LMFDB - Embedded newform 7440.2.a.bu.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7440,2,Mod(1,7440)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7440, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("7440.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7440 = 2^{4} \cdot 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7440.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:	$59.4086991038$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.404.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x - 1 $ x^3 - x^2 - 5*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 3720)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.65544$ of defining polynomial
Character	$\chi$	$=$	7440.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} -0.259511 q^{7} +1.00000 q^{9} -4.00000 q^{11} +2.25951 q^{13} -1.00000 q^{15} +3.31088 q^{17} -6.10275 q^{19} -0.259511 q^{21} +0.791864 q^{23} +1.00000 q^{25} +1.00000 q^{27} -2.53235 q^{29} +1.00000 q^{31} -4.00000 q^{33} +0.259511 q^{35} -4.36226 q^{37} +2.25951 q^{39} +10.2055 q^{41} +10.1027 q^{43} -1.00000 q^{45} +7.41363 q^{47} -6.93265 q^{49} +3.31088 q^{51} -0.689115 q^{53} +4.00000 q^{55} -6.10275 q^{57} -5.05137 q^{59} -1.48098 q^{61} -0.259511 q^{63} -2.25951 q^{65} -5.84324 q^{67} +0.791864 q^{69} -9.05137 q^{71} +0.362259 q^{73} +1.00000 q^{75} +1.03804 q^{77} -4.79186 q^{79} +1.00000 q^{81} -18.0354 q^{83} -3.31088 q^{85} -2.53235 q^{87} -9.15412 q^{89} -0.586367 q^{91} +1.00000 q^{93} +6.10275 q^{95} +9.68648 q^{97} -4.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} - 3 q^{5} + 2 q^{7} + 3 q^{9} - 12 q^{11} + 4 q^{13} - 3 q^{15} - 2 q^{17} + 2 q^{21} - 4 q^{23} + 3 q^{25} + 3 q^{27} - 4 q^{29} + 3 q^{31} - 12 q^{33} - 2 q^{35} + 8 q^{37} + 4 q^{39} - 6 q^{41}+ \cdots - 12 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 - 3 * q^5 + 2 * q^7 + 3 * q^9 - 12 * q^11 + 4 * q^13 - 3 * q^15 - 2 * q^17 + 2 * q^21 - 4 * q^23 + 3 * q^25 + 3 * q^27 - 4 * q^29 + 3 * q^31 - 12 * q^33 - 2 * q^35 + 8 * q^37 + 4 * q^39 - 6 * q^41 + 12 * q^43 - 3 * q^45 - 8 * q^47 + 15 * q^49 - 2 * q^51 - 14 * q^53 + 12 * q^55 - 6 * q^59 - 10 * q^61 + 2 * q^63 - 4 * q^65 - 2 * q^67 - 4 * q^69 - 18 * q^71 - 20 * q^73 + 3 * q^75 - 8 * q^77 - 8 * q^79 + 3 * q^81 + 2 * q^85 - 4 * q^87 - 32 * q^91 + 3 * q^93 - 2 * q^97 - 12 * 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 $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−0.259511	−0.0980858	−0.0490429	−	0.998797i	$-0.515617\pi$
$7$	−0.259511	−0.0980858	−0.0490429	+	0.998797i	$0.515617\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	2.25951	0.626675	0.313338	−	0.949642i	$-0.398553\pi$
$13$	2.25951	0.626675	0.313338	+	0.949642i	$0.398553\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	3.31088	0.803008	0.401504	−	0.915857i	$-0.368488\pi$
$17$	3.31088	0.803008	0.401504	+	0.915857i	$0.368488\pi$
$18$	0	0
$19$	−6.10275	−1.40007	−0.700033	−	0.714110i	$-0.746832\pi$
$19$	−6.10275	−1.40007	−0.700033	+	0.714110i	$0.746832\pi$
$20$	0	0
$21$	−0.259511	−0.0566298
$22$	0	0
$23$	0.791864	0.165115	0.0825575	−	0.996586i	$-0.473691\pi$
$23$	0.791864	0.165115	0.0825575	+	0.996586i	$0.473691\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−2.53235	−0.470246	−0.235123	−	0.971966i	$-0.575549\pi$
$29$	−2.53235	−0.470246	−0.235123	+	0.971966i	$0.575549\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	0	0
$33$	−4.00000	−0.696311
$34$	0	0
$35$	0.259511	0.0438653
$36$	0	0
$37$	−4.36226	−0.717151	−0.358575	−	0.933501i	$-0.616737\pi$
$37$	−4.36226	−0.717151	−0.358575	+	0.933501i	$0.616737\pi$
$38$	0	0
$39$	2.25951	0.361811
$40$	0	0
$41$	10.2055	1.59383	0.796915	−	0.604091i	$-0.206464\pi$
$41$	10.2055	1.59383	0.796915	+	0.604091i	$0.206464\pi$
$42$	0	0
$43$	10.1027	1.54065	0.770327	−	0.637649i	$-0.220093\pi$
$43$	10.1027	1.54065	0.770327	+	0.637649i	$0.220093\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	7.41363	1.08139	0.540695	−	0.841219i	$-0.318161\pi$
$47$	7.41363	1.08139	0.540695	+	0.841219i	$0.318161\pi$
$48$	0	0
$49$	−6.93265	−0.990379
$50$	0	0
$51$	3.31088	0.463617
$52$	0	0
$53$	−0.689115	−0.0946573	−0.0473286	−	0.998879i	$-0.515071\pi$
$53$	−0.689115	−0.0946573	−0.0473286	+	0.998879i	$0.515071\pi$
$54$	0	0
$55$	4.00000	0.539360
$56$	0	0
$57$	−6.10275	−0.808329
$58$	0	0
$59$	−5.05137	−0.657633	−0.328816	−	0.944394i	$-0.606650\pi$
$59$	−5.05137	−0.657633	−0.328816	+	0.944394i	$0.606650\pi$
$60$	0	0
$61$	−1.48098	−0.189620	−0.0948100	−	0.995495i	$-0.530224\pi$
$61$	−1.48098	−0.189620	−0.0948100	+	0.995495i	$0.530224\pi$
$62$	0	0
$63$	−0.259511	−0.0326953
$64$	0	0
$65$	−2.25951	−0.280258
$66$	0	0
$67$	−5.84324	−0.713865	−0.356933	−	0.934130i	$-0.616177\pi$
$67$	−5.84324	−0.713865	−0.356933	+	0.934130i	$0.616177\pi$
$68$	0	0
$69$	0.791864	0.0953292
$70$	0	0
$71$	−9.05137	−1.07420	−0.537100	−	0.843518i	$-0.680480\pi$
$71$	−9.05137	−1.07420	−0.537100	+	0.843518i	$0.680480\pi$
$72$	0	0
$73$	0.362259	0.0423992	0.0211996	−	0.999775i	$-0.493251\pi$
$73$	0.362259	0.0423992	0.0211996	+	0.999775i	$0.493251\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	1.03804	0.118296
$78$	0	0
$79$	−4.79186	−0.539127	−0.269563	−	0.962983i	$-0.586879\pi$
$79$	−4.79186	−0.539127	−0.269563	+	0.962983i	$0.586879\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−18.0354	−1.97964	−0.989821	−	0.142316i	$-0.954545\pi$
$83$	−18.0354	−1.97964	−0.989821	+	0.142316i	$0.954545\pi$
$84$	0	0
$85$	−3.31088	−0.359116
$86$	0	0
$87$	−2.53235	−0.271497
$88$	0	0
$89$	−9.15412	−0.970335	−0.485168	−	0.874421i	$-0.661241\pi$
$89$	−9.15412	−0.970335	−0.485168	+	0.874421i	$0.661241\pi$
$90$	0	0
$91$	−0.586367	−0.0614679
$92$	0	0
$93$	1.00000	0.103695
$94$	0	0
$95$	6.10275	0.626129
$96$	0	0
$97$	9.68648	0.983513	0.491756	−	0.870733i	$-0.336355\pi$
$97$	9.68648	0.983513	0.491756	+	0.870733i	$0.336355\pi$
$98$	0	0
$99$	−4.00000	−0.402015
$100$	0	0
$101$	−16.8273	−1.67438	−0.837188	−	0.546916i	$-0.815802\pi$
$101$	−16.8273	−1.67438	−0.837188	+	0.546916i	$0.815802\pi$
$102$	0	0
$103$	−7.94599	−0.782941	−0.391471	−	0.920191i	$-0.628034\pi$
$103$	−7.94599	−0.782941	−0.391471	+	0.920191i	$0.628034\pi$
$104$	0	0
$105$	0.259511	0.0253256
$106$	0	0
$107$	−7.20814	−0.696837	−0.348418	−	0.937339i	$-0.613281\pi$
$107$	−7.20814	−0.696837	−0.348418	+	0.937339i	$0.613281\pi$
$108$	0	0
$109$	3.31088	0.317125	0.158563	−	0.987349i	$-0.449314\pi$
$109$	3.31088	0.317125	0.158563	+	0.987349i	$0.449314\pi$
$110$	0	0
$111$	−4.36226	−0.414047
$112$	0	0
$113$	−0.416273	−0.0391596	−0.0195798	−	0.999808i	$-0.506233\pi$
$113$	−0.416273	−0.0391596	−0.0195798	+	0.999808i	$0.506233\pi$
$114$	0	0
$115$	−0.791864	−0.0738417
$116$	0	0
$117$	2.25951	0.208892
$118$	0	0
$119$	−0.859209	−0.0787636
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	10.2055	0.920199
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	8.72452	0.774176	0.387088	−	0.922043i	$-0.373481\pi$
$127$	8.72452	0.774176	0.387088	+	0.922043i	$0.373481\pi$
$128$	0	0
$129$	10.1027	0.889497
$130$	0	0
$131$	−21.0514	−1.83927	−0.919634	−	0.392777i	$-0.871514\pi$
$131$	−21.0514	−1.83927	−0.919634	+	0.392777i	$0.871514\pi$
$132$	0	0
$133$	1.58373	0.137327
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	9.20814	0.786704	0.393352	−	0.919388i	$-0.371315\pi$
$137$	9.20814	0.786704	0.393352	+	0.919388i	$0.371315\pi$
$138$	0	0
$139$	−13.5837	−1.15216	−0.576078	−	0.817394i	$-0.695418\pi$
$139$	−13.5837	−1.15216	−0.576078	+	0.817394i	$0.695418\pi$
$140$	0	0
$141$	7.41363	0.624341
$142$	0	0
$143$	−9.03804	−0.755799
$144$	0	0
$145$	2.53235	0.210300
$146$	0	0
$147$	−6.93265	−0.571796
$148$	0	0
$149$	−0.416273	−0.0341024	−0.0170512	−	0.999855i	$-0.505428\pi$
$149$	−0.416273	−0.0341024	−0.0170512	+	0.999855i	$0.505428\pi$
$150$	0	0
$151$	−3.20814	−0.261074	−0.130537	−	0.991443i	$-0.541670\pi$
$151$	−3.20814	−0.261074	−0.130537	+	0.991443i	$0.541670\pi$
$152$	0	0
$153$	3.31088	0.267669
$154$	0	0
$155$	−1.00000	−0.0803219
$156$	0	0
$157$	10.7245	0.855910	0.427955	−	0.903800i	$-0.359234\pi$
$157$	10.7245	0.855910	0.427955	+	0.903800i	$0.359234\pi$
$158$	0	0
$159$	−0.689115	−0.0546504
$160$	0	0
$161$	−0.205497	−0.0161954
$162$	0	0
$163$	15.9460	1.24899	0.624493	−	0.781030i	$-0.285306\pi$
$163$	15.9460	1.24899	0.624493	+	0.781030i	$0.285306\pi$
$164$	0	0
$165$	4.00000	0.311400
$166$	0	0
$167$	−1.58373	−0.122553	−0.0612763	−	0.998121i	$-0.519517\pi$
$167$	−1.58373	−0.122553	−0.0612763	+	0.998121i	$0.519517\pi$
$168$	0	0
$169$	−7.89461	−0.607278
$170$	0	0
$171$	−6.10275	−0.466689
$172$	0	0
$173$	−0.621770	−0.0472723	−0.0236361	−	0.999721i	$-0.507524\pi$
$173$	−0.621770	−0.0472723	−0.0236361	+	0.999721i	$0.507524\pi$
$174$	0	0
$175$	−0.259511	−0.0196172
$176$	0	0
$177$	−5.05137	−0.379685
$178$	0	0
$179$	−5.37823	−0.401988	−0.200994	−	0.979592i	$-0.564417\pi$
$179$	−5.37823	−0.401988	−0.200994	+	0.979592i	$0.564417\pi$
$180$	0	0
$181$	6.51902	0.484555	0.242278	−	0.970207i	$-0.422106\pi$
$181$	6.51902	0.484555	0.242278	+	0.970207i	$0.422106\pi$
$182$	0	0
$183$	−1.48098	−0.109477
$184$	0	0
$185$	4.36226	0.320720
$186$	0	0
$187$	−13.2435	−0.968463
$188$	0	0
$189$	−0.259511	−0.0188766
$190$	0	0
$191$	−24.1922	−1.75048	−0.875242	−	0.483686i	$-0.839298\pi$
$191$	−24.1922	−1.75048	−0.875242	+	0.483686i	$0.839298\pi$
$192$	0	0
$193$	−14.7245	−1.05989	−0.529947	−	0.848031i	$-0.677788\pi$
$193$	−14.7245	−1.05989	−0.529947	+	0.848031i	$0.677788\pi$
$194$	0	0
$195$	−2.25951	−0.161807
$196$	0	0
$197$	−4.37559	−0.311748	−0.155874	−	0.987777i	$-0.549819\pi$
$197$	−4.37559	−0.311748	−0.155874	+	0.987777i	$0.549819\pi$
$198$	0	0
$199$	15.9593	1.13133	0.565663	−	0.824636i	$-0.308620\pi$
$199$	15.9593	1.13133	0.565663	+	0.824636i	$0.308620\pi$
$200$	0	0
$201$	−5.84324	−0.412150
$202$	0	0
$203$	0.657172	0.0461245
$204$	0	0
$205$	−10.2055	−0.712783
$206$	0	0
$207$	0.791864	0.0550383
$208$	0	0
$209$	24.4110	1.68854
$210$	0	0
$211$	−9.89725	−0.681355	−0.340677	−	0.940180i	$-0.610656\pi$
$211$	−9.89725	−0.681355	−0.340677	+	0.940180i	$0.610656\pi$
$212$	0	0
$213$	−9.05137	−0.620190
$214$	0	0
$215$	−10.1027	−0.689002
$216$	0	0
$217$	−0.259511	−0.0176167
$218$	0	0
$219$	0.362259	0.0244792
$220$	0	0
$221$	7.48098	0.503225
$222$	0	0
$223$	13.7892	0.923395	0.461697	−	0.887038i	$-0.347241\pi$
$223$	13.7892	0.923395	0.461697	+	0.887038i	$0.347241\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	16.1382	1.07113	0.535563	−	0.844495i	$-0.320099\pi$
$227$	16.1382	1.07113	0.535563	+	0.844495i	$0.320099\pi$
$228$	0	0
$229$	2.20550	0.145743	0.0728717	−	0.997341i	$-0.476784\pi$
$229$	2.20550	0.145743	0.0728717	+	0.997341i	$0.476784\pi$
$230$	0	0
$231$	1.03804	0.0682982
$232$	0	0
$233$	−13.9327	−0.912759	−0.456379	−	0.889785i	$-0.650854\pi$
$233$	−13.9327	−0.912759	−0.456379	+	0.889785i	$0.650854\pi$
$234$	0	0
$235$	−7.41363	−0.483612
$236$	0	0
$237$	−4.79186	−0.311265
$238$	0	0
$239$	−20.9300	−1.35385	−0.676925	−	0.736052i	$-0.736688\pi$
$239$	−20.9300	−1.35385	−0.676925	+	0.736052i	$0.736688\pi$
$240$	0	0
$241$	0.416273	0.0268145	0.0134072	−	0.999910i	$-0.495732\pi$
$241$	0.416273	0.0268145	0.0134072	+	0.999910i	$0.495732\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	6.93265	0.442911
$246$	0	0
$247$	−13.7892	−0.877387
$248$	0	0
$249$	−18.0354	−1.14295
$250$	0	0
$251$	−28.2055	−1.78032	−0.890158	−	0.455653i	$-0.849406\pi$
$251$	−28.2055	−1.78032	−0.890158	+	0.455653i	$0.849406\pi$
$252$	0	0
$253$	−3.16745	−0.199136
$254$	0	0
$255$	−3.31088	−0.207336
$256$	0	0
$257$	−12.3489	−0.770305	−0.385152	−	0.922853i	$-0.625851\pi$
$257$	−12.3489	−0.770305	−0.385152	+	0.922853i	$0.625851\pi$
$258$	0	0
$259$	1.13205	0.0703423
$260$	0	0
$261$	−2.53235	−0.156749
$262$	0	0
$263$	−21.2435	−1.30993	−0.654966	−	0.755658i	$-0.727317\pi$
$263$	−21.2435	−1.30993	−0.654966	+	0.755658i	$0.727317\pi$
$264$	0	0
$265$	0.689115	0.0423320
$266$	0	0
$267$	−9.15412	−0.560223
$268$	0	0
$269$	−1.98667	−0.121129	−0.0605646	−	0.998164i	$-0.519290\pi$
$269$	−1.98667	−0.121129	−0.0605646	+	0.998164i	$0.519290\pi$
$270$	0	0
$271$	−2.62177	−0.159261	−0.0796306	−	0.996824i	$-0.525374\pi$
$271$	−2.62177	−0.159261	−0.0796306	+	0.996824i	$0.525374\pi$
$272$	0	0
$273$	−0.586367	−0.0354885
$274$	0	0
$275$	−4.00000	−0.241209
$276$	0	0
$277$	16.9079	1.01590	0.507950	−	0.861387i	$-0.330403\pi$
$277$	16.9079	1.01590	0.507950	+	0.861387i	$0.330403\pi$
$278$	0	0
$279$	1.00000	0.0598684
$280$	0	0
$281$	25.3463	1.51203	0.756016	−	0.654553i	$-0.227143\pi$
$281$	25.3463	1.51203	0.756016	+	0.654553i	$0.227143\pi$
$282$	0	0
$283$	−0.805196	−0.0478639	−0.0239320	−	0.999714i	$-0.507619\pi$
$283$	−0.805196	−0.0478639	−0.0239320	+	0.999714i	$0.507619\pi$
$284$	0	0
$285$	6.10275	0.361496
$286$	0	0
$287$	−2.64843	−0.156332
$288$	0	0
$289$	−6.03804	−0.355179
$290$	0	0
$291$	9.68648	0.567831
$292$	0	0
$293$	−7.31088	−0.427106	−0.213553	−	0.976931i	$-0.568504\pi$
$293$	−7.31088	−0.427106	−0.213553	+	0.976931i	$0.568504\pi$
$294$	0	0
$295$	5.05137	0.294102
$296$	0	0
$297$	−4.00000	−0.232104
$298$	0	0
$299$	1.78922	0.103474
$300$	0	0
$301$	−2.62177	−0.151116
$302$	0	0
$303$	−16.8273	−0.966701
$304$	0	0
$305$	1.48098	0.0848006
$306$	0	0
$307$	11.4270	0.652171	0.326086	−	0.945340i	$-0.394270\pi$
$307$	11.4270	0.652171	0.326086	+	0.945340i	$0.394270\pi$
$308$	0	0
$309$	−7.94599	−0.452031
$310$	0	0
$311$	18.6351	1.05670	0.528350	−	0.849027i	$-0.322811\pi$
$311$	18.6351	1.05670	0.528350	+	0.849027i	$0.322811\pi$
$312$	0	0
$313$	−15.2976	−0.864669	−0.432334	−	0.901713i	$-0.642310\pi$
$313$	−15.2976	−0.864669	−0.432334	+	0.901713i	$0.642310\pi$
$314$	0	0
$315$	0.259511	0.0146218
$316$	0	0
$317$	34.6838	1.94804	0.974019	−	0.226466i	$-0.0727171\pi$
$317$	34.6838	1.94804	0.974019	+	0.226466i	$0.0727171\pi$
$318$	0	0
$319$	10.1294	0.567138
$320$	0	0
$321$	−7.20814	−0.402319
$322$	0	0
$323$	−20.2055	−1.12426
$324$	0	0
$325$	2.25951	0.125335
$326$	0	0
$327$	3.31088	0.183092
$328$	0	0
$329$	−1.92392	−0.106069
$330$	0	0
$331$	−6.17009	−0.339139	−0.169570	−	0.985518i	$-0.554238\pi$
$331$	−6.17009	−0.339139	−0.169570	+	0.985518i	$0.554238\pi$
$332$	0	0
$333$	−4.36226	−0.239050
$334$	0	0
$335$	5.84324	0.319250
$336$	0	0
$337$	−20.3623	−1.10920	−0.554601	−	0.832116i	$-0.687129\pi$
$337$	−20.3623	−1.10920	−0.554601	+	0.832116i	$0.687129\pi$
$338$	0	0
$339$	−0.416273	−0.0226088
$340$	0	0
$341$	−4.00000	−0.216612
$342$	0	0
$343$	3.61567	0.195228
$344$	0	0
$345$	−0.791864	−0.0426325
$346$	0	0
$347$	−27.8653	−1.49589	−0.747944	−	0.663762i	$-0.768959\pi$
$347$	−27.8653	−1.49589	−0.747944	+	0.663762i	$0.768959\pi$
$348$	0	0
$349$	−11.1054	−0.594458	−0.297229	−	0.954806i	$-0.596062\pi$
$349$	−11.1054	−0.594458	−0.297229	+	0.954806i	$0.596062\pi$
$350$	0	0
$351$	2.25951	0.120604
$352$	0	0
$353$	−16.8946	−0.899209	−0.449605	−	0.893228i	$-0.648435\pi$
$353$	−16.8946	−0.899209	−0.449605	+	0.893228i	$0.648435\pi$
$354$	0	0
$355$	9.05137	0.480397
$356$	0	0
$357$	−0.859209	−0.0454742
$358$	0	0
$359$	1.88392	0.0994295	0.0497147	−	0.998763i	$-0.484169\pi$
$359$	1.88392	0.0994295	0.0497147	+	0.998763i	$0.484169\pi$
$360$	0	0
$361$	18.2435	0.960186
$362$	0	0
$363$	5.00000	0.262432
$364$	0	0
$365$	−0.362259	−0.0189615
$366$	0	0
$367$	5.58373	0.291468	0.145734	−	0.989324i	$-0.453446\pi$
$367$	5.58373	0.291468	0.145734	+	0.989324i	$0.453446\pi$
$368$	0	0
$369$	10.2055	0.531277
$370$	0	0
$371$	0.178833	0.00928453
$372$	0	0
$373$	37.5518	1.94436	0.972179	−	0.234240i	$-0.0752601\pi$
$373$	37.5518	1.94436	0.972179	+	0.234240i	$0.0752601\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−5.72188	−0.294692
$378$	0	0
$379$	−14.1027	−0.724410	−0.362205	−	0.932099i	$-0.617976\pi$
$379$	−14.1027	−0.724410	−0.362205	+	0.932099i	$0.617976\pi$
$380$	0	0
$381$	8.72452	0.446971
$382$	0	0
$383$	23.6191	1.20688	0.603441	−	0.797408i	$-0.293796\pi$
$383$	23.6191	1.20688	0.603441	+	0.797408i	$0.293796\pi$
$384$	0	0
$385$	−1.03804	−0.0529035
$386$	0	0
$387$	10.1027	0.513552
$388$	0	0
$389$	−14.3977	−0.729990	−0.364995	−	0.931009i	$-0.618929\pi$
$389$	−14.3977	−0.729990	−0.364995	+	0.931009i	$0.618929\pi$
$390$	0	0
$391$	2.62177	0.132589
$392$	0	0
$393$	−21.0514	−1.06190
$394$	0	0
$395$	4.79186	0.241105
$396$	0	0
$397$	−21.3730	−1.07268	−0.536339	−	0.844003i	$-0.680193\pi$
$397$	−21.3730	−1.07268	−0.536339	+	0.844003i	$0.680193\pi$
$398$	0	0
$399$	1.58373	0.0792855
$400$	0	0
$401$	6.32686	0.315948	0.157974	−	0.987443i	$-0.449504\pi$
$401$	6.32686	0.315948	0.157974	+	0.987443i	$0.449504\pi$
$402$	0	0
$403$	2.25951	0.112554
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	17.4490	0.864917
$408$	0	0
$409$	15.7626	0.779408	0.389704	−	0.920940i	$-0.372577\pi$
$409$	15.7626	0.779408	0.389704	+	0.920940i	$0.372577\pi$
$410$	0	0
$411$	9.20814	0.454204
$412$	0	0
$413$	1.31088	0.0645044
$414$	0	0
$415$	18.0354	0.885323
$416$	0	0
$417$	−13.5837	−0.665198
$418$	0	0
$419$	−13.7759	−0.672996	−0.336498	−	0.941684i	$-0.609243\pi$
$419$	−13.7759	−0.672996	−0.336498	+	0.941684i	$0.609243\pi$
$420$	0	0
$421$	16.5544	0.806813	0.403407	−	0.915021i	$-0.367826\pi$
$421$	16.5544	0.806813	0.403407	+	0.915021i	$0.367826\pi$
$422$	0	0
$423$	7.41363	0.360463
$424$	0	0
$425$	3.31088	0.160602
$426$	0	0
$427$	0.384330	0.0185990
$428$	0	0
$429$	−9.03804	−0.436361
$430$	0	0
$431$	33.4624	1.61183	0.805913	−	0.592034i	$-0.201675\pi$
$431$	33.4624	1.61183	0.805913	+	0.592034i	$0.201675\pi$
$432$	0	0
$433$	−2.25951	−0.108585	−0.0542926	−	0.998525i	$-0.517290\pi$
$433$	−2.25951	−0.108585	−0.0542926	+	0.998525i	$0.517290\pi$
$434$	0	0
$435$	2.53235	0.121417
$436$	0	0
$437$	−4.83255	−0.231172
$438$	0	0
$439$	−12.2055	−0.582537	−0.291268	−	0.956641i	$-0.594077\pi$
$439$	−12.2055	−0.582537	−0.291268	+	0.956641i	$0.594077\pi$
$440$	0	0
$441$	−6.93265	−0.330126
$442$	0	0
$443$	−32.1382	−1.52693	−0.763465	−	0.645849i	$-0.776503\pi$
$443$	−32.1382	−1.52693	−0.763465	+	0.645849i	$0.776503\pi$
$444$	0	0
$445$	9.15412	0.433947
$446$	0	0
$447$	−0.416273	−0.0196890
$448$	0	0
$449$	−33.0461	−1.55954	−0.779771	−	0.626065i	$-0.784664\pi$
$449$	−33.0461	−1.55954	−0.779771	+	0.626065i	$0.784664\pi$
$450$	0	0
$451$	−40.8220	−1.92223
$452$	0	0
$453$	−3.20814	−0.150731
$454$	0	0
$455$	0.586367	0.0274893
$456$	0	0
$457$	−0.675783	−0.0316118	−0.0158059	−	0.999875i	$-0.505031\pi$
$457$	−0.675783	−0.0316118	−0.0158059	+	0.999875i	$0.505031\pi$
$458$	0	0
$459$	3.31088	0.154539
$460$	0	0
$461$	21.9867	1.02402	0.512011	−	0.858979i	$-0.328901\pi$
$461$	21.9867	1.02402	0.512011	+	0.858979i	$0.328901\pi$
$462$	0	0
$463$	−26.1027	−1.21310	−0.606549	−	0.795046i	$-0.707447\pi$
$463$	−26.1027	−1.21310	−0.606549	+	0.795046i	$0.707447\pi$
$464$	0	0
$465$	−1.00000	−0.0463739
$466$	0	0
$467$	−16.6572	−0.770802	−0.385401	−	0.922749i	$-0.625937\pi$
$467$	−16.6572	−0.770802	−0.385401	+	0.922749i	$0.625937\pi$
$468$	0	0
$469$	1.51638	0.0700200
$470$	0	0
$471$	10.7245	0.494160
$472$	0	0
$473$	−40.4110	−1.85810
$474$	0	0
$475$	−6.10275	−0.280013
$476$	0	0
$477$	−0.689115	−0.0315524
$478$	0	0
$479$	−40.4243	−1.84703	−0.923517	−	0.383557i	$-0.874699\pi$
$479$	−40.4243	−1.84703	−0.923517	+	0.383557i	$0.874699\pi$
$480$	0	0
$481$	−9.85657	−0.449421
$482$	0	0
$483$	−0.205497	−0.00935044
$484$	0	0
$485$	−9.68648	−0.439840
$486$	0	0
$487$	17.7626	0.804898	0.402449	−	0.915442i	$-0.368159\pi$
$487$	17.7626	0.804898	0.402449	+	0.915442i	$0.368159\pi$
$488$	0	0
$489$	15.9460	0.721102
$490$	0	0
$491$	1.06471	0.0480495	0.0240248	−	0.999711i	$-0.492352\pi$
$491$	1.06471	0.0480495	0.0240248	+	0.999711i	$0.492352\pi$
$492$	0	0
$493$	−8.38433	−0.377611
$494$	0	0
$495$	4.00000	0.179787
$496$	0	0
$497$	2.34893	0.105364
$498$	0	0
$499$	30.4871	1.36479	0.682395	−	0.730984i	$-0.260939\pi$
$499$	30.4871	1.36479	0.682395	+	0.730984i	$0.260939\pi$
$500$	0	0
$501$	−1.58373	−0.0707557
$502$	0	0
$503$	−3.75382	−0.167375	−0.0836873	−	0.996492i	$-0.526670\pi$
$503$	−3.75382	−0.167375	−0.0836873	+	0.996492i	$0.526670\pi$
$504$	0	0
$505$	16.8273	0.748804
$506$	0	0
$507$	−7.89461	−0.350612
$508$	0	0
$509$	12.6084	0.558859	0.279430	−	0.960166i	$-0.409855\pi$
$509$	12.6084	0.558859	0.279430	+	0.960166i	$0.409855\pi$
$510$	0	0
$511$	−0.0940100	−0.00415876
$512$	0	0
$513$	−6.10275	−0.269443
$514$	0	0
$515$	7.94599	0.350142
$516$	0	0
$517$	−29.6545	−1.30420
$518$	0	0
$519$	−0.621770	−0.0272927
$520$	0	0
$521$	15.9947	0.700741	0.350371	−	0.936611i	$-0.386056\pi$
$521$	15.9947	0.700741	0.350371	+	0.936611i	$0.386056\pi$
$522$	0	0
$523$	10.9353	0.478167	0.239084	−	0.970999i	$-0.423153\pi$
$523$	10.9353	0.478167	0.239084	+	0.970999i	$0.423153\pi$
$524$	0	0
$525$	−0.259511	−0.0113260
$526$	0	0
$527$	3.31088	0.144224
$528$	0	0
$529$	−22.3730	−0.972737
$530$	0	0
$531$	−5.05137	−0.219211
$532$	0	0
$533$	23.0594	0.998815
$534$	0	0
$535$	7.20814	0.311635
$536$	0	0
$537$	−5.37823	−0.232088
$538$	0	0
$539$	27.7306	1.19444
$540$	0	0
$541$	−14.1382	−0.607847	−0.303923	−	0.952697i	$-0.598297\pi$
$541$	−14.1382	−0.607847	−0.303923	+	0.952697i	$0.598297\pi$
$542$	0	0
$543$	6.51902	0.279758
$544$	0	0
$545$	−3.31088	−0.141823
$546$	0	0
$547$	−12.8052	−0.547511	−0.273755	−	0.961799i	$-0.588266\pi$
$547$	−12.8052	−0.547511	−0.273755	+	0.961799i	$0.588266\pi$
$548$	0	0
$549$	−1.48098	−0.0632066
$550$	0	0
$551$	15.4543	0.658376
$552$	0	0
$553$	1.24354	0.0528807
$554$	0	0
$555$	4.36226	0.185168
$556$	0	0
$557$	28.0087	1.18677	0.593384	−	0.804919i	$-0.297792\pi$
$557$	28.0087	1.18677	0.593384	+	0.804919i	$0.297792\pi$
$558$	0	0
$559$	22.8273	0.965491
$560$	0	0
$561$	−13.2435	−0.559143
$562$	0	0
$563$	−1.82991	−0.0771213	−0.0385607	−	0.999256i	$-0.512277\pi$
$563$	−1.82991	−0.0771213	−0.0385607	+	0.999256i	$0.512277\pi$
$564$	0	0
$565$	0.416273	0.0175127
$566$	0	0
$567$	−0.259511	−0.0108984
$568$	0	0
$569$	15.9814	0.669975	0.334987	−	0.942223i	$-0.391268\pi$
$569$	15.9814	0.669975	0.334987	+	0.942223i	$0.391268\pi$
$570$	0	0
$571$	3.45431	0.144559	0.0722793	−	0.997384i	$-0.476973\pi$
$571$	3.45431	0.144559	0.0722793	+	0.997384i	$0.476973\pi$
$572$	0	0
$573$	−24.1922	−1.01064
$574$	0	0
$575$	0.791864	0.0330230
$576$	0	0
$577$	−35.4490	−1.47576	−0.737881	−	0.674930i	$-0.764174\pi$
$577$	−35.4490	−1.47576	−0.737881	+	0.674930i	$0.764174\pi$
$578$	0	0
$579$	−14.7245	−0.611930
$580$	0	0
$581$	4.68038	0.194175
$582$	0	0
$583$	2.75646	0.114161
$584$	0	0
$585$	−2.25951	−0.0934193
$586$	0	0
$587$	−28.0000	−1.15568	−0.577842	−	0.816149i	$-0.696105\pi$
$587$	−28.0000	−1.15568	−0.577842	+	0.816149i	$0.696105\pi$
$588$	0	0
$589$	−6.10275	−0.251459
$590$	0	0
$591$	−4.37559	−0.179988
$592$	0	0
$593$	−14.8325	−0.609100	−0.304550	−	0.952496i	$-0.598506\pi$
$593$	−14.8325	−0.609100	−0.304550	+	0.952496i	$0.598506\pi$
$594$	0	0
$595$	0.859209	0.0352242
$596$	0	0
$597$	15.9593	0.653171
$598$	0	0
$599$	20.7112	0.846236	0.423118	−	0.906075i	$-0.360936\pi$
$599$	20.7112	0.846236	0.423118	+	0.906075i	$0.360936\pi$
$600$	0	0
$601$	30.7245	1.25328	0.626640	−	0.779309i	$-0.284430\pi$
$601$	30.7245	1.25328	0.626640	+	0.779309i	$0.284430\pi$
$602$	0	0
$603$	−5.84324	−0.237955
$604$	0	0
$605$	−5.00000	−0.203279
$606$	0	0
$607$	−25.3950	−1.03075	−0.515376	−	0.856964i	$-0.672348\pi$
$607$	−25.3950	−1.03075	−0.515376	+	0.856964i	$0.672348\pi$
$608$	0	0
$609$	0.657172	0.0266300
$610$	0	0
$611$	16.7512	0.677680
$612$	0	0
$613$	30.4383	1.22939	0.614697	−	0.788764i	$-0.289278\pi$
$613$	30.4383	1.22939	0.614697	+	0.788764i	$0.289278\pi$
$614$	0	0
$615$	−10.2055	−0.411525
$616$	0	0
$617$	−30.2055	−1.21603	−0.608014	−	0.793926i	$-0.708033\pi$
$617$	−30.2055	−1.21603	−0.608014	+	0.793926i	$0.708033\pi$
$618$	0	0
$619$	−10.8679	−0.436820	−0.218410	−	0.975857i	$-0.570087\pi$
$619$	−10.8679	−0.436820	−0.218410	+	0.975857i	$0.570087\pi$
$620$	0	0
$621$	0.791864	0.0317764
$622$	0	0
$623$	2.37559	0.0951761
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	24.4110	0.974881
$628$	0	0
$629$	−14.4429	−0.575878
$630$	0	0
$631$	44.2055	1.75979	0.879897	−	0.475165i	$-0.157612\pi$
$631$	44.2055	1.75979	0.879897	+	0.475165i	$0.157612\pi$
$632$	0	0
$633$	−9.89725	−0.393380
$634$	0	0
$635$	−8.72452	−0.346222
$636$	0	0
$637$	−15.6644	−0.620646
$638$	0	0
$639$	−9.05137	−0.358067
$640$	0	0
$641$	29.3596	1.15964	0.579818	−	0.814746i	$-0.303124\pi$
$641$	29.3596	1.15964	0.579818	+	0.814746i	$0.303124\pi$
$642$	0	0
$643$	44.9300	1.77187	0.885933	−	0.463812i	$-0.153519\pi$
$643$	44.9300	1.77187	0.885933	+	0.463812i	$0.153519\pi$
$644$	0	0
$645$	−10.1027	−0.397795
$646$	0	0
$647$	−36.0708	−1.41809	−0.709045	−	0.705163i	$-0.750874\pi$
$647$	−36.0708	−1.41809	−0.709045	+	0.705163i	$0.750874\pi$
$648$	0	0
$649$	20.2055	0.793135
$650$	0	0
$651$	−0.259511	−0.0101710
$652$	0	0
$653$	−22.1382	−0.866333	−0.433166	−	0.901314i	$-0.642604\pi$
$653$	−22.1382	−0.866333	−0.433166	+	0.901314i	$0.642604\pi$
$654$	0	0
$655$	21.0514	0.822545
$656$	0	0
$657$	0.362259	0.0141331
$658$	0	0
$659$	−14.9486	−0.582316	−0.291158	−	0.956675i	$-0.594041\pi$
$659$	−14.9486	−0.582316	−0.291158	+	0.956675i	$0.594041\pi$
$660$	0	0
$661$	22.0000	0.855701	0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000	0.855701	0.427850	+	0.903850i	$0.359271\pi$
$662$	0	0
$663$	7.48098	0.290537
$664$	0	0
$665$	−1.58373	−0.0614143
$666$	0	0
$667$	−2.00528	−0.0776447
$668$	0	0
$669$	13.7892	0.533122
$670$	0	0
$671$	5.92392	0.228690
$672$	0	0
$673$	−17.5350	−0.675924	−0.337962	−	0.941160i	$-0.609738\pi$
$673$	−17.5350	−0.675924	−0.337962	+	0.941160i	$0.609738\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	7.72188	0.296776	0.148388	−	0.988929i	$-0.452592\pi$
$677$	7.72188	0.296776	0.148388	+	0.988929i	$0.452592\pi$
$678$	0	0
$679$	−2.51374	−0.0964686
$680$	0	0
$681$	16.1382	0.618415
$682$	0	0
$683$	46.5544	1.78136	0.890678	−	0.454635i	$-0.150230\pi$
$683$	46.5544	1.78136	0.890678	+	0.454635i	$0.150230\pi$
$684$	0	0
$685$	−9.20814	−0.351825
$686$	0	0
$687$	2.20550	0.0841450
$688$	0	0
$689$	−1.55706	−0.0593194
$690$	0	0
$691$	−26.2816	−0.999798	−0.499899	−	0.866084i	$-0.666630\pi$
$691$	−26.2816	−0.999798	−0.499899	+	0.866084i	$0.666630\pi$
$692$	0	0
$693$	1.03804	0.0394320
$694$	0	0
$695$	13.5837	0.515260
$696$	0	0
$697$	33.7892	1.27986
$698$	0	0
$699$	−13.9327	−0.526981
$700$	0	0
$701$	35.2702	1.33214	0.666069	−	0.745890i	$-0.267976\pi$
$701$	35.2702	1.33214	0.666069	+	0.745890i	$0.267976\pi$
$702$	0	0
$703$	26.6218	1.00406
$704$	0	0
$705$	−7.41363	−0.279214
$706$	0	0
$707$	4.36685	0.164232
$708$	0	0
$709$	−46.2055	−1.73528	−0.867642	−	0.497190i	$-0.834365\pi$
$709$	−46.2055	−1.73528	−0.867642	+	0.497190i	$0.834365\pi$
$710$	0	0
$711$	−4.79186	−0.179709
$712$	0	0
$713$	0.791864	0.0296555
$714$	0	0
$715$	9.03804	0.338004
$716$	0	0
$717$	−20.9300	−0.781646
$718$	0	0
$719$	21.4490	0.799914	0.399957	−	0.916534i	$-0.369025\pi$
$719$	21.4490	0.799914	0.399957	+	0.916534i	$0.369025\pi$
$720$	0	0
$721$	2.06207	0.0767954
$722$	0	0
$723$	0.416273	0.0154813
$724$	0	0
$725$	−2.53235	−0.0940492
$726$	0	0
$727$	16.4917	0.611642	0.305821	−	0.952089i	$-0.401069\pi$
$727$	16.4917	0.611642	0.305821	+	0.952089i	$0.401069\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	33.4490	1.23716
$732$	0	0
$733$	−37.2383	−1.37543	−0.687713	−	0.725982i	$-0.741385\pi$
$733$	−37.2383	−1.37543	−0.687713	+	0.725982i	$0.741385\pi$
$734$	0	0
$735$	6.93265	0.255715
$736$	0	0
$737$	23.3730	0.860954
$738$	0	0
$739$	13.6244	0.501182	0.250591	−	0.968093i	$-0.419375\pi$
$739$	13.6244	0.501182	0.250591	+	0.968093i	$0.419375\pi$
$740$	0	0
$741$	−13.7892	−0.506560
$742$	0	0
$743$	48.4110	1.77603	0.888014	−	0.459817i	$-0.152085\pi$
$743$	48.4110	1.77603	0.888014	+	0.459817i	$0.152085\pi$
$744$	0	0
$745$	0.416273	0.0152510
$746$	0	0
$747$	−18.0354	−0.659881
$748$	0	0
$749$	1.87059	0.0683498
$750$	0	0
$751$	−34.6218	−1.26337	−0.631683	−	0.775227i	$-0.717636\pi$
$751$	−34.6218	−1.26337	−0.631683	+	0.775227i	$0.717636\pi$
$752$	0	0
$753$	−28.2055	−1.02787
$754$	0	0
$755$	3.20814	0.116756
$756$	0	0
$757$	34.6705	1.26012	0.630060	−	0.776546i	$-0.283030\pi$
$757$	34.6705	1.26012	0.630060	+	0.776546i	$0.283030\pi$
$758$	0	0
$759$	−3.16745	−0.114971
$760$	0	0
$761$	−4.63510	−0.168022	−0.0840112	−	0.996465i	$-0.526773\pi$
$761$	−4.63510	−0.168022	−0.0840112	+	0.996465i	$0.526773\pi$
$762$	0	0
$763$	−0.859209	−0.0311055
$764$	0	0
$765$	−3.31088	−0.119705
$766$	0	0
$767$	−11.4136	−0.412122
$768$	0	0
$769$	−25.1001	−0.905133	−0.452567	−	0.891731i	$-0.649492\pi$
$769$	−25.1001	−0.905133	−0.452567	+	0.891731i	$0.649492\pi$
$770$	0	0
$771$	−12.3489	−0.444736
$772$	0	0
$773$	−21.9327	−0.788863	−0.394431	−	0.918925i	$-0.629058\pi$
$773$	−21.9327	−0.788863	−0.394431	+	0.918925i	$0.629058\pi$
$774$	0	0
$775$	1.00000	0.0359211
$776$	0	0
$777$	1.13205	0.0406121
$778$	0	0
$779$	−62.2816	−2.23147
$780$	0	0
$781$	36.2055	1.29553
$782$	0	0
$783$	−2.53235	−0.0904989
$784$	0	0
$785$	−10.7245	−0.382774
$786$	0	0
$787$	8.49236	0.302720	0.151360	−	0.988479i	$-0.451635\pi$
$787$	8.49236	0.302720	0.151360	+	0.988479i	$0.451635\pi$
$788$	0	0
$789$	−21.2435	−0.756290
$790$	0	0
$791$	0.108027	0.00384100
$792$	0	0
$793$	−3.34629	−0.118830
$794$	0	0
$795$	0.689115	0.0244404
$796$	0	0
$797$	−34.1115	−1.20829	−0.604145	−	0.796874i	$-0.706485\pi$
$797$	−34.1115	−1.20829	−0.604145	+	0.796874i	$0.706485\pi$
$798$	0	0
$799$	24.5457	0.868364
$800$	0	0
$801$	−9.15412	−0.323445
$802$	0	0
$803$	−1.44904	−0.0511354
$804$	0	0
$805$	0.205497	0.00724282
$806$	0	0
$807$	−1.98667	−0.0699340
$808$	0	0
$809$	24.2682	0.853226	0.426613	−	0.904434i	$-0.359707\pi$
$809$	24.2682	0.853226	0.426613	+	0.904434i	$0.359707\pi$
$810$	0	0
$811$	30.6484	1.07621	0.538106	−	0.842877i	$-0.319140\pi$
$811$	30.6484	1.07621	0.538106	+	0.842877i	$0.319140\pi$
$812$	0	0
$813$	−2.62177	−0.0919495
$814$	0	0
$815$	−15.9460	−0.558564
$816$	0	0
$817$	−61.6545	−2.15702
$818$	0	0
$819$	−0.586367	−0.0204893
$820$	0	0
$821$	8.42960	0.294195	0.147098	−	0.989122i	$-0.453007\pi$
$821$	8.42960	0.294195	0.147098	+	0.989122i	$0.453007\pi$
$822$	0	0
$823$	10.1027	0.352160	0.176080	−	0.984376i	$-0.443658\pi$
$823$	10.1027	0.352160	0.176080	+	0.984376i	$0.443658\pi$
$824$	0	0
$825$	−4.00000	−0.139262
$826$	0	0
$827$	−16.5864	−0.576764	−0.288382	−	0.957515i	$-0.593117\pi$
$827$	−16.5864	−0.576764	−0.288382	+	0.957515i	$0.593117\pi$
$828$	0	0
$829$	−33.3463	−1.15816	−0.579082	−	0.815269i	$-0.696589\pi$
$829$	−33.3463	−1.15816	−0.579082	+	0.815269i	$0.696589\pi$
$830$	0	0
$831$	16.9079	0.586530
$832$	0	0
$833$	−22.9532	−0.795282
$834$	0	0
$835$	1.58373	0.0548071
$836$	0	0
$837$	1.00000	0.0345651
$838$	0	0
$839$	18.2949	0.631611	0.315805	−	0.948824i	$-0.397725\pi$
$839$	18.2949	0.631611	0.315805	+	0.948824i	$0.397725\pi$
$840$	0	0
$841$	−22.5872	−0.778869
$842$	0	0
$843$	25.3463	0.872973
$844$	0	0
$845$	7.89461	0.271583
$846$	0	0
$847$	−1.29755	−0.0445844
$848$	0	0
$849$	−0.805196	−0.0276343
$850$	0	0
$851$	−3.45431	−0.118412
$852$	0	0
$853$	20.4163	0.699040	0.349520	−	0.936929i	$-0.386345\pi$
$853$	20.4163	0.699040	0.349520	+	0.936929i	$0.386345\pi$
$854$	0	0
$855$	6.10275	0.208710
$856$	0	0
$857$	−4.62177	−0.157877	−0.0789383	−	0.996880i	$-0.525153\pi$
$857$	−4.62177	−0.157877	−0.0789383	+	0.996880i	$0.525153\pi$
$858$	0	0
$859$	18.9620	0.646974	0.323487	−	0.946233i	$-0.395145\pi$
$859$	18.9620	0.646974	0.323487	+	0.946233i	$0.395145\pi$
$860$	0	0
$861$	−2.64843	−0.0902584
$862$	0	0
$863$	−12.6165	−0.429470	−0.214735	−	0.976672i	$-0.568889\pi$
$863$	−12.6165	−0.429470	−0.214735	+	0.976672i	$0.568889\pi$
$864$	0	0
$865$	0.621770	0.0211408
$866$	0	0
$867$	−6.03804	−0.205063
$868$	0	0
$869$	19.1675	0.650211
$870$	0	0
$871$	−13.2029	−0.447362
$872$	0	0
$873$	9.68648	0.327838
$874$	0	0
$875$	0.259511	0.00877306
$876$	0	0
$877$	−13.7945	−0.465807	−0.232904	−	0.972500i	$-0.574823\pi$
$877$	−13.7945	−0.465807	−0.232904	+	0.972500i	$0.574823\pi$
$878$	0	0
$879$	−7.31088	−0.246590
$880$	0	0
$881$	−24.5271	−0.826338	−0.413169	−	0.910654i	$-0.635578\pi$
$881$	−24.5271	−0.826338	−0.413169	+	0.910654i	$0.635578\pi$
$882$	0	0
$883$	15.4543	0.520079	0.260040	−	0.965598i	$-0.416264\pi$
$883$	15.4543	0.520079	0.260040	+	0.965598i	$0.416264\pi$
$884$	0	0
$885$	5.05137	0.169800
$886$	0	0
$887$	4.68384	0.157268	0.0786339	−	0.996904i	$-0.474944\pi$
$887$	4.68384	0.157268	0.0786339	+	0.996904i	$0.474944\pi$
$888$	0	0
$889$	−2.26410	−0.0759356
$890$	0	0
$891$	−4.00000	−0.134005
$892$	0	0
$893$	−45.2435	−1.51402
$894$	0	0
$895$	5.37823	0.179774
$896$	0	0
$897$	1.78922	0.0597405
$898$	0	0
$899$	−2.53235	−0.0844587
$900$	0	0
$901$	−2.28158	−0.0760105
$902$	0	0
$903$	−2.62177	−0.0872470
$904$	0	0
$905$	−6.51902	−0.216700
$906$	0	0
$907$	−35.3189	−1.17275	−0.586373	−	0.810041i	$-0.699445\pi$
$907$	−35.3189	−1.17275	−0.586373	+	0.810041i	$0.699445\pi$
$908$	0	0
$909$	−16.8273	−0.558125
$910$	0	0
$911$	−53.8600	−1.78446	−0.892231	−	0.451579i	$-0.850861\pi$
$911$	−53.8600	−1.78446	−0.892231	+	0.451579i	$0.850861\pi$
$912$	0	0
$913$	72.1416	2.38754
$914$	0	0
$915$	1.48098	0.0489597
$916$	0	0
$917$	5.46305	0.180406
$918$	0	0
$919$	−8.93001	−0.294574	−0.147287	−	0.989094i	$-0.547054\pi$
$919$	−8.93001	−0.294574	−0.147287	+	0.989094i	$0.547054\pi$
$920$	0	0
$921$	11.4270	0.376531
$922$	0	0
$923$	−20.4517	−0.673175
$924$	0	0
$925$	−4.36226	−0.143430
$926$	0	0
$927$	−7.94599	−0.260980
$928$	0	0
$929$	13.4676	0.441859	0.220930	−	0.975290i	$-0.429091\pi$
$929$	13.4676	0.441859	0.220930	+	0.975290i	$0.429091\pi$
$930$	0	0
$931$	42.3082	1.38660
$932$	0	0
$933$	18.6351	0.610086
$934$	0	0
$935$	13.2435	0.433110
$936$	0	0
$937$	44.3082	1.44749	0.723744	−	0.690069i	$-0.242420\pi$
$937$	44.3082	1.44749	0.723744	+	0.690069i	$0.242420\pi$
$938$	0	0
$939$	−15.2976	−0.499217
$940$	0	0
$941$	7.25687	0.236567	0.118284	−	0.992980i	$-0.462261\pi$
$941$	7.25687	0.236567	0.118284	+	0.992980i	$0.462261\pi$
$942$	0	0
$943$	8.08136	0.263165
$944$	0	0
$945$	0.259511	0.00844188
$946$	0	0
$947$	13.2029	0.429035	0.214518	−	0.976720i	$-0.431182\pi$
$947$	13.2029	0.429035	0.214518	+	0.976720i	$0.431182\pi$
$948$	0	0
$949$	0.818528	0.0265705
$950$	0	0
$951$	34.6838	1.12470
$952$	0	0
$953$	−22.2462	−0.720624	−0.360312	−	0.932832i	$-0.617330\pi$
$953$	−22.2462	−0.720624	−0.360312	+	0.932832i	$0.617330\pi$
$954$	0	0
$955$	24.1922	0.782840
$956$	0	0
$957$	10.1294	0.327437
$958$	0	0
$959$	−2.38961	−0.0771645
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	−7.20814	−0.232279
$964$	0	0
$965$	14.7245	0.473999
$966$	0	0
$967$	4.69785	0.151073	0.0755364	−	0.997143i	$-0.475933\pi$
$967$	4.69785	0.151073	0.0755364	+	0.997143i	$0.475933\pi$
$968$	0	0
$969$	−20.2055	−0.649094
$970$	0	0
$971$	8.01333	0.257160	0.128580	−	0.991699i	$-0.458958\pi$
$971$	8.01333	0.257160	0.128580	+	0.991699i	$0.458958\pi$
$972$	0	0
$973$	3.52512	0.113010
$974$	0	0
$975$	2.25951	0.0723622
$976$	0	0
$977$	58.6165	1.87531	0.937654	−	0.347571i	$-0.112993\pi$
$977$	58.6165	1.87531	0.937654	+	0.347571i	$0.112993\pi$
$978$	0	0
$979$	36.6165	1.17027
$980$	0	0
$981$	3.31088	0.105708
$982$	0	0
$983$	40.2763	1.28461	0.642307	−	0.766447i	$-0.277977\pi$
$983$	40.2763	1.28461	0.642307	+	0.766447i	$0.277977\pi$
$984$	0	0
$985$	4.37559	0.139418
$986$	0	0
$987$	−1.92392	−0.0612389
$988$	0	0
$989$	8.00000	0.254385
$990$	0	0
$991$	47.8653	1.52049	0.760246	−	0.649636i	$-0.225078\pi$
$991$	47.8653	1.52049	0.760246	+	0.649636i	$0.225078\pi$
$992$	0	0
$993$	−6.17009	−0.195802
$994$	0	0
$995$	−15.9593	−0.505944
$996$	0	0
$997$	21.5518	0.682552	0.341276	−	0.939963i	$-0.389141\pi$
$997$	21.5518	0.682552	0.341276	+	0.939963i	$0.389141\pi$
$998$	0	0
$999$	−4.36226	−0.138016

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7440.2.a.bu.1.2		3
4.3	odd	2		3720.2.a.l.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3720.2.a.l.1.2	✓	3	4.3	odd	2
7440.2.a.bu.1.2		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:	\( N \)	\(=\)	\( 7440 = 2^{4} \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7440.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} -0.259511 q^{7} +1.00000 q^{9} -4.00000 q^{11} +2.25951 q^{13} -1.00000 q^{15} +3.31088 q^{17} -6.10275 q^{19} -0.259511 q^{21} +0.791864 q^{23} +1.00000 q^{25} +1.00000 q^{27} -2.53235 q^{29} +1.00000 q^{31} -4.00000 q^{33} +0.259511 q^{35} -4.36226 q^{37} +2.25951 q^{39} +10.2055 q^{41} +10.1027 q^{43} -1.00000 q^{45} +7.41363 q^{47} -6.93265 q^{49} +3.31088 q^{51} -0.689115 q^{53} +4.00000 q^{55} -6.10275 q^{57} -5.05137 q^{59} -1.48098 q^{61} -0.259511 q^{63} -2.25951 q^{65} -5.84324 q^{67} +0.791864 q^{69} -9.05137 q^{71} +0.362259 q^{73} +1.00000 q^{75} +1.03804 q^{77} -4.79186 q^{79} +1.00000 q^{81} -18.0354 q^{83} -3.31088 q^{85} -2.53235 q^{87} -9.15412 q^{89} -0.586367 q^{91} +1.00000 q^{93} +6.10275 q^{95} +9.68648 q^{97} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} - 3 q^{5} + 2 q^{7} + 3 q^{9} - 12 q^{11} + 4 q^{13} - 3 q^{15} - 2 q^{17} + 2 q^{21} - 4 q^{23} + 3 q^{25} + 3 q^{27} - 4 q^{29} + 3 q^{31} - 12 q^{33} - 2 q^{35} + 8 q^{37} + 4 q^{39} - 6 q^{41}+ \cdots - 12 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 - 3 * q^5 + 2 * q^7 + 3 * q^9 - 12 * q^11 + 4 * q^13 - 3 * q^15 - 2 * q^17 + 2 * q^21 - 4 * q^23 + 3 * q^25 + 3 * q^27 - 4 * q^29 + 3 * q^31 - 12 * q^33 - 2 * q^35 + 8 * q^37 + 4 * q^39 - 6 * q^41 + 12 * q^43 - 3 * q^45 - 8 * q^47 + 15 * q^49 - 2 * q^51 - 14 * q^53 + 12 * q^55 - 6 * q^59 - 10 * q^61 + 2 * q^63 - 4 * q^65 - 2 * q^67 - 4 * q^69 - 18 * q^71 - 20 * q^73 + 3 * q^75 - 8 * q^77 - 8 * q^79 + 3 * q^81 + 2 * q^85 - 4 * q^87 - 32 * q^91 + 3 * q^93 - 2 * q^97 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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