LMFDB - Embedded newform 3465.2.a.bl.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3465,2,Mod(1,3465)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3465, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3465.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3465 = 3^{2} \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3465.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:	$27.6681643004$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.13448.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 7x^{2} + 2 $ x^4 - 7*x^2 + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1155)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.58874$ of defining polynomial
Character	$\chi$	$=$	3465.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.58874 q^{2} +4.70156 q^{4} +1.00000 q^{5} -1.00000 q^{7} +6.99364 q^{8} +O(q^{10})$	\(q+2.58874 q^{2} +4.70156 q^{4} +1.00000 q^{5} -1.00000 q^{7} +6.99364 q^{8} +2.58874 q^{10} +1.00000 q^{11} -2.40490 q^{13} -2.58874 q^{14} +8.70156 q^{16} +3.70156 q^{17} +4.15641 q^{19} +4.70156 q^{20} +2.58874 q^{22} +0.0692417 q^{23} +1.00000 q^{25} -6.22565 q^{26} -4.70156 q^{28} -0.703336 q^{29} +5.22742 q^{31} +8.53879 q^{32} +9.58237 q^{34} -1.00000 q^{35} +7.95005 q^{37} +10.7598 q^{38} +6.99364 q^{40} -2.31773 q^{41} +4.24849 q^{43} +4.70156 q^{44} +0.179249 q^{46} -13.3532 q^{47} +1.00000 q^{49} +2.58874 q^{50} -11.3068 q^{52} +8.51136 q^{53} +1.00000 q^{55} -6.99364 q^{56} -1.82075 q^{58} +4.47414 q^{59} +5.02107 q^{61} +13.5324 q^{62} +4.70156 q^{64} -2.40490 q^{65} +4.72263 q^{67} +17.4031 q^{68} -2.58874 q^{70} -4.40490 q^{71} -14.2129 q^{73} +20.5806 q^{74} +19.5416 q^{76} -1.00000 q^{77} -4.40490 q^{79} +8.70156 q^{80} -6.00000 q^{82} -5.56308 q^{83} +3.70156 q^{85} +10.9982 q^{86} +6.99364 q^{88} -15.1968 q^{89} +2.40490 q^{91} +0.325544 q^{92} -34.5679 q^{94} +4.15641 q^{95} -8.24494 q^{97} +2.58874 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 6 q^{4} + 4 q^{5} - 4 q^{7}+O(q^{10}) $ 4 * q + 6 * q^4 + 4 * q^5 - 4 * q^7	$ 4 q + 6 q^{4} + 4 q^{5} - 4 q^{7} + 4 q^{11} + 8 q^{13} + 22 q^{16} + 2 q^{17} + 10 q^{19} + 6 q^{20} + 2 q^{23} + 4 q^{25} - 20 q^{26} - 6 q^{28} + 2 q^{29} + 24 q^{31} - 4 q^{35} + 8 q^{37} - 16 q^{38} + 6 q^{43} + 6 q^{44} - 12 q^{46} - 4 q^{47} + 4 q^{49} + 12 q^{52} - 14 q^{53} + 4 q^{55} - 20 q^{58} + 2 q^{59} + 6 q^{61} - 8 q^{62} + 6 q^{64} + 8 q^{65} - 8 q^{67} + 44 q^{68} + 4 q^{73} + 36 q^{74} + 56 q^{76} - 4 q^{77} + 22 q^{80} - 24 q^{82} - 6 q^{83} + 2 q^{85} + 36 q^{86} - 18 q^{89} - 8 q^{91} + 44 q^{92} - 36 q^{94} + 10 q^{95} - 6 q^{97}+O(q^{100}) $ 4 * q + 6 * q^4 + 4 * q^5 - 4 * q^7 + 4 * q^11 + 8 * q^13 + 22 * q^16 + 2 * q^17 + 10 * q^19 + 6 * q^20 + 2 * q^23 + 4 * q^25 - 20 * q^26 - 6 * q^28 + 2 * q^29 + 24 * q^31 - 4 * q^35 + 8 * q^37 - 16 * q^38 + 6 * q^43 + 6 * q^44 - 12 * q^46 - 4 * q^47 + 4 * q^49 + 12 * q^52 - 14 * q^53 + 4 * q^55 - 20 * q^58 + 2 * q^59 + 6 * q^61 - 8 * q^62 + 6 * q^64 + 8 * q^65 - 8 * q^67 + 44 * q^68 + 4 * q^73 + 36 * q^74 + 56 * q^76 - 4 * q^77 + 22 * q^80 - 24 * q^82 - 6 * q^83 + 2 * q^85 + 36 * q^86 - 18 * q^89 - 8 * q^91 + 44 * q^92 - 36 * q^94 + 10 * q^95 - 6 * 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 $

$2$	2.58874	1.83051	0.915257	−	0.402871i	$-0.131988\pi$
$2$	2.58874	1.83051	0.915257	+	0.402871i	$0.131988\pi$
$3$	0	0
$3$	0	0
$4$	4.70156	2.35078
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	6.99364	2.47262
$9$	0	0
$10$	2.58874	0.818631
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−2.40490	−0.666999	−0.333499	−	0.942750i	$-0.608229\pi$
$13$	−2.40490	−0.666999	−0.333499	+	0.942750i	$0.608229\pi$
$14$	−2.58874	−0.691869
$15$	0	0
$16$	8.70156	2.17539
$17$	3.70156	0.897761	0.448880	−	0.893592i	$-0.351823\pi$
$17$	3.70156	0.897761	0.448880	+	0.893592i	$0.351823\pi$
$18$	0	0
$19$	4.15641	0.953545	0.476773	−	0.879027i	$-0.341807\pi$
$19$	4.15641	0.953545	0.476773	+	0.879027i	$0.341807\pi$
$20$	4.70156	1.05130
$21$	0	0
$22$	2.58874	0.551921
$23$	0.0692417	0.0144379	0.00721895	−	0.999974i	$-0.497702\pi$
$23$	0.0692417	0.0144379	0.00721895	+	0.999974i	$0.497702\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−6.22565	−1.22095
$27$	0	0
$28$	−4.70156	−0.888512
$29$	−0.703336	−0.130606	−0.0653031	−	0.997865i	$-0.520801\pi$
$29$	−0.703336	−0.130606	−0.0653031	+	0.997865i	$0.520801\pi$
$30$	0	0
$31$	5.22742	0.938873	0.469436	−	0.882966i	$-0.344457\pi$
$31$	5.22742	0.938873	0.469436	+	0.882966i	$0.344457\pi$
$32$	8.53879	1.50946
$33$	0	0
$34$	9.58237	1.64336
$35$	−1.00000	−0.169031
$36$	0	0
$37$	7.95005	1.30698	0.653490	−	0.756935i	$-0.273304\pi$
$37$	7.95005	1.30698	0.653490	+	0.756935i	$0.273304\pi$
$38$	10.7598	1.74548
$39$	0	0
$40$	6.99364	1.10579
$41$	−2.31773	−0.361969	−0.180984	−	0.983486i	$-0.557928\pi$
$41$	−2.31773	−0.361969	−0.180984	+	0.983486i	$0.557928\pi$
$42$	0	0
$43$	4.24849	0.647889	0.323944	−	0.946076i	$-0.394991\pi$
$43$	4.24849	0.647889	0.323944	+	0.946076i	$0.394991\pi$
$44$	4.70156	0.708787
$45$	0	0
$46$	0.179249	0.0264288
$47$	−13.3532	−1.94776	−0.973881	−	0.227061i	$-0.927088\pi$
$47$	−13.3532	−1.94776	−0.973881	+	0.227061i	$0.927088\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	2.58874	0.366103
$51$	0	0
$52$	−11.3068	−1.56797
$53$	8.51136	1.16912	0.584562	−	0.811349i	$-0.301266\pi$
$53$	8.51136	1.16912	0.584562	+	0.811349i	$0.301266\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	−6.99364	−0.934564
$57$	0	0
$58$	−1.82075	−0.239076
$59$	4.47414	0.582483	0.291242	−	0.956650i	$-0.405932\pi$
$59$	4.47414	0.582483	0.291242	+	0.956650i	$0.405932\pi$
$60$	0	0
$61$	5.02107	0.642882	0.321441	−	0.946930i	$-0.395833\pi$
$61$	5.02107	0.642882	0.321441	+	0.946930i	$0.395833\pi$
$62$	13.5324	1.71862
$63$	0	0
$64$	4.70156	0.587695
$65$	−2.40490	−0.298291
$66$	0	0
$67$	4.72263	0.576961	0.288481	−	0.957486i	$-0.406850\pi$
$67$	4.72263	0.576961	0.288481	+	0.957486i	$0.406850\pi$
$68$	17.4031	2.11044
$69$	0	0
$70$	−2.58874	−0.309413
$71$	−4.40490	−0.522765	−0.261383	−	0.965235i	$-0.584178\pi$
$71$	−4.40490	−0.522765	−0.261383	+	0.965235i	$0.584178\pi$
$72$	0	0
$73$	−14.2129	−1.66350	−0.831748	−	0.555153i	$-0.812660\pi$
$73$	−14.2129	−1.66350	−0.831748	+	0.555153i	$0.812660\pi$
$74$	20.5806	2.39245
$75$	0	0
$76$	19.5416	2.24158
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−4.40490	−0.495590	−0.247795	−	0.968813i	$-0.579706\pi$
$79$	−4.40490	−0.495590	−0.247795	+	0.968813i	$0.579706\pi$
$80$	8.70156	0.972864
$81$	0	0
$82$	−6.00000	−0.662589
$83$	−5.56308	−0.610627	−0.305314	−	0.952252i	$-0.598761\pi$
$83$	−5.56308	−0.610627	−0.305314	+	0.952252i	$0.598761\pi$
$84$	0	0
$85$	3.70156	0.401491
$86$	10.9982	1.18597
$87$	0	0
$88$	6.99364	0.745524
$89$	−15.1968	−1.61085	−0.805427	−	0.592695i	$-0.798064\pi$
$89$	−15.1968	−1.61085	−0.805427	+	0.592695i	$0.798064\pi$
$90$	0	0
$91$	2.40490	0.252102
$92$	0.325544	0.0339403
$93$	0	0
$94$	−34.5679	−3.56540
$95$	4.15641	0.426438
$96$	0	0
$97$	−8.24494	−0.837147	−0.418574	−	0.908183i	$-0.637470\pi$
$97$	−8.24494	−0.837147	−0.418574	+	0.908183i	$0.637470\pi$
$98$	2.58874	0.261502
$99$	0	0
$100$	4.70156	0.470156
$101$	−11.0354	−1.09807	−0.549034	−	0.835800i	$-0.685004\pi$
$101$	−11.0354	−1.09807	−0.549034	+	0.835800i	$0.685004\pi$
$102$	0	0
$103$	−2.84182	−0.280013	−0.140006	−	0.990151i	$-0.544712\pi$
$103$	−2.84182	−0.280013	−0.140006	+	0.990151i	$0.544712\pi$
$104$	−16.8190	−1.64924
$105$	0	0
$106$	22.0337	2.14010
$107$	15.3567	1.48459	0.742295	−	0.670073i	$-0.233737\pi$
$107$	15.3567	1.48459	0.742295	+	0.670073i	$0.233737\pi$
$108$	0	0
$109$	12.8597	1.23174	0.615870	−	0.787848i	$-0.288805\pi$
$109$	12.8597	1.23174	0.615870	+	0.787848i	$0.288805\pi$
$110$	2.58874	0.246826
$111$	0	0
$112$	−8.70156	−0.822220
$113$	−16.5114	−1.55326	−0.776629	−	0.629958i	$-0.783072\pi$
$113$	−16.5114	−1.55326	−0.776629	+	0.629958i	$0.783072\pi$
$114$	0	0
$115$	0.0692417	0.00645682
$116$	−3.30678	−0.307026
$117$	0	0
$118$	11.5824	1.06624
$119$	−3.70156	−0.339322
$120$	0	0
$121$	1.00000	0.0909091
$122$	12.9982	1.17680
$123$	0	0
$124$	24.5771	2.20708
$125$	1.00000	0.0894427
$126$	0	0
$127$	−0.248490	−0.0220500	−0.0110250	−	0.999939i	$-0.503509\pi$
$127$	−0.248490	−0.0220500	−0.0110250	+	0.999939i	$0.503509\pi$
$128$	−4.90647	−0.433675
$129$	0	0
$130$	−6.22565	−0.546026
$131$	−12.6178	−1.10242	−0.551212	−	0.834365i	$-0.685834\pi$
$131$	−12.6178	−1.10242	−0.551212	+	0.834365i	$0.685834\pi$
$132$	0	0
$133$	−4.15641	−0.360406
$134$	12.2256	1.05614
$135$	0	0
$136$	25.8874	2.21982
$137$	−9.40312	−0.803363	−0.401682	−	0.915779i	$-0.631574\pi$
$137$	−9.40312	−0.803363	−0.401682	+	0.915779i	$0.631574\pi$
$138$	0	0
$139$	22.4934	1.90787	0.953934	−	0.300016i	$-0.0969921\pi$
$139$	22.4934	1.90787	0.953934	+	0.300016i	$0.0969921\pi$
$140$	−4.70156	−0.397355
$141$	0	0
$142$	−11.4031	−0.956929
$143$	−2.40490	−0.201108
$144$	0	0
$145$	−0.703336	−0.0584088
$146$	−36.7935	−3.04505
$147$	0	0
$148$	37.3777	3.07243
$149$	13.9001	1.13874	0.569370	−	0.822081i	$-0.307187\pi$
$149$	13.9001	1.13874	0.569370	+	0.822081i	$0.307187\pi$
$150$	0	0
$151$	2.95183	0.240216	0.120108	−	0.992761i	$-0.461676\pi$
$151$	2.95183	0.240216	0.120108	+	0.992761i	$0.461676\pi$
$152$	29.0684	2.35776
$153$	0	0
$154$	−2.58874	−0.208606
$155$	5.22742	0.419877
$156$	0	0
$157$	5.15463	0.411385	0.205692	−	0.978617i	$-0.434055\pi$
$157$	5.15463	0.411385	0.205692	+	0.978617i	$0.434055\pi$
$158$	−11.4031	−0.907184
$159$	0	0
$160$	8.53879	0.675051
$161$	−0.0692417	−0.00545701
$162$	0	0
$163$	2.32128	0.181817	0.0909083	−	0.995859i	$-0.471023\pi$
$163$	2.32128	0.181817	0.0909083	+	0.995859i	$0.471023\pi$
$164$	−10.8970	−0.850910
$165$	0	0
$166$	−14.4014	−1.11776
$167$	−17.7581	−1.37416	−0.687081	−	0.726581i	$-0.741108\pi$
$167$	−17.7581	−1.37416	−0.687081	+	0.726581i	$0.741108\pi$
$168$	0	0
$169$	−7.21647	−0.555113
$170$	9.58237	0.734934
$171$	0	0
$172$	19.9745	1.52304
$173$	−20.2094	−1.53649	−0.768245	−	0.640156i	$-0.778870\pi$
$173$	−20.2094	−1.53649	−0.768245	+	0.640156i	$0.778870\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	8.70156	0.655905
$177$	0	0
$178$	−39.3404	−2.94869
$179$	5.95005	0.444728	0.222364	−	0.974964i	$-0.428623\pi$
$179$	5.95005	0.444728	0.222364	+	0.974964i	$0.428623\pi$
$180$	0	0
$181$	−5.54161	−0.411904	−0.205952	−	0.978562i	$-0.566029\pi$
$181$	−5.54161	−0.411904	−0.205952	+	0.978562i	$0.566029\pi$
$182$	6.22565	0.461476
$183$	0	0
$184$	0.484251	0.0356995
$185$	7.95005	0.584499
$186$	0	0
$187$	3.70156	0.270685
$188$	−62.7808	−4.57876
$189$	0	0
$190$	10.7598	0.780601
$191$	−25.8966	−1.87381	−0.936905	−	0.349585i	$-0.886323\pi$
$191$	−25.8966	−1.87381	−0.936905	+	0.349585i	$0.886323\pi$
$192$	0	0
$193$	−1.31596	−0.0947248	−0.0473624	−	0.998878i	$-0.515082\pi$
$193$	−1.31596	−0.0947248	−0.0473624	+	0.998878i	$0.515082\pi$
$194$	−21.3440	−1.53241
$195$	0	0
$196$	4.70156	0.335826
$197$	4.44212	0.316488	0.158244	−	0.987400i	$-0.449417\pi$
$197$	4.44212	0.316488	0.158244	+	0.987400i	$0.449417\pi$
$198$	0	0
$199$	−25.7453	−1.82504	−0.912520	−	0.409033i	$-0.865866\pi$
$199$	−25.7453	−1.82504	−0.912520	+	0.409033i	$0.865866\pi$
$200$	6.99364	0.494525
$201$	0	0
$202$	−28.5679	−2.01003
$203$	0.703336	0.0493645
$204$	0	0
$205$	−2.31773	−0.161877
$206$	−7.35672	−0.512567
$207$	0	0
$208$	−20.9264	−1.45098
$209$	4.15641	0.287505
$210$	0	0
$211$	−24.2129	−1.66689	−0.833443	−	0.552605i	$-0.813634\pi$
$211$	−24.2129	−1.66689	−0.833443	+	0.552605i	$0.813634\pi$
$212$	40.0167	2.74836
$213$	0	0
$214$	39.7545	2.71756
$215$	4.24849	0.289745
$216$	0	0
$217$	−5.22742	−0.354861
$218$	33.2905	2.25472
$219$	0	0
$220$	4.70156	0.316979
$221$	−8.90188	−0.598805
$222$	0	0
$223$	8.38697	0.561633	0.280817	−	0.959761i	$-0.409395\pi$
$223$	8.38697	0.561633	0.280817	+	0.959761i	$0.409395\pi$
$224$	−8.53879	−0.570522
$225$	0	0
$226$	−42.7436	−2.84326
$227$	−2.79542	−0.185538	−0.0927692	−	0.995688i	$-0.529572\pi$
$227$	−2.79542	−0.185538	−0.0927692	+	0.995688i	$0.529572\pi$
$228$	0	0
$229$	9.95360	0.657752	0.328876	−	0.944373i	$-0.393330\pi$
$229$	9.95360	0.657752	0.328876	+	0.944373i	$0.393330\pi$
$230$	0.179249	0.0118193
$231$	0	0
$232$	−4.91887	−0.322940
$233$	2.77612	0.181870	0.0909350	−	0.995857i	$-0.471014\pi$
$233$	2.77612	0.181870	0.0909350	+	0.995857i	$0.471014\pi$
$234$	0	0
$235$	−13.3532	−0.871065
$236$	21.0354	1.36929
$237$	0	0
$238$	−9.58237	−0.621133
$239$	14.6034	0.944618	0.472309	−	0.881433i	$-0.343421\pi$
$239$	14.6034	0.944618	0.472309	+	0.881433i	$0.343421\pi$
$240$	0	0
$241$	10.5385	0.678842	0.339421	−	0.940635i	$-0.389769\pi$
$241$	10.5385	0.678842	0.339421	+	0.940635i	$0.389769\pi$
$242$	2.58874	0.166410
$243$	0	0
$244$	23.6069	1.51127
$245$	1.00000	0.0638877
$246$	0	0
$247$	−9.99573	−0.636013
$248$	36.5587	2.32148
$249$	0	0
$250$	2.58874	0.163726
$251$	19.3904	1.22391	0.611955	−	0.790892i	$-0.290383\pi$
$251$	19.3904	1.22391	0.611955	+	0.790892i	$0.290383\pi$
$252$	0	0
$253$	0.0692417	0.00435319
$254$	−0.643276	−0.0403627
$255$	0	0
$256$	−22.1047	−1.38154
$257$	−2.71771	−0.169526	−0.0847631	−	0.996401i	$-0.527013\pi$
$257$	−2.71771	−0.169526	−0.0847631	+	0.996401i	$0.527013\pi$
$258$	0	0
$259$	−7.95005	−0.493992
$260$	−11.3068	−0.701216
$261$	0	0
$262$	−32.6642	−2.01800
$263$	14.4548	0.891324	0.445662	−	0.895201i	$-0.352968\pi$
$263$	14.4548	0.891324	0.445662	+	0.895201i	$0.352968\pi$
$264$	0	0
$265$	8.51136	0.522849
$266$	−10.7598	−0.659729
$267$	0	0
$268$	22.2037	1.35631
$269$	17.2677	1.05283	0.526414	−	0.850228i	$-0.323536\pi$
$269$	17.2677	1.05283	0.526414	+	0.850228i	$0.323536\pi$
$270$	0	0
$271$	15.7051	0.954017	0.477009	−	0.878899i	$-0.341721\pi$
$271$	15.7051	0.954017	0.477009	+	0.878899i	$0.341721\pi$
$272$	32.2094	1.95298
$273$	0	0
$274$	−24.3422	−1.47057
$275$	1.00000	0.0603023
$276$	0	0
$277$	15.9837	0.960369	0.480184	−	0.877168i	$-0.340570\pi$
$277$	15.9837	0.960369	0.480184	+	0.877168i	$0.340570\pi$
$278$	58.2296	3.49238
$279$	0	0
$280$	−6.99364	−0.417950
$281$	5.50302	0.328283	0.164141	−	0.986437i	$-0.447515\pi$
$281$	5.50302	0.328283	0.164141	+	0.986437i	$0.447515\pi$
$282$	0	0
$283$	−8.09208	−0.481024	−0.240512	−	0.970646i	$-0.577315\pi$
$283$	−8.09208	−0.481024	−0.240512	+	0.970646i	$0.577315\pi$
$284$	−20.7099	−1.22891
$285$	0	0
$286$	−6.22565	−0.368130
$287$	2.31773	0.136811
$288$	0	0
$289$	−3.29844	−0.194026
$290$	−1.82075	−0.106918
$291$	0	0
$292$	−66.8229	−3.91052
$293$	−6.51490	−0.380605	−0.190302	−	0.981726i	$-0.560947\pi$
$293$	−6.51490	−0.380605	−0.190302	+	0.981726i	$0.560947\pi$
$294$	0	0
$295$	4.47414	0.260494
$296$	55.5998	3.23167
$297$	0	0
$298$	35.9837	2.08448
$299$	−0.166519	−0.00963006
$300$	0	0
$301$	−4.24849	−0.244879
$302$	7.64150	0.439719
$303$	0	0
$304$	36.1672	2.07433
$305$	5.02107	0.287505
$306$	0	0
$307$	−22.5679	−1.28802	−0.644008	−	0.765019i	$-0.722730\pi$
$307$	−22.5679	−1.28802	−0.644008	+	0.765019i	$0.722730\pi$
$308$	−4.70156	−0.267896
$309$	0	0
$310$	13.5324	0.768590
$311$	−0.0513177	−0.00290996	−0.00145498	−	0.999999i	$-0.500463\pi$
$311$	−0.0513177	−0.00290996	−0.00145498	+	0.999999i	$0.500463\pi$
$312$	0	0
$313$	−3.44224	−0.194567	−0.0972835	−	0.995257i	$-0.531015\pi$
$313$	−3.44224	−0.194567	−0.0972835	+	0.995257i	$0.531015\pi$
$314$	13.3440	0.753045
$315$	0	0
$316$	−20.7099	−1.16502
$317$	−31.6582	−1.77810	−0.889050	−	0.457810i	$-0.848634\pi$
$317$	−31.6582	−1.77810	−0.889050	+	0.457810i	$0.848634\pi$
$318$	0	0
$319$	−0.703336	−0.0393792
$320$	4.70156	0.262825
$321$	0	0
$322$	−0.179249	−0.00998914
$323$	15.3852	0.856055
$324$	0	0
$325$	−2.40490	−0.133400
$326$	6.00918	0.332818
$327$	0	0
$328$	−16.2094	−0.895013
$329$	13.3532	0.736184
$330$	0	0
$331$	−3.55953	−0.195650	−0.0978248	−	0.995204i	$-0.531188\pi$
$331$	−3.55953	−0.195650	−0.0978248	+	0.995204i	$0.531188\pi$
$332$	−26.1552	−1.43545
$333$	0	0
$334$	−45.9710	−2.51542
$335$	4.72263	0.258025
$336$	0	0
$337$	4.23221	0.230543	0.115272	−	0.993334i	$-0.463226\pi$
$337$	4.23221	0.230543	0.115272	+	0.993334i	$0.463226\pi$
$338$	−18.6815	−1.01614
$339$	0	0
$340$	17.4031	0.943817
$341$	5.22742	0.283081
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	29.7124	1.60198
$345$	0	0
$346$	−52.3168	−2.81257
$347$	−15.6160	−0.838313	−0.419157	−	0.907914i	$-0.637674\pi$
$347$	−15.6160	−0.838313	−0.419157	+	0.907914i	$0.637674\pi$
$348$	0	0
$349$	2.24357	0.120096	0.0600479	−	0.998195i	$-0.480875\pi$
$349$	2.24357	0.120096	0.0600479	+	0.998195i	$0.480875\pi$
$350$	−2.58874	−0.138374
$351$	0	0
$352$	8.53879	0.455119
$353$	−11.9965	−0.638507	−0.319253	−	0.947669i	$-0.603432\pi$
$353$	−11.9965	−0.638507	−0.319253	+	0.947669i	$0.603432\pi$
$354$	0	0
$355$	−4.40490	−0.233788
$356$	−71.4486	−3.78677
$357$	0	0
$358$	15.4031	0.814080
$359$	−24.6743	−1.30226	−0.651131	−	0.758966i	$-0.725705\pi$
$359$	−24.6743	−1.30226	−0.651131	+	0.758966i	$0.725705\pi$
$360$	0	0
$361$	−1.72428	−0.0907514
$362$	−14.3458	−0.753997
$363$	0	0
$364$	11.3068	0.592636
$365$	−14.2129	−0.743938
$366$	0	0
$367$	22.2808	1.16305	0.581524	−	0.813529i	$-0.302457\pi$
$367$	22.2808	1.16305	0.581524	+	0.813529i	$0.302457\pi$
$368$	0.602511	0.0314081
$369$	0	0
$370$	20.5806	1.06993
$371$	−8.51136	−0.441888
$372$	0	0
$373$	13.6389	0.706195	0.353097	−	0.935587i	$-0.385128\pi$
$373$	13.6389	0.706195	0.353097	+	0.935587i	$0.385128\pi$
$374$	9.58237	0.495493
$375$	0	0
$376$	−93.3872	−4.81608
$377$	1.69145	0.0871141
$378$	0	0
$379$	14.1985	0.729330	0.364665	−	0.931139i	$-0.381183\pi$
$379$	14.1985	0.729330	0.364665	+	0.931139i	$0.381183\pi$
$380$	19.5416	1.00246
$381$	0	0
$382$	−67.0394	−3.43003
$383$	10.1244	0.517332	0.258666	−	0.965967i	$-0.416717\pi$
$383$	10.1244	0.517332	0.258666	+	0.965967i	$0.416717\pi$
$384$	0	0
$385$	−1.00000	−0.0509647
$386$	−3.40667	−0.173395
$387$	0	0
$388$	−38.7641	−1.96795
$389$	20.9342	1.06141	0.530703	−	0.847558i	$-0.321928\pi$
$389$	20.9342	1.06141	0.530703	+	0.847558i	$0.321928\pi$
$390$	0	0
$391$	0.256303	0.0129618
$392$	6.99364	0.353232
$393$	0	0
$394$	11.4995	0.579335
$395$	−4.40490	−0.221634
$396$	0	0
$397$	−24.5679	−1.23303	−0.616513	−	0.787345i	$-0.711455\pi$
$397$	−24.5679	−1.23303	−0.616513	+	0.787345i	$0.711455\pi$
$398$	−66.6479	−3.34076
$399$	0	0
$400$	8.70156	0.435078
$401$	38.4180	1.91850	0.959252	−	0.282551i	$-0.0911805\pi$
$401$	38.4180	1.91850	0.959252	+	0.282551i	$0.0911805\pi$
$402$	0	0
$403$	−12.5714	−0.626227
$404$	−51.8838	−2.58132
$405$	0	0
$406$	1.82075	0.0903624
$407$	7.95005	0.394069
$408$	0	0
$409$	−1.17748	−0.0582224	−0.0291112	−	0.999576i	$-0.509268\pi$
$409$	−1.17748	−0.0582224	−0.0291112	+	0.999576i	$0.509268\pi$
$410$	−6.00000	−0.296319
$411$	0	0
$412$	−13.3610	−0.658249
$413$	−4.47414	−0.220158
$414$	0	0
$415$	−5.56308	−0.273081
$416$	−20.5349	−1.00681
$417$	0	0
$418$	10.7598	0.526281
$419$	37.2585	1.82020	0.910098	−	0.414393i	$-0.136006\pi$
$419$	37.2585	1.82020	0.910098	+	0.414393i	$0.136006\pi$
$420$	0	0
$421$	3.10469	0.151313	0.0756566	−	0.997134i	$-0.475895\pi$
$421$	3.10469	0.151313	0.0756566	+	0.997134i	$0.475895\pi$
$422$	−62.6809	−3.05126
$423$	0	0
$424$	59.5253	2.89081
$425$	3.70156	0.179552
$426$	0	0
$427$	−5.02107	−0.242986
$428$	72.2006	3.48995
$429$	0	0
$430$	10.9982	0.530382
$431$	13.5452	0.652447	0.326224	−	0.945293i	$-0.394224\pi$
$431$	13.5452	0.652447	0.326224	+	0.945293i	$0.394224\pi$
$432$	0	0
$433$	−33.7904	−1.62386	−0.811931	−	0.583754i	$-0.801583\pi$
$433$	−33.7904	−1.62386	−0.811931	+	0.583754i	$0.801583\pi$
$434$	−13.5324	−0.649577
$435$	0	0
$436$	60.4609	2.89555
$437$	0.287797	0.0137672
$438$	0	0
$439$	11.6052	0.553887	0.276943	−	0.960886i	$-0.410679\pi$
$439$	11.6052	0.553887	0.276943	+	0.960886i	$0.410679\pi$
$440$	6.99364	0.333408
$441$	0	0
$442$	−23.0446	−1.09612
$443$	24.7226	1.17461	0.587304	−	0.809367i	$-0.300189\pi$
$443$	24.7226	1.17461	0.587304	+	0.809367i	$0.300189\pi$
$444$	0	0
$445$	−15.1968	−0.720396
$446$	21.7117	1.02808
$447$	0	0
$448$	−4.70156	−0.222128
$449$	−31.5275	−1.48788	−0.743938	−	0.668249i	$-0.767044\pi$
$449$	−31.5275	−1.48788	−0.743938	+	0.668249i	$0.767044\pi$
$450$	0	0
$451$	−2.31773	−0.109138
$452$	−77.6292	−3.65137
$453$	0	0
$454$	−7.23660	−0.339631
$455$	2.40490	0.112743
$456$	0	0
$457$	−13.3874	−0.626235	−0.313118	−	0.949714i	$-0.601373\pi$
$457$	−13.3874	−0.626235	−0.313118	+	0.949714i	$0.601373\pi$
$458$	25.7673	1.20402
$459$	0	0
$460$	0.325544	0.0151786
$461$	−3.87070	−0.180276	−0.0901382	−	0.995929i	$-0.528731\pi$
$461$	−3.87070	−0.180276	−0.0901382	+	0.995929i	$0.528731\pi$
$462$	0	0
$463$	29.2484	1.35929	0.679643	−	0.733543i	$-0.262135\pi$
$463$	29.2484	1.35929	0.679643	+	0.733543i	$0.262135\pi$
$464$	−6.12012	−0.284119
$465$	0	0
$466$	7.18666	0.332915
$467$	−14.2551	−0.659645	−0.329823	−	0.944043i	$-0.606989\pi$
$467$	−14.2551	−0.659645	−0.329823	+	0.944043i	$0.606989\pi$
$468$	0	0
$469$	−4.72263	−0.218071
$470$	−34.5679	−1.59450
$471$	0	0
$472$	31.2905	1.44026
$473$	4.24849	0.195346
$474$	0	0
$475$	4.15641	0.190709
$476$	−17.4031	−0.797671
$477$	0	0
$478$	37.8045	1.72914
$479$	32.2339	1.47280	0.736401	−	0.676545i	$-0.236524\pi$
$479$	32.2339	1.47280	0.736401	+	0.676545i	$0.236524\pi$
$480$	0	0
$481$	−19.1191	−0.871754
$482$	27.2813	1.24263
$483$	0	0
$484$	4.70156	0.213707
$485$	−8.24494	−0.374384
$486$	0	0
$487$	5.27382	0.238980	0.119490	−	0.992835i	$-0.461874\pi$
$487$	5.27382	0.238980	0.119490	+	0.992835i	$0.461874\pi$
$488$	35.1155	1.58960
$489$	0	0
$490$	2.58874	0.116947
$491$	−35.6647	−1.60953	−0.804764	−	0.593595i	$-0.797708\pi$
$491$	−35.6647	−1.60953	−0.804764	+	0.593595i	$0.797708\pi$
$492$	0	0
$493$	−2.60344	−0.117253
$494$	−25.8763	−1.16423
$495$	0	0
$496$	45.4867	2.04242
$497$	4.40490	0.197587
$498$	0	0
$499$	−39.0083	−1.74625	−0.873127	−	0.487494i	$-0.837911\pi$
$499$	−39.0083	−1.74625	−0.873127	+	0.487494i	$0.837911\pi$
$500$	4.70156	0.210260
$501$	0	0
$502$	50.1966	2.24039
$503$	32.6821	1.45722	0.728612	−	0.684926i	$-0.240166\pi$
$503$	32.6821	1.45722	0.728612	+	0.684926i	$0.240166\pi$
$504$	0	0
$505$	−11.0354	−0.491071
$506$	0.179249	0.00796857
$507$	0	0
$508$	−1.16829	−0.0518346
$509$	−3.25808	−0.144412	−0.0722058	−	0.997390i	$-0.523004\pi$
$509$	−3.25808	−0.144412	−0.0722058	+	0.997390i	$0.523004\pi$
$510$	0	0
$511$	14.2129	0.628743
$512$	−47.4103	−2.09526
$513$	0	0
$514$	−7.03544	−0.310320
$515$	−2.84182	−0.125226
$516$	0	0
$517$	−13.3532	−0.587272
$518$	−20.5806	−0.904260
$519$	0	0
$520$	−16.8190	−0.737561
$521$	−7.75506	−0.339755	−0.169878	−	0.985465i	$-0.554337\pi$
$521$	−7.75506	−0.339755	−0.169878	+	0.985465i	$0.554337\pi$
$522$	0	0
$523$	27.1647	1.18783	0.593916	−	0.804527i	$-0.297581\pi$
$523$	27.1647	1.18783	0.593916	+	0.804527i	$0.297581\pi$
$524$	−59.3235	−2.59156
$525$	0	0
$526$	37.4198	1.63158
$527$	19.3496	0.842883
$528$	0	0
$529$	−22.9952	−0.999792
$530$	22.0337	0.957082
$531$	0	0
$532$	−19.5416	−0.847236
$533$	5.57391	0.241433
$534$	0	0
$535$	15.3567	0.663929
$536$	33.0284	1.42661
$537$	0	0
$538$	44.7014	1.92722
$539$	1.00000	0.0430730
$540$	0	0
$541$	36.2087	1.55673	0.778366	−	0.627811i	$-0.216049\pi$
$541$	36.2087	1.55673	0.778366	+	0.627811i	$0.216049\pi$
$542$	40.6564	1.74634
$543$	0	0
$544$	31.6069	1.35513
$545$	12.8597	0.550851
$546$	0	0
$547$	−11.1160	−0.475288	−0.237644	−	0.971352i	$-0.576375\pi$
$547$	−11.1160	−0.475288	−0.237644	+	0.971352i	$0.576375\pi$
$548$	−44.2094	−1.88853
$549$	0	0
$550$	2.58874	0.110384
$551$	−2.92335	−0.124539
$552$	0	0
$553$	4.40490	0.187315
$554$	41.3777	1.75797
$555$	0	0
$556$	105.754	4.48498
$557$	−34.4772	−1.46084	−0.730422	−	0.682996i	$-0.760677\pi$
$557$	−34.4772	−1.46084	−0.730422	+	0.682996i	$0.760677\pi$
$558$	0	0
$559$	−10.2172	−0.432141
$560$	−8.70156	−0.367708
$561$	0	0
$562$	14.2459	0.600926
$563$	−16.2129	−0.683293	−0.341647	−	0.939829i	$-0.610985\pi$
$563$	−16.2129	−0.683293	−0.341647	+	0.939829i	$0.610985\pi$
$564$	0	0
$565$	−16.5114	−0.694638
$566$	−20.9483	−0.880522
$567$	0	0
$568$	−30.8062	−1.29260
$569$	−3.64807	−0.152935	−0.0764675	−	0.997072i	$-0.524364\pi$
$569$	−3.64807	−0.152935	−0.0764675	+	0.997072i	$0.524364\pi$
$570$	0	0
$571$	−36.7563	−1.53820	−0.769102	−	0.639126i	$-0.779296\pi$
$571$	−36.7563	−1.53820	−0.769102	+	0.639126i	$0.779296\pi$
$572$	−11.3068	−0.472760
$573$	0	0
$574$	6.00000	0.250435
$575$	0.0692417	0.00288758
$576$	0	0
$577$	−2.17433	−0.0905186	−0.0452593	−	0.998975i	$-0.514411\pi$
$577$	−2.17433	−0.0905186	−0.0452593	+	0.998975i	$0.514411\pi$
$578$	−8.53879	−0.355167
$579$	0	0
$580$	−3.30678	−0.137306
$581$	5.56308	0.230795
$582$	0	0
$583$	8.51136	0.352504
$584$	−99.4000	−4.11320
$585$	0	0
$586$	−16.8654	−0.696702
$587$	−13.9501	−0.575780	−0.287890	−	0.957663i	$-0.592954\pi$
$587$	−13.9501	−0.575780	−0.287890	+	0.957663i	$0.592954\pi$
$588$	0	0
$589$	21.7273	0.895258
$590$	11.5824	0.476839
$591$	0	0
$592$	69.1779	2.84319
$593$	10.9483	0.449592	0.224796	−	0.974406i	$-0.427828\pi$
$593$	10.9483	0.449592	0.224796	+	0.974406i	$0.427828\pi$
$594$	0	0
$595$	−3.70156	−0.151749
$596$	65.3522	2.67693
$597$	0	0
$598$	−0.431075	−0.0176280
$599$	41.1998	1.68338	0.841689	−	0.539963i	$-0.181562\pi$
$599$	41.1998	1.68338	0.841689	+	0.539963i	$0.181562\pi$
$600$	0	0
$601$	21.5110	0.877450	0.438725	−	0.898621i	$-0.355430\pi$
$601$	21.5110	0.877450	0.438725	+	0.898621i	$0.355430\pi$
$602$	−10.9982	−0.448254
$603$	0	0
$604$	13.8782	0.564696
$605$	1.00000	0.0406558
$606$	0	0
$607$	−13.8580	−0.562478	−0.281239	−	0.959638i	$-0.590745\pi$
$607$	−13.8580	−0.562478	−0.281239	+	0.959638i	$0.590745\pi$
$608$	35.4907	1.43934
$609$	0	0
$610$	12.9982	0.526283
$611$	32.1130	1.29915
$612$	0	0
$613$	−32.0223	−1.29337	−0.646684	−	0.762758i	$-0.723845\pi$
$613$	−32.0223	−1.29337	−0.646684	+	0.762758i	$0.723845\pi$
$614$	−58.4223	−2.35773
$615$	0	0
$616$	−6.99364	−0.281782
$617$	3.39958	0.136862	0.0684309	−	0.997656i	$-0.478201\pi$
$617$	3.39958	0.136862	0.0684309	+	0.997656i	$0.478201\pi$
$618$	0	0
$619$	−1.91638	−0.0770259	−0.0385129	−	0.999258i	$-0.512262\pi$
$619$	−1.91638	−0.0770259	−0.0385129	+	0.999258i	$0.512262\pi$
$620$	24.5771	0.987038
$621$	0	0
$622$	−0.132848	−0.00532672
$623$	15.1968	0.608846
$624$	0	0
$625$	1.00000	0.0400000
$626$	−8.91106	−0.356158
$627$	0	0
$628$	24.2348	0.967075
$629$	29.4276	1.17336
$630$	0	0
$631$	−37.4019	−1.48895	−0.744473	−	0.667653i	$-0.767299\pi$
$631$	−37.4019	−1.48895	−0.744473	+	0.667653i	$0.767299\pi$
$632$	−30.8062	−1.22541
$633$	0	0
$634$	−81.9547	−3.25484
$635$	−0.248490	−0.00986104
$636$	0	0
$637$	−2.40490	−0.0952855
$638$	−1.82075	−0.0720842
$639$	0	0
$640$	−4.90647	−0.193945
$641$	−8.52928	−0.336886	−0.168443	−	0.985711i	$-0.553874\pi$
$641$	−8.52928	−0.336886	−0.168443	+	0.985711i	$0.553874\pi$
$642$	0	0
$643$	36.3615	1.43396	0.716979	−	0.697095i	$-0.245524\pi$
$643$	36.3615	1.43396	0.716979	+	0.697095i	$0.245524\pi$
$644$	−0.325544	−0.0128282
$645$	0	0
$646$	39.8282	1.56702
$647$	−42.2551	−1.66122	−0.830609	−	0.556856i	$-0.812007\pi$
$647$	−42.2551	−1.66122	−0.830609	+	0.556856i	$0.812007\pi$
$648$	0	0
$649$	4.47414	0.175625
$650$	−6.22565	−0.244190
$651$	0	0
$652$	10.9136	0.427411
$653$	21.8110	0.853532	0.426766	−	0.904362i	$-0.359653\pi$
$653$	21.8110	0.853532	0.426766	+	0.904362i	$0.359653\pi$
$654$	0	0
$655$	−12.6178	−0.493019
$656$	−20.1679	−0.787424
$657$	0	0
$658$	34.5679	1.34760
$659$	46.8164	1.82371	0.911853	−	0.410516i	$-0.134652\pi$
$659$	46.8164	1.82371	0.911853	+	0.410516i	$0.134652\pi$
$660$	0	0
$661$	−17.1113	−0.665551	−0.332775	−	0.943006i	$-0.607985\pi$
$661$	−17.1113	−0.665551	−0.332775	+	0.943006i	$0.607985\pi$
$662$	−9.21469	−0.358139
$663$	0	0
$664$	−38.9061	−1.50985
$665$	−4.15641	−0.161179
$666$	0	0
$667$	−0.0487002	−0.00188568
$668$	−83.4907	−3.23035
$669$	0	0
$670$	12.2256	0.472318
$671$	5.02107	0.193836
$672$	0	0
$673$	−48.3707	−1.86455	−0.932277	−	0.361746i	$-0.882181\pi$
$673$	−48.3707	−1.86455	−0.932277	+	0.361746i	$0.882181\pi$
$674$	10.9561	0.422013
$675$	0	0
$676$	−33.9287	−1.30495
$677$	−26.6463	−1.02410	−0.512050	−	0.858956i	$-0.671114\pi$
$677$	−26.6463	−1.02410	−0.512050	+	0.858956i	$0.671114\pi$
$678$	0	0
$679$	8.24494	0.316412
$680$	25.8874	0.992736
$681$	0	0
$682$	13.5324	0.518183
$683$	−5.94514	−0.227484	−0.113742	−	0.993510i	$-0.536284\pi$
$683$	−5.94514	−0.227484	−0.113742	+	0.993510i	$0.536284\pi$
$684$	0	0
$685$	−9.40312	−0.359275
$686$	−2.58874	−0.0988385
$687$	0	0
$688$	36.9685	1.40941
$689$	−20.4689	−0.779805
$690$	0	0
$691$	7.72867	0.294012	0.147006	−	0.989136i	$-0.453036\pi$
$691$	7.72867	0.294012	0.147006	+	0.989136i	$0.453036\pi$
$692$	−95.0156	−3.61195
$693$	0	0
$694$	−40.4258	−1.53454
$695$	22.4934	0.853225
$696$	0	0
$697$	−8.57923	−0.324961
$698$	5.80802	0.219837
$699$	0	0
$700$	−4.70156	−0.177702
$701$	4.70334	0.177643	0.0888213	−	0.996048i	$-0.471690\pi$
$701$	4.70334	0.177643	0.0888213	+	0.996048i	$0.471690\pi$
$702$	0	0
$703$	33.0437	1.24627
$704$	4.70156	0.177197
$705$	0	0
$706$	−31.0557	−1.16880
$707$	11.0354	0.415031
$708$	0	0
$709$	21.8822	0.821803	0.410901	−	0.911680i	$-0.365214\pi$
$709$	21.8822	0.821803	0.410901	+	0.911680i	$0.365214\pi$
$710$	−11.4031	−0.427952
$711$	0	0
$712$	−106.281	−3.98304
$713$	0.361956	0.0135553
$714$	0	0
$715$	−2.40490	−0.0899381
$716$	27.9745	1.04546
$717$	0	0
$718$	−63.8754	−2.38381
$719$	19.2875	0.719302	0.359651	−	0.933087i	$-0.382896\pi$
$719$	19.2875	0.719302	0.359651	+	0.933087i	$0.382896\pi$
$720$	0	0
$721$	2.84182	0.105835
$722$	−4.46370	−0.166122
$723$	0	0
$724$	−26.0542	−0.968297
$725$	−0.703336	−0.0261212
$726$	0	0
$727$	35.4877	1.31616	0.658082	−	0.752946i	$-0.271368\pi$
$727$	35.4877	1.31616	0.658082	+	0.752946i	$0.271368\pi$
$728$	16.8190	0.623353
$729$	0	0
$730$	−36.7935	−1.36179
$731$	15.7261	0.581649
$732$	0	0
$733$	49.4163	1.82523	0.912615	−	0.408819i	$-0.134059\pi$
$733$	49.4163	1.82523	0.912615	+	0.408819i	$0.134059\pi$
$734$	57.6791	2.12898
$735$	0	0
$736$	0.591240	0.0217934
$737$	4.72263	0.173960
$738$	0	0
$739$	18.1095	0.666168	0.333084	−	0.942897i	$-0.391911\pi$
$739$	18.1095	0.666168	0.333084	+	0.942897i	$0.391911\pi$
$740$	37.3777	1.37403
$741$	0	0
$742$	−22.0337	−0.808882
$743$	45.0972	1.65445	0.827227	−	0.561868i	$-0.189917\pi$
$743$	45.0972	1.65445	0.827227	+	0.561868i	$0.189917\pi$
$744$	0	0
$745$	13.9001	0.509260
$746$	35.3075	1.29270
$747$	0	0
$748$	17.4031	0.636321
$749$	−15.3567	−0.561122
$750$	0	0
$751$	47.9495	1.74970	0.874851	−	0.484391i	$-0.160959\pi$
$751$	47.9495	1.74970	0.874851	+	0.484391i	$0.160959\pi$
$752$	−116.193	−4.23714
$753$	0	0
$754$	4.37872	0.159464
$755$	2.95183	0.107428
$756$	0	0
$757$	13.8158	0.502145	0.251073	−	0.967968i	$-0.419217\pi$
$757$	13.8158	0.502145	0.251073	+	0.967968i	$0.419217\pi$
$758$	36.7563	1.33505
$759$	0	0
$760$	29.0684	1.05442
$761$	−12.2221	−0.443051	−0.221525	−	0.975155i	$-0.571104\pi$
$761$	−12.2221	−0.443051	−0.221525	+	0.975155i	$0.571104\pi$
$762$	0	0
$763$	−12.8597	−0.465554
$764$	−121.754	−4.40492
$765$	0	0
$766$	26.2094	0.946983
$767$	−10.7598	−0.388516
$768$	0	0
$769$	29.3013	1.05663	0.528316	−	0.849048i	$-0.322823\pi$
$769$	29.3013	1.05663	0.528316	+	0.849048i	$0.322823\pi$
$770$	−2.58874	−0.0932916
$771$	0	0
$772$	−6.18706	−0.222677
$773$	−43.2356	−1.55508	−0.777539	−	0.628835i	$-0.783532\pi$
$773$	−43.2356	−1.55508	−0.777539	+	0.628835i	$0.783532\pi$
$774$	0	0
$775$	5.22742	0.187775
$776$	−57.6621	−2.06995
$777$	0	0
$778$	54.1931	1.94292
$779$	−9.63344	−0.345154
$780$	0	0
$781$	−4.40490	−0.157620
$782$	0.663500	0.0237267
$783$	0	0
$784$	8.70156	0.310770
$785$	5.15463	0.183977
$786$	0	0
$787$	−20.1489	−0.718230	−0.359115	−	0.933293i	$-0.616921\pi$
$787$	−20.1489	−0.718230	−0.359115	+	0.933293i	$0.616921\pi$
$788$	20.8849	0.743993
$789$	0	0
$790$	−11.4031	−0.405705
$791$	16.5114	0.587076
$792$	0	0
$793$	−12.0752	−0.428801
$794$	−63.5998	−2.25707
$795$	0	0
$796$	−121.043	−4.29027
$797$	7.08676	0.251026	0.125513	−	0.992092i	$-0.459942\pi$
$797$	7.08676	0.251026	0.125513	+	0.992092i	$0.459942\pi$
$798$	0	0
$799$	−49.4276	−1.74862
$800$	8.53879	0.301892
$801$	0	0
$802$	99.4542	3.51185
$803$	−14.2129	−0.501563
$804$	0	0
$805$	−0.0692417	−0.00244045
$806$	−32.5441	−1.14632
$807$	0	0
$808$	−77.1779	−2.71511
$809$	12.5391	0.440852	0.220426	−	0.975404i	$-0.429255\pi$
$809$	12.5391	0.440852	0.220426	+	0.975404i	$0.429255\pi$
$810$	0	0
$811$	−8.27697	−0.290644	−0.145322	−	0.989384i	$-0.546422\pi$
$811$	−8.27697	−0.290644	−0.145322	+	0.989384i	$0.546422\pi$
$812$	3.30678	0.116045
$813$	0	0
$814$	20.5806	0.721350
$815$	2.32128	0.0813109
$816$	0	0
$817$	17.6585	0.617791
$818$	−3.04817	−0.106577
$819$	0	0
$820$	−10.8970	−0.380538
$821$	−50.1133	−1.74897	−0.874483	−	0.485056i	$-0.838799\pi$
$821$	−50.1133	−1.74897	−0.874483	+	0.485056i	$0.838799\pi$
$822$	0	0
$823$	−52.3851	−1.82603	−0.913014	−	0.407927i	$-0.866252\pi$
$823$	−52.3851	−1.82603	−0.913014	+	0.407927i	$0.866252\pi$
$824$	−19.8746	−0.692366
$825$	0	0
$826$	−11.5824	−0.403002
$827$	−27.7003	−0.963234	−0.481617	−	0.876382i	$-0.659950\pi$
$827$	−27.7003	−0.963234	−0.481617	+	0.876382i	$0.659950\pi$
$828$	0	0
$829$	−7.51970	−0.261170	−0.130585	−	0.991437i	$-0.541686\pi$
$829$	−7.51970	−0.261170	−0.130585	+	0.991437i	$0.541686\pi$
$830$	−14.4014	−0.499878
$831$	0	0
$832$	−11.3068	−0.391992
$833$	3.70156	0.128252
$834$	0	0
$835$	−17.7581	−0.614544
$836$	19.5416	0.675861
$837$	0	0
$838$	96.4524	3.33189
$839$	41.0385	1.41681	0.708403	−	0.705809i	$-0.249416\pi$
$839$	41.0385	1.41681	0.708403	+	0.705809i	$0.249416\pi$
$840$	0	0
$841$	−28.5053	−0.982942
$842$	8.03722	0.276981
$843$	0	0
$844$	−113.839	−3.91848
$845$	−7.21647	−0.248254
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	74.0621	2.54330
$849$	0	0
$850$	9.58237	0.328673
$851$	0.550475	0.0188700
$852$	0	0
$853$	42.8693	1.46782	0.733909	−	0.679248i	$-0.237694\pi$
$853$	42.8693	1.46782	0.733909	+	0.679248i	$0.237694\pi$
$854$	−12.9982	−0.444790
$855$	0	0
$856$	107.399	3.67083
$857$	−18.7354	−0.639988	−0.319994	−	0.947420i	$-0.603681\pi$
$857$	−18.7354	−0.639988	−0.319994	+	0.947420i	$0.603681\pi$
$858$	0	0
$859$	−47.5569	−1.62262	−0.811310	−	0.584616i	$-0.801245\pi$
$859$	−47.5569	−1.62262	−0.811310	+	0.584616i	$0.801245\pi$
$860$	19.9745	0.681126
$861$	0	0
$862$	35.0649	1.19431
$863$	49.6177	1.68901	0.844503	−	0.535551i	$-0.179896\pi$
$863$	49.6177	1.68901	0.844503	+	0.535551i	$0.179896\pi$
$864$	0	0
$865$	−20.2094	−0.687139
$866$	−87.4744	−2.97250
$867$	0	0
$868$	−24.5771	−0.834200
$869$	−4.40490	−0.149426
$870$	0	0
$871$	−11.3574	−0.384832
$872$	89.9364	3.04563
$873$	0	0
$874$	0.745030	0.0252010
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	29.0962	0.982510	0.491255	−	0.871016i	$-0.336538\pi$
$877$	29.0962	0.982510	0.491255	+	0.871016i	$0.336538\pi$
$878$	30.0429	1.01390
$879$	0	0
$880$	8.70156	0.293330
$881$	−10.8454	−0.365390	−0.182695	−	0.983170i	$-0.558482\pi$
$881$	−10.8454	−0.365390	−0.182695	+	0.983170i	$0.558482\pi$
$882$	0	0
$883$	12.4000	0.417293	0.208646	−	0.977991i	$-0.433094\pi$
$883$	12.4000	0.417293	0.208646	+	0.977991i	$0.433094\pi$
$884$	−41.8527	−1.40766
$885$	0	0
$886$	64.0004	2.15014
$887$	−19.7823	−0.664224	−0.332112	−	0.943240i	$-0.607761\pi$
$887$	−19.7823	−0.664224	−0.332112	+	0.943240i	$0.607761\pi$
$888$	0	0
$889$	0.248490	0.00833410
$890$	−39.3404	−1.31869
$891$	0	0
$892$	39.4319	1.32028
$893$	−55.5012	−1.85728
$894$	0	0
$895$	5.95005	0.198888
$896$	4.90647	0.163914
$897$	0	0
$898$	−81.6164	−2.72358
$899$	−3.67663	−0.122623
$900$	0	0
$901$	31.5053	1.04959
$902$	−6.00000	−0.199778
$903$	0	0
$904$	−115.474	−3.84062
$905$	−5.54161	−0.184209
$906$	0	0
$907$	−9.96278	−0.330809	−0.165404	−	0.986226i	$-0.552893\pi$
$907$	−9.96278	−0.330809	−0.165404	+	0.986226i	$0.552893\pi$
$908$	−13.1428	−0.436160
$909$	0	0
$910$	6.22565	0.206378
$911$	−57.6869	−1.91125	−0.955627	−	0.294580i	$-0.904820\pi$
$911$	−57.6869	−1.91125	−0.955627	+	0.294580i	$0.904820\pi$
$912$	0	0
$913$	−5.56308	−0.184111
$914$	−34.6564	−1.14633
$915$	0	0
$916$	46.7975	1.54623
$917$	12.6178	0.416677
$918$	0	0
$919$	19.2033	0.633460	0.316730	−	0.948516i	$-0.397415\pi$
$919$	19.2033	0.633460	0.316730	+	0.948516i	$0.397415\pi$
$920$	0.484251	0.0159653
$921$	0	0
$922$	−10.0202	−0.329998
$923$	10.5933	0.348684
$924$	0	0
$925$	7.95005	0.261396
$926$	75.7163	2.48819
$927$	0	0
$928$	−6.00564	−0.197145
$929$	−39.9422	−1.31046	−0.655231	−	0.755428i	$-0.727429\pi$
$929$	−39.9422	−1.31046	−0.655231	+	0.755428i	$0.727429\pi$
$930$	0	0
$931$	4.15641	0.136221
$932$	13.0521	0.427536
$933$	0	0
$934$	−36.9026	−1.20749
$935$	3.70156	0.121054
$936$	0	0
$937$	29.7826	0.972954	0.486477	−	0.873693i	$-0.338282\pi$
$937$	29.7826	0.972954	0.486477	+	0.873693i	$0.338282\pi$
$938$	−12.2256	−0.399182
$939$	0	0
$940$	−62.7808	−2.04768
$941$	−26.5208	−0.864554	−0.432277	−	0.901741i	$-0.642290\pi$
$941$	−26.5208	−0.864554	−0.432277	+	0.901741i	$0.642290\pi$
$942$	0	0
$943$	−0.160484	−0.00522607
$944$	38.9320	1.26713
$945$	0	0
$946$	10.9982	0.357583
$947$	34.6659	1.12649	0.563245	−	0.826290i	$-0.309553\pi$
$947$	34.6659	1.12649	0.563245	+	0.826290i	$0.309553\pi$
$948$	0	0
$949$	34.1806	1.10955
$950$	10.7598	0.349096
$951$	0	0
$952$	−25.8874	−0.839015
$953$	31.9451	1.03480	0.517402	−	0.855742i	$-0.326899\pi$
$953$	31.9451	1.03480	0.517402	+	0.855742i	$0.326899\pi$
$954$	0	0
$955$	−25.8966	−0.837993
$956$	68.6590	2.22059
$957$	0	0
$958$	83.4450	2.69599
$959$	9.40312	0.303643
$960$	0	0
$961$	−3.67405	−0.118518
$962$	−49.4942	−1.59576
$963$	0	0
$964$	49.5472	1.59581
$965$	−1.31596	−0.0423622
$966$	0	0
$967$	5.97781	0.192233	0.0961167	−	0.995370i	$-0.469358\pi$
$967$	5.97781	0.192233	0.0961167	+	0.995370i	$0.469358\pi$
$968$	6.99364	0.224784
$969$	0	0
$970$	−21.3440	−0.685314
$971$	6.61617	0.212323	0.106161	−	0.994349i	$-0.466144\pi$
$971$	6.61617	0.212323	0.106161	+	0.994349i	$0.466144\pi$
$972$	0	0
$973$	−22.4934	−0.721106
$974$	13.6525	0.437456
$975$	0	0
$976$	43.6911	1.39852
$977$	−14.9276	−0.477577	−0.238788	−	0.971072i	$-0.576750\pi$
$977$	−14.9276	−0.477577	−0.238788	+	0.971072i	$0.576750\pi$
$978$	0	0
$979$	−15.1968	−0.485691
$980$	4.70156	0.150186
$981$	0	0
$982$	−92.3267	−2.94626
$983$	15.6774	0.500030	0.250015	−	0.968242i	$-0.419564\pi$
$983$	15.6774	0.500030	0.250015	+	0.968242i	$0.419564\pi$
$984$	0	0
$985$	4.44212	0.141538
$986$	−6.73962	−0.214633
$987$	0	0
$988$	−46.9956	−1.49513
$989$	0.294173	0.00935415
$990$	0	0
$991$	−33.8504	−1.07529	−0.537647	−	0.843170i	$-0.680687\pi$
$991$	−33.8504	−1.07529	−0.537647	+	0.843170i	$0.680687\pi$
$992$	44.6359	1.41719
$993$	0	0
$994$	11.4031	0.361685
$995$	−25.7453	−0.816182
$996$	0	0
$997$	−6.45130	−0.204315	−0.102157	−	0.994768i	$-0.532575\pi$
$997$	−6.45130	−0.204315	−0.102157	+	0.994768i	$0.532575\pi$
$998$	−100.982	−3.19654
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3465.2.a.bl.1.4		4
3.2	odd	2		1155.2.a.u.1.1	✓	4
15.14	odd	2		5775.2.a.bz.1.4		4
21.20	even	2		8085.2.a.bn.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1155.2.a.u.1.1	✓	4	3.2	odd	2
3465.2.a.bl.1.4		4	1.1	even	1	trivial
5775.2.a.bz.1.4		4	15.14	odd	2
8085.2.a.bn.1.1		4	21.20	even	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:	\( N \)	\(=\)	\( 3465 = 3^{2} \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3465.a (trivial)

\(f(q)\)	\(=\)	\(q+2.58874 q^{2} +4.70156 q^{4} +1.00000 q^{5} -1.00000 q^{7} +6.99364 q^{8} +O(q^{10})\)	\(q+2.58874 q^{2} +4.70156 q^{4} +1.00000 q^{5} -1.00000 q^{7} +6.99364 q^{8} +2.58874 q^{10} +1.00000 q^{11} -2.40490 q^{13} -2.58874 q^{14} +8.70156 q^{16} +3.70156 q^{17} +4.15641 q^{19} +4.70156 q^{20} +2.58874 q^{22} +0.0692417 q^{23} +1.00000 q^{25} -6.22565 q^{26} -4.70156 q^{28} -0.703336 q^{29} +5.22742 q^{31} +8.53879 q^{32} +9.58237 q^{34} -1.00000 q^{35} +7.95005 q^{37} +10.7598 q^{38} +6.99364 q^{40} -2.31773 q^{41} +4.24849 q^{43} +4.70156 q^{44} +0.179249 q^{46} -13.3532 q^{47} +1.00000 q^{49} +2.58874 q^{50} -11.3068 q^{52} +8.51136 q^{53} +1.00000 q^{55} -6.99364 q^{56} -1.82075 q^{58} +4.47414 q^{59} +5.02107 q^{61} +13.5324 q^{62} +4.70156 q^{64} -2.40490 q^{65} +4.72263 q^{67} +17.4031 q^{68} -2.58874 q^{70} -4.40490 q^{71} -14.2129 q^{73} +20.5806 q^{74} +19.5416 q^{76} -1.00000 q^{77} -4.40490 q^{79} +8.70156 q^{80} -6.00000 q^{82} -5.56308 q^{83} +3.70156 q^{85} +10.9982 q^{86} +6.99364 q^{88} -15.1968 q^{89} +2.40490 q^{91} +0.325544 q^{92} -34.5679 q^{94} +4.15641 q^{95} -8.24494 q^{97} +2.58874 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 6 q^{4} + 4 q^{5} - 4 q^{7}+O(q^{10}) \) 4 * q + 6 * q^4 + 4 * q^5 - 4 * q^7	\( 4 q + 6 q^{4} + 4 q^{5} - 4 q^{7} + 4 q^{11} + 8 q^{13} + 22 q^{16} + 2 q^{17} + 10 q^{19} + 6 q^{20} + 2 q^{23} + 4 q^{25} - 20 q^{26} - 6 q^{28} + 2 q^{29} + 24 q^{31} - 4 q^{35} + 8 q^{37} - 16 q^{38} + 6 q^{43} + 6 q^{44} - 12 q^{46} - 4 q^{47} + 4 q^{49} + 12 q^{52} - 14 q^{53} + 4 q^{55} - 20 q^{58} + 2 q^{59} + 6 q^{61} - 8 q^{62} + 6 q^{64} + 8 q^{65} - 8 q^{67} + 44 q^{68} + 4 q^{73} + 36 q^{74} + 56 q^{76} - 4 q^{77} + 22 q^{80} - 24 q^{82} - 6 q^{83} + 2 q^{85} + 36 q^{86} - 18 q^{89} - 8 q^{91} + 44 q^{92} - 36 q^{94} + 10 q^{95} - 6 q^{97}+O(q^{100}) \) 4 * q + 6 * q^4 + 4 * q^5 - 4 * q^7 + 4 * q^11 + 8 * q^13 + 22 * q^16 + 2 * q^17 + 10 * q^19 + 6 * q^20 + 2 * q^23 + 4 * q^25 - 20 * q^26 - 6 * q^28 + 2 * q^29 + 24 * q^31 - 4 * q^35 + 8 * q^37 - 16 * q^38 + 6 * q^43 + 6 * q^44 - 12 * q^46 - 4 * q^47 + 4 * q^49 + 12 * q^52 - 14 * q^53 + 4 * q^55 - 20 * q^58 + 2 * q^59 + 6 * q^61 - 8 * q^62 + 6 * q^64 + 8 * q^65 - 8 * q^67 + 44 * q^68 + 4 * q^73 + 36 * q^74 + 56 * q^76 - 4 * q^77 + 22 * q^80 - 24 * q^82 - 6 * q^83 + 2 * q^85 + 36 * q^86 - 18 * q^89 - 8 * q^91 + 44 * q^92 - 36 * q^94 + 10 * q^95 - 6 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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