638\pi$
$971$	49.6446	1.59317	0.796585	+	0.604527i	$0.206638\pi$
$972$	−9.63429	−0.309020
$973$	27.0429	0.866954
$974$	8.00000	0.256337
$975$	0	0
$976$	−0.817143	−0.0261561
$977$	−39.6066	−1.26713	−0.633564	−	0.773690i	$-0.718409\pi$
$977$	−39.6066	−1.26713	−0.633564	+	0.773690i	$0.718409\pi$
$978$	−36.4996	−1.16713
$979$	1.94614	0.0621989
$980$	0	0
$981$	3.68814	0.117753
$982$	−19.8061	−0.632039
$983$	−0.725351	−0.0231351	−0.0115676	−	0.999933i	$-0.503682\pi$
$983$	−0.725351	−0.0231351	−0.0115676	+	0.999933i	$0.503682\pi$
$984$	−13.9461	−0.444587
$985$	0	0
$986$	−5.71015	−0.181848
$987$	−4.84125	−0.154099
$988$	0	0
$989$	36.4996	1.16062
$990$	0	0
$991$	−35.5046	−1.12784	−0.563920	−	0.825830i	$-0.690707\pi$
$991$	−35.5046	−1.12784	−0.563920	+	0.825830i	$0.690707\pi$
$992$	1.79021	0.0568393
$993$	−62.0262	−1.96834
$994$	−17.1850	−0.545074
$995$	0	0
$996$	−20.1878	−0.639674
$997$	−10.4196	−0.329991	−0.164996	−	0.986294i	$-0.552761\pi$
$997$	−10.4196	−0.329991	−0.164996	+	0.986294i	$0.552761\pi$
$998$	−9.12900	−0.288973
$999$	−4.07796	−0.129021

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8450.2.a.cc.1.3		3
5.2	odd	4		1690.2.b.a.339.4		6
5.3	odd	4		1690.2.b.a.339.3		6
5.4	even	2		8450.2.a.bs.1.1		3
13.12	even	2		650.2.a.n.1.3		3
39.38	odd	2		5850.2.a.cs.1.3		3
52.51	odd	2		5200.2.a.ce.1.1		3
65.8	even	4		1690.2.c.d.1689.6		6
65.12	odd	4		130.2.b.a.79.1	✓	6
65.18	even	4		1690.2.c.a.1689.6		6
65.38	odd	4		130.2.b.a.79.6	yes	6
65.47	even	4		1690.2.c.a.1689.1		6
65.57	even	4		1690.2.c.d.1689.1		6
65.64	even	2		650.2.a.o.1.1		3
195.38	even	4		1170.2.e.f.469.1		6
195.77	even	4		1170.2.e.f.469.4		6
195.194	odd	2		5850.2.a.cp.1.1		3
260.103	even	4		1040.2.d.b.209.1		6
260.207	even	4		1040.2.d.b.209.6		6
260.259	odd	2		5200.2.a.cf.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
130.2.b.a.79.1	✓	6	65.12	odd	4
130.2.b.a.79.6	yes	6	65.38	odd	4
650.2.a.n.1.3		3	13.12	even	2
650.2.a.o.1.1		3	65.64	even	2
1040.2.d.b.209.1		6	260.103	even	4
1040.2.d.b.209.6		6	260.207	even	4
1170.2.e.f.469.1		6	195.38	even	4
1170.2.e.f.469.4		6	195.77	even	4
1690.2.b.a.339.3		6	5.3	odd	4
1690.2.b.a.339.4		6	5.2	odd	4
1690.2.c.a.1689.1		6	65.47	even	4
1690.2.c.a.1689.6		6	65.18	even	4
1690.2.c.d.1689.1		6	65.57	even	4
1690.2.c.d.1689.6		6	65.8	even	4
5200.2.a.ce.1.1		3	52.51	odd	2
5200.2.a.cf.1.3		3	260.259	odd	2
5850.2.a.cp.1.1		3	195.194	odd	2
5850.2.a.cs.1.3		3	39.38	odd	2
8450.2.a.bs.1.1		3	5.4	even	2
8450.2.a.cc.1.3		3	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:	\( N \)	\(=\)	\( 8450 = 2 \cdot 5^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8450.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +2.89511 q^{3} +1.00000 q^{4} +2.89511 q^{6} -4.38164 q^{7} +1.00000 q^{8} +5.38164 q^{9} -2.00000 q^{11} +2.89511 q^{12} -4.38164 q^{14} +1.00000 q^{16} -5.86818 q^{17} +5.38164 q^{18} +0.973070 q^{19} -12.6853 q^{21} -2.00000 q^{22} -7.79021 q^{23} +2.89511 q^{24} +6.89511 q^{27} -4.38164 q^{28} +0.973070 q^{29} +1.79021 q^{31} +1.00000 q^{32} -5.79021 q^{33} -5.86818 q^{34} +5.38164 q^{36} -0.591429 q^{37} +0.973070 q^{38} -4.81714 q^{41} -12.6853 q^{42} -4.68532 q^{43} -2.00000 q^{44} -7.79021 q^{46} +0.381642 q^{47} +2.89511 q^{48} +12.1988 q^{49} -16.9890 q^{51} -7.79021 q^{53} +6.89511 q^{54} -4.38164 q^{56} +2.81714 q^{57} +0.973070 q^{58} +0.973070 q^{59} -0.817143 q^{61} +1.79021 q^{62} -23.5804 q^{63} +1.00000 q^{64} -5.79021 q^{66} +1.79021 q^{67} -5.86818 q^{68} -22.5535 q^{69} +3.92204 q^{71} +5.38164 q^{72} -6.00000 q^{73} -0.591429 q^{74} +0.973070 q^{76} +8.76328 q^{77} +10.9731 q^{79} +3.81714 q^{81} -4.81714 q^{82} -6.97307 q^{83} -12.6853 q^{84} -4.68532 q^{86} +2.81714 q^{87} -2.00000 q^{88} -0.973070 q^{89} -7.79021 q^{92} +5.18286 q^{93} +0.381642 q^{94} +2.89511 q^{96} -18.6074 q^{97} +12.1988 q^{98} -10.7633 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} - 2 q^{7} + 3 q^{8} + 5 q^{9} - 6 q^{11} - 2 q^{14} + 3 q^{16} - 4 q^{17} + 5 q^{18} - 2 q^{19} - 12 q^{21} - 6 q^{22} - 6 q^{23} + 12 q^{27} - 2 q^{28} - 2 q^{29} - 12 q^{31} + 3 q^{32}+ \cdots - 10 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 - 2 * q^7 + 3 * q^8 + 5 * q^9 - 6 * q^11 - 2 * q^14 + 3 * q^16 - 4 * q^17 + 5 * q^18 - 2 * q^19 - 12 * q^21 - 6 * q^22 - 6 * q^23 + 12 * q^27 - 2 * q^28 - 2 * q^29 - 12 * q^31 + 3 * q^32 - 4 * q^34 + 5 * q^36 - 8 * q^37 - 2 * q^38 - 2 * q^41 - 12 * q^42 + 12 * q^43 - 6 * q^44 - 6 * q^46 - 10 * q^47 + 13 * q^49 - 10 * q^51 - 6 * q^53 + 12 * q^54 - 2 * q^56 - 4 * q^57 - 2 * q^58 - 2 * q^59 + 10 * q^61 - 12 * q^62 - 36 * q^63 + 3 * q^64 - 12 * q^67 - 4 * q^68 - 28 * q^69 + 8 * q^71 + 5 * q^72 - 18 * q^73 - 8 * q^74 - 2 * q^76 + 4 * q^77 + 28 * q^79 - q^81 - 2 * q^82 - 16 * q^83 - 12 * q^84 + 12 * q^86 - 4 * q^87 - 6 * q^88 + 2 * q^89 - 6 * q^92 + 28 * q^93 - 10 * q^94 - 26 * q^97 + 13 * q^98 - 10 * q^99

Introduction

L-functions

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Database

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