LMFDB - Embedded newform 2475.2.a.y.1.1

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:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.48119$ 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)

$f(q)$	$=$	$q-2.67513 q^{2} +5.15633 q^{4} +2.80606 q^{7} -8.44358 q^{8} +O(q^{10})$	$q-2.67513 q^{2} +5.15633 q^{4} +2.80606 q^{7} -8.44358 q^{8} +1.00000 q^{11} -5.11871 q^{13} -7.50659 q^{14} +12.2750 q^{16} -4.54420 q^{17} -4.57452 q^{19} -2.67513 q^{22} -4.00000 q^{23} +13.6932 q^{26} +14.4690 q^{28} +2.38787 q^{29} -0.962389 q^{31} -15.9502 q^{32} +12.1563 q^{34} +1.61213 q^{37} +12.2374 q^{38} +2.38787 q^{41} +2.80606 q^{43} +5.15633 q^{44} +10.7005 q^{46} +4.31265 q^{47} +0.873992 q^{49} -26.3938 q^{52} +6.57452 q^{53} -23.6932 q^{56} -6.38787 q^{58} +13.2750 q^{59} +7.92478 q^{61} +2.57452 q^{62} +18.1187 q^{64} +10.7005 q^{67} -23.4314 q^{68} +7.35026 q^{71} +6.41819 q^{73} -4.31265 q^{74} -23.5877 q^{76} +2.80606 q^{77} +1.35026 q^{79} -6.38787 q^{82} +0.806063 q^{83} -7.50659 q^{86} -8.44358 q^{88} +2.96239 q^{89} -14.3634 q^{91} -20.6253 q^{92} -11.5369 q^{94} -9.92478 q^{97} -2.33804 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 5 q^{4} + 8 q^{7} - 9 q^{8}+O(q^{10}) $ 3 * q - 3 * q^2 + 5 * q^4 + 8 * q^7 - 9 * q^8	$ 3 q - 3 q^{2} + 5 q^{4} + 8 q^{7} - 9 q^{8} + 3 q^{11} + 6 q^{13} - 2 q^{14} + 5 q^{16} - 4 q^{17} - 2 q^{19} - 3 q^{22} - 12 q^{23} + 8 q^{26} + 12 q^{28} + 8 q^{29} + 8 q^{31} - 11 q^{32} + 26 q^{34} + 4 q^{37} - 6 q^{38} + 8 q^{41} + 8 q^{43} + 5 q^{44} + 12 q^{46} - 8 q^{47} + 11 q^{49} - 26 q^{52} + 8 q^{53} - 38 q^{56} - 20 q^{58} + 8 q^{59} + 2 q^{61} - 4 q^{62} + 33 q^{64} + 12 q^{67} - 28 q^{68} + 12 q^{71} + 18 q^{73} + 8 q^{74} - 18 q^{76} + 8 q^{77} - 6 q^{79} - 20 q^{82} + 2 q^{83} - 2 q^{86} - 9 q^{88} - 2 q^{89} + 8 q^{91} - 20 q^{92} - 12 q^{94} - 8 q^{97} + 29 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 + 5 * q^4 + 8 * q^7 - 9 * q^8 + 3 * q^11 + 6 * q^13 - 2 * q^14 + 5 * q^16 - 4 * q^17 - 2 * q^19 - 3 * q^22 - 12 * q^23 + 8 * q^26 + 12 * q^28 + 8 * q^29 + 8 * q^31 - 11 * q^32 + 26 * q^34 + 4 * q^37 - 6 * q^38 + 8 * q^41 + 8 * q^43 + 5 * q^44 + 12 * q^46 - 8 * q^47 + 11 * q^49 - 26 * q^52 + 8 * q^53 - 38 * q^56 - 20 * q^58 + 8 * q^59 + 2 * q^61 - 4 * q^62 + 33 * q^64 + 12 * q^67 - 28 * q^68 + 12 * q^71 + 18 * q^73 + 8 * q^74 - 18 * q^76 + 8 * q^77 - 6 * q^79 - 20 * q^82 + 2 * q^83 - 2 * q^86 - 9 * q^88 - 2 * q^89 + 8 * q^91 - 20 * q^92 - 12 * q^94 - 8 * q^97 + 29 * 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 $

$2$	−2.67513	−1.89160	−0.945802	−	0.324745i	$-0.894721\pi$
$2$	−2.67513	−1.89160	−0.945802	+	0.324745i	$0.894721\pi$
$3$	0	0
$3$	0	0
$4$	5.15633	2.57816
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.80606	1.06059	0.530296	−	0.847812i	$-0.322081\pi$
$7$	2.80606	1.06059	0.530296	+	0.847812i	$0.322081\pi$
$8$	−8.44358	−2.98526
$9$	0	0
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−5.11871	−1.41968	−0.709838	−	0.704365i	$-0.751232\pi$
$13$	−5.11871	−1.41968	−0.709838	+	0.704365i	$0.751232\pi$
$14$	−7.50659	−2.00622
$15$	0	0
$16$	12.2750	3.06876
$17$	−4.54420	−1.10213	−0.551065	−	0.834462i	$-0.685778\pi$
$17$	−4.54420	−1.10213	−0.551065	+	0.834462i	$0.685778\pi$
$18$	0	0
$19$	−4.57452	−1.04947	−0.524733	−	0.851267i	$-0.675835\pi$
$19$	−4.57452	−1.04947	−0.524733	+	0.851267i	$0.675835\pi$
$20$	0	0
$21$	0	0
$22$	−2.67513	−0.570340
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	0	0
$26$	13.6932	2.68546
$27$	0	0
$28$	14.4690	2.73438
$29$	2.38787	0.443417	0.221708	−	0.975113i	$-0.428837\pi$
$29$	2.38787	0.443417	0.221708	+	0.975113i	$0.428837\pi$
$30$	0	0
$31$	−0.962389	−0.172850	−0.0864250	−	0.996258i	$-0.527544\pi$
$31$	−0.962389	−0.172850	−0.0864250	+	0.996258i	$0.527544\pi$
$32$	−15.9502	−2.81962
$33$	0	0
$34$	12.1563	2.08479
$35$	0	0
$36$	0	0
$37$	1.61213	0.265032	0.132516	−	0.991181i	$-0.457694\pi$
$37$	1.61213	0.265032	0.132516	+	0.991181i	$0.457694\pi$
$38$	12.2374	1.98517
$39$	0	0
$40$	0	0
$41$	2.38787	0.372923	0.186462	−	0.982462i	$-0.440298\pi$
$41$	2.38787	0.372923	0.186462	+	0.982462i	$0.440298\pi$
$42$	0	0
$43$	2.80606	0.427921	0.213960	−	0.976842i	$-0.431364\pi$
$43$	2.80606	0.427921	0.213960	+	0.976842i	$0.431364\pi$
$44$	5.15633	0.777345
$45$	0	0
$46$	10.7005	1.57771
$47$	4.31265	0.629065	0.314532	−	0.949247i	$-0.398152\pi$
$47$	4.31265	0.629065	0.314532	+	0.949247i	$0.398152\pi$
$48$	0	0
$49$	0.873992	0.124856
$50$	0	0
$51$	0	0
$52$	−26.3938	−3.66015
$53$	6.57452	0.903079	0.451540	−	0.892251i	$-0.350875\pi$
$53$	6.57452	0.903079	0.451540	+	0.892251i	$0.350875\pi$
$54$	0	0
$55$	0	0
$56$	−23.6932	−3.16614
$57$	0	0
$58$	−6.38787	−0.838769
$59$	13.2750	1.72826	0.864131	−	0.503266i	$-0.167868\pi$
$59$	13.2750	1.72826	0.864131	+	0.503266i	$0.167868\pi$
$60$	0	0
$61$	7.92478	1.01466	0.507332	−	0.861751i	$-0.330632\pi$
$61$	7.92478	1.01466	0.507332	+	0.861751i	$0.330632\pi$
$62$	2.57452	0.326964
$63$	0	0
$64$	18.1187	2.26484
$65$	0	0
$66$	0	0
$67$	10.7005	1.30728	0.653639	−	0.756807i	$-0.273242\pi$
$67$	10.7005	1.30728	0.653639	+	0.756807i	$0.273242\pi$
$68$	−23.4314	−2.84147
$69$	0	0
$70$	0	0
$71$	7.35026	0.872316	0.436158	−	0.899870i	$-0.356339\pi$
$71$	7.35026	0.872316	0.436158	+	0.899870i	$0.356339\pi$
$72$	0	0
$73$	6.41819	0.751192	0.375596	−	0.926783i	$-0.377438\pi$
$73$	6.41819	0.751192	0.375596	+	0.926783i	$0.377438\pi$
$74$	−4.31265	−0.501335
$75$	0	0
$76$	−23.5877	−2.70569
$77$	2.80606	0.319781
$78$	0	0
$79$	1.35026	0.151916	0.0759582	−	0.997111i	$-0.475798\pi$
$79$	1.35026	0.151916	0.0759582	+	0.997111i	$0.475798\pi$
$80$	0	0
$81$	0	0
$82$	−6.38787	−0.705423
$83$	0.806063	0.0884770	0.0442385	−	0.999021i	$-0.485914\pi$
$83$	0.806063	0.0884770	0.0442385	+	0.999021i	$0.485914\pi$
$84$	0	0
$85$	0	0
$86$	−7.50659	−0.809456
$87$	0	0
$88$	−8.44358	−0.900089
$89$	2.96239	0.314013	0.157006	−	0.987598i	$-0.449816\pi$
$89$	2.96239	0.314013	0.157006	+	0.987598i	$0.449816\pi$
$90$	0	0
$91$	−14.3634	−1.50570
$92$	−20.6253	−2.15034
$93$	0	0
$94$	−11.5369	−1.18994
$95$	0	0
$96$	0	0
$97$	−9.92478	−1.00771	−0.503854	−	0.863789i	$-0.668085\pi$
$97$	−9.92478	−1.00771	−0.503854	+	0.863789i	$0.668085\pi$
$98$	−2.33804	−0.236178
$99$	0	0
$100$	0	0
$101$	13.6121	1.35446	0.677229	−	0.735773i	$-0.263181\pi$
$101$	13.6121	1.35446	0.677229	+	0.735773i	$0.263181\pi$
$102$	0	0
$103$	16.3127	1.60733	0.803667	−	0.595080i	$-0.202880\pi$
$103$	16.3127	1.60733	0.803667	+	0.595080i	$0.202880\pi$
$104$	43.2203	4.23810
$105$	0	0
$106$	−17.5877	−1.70827
$107$	9.43136	0.911764	0.455882	−	0.890040i	$-0.349324\pi$
$107$	9.43136	0.911764	0.455882	+	0.890040i	$0.349324\pi$
$108$	0	0
$109$	−15.4010	−1.47515	−0.737576	−	0.675264i	$-0.764030\pi$
$109$	−15.4010	−1.47515	−0.737576	+	0.675264i	$0.764030\pi$
$110$	0	0
$111$	0	0
$112$	34.4445	3.25470
$113$	13.7381	1.29238	0.646188	−	0.763179i	$-0.276362\pi$
$113$	13.7381	1.29238	0.646188	+	0.763179i	$0.276362\pi$
$114$	0	0
$115$	0	0
$116$	12.3127	1.14320
$117$	0	0
$118$	−35.5125	−3.26919
$119$	−12.7513	−1.16891
$120$	0	0
$121$	1.00000	0.0909091
$122$	−21.1998	−1.91934
$123$	0	0
$124$	−4.96239	−0.445636
$125$	0	0
$126$	0	0
$127$	12.4182	1.10194	0.550968	−	0.834526i	$-0.314259\pi$
$127$	12.4182	1.10194	0.550968	+	0.834526i	$0.314259\pi$
$128$	−16.5696	−1.46456
$129$	0	0
$130$	0	0
$131$	−5.92478	−0.517650	−0.258825	−	0.965924i	$-0.583335\pi$
$131$	−5.92478	−0.517650	−0.258825	+	0.965924i	$0.583335\pi$
$132$	0	0
$133$	−12.8364	−1.11306
$134$	−28.6253	−2.47285
$135$	0	0
$136$	38.3693	3.29014
$137$	−9.79877	−0.837165	−0.418583	−	0.908179i	$-0.637473\pi$
$137$	−9.79877	−0.837165	−0.418583	+	0.908179i	$0.637473\pi$
$138$	0	0
$139$	−2.12601	−0.180326	−0.0901628	−	0.995927i	$-0.528739\pi$
$139$	−2.12601	−0.180326	−0.0901628	+	0.995927i	$0.528739\pi$
$140$	0	0
$141$	0	0
$142$	−19.6629	−1.65007
$143$	−5.11871	−0.428048
$144$	0	0
$145$	0	0
$146$	−17.1695	−1.42096
$147$	0	0
$148$	8.31265	0.683296
$149$	−8.31265	−0.680999	−0.340499	−	0.940245i	$-0.610596\pi$
$149$	−8.31265	−0.680999	−0.340499	+	0.940245i	$0.610596\pi$
$150$	0	0
$151$	−15.2750	−1.24307	−0.621533	−	0.783388i	$-0.713490\pi$
$151$	−15.2750	−1.24307	−0.621533	+	0.783388i	$0.713490\pi$
$152$	38.6253	3.13293
$153$	0	0
$154$	−7.50659	−0.604898
$155$	0	0
$156$	0	0
$157$	7.01317	0.559712	0.279856	−	0.960042i	$-0.409713\pi$
$157$	7.01317	0.559712	0.279856	+	0.960042i	$0.409713\pi$
$158$	−3.61213	−0.287365
$159$	0	0
$160$	0	0
$161$	−11.2243	−0.884595
$162$	0	0
$163$	6.76116	0.529575	0.264787	−	0.964307i	$-0.414698\pi$
$163$	6.76116	0.529575	0.264787	+	0.964307i	$0.414698\pi$
$164$	12.3127	0.961456
$165$	0	0
$166$	−2.15633	−0.167363
$167$	11.8192	0.914600	0.457300	−	0.889312i	$-0.348817\pi$
$167$	11.8192	0.914600	0.457300	+	0.889312i	$0.348817\pi$
$168$	0	0
$169$	13.2012	1.01548
$170$	0	0
$171$	0	0
$172$	14.4690	1.10325
$173$	−6.99271	−0.531646	−0.265823	−	0.964022i	$-0.585644\pi$
$173$	−6.99271	−0.531646	−0.265823	+	0.964022i	$0.585644\pi$
$174$	0	0
$175$	0	0
$176$	12.2750	0.925266
$177$	0	0
$178$	−7.92478	−0.593987
$179$	2.70052	0.201847	0.100923	−	0.994894i	$-0.467820\pi$
$179$	2.70052	0.201847	0.100923	+	0.994894i	$0.467820\pi$
$180$	0	0
$181$	−21.1998	−1.57577	−0.787885	−	0.615822i	$-0.788824\pi$
$181$	−21.1998	−1.57577	−0.787885	+	0.615822i	$0.788824\pi$
$182$	38.4241	2.84818
$183$	0	0
$184$	33.7743	2.48988
$185$	0	0
$186$	0	0
$187$	−4.54420	−0.332305
$188$	22.2374	1.62183
$189$	0	0
$190$	0	0
$191$	−13.1490	−0.951430	−0.475715	−	0.879599i	$-0.657811\pi$
$191$	−13.1490	−0.951430	−0.475715	+	0.879599i	$0.657811\pi$
$192$	0	0
$193$	5.89446	0.424293	0.212146	−	0.977238i	$-0.431955\pi$
$193$	5.89446	0.424293	0.212146	+	0.977238i	$0.431955\pi$
$194$	26.5501	1.90618
$195$	0	0
$196$	4.50659	0.321899
$197$	−20.3938	−1.45299	−0.726497	−	0.687169i	$-0.758853\pi$
$197$	−20.3938	−1.45299	−0.726497	+	0.687169i	$0.758853\pi$
$198$	0	0
$199$	17.4010	1.23353	0.616764	−	0.787148i	$-0.288443\pi$
$199$	17.4010	1.23353	0.616764	+	0.787148i	$0.288443\pi$
$200$	0	0
$201$	0	0
$202$	−36.4142	−2.56210
$203$	6.70052	0.470285
$204$	0	0
$205$	0	0
$206$	−43.6385	−3.04044
$207$	0	0
$208$	−62.8324	−4.35664
$209$	−4.57452	−0.316426
$210$	0	0
$211$	18.1260	1.24785	0.623923	−	0.781486i	$-0.285538\pi$
$211$	18.1260	1.24785	0.623923	+	0.781486i	$0.285538\pi$
$212$	33.9003	2.32828
$213$	0	0
$214$	−25.2301	−1.72470
$215$	0	0
$216$	0	0
$217$	−2.70052	−0.183323
$218$	41.1998	2.79040
$219$	0	0
$220$	0	0
$221$	23.2605	1.56467
$222$	0	0
$223$	23.6385	1.58295	0.791475	−	0.611202i	$-0.209314\pi$
$223$	23.6385	1.58295	0.791475	+	0.611202i	$0.209314\pi$
$224$	−44.7572	−2.99047
$225$	0	0
$226$	−36.7513	−2.44466
$227$	1.26916	0.0842371	0.0421185	−	0.999113i	$-0.486589\pi$
$227$	1.26916	0.0842371	0.0421185	+	0.999113i	$0.486589\pi$
$228$	0	0
$229$	16.1768	1.06899	0.534496	−	0.845171i	$-0.320501\pi$
$229$	16.1768	1.06899	0.534496	+	0.845171i	$0.320501\pi$
$230$	0	0
$231$	0	0
$232$	−20.1622	−1.32371
$233$	18.4690	1.20994	0.604971	−	0.796247i	$-0.293185\pi$
$233$	18.4690	1.20994	0.604971	+	0.796247i	$0.293185\pi$
$234$	0	0
$235$	0	0
$236$	68.4504	4.45574
$237$	0	0
$238$	34.1114	2.21111
$239$	−19.3258	−1.25008	−0.625042	−	0.780591i	$-0.714918\pi$
$239$	−19.3258	−1.25008	−0.625042	+	0.780591i	$0.714918\pi$
$240$	0	0
$241$	28.5501	1.83907	0.919536	−	0.393006i	$-0.128565\pi$
$241$	28.5501	1.83907	0.919536	+	0.393006i	$0.128565\pi$
$242$	−2.67513	−0.171964
$243$	0	0
$244$	40.8627	2.61597
$245$	0	0
$246$	0	0
$247$	23.4156	1.48990
$248$	8.12601	0.516002
$249$	0	0
$250$	0	0
$251$	−23.1998	−1.46436	−0.732180	−	0.681112i	$-0.761497\pi$
$251$	−23.1998	−1.46436	−0.732180	+	0.681112i	$0.761497\pi$
$252$	0	0
$253$	−4.00000	−0.251478
$254$	−33.2203	−2.08443
$255$	0	0
$256$	8.08840	0.505525
$257$	10.8872	0.679123	0.339561	−	0.940584i	$-0.389721\pi$
$257$	10.8872	0.679123	0.339561	+	0.940584i	$0.389721\pi$
$258$	0	0
$259$	4.52373	0.281091
$260$	0	0
$261$	0	0
$262$	15.8496	0.979189
$263$	−9.11871	−0.562284	−0.281142	−	0.959666i	$-0.590713\pi$
$263$	−9.11871	−0.562284	−0.281142	+	0.959666i	$0.590713\pi$
$264$	0	0
$265$	0	0
$266$	34.3390	2.10546
$267$	0	0
$268$	55.1754	3.37037
$269$	11.2995	0.688941	0.344471	−	0.938797i	$-0.388058\pi$
$269$	11.2995	0.688941	0.344471	+	0.938797i	$0.388058\pi$
$270$	0	0
$271$	25.1998	1.53078	0.765390	−	0.643567i	$-0.222546\pi$
$271$	25.1998	1.53078	0.765390	+	0.643567i	$0.222546\pi$
$272$	−55.7802	−3.38217
$273$	0	0
$274$	26.2130	1.58358
$275$	0	0
$276$	0	0
$277$	−2.41819	−0.145295	−0.0726475	−	0.997358i	$-0.523145\pi$
$277$	−2.41819	−0.145295	−0.0726475	+	0.997358i	$0.523145\pi$
$278$	5.68735	0.341105
$279$	0	0
$280$	0	0
$281$	−30.4894	−1.81885	−0.909424	−	0.415870i	$-0.863477\pi$
$281$	−30.4894	−1.81885	−0.909424	+	0.415870i	$0.863477\pi$
$282$	0	0
$283$	8.35756	0.496805	0.248403	−	0.968657i	$-0.420094\pi$
$283$	8.35756	0.496805	0.248403	+	0.968657i	$0.420094\pi$
$284$	37.9003	2.24897
$285$	0	0
$286$	13.6932	0.809698
$287$	6.70052	0.395519
$288$	0	0
$289$	3.64974	0.214690
$290$	0	0
$291$	0	0
$292$	33.0943	1.93670
$293$	23.0943	1.34918	0.674591	−	0.738192i	$-0.264320\pi$
$293$	23.0943	1.34918	0.674591	+	0.738192i	$0.264320\pi$
$294$	0	0
$295$	0	0
$296$	−13.6121	−0.791189
$297$	0	0
$298$	22.2374	1.28818
$299$	20.4749	1.18409
$300$	0	0
$301$	7.87399	0.453849
$302$	40.8627	2.35139
$303$	0	0
$304$	−56.1524	−3.22056
$305$	0	0
$306$	0	0
$307$	2.65562	0.151564	0.0757821	−	0.997124i	$-0.475855\pi$
$307$	2.65562	0.151564	0.0757821	+	0.997124i	$0.475855\pi$
$308$	14.4690	0.824446
$309$	0	0
$310$	0	0
$311$	15.9756	0.905891	0.452946	−	0.891538i	$-0.350373\pi$
$311$	15.9756	0.905891	0.452946	+	0.891538i	$0.350373\pi$
$312$	0	0
$313$	−0.0606343	−0.00342726	−0.00171363	−	0.999999i	$-0.500545\pi$
$313$	−0.0606343	−0.00342726	−0.00171363	+	0.999999i	$0.500545\pi$
$314$	−18.7612	−1.05875
$315$	0	0
$316$	6.96239	0.391665
$317$	−3.81336	−0.214180	−0.107090	−	0.994249i	$-0.534153\pi$
$317$	−3.81336	−0.214180	−0.107090	+	0.994249i	$0.534153\pi$
$318$	0	0
$319$	2.38787	0.133695
$320$	0	0
$321$	0	0
$322$	30.0263	1.67330
$323$	20.7875	1.15665
$324$	0	0
$325$	0	0
$326$	−18.0870	−1.00175
$327$	0	0
$328$	−20.1622	−1.11327
$329$	12.1016	0.667181
$330$	0	0
$331$	24.2882	1.33500	0.667500	−	0.744609i	$-0.267364\pi$
$331$	24.2882	1.33500	0.667500	+	0.744609i	$0.267364\pi$
$332$	4.15633	0.228108
$333$	0	0
$334$	−31.6180	−1.73006
$335$	0	0
$336$	0	0
$337$	−22.5804	−1.23003	−0.615016	−	0.788514i	$-0.710851\pi$
$337$	−22.5804	−1.23003	−0.615016	+	0.788514i	$0.710851\pi$
$338$	−35.3150	−1.92088
$339$	0	0
$340$	0	0
$341$	−0.962389	−0.0521163
$342$	0	0
$343$	−17.1900	−0.928171
$344$	−23.6932	−1.27745
$345$	0	0
$346$	18.7064	1.00566
$347$	−20.1925	−1.08399	−0.541996	−	0.840381i	$-0.682331\pi$
$347$	−20.1925	−1.08399	−0.541996	+	0.840381i	$0.682331\pi$
$348$	0	0
$349$	−27.2506	−1.45869	−0.729346	−	0.684145i	$-0.760175\pi$
$349$	−27.2506	−1.45869	−0.729346	+	0.684145i	$0.760175\pi$
$350$	0	0
$351$	0	0
$352$	−15.9502	−0.850147
$353$	−5.02302	−0.267349	−0.133674	−	0.991025i	$-0.542678\pi$
$353$	−5.02302	−0.267349	−0.133674	+	0.991025i	$0.542678\pi$
$354$	0	0
$355$	0	0
$356$	15.2750	0.809575
$357$	0	0
$358$	−7.22425	−0.381814
$359$	18.1768	0.959334	0.479667	−	0.877450i	$-0.340757\pi$
$359$	18.1768	0.959334	0.479667	+	0.877450i	$0.340757\pi$
$360$	0	0
$361$	1.92619	0.101379
$362$	56.7123	2.98073
$363$	0	0
$364$	−74.0625	−3.88193
$365$	0	0
$366$	0	0
$367$	−8.56467	−0.447072	−0.223536	−	0.974696i	$-0.571760\pi$
$367$	−8.56467	−0.447072	−0.223536	+	0.974696i	$0.571760\pi$
$368$	−49.1002	−2.55952
$369$	0	0
$370$	0	0
$371$	18.4485	0.957799
$372$	0	0
$373$	7.81924	0.404865	0.202432	−	0.979296i	$-0.435115\pi$
$373$	7.81924	0.404865	0.202432	+	0.979296i	$0.435115\pi$
$374$	12.1563	0.628589
$375$	0	0
$376$	−36.4142	−1.87792
$377$	−12.2228	−0.629508
$378$	0	0
$379$	1.14903	0.0590218	0.0295109	−	0.999564i	$-0.490605\pi$
$379$	1.14903	0.0590218	0.0295109	+	0.999564i	$0.490605\pi$
$380$	0	0
$381$	0	0
$382$	35.1754	1.79973
$383$	−34.0870	−1.74176	−0.870882	−	0.491493i	$-0.836451\pi$
$383$	−34.0870	−1.74176	−0.870882	+	0.491493i	$0.836451\pi$
$384$	0	0
$385$	0	0
$386$	−15.7685	−0.802593
$387$	0	0
$388$	−51.1754	−2.59804
$389$	−23.7743	−1.20541	−0.602703	−	0.797965i	$-0.705910\pi$
$389$	−23.7743	−1.20541	−0.602703	+	0.797965i	$0.705910\pi$
$390$	0	0
$391$	18.1768	0.919240
$392$	−7.37962	−0.372727
$393$	0	0
$394$	54.5560	2.74849
$395$	0	0
$396$	0	0
$397$	−4.15045	−0.208305	−0.104152	−	0.994561i	$-0.533213\pi$
$397$	−4.15045	−0.208305	−0.104152	+	0.994561i	$0.533213\pi$
$398$	−46.5501	−2.33334
$399$	0	0
$400$	0	0
$401$	33.9149	1.69363	0.846815	−	0.531887i	$-0.178517\pi$
$401$	33.9149	1.69363	0.846815	+	0.531887i	$0.178517\pi$
$402$	0	0
$403$	4.92619	0.245391
$404$	70.1886	3.49201
$405$	0	0
$406$	−17.9248	−0.889592
$407$	1.61213	0.0799102
$408$	0	0
$409$	8.55008	0.422774	0.211387	−	0.977402i	$-0.432202\pi$
$409$	8.55008	0.422774	0.211387	+	0.977402i	$0.432202\pi$
$410$	0	0
$411$	0	0
$412$	84.1133	4.14397
$413$	37.2506	1.83298
$414$	0	0
$415$	0	0
$416$	81.6444	4.00294
$417$	0	0
$418$	12.2374	0.598552
$419$	−13.2995	−0.649722	−0.324861	−	0.945762i	$-0.605318\pi$
$419$	−13.2995	−0.649722	−0.324861	+	0.945762i	$0.605318\pi$
$420$	0	0
$421$	33.3014	1.62301	0.811505	−	0.584345i	$-0.198649\pi$
$421$	33.3014	1.62301	0.811505	+	0.584345i	$0.198649\pi$
$422$	−48.4894	−2.36043
$423$	0	0
$424$	−55.5125	−2.69592
$425$	0	0
$426$	0	0
$427$	22.2374	1.07614
$428$	48.6312	2.35068
$429$	0	0
$430$	0	0
$431$	−2.07522	−0.0999600	−0.0499800	−	0.998750i	$-0.515916\pi$
$431$	−2.07522	−0.0999600	−0.0499800	+	0.998750i	$0.515916\pi$
$432$	0	0
$433$	−37.4010	−1.79738	−0.898690	−	0.438585i	$-0.855480\pi$
$433$	−37.4010	−1.79738	−0.898690	+	0.438585i	$0.855480\pi$
$434$	7.22425	0.346775
$435$	0	0
$436$	−79.4128	−3.80318
$437$	18.2981	0.875315
$438$	0	0
$439$	−6.02444	−0.287531	−0.143765	−	0.989612i	$-0.545921\pi$
$439$	−6.02444	−0.287531	−0.143765	+	0.989612i	$0.545921\pi$
$440$	0	0
$441$	0	0
$442$	−62.2247	−2.95973
$443$	10.1359	0.481569	0.240785	−	0.970579i	$-0.422595\pi$
$443$	10.1359	0.481569	0.240785	+	0.970579i	$0.422595\pi$
$444$	0	0
$445$	0	0
$446$	−63.2360	−2.99431
$447$	0	0
$448$	50.8423	2.40207
$449$	11.4861	0.542063	0.271032	−	0.962570i	$-0.412635\pi$
$449$	11.4861	0.542063	0.271032	+	0.962570i	$0.412635\pi$
$450$	0	0
$451$	2.38787	0.112441
$452$	70.8383	3.33195
$453$	0	0
$454$	−3.39517	−0.159343
$455$	0	0
$456$	0	0
$457$	−15.0435	−0.703705	−0.351852	−	0.936055i	$-0.614448\pi$
$457$	−15.0435	−0.703705	−0.351852	+	0.936055i	$0.614448\pi$
$458$	−43.2750	−2.02211
$459$	0	0
$460$	0	0
$461$	26.2374	1.22200	0.610999	−	0.791631i	$-0.290768\pi$
$461$	26.2374	1.22200	0.610999	+	0.791631i	$0.290768\pi$
$462$	0	0
$463$	−5.46168	−0.253826	−0.126913	−	0.991914i	$-0.540507\pi$
$463$	−5.46168	−0.253826	−0.126913	+	0.991914i	$0.540507\pi$
$464$	29.3112	1.36074
$465$	0	0
$466$	−49.4069	−2.28873
$467$	2.70052	0.124965	0.0624827	−	0.998046i	$-0.480098\pi$
$467$	2.70052	0.124965	0.0624827	+	0.998046i	$0.480098\pi$
$468$	0	0
$469$	30.0263	1.38649
$470$	0	0
$471$	0	0
$472$	−112.089	−5.15931
$473$	2.80606	0.129023
$474$	0	0
$475$	0	0
$476$	−65.7499	−3.01364
$477$	0	0
$478$	51.6991	2.36466
$479$	−16.7757	−0.766503	−0.383252	−	0.923644i	$-0.625196\pi$
$479$	−16.7757	−0.766503	−0.383252	+	0.923644i	$0.625196\pi$
$480$	0	0
$481$	−8.25202	−0.376260
$482$	−76.3752	−3.47879
$483$	0	0
$484$	5.15633	0.234378
$485$	0	0
$486$	0	0
$487$	−34.3996	−1.55880	−0.779398	−	0.626529i	$-0.784475\pi$
$487$	−34.3996	−1.55880	−0.779398	+	0.626529i	$0.784475\pi$
$488$	−66.9135	−3.02903
$489$	0	0
$490$	0	0
$491$	−14.9525	−0.674799	−0.337399	−	0.941362i	$-0.609547\pi$
$491$	−14.9525	−0.674799	−0.337399	+	0.941362i	$0.609547\pi$
$492$	0	0
$493$	−10.8510	−0.488703
$494$	−62.6399	−2.81830
$495$	0	0
$496$	−11.8134	−0.530435
$497$	20.6253	0.925171
$498$	0	0
$499$	−2.85097	−0.127627	−0.0638135	−	0.997962i	$-0.520326\pi$
$499$	−2.85097	−0.127627	−0.0638135	+	0.997962i	$0.520326\pi$
$500$	0	0
$501$	0	0
$502$	62.0625	2.76999
$503$	−1.48024	−0.0660006	−0.0330003	−	0.999455i	$-0.510506\pi$
$503$	−1.48024	−0.0660006	−0.0330003	+	0.999455i	$0.510506\pi$
$504$	0	0
$505$	0	0
$506$	10.7005	0.475696
$507$	0	0
$508$	64.0322	2.84097
$509$	23.2995	1.03273	0.516366	−	0.856368i	$-0.327285\pi$
$509$	23.2995	1.03273	0.516366	+	0.856368i	$0.327285\pi$
$510$	0	0
$511$	18.0098	0.796709
$512$	11.5017	0.508306
$513$	0	0
$514$	−29.1246	−1.28463
$515$	0	0
$516$	0	0
$517$	4.31265	0.189670
$518$	−12.1016	−0.531712
$519$	0	0
$520$	0	0
$521$	6.81194	0.298437	0.149218	−	0.988804i	$-0.452324\pi$
$521$	6.81194	0.298437	0.149218	+	0.988804i	$0.452324\pi$
$522$	0	0
$523$	−16.0567	−0.702109	−0.351054	−	0.936355i	$-0.614177\pi$
$523$	−16.0567	−0.702109	−0.351054	+	0.936355i	$0.614177\pi$
$524$	−30.5501	−1.33459
$525$	0	0
$526$	24.3938	1.06362
$527$	4.37328	0.190503
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	0	0
$532$	−66.1886	−2.86964
$533$	−12.2228	−0.529430
$534$	0	0
$535$	0	0
$536$	−90.3508	−3.90256
$537$	0	0
$538$	−30.2276	−1.30320
$539$	0.873992	0.0376455
$540$	0	0
$541$	6.62530	0.284844	0.142422	−	0.989806i	$-0.454511\pi$
$541$	6.62530	0.284844	0.142422	+	0.989806i	$0.454511\pi$
$542$	−67.4128	−2.89563
$543$	0	0
$544$	72.4807	3.10759
$545$	0	0
$546$	0	0
$547$	−5.75860	−0.246220	−0.123110	−	0.992393i	$-0.539287\pi$
$547$	−5.75860	−0.246220	−0.123110	+	0.992393i	$0.539287\pi$
$548$	−50.5256	−2.15835
$549$	0	0
$550$	0	0
$551$	−10.9234	−0.465351
$552$	0	0
$553$	3.78892	0.161121
$554$	6.46898	0.274840
$555$	0	0
$556$	−10.9624	−0.464909
$557$	−19.8700	−0.841920	−0.420960	−	0.907079i	$-0.638307\pi$
$557$	−19.8700	−0.841920	−0.420960	+	0.907079i	$0.638307\pi$
$558$	0	0
$559$	−14.3634	−0.607509
$560$	0	0
$561$	0	0
$562$	81.5633	3.44054
$563$	−5.83383	−0.245866	−0.122933	−	0.992415i	$-0.539230\pi$
$563$	−5.83383	−0.245866	−0.122933	+	0.992415i	$0.539230\pi$
$564$	0	0
$565$	0	0
$566$	−22.3576	−0.939758
$567$	0	0
$568$	−62.0625	−2.60409
$569$	−46.7123	−1.95828	−0.979140	−	0.203185i	$-0.934871\pi$
$569$	−46.7123	−1.95828	−0.979140	+	0.203185i	$0.934871\pi$
$570$	0	0
$571$	−24.4241	−1.02212	−0.511058	−	0.859546i	$-0.670746\pi$
$571$	−24.4241	−1.02212	−0.511058	+	0.859546i	$0.670746\pi$
$572$	−26.3938	−1.10358
$573$	0	0
$574$	−17.9248	−0.748166
$575$	0	0
$576$	0	0
$577$	16.5647	0.689596	0.344798	−	0.938677i	$-0.387947\pi$
$577$	16.5647	0.689596	0.344798	+	0.938677i	$0.387947\pi$
$578$	−9.76353	−0.406109
$579$	0	0
$580$	0	0
$581$	2.26187	0.0938380
$582$	0	0
$583$	6.57452	0.272289
$584$	−54.1925	−2.24250
$585$	0	0
$586$	−61.7802	−2.55212
$587$	−27.3258	−1.12786	−0.563929	−	0.825823i	$-0.690711\pi$
$587$	−27.3258	−1.12786	−0.563929	+	0.825823i	$0.690711\pi$
$588$	0	0
$589$	4.40246	0.181400
$590$	0	0
$591$	0	0
$592$	19.7889	0.813320
$593$	12.6048	0.517618	0.258809	−	0.965928i	$-0.416670\pi$
$593$	12.6048	0.517618	0.258809	+	0.965928i	$0.416670\pi$
$594$	0	0
$595$	0	0
$596$	−42.8627	−1.75573
$597$	0	0
$598$	−54.7729	−2.23983
$599$	−10.5990	−0.433061	−0.216531	−	0.976276i	$-0.569474\pi$
$599$	−10.5990	−0.433061	−0.216531	+	0.976276i	$0.569474\pi$
$600$	0	0
$601$	−22.4749	−0.916768	−0.458384	−	0.888754i	$-0.651572\pi$
$601$	−22.4749	−0.916768	−0.458384	+	0.888754i	$0.651572\pi$
$602$	−21.0640	−0.858503
$603$	0	0
$604$	−78.7631	−3.20482
$605$	0	0
$606$	0	0
$607$	33.5183	1.36047	0.680234	−	0.732995i	$-0.261878\pi$
$607$	33.5183	1.36047	0.680234	+	0.732995i	$0.261878\pi$
$608$	72.9643	2.95909
$609$	0	0
$610$	0	0
$611$	−22.0752	−0.893068
$612$	0	0
$613$	11.5672	0.467196	0.233598	−	0.972333i	$-0.424950\pi$
$613$	11.5672	0.467196	0.233598	+	0.972333i	$0.424950\pi$
$614$	−7.10413	−0.286699
$615$	0	0
$616$	−23.6932	−0.954627
$617$	33.3357	1.34204	0.671022	−	0.741438i	$-0.265856\pi$
$617$	33.3357	1.34204	0.671022	+	0.741438i	$0.265856\pi$
$618$	0	0
$619$	4.43866	0.178405	0.0892024	−	0.996014i	$-0.471568\pi$
$619$	4.43866	0.178405	0.0892024	+	0.996014i	$0.471568\pi$
$620$	0	0
$621$	0	0
$622$	−42.7367	−1.71359
$623$	8.31265	0.333039
$624$	0	0
$625$	0	0
$626$	0.162205	0.00648301
$627$	0	0
$628$	36.1622	1.44303
$629$	−7.32582	−0.292100
$630$	0	0
$631$	−39.8496	−1.58639	−0.793193	−	0.608971i	$-0.791583\pi$
$631$	−39.8496	−1.58639	−0.793193	+	0.608971i	$0.791583\pi$
$632$	−11.4010	−0.453509
$633$	0	0
$634$	10.2012	0.405143
$635$	0	0
$636$	0	0
$637$	−4.47371	−0.177255
$638$	−6.38787	−0.252898
$639$	0	0
$640$	0	0
$641$	3.88858	0.153590	0.0767948	−	0.997047i	$-0.475531\pi$
$641$	3.88858	0.153590	0.0767948	+	0.997047i	$0.475531\pi$
$642$	0	0
$643$	40.9380	1.61444	0.807218	−	0.590254i	$-0.200972\pi$
$643$	40.9380	1.61444	0.807218	+	0.590254i	$0.200972\pi$
$644$	−57.8759	−2.28063
$645$	0	0
$646$	−55.6093	−2.18792
$647$	−1.76257	−0.0692939	−0.0346469	−	0.999400i	$-0.511031\pi$
$647$	−1.76257	−0.0692939	−0.0346469	+	0.999400i	$0.511031\pi$
$648$	0	0
$649$	13.2750	0.521091
$650$	0	0
$651$	0	0
$652$	34.8627	1.36533
$653$	7.03761	0.275403	0.137702	−	0.990474i	$-0.456029\pi$
$653$	7.03761	0.275403	0.137702	+	0.990474i	$0.456029\pi$
$654$	0	0
$655$	0	0
$656$	29.3112	1.14441
$657$	0	0
$658$	−32.3733	−1.26204
$659$	12.6253	0.491812	0.245906	−	0.969294i	$-0.420915\pi$
$659$	12.6253	0.491812	0.245906	+	0.969294i	$0.420915\pi$
$660$	0	0
$661$	13.2243	0.514364	0.257182	−	0.966363i	$-0.417206\pi$
$661$	13.2243	0.514364	0.257182	+	0.966363i	$0.417206\pi$
$662$	−64.9741	−2.52529
$663$	0	0
$664$	−6.80606	−0.264126
$665$	0	0
$666$	0	0
$667$	−9.55149	−0.369835
$668$	60.9438	2.35799
$669$	0	0
$670$	0	0
$671$	7.92478	0.305933
$672$	0	0
$673$	18.2677	0.704170	0.352085	−	0.935968i	$-0.385473\pi$
$673$	18.2677	0.704170	0.352085	+	0.935968i	$0.385473\pi$
$674$	60.4055	2.32673
$675$	0	0
$676$	68.0698	2.61807
$677$	33.0191	1.26903	0.634513	−	0.772912i	$-0.281201\pi$
$677$	33.0191	1.26903	0.634513	+	0.772912i	$0.281201\pi$
$678$	0	0
$679$	−27.8496	−1.06877
$680$	0	0
$681$	0	0
$682$	2.57452	0.0985833
$683$	−30.8627	−1.18093	−0.590465	−	0.807063i	$-0.701056\pi$
$683$	−30.8627	−1.18093	−0.590465	+	0.807063i	$0.701056\pi$
$684$	0	0
$685$	0	0
$686$	45.9854	1.75573
$687$	0	0
$688$	34.4445	1.31319
$689$	−33.6531	−1.28208
$690$	0	0
$691$	27.6991	1.05372	0.526862	−	0.849951i	$-0.323369\pi$
$691$	27.6991	1.05372	0.526862	+	0.849951i	$0.323369\pi$
$692$	−36.0567	−1.37067
$693$	0	0
$694$	54.0176	2.05048
$695$	0	0
$696$	0	0
$697$	−10.8510	−0.411010
$698$	72.8989	2.75927
$699$	0	0
$700$	0	0
$701$	38.9643	1.47166	0.735831	−	0.677166i	$-0.236792\pi$
$701$	38.9643	1.47166	0.735831	+	0.677166i	$0.236792\pi$
$702$	0	0
$703$	−7.37470	−0.278142
$704$	18.1187	0.682875
$705$	0	0
$706$	13.4372	0.505717
$707$	38.1965	1.43653
$708$	0	0
$709$	−4.32250	−0.162335	−0.0811674	−	0.996700i	$-0.525865\pi$
$709$	−4.32250	−0.162335	−0.0811674	+	0.996700i	$0.525865\pi$
$710$	0	0
$711$	0	0
$712$	−25.0132	−0.937408
$713$	3.84955	0.144167
$714$	0	0
$715$	0	0
$716$	13.9248	0.520393
$717$	0	0
$718$	−48.6253	−1.81468
$719$	38.3996	1.43206	0.716032	−	0.698067i	$-0.245956\pi$
$719$	38.3996	1.43206	0.716032	+	0.698067i	$0.245956\pi$
$720$	0	0
$721$	45.7743	1.70473
$722$	−5.15282	−0.191768
$723$	0	0
$724$	−109.313	−4.06259
$725$	0	0
$726$	0	0
$727$	−21.6728	−0.803798	−0.401899	−	0.915684i	$-0.631650\pi$
$727$	−21.6728	−0.803798	−0.401899	+	0.915684i	$0.631650\pi$
$728$	121.279	4.49489
$729$	0	0
$730$	0	0
$731$	−12.7513	−0.471624
$732$	0	0
$733$	25.0698	0.925976	0.462988	−	0.886365i	$-0.346777\pi$
$733$	25.0698	0.925976	0.462988	+	0.886365i	$0.346777\pi$
$734$	22.9116	0.845683
$735$	0	0
$736$	63.8007	2.35172
$737$	10.7005	0.394159
$738$	0	0
$739$	−18.9018	−0.695312	−0.347656	−	0.937622i	$-0.613022\pi$
$739$	−18.9018	−0.695312	−0.347656	+	0.937622i	$0.613022\pi$
$740$	0	0
$741$	0	0
$742$	−49.3522	−1.81178
$743$	−43.6688	−1.60205	−0.801026	−	0.598629i	$-0.795712\pi$
$743$	−43.6688	−1.60205	−0.801026	+	0.598629i	$0.795712\pi$
$744$	0	0
$745$	0	0
$746$	−20.9175	−0.765843
$747$	0	0
$748$	−23.4314	−0.856736
$749$	26.4650	0.967010
$750$	0	0
$751$	−47.0541	−1.71703	−0.858514	−	0.512789i	$-0.828612\pi$
$751$	−47.0541	−1.71703	−0.858514	+	0.512789i	$0.828612\pi$
$752$	52.9380	1.93045
$753$	0	0
$754$	32.6977	1.19078
$755$	0	0
$756$	0	0
$757$	−26.9116	−0.978119	−0.489059	−	0.872250i	$-0.662660\pi$
$757$	−26.9116	−0.978119	−0.489059	+	0.872250i	$0.662660\pi$
$758$	−3.07381	−0.111646
$759$	0	0
$760$	0	0
$761$	−16.6859	−0.604865	−0.302432	−	0.953171i	$-0.597799\pi$
$761$	−16.6859	−0.604865	−0.302432	+	0.953171i	$0.597799\pi$
$762$	0	0
$763$	−43.2163	−1.56454
$764$	−67.8007	−2.45294
$765$	0	0
$766$	91.1871	3.29473
$767$	−67.9511	−2.45357
$768$	0	0
$769$	23.2995	0.840201	0.420100	−	0.907478i	$-0.361995\pi$
$769$	23.2995	0.840201	0.420100	+	0.907478i	$0.361995\pi$
$770$	0	0
$771$	0	0
$772$	30.3938	1.09390
$773$	1.63656	0.0588631	0.0294316	−	0.999567i	$-0.490630\pi$
$773$	1.63656	0.0588631	0.0294316	+	0.999567i	$0.490630\pi$
$774$	0	0
$775$	0	0
$776$	83.8007	3.00827
$777$	0	0
$778$	63.5994	2.28015
$779$	−10.9234	−0.391370
$780$	0	0
$781$	7.35026	0.263013
$782$	−48.6253	−1.73884
$783$	0	0
$784$	10.7283	0.383153
$785$	0	0
$786$	0	0
$787$	−2.09095	−0.0745344	−0.0372672	−	0.999305i	$-0.511865\pi$
$787$	−2.09095	−0.0745344	−0.0372672	+	0.999305i	$0.511865\pi$
$788$	−105.157	−3.74606
$789$	0	0
$790$	0	0
$791$	38.5501	1.37068
$792$	0	0
$793$	−40.5647	−1.44049
$794$	11.1030	0.394030
$795$	0	0
$796$	89.7255	3.18023
$797$	26.8872	0.952392	0.476196	−	0.879339i	$-0.342015\pi$
$797$	26.8872	0.952392	0.476196	+	0.879339i	$0.342015\pi$
$798$	0	0
$799$	−19.5975	−0.693311
$800$	0	0
$801$	0	0
$802$	−90.7269	−3.20368
$803$	6.41819	0.226493
$804$	0	0
$805$	0	0
$806$	−13.1782	−0.464183
$807$	0	0
$808$	−114.935	−4.04340
$809$	45.4128	1.59663	0.798315	−	0.602241i	$-0.205725\pi$
$809$	45.4128	1.59663	0.798315	+	0.602241i	$0.205725\pi$
$810$	0	0
$811$	34.3488	1.20615	0.603076	−	0.797684i	$-0.293942\pi$
$811$	34.3488	1.20615	0.603076	+	0.797684i	$0.293942\pi$
$812$	34.5501	1.21247
$813$	0	0
$814$	−4.31265	−0.151158
$815$	0	0
$816$	0	0
$817$	−12.8364	−0.449088
$818$	−22.8726	−0.799721
$819$	0	0
$820$	0	0
$821$	−18.7612	−0.654769	−0.327384	−	0.944891i	$-0.606167\pi$
$821$	−18.7612	−0.654769	−0.327384	+	0.944891i	$0.606167\pi$
$822$	0	0
$823$	−33.0592	−1.15237	−0.576186	−	0.817319i	$-0.695460\pi$
$823$	−33.0592	−1.15237	−0.576186	+	0.817319i	$0.695460\pi$
$824$	−137.737	−4.79830
$825$	0	0
$826$	−99.6502	−3.46728
$827$	10.8813	0.378379	0.189190	−	0.981941i	$-0.439414\pi$
$827$	10.8813	0.378379	0.189190	+	0.981941i	$0.439414\pi$
$828$	0	0
$829$	42.7221	1.48380	0.741900	−	0.670510i	$-0.233925\pi$
$829$	42.7221	1.48380	0.741900	+	0.670510i	$0.233925\pi$
$830$	0	0
$831$	0	0
$832$	−92.7445	−3.21534
$833$	−3.97159	−0.137608
$834$	0	0
$835$	0	0
$836$	−23.5877	−0.815797
$837$	0	0
$838$	35.5778	1.22902
$839$	21.1735	0.730989	0.365495	−	0.930813i	$-0.380900\pi$
$839$	21.1735	0.730989	0.365495	+	0.930813i	$0.380900\pi$
$840$	0	0
$841$	−23.2981	−0.803381
$842$	−89.0856	−3.07009
$843$	0	0
$844$	93.4636	3.21715
$845$	0	0
$846$	0	0
$847$	2.80606	0.0964175
$848$	80.7024	2.77133
$849$	0	0
$850$	0	0
$851$	−6.44851	−0.221052
$852$	0	0
$853$	54.4709	1.86505	0.932524	−	0.361109i	$-0.117602\pi$
$853$	54.4709	1.86505	0.932524	+	0.361109i	$0.117602\pi$
$854$	−59.4880	−2.03564
$855$	0	0
$856$	−79.6345	−2.72185
$857$	−38.5705	−1.31754	−0.658772	−	0.752342i	$-0.728924\pi$
$857$	−38.5705	−1.31754	−0.658772	+	0.752342i	$0.728924\pi$
$858$	0	0
$859$	−18.5139	−0.631685	−0.315843	−	0.948812i	$-0.602287\pi$
$859$	−18.5139	−0.631685	−0.315843	+	0.948812i	$0.602287\pi$
$860$	0	0
$861$	0	0
$862$	5.55149	0.189085
$863$	−22.5383	−0.767213	−0.383607	−	0.923497i	$-0.625318\pi$
$863$	−22.5383	−0.767213	−0.383607	+	0.923497i	$0.625318\pi$
$864$	0	0
$865$	0	0
$866$	100.053	3.39993
$867$	0	0
$868$	−13.9248	−0.472638
$869$	1.35026	0.0458045
$870$	0	0
$871$	−54.7729	−1.85591
$872$	130.040	4.40371
$873$	0	0
$874$	−48.9497	−1.65575
$875$	0	0
$876$	0	0
$877$	18.9419	0.639623	0.319812	−	0.947481i	$-0.396380\pi$
$877$	18.9419	0.639623	0.319812	+	0.947481i	$0.396380\pi$
$878$	16.1162	0.543894
$879$	0	0
$880$	0	0
$881$	20.0263	0.674705	0.337352	−	0.941378i	$-0.390469\pi$
$881$	20.0263	0.674705	0.337352	+	0.941378i	$0.390469\pi$
$882$	0	0
$883$	22.2981	0.750390	0.375195	−	0.926946i	$-0.377576\pi$
$883$	22.2981	0.750390	0.375195	+	0.926946i	$0.377576\pi$
$884$	119.938	4.03397
$885$	0	0
$886$	−27.1147	−0.910938
$887$	−3.10413	−0.104226	−0.0521132	−	0.998641i	$-0.516596\pi$
$887$	−3.10413	−0.104226	−0.0521132	+	0.998641i	$0.516596\pi$
$888$	0	0
$889$	34.8462	1.16871
$890$	0	0
$891$	0	0
$892$	121.888	4.08110
$893$	−19.7283	−0.660182
$894$	0	0
$895$	0	0
$896$	−46.4953	−1.55330
$897$	0	0
$898$	−30.7269	−1.02537
$899$	−2.29806	−0.0766447
$900$	0	0
$901$	−29.8759	−0.995311
$902$	−6.38787	−0.212693
$903$	0	0
$904$	−115.999	−3.85807
$905$	0	0
$906$	0	0
$907$	10.8627	0.360691	0.180345	−	0.983603i	$-0.442278\pi$
$907$	10.8627	0.360691	0.180345	+	0.983603i	$0.442278\pi$
$908$	6.54420	0.217177
$909$	0	0
$910$	0	0
$911$	5.82321	0.192931	0.0964657	−	0.995336i	$-0.469246\pi$
$911$	5.82321	0.192931	0.0964657	+	0.995336i	$0.469246\pi$
$912$	0	0
$913$	0.806063	0.0266768
$914$	40.2433	1.33113
$915$	0	0
$916$	83.4128	2.75604
$917$	−16.6253	−0.549016
$918$	0	0
$919$	−31.2750	−1.03167	−0.515834	−	0.856688i	$-0.672518\pi$
$919$	−31.2750	−1.03167	−0.515834	+	0.856688i	$0.672518\pi$
$920$	0	0
$921$	0	0
$922$	−70.1886	−2.31154
$923$	−37.6239	−1.23841
$924$	0	0
$925$	0	0
$926$	14.6107	0.480138
$927$	0	0
$928$	−38.0870	−1.25027
$929$	21.2243	0.696345	0.348173	−	0.937430i	$-0.386802\pi$
$929$	21.2243	0.696345	0.348173	+	0.937430i	$0.386802\pi$
$930$	0	0
$931$	−3.99809	−0.131032
$932$	95.2320	3.11943
$933$	0	0
$934$	−7.22425	−0.236385
$935$	0	0
$936$	0	0
$937$	16.4328	0.536835	0.268418	−	0.963303i	$-0.413499\pi$
$937$	16.4328	0.536835	0.268418	+	0.963303i	$0.413499\pi$
$938$	−80.3244	−2.62268
$939$	0	0
$940$	0	0
$941$	−5.86414	−0.191166	−0.0955828	−	0.995421i	$-0.530471\pi$
$941$	−5.86414	−0.191166	−0.0955828	+	0.995421i	$0.530471\pi$
$942$	0	0
$943$	−9.55149	−0.311039
$944$	162.952	5.30362
$945$	0	0
$946$	−7.50659	−0.244060
$947$	38.7415	1.25893	0.629464	−	0.777030i	$-0.283274\pi$
$947$	38.7415	1.25893	0.629464	+	0.777030i	$0.283274\pi$
$948$	0	0
$949$	−32.8529	−1.06645
$950$	0	0
$951$	0	0
$952$	107.667	3.48950
$953$	6.09569	0.197459	0.0987294	−	0.995114i	$-0.468522\pi$
$953$	6.09569	0.197459	0.0987294	+	0.995114i	$0.468522\pi$
$954$	0	0
$955$	0	0
$956$	−99.6502	−3.22292
$957$	0	0
$958$	44.8773	1.44992
$959$	−27.4960	−0.887891
$960$	0	0
$961$	−30.0738	−0.970123
$962$	22.0752	0.711734
$963$	0	0
$964$	147.213	4.74143
$965$	0	0
$966$	0	0
$967$	37.9697	1.22102	0.610511	−	0.792008i	$-0.290964\pi$
$967$	37.9697	1.22102	0.610511	+	0.792008i	$0.290964\pi$
$968$	−8.44358	−0.271387
$969$	0	0
$970$	0	0
$971$	60.8773	1.95365	0.976823	−	0.214049i	$-0.0686651\pi$
$971$	60.8773	1.95365	0.976823	+	0.214049i	$0.0686651\pi$
$972$	0	0
$973$	−5.96571	−0.191252
$974$	92.0235	2.94862
$975$	0	0
$976$	97.2769	3.11376
$977$	−21.3963	−0.684529	−0.342264	−	0.939604i	$-0.611194\pi$
$977$	−21.3963	−0.684529	−0.342264	+	0.939604i	$0.611194\pi$
$978$	0	0
$979$	2.96239	0.0946784
$980$	0	0
$981$	0	0
$982$	40.0000	1.27645
$983$	−18.8919	−0.602558	−0.301279	−	0.953536i	$-0.597414\pi$
$983$	−18.8919	−0.602558	−0.301279	+	0.953536i	$0.597414\pi$
$984$	0	0
$985$	0	0
$986$	29.0278	0.924432
$987$	0	0
$988$	120.739	3.84121
$989$	−11.2243	−0.356911
$990$	0	0
$991$	−1.33567	−0.0424291	−0.0212145	−	0.999775i	$-0.506753\pi$
$991$	−1.33567	−0.0424291	−0.0212145	+	0.999775i	$0.506753\pi$
$992$	15.3503	0.487371
$993$	0	0
$994$	−55.1754	−1.75006
$995$	0	0
$996$	0	0
$997$	−48.9076	−1.54892	−0.774460	−	0.632623i	$-0.781978\pi$
$997$	−48.9076	−1.54892	−0.774460	+	0.632623i	$0.781978\pi$
$998$	7.62672	0.241419
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.2.a.y.1.1		3
3.2	odd	2		825.2.a.n.1.3		3
5.2	odd	4		495.2.c.d.199.1		6
5.3	odd	4		495.2.c.d.199.6		6
5.4	even	2		2475.2.a.be.1.3		3
15.2	even	4		165.2.c.a.34.6	yes	6
15.8	even	4		165.2.c.a.34.1	✓	6
15.14	odd	2		825.2.a.h.1.1		3
33.32	even	2		9075.2.a.cc.1.1		3
60.23	odd	4		2640.2.d.i.529.4		6
60.47	odd	4		2640.2.d.i.529.1		6
165.32	odd	4		1815.2.c.d.364.1		6
165.98	odd	4		1815.2.c.d.364.6		6
165.164	even	2		9075.2.a.ck.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.c.a.34.1	✓	6	15.8	even	4
165.2.c.a.34.6	yes	6	15.2	even	4
495.2.c.d.199.1		6	5.2	odd	4
495.2.c.d.199.6		6	5.3	odd	4
825.2.a.h.1.1		3	15.14	odd	2
825.2.a.n.1.3		3	3.2	odd	2
1815.2.c.d.364.1		6	165.32	odd	4
1815.2.c.d.364.6		6	165.98	odd	4
2475.2.a.y.1.1		3	1.1	even	1	trivial
2475.2.a.be.1.3		3	5.4	even	2
2640.2.d.i.529.1		6	60.47	odd	4
2640.2.d.i.529.4		6	60.23	odd	4
9075.2.a.cc.1.1		3	33.32	even	2
9075.2.a.ck.1.3		3	165.164	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:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2475.a (trivial)

\(f(q)\)	\(=\)	\(q-2.67513 q^{2} +5.15633 q^{4} +2.80606 q^{7} -8.44358 q^{8} +O(q^{10})\)	\(q-2.67513 q^{2} +5.15633 q^{4} +2.80606 q^{7} -8.44358 q^{8} +1.00000 q^{11} -5.11871 q^{13} -7.50659 q^{14} +12.2750 q^{16} -4.54420 q^{17} -4.57452 q^{19} -2.67513 q^{22} -4.00000 q^{23} +13.6932 q^{26} +14.4690 q^{28} +2.38787 q^{29} -0.962389 q^{31} -15.9502 q^{32} +12.1563 q^{34} +1.61213 q^{37} +12.2374 q^{38} +2.38787 q^{41} +2.80606 q^{43} +5.15633 q^{44} +10.7005 q^{46} +4.31265 q^{47} +0.873992 q^{49} -26.3938 q^{52} +6.57452 q^{53} -23.6932 q^{56} -6.38787 q^{58} +13.2750 q^{59} +7.92478 q^{61} +2.57452 q^{62} +18.1187 q^{64} +10.7005 q^{67} -23.4314 q^{68} +7.35026 q^{71} +6.41819 q^{73} -4.31265 q^{74} -23.5877 q^{76} +2.80606 q^{77} +1.35026 q^{79} -6.38787 q^{82} +0.806063 q^{83} -7.50659 q^{86} -8.44358 q^{88} +2.96239 q^{89} -14.3634 q^{91} -20.6253 q^{92} -11.5369 q^{94} -9.92478 q^{97} -2.33804 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 5 q^{4} + 8 q^{7} - 9 q^{8}+O(q^{10}) \) 3 * q - 3 * q^2 + 5 * q^4 + 8 * q^7 - 9 * q^8	\( 3 q - 3 q^{2} + 5 q^{4} + 8 q^{7} - 9 q^{8} + 3 q^{11} + 6 q^{13} - 2 q^{14} + 5 q^{16} - 4 q^{17} - 2 q^{19} - 3 q^{22} - 12 q^{23} + 8 q^{26} + 12 q^{28} + 8 q^{29} + 8 q^{31} - 11 q^{32} + 26 q^{34} + 4 q^{37} - 6 q^{38} + 8 q^{41} + 8 q^{43} + 5 q^{44} + 12 q^{46} - 8 q^{47} + 11 q^{49} - 26 q^{52} + 8 q^{53} - 38 q^{56} - 20 q^{58} + 8 q^{59} + 2 q^{61} - 4 q^{62} + 33 q^{64} + 12 q^{67} - 28 q^{68} + 12 q^{71} + 18 q^{73} + 8 q^{74} - 18 q^{76} + 8 q^{77} - 6 q^{79} - 20 q^{82} + 2 q^{83} - 2 q^{86} - 9 q^{88} - 2 q^{89} + 8 q^{91} - 20 q^{92} - 12 q^{94} - 8 q^{97} + 29 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + 5 * q^4 + 8 * q^7 - 9 * q^8 + 3 * q^11 + 6 * q^13 - 2 * q^14 + 5 * q^16 - 4 * q^17 - 2 * q^19 - 3 * q^22 - 12 * q^23 + 8 * q^26 + 12 * q^28 + 8 * q^29 + 8 * q^31 - 11 * q^32 + 26 * q^34 + 4 * q^37 - 6 * q^38 + 8 * q^41 + 8 * q^43 + 5 * q^44 + 12 * q^46 - 8 * q^47 + 11 * q^49 - 26 * q^52 + 8 * q^53 - 38 * q^56 - 20 * q^58 + 8 * q^59 + 2 * q^61 - 4 * q^62 + 33 * q^64 + 12 * q^67 - 28 * q^68 + 12 * q^71 + 18 * q^73 + 8 * q^74 - 18 * q^76 + 8 * q^77 - 6 * q^79 - 20 * q^82 + 2 * q^83 - 2 * q^86 - 9 * q^88 - 2 * q^89 + 8 * q^91 - 20 * q^92 - 12 * q^94 - 8 * q^97 + 29 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more