LMFDB - Embedded newform 5445.2.a.bz.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5445,2,Mod(1,5445)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5445, 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("5445.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5445 = 3^{2} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5445.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:	$43.4785439006$
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	$\chi$	$=$	5445.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.63162 q^{2} +4.92542 q^{4} -1.00000 q^{5} +4.16741 q^{7} +7.69860 q^{8} +O(q^{10})$	$q+2.63162 q^{2} +4.92542 q^{4} -1.00000 q^{5} +4.16741 q^{7} +7.69860 q^{8} -2.63162 q^{10} -5.26324 q^{13} +10.9670 q^{14} +10.4089 q^{16} +4.23450 q^{17} +2.50245 q^{19} -4.92542 q^{20} -5.48352 q^{23} +1.00000 q^{25} -13.8508 q^{26} +20.5263 q^{28} +5.26324 q^{29} +10.1162 q^{31} +11.9952 q^{32} +11.1436 q^{34} -4.16741 q^{35} -4.48352 q^{37} +6.58549 q^{38} -7.69860 q^{40} -3.46410 q^{41} +6.66986 q^{43} -14.4305 q^{46} -4.21817 q^{47} +10.3673 q^{49} +2.63162 q^{50} -25.9237 q^{52} +3.63268 q^{53} +32.0832 q^{56} +13.8508 q^{58} +6.58549 q^{59} +12.2585 q^{61} +26.6220 q^{62} +10.7489 q^{64} +5.26324 q^{65} -4.48352 q^{67} +20.8567 q^{68} -10.9670 q^{70} -1.26535 q^{71} -4.55993 q^{73} -11.7989 q^{74} +12.3256 q^{76} +6.86112 q^{79} -10.4089 q^{80} -9.11620 q^{82} -4.87072 q^{83} -4.23450 q^{85} +17.5525 q^{86} -15.1162 q^{89} -21.9341 q^{91} -27.0087 q^{92} -11.1006 q^{94} -2.50245 q^{95} -16.1852 q^{97} +27.2829 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 14 q^{4} - 6 q^{5}+O(q^{10}) $ 6 * q + 14 * q^4 - 6 * q^5	$ 6 q + 14 q^{4} - 6 q^{5} - 8 q^{14} + 10 q^{16} - 14 q^{20} + 4 q^{23} + 6 q^{25} - 52 q^{26} + 18 q^{31} + 26 q^{34} + 10 q^{37} + 20 q^{38} + 68 q^{49} + 16 q^{53} + 76 q^{56} + 52 q^{58} + 20 q^{59} + 16 q^{64} + 10 q^{67} + 8 q^{70} + 4 q^{71} - 10 q^{80} - 12 q^{82} + 12 q^{86} - 48 q^{89} + 16 q^{91} - 30 q^{92} + 2 q^{97}+O(q^{100}) $ 6 * q + 14 * q^4 - 6 * q^5 - 8 * q^14 + 10 * q^16 - 14 * q^20 + 4 * q^23 + 6 * q^25 - 52 * q^26 + 18 * q^31 + 26 * q^34 + 10 * q^37 + 20 * q^38 + 68 * q^49 + 16 * q^53 + 76 * q^56 + 52 * q^58 + 20 * q^59 + 16 * q^64 + 10 * q^67 + 8 * q^70 + 4 * q^71 - 10 * q^80 - 12 * q^82 + 12 * q^86 - 48 * q^89 + 16 * q^91 - 30 * q^92 + 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	$0.119444\pi$
$3$	0	0
$3$	0	0
$4$	4.92542	2.46271
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	4.16741	1.57513	0.787567	−	0.616229i	$-0.211341\pi$
$7$	4.16741	1.57513	0.787567	+	0.616229i	$0.211341\pi$
$8$	7.69860	2.72187
$9$	0	0
$10$	−2.63162	−0.832191
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−5.26324	−1.45976	−0.729880	−	0.683575i	$-0.760424\pi$
$13$	−5.26324	−1.45976	−0.729880	+	0.683575i	$0.760424\pi$
$14$	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	$0.328345\pi$
$18$	0	0
$19$	2.50245	0.574101	0.287051	−	0.957915i	$-0.407325\pi$
$19$	2.50245	0.574101	0.287051	+	0.957915i	$0.407325\pi$
$20$	−4.92542	−1.10136
$21$	0	0
$22$	0	0
$23$	−5.48352	−1.14339	−0.571697	−	0.820465i	$-0.693715\pi$
$23$	−5.48352	−1.14339	−0.571697	+	0.820465i	$0.693715\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−13.8508	−2.71637
$27$	0	0
$28$	20.5263	3.87910
$29$	5.26324	0.977359	0.488680	−	0.872463i	$-0.337479\pi$
$29$	5.26324	0.977359	0.488680	+	0.872463i	$0.337479\pi$
$30$	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
$35$	−4.16741	−0.704421
$36$	0	0
$37$	−4.48352	−0.737087	−0.368543	−	0.929611i	$-0.620143\pi$
$37$	−4.48352	−0.737087	−0.368543	+	0.929611i	$0.620143\pi$
$38$	6.58549	1.06831
$39$	0	0
$40$	−7.69860	−1.21726
$41$	−3.46410	−0.541002	−0.270501	−	0.962720i	$-0.587189\pi$
$41$	−3.46410	−0.541002	−0.270501	+	0.962720i	$0.587189\pi$
$42$	0	0
$43$	6.66986	1.01714	0.508572	−	0.861019i	$-0.330174\pi$
$43$	6.66986	1.01714	0.508572	+	0.861019i	$0.330174\pi$
$44$	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	$0.599540\pi$
$48$	0	0
$49$	10.3673	1.48105
$50$	2.63162	0.372167
$51$	0	0
$52$	−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	$0.419736\pi$
$54$	0	0
$55$	0	0
$56$	32.0832	4.28730
$57$	0	0
$58$	13.8508	1.81871
$59$	6.58549	0.857358	0.428679	−	0.903457i	$-0.358979\pi$
$59$	6.58549	0.857358	0.428679	+	0.903457i	$0.358979\pi$
$60$	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
$63$	0	0
$64$	10.7489	1.34361
$65$	5.26324	0.652825
$66$	0	0
$67$	−4.48352	−0.547749	−0.273875	−	0.961765i	$-0.588305\pi$
$67$	−4.48352	−0.547749	−0.273875	+	0.961765i	$0.588305\pi$
$68$	20.8567	2.52925
$69$	0	0
$70$	−10.9670	−1.31081
$71$	−1.26535	−0.150170	−0.0750849	−	0.997177i	$-0.523923\pi$
$71$	−1.26535	−0.150170	−0.0750849	+	0.997177i	$0.523923\pi$
$72$	0	0
$73$	−4.55993	−0.533699	−0.266850	−	0.963738i	$-0.585983\pi$
$73$	−4.55993	−0.533699	−0.266850	+	0.963738i	$0.585983\pi$
$74$	−11.7989	−1.37160
$75$	0	0
$76$	12.3256	1.41385
$77$	0	0
$78$	0	0
$79$	6.86112	0.771936	0.385968	−	0.922512i	$-0.373868\pi$
$79$	6.86112	0.771936	0.385968	+	0.922512i	$0.373868\pi$
$80$	−10.4089	−1.16376
$81$	0	0
$82$	−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	$0.586137\pi$
$84$	0	0
$85$	−4.23450	−0.459296
$86$	17.5525	1.89274
$87$	0	0
$88$	0	0
$89$	−15.1162	−1.60231	−0.801157	−	0.598454i	$-0.795782\pi$
$89$	−15.1162	−1.60231	−0.801157	+	0.598454i	$0.795782\pi$
$90$	0	0
$91$	−21.9341	−2.29932
$92$	−27.0087	−2.81585
$93$	0	0
$94$	−11.1006	−1.14494
$95$	−2.50245	−0.256746
$96$	0	0
$97$	−16.1852	−1.64336	−0.821680	−	0.569949i	$-0.806963\pi$
$97$	−16.1852	−1.64336	−0.821680	+	0.569949i	$0.806963\pi$
$98$	27.2829	2.75598
$99$	0	0
$100$	4.92542	0.492542
$101$	−3.59828	−0.358042	−0.179021	−	0.983845i	$-0.557293\pi$
$101$	−3.59828	−0.358042	−0.179021	+	0.983845i	$0.557293\pi$
$102$	0	0
$103$	1.36732	0.134726	0.0673632	−	0.997729i	$-0.478541\pi$
$103$	1.36732	0.134726	0.0673632	+	0.997729i	$0.478541\pi$
$104$	−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	$0.715527\pi$
$108$	0	0
$109$	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
$111$	0	0
$112$	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	$0.407853\pi$
$114$	0	0
$115$	5.48352	0.511341
$116$	25.9237	2.40695
$117$	0	0
$118$	17.3305	1.59540
$119$	17.6469	1.61769
$120$	0	0
$121$	0	0
$122$	32.2598	2.92066
$123$	0	0
$124$	49.8266	4.47456
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	9.82317	0.871665	0.435833	−	0.900028i	$-0.356454\pi$
$127$	9.82317	0.871665	0.435833	+	0.900028i	$0.356454\pi$
$128$	4.29658	0.379768
$129$	0	0
$130$	13.8508	1.21480
$131$	−3.46410	−0.302660	−0.151330	−	0.988483i	$-0.548356\pi$
$131$	−3.46410	−0.302660	−0.151330	+	0.988483i	$0.548356\pi$
$132$	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	$0.842442\pi$
$138$	0	0
$139$	−17.6984	−1.50116	−0.750579	−	0.660781i	$-0.770225\pi$
$139$	−17.6984	−1.50116	−0.750579	+	0.660781i	$0.770225\pi$
$140$	−20.5263	−1.73479
$141$	0	0
$142$	−3.32993	−0.279441
$143$	0	0
$144$	0	0
$145$	−5.26324	−0.437088
$146$	−12.0000	−0.993127
$147$	0	0
$148$	−22.0832	−1.81523
$149$	20.5263	1.68158	0.840789	−	0.541363i	$-0.182092\pi$
$149$	20.5263	1.68158	0.840789	+	0.541363i	$0.182092\pi$
$150$	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
$153$	0	0
$154$	0	0
$155$	−10.1162	−0.812553
$156$	0	0
$157$	−0.781830	−0.0623969	−0.0311984	−	0.999513i	$-0.509932\pi$
$157$	−0.781830	−0.0623969	−0.0311984	+	0.999513i	$0.509932\pi$
$158$	18.0558	1.43645
$159$	0	0
$160$	−11.9952	−0.948303
$161$	−22.8521	−1.80100
$162$	0	0
$163$	−7.74887	−0.606939	−0.303469	−	0.952841i	$-0.598145\pi$
$163$	−7.74887	−0.606939	−0.303469	+	0.952841i	$0.598145\pi$
$164$	−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	$0.695357\pi$
$168$	0	0
$169$	14.7017	1.13090
$170$	−11.1436	−0.854675
$171$	0	0
$172$	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	$0.526551\pi$
$174$	0	0
$175$	4.16741	0.315027
$176$	0	0
$177$	0	0
$178$	−39.7801	−2.98164
$179$	13.8508	1.03526	0.517630	−	0.855604i	$-0.326814\pi$
$179$	13.8508	1.03526	0.517630	+	0.855604i	$0.326814\pi$
$180$	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
$183$	0	0
$184$	−42.2154	−3.11216
$185$	4.48352	0.329635
$186$	0	0
$187$	0	0
$188$	−20.7763	−1.51527
$189$	0	0
$190$	−6.58549	−0.477762
$191$	−0.679859	−0.0491929	−0.0245965	−	0.999697i	$-0.507830\pi$
$191$	−0.679859	−0.0491929	−0.0245965	+	0.999697i	$0.507830\pi$
$192$	0	0
$193$	4.95245	0.356485	0.178242	−	0.983987i	$-0.442959\pi$
$193$	4.95245	0.356485	0.178242	+	0.983987i	$0.442959\pi$
$194$	−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	$0.456043\pi$
$200$	7.69860	0.544373
$201$	0	0
$202$	−9.46929	−0.666257
$203$	21.9341	1.53947
$204$	0	0
$205$	3.46410	0.241943
$206$	3.59828	0.250704
$207$	0	0
$208$	−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
$213$	0	0
$214$	−34.1106	−2.33176
$215$	−6.66986	−0.454881
$216$	0	0
$217$	42.1584	2.86190
$218$	15.7017	1.06345
$219$	0	0
$220$	0	0
$221$	−22.2872	−1.49920
$222$	0	0
$223$	1.59972	0.107125	0.0535626	−	0.998564i	$-0.482942\pi$
$223$	1.59972	0.107125	0.0535626	+	0.998564i	$0.482942\pi$
$224$	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	$0.711995\pi$
$228$	0	0
$229$	23.3344	1.54198	0.770989	−	0.636848i	$-0.219762\pi$
$229$	23.3344	1.54198	0.770989	+	0.636848i	$0.219762\pi$
$230$	14.4305	0.951522
$231$	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	$0.603539\pi$
$234$	0	0
$235$	4.21817	0.275163
$236$	32.4363	2.11143
$237$	0	0
$238$	46.4399	3.01025
$239$	8.59317	0.555846	0.277923	−	0.960603i	$-0.410354\pi$
$239$	8.59317	0.555846	0.277923	+	0.960603i	$0.410354\pi$
$240$	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
$243$	0	0
$244$	60.3784	3.86533
$245$	−10.3673	−0.662344
$246$	0	0
$247$	−13.1710	−0.838050
$248$	77.8806	4.94542
$249$	0	0
$250$	−2.63162	−0.166438
$251$	−5.32014	−0.335804	−0.167902	−	0.985804i	$-0.553699\pi$
$251$	−5.32014	−0.335804	−0.167902	+	0.985804i	$0.553699\pi$
$252$	0	0
$253$	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	$0.833409\pi$
$258$	0	0
$259$	−18.6847	−1.16101
$260$	25.9237	1.60772
$261$	0	0
$262$	−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
$265$	−3.63268	−0.223154
$266$	27.4445	1.68273
$267$	0	0
$268$	−22.0832	−1.34895
$269$	6.23240	0.379996	0.189998	−	0.981784i	$-0.439152\pi$
$269$	6.23240	0.379996	0.189998	+	0.981784i	$0.439152\pi$
$270$	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
$273$	0	0
$274$	−54.2106	−3.27498
$275$	0	0
$276$	0	0
$277$	9.29648	0.558571	0.279286	−	0.960208i	$-0.409902\pi$
$277$	9.29648	0.558571	0.279286	+	0.960208i	$0.409902\pi$
$278$	−46.5754	−2.79341
$279$	0	0
$280$	−32.0832	−1.91734
$281$	−11.5406	−0.688453	−0.344227	−	0.938887i	$-0.611859\pi$
$281$	−11.5406	−0.688453	−0.344227	+	0.938887i	$0.611859\pi$
$282$	0	0
$283$	−7.50735	−0.446265	−0.223133	−	0.974788i	$-0.571628\pi$
$283$	−7.50735	−0.446265	−0.223133	+	0.974788i	$0.571628\pi$
$284$	−6.23240	−0.369825
$285$	0	0
$286$	0	0
$287$	−14.4363	−0.852150
$288$	0	0
$289$	0.930985	0.0547638
$290$	−13.8508	−0.813350
$291$	0	0
$292$	−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	$0.322777\pi$
$294$	0	0
$295$	−6.58549	−0.383422
$296$	−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
$303$	0	0
$304$	26.0478	1.49395
$305$	−12.2585	−0.701921
$306$	0	0
$307$	3.90907	0.223102	0.111551	−	0.993759i	$-0.464418\pi$
$307$	3.90907	0.223102	0.111551	+	0.993759i	$0.464418\pi$
$308$	0	0
$309$	0	0
$310$	−26.6220	−1.51203
$311$	−22.5196	−1.27697	−0.638484	−	0.769635i	$-0.720438\pi$
$311$	−22.5196	−1.27697	−0.638484	+	0.769635i	$0.720438\pi$
$312$	0	0
$313$	19.2542	1.08831	0.544157	−	0.838984i	$-0.316850\pi$
$313$	19.2542	1.08831	0.544157	+	0.838984i	$0.316850\pi$
$314$	−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	$0.424498\pi$
$318$	0	0
$319$	0	0
$320$	−10.7489	−0.600880
$321$	0	0
$322$	−60.1380	−3.35136
$323$	10.5966	0.589612
$324$	0	0
$325$	−5.26324	−0.291952
$326$	−20.3921	−1.12941
$327$	0	0
$328$	−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
$333$	0	0
$334$	−39.1720	−2.14340
$335$	4.48352	0.244961
$336$	0	0
$337$	21.3638	1.16376	0.581879	−	0.813275i	$-0.302318\pi$
$337$	21.3638	1.16376	0.581879	+	0.813275i	$0.302318\pi$
$338$	38.6893	2.10442
$339$	0	0
$340$	−20.8567	−1.13111
$341$	0	0
$342$	0	0
$343$	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	$0.570860\pi$
$348$	0	0
$349$	−6.11536	−0.327348	−0.163674	−	0.986515i	$-0.552334\pi$
$349$	−6.11536	−0.327348	−0.163674	+	0.986515i	$0.552334\pi$
$350$	10.9670	0.586213
$351$	0	0
$352$	0	0
$353$	23.0361	1.22609	0.613043	−	0.790050i	$-0.289945\pi$
$353$	23.0361	1.22609	0.613043	+	0.790050i	$0.289945\pi$
$354$	0	0
$355$	1.26535	0.0671579
$356$	−74.4537	−3.94604
$357$	0	0
$358$	36.4502	1.92645
$359$	−15.2630	−0.805552	−0.402776	−	0.915299i	$-0.631955\pi$
$359$	−15.2630	−0.805552	−0.402776	+	0.915299i	$0.631955\pi$
$360$	0	0
$361$	−12.7378	−0.670408
$362$	19.3880	1.01901
$363$	0	0
$364$	−108.035	−5.66255
$365$	4.55993	0.238678
$366$	0	0
$367$	−32.3816	−1.69030	−0.845152	−	0.534527i	$-0.820490\pi$
$367$	−32.3816	−1.69030	−0.845152	+	0.534527i	$0.820490\pi$
$368$	−57.0777	−2.97538
$369$	0	0
$370$	11.7989	0.613397
$371$	15.1389	0.785970
$372$	0	0
$373$	8.15821	0.422416	0.211208	−	0.977441i	$-0.432260\pi$
$373$	8.15821	0.422416	0.211208	+	0.977441i	$0.432260\pi$
$374$	0	0
$375$	0	0
$376$	−32.4740	−1.67472
$377$	−27.7017	−1.42671
$378$	0	0
$379$	7.48352	0.384403	0.192201	−	0.981356i	$-0.438437\pi$
$379$	7.48352	0.384403	0.192201	+	0.981356i	$0.438437\pi$
$380$	−12.3256	−0.632291
$381$	0	0
$382$	−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	$0.346628\pi$
$384$	0	0
$385$	0	0
$386$	13.0330	0.663360
$387$	0	0
$388$	−79.7190	−4.04712
$389$	−17.7849	−0.901732	−0.450866	−	0.892592i	$-0.648885\pi$
$389$	−17.7849	−0.901732	−0.450866	+	0.892592i	$0.648885\pi$
$390$	0	0
$391$	−23.2200	−1.17428
$392$	79.8139	4.03121
$393$	0	0
$394$	−9.46929	−0.477056
$395$	−6.86112	−0.345220
$396$	0	0
$397$	−8.33437	−0.418290	−0.209145	−	0.977885i	$-0.567068\pi$
$397$	−8.33437	−0.418290	−0.209145	+	0.977885i	$0.567068\pi$
$398$	10.2207	0.512317
$399$	0	0
$400$	10.4089	0.520447
$401$	13.2654	0.662440	0.331220	−	0.943554i	$-0.392540\pi$
$401$	13.2654	0.662440	0.331220	+	0.943554i	$0.392540\pi$
$402$	0	0
$403$	−53.2440	−2.65227
$404$	−17.7230	−0.881754
$405$	0	0
$406$	57.7222	2.86470
$407$	0	0
$408$	0	0
$409$	−16.8609	−0.833718	−0.416859	−	0.908971i	$-0.636869\pi$
$409$	−16.8609	−0.833718	−0.416859	+	0.908971i	$0.636869\pi$
$410$	9.11620	0.450217
$411$	0	0
$412$	6.73465	0.331792
$413$	27.4445	1.35045
$414$	0	0
$415$	4.87072	0.239094
$416$	−63.1335	−3.09538
$417$	0	0
$418$	0	0
$419$	−9.79606	−0.478569	−0.239284	−	0.970950i	$-0.576913\pi$
$419$	−9.79606	−0.478569	−0.239284	+	0.970950i	$0.576913\pi$
$420$	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
$423$	0	0
$424$	27.9665	1.35817
$425$	4.23450	0.205403
$426$	0	0
$427$	51.0863	2.47224
$428$	−63.8425	−3.08595
$429$	0	0
$430$	−17.5525	−0.846459
$431$	−37.1959	−1.79166	−0.895832	−	0.444393i	$-0.853420\pi$
$431$	−37.1959	−1.79166	−0.895832	+	0.444393i	$0.853420\pi$
$432$	0	0
$433$	−10.1852	−0.489470	−0.244735	−	0.969590i	$-0.578701\pi$
$433$	−10.1852	−0.489470	−0.244735	+	0.969590i	$0.578701\pi$
$434$	110.945	5.32552
$435$	0	0
$436$	29.3878	1.40742
$437$	−13.7222	−0.656423
$438$	0	0
$439$	−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
$441$	0	0
$442$	−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	$0.283868\pi$
$444$	0	0
$445$	15.1162	0.716577
$446$	4.20986	0.199342
$447$	0	0
$448$	44.7950	2.11636
$449$	21.7017	1.02417	0.512083	−	0.858936i	$-0.328874\pi$
$449$	21.7017	1.02417	0.512083	+	0.858936i	$0.328874\pi$
$450$	0	0
$451$	0	0
$452$	29.8925	1.40602
$453$	0	0
$454$	−48.9944	−2.29942
$455$	21.9341	1.02829
$456$	0	0
$457$	11.7989	0.551930	0.275965	−	0.961168i	$-0.411003\pi$
$457$	11.7989	0.551930	0.275965	+	0.961168i	$0.411003\pi$
$458$	61.4072	2.86937
$459$	0	0
$460$	27.0087	1.25929
$461$	−31.0527	−1.44627	−0.723135	−	0.690706i	$-0.757300\pi$
$461$	−31.0527	−1.44627	−0.723135	+	0.690706i	$0.757300\pi$
$462$	0	0
$463$	38.3704	1.78323	0.891613	−	0.452799i	$-0.149575\pi$
$463$	38.3704	1.78323	0.891613	+	0.452799i	$0.149575\pi$
$464$	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	$0.612449\pi$
$468$	0	0
$469$	−18.6847	−0.862779
$470$	11.1006	0.512033
$471$	0	0
$472$	50.6991	2.33361
$473$	0	0
$474$	0	0
$475$	2.50245	0.114820
$476$	86.9185	3.98390
$477$	0	0
$478$	22.6139	1.03434
$479$	−12.1914	−0.557041	−0.278521	−	0.960430i	$-0.589844\pi$
$479$	−12.1914	−0.557041	−0.278521	+	0.960430i	$0.589844\pi$
$480$	0	0
$481$	23.5979	1.07597
$482$	−23.6084	−1.07533
$483$	0	0
$484$	0	0
$485$	16.1852	0.734933
$486$	0	0
$487$	14.2872	0.647414	0.323707	−	0.946157i	$-0.395071\pi$
$487$	14.2872	0.647414	0.323707	+	0.946157i	$0.395071\pi$
$488$	94.3735	4.27209
$489$	0	0
$490$	−27.2829	−1.23251
$491$	26.0579	1.17597	0.587987	−	0.808870i	$-0.299921\pi$
$491$	26.0579	1.17597	0.587987	+	0.808870i	$0.299921\pi$
$492$	0	0
$493$	22.2872	1.00376
$494$	−34.6610	−1.55947
$495$	0	0
$496$	105.299	4.72806
$497$	−5.27325	−0.236537
$498$	0	0
$499$	28.8651	1.29218	0.646089	−	0.763262i	$-0.276403\pi$
$499$	28.8651	1.29218	0.646089	+	0.763262i	$0.276403\pi$
$500$	−4.92542	−0.220272
$501$	0	0
$502$	−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	$0.840974\pi$
$504$	0	0
$505$	3.59828	0.160121
$506$	0	0
$507$	0	0
$508$	48.3833	2.14666
$509$	−20.0832	−0.890174	−0.445087	−	0.895487i	$-0.646827\pi$
$509$	−20.0832	−0.890174	−0.445087	+	0.895487i	$0.646827\pi$
$510$	0	0
$511$	−19.0031	−0.840648
$512$	−35.4114	−1.56498
$513$	0	0
$514$	−73.0819	−3.22351
$515$	−1.36732	−0.0602515
$516$	0	0
$517$	0	0
$518$	−49.1710	−2.16045
$519$	0	0
$520$	40.5196	1.77690
$521$	34.7520	1.52251	0.761256	−	0.648452i	$-0.224583\pi$
$521$	34.7520	1.52251	0.761256	+	0.648452i	$0.224583\pi$
$522$	0	0
$523$	14.6939	0.642519	0.321260	−	0.946991i	$-0.395894\pi$
$523$	14.6939	0.642519	0.321260	+	0.946991i	$0.395894\pi$
$524$	−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
$530$	−9.55982	−0.415252
$531$	0	0
$532$	51.3659	2.22700
$533$	18.2324	0.789733
$534$	0	0
$535$	12.9618	0.560389
$536$	−34.5168	−1.49090
$537$	0	0
$538$	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
$543$	0	0
$544$	50.7936	2.17776
$545$	−5.96655	−0.255579
$546$	0	0
$547$	−27.4545	−1.17387	−0.586934	−	0.809635i	$-0.699665\pi$
$547$	−27.4545	−1.17387	−0.586934	+	0.809635i	$0.699665\pi$
$548$	−101.462	−4.33426
$549$	0	0
$550$	0	0
$551$	13.1710	0.561103
$552$	0	0
$553$	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	$0.692428\pi$
$558$	0	0
$559$	−35.1051	−1.48479
$560$	−43.3784	−1.83307
$561$	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	$0.784264\pi$
$564$	0	0
$565$	−6.06902	−0.255325
$566$	−19.7565	−0.830427
$567$	0	0
$568$	−9.74145	−0.408742
$569$	32.4594	1.36077	0.680384	−	0.732856i	$-0.261813\pi$
$569$	32.4594	1.36077	0.680384	+	0.732856i	$0.261813\pi$
$570$	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	0
$574$	−37.9910	−1.58571
$575$	−5.48352	−0.228679
$576$	0	0
$577$	39.5997	1.64856	0.824279	−	0.566184i	$-0.191581\pi$
$577$	39.5997	1.64856	0.824279	+	0.566184i	$0.191581\pi$
$578$	2.45000	0.101906
$579$	0	0
$580$	−25.9237	−1.07642
$581$	−20.2983	−0.842116
$582$	0	0
$583$	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	$0.327590\pi$
$588$	0	0
$589$	25.3153	1.04310
$590$	−17.3305	−0.713486
$591$	0	0
$592$	−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
$595$	−17.6469	−0.723453
$596$	101.101	4.14124
$597$	0	0
$598$	75.9514	3.10588
$599$	−19.9889	−0.816723	−0.408362	−	0.912820i	$-0.633900\pi$
$599$	−19.9889	−0.816723	−0.408362	+	0.912820i	$0.633900\pi$
$600$	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
$603$	0	0
$604$	−18.9234	−0.769982
$605$	0	0
$606$	0	0
$607$	−11.8089	−0.479310	−0.239655	−	0.970858i	$-0.577034\pi$
$607$	−11.8089	−0.479310	−0.239655	+	0.970858i	$0.577034\pi$
$608$	30.0173	1.21736
$609$	0	0
$610$	−32.2598	−1.30616
$611$	22.2012	0.898166
$612$	0	0
$613$	13.4114	0.541683	0.270841	−	0.962624i	$-0.412698\pi$
$613$	13.4114	0.541683	0.270841	+	0.962624i	$0.412698\pi$
$614$	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	$0.309920\pi$
$618$	0	0
$619$	38.2324	1.53669	0.768345	−	0.640036i	$-0.221081\pi$
$619$	38.2324	1.53669	0.768345	+	0.640036i	$0.221081\pi$
$620$	−49.8266	−2.00108
$621$	0	0
$622$	−59.2630	−2.37623
$623$	−62.9954	−2.52386
$624$	0	0
$625$	1.00000	0.0400000
$626$	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
$633$	0	0
$634$	22.0196	0.874511
$635$	−9.82317	−0.389821
$636$	0	0
$637$	−54.5657	−2.16197
$638$	0	0
$639$	0	0
$640$	−4.29658	−0.169837
$641$	−4.47592	−0.176788	−0.0883941	−	0.996086i	$-0.528173\pi$
$641$	−4.47592	−0.176788	−0.0883941	+	0.996086i	$0.528173\pi$
$642$	0	0
$643$	21.8980	0.863574	0.431787	−	0.901976i	$-0.357883\pi$
$643$	21.8980	0.863574	0.431787	+	0.901976i	$0.357883\pi$
$644$	−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	$0.637372\pi$
$648$	0	0
$649$	0	0
$650$	−13.8508	−0.543275
$651$	0	0
$652$	−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	$0.631586\pi$
$654$	0	0
$655$	3.46410	0.135354
$656$	−36.0576	−1.40781
$657$	0	0
$658$	−46.2609	−1.80344
$659$	18.9855	0.739569	0.369784	−	0.929118i	$-0.379432\pi$
$659$	18.9855	0.739569	0.369784	+	0.929118i	$0.379432\pi$
$660$	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
$663$	0	0
$664$	−37.4977	−1.45519
$665$	−10.4287	−0.404409
$666$	0	0
$667$	−28.8611	−1.11751
$668$	−73.3156	−2.83667
$669$	0	0
$670$	11.7989	0.455832
$671$	0	0
$672$	0	0
$673$	−49.6033	−1.91207	−0.956033	−	0.293261i	$-0.905260\pi$
$673$	−49.6033	−1.91207	−0.956033	+	0.293261i	$0.905260\pi$
$674$	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	$0.270171\pi$
$678$	0	0
$679$	−67.4505	−2.58851
$680$	−32.5997	−1.25014
$681$	0	0
$682$	0	0
$683$	−8.20394	−0.313915	−0.156958	−	0.987605i	$-0.550169\pi$
$683$	−8.20394	−0.313915	−0.156958	+	0.987605i	$0.550169\pi$
$684$	0	0
$685$	20.5997	0.787075
$686$	36.9296	1.40998
$687$	0	0
$688$	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
$695$	17.6984	0.671338
$696$	0	0
$697$	−14.6687	−0.555618
$698$	−16.0933	−0.609141
$699$	0	0
$700$	20.5263	0.775820
$701$	−22.0671	−0.833461	−0.416731	−	0.909030i	$-0.636824\pi$
$701$	−22.0671	−0.833461	−0.416731	+	0.909030i	$0.636824\pi$
$702$	0	0
$703$	−11.2198	−0.423162
$704$	0	0
$705$	0	0
$706$	60.6222	2.28154
$707$	−14.9955	−0.563964
$708$	0	0
$709$	−39.7992	−1.49469	−0.747344	−	0.664437i	$-0.768671\pi$
$709$	−39.7992	−1.49469	−0.747344	+	0.664437i	$0.768671\pi$
$710$	3.32993	0.124970
$711$	0	0
$712$	−116.374	−4.36128
$713$	−55.4724	−2.07746
$714$	0	0
$715$	0	0
$716$	68.2213	2.54955
$717$	0	0
$718$	−40.1665	−1.49900
$719$	−12.1380	−0.452672	−0.226336	−	0.974049i	$-0.572675\pi$
$719$	−12.1380	−0.452672	−0.226336	+	0.974049i	$0.572675\pi$
$720$	0	0
$721$	5.69820	0.212212
$722$	−33.5209	−1.24752
$723$	0	0
$724$	36.2872	1.34860
$725$	5.26324	0.195472
$726$	0	0
$727$	38.9670	1.44521	0.722604	−	0.691262i	$-0.242945\pi$
$727$	38.9670	1.44521	0.722604	+	0.691262i	$0.242945\pi$
$728$	−168.862	−6.25843
$729$	0	0
$730$	12.0000	0.444140
$731$	28.2435	1.04462
$732$	0	0
$733$	5.65576	0.208900	0.104450	−	0.994530i	$-0.466692\pi$
$733$	5.65576	0.208900	0.104450	+	0.994530i	$0.466692\pi$
$734$	−85.2159	−3.14538
$735$	0	0
$736$	−65.7759	−2.42453
$737$	0	0
$738$	0	0
$739$	−28.8040	−1.05957	−0.529786	−	0.848131i	$-0.677728\pi$
$739$	−28.8040	−1.05957	−0.529786	+	0.848131i	$0.677728\pi$
$740$	22.0832	0.811796
$741$	0	0
$742$	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	$0.208157\pi$
$744$	0	0
$745$	−20.5263	−0.752024
$746$	21.4693	0.786047
$747$	0	0
$748$	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
$753$	0	0
$754$	−72.9003	−2.65487
$755$	3.84198	0.139824
$756$	0	0
$757$	42.1741	1.53284	0.766422	−	0.642338i	$-0.222035\pi$
$757$	42.1741	1.53284	0.766422	+	0.642338i	$0.222035\pi$
$758$	19.6938	0.715310
$759$	0	0
$760$	−19.2654	−0.698828
$761$	−6.15318	−0.223052	−0.111526	−	0.993761i	$-0.535574\pi$
$761$	−6.15318	−0.223052	−0.111526	+	0.993761i	$0.535574\pi$
$762$	0	0
$763$	24.8651	0.900176
$764$	−3.34860	−0.121148
$765$	0	0
$766$	47.7324	1.72464
$767$	−34.6610	−1.25154
$768$	0	0
$769$	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
$771$	0	0
$772$	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	$0.711120\pi$
$774$	0	0
$775$	10.1162	0.363385
$776$	−124.604	−4.47301
$777$	0	0
$778$	−46.8032	−1.67798
$779$	−8.66874	−0.310590
$780$	0	0
$781$	0	0
$782$	−61.1061	−2.18515
$783$	0	0
$784$	107.913	3.85403
$785$	0.781830	0.0279047
$786$	0	0
$787$	−4.73655	−0.168840	−0.0844199	−	0.996430i	$-0.526904\pi$
$787$	−4.73655	−0.168840	−0.0844199	+	0.996430i	$0.526904\pi$
$788$	−17.7230	−0.631357
$789$	0	0
$790$	−18.0558	−0.642398
$791$	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	$0.381542\pi$
$798$	0	0
$799$	−17.8618	−0.631906
$800$	11.9952	0.424094
$801$	0	0
$802$	34.9094	1.23269
$803$	0	0
$804$	0	0
$805$	22.8521	0.805431
$806$	−140.118	−4.93544
$807$	0	0
$808$	−27.7017	−0.974542
$809$	48.8707	1.71820	0.859101	−	0.511807i	$-0.171024\pi$
$809$	48.8707	1.71820	0.859101	+	0.511807i	$0.171024\pi$
$810$	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
$813$	0	0
$814$	0	0
$815$	7.74887	0.271431
$816$	0	0
$817$	16.6910	0.583944
$818$	−44.3715	−1.55141
$819$	0	0
$820$	17.0622	0.595837
$821$	26.8136	0.935802	0.467901	−	0.883781i	$-0.345010\pi$
$821$	26.8136	0.935802	0.467901	+	0.883781i	$0.345010\pi$
$822$	0	0
$823$	−20.1020	−0.700711	−0.350355	−	0.936617i	$-0.613939\pi$
$823$	−20.1020	−0.700711	−0.350355	+	0.936617i	$0.613939\pi$
$824$	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	$0.631620\pi$
$828$	0	0
$829$	−41.0721	−1.42649	−0.713247	−	0.700913i	$-0.752776\pi$
$829$	−41.0721	−1.42649	−0.713247	+	0.700913i	$0.752776\pi$
$830$	12.8179	0.444915
$831$	0	0
$832$	−56.5739	−1.96135
$833$	43.9004	1.52106
$834$	0	0
$835$	14.8851	0.515122
$836$	0	0
$837$	0	0
$838$	−25.7795	−0.890538
$839$	7.40338	0.255593	0.127797	−	0.991800i	$-0.459210\pi$
$839$	7.40338	0.255593	0.127797	+	0.991800i	$0.459210\pi$
$840$	0	0
$841$	−1.29831	−0.0447693
$842$	−47.0141	−1.62021
$843$	0	0
$844$	120.426	4.14525
$845$	−14.7017	−0.505754
$846$	0	0
$847$	0	0
$848$	37.8123	1.29848
$849$	0	0
$850$	11.1436	0.382222
$851$	24.5855	0.842780
$852$	0	0
$853$	−26.3686	−0.902845	−0.451423	−	0.892310i	$-0.649083\pi$
$853$	−26.3686	−0.902845	−0.451423	+	0.892310i	$0.649083\pi$
$854$	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	$0.486212\pi$
$860$	−32.8519	−1.12024
$861$	0	0
$862$	−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	$0.434527\pi$
$864$	0	0
$865$	2.19166	0.0745185
$866$	−26.8036	−0.910824
$867$	0	0
$868$	207.648	7.04803
$869$	0	0
$870$	0	0
$871$	23.5979	0.799583
$872$	45.9341	1.55552
$873$	0	0
$874$	−36.1117	−1.22150
$875$	−4.16741	−0.140884
$876$	0	0
$877$	32.1386	1.08524	0.542621	−	0.839978i	$-0.317432\pi$
$877$	32.1386	1.08524	0.542621	+	0.839978i	$0.317432\pi$
$878$	−51.8782	−1.75081
$879$	0	0
$880$	0	0
$881$	−47.1051	−1.58701	−0.793505	−	0.608564i	$-0.791746\pi$
$881$	−47.1051	−1.58701	−0.793505	+	0.608564i	$0.791746\pi$
$882$	0	0
$883$	−19.9528	−0.671466	−0.335733	−	0.941957i	$-0.608984\pi$
$883$	−19.9528	−0.671466	−0.335733	+	0.941957i	$0.608984\pi$
$884$	−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	$0.804058\pi$
$888$	0	0
$889$	40.9372	1.37299
$890$	39.7801	1.33343
$891$	0	0
$892$	7.87930	0.263819
$893$	−10.5558	−0.353235
$894$	0	0
$895$	−13.8508	−0.462983
$896$	17.9056	0.598185
$897$	0	0
$898$	57.1106	1.90581
$899$	53.2440	1.77579
$900$	0	0
$901$	15.3826	0.512468
$902$	0	0
$903$	0	0
$904$	46.7229	1.55398
$905$	−7.36732	−0.244898
$906$	0	0
$907$	40.0361	1.32938	0.664688	−	0.747121i	$-0.268565\pi$
$907$	40.0361	1.32938	0.664688	+	0.747121i	$0.268565\pi$
$908$	−91.6995	−3.04316
$909$	0	0
$910$	57.7222	1.91347
$911$	22.1492	0.733834	0.366917	−	0.930254i	$-0.380413\pi$
$911$	22.1492	0.733834	0.366917	+	0.930254i	$0.380413\pi$
$912$	0	0
$913$	0	0
$914$	31.0503	1.02705
$915$	0	0
$916$	114.932	3.79745
$917$	−14.4363	−0.476730
$918$	0	0
$919$	−24.4453	−0.806377	−0.403189	−	0.915117i	$-0.632098\pi$
$919$	−24.4453	−0.806377	−0.403189	+	0.915117i	$0.632098\pi$
$920$	42.2154	1.39180
$921$	0	0
$922$	−81.7190	−2.69127
$923$	6.65985	0.219212
$924$	0	0
$925$	−4.48352	−0.147417
$926$	100.976	3.31829
$927$	0	0
$928$	63.1335	2.07246
$929$	57.2716	1.87902	0.939509	−	0.342523i	$-0.111281\pi$
$929$	57.2716	1.87902	0.939509	+	0.342523i	$0.111281\pi$
$930$	0	0
$931$	25.9437	0.850270
$932$	−48.0528	−1.57402
$933$	0	0
$934$	−39.3501	−1.28758
$935$	0	0
$936$	0	0
$937$	−51.0099	−1.66642	−0.833210	−	0.552957i	$-0.813500\pi$
$937$	−51.0099	−1.66642	−0.833210	+	0.552957i	$0.813500\pi$
$938$	−49.1710	−1.60549
$939$	0	0
$940$	20.7763	0.677647
$941$	15.1681	0.494467	0.247233	−	0.968956i	$-0.420479\pi$
$941$	15.1681	0.494467	0.247233	+	0.968956i	$0.420479\pi$
$942$	0	0
$943$	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	$0.137171\pi$
$948$	0	0
$949$	24.0000	0.779073
$950$	6.58549	0.213662
$951$	0	0
$952$	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	$0.322224\pi$
$954$	0	0
$955$	0.679859	0.0219997
$956$	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
$963$	0	0
$964$	−44.1862	−1.42314
$965$	−4.95245	−0.159425
$966$	0	0
$967$	−53.9473	−1.73483	−0.867414	−	0.497587i	$-0.834219\pi$
$967$	−53.9473	−1.73483	−0.867414	+	0.497587i	$0.834219\pi$
$968$	0	0
$969$	0	0
$970$	42.5933	1.36759
$971$	−4.14915	−0.133153	−0.0665763	−	0.997781i	$-0.521208\pi$
$971$	−4.14915	−0.133153	−0.0665763	+	0.997781i	$0.521208\pi$
$972$	0	0
$973$	−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	$0.430366\pi$
$978$	0	0
$979$	0	0
$980$	−51.0635	−1.63116
$981$	0	0
$982$	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	$0.699319\pi$
$984$	0	0
$985$	3.59828	0.114651
$986$	58.6514	1.86784
$987$	0	0
$988$	−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
$993$	0	0
$994$	−13.8772	−0.440157
$995$	−3.88380	−0.123125
$996$	0	0
$997$	−41.7558	−1.32242	−0.661210	−	0.750200i	$-0.729957\pi$
$997$	−41.7558	−1.32242	−0.661210	+	0.750200i	$0.729957\pi$
$998$	75.9619	2.40453
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1815.2.a.y.1.1	✓	6	3.2	odd	2
1815.2.a.y.1.6	yes	6	33.32	even	2
5445.2.a.bz.1.1		6	11.10	odd	2	inner
5445.2.a.bz.1.6		6	1.1	even	1	trivial
9075.2.a.dq.1.1		6	165.164	even	2
9075.2.a.dq.1.6		6	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5445 = 3^{2} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5445.a (trivial)

\(f(q)\)	\(=\)	\(q+2.63162 q^{2} +4.92542 q^{4} -1.00000 q^{5} +4.16741 q^{7} +7.69860 q^{8} +O(q^{10})\)	\(q+2.63162 q^{2} +4.92542 q^{4} -1.00000 q^{5} +4.16741 q^{7} +7.69860 q^{8} -2.63162 q^{10} -5.26324 q^{13} +10.9670 q^{14} +10.4089 q^{16} +4.23450 q^{17} +2.50245 q^{19} -4.92542 q^{20} -5.48352 q^{23} +1.00000 q^{25} -13.8508 q^{26} +20.5263 q^{28} +5.26324 q^{29} +10.1162 q^{31} +11.9952 q^{32} +11.1436 q^{34} -4.16741 q^{35} -4.48352 q^{37} +6.58549 q^{38} -7.69860 q^{40} -3.46410 q^{41} +6.66986 q^{43} -14.4305 q^{46} -4.21817 q^{47} +10.3673 q^{49} +2.63162 q^{50} -25.9237 q^{52} +3.63268 q^{53} +32.0832 q^{56} +13.8508 q^{58} +6.58549 q^{59} +12.2585 q^{61} +26.6220 q^{62} +10.7489 q^{64} +5.26324 q^{65} -4.48352 q^{67} +20.8567 q^{68} -10.9670 q^{70} -1.26535 q^{71} -4.55993 q^{73} -11.7989 q^{74} +12.3256 q^{76} +6.86112 q^{79} -10.4089 q^{80} -9.11620 q^{82} -4.87072 q^{83} -4.23450 q^{85} +17.5525 q^{86} -15.1162 q^{89} -21.9341 q^{91} -27.0087 q^{92} -11.1006 q^{94} -2.50245 q^{95} -16.1852 q^{97} +27.2829 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 14 q^{4} - 6 q^{5}+O(q^{10}) \) 6 * q + 14 * q^4 - 6 * q^5	\( 6 q + 14 q^{4} - 6 q^{5} - 8 q^{14} + 10 q^{16} - 14 q^{20} + 4 q^{23} + 6 q^{25} - 52 q^{26} + 18 q^{31} + 26 q^{34} + 10 q^{37} + 20 q^{38} + 68 q^{49} + 16 q^{53} + 76 q^{56} + 52 q^{58} + 20 q^{59} + 16 q^{64} + 10 q^{67} + 8 q^{70} + 4 q^{71} - 10 q^{80} - 12 q^{82} + 12 q^{86} - 48 q^{89} + 16 q^{91} - 30 q^{92} + 2 q^{97}+O(q^{100}) \) 6 * q + 14 * q^4 - 6 * q^5 - 8 * q^14 + 10 * q^16 - 14 * q^20 + 4 * q^23 + 6 * q^25 - 52 * q^26 + 18 * q^31 + 26 * q^34 + 10 * q^37 + 20 * q^38 + 68 * q^49 + 16 * q^53 + 76 * q^56 + 52 * q^58 + 20 * q^59 + 16 * q^64 + 10 * q^67 + 8 * q^70 + 4 * q^71 - 10 * q^80 - 12 * q^82 + 12 * q^86 - 48 * q^89 + 16 * q^91 - 30 * q^92 + 2 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more