LMFDB - Embedded newform 9075.2.a.dq.1.6

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:	$0$
Dimension:	$6$
Coefficient field:	6.6.437199552.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 13x^{4} + 49x^{2} - 48 $ x^6 - 13x^4 + 49x^2 - 48
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
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			1.6
Root			$2.63162$ 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)

$f(q)$	$=$	$q+2.63162 q^{2} +1.00000 q^{3} +4.92542 q^{4} +2.63162 q^{6} -4.16741 q^{7} +7.69860 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+2.63162 q^{2} +1.00000 q^{3} +4.92542 q^{4} +2.63162 q^{6} -4.16741 q^{7} +7.69860 q^{8} +1.00000 q^{9} +4.92542 q^{12} +5.26324 q^{13} -10.9670 q^{14} +10.4089 q^{16} +4.23450 q^{17} +2.63162 q^{18} +2.50245 q^{19} -4.16741 q^{21} -5.48352 q^{23} +7.69860 q^{24} +13.8508 q^{26} +1.00000 q^{27} -20.5263 q^{28} -5.26324 q^{29} +10.1162 q^{31} +11.9952 q^{32} +11.1436 q^{34} +4.92542 q^{36} +4.48352 q^{37} +6.58549 q^{38} +5.26324 q^{39} +3.46410 q^{41} -10.9670 q^{42} -6.66986 q^{43} -14.4305 q^{46} -4.21817 q^{47} +10.4089 q^{48} +10.3673 q^{49} +4.23450 q^{51} +25.9237 q^{52} +3.63268 q^{53} +2.63162 q^{54} -32.0832 q^{56} +2.50245 q^{57} -13.8508 q^{58} -6.58549 q^{59} +12.2585 q^{61} +26.6220 q^{62} -4.16741 q^{63} +10.7489 q^{64} +4.48352 q^{67} +20.8567 q^{68} -5.48352 q^{69} +1.26535 q^{71} +7.69860 q^{72} +4.55993 q^{73} +11.7989 q^{74} +12.3256 q^{76} +13.8508 q^{78} +6.86112 q^{79} +1.00000 q^{81} +9.11620 q^{82} -4.87072 q^{83} -20.5263 q^{84} -17.5525 q^{86} -5.26324 q^{87} +15.1162 q^{89} -21.9341 q^{91} -27.0087 q^{92} +10.1162 q^{93} -11.1006 q^{94} +11.9952 q^{96} +16.1852 q^{97} +27.2829 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{3} + 14 q^{4} + 6 q^{9}+O(q^{10}) $ 6 * q + 6 * q^3 + 14 * q^4 + 6 * q^9	$ 6 q + 6 q^{3} + 14 q^{4} + 6 q^{9} + 14 q^{12} + 8 q^{14} + 10 q^{16} + 4 q^{23} + 52 q^{26} + 6 q^{27} + 18 q^{31} + 26 q^{34} + 14 q^{36} - 10 q^{37} + 20 q^{38} + 8 q^{42} + 10 q^{48} + 68 q^{49} + 16 q^{53} - 76 q^{56} - 52 q^{58} - 20 q^{59} + 16 q^{64} - 10 q^{67} + 4 q^{69} - 4 q^{71} + 52 q^{78} + 6 q^{81} + 12 q^{82} - 12 q^{86} + 48 q^{89} + 16 q^{91} - 30 q^{92} + 18 q^{93} - 2 q^{97}+O(q^{100}) $ 6 * q + 6 * q^3 + 14 * q^4 + 6 * q^9 + 14 * q^12 + 8 * q^14 + 10 * q^16 + 4 * q^23 + 52 * q^26 + 6 * q^27 + 18 * q^31 + 26 * q^34 + 14 * q^36 - 10 * q^37 + 20 * q^38 + 8 * q^42 + 10 * q^48 + 68 * q^49 + 16 * q^53 - 76 * q^56 - 52 * q^58 - 20 * q^59 + 16 * q^64 - 10 * q^67 + 4 * q^69 - 4 * q^71 + 52 * q^78 + 6 * q^81 + 12 * q^82 - 12 * q^86 + 48 * q^89 + 16 * q^91 - 30 * q^92 + 18 * q^93 - 2 * q^97

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 $

$2$	2.63162	1.86084	0.930418	−	0.366500i	$-0.119444\pi$
$2$	2.63162	1.86084	0.930418	+	0.366500i	$0.119444\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	4.92542	2.46271
$5$	0	0
$5$	0	0
$6$	2.63162	1.07435
$7$	−4.16741	−1.57513	−0.787567	−	0.616229i	$-0.788659\pi$
$7$	−4.16741	−1.57513	−0.787567	+	0.616229i	$0.788659\pi$
$8$	7.69860	2.72187
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	4.92542	1.42185
$13$	5.26324	1.45976	0.729880	−	0.683575i	$-0.239576\pi$
$13$	5.26324	1.45976	0.729880	+	0.683575i	$0.239576\pi$
$14$	−10.9670	−2.93107
$15$	0	0
$16$	10.4089	2.60224
$17$	4.23450	1.02702	0.513508	−	0.858085i	$-0.328345\pi$
$17$	4.23450	1.02702	0.513508	+	0.858085i	$0.328345\pi$
$18$	2.63162	0.620279
$19$	2.50245	0.574101	0.287051	−	0.957915i	$-0.407325\pi$
$19$	2.50245	0.574101	0.287051	+	0.957915i	$0.407325\pi$
$20$	0	0
$21$	−4.16741	−0.909404
$22$	0	0
$23$	−5.48352	−1.14339	−0.571697	−	0.820465i	$-0.693715\pi$
$23$	−5.48352	−1.14339	−0.571697	+	0.820465i	$0.693715\pi$
$24$	7.69860	1.57147
$25$	0	0
$26$	13.8508	2.71637
$27$	1.00000	0.192450
$28$	−20.5263	−3.87910
$29$	−5.26324	−0.977359	−0.488680	−	0.872463i	$-0.662521\pi$
$29$	−5.26324	−0.977359	−0.488680	+	0.872463i	$0.662521\pi$
$30$	0	0
$31$	10.1162	1.81692	0.908461	−	0.417969i	$-0.137258\pi$
$31$	10.1162	1.81692	0.908461	+	0.417969i	$0.137258\pi$
$32$	11.9952	2.12047
$33$	0	0
$34$	11.1436	1.91111
$35$	0	0
$36$	4.92542	0.820904
$37$	4.48352	0.737087	0.368543	−	0.929611i	$-0.379857\pi$
$37$	4.48352	0.737087	0.368543	+	0.929611i	$0.379857\pi$
$38$	6.58549	1.06831
$39$	5.26324	0.842793
$40$	0	0
$41$	3.46410	0.541002	0.270501	−	0.962720i	$-0.412811\pi$
$41$	3.46410	0.541002	0.270501	+	0.962720i	$0.412811\pi$
$42$	−10.9670	−1.69225
$43$	−6.66986	−1.01714	−0.508572	−	0.861019i	$-0.669826\pi$
$43$	−6.66986	−1.01714	−0.508572	+	0.861019i	$0.669826\pi$
$44$	0	0
$45$	0	0
$46$	−14.4305	−2.12767
$47$	−4.21817	−0.615283	−0.307642	−	0.951502i	$-0.599540\pi$
$47$	−4.21817	−0.615283	−0.307642	+	0.951502i	$0.599540\pi$
$48$	10.4089	1.50240
$49$	10.3673	1.48105
$50$	0	0
$51$	4.23450	0.592949
$52$	25.9237	3.59497
$53$	3.63268	0.498986	0.249493	−	0.968377i	$-0.419736\pi$
$53$	3.63268	0.498986	0.249493	+	0.968377i	$0.419736\pi$
$54$	2.63162	0.358118
$55$	0	0
$56$	−32.0832	−4.28730
$57$	2.50245	0.331457
$58$	−13.8508	−1.81871
$59$	−6.58549	−0.857358	−0.428679	−	0.903457i	$-0.641021\pi$
$59$	−6.58549	−0.857358	−0.428679	+	0.903457i	$0.641021\pi$
$60$	0	0
$61$	12.2585	1.56954	0.784772	−	0.619785i	$-0.212780\pi$
$61$	12.2585	1.56954	0.784772	+	0.619785i	$0.212780\pi$
$62$	26.6220	3.38100
$63$	−4.16741	−0.525045
$64$	10.7489	1.34361
$65$	0	0
$66$	0	0
$67$	4.48352	0.547749	0.273875	−	0.961765i	$-0.411695\pi$
$67$	4.48352	0.547749	0.273875	+	0.961765i	$0.411695\pi$
$68$	20.8567	2.52925
$69$	−5.48352	−0.660138
$70$	0	0
$71$	1.26535	0.150170	0.0750849	−	0.997177i	$-0.476077\pi$
$71$	1.26535	0.150170	0.0750849	+	0.997177i	$0.476077\pi$
$72$	7.69860	0.907289
$73$	4.55993	0.533699	0.266850	−	0.963738i	$-0.414017\pi$
$73$	4.55993	0.533699	0.266850	+	0.963738i	$0.414017\pi$
$74$	11.7989	1.37160
$75$	0	0
$76$	12.3256	1.41385
$77$	0	0
$78$	13.8508	1.56830
$79$	6.86112	0.771936	0.385968	−	0.922512i	$-0.373868\pi$
$79$	6.86112	0.771936	0.385968	+	0.922512i	$0.373868\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	9.11620	1.00672
$83$	−4.87072	−0.534631	−0.267316	−	0.963609i	$-0.586137\pi$
$83$	−4.87072	−0.534631	−0.267316	+	0.963609i	$0.586137\pi$
$84$	−20.5263	−2.23960
$85$	0	0
$86$	−17.5525	−1.89274
$87$	−5.26324	−0.564279
$88$	0	0
$89$	15.1162	1.60231	0.801157	−	0.598454i	$-0.204218\pi$
$89$	15.1162	1.60231	0.801157	+	0.598454i	$0.204218\pi$
$90$	0	0
$91$	−21.9341	−2.29932
$92$	−27.0087	−2.81585
$93$	10.1162	1.04900
$94$	−11.1006	−1.14494
$95$	0	0
$96$	11.9952	1.22425
$97$	16.1852	1.64336	0.821680	−	0.569949i	$-0.193037\pi$
$97$	16.1852	1.64336	0.821680	+	0.569949i	$0.193037\pi$
$98$	27.2829	2.75598
$99$	0	0
$100$	0	0
$101$	3.59828	0.358042	0.179021	−	0.983845i	$-0.442707\pi$
$101$	3.59828	0.358042	0.179021	+	0.983845i	$0.442707\pi$
$102$	11.1436	1.10338
$103$	−1.36732	−0.134726	−0.0673632	−	0.997729i	$-0.521459\pi$
$103$	−1.36732	−0.134726	−0.0673632	+	0.997729i	$0.521459\pi$
$104$	40.5196	3.97327
$105$	0	0
$106$	9.55982	0.928532
$107$	−12.9618	−1.25307	−0.626534	−	0.779394i	$-0.715527\pi$
$107$	−12.9618	−1.25307	−0.626534	+	0.779394i	$0.715527\pi$
$108$	4.92542	0.473949
$109$	5.96655	0.571492	0.285746	−	0.958305i	$-0.407759\pi$
$109$	5.96655	0.571492	0.285746	+	0.958305i	$0.407759\pi$
$110$	0	0
$111$	4.48352	0.425557
$112$	−43.3784	−4.09887
$113$	6.06902	0.570925	0.285462	−	0.958390i	$-0.407853\pi$
$113$	6.06902	0.570925	0.285462	+	0.958390i	$0.407853\pi$
$114$	6.58549	0.616788
$115$	0	0
$116$	−25.9237	−2.40695
$117$	5.26324	0.486587
$118$	−17.3305	−1.59540
$119$	−17.6469	−1.61769
$120$	0	0
$121$	0	0
$122$	32.2598	2.92066
$123$	3.46410	0.312348
$124$	49.8266	4.47456
$125$	0	0
$126$	−10.9670	−0.977022
$127$	−9.82317	−0.871665	−0.435833	−	0.900028i	$-0.643546\pi$
$127$	−9.82317	−0.871665	−0.435833	+	0.900028i	$0.643546\pi$
$128$	4.29658	0.379768
$129$	−6.66986	−0.587248
$130$	0	0
$131$	3.46410	0.302660	0.151330	−	0.988483i	$-0.451644\pi$
$131$	3.46410	0.302660	0.151330	+	0.988483i	$0.451644\pi$
$132$	0	0
$133$	−10.4287	−0.904286
$134$	11.7989	1.01927
$135$	0	0
$136$	32.5997	2.79540
$137$	−20.5997	−1.75995	−0.879976	−	0.475017i	$-0.842442\pi$
$137$	−20.5997	−1.75995	−0.879976	+	0.475017i	$0.842442\pi$
$138$	−14.4305	−1.22841
$139$	−17.6984	−1.50116	−0.750579	−	0.660781i	$-0.770225\pi$
$139$	−17.6984	−1.50116	−0.750579	+	0.660781i	$0.770225\pi$
$140$	0	0
$141$	−4.21817	−0.355234
$142$	3.32993	0.279441
$143$	0	0
$144$	10.4089	0.867412
$145$	0	0
$146$	12.0000	0.993127
$147$	10.3673	0.855082
$148$	22.0832	1.81523
$149$	−20.5263	−1.68158	−0.840789	−	0.541363i	$-0.817908\pi$
$149$	−20.5263	−1.68158	−0.840789	+	0.541363i	$0.817908\pi$
$150$	0	0
$151$	−3.84198	−0.312656	−0.156328	−	0.987705i	$-0.549966\pi$
$151$	−3.84198	−0.312656	−0.156328	+	0.987705i	$0.549966\pi$
$152$	19.2654	1.56263
$153$	4.23450	0.342339
$154$	0	0
$155$	0	0
$156$	25.9237	2.07556
$157$	0.781830	0.0623969	0.0311984	−	0.999513i	$-0.490068\pi$
$157$	0.781830	0.0623969	0.0311984	+	0.999513i	$0.490068\pi$
$158$	18.0558	1.43645
$159$	3.63268	0.288090
$160$	0	0
$161$	22.8521	1.80100
$162$	2.63162	0.206760
$163$	7.74887	0.606939	0.303469	−	0.952841i	$-0.401855\pi$
$163$	7.74887	0.606939	0.303469	+	0.952841i	$0.401855\pi$
$164$	17.0622	1.33233
$165$	0	0
$166$	−12.8179	−0.994861
$167$	−14.8851	−1.15185	−0.575924	−	0.817504i	$-0.695357\pi$
$167$	−14.8851	−1.15185	−0.575924	+	0.817504i	$0.695357\pi$
$168$	−32.0832	−2.47528
$169$	14.7017	1.13090
$170$	0	0
$171$	2.50245	0.191367
$172$	−32.8519	−2.50493
$173$	−2.19166	−0.166628	−0.0833142	−	0.996523i	$-0.526551\pi$
$173$	−2.19166	−0.166628	−0.0833142	+	0.996523i	$0.526551\pi$
$174$	−13.8508	−1.05003
$175$	0	0
$176$	0	0
$177$	−6.58549	−0.494996
$178$	39.7801	2.98164
$179$	−13.8508	−1.03526	−0.517630	−	0.855604i	$-0.673186\pi$
$179$	−13.8508	−1.03526	−0.517630	+	0.855604i	$0.673186\pi$
$180$	0	0
$181$	7.36732	0.547609	0.273804	−	0.961785i	$-0.411718\pi$
$181$	7.36732	0.547609	0.273804	+	0.961785i	$0.411718\pi$
$182$	−57.7222	−4.27865
$183$	12.2585	0.906177
$184$	−42.2154	−3.11216
$185$	0	0
$186$	26.6220	1.95202
$187$	0	0
$188$	−20.7763	−1.51527
$189$	−4.16741	−0.303135
$190$	0	0
$191$	0.679859	0.0491929	0.0245965	−	0.999697i	$-0.492170\pi$
$191$	0.679859	0.0491929	0.0245965	+	0.999697i	$0.492170\pi$
$192$	10.7489	0.775733
$193$	−4.95245	−0.356485	−0.178242	−	0.983987i	$-0.557041\pi$
$193$	−4.95245	−0.356485	−0.178242	+	0.983987i	$0.557041\pi$
$194$	42.5933	3.05802
$195$	0	0
$196$	51.0635	3.64739
$197$	−3.59828	−0.256367	−0.128183	−	0.991750i	$-0.540915\pi$
$197$	−3.59828	−0.256367	−0.128183	+	0.991750i	$0.540915\pi$
$198$	0	0
$199$	3.88380	0.275315	0.137658	−	0.990480i	$-0.456043\pi$
$199$	3.88380	0.275315	0.137658	+	0.990480i	$0.456043\pi$
$200$	0	0
$201$	4.48352	0.316243
$202$	9.46929	0.666257
$203$	21.9341	1.53947
$204$	20.8567	1.46026
$205$	0	0
$206$	−3.59828	−0.250704
$207$	−5.48352	−0.381131
$208$	54.7848	3.79864
$209$	0	0
$210$	0	0
$211$	24.4500	1.68321	0.841603	−	0.540097i	$-0.181612\pi$
$211$	24.4500	1.68321	0.841603	+	0.540097i	$0.181612\pi$
$212$	17.8925	1.22886
$213$	1.26535	0.0867005
$214$	−34.1106	−2.33176
$215$	0	0
$216$	7.69860	0.523823
$217$	−42.1584	−2.86190
$218$	15.7017	1.06345
$219$	4.55993	0.308131
$220$	0	0
$221$	22.2872	1.49920
$222$	11.7989	0.791892
$223$	−1.59972	−0.107125	−0.0535626	−	0.998564i	$-0.517058\pi$
$223$	−1.59972	−0.107125	−0.0535626	+	0.998564i	$0.517058\pi$
$224$	−49.9889	−3.34002
$225$	0	0
$226$	15.9713	1.06240
$227$	−18.6176	−1.23569	−0.617847	−	0.786299i	$-0.711995\pi$
$227$	−18.6176	−1.23569	−0.617847	+	0.786299i	$0.711995\pi$
$228$	12.3256	0.816284
$229$	23.3344	1.54198	0.770989	−	0.636848i	$-0.219762\pi$
$229$	23.3344	1.54198	0.770989	+	0.636848i	$0.219762\pi$
$230$	0	0
$231$	0	0
$232$	−40.5196	−2.66024
$233$	−9.75608	−0.639142	−0.319571	−	0.947562i	$-0.603539\pi$
$233$	−9.75608	−0.639142	−0.319571	+	0.947562i	$0.603539\pi$
$234$	13.8508	0.905458
$235$	0	0
$236$	−32.4363	−2.11143
$237$	6.86112	0.445677
$238$	−46.4399	−3.01025
$239$	−8.59317	−0.555846	−0.277923	−	0.960603i	$-0.589646\pi$
$239$	−8.59317	−0.555846	−0.277923	+	0.960603i	$0.589646\pi$
$240$	0	0
$241$	−8.97105	−0.577876	−0.288938	−	0.957348i	$-0.593302\pi$
$241$	−8.97105	−0.577876	−0.288938	+	0.957348i	$0.593302\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	60.3784	3.86533
$245$	0	0
$246$	9.11620	0.581228
$247$	13.1710	0.838050
$248$	77.8806	4.94542
$249$	−4.87072	−0.308670
$250$	0	0
$251$	5.32014	0.335804	0.167902	−	0.985804i	$-0.446301\pi$
$251$	5.32014	0.335804	0.167902	+	0.985804i	$0.446301\pi$
$252$	−20.5263	−1.29303
$253$	0	0
$254$	−25.8508	−1.62203
$255$	0	0
$256$	−10.1908	−0.636923
$257$	−27.7707	−1.73229	−0.866145	−	0.499794i	$-0.833409\pi$
$257$	−27.7707	−1.73229	−0.866145	+	0.499794i	$0.833409\pi$
$258$	−17.5525	−1.09277
$259$	−18.6847	−1.16101
$260$	0	0
$261$	−5.26324	−0.325786
$262$	9.11620	0.563201
$263$	−5.89946	−0.363776	−0.181888	−	0.983319i	$-0.558221\pi$
$263$	−5.89946	−0.363776	−0.181888	+	0.983319i	$0.558221\pi$
$264$	0	0
$265$	0	0
$266$	−27.4445	−1.68273
$267$	15.1162	0.925096
$268$	22.0832	1.34895
$269$	−6.23240	−0.379996	−0.189998	−	0.981784i	$-0.560848\pi$
$269$	−6.23240	−0.379996	−0.189998	+	0.981784i	$0.560848\pi$
$270$	0	0
$271$	−29.7657	−1.80814	−0.904068	−	0.427389i	$-0.859433\pi$
$271$	−29.7657	−1.80814	−0.904068	+	0.427389i	$0.859433\pi$
$272$	44.0767	2.67254
$273$	−21.9341	−1.32751
$274$	−54.2106	−3.27498
$275$	0	0
$276$	−27.0087	−1.62573
$277$	−9.29648	−0.558571	−0.279286	−	0.960208i	$-0.590098\pi$
$277$	−9.29648	−0.558571	−0.279286	+	0.960208i	$0.590098\pi$
$278$	−46.5754	−2.79341
$279$	10.1162	0.605641
$280$	0	0
$281$	11.5406	0.688453	0.344227	−	0.938887i	$-0.388141\pi$
$281$	11.5406	0.688453	0.344227	+	0.938887i	$0.388141\pi$
$282$	−11.1006	−0.661032
$283$	7.50735	0.446265	0.223133	−	0.974788i	$-0.428372\pi$
$283$	7.50735	0.446265	0.223133	+	0.974788i	$0.428372\pi$
$284$	6.23240	0.369825
$285$	0	0
$286$	0	0
$287$	−14.4363	−0.852150
$288$	11.9952	0.706823
$289$	0.930985	0.0547638
$290$	0	0
$291$	16.1852	0.948794
$292$	22.4596	1.31435
$293$	18.0909	1.05688	0.528441	−	0.848970i	$-0.322777\pi$
$293$	18.0909	1.05688	0.528441	+	0.848970i	$0.322777\pi$
$294$	27.2829	1.59117
$295$	0	0
$296$	34.5168	2.00625
$297$	0	0
$298$	−54.0173	−3.12914
$299$	−28.8611	−1.66908
$300$	0	0
$301$	27.7961	1.60214
$302$	−10.1106	−0.581802
$303$	3.59828	0.206716
$304$	26.0478	1.49395
$305$	0	0
$306$	11.1436	0.637037
$307$	−3.90907	−0.223102	−0.111551	−	0.993759i	$-0.535582\pi$
$307$	−3.90907	−0.223102	−0.111551	+	0.993759i	$0.535582\pi$
$308$	0	0
$309$	−1.36732	−0.0777843
$310$	0	0
$311$	22.5196	1.27697	0.638484	−	0.769635i	$-0.279562\pi$
$311$	22.5196	1.27697	0.638484	+	0.769635i	$0.279562\pi$
$312$	40.5196	2.29397
$313$	−19.2542	−1.08831	−0.544157	−	0.838984i	$-0.683150\pi$
$313$	−19.2542	−1.08831	−0.544157	+	0.838984i	$0.683150\pi$
$314$	2.05748	0.116110
$315$	0	0
$316$	33.7939	1.90106
$317$	8.36732	0.469956	0.234978	−	0.972001i	$-0.424498\pi$
$317$	8.36732	0.469956	0.234978	+	0.972001i	$0.424498\pi$
$318$	9.55982	0.536088
$319$	0	0
$320$	0	0
$321$	−12.9618	−0.723459
$322$	60.1380	3.35136
$323$	10.5966	0.589612
$324$	4.92542	0.273635
$325$	0	0
$326$	20.3921	1.12941
$327$	5.96655	0.329951
$328$	26.6687	1.47253
$329$	17.5789	0.969153
$330$	0	0
$331$	−12.5525	−0.689950	−0.344975	−	0.938612i	$-0.612113\pi$
$331$	−12.5525	−0.689950	−0.344975	+	0.938612i	$0.612113\pi$
$332$	−23.9904	−1.31664
$333$	4.48352	0.245696
$334$	−39.1720	−2.14340
$335$	0	0
$336$	−43.3784	−2.36648
$337$	−21.3638	−1.16376	−0.581879	−	0.813275i	$-0.697682\pi$
$337$	−21.3638	−1.16376	−0.581879	+	0.813275i	$0.697682\pi$
$338$	38.6893	2.10442
$339$	6.06902	0.329624
$340$	0	0
$341$	0	0
$342$	6.58549	0.356103
$343$	−14.0330	−0.757712
$344$	−51.3486	−2.76853
$345$	0	0
$346$	−5.76760	−0.310068
$347$	−8.22529	−0.441557	−0.220778	−	0.975324i	$-0.570860\pi$
$347$	−8.22529	−0.441557	−0.220778	+	0.975324i	$0.570860\pi$
$348$	−25.9237	−1.38966
$349$	−6.11536	−0.327348	−0.163674	−	0.986515i	$-0.552334\pi$
$349$	−6.11536	−0.327348	−0.163674	+	0.986515i	$0.552334\pi$
$350$	0	0
$351$	5.26324	0.280931
$352$	0	0
$353$	23.0361	1.22609	0.613043	−	0.790050i	$-0.289945\pi$
$353$	23.0361	1.22609	0.613043	+	0.790050i	$0.289945\pi$
$354$	−17.3305	−0.921106
$355$	0	0
$356$	74.4537	3.94604
$357$	−17.6469	−0.933973
$358$	−36.4502	−1.92645
$359$	15.2630	0.805552	0.402776	−	0.915299i	$-0.368045\pi$
$359$	15.2630	0.805552	0.402776	+	0.915299i	$0.368045\pi$
$360$	0	0
$361$	−12.7378	−0.670408
$362$	19.3880	1.01901
$363$	0	0
$364$	−108.035	−5.66255
$365$	0	0
$366$	32.2598	1.68625
$367$	32.3816	1.69030	0.845152	−	0.534527i	$-0.179510\pi$
$367$	32.3816	1.69030	0.845152	+	0.534527i	$0.179510\pi$
$368$	−57.0777	−2.97538
$369$	3.46410	0.180334
$370$	0	0
$371$	−15.1389	−0.785970
$372$	49.8266	2.58339
$373$	−8.15821	−0.422416	−0.211208	−	0.977441i	$-0.567740\pi$
$373$	−8.15821	−0.422416	−0.211208	+	0.977441i	$0.567740\pi$
$374$	0	0
$375$	0	0
$376$	−32.4740	−1.67472
$377$	−27.7017	−1.42671
$378$	−10.9670	−0.564084
$379$	7.48352	0.384403	0.192201	−	0.981356i	$-0.438437\pi$
$379$	7.48352	0.384403	0.192201	+	0.981356i	$0.438437\pi$
$380$	0	0
$381$	−9.82317	−0.503256
$382$	1.78913	0.0915399
$383$	18.1380	0.926810	0.463405	−	0.886147i	$-0.346628\pi$
$383$	18.1380	0.926810	0.463405	+	0.886147i	$0.346628\pi$
$384$	4.29658	0.219259
$385$	0	0
$386$	−13.0330	−0.663360
$387$	−6.66986	−0.339048
$388$	79.7190	4.04712
$389$	17.7849	0.901732	0.450866	−	0.892592i	$-0.351115\pi$
$389$	17.7849	0.901732	0.450866	+	0.892592i	$0.351115\pi$
$390$	0	0
$391$	−23.2200	−1.17428
$392$	79.8139	4.03121
$393$	3.46410	0.174741
$394$	−9.46929	−0.477056
$395$	0	0
$396$	0	0
$397$	8.33437	0.418290	0.209145	−	0.977885i	$-0.432932\pi$
$397$	8.33437	0.418290	0.209145	+	0.977885i	$0.432932\pi$
$398$	10.2207	0.512317
$399$	−10.4287	−0.522090
$400$	0	0
$401$	−13.2654	−0.662440	−0.331220	−	0.943554i	$-0.607460\pi$
$401$	−13.2654	−0.662440	−0.331220	+	0.943554i	$0.607460\pi$
$402$	11.7989	0.588477
$403$	53.2440	2.65227
$404$	17.7230	0.881754
$405$	0	0
$406$	57.7222	2.86470
$407$	0	0
$408$	32.5997	1.61393
$409$	−16.8609	−0.833718	−0.416859	−	0.908971i	$-0.636869\pi$
$409$	−16.8609	−0.833718	−0.416859	+	0.908971i	$0.636869\pi$
$410$	0	0
$411$	−20.5997	−1.01611
$412$	−6.73465	−0.331792
$413$	27.4445	1.35045
$414$	−14.4305	−0.709223
$415$	0	0
$416$	63.1335	3.09538
$417$	−17.6984	−0.866694
$418$	0	0
$419$	9.79606	0.478569	0.239284	−	0.970950i	$-0.423087\pi$
$419$	9.79606	0.478569	0.239284	+	0.970950i	$0.423087\pi$
$420$	0	0
$421$	−17.8651	−0.870690	−0.435345	−	0.900264i	$-0.643374\pi$
$421$	−17.8651	−0.870690	−0.435345	+	0.900264i	$0.643374\pi$
$422$	64.3430	3.13217
$423$	−4.21817	−0.205094
$424$	27.9665	1.35817
$425$	0	0
$426$	3.32993	0.161335
$427$	−51.0863	−2.47224
$428$	−63.8425	−3.08595
$429$	0	0
$430$	0	0
$431$	37.1959	1.79166	0.895832	−	0.444393i	$-0.146580\pi$
$431$	37.1959	1.79166	0.895832	+	0.444393i	$0.146580\pi$
$432$	10.4089	0.500801
$433$	10.1852	0.489470	0.244735	−	0.969590i	$-0.421299\pi$
$433$	10.1852	0.489470	0.244735	+	0.969590i	$0.421299\pi$
$434$	−110.945	−5.32552
$435$	0	0
$436$	29.3878	1.40742
$437$	−13.7222	−0.656423
$438$	12.0000	0.573382
$439$	−19.7134	−0.940870	−0.470435	−	0.882435i	$-0.655903\pi$
$439$	−19.7134	−0.940870	−0.470435	+	0.882435i	$0.655903\pi$
$440$	0	0
$441$	10.3673	0.493682
$442$	58.6514	2.78976
$443$	26.4363	1.25603	0.628014	−	0.778202i	$-0.283868\pi$
$443$	26.4363	1.25603	0.628014	+	0.778202i	$0.283868\pi$
$444$	22.0832	1.04802
$445$	0	0
$446$	−4.20986	−0.199342
$447$	−20.5263	−0.970859
$448$	−44.7950	−2.11636
$449$	−21.7017	−1.02417	−0.512083	−	0.858936i	$-0.671126\pi$
$449$	−21.7017	−1.02417	−0.512083	+	0.858936i	$0.671126\pi$
$450$	0	0
$451$	0	0
$452$	29.8925	1.40602
$453$	−3.84198	−0.180512
$454$	−48.9944	−2.29942
$455$	0	0
$456$	19.2654	0.902183
$457$	−11.7989	−0.551930	−0.275965	−	0.961168i	$-0.588997\pi$
$457$	−11.7989	−0.551930	−0.275965	+	0.961168i	$0.588997\pi$
$458$	61.4072	2.86937
$459$	4.23450	0.197650
$460$	0	0
$461$	31.0527	1.44627	0.723135	−	0.690706i	$-0.242700\pi$
$461$	31.0527	1.44627	0.723135	+	0.690706i	$0.242700\pi$
$462$	0	0
$463$	−38.3704	−1.78323	−0.891613	−	0.452799i	$-0.850425\pi$
$463$	−38.3704	−1.78323	−0.891613	+	0.452799i	$0.850425\pi$
$464$	−54.7848	−2.54332
$465$	0	0
$466$	−25.6743	−1.18934
$467$	−14.9528	−0.691934	−0.345967	−	0.938247i	$-0.612449\pi$
$467$	−14.9528	−0.691934	−0.345967	+	0.938247i	$0.612449\pi$
$468$	25.9237	1.19832
$469$	−18.6847	−0.862779
$470$	0	0
$471$	0.781830	0.0360248
$472$	−50.6991	−2.33361
$473$	0	0
$474$	18.0558	0.829333
$475$	0	0
$476$	−86.9185	−3.98390
$477$	3.63268	0.166329
$478$	−22.6139	−1.03434
$479$	12.1914	0.557041	0.278521	−	0.960430i	$-0.410156\pi$
$479$	12.1914	0.557041	0.278521	+	0.960430i	$0.410156\pi$
$480$	0	0
$481$	23.5979	1.07597
$482$	−23.6084	−1.07533
$483$	22.8521	1.03981
$484$	0	0
$485$	0	0
$486$	2.63162	0.119373
$487$	−14.2872	−0.647414	−0.323707	−	0.946157i	$-0.604929\pi$
$487$	−14.2872	−0.647414	−0.323707	+	0.946157i	$0.604929\pi$
$488$	94.3735	4.27209
$489$	7.74887	0.350416
$490$	0	0
$491$	−26.0579	−1.17597	−0.587987	−	0.808870i	$-0.700079\pi$
$491$	−26.0579	−1.17597	−0.587987	+	0.808870i	$0.700079\pi$
$492$	17.0622	0.769222
$493$	−22.2872	−1.00376
$494$	34.6610	1.55947
$495$	0	0
$496$	105.299	4.72806
$497$	−5.27325	−0.236537
$498$	−12.8179	−0.574383
$499$	28.8651	1.29218	0.646089	−	0.763262i	$-0.276403\pi$
$499$	28.8651	1.29218	0.646089	+	0.763262i	$0.276403\pi$
$500$	0	0
$501$	−14.8851	−0.665019
$502$	14.0006	0.624877
$503$	−39.3729	−1.75555	−0.877776	−	0.479071i	$-0.840974\pi$
$503$	−39.3729	−1.75555	−0.877776	+	0.479071i	$0.840974\pi$
$504$	−32.0832	−1.42910
$505$	0	0
$506$	0	0
$507$	14.7017	0.652925
$508$	−48.3833	−2.14666
$509$	20.0832	0.890174	0.445087	−	0.895487i	$-0.353173\pi$
$509$	20.0832	0.890174	0.445087	+	0.895487i	$0.353173\pi$
$510$	0	0
$511$	−19.0031	−0.840648
$512$	−35.4114	−1.56498
$513$	2.50245	0.110486
$514$	−73.0819	−3.22351
$515$	0	0
$516$	−32.8519	−1.44622
$517$	0	0
$518$	−49.1710	−2.16045
$519$	−2.19166	−0.0962030
$520$	0	0
$521$	−34.7520	−1.52251	−0.761256	−	0.648452i	$-0.775417\pi$
$521$	−34.7520	−1.52251	−0.761256	+	0.648452i	$0.775417\pi$
$522$	−13.8508	−0.606235
$523$	−14.6939	−0.642519	−0.321260	−	0.946991i	$-0.604106\pi$
$523$	−14.6939	−0.642519	−0.321260	+	0.946991i	$0.604106\pi$
$524$	17.0622	0.745364
$525$	0	0
$526$	−15.5251	−0.676928
$527$	42.8370	1.86601
$528$	0	0
$529$	7.06902	0.307348
$530$	0	0
$531$	−6.58549	−0.285786
$532$	−51.3659	−2.22700
$533$	18.2324	0.789733
$534$	39.7801	1.72145
$535$	0	0
$536$	34.5168	1.49090
$537$	−13.8508	−0.597708
$538$	−16.4013	−0.707110
$539$	0	0
$540$	0	0
$541$	−3.32993	−0.143165	−0.0715824	−	0.997435i	$-0.522805\pi$
$541$	−3.32993	−0.143165	−0.0715824	+	0.997435i	$0.522805\pi$
$542$	−78.3319	−3.36464
$543$	7.36732	0.316162
$544$	50.7936	2.17776
$545$	0	0
$546$	−57.7222	−2.47028
$547$	27.4545	1.17387	0.586934	−	0.809635i	$-0.300335\pi$
$547$	27.4545	1.17387	0.586934	+	0.809635i	$0.300335\pi$
$548$	−101.462	−4.33426
$549$	12.2585	0.523181
$550$	0	0
$551$	−13.1710	−0.561103
$552$	−42.2154	−1.79681
$553$	−28.5931	−1.21590
$554$	−24.4648	−1.03941
$555$	0	0
$556$	−87.1720	−3.69692
$557$	−26.8283	−1.13675	−0.568375	−	0.822770i	$-0.692428\pi$
$557$	−26.8283	−1.13675	−0.568375	+	0.822770i	$0.692428\pi$
$558$	26.6220	1.12700
$559$	−35.1051	−1.48479
$560$	0	0
$561$	0	0
$562$	30.3704	1.28110
$563$	−36.9668	−1.55797	−0.778983	−	0.627045i	$-0.784264\pi$
$563$	−36.9668	−1.55797	−0.778983	+	0.627045i	$0.784264\pi$
$564$	−20.7763	−0.874839
$565$	0	0
$566$	19.7565	0.830427
$567$	−4.16741	−0.175015
$568$	9.74145	0.408742
$569$	−32.4594	−1.36077	−0.680384	−	0.732856i	$-0.738187\pi$
$569$	−32.4594	−1.36077	−0.680384	+	0.732856i	$0.738187\pi$
$570$	0	0
$571$	−18.0092	−0.753661	−0.376830	−	0.926282i	$-0.622986\pi$
$571$	−18.0092	−0.753661	−0.376830	+	0.926282i	$0.622986\pi$
$572$	0	0
$573$	0.679859	0.0284015
$574$	−37.9910	−1.58571
$575$	0	0
$576$	10.7489	0.447870
$577$	−39.5997	−1.64856	−0.824279	−	0.566184i	$-0.808419\pi$
$577$	−39.5997	−1.64856	−0.824279	+	0.566184i	$0.808419\pi$
$578$	2.45000	0.101906
$579$	−4.95245	−0.205817
$580$	0	0
$581$	20.2983	0.842116
$582$	42.5933	1.76555
$583$	0	0
$584$	35.1051	1.45266
$585$	0	0
$586$	47.6084	1.96668
$587$	24.9813	1.03109	0.515544	−	0.856863i	$-0.327590\pi$
$587$	24.9813	1.03109	0.515544	+	0.856863i	$0.327590\pi$
$588$	51.0635	2.10582
$589$	25.3153	1.04310
$590$	0	0
$591$	−3.59828	−0.148013
$592$	46.6687	1.91807
$593$	−1.92331	−0.0789807	−0.0394904	−	0.999220i	$-0.512573\pi$
$593$	−1.92331	−0.0789807	−0.0394904	+	0.999220i	$0.512573\pi$
$594$	0	0
$595$	0	0
$596$	−101.101	−4.14124
$597$	3.88380	0.158953
$598$	−75.9514	−3.10588
$599$	19.9889	0.816723	0.408362	−	0.912820i	$-0.366100\pi$
$599$	19.9889	0.816723	0.408362	+	0.912820i	$0.366100\pi$
$600$	0	0
$601$	0.176619	0.00720444	0.00360222	−	0.999994i	$-0.498853\pi$
$601$	0.176619	0.00720444	0.00360222	+	0.999994i	$0.498853\pi$
$602$	73.1487	2.98132
$603$	4.48352	0.182583
$604$	−18.9234	−0.769982
$605$	0	0
$606$	9.46929	0.384664
$607$	11.8089	0.479310	0.239655	−	0.970858i	$-0.422966\pi$
$607$	11.8089	0.479310	0.239655	+	0.970858i	$0.422966\pi$
$608$	30.0173	1.21736
$609$	21.9341	0.888814
$610$	0	0
$611$	−22.2012	−0.898166
$612$	20.8567	0.843082
$613$	−13.4114	−0.541683	−0.270841	−	0.962624i	$-0.587302\pi$
$613$	−13.4114	−0.541683	−0.270841	+	0.962624i	$0.587302\pi$
$614$	−10.2872	−0.415157
$615$	0	0
$616$	0	0
$617$	27.9341	1.12458	0.562292	−	0.826939i	$-0.309920\pi$
$617$	27.9341	1.12458	0.562292	+	0.826939i	$0.309920\pi$
$618$	−3.59828	−0.144744
$619$	38.2324	1.53669	0.768345	−	0.640036i	$-0.221081\pi$
$619$	38.2324	1.53669	0.768345	+	0.640036i	$0.221081\pi$
$620$	0	0
$621$	−5.48352	−0.220046
$622$	59.2630	2.37623
$623$	−62.9954	−2.52386
$624$	54.7848	2.19315
$625$	0	0
$626$	−50.6698	−2.02517
$627$	0	0
$628$	3.85085	0.153665
$629$	18.9855	0.757000
$630$	0	0
$631$	0.952817	0.0379310	0.0189655	−	0.999820i	$-0.493963\pi$
$631$	0.952817	0.0379310	0.0189655	+	0.999820i	$0.493963\pi$
$632$	52.8210	2.10111
$633$	24.4500	0.971799
$634$	22.0196	0.874511
$635$	0	0
$636$	17.8925	0.709482
$637$	54.5657	2.16197
$638$	0	0
$639$	1.26535	0.0500566
$640$	0	0
$641$	4.47592	0.176788	0.0883941	−	0.996086i	$-0.471827\pi$
$641$	4.47592	0.176788	0.0883941	+	0.996086i	$0.471827\pi$
$642$	−34.1106	−1.34624
$643$	−21.8980	−0.863574	−0.431787	−	0.901976i	$-0.642117\pi$
$643$	−21.8980	−0.863574	−0.431787	+	0.901976i	$0.642117\pi$
$644$	112.556	4.43534
$645$	0	0
$646$	27.8863	1.09717
$647$	−21.2796	−0.836587	−0.418293	−	0.908312i	$-0.637372\pi$
$647$	−21.2796	−0.836587	−0.418293	+	0.908312i	$0.637372\pi$
$648$	7.69860	0.302430
$649$	0	0
$650$	0	0
$651$	−42.1584	−1.65232
$652$	38.1665	1.49471
$653$	−20.5307	−0.803429	−0.401714	−	0.915765i	$-0.631586\pi$
$653$	−20.5307	−0.803429	−0.401714	+	0.915765i	$0.631586\pi$
$654$	15.7017	0.613985
$655$	0	0
$656$	36.0576	1.40781
$657$	4.55993	0.177900
$658$	46.2609	1.80344
$659$	−18.9855	−0.739569	−0.369784	−	0.929118i	$-0.620568\pi$
$659$	−18.9855	−0.739569	−0.369784	+	0.929118i	$0.620568\pi$
$660$	0	0
$661$	−2.86507	−0.111438	−0.0557192	−	0.998446i	$-0.517745\pi$
$661$	−2.86507	−0.111438	−0.0557192	+	0.998446i	$0.517745\pi$
$662$	−33.0335	−1.28388
$663$	22.2872	0.865563
$664$	−37.4977	−1.45519
$665$	0	0
$666$	11.7989	0.457199
$667$	28.8611	1.11751
$668$	−73.3156	−2.83667
$669$	−1.59972	−0.0618488
$670$	0	0
$671$	0	0
$672$	−49.9889	−1.92836
$673$	49.6033	1.91207	0.956033	−	0.293261i	$-0.0947404\pi$
$673$	49.6033	1.91207	0.956033	+	0.293261i	$0.0947404\pi$
$674$	−56.2213	−2.16556
$675$	0	0
$676$	72.4120	2.78508
$677$	34.3927	1.32182	0.660909	−	0.750466i	$-0.270171\pi$
$677$	34.3927	1.32182	0.660909	+	0.750466i	$0.270171\pi$
$678$	15.9713	0.613376
$679$	−67.4505	−2.58851
$680$	0	0
$681$	−18.6176	−0.713428
$682$	0	0
$683$	−8.20394	−0.313915	−0.156958	−	0.987605i	$-0.550169\pi$
$683$	−8.20394	−0.313915	−0.156958	+	0.987605i	$0.550169\pi$
$684$	12.3256	0.471282
$685$	0	0
$686$	−36.9296	−1.40998
$687$	23.3344	0.890262
$688$	−69.4262	−2.64685
$689$	19.1196	0.728401
$690$	0	0
$691$	−21.0832	−0.802044	−0.401022	−	0.916068i	$-0.631345\pi$
$691$	−21.0832	−0.802044	−0.401022	+	0.916068i	$0.631345\pi$
$692$	−10.7948	−0.410358
$693$	0	0
$694$	−21.6458	−0.821665
$695$	0	0
$696$	−40.5196	−1.53589
$697$	14.6687	0.555618
$698$	−16.0933	−0.609141
$699$	−9.75608	−0.369009
$700$	0	0
$701$	22.0671	0.833461	0.416731	−	0.909030i	$-0.363176\pi$
$701$	22.0671	0.833461	0.416731	+	0.909030i	$0.363176\pi$
$702$	13.8508	0.522766
$703$	11.2198	0.423162
$704$	0	0
$705$	0	0
$706$	60.6222	2.28154
$707$	−14.9955	−0.563964
$708$	−32.4363	−1.21903
$709$	−39.7992	−1.49469	−0.747344	−	0.664437i	$-0.768671\pi$
$709$	−39.7992	−1.49469	−0.747344	+	0.664437i	$0.768671\pi$
$710$	0	0
$711$	6.86112	0.257312
$712$	116.374	4.36128
$713$	−55.4724	−2.07746
$714$	−46.4399	−1.73797
$715$	0	0
$716$	−68.2213	−2.54955
$717$	−8.59317	−0.320918
$718$	40.1665	1.49900
$719$	12.1380	0.452672	0.226336	−	0.974049i	$-0.427325\pi$
$719$	12.1380	0.452672	0.226336	+	0.974049i	$0.427325\pi$
$720$	0	0
$721$	5.69820	0.212212
$722$	−33.5209	−1.24752
$723$	−8.97105	−0.333637
$724$	36.2872	1.34860
$725$	0	0
$726$	0	0
$727$	−38.9670	−1.44521	−0.722604	−	0.691262i	$-0.757055\pi$
$727$	−38.9670	−1.44521	−0.722604	+	0.691262i	$0.757055\pi$
$728$	−168.862	−6.25843
$729$	1.00000	0.0370370
$730$	0	0
$731$	−28.2435	−1.04462
$732$	60.3784	2.23165
$733$	−5.65576	−0.208900	−0.104450	−	0.994530i	$-0.533308\pi$
$733$	−5.65576	−0.208900	−0.104450	+	0.994530i	$0.533308\pi$
$734$	85.2159	3.14538
$735$	0	0
$736$	−65.7759	−2.42453
$737$	0	0
$738$	9.11620	0.335572
$739$	−28.8040	−1.05957	−0.529786	−	0.848131i	$-0.677728\pi$
$739$	−28.8040	−1.05957	−0.529786	+	0.848131i	$0.677728\pi$
$740$	0	0
$741$	13.1710	0.483848
$742$	−39.8397	−1.46256
$743$	43.2688	1.58738	0.793690	−	0.608323i	$-0.208157\pi$
$743$	43.2688	1.58738	0.793690	+	0.608323i	$0.208157\pi$
$744$	77.8806	2.85524
$745$	0	0
$746$	−21.4693	−0.786047
$747$	−4.87072	−0.178210
$748$	0	0
$749$	54.0173	1.97375
$750$	0	0
$751$	7.58549	0.276799	0.138399	−	0.990377i	$-0.455804\pi$
$751$	7.58549	0.276799	0.138399	+	0.990377i	$0.455804\pi$
$752$	−43.9067	−1.60111
$753$	5.32014	0.193877
$754$	−72.9003	−2.65487
$755$	0	0
$756$	−20.5263	−0.746533
$757$	−42.1741	−1.53284	−0.766422	−	0.642338i	$-0.777965\pi$
$757$	−42.1741	−1.53284	−0.766422	+	0.642338i	$0.777965\pi$
$758$	19.6938	0.715310
$759$	0	0
$760$	0	0
$761$	6.15318	0.223052	0.111526	−	0.993761i	$-0.464426\pi$
$761$	6.15318	0.223052	0.111526	+	0.993761i	$0.464426\pi$
$762$	−25.8508	−0.936477
$763$	−24.8651	−0.900176
$764$	3.34860	0.121148
$765$	0	0
$766$	47.7324	1.72464
$767$	−34.6610	−1.25154
$768$	−10.1908	−0.367728
$769$	20.4592	0.737777	0.368888	−	0.929474i	$-0.379738\pi$
$769$	20.4592	0.737777	0.368888	+	0.929474i	$0.379738\pi$
$770$	0	0
$771$	−27.7707	−1.00014
$772$	−24.3929	−0.877919
$773$	−34.2355	−1.23137	−0.615683	−	0.787994i	$-0.711120\pi$
$773$	−34.2355	−1.23137	−0.615683	+	0.787994i	$0.711120\pi$
$774$	−17.5525	−0.630913
$775$	0	0
$776$	124.604	4.47301
$777$	−18.6847	−0.670309
$778$	46.8032	1.67798
$779$	8.66874	0.310590
$780$	0	0
$781$	0	0
$782$	−61.1061	−2.18515
$783$	−5.26324	−0.188093
$784$	107.913	3.85403
$785$	0	0
$786$	9.11620	0.325164
$787$	4.73655	0.168840	0.0844199	−	0.996430i	$-0.473096\pi$
$787$	4.73655	0.168840	0.0844199	+	0.996430i	$0.473096\pi$
$788$	−17.7230	−0.631357
$789$	−5.89946	−0.210026
$790$	0	0
$791$	−25.2921	−0.899283
$792$	0	0
$793$	64.5196	2.29116
$794$	21.9329	0.778369
$795$	0	0
$796$	19.1294	0.678022
$797$	20.5307	0.727235	0.363617	−	0.931548i	$-0.381542\pi$
$797$	20.5307	0.727235	0.363617	+	0.931548i	$0.381542\pi$
$798$	−27.4445	−0.971523
$799$	−17.8618	−0.631906
$800$	0	0
$801$	15.1162	0.534105
$802$	−34.9094	−1.23269
$803$	0	0
$804$	22.0832	0.778816
$805$	0	0
$806$	140.118	4.93544
$807$	−6.23240	−0.219391
$808$	27.7017	0.974542
$809$	−48.8707	−1.71820	−0.859101	−	0.511807i	$-0.828976\pi$
$809$	−48.8707	−1.71820	−0.859101	+	0.511807i	$0.828976\pi$
$810$	0	0
$811$	−10.9468	−0.384394	−0.192197	−	0.981356i	$-0.561561\pi$
$811$	−10.9468	−0.384394	−0.192197	+	0.981356i	$0.561561\pi$
$812$	108.035	3.79127
$813$	−29.7657	−1.04393
$814$	0	0
$815$	0	0
$816$	44.0767	1.54299
$817$	−16.6910	−0.583944
$818$	−44.3715	−1.55141
$819$	−21.9341	−0.766439
$820$	0	0
$821$	−26.8136	−0.935802	−0.467901	−	0.883781i	$-0.654990\pi$
$821$	−26.8136	−0.935802	−0.467901	+	0.883781i	$0.654990\pi$
$822$	−54.2106	−1.89081
$823$	20.1020	0.700711	0.350355	−	0.936617i	$-0.386061\pi$
$823$	20.1020	0.700711	0.350355	+	0.936617i	$0.386061\pi$
$824$	−10.5265	−0.366707
$825$	0	0
$826$	72.2234	2.51297
$827$	−23.1104	−0.803629	−0.401814	−	0.915721i	$-0.631620\pi$
$827$	−23.1104	−0.803629	−0.401814	+	0.915721i	$0.631620\pi$
$828$	−27.0087	−0.938616
$829$	−41.0721	−1.42649	−0.713247	−	0.700913i	$-0.752776\pi$
$829$	−41.0721	−1.42649	−0.713247	+	0.700913i	$0.752776\pi$
$830$	0	0
$831$	−9.29648	−0.322491
$832$	56.5739	1.96135
$833$	43.9004	1.52106
$834$	−46.5754	−1.61278
$835$	0	0
$836$	0	0
$837$	10.1162	0.349667
$838$	25.7795	0.890538
$839$	−7.40338	−0.255593	−0.127797	−	0.991800i	$-0.540790\pi$
$839$	−7.40338	−0.255593	−0.127797	+	0.991800i	$0.540790\pi$
$840$	0	0
$841$	−1.29831	−0.0447693
$842$	−47.0141	−1.62021
$843$	11.5406	0.397479
$844$	120.426	4.14525
$845$	0	0
$846$	−11.1006	−0.381647
$847$	0	0
$848$	37.8123	1.29848
$849$	7.50735	0.257651
$850$	0	0
$851$	−24.5855	−0.842780
$852$	6.23240	0.213518
$853$	26.3686	0.902845	0.451423	−	0.892310i	$-0.350917\pi$
$853$	26.3686	0.902845	0.451423	+	0.892310i	$0.350917\pi$
$854$	−134.440	−4.60044
$855$	0	0
$856$	−99.7880	−3.41068
$857$	−38.3588	−1.31031	−0.655156	−	0.755493i	$-0.727397\pi$
$857$	−38.3588	−1.31031	−0.655156	+	0.755493i	$0.727397\pi$
$858$	0	0
$859$	2.53831	0.0866060	0.0433030	−	0.999062i	$-0.486212\pi$
$859$	2.53831	0.0866060	0.0433030	+	0.999062i	$0.486212\pi$
$860$	0	0
$861$	−14.4363	−0.491989
$862$	97.8855	3.33399
$863$	12.0000	0.408485	0.204242	−	0.978920i	$-0.434527\pi$
$863$	12.0000	0.408485	0.204242	+	0.978920i	$0.434527\pi$
$864$	11.9952	0.408084
$865$	0	0
$866$	26.8036	0.910824
$867$	0.930985	0.0316179
$868$	−207.648	−7.04803
$869$	0	0
$870$	0	0
$871$	23.5979	0.799583
$872$	45.9341	1.55552
$873$	16.1852	0.547786
$874$	−36.1117	−1.22150
$875$	0	0
$876$	22.4596	0.758839
$877$	−32.1386	−1.08524	−0.542621	−	0.839978i	$-0.682568\pi$
$877$	−32.1386	−1.08524	−0.542621	+	0.839978i	$0.682568\pi$
$878$	−51.8782	−1.75081
$879$	18.0909	0.610191
$880$	0	0
$881$	47.1051	1.58701	0.793505	−	0.608564i	$-0.208254\pi$
$881$	47.1051	1.58701	0.793505	+	0.608564i	$0.208254\pi$
$882$	27.2829	0.918661
$883$	19.9528	0.671466	0.335733	−	0.941957i	$-0.391016\pi$
$883$	19.9528	0.671466	0.335733	+	0.941957i	$0.391016\pi$
$884$	109.774	3.69209
$885$	0	0
$886$	69.5704	2.33726
$887$	−48.6316	−1.63289	−0.816445	−	0.577424i	$-0.804058\pi$
$887$	−48.6316	−1.63289	−0.816445	+	0.577424i	$0.804058\pi$
$888$	34.5168	1.15831
$889$	40.9372	1.37299
$890$	0	0
$891$	0	0
$892$	−7.87930	−0.263819
$893$	−10.5558	−0.353235
$894$	−54.0173	−1.80661
$895$	0	0
$896$	−17.9056	−0.598185
$897$	−28.8611	−0.963644
$898$	−57.1106	−1.90581
$899$	−53.2440	−1.77579
$900$	0	0
$901$	15.3826	0.512468
$902$	0	0
$903$	27.7961	0.924995
$904$	46.7229	1.55398
$905$	0	0
$906$	−10.1106	−0.335903
$907$	−40.0361	−1.32938	−0.664688	−	0.747121i	$-0.731435\pi$
$907$	−40.0361	−1.32938	−0.664688	+	0.747121i	$0.731435\pi$
$908$	−91.6995	−3.04316
$909$	3.59828	0.119347
$910$	0	0
$911$	−22.1492	−0.733834	−0.366917	−	0.930254i	$-0.619587\pi$
$911$	−22.1492	−0.733834	−0.366917	+	0.930254i	$0.619587\pi$
$912$	26.0478	0.862530
$913$	0	0
$914$	−31.0503	−1.02705
$915$	0	0
$916$	114.932	3.79745
$917$	−14.4363	−0.476730
$918$	11.1436	0.367793
$919$	−24.4453	−0.806377	−0.403189	−	0.915117i	$-0.632098\pi$
$919$	−24.4453	−0.806377	−0.403189	+	0.915117i	$0.632098\pi$
$920$	0	0
$921$	−3.90907	−0.128808
$922$	81.7190	2.69127
$923$	6.65985	0.219212
$924$	0	0
$925$	0	0
$926$	−100.976	−3.31829
$927$	−1.36732	−0.0449088
$928$	−63.1335	−2.07246
$929$	−57.2716	−1.87902	−0.939509	−	0.342523i	$-0.888719\pi$
$929$	−57.2716	−1.87902	−0.939509	+	0.342523i	$0.888719\pi$
$930$	0	0
$931$	25.9437	0.850270
$932$	−48.0528	−1.57402
$933$	22.5196	0.737258
$934$	−39.3501	−1.28758
$935$	0	0
$936$	40.5196	1.32442
$937$	51.0099	1.66642	0.833210	−	0.552957i	$-0.186500\pi$
$937$	51.0099	1.66642	0.833210	+	0.552957i	$0.186500\pi$
$938$	−49.1710	−1.60549
$939$	−19.2542	−0.628338
$940$	0	0
$941$	−15.1681	−0.494467	−0.247233	−	0.968956i	$-0.579521\pi$
$941$	−15.1681	−0.494467	−0.247233	+	0.968956i	$0.579521\pi$
$942$	2.05748	0.0670363
$943$	−18.9955	−0.618578
$944$	−68.5480	−2.23105
$945$	0	0
$946$	0	0
$947$	55.9199	1.81715	0.908576	−	0.417720i	$-0.137171\pi$
$947$	55.9199	1.81715	0.908576	+	0.417720i	$0.137171\pi$
$948$	33.7939	1.09757
$949$	24.0000	0.779073
$950$	0	0
$951$	8.36732	0.271329
$952$	−135.856	−4.40313
$953$	32.7177	1.05983	0.529915	−	0.848051i	$-0.322224\pi$
$953$	32.7177	1.05983	0.529915	+	0.848051i	$0.322224\pi$
$954$	9.55982	0.309511
$955$	0	0
$956$	−42.3250	−1.36889
$957$	0	0
$958$	32.0832	1.03656
$959$	85.8475	2.77216
$960$	0	0
$961$	71.3375	2.30121
$962$	62.1006	2.00220
$963$	−12.9618	−0.417689
$964$	−44.1862	−1.42314
$965$	0	0
$966$	60.1380	1.93491
$967$	53.9473	1.73483	0.867414	−	0.497587i	$-0.165781\pi$
$967$	53.9473	1.73483	0.867414	+	0.497587i	$0.165781\pi$
$968$	0	0
$969$	10.5966	0.340412
$970$	0	0
$971$	4.14915	0.133153	0.0665763	−	0.997781i	$-0.478792\pi$
$971$	4.14915	0.133153	0.0665763	+	0.997781i	$0.478792\pi$
$972$	4.92542	0.157983
$973$	73.7565	2.36452
$974$	−37.5984	−1.20473
$975$	0	0
$976$	127.598	4.08432
$977$	13.5668	0.434039	0.217020	−	0.976167i	$-0.430366\pi$
$977$	13.5668	0.434039	0.217020	+	0.976167i	$0.430366\pi$
$978$	20.3921	0.652067
$979$	0	0
$980$	0	0
$981$	5.96655	0.190497
$982$	−68.5744	−2.18830
$983$	−36.7489	−1.17211	−0.586054	−	0.810272i	$-0.699319\pi$
$983$	−36.7489	−1.17211	−0.586054	+	0.810272i	$0.699319\pi$
$984$	26.6687	0.850168
$985$	0	0
$986$	−58.6514	−1.86784
$987$	17.5789	0.559541
$988$	64.8727	2.06387
$989$	36.5743	1.16300
$990$	0	0
$991$	−48.6830	−1.54647	−0.773233	−	0.634122i	$-0.781362\pi$
$991$	−48.6830	−1.54647	−0.773233	+	0.634122i	$0.781362\pi$
$992$	121.346	3.85273
$993$	−12.5525	−0.398343
$994$	−13.8772	−0.440157
$995$	0	0
$996$	−23.9904	−0.760164
$997$	41.7558	1.32242	0.661210	−	0.750200i	$-0.270043\pi$
$997$	41.7558	1.32242	0.661210	+	0.750200i	$0.270043\pi$
$998$	75.9619	2.40453
$999$	4.48352	0.141852

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9075.2.a.dq.1.6		6
5.4	even	2		1815.2.a.y.1.1	✓	6
11.10	odd	2	inner	9075.2.a.dq.1.1		6
15.14	odd	2		5445.2.a.bz.1.6		6
55.54	odd	2		1815.2.a.y.1.6	yes	6
165.164	even	2		5445.2.a.bz.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1815.2.a.y.1.1	✓	6	5.4	even	2
1815.2.a.y.1.6	yes	6	55.54	odd	2
5445.2.a.bz.1.1		6	165.164	even	2
5445.2.a.bz.1.6		6	15.14	odd	2
9075.2.a.dq.1.1		6	11.10	odd	2	inner
9075.2.a.dq.1.6		6	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)

\(f(q)\)	\(=\)	\(q+2.63162 q^{2} +1.00000 q^{3} +4.92542 q^{4} +2.63162 q^{6} -4.16741 q^{7} +7.69860 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+2.63162 q^{2} +1.00000 q^{3} +4.92542 q^{4} +2.63162 q^{6} -4.16741 q^{7} +7.69860 q^{8} +1.00000 q^{9} +4.92542 q^{12} +5.26324 q^{13} -10.9670 q^{14} +10.4089 q^{16} +4.23450 q^{17} +2.63162 q^{18} +2.50245 q^{19} -4.16741 q^{21} -5.48352 q^{23} +7.69860 q^{24} +13.8508 q^{26} +1.00000 q^{27} -20.5263 q^{28} -5.26324 q^{29} +10.1162 q^{31} +11.9952 q^{32} +11.1436 q^{34} +4.92542 q^{36} +4.48352 q^{37} +6.58549 q^{38} +5.26324 q^{39} +3.46410 q^{41} -10.9670 q^{42} -6.66986 q^{43} -14.4305 q^{46} -4.21817 q^{47} +10.4089 q^{48} +10.3673 q^{49} +4.23450 q^{51} +25.9237 q^{52} +3.63268 q^{53} +2.63162 q^{54} -32.0832 q^{56} +2.50245 q^{57} -13.8508 q^{58} -6.58549 q^{59} +12.2585 q^{61} +26.6220 q^{62} -4.16741 q^{63} +10.7489 q^{64} +4.48352 q^{67} +20.8567 q^{68} -5.48352 q^{69} +1.26535 q^{71} +7.69860 q^{72} +4.55993 q^{73} +11.7989 q^{74} +12.3256 q^{76} +13.8508 q^{78} +6.86112 q^{79} +1.00000 q^{81} +9.11620 q^{82} -4.87072 q^{83} -20.5263 q^{84} -17.5525 q^{86} -5.26324 q^{87} +15.1162 q^{89} -21.9341 q^{91} -27.0087 q^{92} +10.1162 q^{93} -11.1006 q^{94} +11.9952 q^{96} +16.1852 q^{97} +27.2829 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{3} + 14 q^{4} + 6 q^{9}+O(q^{10}) \) 6 * q + 6 * q^3 + 14 * q^4 + 6 * q^9	\( 6 q + 6 q^{3} + 14 q^{4} + 6 q^{9} + 14 q^{12} + 8 q^{14} + 10 q^{16} + 4 q^{23} + 52 q^{26} + 6 q^{27} + 18 q^{31} + 26 q^{34} + 14 q^{36} - 10 q^{37} + 20 q^{38} + 8 q^{42} + 10 q^{48} + 68 q^{49} + 16 q^{53} - 76 q^{56} - 52 q^{58} - 20 q^{59} + 16 q^{64} - 10 q^{67} + 4 q^{69} - 4 q^{71} + 52 q^{78} + 6 q^{81} + 12 q^{82} - 12 q^{86} + 48 q^{89} + 16 q^{91} - 30 q^{92} + 18 q^{93} - 2 q^{97}+O(q^{100}) \) 6 * q + 6 * q^3 + 14 * q^4 + 6 * q^9 + 14 * q^12 + 8 * q^14 + 10 * q^16 + 4 * q^23 + 52 * q^26 + 6 * q^27 + 18 * q^31 + 26 * q^34 + 14 * q^36 - 10 * q^37 + 20 * q^38 + 8 * q^42 + 10 * q^48 + 68 * q^49 + 16 * q^53 - 76 * q^56 - 52 * q^58 - 20 * q^59 + 16 * q^64 - 10 * q^67 + 4 * q^69 - 4 * q^71 + 52 * q^78 + 6 * q^81 + 12 * q^82 - 12 * q^86 + 48 * q^89 + 16 * q^91 - 30 * q^92 + 18 * q^93 - 2 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more