LMFDB - Embedded newform 2475.2.a.bf.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,2,Mod(1,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, 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("2475.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2475.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:	$19.7629745003$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.48704.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 6x^{2} + 4x + 6 $ x^4 - 2x^3 - 6x^2 + 4*x + 6
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 495)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.852061$ of defining polynomial
Character	$\chi$	$=$	2475.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.85206 q^{2} +1.43013 q^{4} -4.90749 q^{7} +1.05543 q^{8} +O(q^{10})$	$q-1.85206 q^{2} +1.43013 q^{4} -4.90749 q^{7} +1.05543 q^{8} -1.00000 q^{11} +4.61162 q^{13} +9.08898 q^{14} -4.81499 q^{16} -3.76776 q^{17} +4.84386 q^{19} +1.85206 q^{22} +0.860262 q^{23} -8.54100 q^{26} -7.01836 q^{28} -10.3794 q^{29} +7.81499 q^{31} +6.80679 q^{32} +6.97811 q^{34} +4.67525 q^{37} -8.97113 q^{38} +2.97113 q^{41} +0.907494 q^{43} -1.43013 q^{44} -1.59326 q^{46} +13.2232 q^{47} +17.0835 q^{49} +6.59522 q^{52} -5.13974 q^{53} -5.17953 q^{56} +19.2232 q^{58} -12.5480 q^{59} -11.2232 q^{61} -14.4738 q^{62} -2.97661 q^{64} -4.26851 q^{67} -5.38838 q^{68} -5.13974 q^{71} +4.61162 q^{73} -8.65885 q^{74} +6.92735 q^{76} +4.90749 q^{77} -0.843861 q^{79} -5.50271 q^{82} +5.75135 q^{83} -1.68073 q^{86} -1.05543 q^{88} +1.40825 q^{89} -22.6315 q^{91} +1.23029 q^{92} -24.4902 q^{94} -8.54798 q^{97} -31.6397 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} + 8 q^{4} - 4 q^{7} - 6 q^{8}+O(q^{10}) $ 4 * q - 2 * q^2 + 8 * q^4 - 4 * q^7 - 6 * q^8	$ 4 q - 2 q^{2} + 8 q^{4} - 4 q^{7} - 6 q^{8} - 4 q^{11} - 8 q^{13} - 8 q^{14} + 12 q^{16} - 4 q^{17} + 4 q^{19} + 2 q^{22} + 8 q^{23} - 16 q^{26} + 8 q^{28} - 4 q^{29} - 14 q^{32} + 4 q^{34} - 8 q^{37} - 20 q^{38} - 4 q^{41} - 12 q^{43} - 8 q^{44} - 16 q^{46} + 20 q^{49} - 20 q^{52} - 16 q^{53} - 48 q^{56} + 24 q^{58} - 24 q^{59} + 8 q^{61} + 20 q^{62} - 48 q^{68} - 16 q^{71} - 8 q^{73} + 12 q^{74} - 36 q^{76} + 4 q^{77} + 12 q^{79} + 40 q^{82} - 8 q^{83} + 16 q^{86} + 6 q^{88} - 16 q^{89} - 16 q^{91} + 72 q^{92} - 40 q^{94} - 8 q^{97} - 62 q^{98}+O(q^{100}) $ 4 * q - 2 * q^2 + 8 * q^4 - 4 * q^7 - 6 * q^8 - 4 * q^11 - 8 * q^13 - 8 * q^14 + 12 * q^16 - 4 * q^17 + 4 * q^19 + 2 * q^22 + 8 * q^23 - 16 * q^26 + 8 * q^28 - 4 * q^29 - 14 * q^32 + 4 * q^34 - 8 * q^37 - 20 * q^38 - 4 * q^41 - 12 * q^43 - 8 * q^44 - 16 * q^46 + 20 * q^49 - 20 * q^52 - 16 * q^53 - 48 * q^56 + 24 * q^58 - 24 * q^59 + 8 * q^61 + 20 * q^62 - 48 * q^68 - 16 * q^71 - 8 * q^73 + 12 * q^74 - 36 * q^76 + 4 * q^77 + 12 * q^79 + 40 * q^82 - 8 * q^83 + 16 * q^86 + 6 * q^88 - 16 * q^89 - 16 * q^91 + 72 * q^92 - 40 * q^94 - 8 * q^97 - 62 * q^98

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 $

$2$	−1.85206	−1.30961	−0.654803	−	0.755800i	$-0.727248\pi$
$2$	−1.85206	−1.30961	−0.654803	+	0.755800i	$0.727248\pi$
$3$	0	0
$3$	0	0
$4$	1.43013	0.715065
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.90749	−1.85486	−0.927429	−	0.373999i	$-0.877986\pi$
$7$	−4.90749	−1.85486	−0.927429	+	0.373999i	$0.877986\pi$
$8$	1.05543	0.373152
$9$	0	0
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	4.61162	1.27903	0.639516	−	0.768778i	$-0.279135\pi$
$13$	4.61162	1.27903	0.639516	+	0.768778i	$0.279135\pi$
$14$	9.08898	2.42913
$15$	0	0
$16$	−4.81499	−1.20375
$17$	−3.76776	−0.913815	−0.456907	−	0.889514i	$-0.651043\pi$
$17$	−3.76776	−0.913815	−0.456907	+	0.889514i	$0.651043\pi$
$18$	0	0
$19$	4.84386	1.11126	0.555629	−	0.831430i	$-0.312478\pi$
$19$	4.84386	1.11126	0.555629	+	0.831430i	$0.312478\pi$
$20$	0	0
$21$	0	0
$22$	1.85206	0.394861
$23$	0.860262	0.179377	0.0896885	−	0.995970i	$-0.471413\pi$
$23$	0.860262	0.179377	0.0896885	+	0.995970i	$0.471413\pi$
$24$	0	0
$25$	0	0
$26$	−8.54100	−1.67503
$27$	0	0
$28$	−7.01836	−1.32635
$29$	−10.3794	−1.92740	−0.963700	−	0.266986i	$-0.913972\pi$
$29$	−10.3794	−1.92740	−0.963700	+	0.266986i	$0.913972\pi$
$30$	0	0
$31$	7.81499	1.40361	0.701807	−	0.712368i	$-0.252377\pi$
$31$	7.81499	1.40361	0.701807	+	0.712368i	$0.252377\pi$
$32$	6.80679	1.20328
$33$	0	0
$34$	6.97811	1.19674
$35$	0	0
$36$	0	0
$37$	4.67525	0.768606	0.384303	−	0.923207i	$-0.374442\pi$
$37$	4.67525	0.768606	0.384303	+	0.923207i	$0.374442\pi$
$38$	−8.97113	−1.45531
$39$	0	0
$40$	0	0
$41$	2.97113	0.464012	0.232006	−	0.972714i	$-0.425471\pi$
$41$	2.97113	0.464012	0.232006	+	0.972714i	$0.425471\pi$
$42$	0	0
$43$	0.907494	0.138391	0.0691957	−	0.997603i	$-0.477957\pi$
$43$	0.907494	0.138391	0.0691957	+	0.997603i	$0.477957\pi$
$44$	−1.43013	−0.215600
$45$	0	0
$46$	−1.59326	−0.234913
$47$	13.2232	1.92881	0.964403	−	0.264436i	$-0.0851857\pi$
$47$	13.2232	1.92881	0.964403	+	0.264436i	$0.0851857\pi$
$48$	0	0
$49$	17.0835	2.44050
$50$	0	0
$51$	0	0
$52$	6.59522	0.914592
$53$	−5.13974	−0.705997	−0.352999	−	0.935624i	$-0.614838\pi$
$53$	−5.13974	−0.705997	−0.352999	+	0.935624i	$0.614838\pi$
$54$	0	0
$55$	0	0
$56$	−5.17953	−0.692144
$57$	0	0
$58$	19.2232	2.52413
$59$	−12.5480	−1.63361	−0.816804	−	0.576915i	$-0.804256\pi$
$59$	−12.5480	−1.63361	−0.816804	+	0.576915i	$0.804256\pi$
$60$	0	0
$61$	−11.2232	−1.43699	−0.718494	−	0.695533i	$-0.755168\pi$
$61$	−11.2232	−1.43699	−0.718494	+	0.695533i	$0.755168\pi$
$62$	−14.4738	−1.83818
$63$	0	0
$64$	−2.97661	−0.372076
$65$	0	0
$66$	0	0
$67$	−4.26851	−0.521481	−0.260741	−	0.965409i	$-0.583967\pi$
$67$	−4.26851	−0.521481	−0.260741	+	0.965409i	$0.583967\pi$
$68$	−5.38838	−0.653438
$69$	0	0
$70$	0	0
$71$	−5.13974	−0.609975	−0.304987	−	0.952356i	$-0.598652\pi$
$71$	−5.13974	−0.609975	−0.304987	+	0.952356i	$0.598652\pi$
$72$	0	0
$73$	4.61162	0.539749	0.269874	−	0.962896i	$-0.413018\pi$
$73$	4.61162	0.539749	0.269874	+	0.962896i	$0.413018\pi$
$74$	−8.65885	−1.00657
$75$	0	0
$76$	6.92735	0.794622
$77$	4.90749	0.559261
$78$	0	0
$79$	−0.843861	−0.0949417	−0.0474709	−	0.998873i	$-0.515116\pi$
$79$	−0.843861	−0.0949417	−0.0474709	+	0.998873i	$0.515116\pi$
$80$	0	0
$81$	0	0
$82$	−5.50271	−0.607673
$83$	5.75135	0.631293	0.315647	−	0.948877i	$-0.397779\pi$
$83$	5.75135	0.631293	0.315647	+	0.948877i	$0.397779\pi$
$84$	0	0
$85$	0	0
$86$	−1.68073	−0.181238
$87$	0	0
$88$	−1.05543	−0.112509
$89$	1.40825	0.149274	0.0746368	−	0.997211i	$-0.476220\pi$
$89$	1.40825	0.149274	0.0746368	+	0.997211i	$0.476220\pi$
$90$	0	0
$91$	−22.6315	−2.37242
$92$	1.23029	0.128266
$93$	0	0
$94$	−24.4902	−2.52598
$95$	0	0
$96$	0	0
$97$	−8.54798	−0.867916	−0.433958	−	0.900933i	$-0.642883\pi$
$97$	−8.54798	−0.867916	−0.433958	+	0.900933i	$0.642883\pi$
$98$	−31.6397	−3.19609
$99$	0	0
$100$	0	0
$101$	−4.56438	−0.454173	−0.227087	−	0.973875i	$-0.572920\pi$
$101$	−4.56438	−0.454173	−0.227087	+	0.973875i	$0.572920\pi$
$102$	0	0
$103$	4.73300	0.466356	0.233178	−	0.972434i	$-0.425088\pi$
$103$	4.73300	0.466356	0.233178	+	0.972434i	$0.425088\pi$
$104$	4.86725	0.477273
$105$	0	0
$106$	9.51911	0.924578
$107$	−7.34461	−0.710030	−0.355015	−	0.934861i	$-0.615524\pi$
$107$	−7.34461	−0.710030	−0.355015	+	0.934861i	$0.615524\pi$
$108$	0	0
$109$	9.40825	0.901146	0.450573	−	0.892739i	$-0.351220\pi$
$109$	9.40825	0.901146	0.450573	+	0.892739i	$0.351220\pi$
$110$	0	0
$111$	0	0
$112$	23.6295	2.23278
$113$	−10.9547	−1.03053	−0.515267	−	0.857030i	$-0.672307\pi$
$113$	−10.9547	−1.03053	−0.515267	+	0.857030i	$0.672307\pi$
$114$	0	0
$115$	0	0
$116$	−14.8439	−1.37822
$117$	0	0
$118$	23.2396	2.13938
$119$	18.4902	1.69500
$120$	0	0
$121$	1.00000	0.0909091
$122$	20.7861	1.88189
$123$	0	0
$124$	11.1765	1.00368
$125$	0	0
$126$	0	0
$127$	−6.62802	−0.588141	−0.294071	−	0.955784i	$-0.595010\pi$
$127$	−6.62802	−0.588141	−0.294071	+	0.955784i	$0.595010\pi$
$128$	−8.10071	−0.716008
$129$	0	0
$130$	0	0
$131$	13.2232	1.15532	0.577660	−	0.816278i	$-0.303966\pi$
$131$	13.2232	1.15532	0.577660	+	0.816278i	$0.303966\pi$
$132$	0	0
$133$	−23.7712	−2.06123
$134$	7.90554	0.682934
$135$	0	0
$136$	−3.97661	−0.340992
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	0	0
$139$	−12.9711	−1.10020	−0.550098	−	0.835100i	$-0.685410\pi$
$139$	−12.9711	−1.10020	−0.550098	+	0.835100i	$0.685410\pi$
$140$	0	0
$141$	0	0
$142$	9.51911	0.798826
$143$	−4.61162	−0.385643
$144$	0	0
$145$	0	0
$146$	−8.54100	−0.706858
$147$	0	0
$148$	6.68622	0.549604
$149$	−0.0273705	−0.00224228	−0.00112114	−	0.999999i	$-0.500357\pi$
$149$	−0.0273705	−0.00224228	−0.00112114	+	0.999999i	$0.500357\pi$
$150$	0	0
$151$	4.84386	0.394188	0.197094	−	0.980385i	$-0.436850\pi$
$151$	4.84386	0.394188	0.197094	+	0.980385i	$0.436850\pi$
$152$	5.11237	0.414668
$153$	0	0
$154$	−9.08898	−0.732411
$155$	0	0
$156$	0	0
$157$	−2.12727	−0.169774	−0.0848872	−	0.996391i	$-0.527053\pi$
$157$	−2.12727	−0.169774	−0.0848872	+	0.996391i	$0.527053\pi$
$158$	1.56288	0.124336
$159$	0	0
$160$	0	0
$161$	−4.22173	−0.332719
$162$	0	0
$163$	−10.0835	−0.789800	−0.394900	−	0.918724i	$-0.629221\pi$
$163$	−10.0835	−0.789800	−0.394900	+	0.918724i	$0.629221\pi$
$164$	4.24910	0.331799
$165$	0	0
$166$	−10.6519	−0.826745
$167$	−17.7514	−1.37364	−0.686821	−	0.726827i	$-0.740994\pi$
$167$	−17.7514	−1.37364	−0.686821	+	0.726827i	$0.740994\pi$
$168$	0	0
$169$	8.26700	0.635923
$170$	0	0
$171$	0	0
$172$	1.29783	0.0989590
$173$	−15.7678	−1.19880	−0.599400	−	0.800450i	$-0.704594\pi$
$173$	−15.7678	−1.19880	−0.599400	+	0.800450i	$0.704594\pi$
$174$	0	0
$175$	0	0
$176$	4.81499	0.362943
$177$	0	0
$178$	−2.60816	−0.195490
$179$	7.40825	0.553718	0.276859	−	0.960911i	$-0.410706\pi$
$179$	7.40825	0.553718	0.276859	+	0.960911i	$0.410706\pi$
$180$	0	0
$181$	7.26700	0.540152	0.270076	−	0.962839i	$-0.412951\pi$
$181$	7.26700	0.540152	0.270076	+	0.962839i	$0.412951\pi$
$182$	41.9149	3.10694
$183$	0	0
$184$	0.907948	0.0669348
$185$	0	0
$186$	0	0
$187$	3.76776	0.275526
$188$	18.9110	1.37922
$189$	0	0
$190$	0	0
$191$	−2.87123	−0.207755	−0.103877	−	0.994590i	$-0.533125\pi$
$191$	−2.87123	−0.207755	−0.103877	+	0.994590i	$0.533125\pi$
$192$	0	0
$193$	−18.8911	−1.35981	−0.679905	−	0.733300i	$-0.737979\pi$
$193$	−18.8911	−1.35981	−0.679905	+	0.733300i	$0.737979\pi$
$194$	15.8314	1.13663
$195$	0	0
$196$	24.4316	1.74512
$197$	−14.0472	−1.00082	−0.500412	−	0.865787i	$-0.666818\pi$
$197$	−14.0472	−1.00082	−0.500412	+	0.865787i	$0.666818\pi$
$198$	0	0
$199$	3.40825	0.241604	0.120802	−	0.992677i	$-0.461453\pi$
$199$	3.40825	0.241604	0.120802	+	0.992677i	$0.461453\pi$
$200$	0	0
$201$	0	0
$202$	8.45352	0.594788
$203$	50.9367	3.57506
$204$	0	0
$205$	0	0
$206$	−8.76580	−0.610742
$207$	0	0
$208$	−22.2049	−1.53963
$209$	−4.84386	−0.335057
$210$	0	0
$211$	−24.4738	−1.68485	−0.842424	−	0.538815i	$-0.818872\pi$
$211$	−24.4738	−1.68485	−0.842424	+	0.538815i	$0.818872\pi$
$212$	−7.35050	−0.504834
$213$	0	0
$214$	13.6027	0.929859
$215$	0	0
$216$	0	0
$217$	−38.3520	−2.60350
$218$	−17.4246	−1.18015
$219$	0	0
$220$	0	0
$221$	−17.3754	−1.16880
$222$	0	0
$223$	11.5355	0.772475	0.386237	−	0.922399i	$-0.373775\pi$
$223$	11.5355	0.772475	0.386237	+	0.922399i	$0.373775\pi$
$224$	−33.4043	−2.23192
$225$	0	0
$226$	20.2888	1.34959
$227$	1.15960	0.0769653	0.0384827	−	0.999259i	$-0.487748\pi$
$227$	1.15960	0.0769653	0.0384827	+	0.999259i	$0.487748\pi$
$228$	0	0
$229$	−24.4465	−1.61547	−0.807734	−	0.589547i	$-0.799306\pi$
$229$	−24.4465	−1.61547	−0.807734	+	0.589547i	$0.799306\pi$
$230$	0	0
$231$	0	0
$232$	−10.9547	−0.719213
$233$	−9.45548	−0.619449	−0.309724	−	0.950826i	$-0.600237\pi$
$233$	−9.45548	−0.619449	−0.309724	+	0.950826i	$0.600237\pi$
$234$	0	0
$235$	0	0
$236$	−17.9453	−1.16814
$237$	0	0
$238$	−34.2451	−2.21978
$239$	−14.9438	−0.966631	−0.483316	−	0.875446i	$-0.660568\pi$
$239$	−14.9438	−0.966631	−0.483316	+	0.875446i	$0.660568\pi$
$240$	0	0
$241$	7.81499	0.503408	0.251704	−	0.967804i	$-0.419009\pi$
$241$	7.81499	0.503408	0.251704	+	0.967804i	$0.419009\pi$
$242$	−1.85206	−0.119055
$243$	0	0
$244$	−16.0507	−1.02754
$245$	0	0
$246$	0	0
$247$	22.3380	1.42133
$248$	8.24819	0.523761
$249$	0	0
$250$	0	0
$251$	3.41921	0.215819	0.107909	−	0.994161i	$-0.465584\pi$
$251$	3.41921	0.215819	0.107909	+	0.994161i	$0.465584\pi$
$252$	0	0
$253$	−0.860262	−0.0540842
$254$	12.2755	0.770233
$255$	0	0
$256$	20.9562	1.30976
$257$	−1.53551	−0.0957826	−0.0478913	−	0.998853i	$-0.515250\pi$
$257$	−1.53551	−0.0957826	−0.0478913	+	0.998853i	$0.515250\pi$
$258$	0	0
$259$	−22.9438	−1.42566
$260$	0	0
$261$	0	0
$262$	−24.4902	−1.51301
$263$	0.190899	0.0117713	0.00588567	−	0.999983i	$-0.498127\pi$
$263$	0.190899	0.0117713	0.00588567	+	0.999983i	$0.498127\pi$
$264$	0	0
$265$	0	0
$266$	44.0257	2.69939
$267$	0	0
$268$	−6.10452	−0.372893
$269$	14.4465	0.880817	0.440408	−	0.897798i	$-0.354834\pi$
$269$	14.4465	0.880817	0.440408	+	0.897798i	$0.354834\pi$
$270$	0	0
$271$	−9.97263	−0.605794	−0.302897	−	0.953023i	$-0.597954\pi$
$271$	−9.97263	−0.605794	−0.302897	+	0.953023i	$0.597954\pi$
$272$	18.1417	1.10000
$273$	0	0
$274$	−11.1124	−0.671323
$275$	0	0
$276$	0	0
$277$	−12.5788	−0.755788	−0.377894	−	0.925849i	$-0.623352\pi$
$277$	−12.5788	−0.755788	−0.377894	+	0.925849i	$0.623352\pi$
$278$	24.0233	1.44082
$279$	0	0
$280$	0	0
$281$	−16.6917	−0.995740	−0.497870	−	0.867252i	$-0.665884\pi$
$281$	−16.6917	−0.995740	−0.497870	+	0.867252i	$0.665884\pi$
$282$	0	0
$283$	21.5937	1.28361	0.641806	−	0.766867i	$-0.278185\pi$
$283$	21.5937	1.28361	0.641806	+	0.766867i	$0.278185\pi$
$284$	−7.35050	−0.436172
$285$	0	0
$286$	8.54100	0.505040
$287$	−14.5808	−0.860677
$288$	0	0
$289$	−2.80402	−0.164942
$290$	0	0
$291$	0	0
$292$	6.59522	0.385956
$293$	−33.0855	−1.93287	−0.966436	−	0.256906i	$-0.917297\pi$
$293$	−33.0855	−1.93287	−0.966436	+	0.256906i	$0.917297\pi$
$294$	0	0
$295$	0	0
$296$	4.93441	0.286807
$297$	0	0
$298$	0.0506919	0.00293650
$299$	3.96720	0.229429
$300$	0	0
$301$	−4.45352	−0.256697
$302$	−8.97113	−0.516230
$303$	0	0
$304$	−23.3231	−1.33767
$305$	0	0
$306$	0	0
$307$	−7.72398	−0.440831	−0.220416	−	0.975406i	$-0.570741\pi$
$307$	−7.72398	−0.440831	−0.220416	+	0.975406i	$0.570741\pi$
$308$	7.01836	0.399908
$309$	0	0
$310$	0	0
$311$	−12.1780	−0.690549	−0.345274	−	0.938502i	$-0.612214\pi$
$311$	−12.1780	−0.690549	−0.345274	+	0.938502i	$0.612214\pi$
$312$	0	0
$313$	2.95473	0.167011	0.0835055	−	0.996507i	$-0.473388\pi$
$313$	2.95473	0.167011	0.0835055	+	0.996507i	$0.473388\pi$
$314$	3.93983	0.222337
$315$	0	0
$316$	−1.20683	−0.0678896
$317$	−27.4917	−1.54409	−0.772045	−	0.635568i	$-0.780766\pi$
$317$	−27.4917	−1.54409	−0.772045	+	0.635568i	$0.780766\pi$
$318$	0	0
$319$	10.3794	0.581133
$320$	0	0
$321$	0	0
$322$	7.81890	0.435730
$323$	−18.2505	−1.01548
$324$	0	0
$325$	0	0
$326$	18.6752	1.03433
$327$	0	0
$328$	3.13582	0.173147
$329$	−64.8929	−3.57766
$330$	0	0
$331$	16.5737	0.910975	0.455487	−	0.890242i	$-0.349465\pi$
$331$	16.5737	0.910975	0.455487	+	0.890242i	$0.349465\pi$
$332$	8.22519	0.451416
$333$	0	0
$334$	32.8766	1.79893
$335$	0	0
$336$	0	0
$337$	−4.64442	−0.252998	−0.126499	−	0.991967i	$-0.540374\pi$
$337$	−4.64442	−0.252998	−0.126499	+	0.991967i	$0.540374\pi$
$338$	−15.3110	−0.832809
$339$	0	0
$340$	0	0
$341$	−7.81499	−0.423205
$342$	0	0
$343$	−49.4847	−2.67192
$344$	0.957798	0.0516410
$345$	0	0
$346$	29.2028	1.56995
$347$	−23.6936	−1.27194	−0.635970	−	0.771714i	$-0.719400\pi$
$347$	−23.6936	−1.27194	−0.635970	+	0.771714i	$0.719400\pi$
$348$	0	0
$349$	−21.7572	−1.16464	−0.582319	−	0.812960i	$-0.697855\pi$
$349$	−21.7572	−1.16464	−0.582319	+	0.812960i	$0.697855\pi$
$350$	0	0
$351$	0	0
$352$	−6.80679	−0.362803
$353$	−6.49024	−0.345440	−0.172720	−	0.984971i	$-0.555256\pi$
$353$	−6.49024	−0.345440	−0.172720	+	0.984971i	$0.555256\pi$
$354$	0	0
$355$	0	0
$356$	2.01397	0.106740
$357$	0	0
$358$	−13.7205	−0.725152
$359$	19.1655	1.01152	0.505758	−	0.862676i	$-0.331213\pi$
$359$	19.1655	1.01152	0.505758	+	0.862676i	$0.331213\pi$
$360$	0	0
$361$	4.46299	0.234894
$362$	−13.4589	−0.707386
$363$	0	0
$364$	−32.3660	−1.69644
$365$	0	0
$366$	0	0
$367$	10.1850	0.531653	0.265827	−	0.964021i	$-0.414355\pi$
$367$	10.1850	0.531653	0.265827	+	0.964021i	$0.414355\pi$
$368$	−4.14215	−0.215924
$369$	0	0
$370$	0	0
$371$	25.2232	1.30952
$372$	0	0
$373$	−19.0184	−0.984733	−0.492367	−	0.870388i	$-0.663868\pi$
$373$	−19.0184	−0.984733	−0.492367	+	0.870388i	$0.663868\pi$
$374$	−6.97811	−0.360830
$375$	0	0
$376$	13.9562	0.719738
$377$	−47.8657	−2.46521
$378$	0	0
$379$	11.0710	0.568680	0.284340	−	0.958723i	$-0.408226\pi$
$379$	11.0710	0.568680	0.284340	+	0.958723i	$0.408226\pi$
$380$	0	0
$381$	0	0
$382$	5.31770	0.272077
$383$	12.7330	0.650626	0.325313	−	0.945606i	$-0.394530\pi$
$383$	12.7330	0.650626	0.325313	+	0.945606i	$0.394530\pi$
$384$	0	0
$385$	0	0
$386$	34.9875	1.78081
$387$	0	0
$388$	−12.2247	−0.620617
$389$	−32.7587	−1.66093	−0.830467	−	0.557068i	$-0.811926\pi$
$389$	−32.7587	−1.66093	−0.830467	+	0.557068i	$0.811926\pi$
$390$	0	0
$391$	−3.24126	−0.163917
$392$	18.0305	0.910676
$393$	0	0
$394$	26.0163	1.31068
$395$	0	0
$396$	0	0
$397$	15.0820	0.756943	0.378472	−	0.925613i	$-0.376450\pi$
$397$	15.0820	0.756943	0.378472	+	0.925613i	$0.376450\pi$
$398$	−6.31228	−0.316406
$399$	0	0
$400$	0	0
$401$	18.7259	0.935129	0.467564	−	0.883959i	$-0.345132\pi$
$401$	18.7259	0.935129	0.467564	+	0.883959i	$0.345132\pi$
$402$	0	0
$403$	36.0397	1.79527
$404$	−6.52767	−0.324764
$405$	0	0
$406$	−94.3379	−4.68191
$407$	−4.67525	−0.231744
$408$	0	0
$409$	24.4793	1.21042	0.605211	−	0.796065i	$-0.293089\pi$
$409$	24.4793	1.21042	0.605211	+	0.796065i	$0.293089\pi$
$410$	0	0
$411$	0	0
$412$	6.76880	0.333475
$413$	61.5791	3.03011
$414$	0	0
$415$	0	0
$416$	31.3903	1.53904
$417$	0	0
$418$	8.97113	0.438792
$419$	9.12877	0.445970	0.222985	−	0.974822i	$-0.428420\pi$
$419$	9.12877	0.445970	0.222985	+	0.974822i	$0.428420\pi$
$420$	0	0
$421$	−13.5465	−0.660215	−0.330108	−	0.943943i	$-0.607085\pi$
$421$	−13.5465	−0.660215	−0.330108	+	0.943943i	$0.607085\pi$
$422$	45.3270	2.20649
$423$	0	0
$424$	−5.42465	−0.263444
$425$	0	0
$426$	0	0
$427$	55.0779	2.66541
$428$	−10.5038	−0.507718
$429$	0	0
$430$	0	0
$431$	−26.4465	−1.27388	−0.636941	−	0.770913i	$-0.719800\pi$
$431$	−26.4465	−1.27388	−0.636941	+	0.770913i	$0.719800\pi$
$432$	0	0
$433$	16.1780	0.777463	0.388732	−	0.921351i	$-0.372913\pi$
$433$	16.1780	0.777463	0.388732	+	0.921351i	$0.372913\pi$
$434$	71.0303	3.40956
$435$	0	0
$436$	13.4550	0.644379
$437$	4.16699	0.199334
$438$	0	0
$439$	−7.02887	−0.335470	−0.167735	−	0.985832i	$-0.553645\pi$
$439$	−7.02887	−0.335470	−0.167735	+	0.985832i	$0.553645\pi$
$440$	0	0
$441$	0	0
$442$	32.1804	1.53066
$443$	24.3630	1.15752	0.578760	−	0.815498i	$-0.303537\pi$
$443$	24.3630	1.15752	0.578760	+	0.815498i	$0.303537\pi$
$444$	0	0
$445$	0	0
$446$	−21.3645	−1.01164
$447$	0	0
$448$	14.6077	0.690149
$449$	−38.3887	−1.81168	−0.905838	−	0.423625i	$-0.860758\pi$
$449$	−38.3887	−1.81168	−0.905838	+	0.423625i	$0.860758\pi$
$450$	0	0
$451$	−2.97113	−0.139905
$452$	−15.6667	−0.736899
$453$	0	0
$454$	−2.14765	−0.100794
$455$	0	0
$456$	0	0
$457$	−21.9621	−1.02734	−0.513672	−	0.857987i	$-0.671715\pi$
$457$	−21.9621	−1.02734	−0.513672	+	0.857987i	$0.671715\pi$
$458$	45.2764	2.11562
$459$	0	0
$460$	0	0
$461$	27.5698	1.28405	0.642027	−	0.766682i	$-0.278094\pi$
$461$	27.5698	1.28405	0.642027	+	0.766682i	$0.278094\pi$
$462$	0	0
$463$	24.8860	1.15655	0.578275	−	0.815842i	$-0.303726\pi$
$463$	24.8860	1.15655	0.578275	+	0.815842i	$0.303726\pi$
$464$	49.9765	2.32010
$465$	0	0
$466$	17.5121	0.811233
$467$	29.0273	1.34322	0.671610	−	0.740904i	$-0.265603\pi$
$467$	29.0273	1.34322	0.671610	+	0.740904i	$0.265603\pi$
$468$	0	0
$469$	20.9477	0.967274
$470$	0	0
$471$	0	0
$472$	−13.2435	−0.609584
$473$	−0.907494	−0.0417266
$474$	0	0
$475$	0	0
$476$	26.4435	1.21203
$477$	0	0
$478$	27.6768	1.26591
$479$	29.4450	1.34537	0.672687	−	0.739927i	$-0.265140\pi$
$479$	29.4450	1.34537	0.672687	+	0.739927i	$0.265140\pi$
$480$	0	0
$481$	21.5605	0.983072
$482$	−14.4738	−0.659265
$483$	0	0
$484$	1.43013	0.0650060
$485$	0	0
$486$	0	0
$487$	−27.5285	−1.24743	−0.623717	−	0.781650i	$-0.714378\pi$
$487$	−27.5285	−1.24743	−0.623717	+	0.781650i	$0.714378\pi$
$488$	−11.8454	−0.536214
$489$	0	0
$490$	0	0
$491$	22.2795	1.00546	0.502729	−	0.864444i	$-0.332329\pi$
$491$	22.2795	1.00546	0.502729	+	0.864444i	$0.332329\pi$
$492$	0	0
$493$	39.1069	1.76129
$494$	−41.3714	−1.86139
$495$	0	0
$496$	−37.6291	−1.68959
$497$	25.2232	1.13142
$498$	0	0
$499$	−35.0382	−1.56853	−0.784263	−	0.620428i	$-0.786959\pi$
$499$	−35.0382	−1.56853	−0.784263	+	0.620428i	$0.786959\pi$
$500$	0	0
$501$	0	0
$502$	−6.33259	−0.282638
$503$	33.9731	1.51478	0.757392	−	0.652960i	$-0.226473\pi$
$503$	33.9731	1.51478	0.757392	+	0.652960i	$0.226473\pi$
$504$	0	0
$505$	0	0
$506$	1.59326	0.0708289
$507$	0	0
$508$	−9.47893	−0.420560
$509$	13.3505	0.591750	0.295875	−	0.955227i	$-0.404389\pi$
$509$	13.3505	0.591750	0.295875	+	0.955227i	$0.404389\pi$
$510$	0	0
$511$	−22.6315	−1.00116
$512$	−22.6108	−0.999266
$513$	0	0
$514$	2.84386	0.125437
$515$	0	0
$516$	0	0
$517$	−13.2232	−0.581557
$518$	42.4932	1.86705
$519$	0	0
$520$	0	0
$521$	−7.46599	−0.327091	−0.163546	−	0.986536i	$-0.552293\pi$
$521$	−7.46599	−0.327091	−0.163546	+	0.986536i	$0.552293\pi$
$522$	0	0
$523$	15.9785	0.698692	0.349346	−	0.936994i	$-0.386404\pi$
$523$	15.9785	0.698692	0.349346	+	0.936994i	$0.386404\pi$
$524$	18.9110	0.826129
$525$	0	0
$526$	−0.353557	−0.0154158
$527$	−29.4450	−1.28264
$528$	0	0
$529$	−22.2599	−0.967824
$530$	0	0
$531$	0	0
$532$	−33.9960	−1.47391
$533$	13.7017	0.593486
$534$	0	0
$535$	0	0
$536$	−4.50512	−0.194592
$537$	0	0
$538$	−26.7557	−1.15352
$539$	−17.0835	−0.735838
$540$	0	0
$541$	−5.46299	−0.234872	−0.117436	−	0.993080i	$-0.537468\pi$
$541$	−5.46299	−0.234872	−0.117436	+	0.993080i	$0.537468\pi$
$542$	18.4699	0.793351
$543$	0	0
$544$	−25.6463	−1.09958
$545$	0	0
$546$	0	0
$547$	−38.8895	−1.66279	−0.831397	−	0.555679i	$-0.812458\pi$
$547$	−38.8895	−1.66279	−0.831397	+	0.555679i	$0.812458\pi$
$548$	8.58079	0.366553
$549$	0	0
$550$	0	0
$551$	−50.2762	−2.14184
$552$	0	0
$553$	4.14124	0.176103
$554$	23.2967	0.989783
$555$	0	0
$556$	−18.5504	−0.786713
$557$	6.58425	0.278983	0.139492	−	0.990223i	$-0.455453\pi$
$557$	6.58425	0.278983	0.139492	+	0.990223i	$0.455453\pi$
$558$	0	0
$559$	4.18501	0.177007
$560$	0	0
$561$	0	0
$562$	30.9140	1.30403
$563$	25.0323	1.05499	0.527494	−	0.849559i	$-0.323132\pi$
$563$	25.0323	1.05499	0.527494	+	0.849559i	$0.323132\pi$
$564$	0	0
$565$	0	0
$566$	−39.9929	−1.68103
$567$	0	0
$568$	−5.42465	−0.227613
$569$	34.5066	1.44659	0.723297	−	0.690537i	$-0.242626\pi$
$569$	34.5066	1.44659	0.723297	+	0.690537i	$0.242626\pi$
$570$	0	0
$571$	−18.6588	−0.780848	−0.390424	−	0.920635i	$-0.627672\pi$
$571$	−18.6588	−0.780848	−0.390424	+	0.920635i	$0.627672\pi$
$572$	−6.59522	−0.275760
$573$	0	0
$574$	27.0045	1.12715
$575$	0	0
$576$	0	0
$577$	37.6697	1.56821	0.784105	−	0.620628i	$-0.213122\pi$
$577$	37.6697	1.56821	0.784105	+	0.620628i	$0.213122\pi$
$578$	5.19321	0.216009
$579$	0	0
$580$	0	0
$581$	−28.2247	−1.17096
$582$	0	0
$583$	5.13974	0.212866
$584$	4.86725	0.201408
$585$	0	0
$586$	61.2763	2.53130
$587$	11.0242	0.455019	0.227510	−	0.973776i	$-0.426942\pi$
$587$	11.0242	0.455019	0.227510	+	0.973776i	$0.426942\pi$
$588$	0	0
$589$	37.8547	1.55978
$590$	0	0
$591$	0	0
$592$	−22.5113	−0.925207
$593$	−39.4703	−1.62085	−0.810425	−	0.585843i	$-0.800764\pi$
$593$	−39.4703	−1.62085	−0.810425	+	0.585843i	$0.800764\pi$
$594$	0	0
$595$	0	0
$596$	−0.0391434	−0.00160338
$597$	0	0
$598$	−7.34749	−0.300461
$599$	2.44646	0.0999598	0.0499799	−	0.998750i	$-0.484084\pi$
$599$	2.44646	0.0999598	0.0499799	+	0.998750i	$0.484084\pi$
$600$	0	0
$601$	−37.6697	−1.53658	−0.768290	−	0.640103i	$-0.778892\pi$
$601$	−37.6697	−1.53658	−0.768290	+	0.640103i	$0.778892\pi$
$602$	8.24819	0.336171
$603$	0	0
$604$	6.92735	0.281870
$605$	0	0
$606$	0	0
$607$	−6.62802	−0.269023	−0.134511	−	0.990912i	$-0.542947\pi$
$607$	−6.62802	−0.269023	−0.134511	+	0.990912i	$0.542947\pi$
$608$	32.9711	1.33716
$609$	0	0
$610$	0	0
$611$	60.9805	2.46701
$612$	0	0
$613$	−15.6498	−0.632091	−0.316045	−	0.948744i	$-0.602355\pi$
$613$	−15.6498	−0.632091	−0.316045	+	0.948744i	$0.602355\pi$
$614$	14.3053	0.577315
$615$	0	0
$616$	5.17953	0.208689
$617$	27.7463	1.11702	0.558511	−	0.829497i	$-0.311373\pi$
$617$	27.7463	1.11702	0.558511	+	0.829497i	$0.311373\pi$
$618$	0	0
$619$	16.5737	0.666154	0.333077	−	0.942900i	$-0.391913\pi$
$619$	16.5737	0.666154	0.333077	+	0.942900i	$0.391913\pi$
$620$	0	0
$621$	0	0
$622$	22.5543	0.904346
$623$	−6.91095	−0.276882
$624$	0	0
$625$	0	0
$626$	−5.47233	−0.218718
$627$	0	0
$628$	−3.04227	−0.121400
$629$	−17.6152	−0.702364
$630$	0	0
$631$	−37.8547	−1.50697	−0.753486	−	0.657464i	$-0.771629\pi$
$631$	−37.8547	−1.50697	−0.753486	+	0.657464i	$0.771629\pi$
$632$	−0.890638	−0.0354277
$633$	0	0
$634$	50.9164	2.02215
$635$	0	0
$636$	0	0
$637$	78.7825	3.12148
$638$	−19.2232	−0.761055
$639$	0	0
$640$	0	0
$641$	11.9423	0.471691	0.235845	−	0.971791i	$-0.424214\pi$
$641$	11.9423	0.471691	0.235845	+	0.971791i	$0.424214\pi$
$642$	0	0
$643$	−10.3380	−0.407692	−0.203846	−	0.979003i	$-0.565344\pi$
$643$	−10.3380	−0.407692	−0.203846	+	0.979003i	$0.565344\pi$
$644$	−6.03763	−0.237916
$645$	0	0
$646$	33.8010	1.32988
$647$	−11.0125	−0.432945	−0.216472	−	0.976289i	$-0.569455\pi$
$647$	−11.0125	−0.432945	−0.216472	+	0.976289i	$0.569455\pi$
$648$	0	0
$649$	12.5480	0.492551
$650$	0	0
$651$	0	0
$652$	−14.4207	−0.564759
$653$	−13.2810	−0.519725	−0.259862	−	0.965646i	$-0.583677\pi$
$653$	−13.2810	−0.519725	−0.259862	+	0.965646i	$0.583677\pi$
$654$	0	0
$655$	0	0
$656$	−14.3059	−0.558553
$657$	0	0
$658$	120.186	4.68533
$659$	−10.2247	−0.398299	−0.199150	−	0.979969i	$-0.563818\pi$
$659$	−10.2247	−0.398299	−0.199150	+	0.979969i	$0.563818\pi$
$660$	0	0
$661$	17.0710	0.663986	0.331993	−	0.943282i	$-0.392279\pi$
$661$	17.0710	0.663986	0.331993	+	0.943282i	$0.392279\pi$
$662$	−30.6956	−1.19302
$663$	0	0
$664$	6.07017	0.235568
$665$	0	0
$666$	0	0
$667$	−8.92898	−0.345731
$668$	−25.3868	−0.982243
$669$	0	0
$670$	0	0
$671$	11.2232	0.433268
$672$	0	0
$673$	−27.8926	−1.07518	−0.537590	−	0.843206i	$-0.680665\pi$
$673$	−27.8926	−1.07518	−0.537590	+	0.843206i	$0.680665\pi$
$674$	8.60175	0.331327
$675$	0	0
$676$	11.8229	0.454727
$677$	19.4922	0.749146	0.374573	−	0.927197i	$-0.377789\pi$
$677$	19.4922	0.749146	0.374573	+	0.927197i	$0.377789\pi$
$678$	0	0
$679$	41.9492	1.60986
$680$	0	0
$681$	0	0
$682$	14.4738	0.554232
$683$	26.9438	1.03097	0.515487	−	0.856897i	$-0.327611\pi$
$683$	26.9438	1.03097	0.515487	+	0.856897i	$0.327611\pi$
$684$	0	0
$685$	0	0
$686$	91.6487	3.49916
$687$	0	0
$688$	−4.36957	−0.166588
$689$	−23.7025	−0.902993
$690$	0	0
$691$	41.2849	1.57055	0.785276	−	0.619146i	$-0.212521\pi$
$691$	41.2849	1.57055	0.785276	+	0.619146i	$0.212521\pi$
$692$	−22.5500	−0.857221
$693$	0	0
$694$	43.8820	1.66574
$695$	0	0
$696$	0	0
$697$	−11.1945	−0.424021
$698$	40.2957	1.52522
$699$	0	0
$700$	0	0
$701$	41.1631	1.55471	0.777354	−	0.629064i	$-0.216562\pi$
$701$	41.1631	1.55471	0.777354	+	0.629064i	$0.216562\pi$
$702$	0	0
$703$	22.6463	0.854120
$704$	2.97661	0.112185
$705$	0	0
$706$	12.0203	0.452391
$707$	22.3997	0.842427
$708$	0	0
$709$	−19.8040	−0.743756	−0.371878	−	0.928282i	$-0.621286\pi$
$709$	−19.8040	−0.743756	−0.371878	+	0.928282i	$0.621286\pi$
$710$	0	0
$711$	0	0
$712$	1.48631	0.0557017
$713$	6.72294	0.251776
$714$	0	0
$715$	0	0
$716$	10.5948	0.395945
$717$	0	0
$718$	−35.4957	−1.32469
$719$	−42.6682	−1.59126	−0.795628	−	0.605786i	$-0.792859\pi$
$719$	−42.6682	−1.59126	−0.795628	+	0.605786i	$0.792859\pi$
$720$	0	0
$721$	−23.2271	−0.865024
$722$	−8.26572	−0.307618
$723$	0	0
$724$	10.3928	0.386244
$725$	0	0
$726$	0	0
$727$	39.0054	1.44663	0.723315	−	0.690518i	$-0.242617\pi$
$727$	39.0054	1.44663	0.723315	+	0.690518i	$0.242617\pi$
$728$	−23.8860	−0.885274
$729$	0	0
$730$	0	0
$731$	−3.41921	−0.126464
$732$	0	0
$733$	12.2744	0.453365	0.226683	−	0.973969i	$-0.427212\pi$
$733$	12.2744	0.453365	0.226683	+	0.973969i	$0.427212\pi$
$734$	−18.8633	−0.696256
$735$	0	0
$736$	5.85562	0.215841
$737$	4.26851	0.157232
$738$	0	0
$739$	34.0343	1.25197	0.625986	−	0.779834i	$-0.284697\pi$
$739$	34.0343	1.25197	0.625986	+	0.779834i	$0.284697\pi$
$740$	0	0
$741$	0	0
$742$	−46.7150	−1.71496
$743$	−16.2854	−0.597452	−0.298726	−	0.954339i	$-0.596562\pi$
$743$	−16.2854	−0.597452	−0.298726	+	0.954339i	$0.596562\pi$
$744$	0	0
$745$	0	0
$746$	35.2232	1.28961
$747$	0	0
$748$	5.38838	0.197019
$749$	36.0436	1.31701
$750$	0	0
$751$	14.0577	0.512974	0.256487	−	0.966548i	$-0.417435\pi$
$751$	14.0577	0.512974	0.256487	+	0.966548i	$0.417435\pi$
$752$	−63.6697	−2.32179
$753$	0	0
$754$	88.6502	3.22845
$755$	0	0
$756$	0	0
$757$	−38.2286	−1.38944	−0.694722	−	0.719278i	$-0.744473\pi$
$757$	−38.2286	−1.38944	−0.694722	+	0.719278i	$0.744473\pi$
$758$	−20.5042	−0.744746
$759$	0	0
$760$	0	0
$761$	−9.52616	−0.345323	−0.172662	−	0.984981i	$-0.555237\pi$
$761$	−9.52616	−0.345323	−0.172662	+	0.984981i	$0.555237\pi$
$762$	0	0
$763$	−46.1709	−1.67150
$764$	−4.10624	−0.148558
$765$	0	0
$766$	−23.5823	−0.852063
$767$	−57.8665	−2.08944
$768$	0	0
$769$	2.24276	0.0808760	0.0404380	−	0.999182i	$-0.487125\pi$
$769$	2.24276	0.0808760	0.0404380	+	0.999182i	$0.487125\pi$
$770$	0	0
$771$	0	0
$772$	−27.0167	−0.972354
$773$	17.7025	0.636715	0.318357	−	0.947971i	$-0.396869\pi$
$773$	17.7025	0.636715	0.318357	+	0.947971i	$0.396869\pi$
$774$	0	0
$775$	0	0
$776$	−9.02182	−0.323864
$777$	0	0
$778$	60.6712	2.17517
$779$	14.3917	0.515637
$780$	0	0
$781$	5.13974	0.183914
$782$	6.00301	0.214667
$783$	0	0
$784$	−82.2568	−2.93774
$785$	0	0
$786$	0	0
$787$	17.4445	0.621830	0.310915	−	0.950438i	$-0.399365\pi$
$787$	17.4445	0.621830	0.310915	+	0.950438i	$0.399365\pi$
$788$	−20.0894	−0.715655
$789$	0	0
$790$	0	0
$791$	53.7602	1.91149
$792$	0	0
$793$	−51.7572	−1.83795
$794$	−27.9328	−0.991297
$795$	0	0
$796$	4.87424	0.172763
$797$	−11.2093	−0.397052	−0.198526	−	0.980096i	$-0.563615\pi$
$797$	−11.2093	−0.397052	−0.198526	+	0.980096i	$0.563615\pi$
$798$	0	0
$799$	−49.8219	−1.76257
$800$	0	0
$801$	0	0
$802$	−34.6816	−1.22465
$803$	−4.61162	−0.162740
$804$	0	0
$805$	0	0
$806$	−66.7478	−2.35109
$807$	0	0
$808$	−4.81740	−0.169476
$809$	43.1381	1.51666	0.758328	−	0.651874i	$-0.226017\pi$
$809$	43.1381	1.51666	0.758328	+	0.651874i	$0.226017\pi$
$810$	0	0
$811$	11.0289	0.387276	0.193638	−	0.981073i	$-0.437971\pi$
$811$	11.0289	0.387276	0.193638	+	0.981073i	$0.437971\pi$
$812$	72.8462	2.55640
$813$	0	0
$814$	8.65885	0.303492
$815$	0	0
$816$	0	0
$817$	4.39577	0.153789
$818$	−45.3371	−1.58517
$819$	0	0
$820$	0	0
$821$	14.9164	0.520585	0.260293	−	0.965530i	$-0.416181\pi$
$821$	14.9164	0.520585	0.260293	+	0.965530i	$0.416181\pi$
$822$	0	0
$823$	41.4589	1.44517	0.722584	−	0.691283i	$-0.242954\pi$
$823$	41.4589	1.44517	0.722584	+	0.691283i	$0.242954\pi$
$824$	4.99536	0.174022
$825$	0	0
$826$	−114.048	−3.96825
$827$	−19.4171	−0.675200	−0.337600	−	0.941290i	$-0.609615\pi$
$827$	−19.4171	−0.675200	−0.337600	+	0.941290i	$0.609615\pi$
$828$	0	0
$829$	−17.8290	−0.619225	−0.309613	−	0.950863i	$-0.600199\pi$
$829$	−17.8290	−0.619225	−0.309613	+	0.950863i	$0.600199\pi$
$830$	0	0
$831$	0	0
$832$	−13.7270	−0.475898
$833$	−64.3664	−2.23016
$834$	0	0
$835$	0	0
$836$	−6.92735	−0.239588
$837$	0	0
$838$	−16.9070	−0.584044
$839$	12.2935	0.424417	0.212209	−	0.977224i	$-0.431934\pi$
$839$	12.2935	0.424417	0.212209	+	0.977224i	$0.431934\pi$
$840$	0	0
$841$	78.7314	2.71487
$842$	25.0889	0.864621
$843$	0	0
$844$	−35.0008	−1.20478
$845$	0	0
$846$	0	0
$847$	−4.90749	−0.168623
$848$	24.7478	0.849842
$849$	0	0
$850$	0	0
$851$	4.02194	0.137870
$852$	0	0
$853$	18.0776	0.618966	0.309483	−	0.950905i	$-0.399844\pi$
$853$	18.0776	0.618966	0.309483	+	0.950905i	$0.399844\pi$
$854$	−102.008	−3.49063
$855$	0	0
$856$	−7.75174	−0.264949
$857$	55.3649	1.89123	0.945615	−	0.325288i	$-0.105461\pi$
$857$	55.3649	1.89123	0.945615	+	0.325288i	$0.105461\pi$
$858$	0	0
$859$	−0.943756	−0.0322005	−0.0161003	−	0.999870i	$-0.505125\pi$
$859$	−0.943756	−0.0322005	−0.0161003	+	0.999870i	$0.505125\pi$
$860$	0	0
$861$	0	0
$862$	48.9805	1.66828
$863$	23.6370	0.804614	0.402307	−	0.915505i	$-0.368208\pi$
$863$	23.6370	0.804614	0.402307	+	0.915505i	$0.368208\pi$
$864$	0	0
$865$	0	0
$866$	−29.9626	−1.01817
$867$	0	0
$868$	−54.8484	−1.86168
$869$	0.843861	0.0286260
$870$	0	0
$871$	−19.6847	−0.666991
$872$	9.92977	0.336264
$873$	0	0
$874$	−7.71752	−0.261049
$875$	0	0
$876$	0	0
$877$	16.6841	0.563383	0.281692	−	0.959505i	$-0.409104\pi$
$877$	16.6841	0.563383	0.281692	+	0.959505i	$0.409104\pi$
$878$	13.0179	0.439333
$879$	0	0
$880$	0	0
$881$	−16.2217	−0.546524	−0.273262	−	0.961940i	$-0.588103\pi$
$881$	−16.2217	−0.546524	−0.273262	+	0.961940i	$0.588103\pi$
$882$	0	0
$883$	−35.0710	−1.18023	−0.590117	−	0.807318i	$-0.700918\pi$
$883$	−35.0710	−1.18023	−0.590117	+	0.807318i	$0.700918\pi$
$884$	−24.8492	−0.835768
$885$	0	0
$886$	−45.1217	−1.51589
$887$	−16.1581	−0.542536	−0.271268	−	0.962504i	$-0.587443\pi$
$887$	−16.1581	−0.542536	−0.271268	+	0.962504i	$0.587443\pi$
$888$	0	0
$889$	32.5270	1.09092
$890$	0	0
$891$	0	0
$892$	16.4973	0.552370
$893$	64.0515	2.14340
$894$	0	0
$895$	0	0
$896$	39.7542	1.32809
$897$	0	0
$898$	71.0983	2.37258
$899$	−81.1147	−2.70533
$900$	0	0
$901$	19.3653	0.645151
$902$	5.50271	0.183220
$903$	0	0
$904$	−11.5620	−0.384545
$905$	0	0
$906$	0	0
$907$	−41.9820	−1.39399	−0.696994	−	0.717077i	$-0.745480\pi$
$907$	−41.9820	−1.39399	−0.696994	+	0.717077i	$0.745480\pi$
$908$	1.65838	0.0550352
$909$	0	0
$910$	0	0
$911$	−57.2302	−1.89612	−0.948060	−	0.318092i	$-0.896958\pi$
$911$	−57.2302	−1.89612	−0.948060	+	0.318092i	$0.896958\pi$
$912$	0	0
$913$	−5.75135	−0.190342
$914$	40.6752	1.34542
$915$	0	0
$916$	−34.9616	−1.15517
$917$	−64.8929	−2.14295
$918$	0	0
$919$	−0.346569	−0.0114323	−0.00571613	−	0.999984i	$-0.501820\pi$
$919$	−0.346569	−0.0114323	−0.00571613	+	0.999984i	$0.501820\pi$
$920$	0	0
$921$	0	0
$922$	−51.0610	−1.68160
$923$	−23.7025	−0.780177
$924$	0	0
$925$	0	0
$926$	−46.0904	−1.51462
$927$	0	0
$928$	−70.6502	−2.31921
$929$	−7.66278	−0.251408	−0.125704	−	0.992068i	$-0.540119\pi$
$929$	−7.66278	−0.251408	−0.125704	+	0.992068i	$0.540119\pi$
$930$	0	0
$931$	82.7501	2.71202
$932$	−13.5226	−0.442947
$933$	0	0
$934$	−53.7602	−1.75909
$935$	0	0
$936$	0	0
$937$	−34.0894	−1.11365	−0.556826	−	0.830629i	$-0.687981\pi$
$937$	−34.0894	−1.11365	−0.556826	+	0.830629i	$0.687981\pi$
$938$	−38.7964	−1.26675
$939$	0	0
$940$	0	0
$941$	−28.1944	−0.919110	−0.459555	−	0.888149i	$-0.651991\pi$
$941$	−28.1944	−0.919110	−0.459555	+	0.888149i	$0.651991\pi$
$942$	0	0
$943$	2.55595	0.0832331
$944$	60.4184	1.96645
$945$	0	0
$946$	1.68073	0.0546454
$947$	−29.7602	−0.967078	−0.483539	−	0.875323i	$-0.660649\pi$
$947$	−29.7602	−0.967078	−0.483539	+	0.875323i	$0.660649\pi$
$948$	0	0
$949$	21.2670	0.690356
$950$	0	0
$951$	0	0
$952$	19.5152	0.632491
$953$	20.8309	0.674780	0.337390	−	0.941365i	$-0.390456\pi$
$953$	20.8309	0.674780	0.337390	+	0.941365i	$0.390456\pi$
$954$	0	0
$955$	0	0
$956$	−21.3715	−0.691205
$957$	0	0
$958$	−54.5339	−1.76191
$959$	−29.4450	−0.950827
$960$	0	0
$961$	30.0740	0.970130
$962$	−39.9313	−1.28744
$963$	0	0
$964$	11.1765	0.359969
$965$	0	0
$966$	0	0
$967$	30.7225	0.987968	0.493984	−	0.869471i	$-0.335540\pi$
$967$	30.7225	0.987968	0.493984	+	0.869471i	$0.335540\pi$
$968$	1.05543	0.0339229
$969$	0	0
$970$	0	0
$971$	21.9095	0.703108	0.351554	−	0.936168i	$-0.385653\pi$
$971$	21.9095	0.703108	0.351554	+	0.936168i	$0.385653\pi$
$972$	0	0
$973$	63.6557	2.04071
$974$	50.9844	1.63365
$975$	0	0
$976$	54.0397	1.72977
$977$	14.1889	0.453944	0.226972	−	0.973901i	$-0.427117\pi$
$977$	14.1889	0.453944	0.226972	+	0.973901i	$0.427117\pi$
$978$	0	0
$979$	−1.40825	−0.0450077
$980$	0	0
$981$	0	0
$982$	−41.2630	−1.31675
$983$	5.52454	0.176206	0.0881028	−	0.996111i	$-0.471920\pi$
$983$	5.52454	0.176206	0.0881028	+	0.996111i	$0.471920\pi$
$984$	0	0
$985$	0	0
$986$	−72.4284	−2.30659
$987$	0	0
$988$	31.9463	1.01635
$989$	0.780682	0.0248243
$990$	0	0
$991$	7.93048	0.251920	0.125960	−	0.992035i	$-0.459799\pi$
$991$	7.93048	0.251920	0.125960	+	0.992035i	$0.459799\pi$
$992$	53.1950	1.68894
$993$	0	0
$994$	−46.7150	−1.48171
$995$	0	0
$996$	0	0
$997$	52.0596	1.64874	0.824372	−	0.566049i	$-0.191529\pi$
$997$	52.0596	1.64874	0.824372	+	0.566049i	$0.191529\pi$
$998$	64.8929	2.05415
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.2.a.bf.1.2		4
3.2	odd	2		2475.2.a.bj.1.3		4
5.2	odd	4		2475.2.c.s.199.3		8
5.3	odd	4		2475.2.c.s.199.6		8
5.4	even	2		495.2.a.g.1.3	yes	4
15.2	even	4		2475.2.c.t.199.6		8
15.8	even	4		2475.2.c.t.199.3		8
15.14	odd	2		495.2.a.f.1.2	✓	4
20.19	odd	2		7920.2.a.cn.1.1		4
55.54	odd	2		5445.2.a.bh.1.2		4
60.59	even	2		7920.2.a.cm.1.1		4
165.164	even	2		5445.2.a.bs.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
495.2.a.f.1.2	✓	4	15.14	odd	2
495.2.a.g.1.3	yes	4	5.4	even	2
2475.2.a.bf.1.2		4	1.1	even	1	trivial
2475.2.a.bj.1.3		4	3.2	odd	2
2475.2.c.s.199.3		8	5.2	odd	4
2475.2.c.s.199.6		8	5.3	odd	4
2475.2.c.t.199.3		8	15.8	even	4
2475.2.c.t.199.6		8	15.2	even	4
5445.2.a.bh.1.2		4	55.54	odd	2
5445.2.a.bs.1.3		4	165.164	even	2
7920.2.a.cm.1.1		4	60.59	even	2
7920.2.a.cn.1.1		4	20.19	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:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2475.a (trivial)

\(f(q)\)	\(=\)	\(q-1.85206 q^{2} +1.43013 q^{4} -4.90749 q^{7} +1.05543 q^{8} +O(q^{10})\)	\(q-1.85206 q^{2} +1.43013 q^{4} -4.90749 q^{7} +1.05543 q^{8} -1.00000 q^{11} +4.61162 q^{13} +9.08898 q^{14} -4.81499 q^{16} -3.76776 q^{17} +4.84386 q^{19} +1.85206 q^{22} +0.860262 q^{23} -8.54100 q^{26} -7.01836 q^{28} -10.3794 q^{29} +7.81499 q^{31} +6.80679 q^{32} +6.97811 q^{34} +4.67525 q^{37} -8.97113 q^{38} +2.97113 q^{41} +0.907494 q^{43} -1.43013 q^{44} -1.59326 q^{46} +13.2232 q^{47} +17.0835 q^{49} +6.59522 q^{52} -5.13974 q^{53} -5.17953 q^{56} +19.2232 q^{58} -12.5480 q^{59} -11.2232 q^{61} -14.4738 q^{62} -2.97661 q^{64} -4.26851 q^{67} -5.38838 q^{68} -5.13974 q^{71} +4.61162 q^{73} -8.65885 q^{74} +6.92735 q^{76} +4.90749 q^{77} -0.843861 q^{79} -5.50271 q^{82} +5.75135 q^{83} -1.68073 q^{86} -1.05543 q^{88} +1.40825 q^{89} -22.6315 q^{91} +1.23029 q^{92} -24.4902 q^{94} -8.54798 q^{97} -31.6397 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} + 8 q^{4} - 4 q^{7} - 6 q^{8}+O(q^{10}) \) 4 * q - 2 * q^2 + 8 * q^4 - 4 * q^7 - 6 * q^8	\( 4 q - 2 q^{2} + 8 q^{4} - 4 q^{7} - 6 q^{8} - 4 q^{11} - 8 q^{13} - 8 q^{14} + 12 q^{16} - 4 q^{17} + 4 q^{19} + 2 q^{22} + 8 q^{23} - 16 q^{26} + 8 q^{28} - 4 q^{29} - 14 q^{32} + 4 q^{34} - 8 q^{37} - 20 q^{38} - 4 q^{41} - 12 q^{43} - 8 q^{44} - 16 q^{46} + 20 q^{49} - 20 q^{52} - 16 q^{53} - 48 q^{56} + 24 q^{58} - 24 q^{59} + 8 q^{61} + 20 q^{62} - 48 q^{68} - 16 q^{71} - 8 q^{73} + 12 q^{74} - 36 q^{76} + 4 q^{77} + 12 q^{79} + 40 q^{82} - 8 q^{83} + 16 q^{86} + 6 q^{88} - 16 q^{89} - 16 q^{91} + 72 q^{92} - 40 q^{94} - 8 q^{97} - 62 q^{98}+O(q^{100}) \) 4 * q - 2 * q^2 + 8 * q^4 - 4 * q^7 - 6 * q^8 - 4 * q^11 - 8 * q^13 - 8 * q^14 + 12 * q^16 - 4 * q^17 + 4 * q^19 + 2 * q^22 + 8 * q^23 - 16 * q^26 + 8 * q^28 - 4 * q^29 - 14 * q^32 + 4 * q^34 - 8 * q^37 - 20 * q^38 - 4 * q^41 - 12 * q^43 - 8 * q^44 - 16 * q^46 + 20 * q^49 - 20 * q^52 - 16 * q^53 - 48 * q^56 + 24 * q^58 - 24 * q^59 + 8 * q^61 + 20 * q^62 - 48 * q^68 - 16 * q^71 - 8 * q^73 + 12 * q^74 - 36 * q^76 + 4 * q^77 + 12 * q^79 + 40 * q^82 - 8 * q^83 + 16 * q^86 + 6 * q^88 - 16 * q^89 - 16 * q^91 + 72 * q^92 - 40 * q^94 - 8 * q^97 - 62 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more