LMFDB - Embedded newform 2601.2.a.bj.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2601,2,Mod(1,2601)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2601, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2601.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2601 = 3^{2} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2601.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:	$20.7690895657$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.45769536.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 15x^{4} + 72x^{2} - 109 $ x^6 - 15x^4 + 72x^2 - 109
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$2.55580$ of defining polynomial
Character	$\chi$	$=$	2601.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.55580 q^{2} +4.53209 q^{4} -3.91571 q^{5} -1.46791 q^{7} +6.47150 q^{8} +O(q^{10})$	$q+2.55580 q^{2} +4.53209 q^{4} -3.91571 q^{5} -1.46791 q^{7} +6.47150 q^{8} -10.0077 q^{10} +2.24753 q^{11} -5.10607 q^{13} -3.75168 q^{14} +7.47565 q^{16} -3.83750 q^{19} -17.7463 q^{20} +5.74422 q^{22} -4.33103 q^{23} +10.3327 q^{25} -13.0501 q^{26} -6.65270 q^{28} -3.02809 q^{29} -7.87939 q^{31} +6.16323 q^{32} +5.74791 q^{35} +6.29086 q^{37} -9.80785 q^{38} -25.3405 q^{40} -0.415326 q^{41} -1.06418 q^{43} +10.1860 q^{44} -11.0692 q^{46} -9.91491 q^{47} -4.84524 q^{49} +26.4084 q^{50} -23.1411 q^{52} +1.05164 q^{53} -8.80066 q^{55} -9.49959 q^{56} -7.73917 q^{58} +10.6955 q^{59} -1.58853 q^{61} -20.1381 q^{62} +0.800660 q^{64} +19.9939 q^{65} -10.5817 q^{67} +14.6905 q^{70} +6.94379 q^{71} -0.879385 q^{73} +16.0781 q^{74} -17.3919 q^{76} -3.29917 q^{77} -3.93582 q^{79} -29.2725 q^{80} -1.06149 q^{82} +9.29838 q^{83} -2.71982 q^{86} +14.5449 q^{88} -4.38800 q^{89} +7.49525 q^{91} -19.6286 q^{92} -25.3405 q^{94} +15.0265 q^{95} +12.2763 q^{97} -12.3834 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 18 q^{4} - 18 q^{7}+O(q^{10}) $ 6 * q + 18 * q^4 - 18 * q^7	$ 6 q + 18 q^{4} - 18 q^{7} - 12 q^{10} - 6 q^{13} + 6 q^{16} - 18 q^{19} - 24 q^{22} + 24 q^{25} - 42 q^{28} - 36 q^{31} + 6 q^{37} - 66 q^{40} + 12 q^{43} - 18 q^{46} + 24 q^{49} - 24 q^{52} - 24 q^{55} - 18 q^{58} - 30 q^{61} - 24 q^{64} - 18 q^{70} + 6 q^{73} - 78 q^{76} - 42 q^{79} - 6 q^{82} + 12 q^{91} - 66 q^{94} + 6 q^{97}+O(q^{100}) $ 6 * q + 18 * q^4 - 18 * q^7 - 12 * q^10 - 6 * q^13 + 6 * q^16 - 18 * q^19 - 24 * q^22 + 24 * q^25 - 42 * q^28 - 36 * q^31 + 6 * q^37 - 66 * q^40 + 12 * q^43 - 18 * q^46 + 24 * q^49 - 24 * q^52 - 24 * q^55 - 18 * q^58 - 30 * q^61 - 24 * q^64 - 18 * q^70 + 6 * q^73 - 78 * q^76 - 42 * q^79 - 6 * q^82 + 12 * q^91 - 66 * q^94 + 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 $

$2$	2.55580	1.80722	0.903610	−	0.428356i	$-0.140907\pi$
$2$	2.55580	1.80722	0.903610	+	0.428356i	$0.140907\pi$
$3$	0	0
$3$	0	0
$4$	4.53209	2.26604
$5$	−3.91571	−1.75116	−0.875578	−	0.483076i	$-0.839519\pi$
$5$	−3.91571	−1.75116	−0.875578	+	0.483076i	$0.839519\pi$
$6$	0	0
$7$	−1.46791	−0.554818	−0.277409	−	0.960752i	$-0.589476\pi$
$7$	−1.46791	−0.554818	−0.277409	+	0.960752i	$0.589476\pi$
$8$	6.47150	2.28802
$9$	0	0
$10$	−10.0077	−3.16473
$11$	2.24753	0.677655	0.338828	−	0.940848i	$-0.389970\pi$
$11$	2.24753	0.677655	0.338828	+	0.940848i	$0.389970\pi$
$12$	0	0
$13$	−5.10607	−1.41617	−0.708084	−	0.706128i	$-0.750440\pi$
$13$	−5.10607	−1.41617	−0.708084	+	0.706128i	$0.750440\pi$
$14$	−3.75168	−1.00268
$15$	0	0
$16$	7.47565	1.86891
$17$	0	0
$17$	0	0
$18$	0	0
$19$	−3.83750	−0.880382	−0.440191	−	0.897904i	$-0.645089\pi$
$19$	−3.83750	−0.880382	−0.440191	+	0.897904i	$0.645089\pi$
$20$	−17.7463	−3.96820
$21$	0	0
$22$	5.74422	1.22467
$23$	−4.33103	−0.903083	−0.451541	−	0.892250i	$-0.649126\pi$
$23$	−4.33103	−0.903083	−0.451541	+	0.892250i	$0.649126\pi$
$24$	0	0
$25$	10.3327	2.06655
$26$	−13.0501	−2.55933
$27$	0	0
$28$	−6.65270	−1.25724
$29$	−3.02809	−0.562302	−0.281151	−	0.959664i	$-0.590716\pi$
$29$	−3.02809	−0.562302	−0.281151	+	0.959664i	$0.590716\pi$
$30$	0	0
$31$	−7.87939	−1.41518	−0.707590	−	0.706624i	$-0.750217\pi$
$31$	−7.87939	−1.41518	−0.707590	+	0.706624i	$0.750217\pi$
$32$	6.16323	1.08952
$33$	0	0
$34$	0	0
$35$	5.74791	0.971574
$36$	0	0
$37$	6.29086	1.03421	0.517105	−	0.855922i	$-0.327009\pi$
$37$	6.29086	1.03421	0.517105	+	0.855922i	$0.327009\pi$
$38$	−9.80785	−1.59104
$39$	0	0
$40$	−25.3405	−4.00668
$41$	−0.415326	−0.0648631	−0.0324315	−	0.999474i	$-0.510325\pi$
$41$	−0.415326	−0.0648631	−0.0324315	+	0.999474i	$0.510325\pi$
$42$	0	0
$43$	−1.06418	−0.162286	−0.0811428	−	0.996702i	$-0.525857\pi$
$43$	−1.06418	−0.162286	−0.0811428	+	0.996702i	$0.525857\pi$
$44$	10.1860	1.53560
$45$	0	0
$46$	−11.0692	−1.63207
$47$	−9.91491	−1.44624	−0.723119	−	0.690723i	$-0.757292\pi$
$47$	−9.91491	−1.44624	−0.723119	+	0.690723i	$0.757292\pi$
$48$	0	0
$49$	−4.84524	−0.692177
$50$	26.4084	3.73471
$51$	0	0
$52$	−23.1411	−3.20910
$53$	1.05164	0.144454	0.0722272	−	0.997388i	$-0.476989\pi$
$53$	1.05164	0.144454	0.0722272	+	0.997388i	$0.476989\pi$
$54$	0	0
$55$	−8.80066	−1.18668
$56$	−9.49959	−1.26944
$57$	0	0
$58$	−7.73917	−1.01620
$59$	10.6955	1.39243	0.696216	−	0.717832i	$-0.254866\pi$
$59$	10.6955	1.39243	0.696216	+	0.717832i	$0.254866\pi$
$60$	0	0
$61$	−1.58853	−0.203390	−0.101695	−	0.994816i	$-0.532427\pi$
$61$	−1.58853	−0.203390	−0.101695	+	0.994816i	$0.532427\pi$
$62$	−20.1381	−2.55754
$63$	0	0
$64$	0.800660	0.100082
$65$	19.9939	2.47993
$66$	0	0
$67$	−10.5817	−1.29276	−0.646381	−	0.763015i	$-0.723718\pi$
$67$	−10.5817	−1.29276	−0.646381	+	0.763015i	$0.723718\pi$
$68$	0	0
$69$	0	0
$70$	14.6905	1.75585
$71$	6.94379	0.824077	0.412038	−	0.911167i	$-0.364817\pi$
$71$	6.94379	0.824077	0.412038	+	0.911167i	$0.364817\pi$
$72$	0	0
$73$	−0.879385	−0.102924	−0.0514621	−	0.998675i	$-0.516388\pi$
$73$	−0.879385	−0.102924	−0.0514621	+	0.998675i	$0.516388\pi$
$74$	16.0781	1.86905
$75$	0	0
$76$	−17.3919	−1.99498
$77$	−3.29917	−0.375976
$78$	0	0
$79$	−3.93582	−0.442815	−0.221407	−	0.975181i	$-0.571065\pi$
$79$	−3.93582	−0.442815	−0.221407	+	0.975181i	$0.571065\pi$
$80$	−29.2725	−3.27276
$81$	0	0
$82$	−1.06149	−0.117222
$83$	9.29838	1.02063	0.510315	−	0.859988i	$-0.329529\pi$
$83$	9.29838	1.02063	0.510315	+	0.859988i	$0.329529\pi$
$84$	0	0
$85$	0	0
$86$	−2.71982	−0.293286
$87$	0	0
$88$	14.5449	1.55049
$89$	−4.38800	−0.465127	−0.232563	−	0.972581i	$-0.574711\pi$
$89$	−4.38800	−0.465127	−0.232563	+	0.972581i	$0.574711\pi$
$90$	0	0
$91$	7.49525	0.785716
$92$	−19.6286	−2.04643
$93$	0	0
$94$	−25.3405	−2.61367
$95$	15.0265	1.54169
$96$	0	0
$97$	12.2763	1.24647	0.623235	−	0.782034i	$-0.285818\pi$
$97$	12.2763	1.24647	0.623235	+	0.782034i	$0.285818\pi$
$98$	−12.3834	−1.25092
$99$	0	0
$100$	46.8289	4.68289
$101$	−0.743377	−0.0739688	−0.0369844	−	0.999316i	$-0.511775\pi$
$101$	−0.743377	−0.0739688	−0.0369844	+	0.999316i	$0.511775\pi$
$102$	0	0
$103$	5.73648	0.565232	0.282616	−	0.959233i	$-0.408798\pi$
$103$	5.73648	0.565232	0.282616	+	0.959233i	$0.408798\pi$
$104$	−33.0439	−3.24022
$105$	0	0
$106$	2.68779	0.261061
$107$	−1.35991	−0.131467	−0.0657337	−	0.997837i	$-0.520939\pi$
$107$	−1.35991	−0.131467	−0.0657337	+	0.997837i	$0.520939\pi$
$108$	0	0
$109$	−6.62361	−0.634427	−0.317213	−	0.948354i	$-0.602747\pi$
$109$	−6.62361	−0.634427	−0.317213	+	0.948354i	$0.602747\pi$
$110$	−22.4927	−2.14459
$111$	0	0
$112$	−10.9736	−1.03691
$113$	18.5770	1.74758	0.873788	−	0.486308i	$-0.161657\pi$
$113$	18.5770	1.74758	0.873788	+	0.486308i	$0.161657\pi$
$114$	0	0
$115$	16.9590	1.58144
$116$	−13.7236	−1.27420
$117$	0	0
$118$	27.3354	2.51643
$119$	0	0
$120$	0	0
$121$	−5.94862	−0.540783
$122$	−4.05995	−0.367570
$123$	0	0
$124$	−35.7101	−3.20686
$125$	−20.8815	−1.86770
$126$	0	0
$127$	−18.4388	−1.63618	−0.818090	−	0.575090i	$-0.804967\pi$
$127$	−18.4388	−1.63618	−0.818090	+	0.575090i	$0.804967\pi$
$128$	−10.2801	−0.908645
$129$	0	0
$130$	51.1002	4.48178
$131$	10.2232	0.893203	0.446602	−	0.894733i	$-0.352634\pi$
$131$	10.2232	0.893203	0.446602	+	0.894733i	$0.352634\pi$
$132$	0	0
$133$	5.63310	0.488452
$134$	−27.0447	−2.33631
$135$	0	0
$136$	0	0
$137$	12.1624	1.03911	0.519554	−	0.854438i	$-0.326098\pi$
$137$	12.1624	1.03911	0.519554	+	0.854438i	$0.326098\pi$
$138$	0	0
$139$	−11.9581	−1.01427	−0.507137	−	0.861866i	$-0.669296\pi$
$139$	−11.9581	−1.01427	−0.507137	+	0.861866i	$0.669296\pi$
$140$	26.0500	2.20163
$141$	0	0
$142$	17.7469	1.48929
$143$	−11.4760	−0.959674
$144$	0	0
$145$	11.8571	0.984678
$146$	−2.24753	−0.186007
$147$	0	0
$148$	28.5107	2.34357
$149$	−11.2179	−0.919003	−0.459501	−	0.888177i	$-0.651972\pi$
$149$	−11.2179	−0.919003	−0.459501	+	0.888177i	$0.651972\pi$
$150$	0	0
$151$	7.88713	0.641845	0.320923	−	0.947105i	$-0.396007\pi$
$151$	7.88713	0.641845	0.320923	+	0.947105i	$0.396007\pi$
$152$	−24.8344	−2.01433
$153$	0	0
$154$	−8.43201	−0.679471
$155$	30.8534	2.47820
$156$	0	0
$157$	3.75103	0.299365	0.149682	−	0.988734i	$-0.452175\pi$
$157$	3.75103	0.299365	0.149682	+	0.988734i	$0.452175\pi$
$158$	−10.0592	−0.800263
$159$	0	0
$160$	−24.1334	−1.90791
$161$	6.35757	0.501047
$162$	0	0
$163$	16.7219	1.30976	0.654881	−	0.755732i	$-0.272719\pi$
$163$	16.7219	1.30976	0.654881	+	0.755732i	$0.272719\pi$
$164$	−1.88230	−0.146983
$165$	0	0
$166$	23.7648	1.84450
$167$	−5.85497	−0.453071	−0.226535	−	0.974003i	$-0.572740\pi$
$167$	−5.85497	−0.453071	−0.226535	+	0.974003i	$0.572740\pi$
$168$	0	0
$169$	13.0719	1.00553
$170$	0	0
$171$	0	0
$172$	−4.82295	−0.367746
$173$	15.9141	1.20993	0.604964	−	0.796253i	$-0.293187\pi$
$173$	15.9141	1.20993	0.604964	+	0.796253i	$0.293187\pi$
$174$	0	0
$175$	−15.1676	−1.14656
$176$	16.8017	1.26648
$177$	0	0
$178$	−11.2148	−0.840586
$179$	18.6339	1.39277	0.696383	−	0.717670i	$-0.254791\pi$
$179$	18.6339	1.39277	0.696383	+	0.717670i	$0.254791\pi$
$180$	0	0
$181$	−12.3523	−0.918143	−0.459071	−	0.888399i	$-0.651818\pi$
$181$	−12.3523	−0.918143	−0.459071	+	0.888399i	$0.651818\pi$
$182$	19.1563	1.41996
$183$	0	0
$184$	−28.0283	−2.06627
$185$	−24.6332	−1.81107
$186$	0	0
$187$	0	0
$188$	−44.9353	−3.27724
$189$	0	0
$190$	38.4047	2.78617
$191$	10.4943	0.759338	0.379669	−	0.925122i	$-0.376038\pi$
$191$	10.4943	0.759338	0.379669	+	0.925122i	$0.376038\pi$
$192$	0	0
$193$	−23.2053	−1.67036	−0.835178	−	0.549980i	$-0.814635\pi$
$193$	−23.2053	−1.67036	−0.835178	+	0.549980i	$0.814635\pi$
$194$	31.3757	2.25265
$195$	0	0
$196$	−21.9590	−1.56850
$197$	−18.2687	−1.30159	−0.650796	−	0.759253i	$-0.725565\pi$
$197$	−18.2687	−1.30159	−0.650796	+	0.759253i	$0.725565\pi$
$198$	0	0
$199$	−7.41147	−0.525385	−0.262693	−	0.964880i	$-0.584611\pi$
$199$	−7.41147	−0.525385	−0.262693	+	0.964880i	$0.584611\pi$
$200$	66.8684	4.72831
$201$	0	0
$202$	−1.89992	−0.133678
$203$	4.44496	0.311975
$204$	0	0
$205$	1.62630	0.113585
$206$	14.6613	1.02150
$207$	0	0
$208$	−38.1712	−2.64670
$209$	−8.62488	−0.596596
$210$	0	0
$211$	−10.6750	−0.734897	−0.367448	−	0.930044i	$-0.619769\pi$
$211$	−10.6750	−0.734897	−0.367448	+	0.930044i	$0.619769\pi$
$212$	4.76614	0.327340
$213$	0	0
$214$	−3.47565	−0.237591
$215$	4.16701	0.284187
$216$	0	0
$217$	11.5662	0.785167
$218$	−16.9286	−1.14655
$219$	0	0
$220$	−39.8854	−2.68907
$221$	0	0
$222$	0	0
$223$	23.0205	1.54157	0.770784	−	0.637096i	$-0.219865\pi$
$223$	23.0205	1.54157	0.770784	+	0.637096i	$0.219865\pi$
$224$	−9.04708	−0.604483
$225$	0	0
$226$	47.4789	3.15825
$227$	−5.27562	−0.350155	−0.175077	−	0.984555i	$-0.556018\pi$
$227$	−5.27562	−0.350155	−0.175077	+	0.984555i	$0.556018\pi$
$228$	0	0
$229$	−0.467911	−0.0309204	−0.0154602	−	0.999880i	$-0.504921\pi$
$229$	−0.467911	−0.0309204	−0.0154602	+	0.999880i	$0.504921\pi$
$230$	43.3438	2.85801
$231$	0	0
$232$	−19.5963	−1.28656
$233$	16.7516	1.09744	0.548718	−	0.836007i	$-0.315116\pi$
$233$	16.7516	1.09744	0.548718	+	0.836007i	$0.315116\pi$
$234$	0	0
$235$	38.8239	2.53259
$236$	48.4728	3.15531
$237$	0	0
$238$	0	0
$239$	23.8526	1.54290	0.771448	−	0.636292i	$-0.219533\pi$
$239$	23.8526	1.54290	0.771448	+	0.636292i	$0.219533\pi$
$240$	0	0
$241$	−25.1584	−1.62059	−0.810297	−	0.586019i	$-0.800694\pi$
$241$	−25.1584	−1.62059	−0.810297	+	0.586019i	$0.800694\pi$
$242$	−15.2034	−0.977314
$243$	0	0
$244$	−7.19934	−0.460891
$245$	18.9725	1.21211
$246$	0	0
$247$	19.5945	1.24677
$248$	−50.9914	−3.23796
$249$	0	0
$250$	−53.3688	−3.37534
$251$	−20.2452	−1.27786	−0.638931	−	0.769264i	$-0.720623\pi$
$251$	−20.2452	−1.27786	−0.638931	+	0.769264i	$0.720623\pi$
$252$	0	0
$253$	−9.73412	−0.611979
$254$	−47.1258	−2.95694
$255$	0	0
$256$	−27.8753	−1.74220
$257$	−2.19743	−0.137072	−0.0685361	−	0.997649i	$-0.521833\pi$
$257$	−2.19743	−0.137072	−0.0685361	+	0.997649i	$0.521833\pi$
$258$	0	0
$259$	−9.23442	−0.573799
$260$	90.6139	5.61964
$261$	0	0
$262$	26.1284	1.61422
$263$	−24.3051	−1.49872	−0.749358	−	0.662165i	$-0.769638\pi$
$263$	−24.3051	−1.49872	−0.749358	+	0.662165i	$0.769638\pi$
$264$	0	0
$265$	−4.11793	−0.252962
$266$	14.3971	0.882740
$267$	0	0
$268$	−47.9573	−2.92946
$269$	26.4654	1.61362	0.806811	−	0.590810i	$-0.201192\pi$
$269$	26.4654	1.61362	0.806811	+	0.590810i	$0.201192\pi$
$270$	0	0
$271$	5.87258	0.356734	0.178367	−	0.983964i	$-0.442919\pi$
$271$	5.87258	0.356734	0.178367	+	0.983964i	$0.442919\pi$
$272$	0	0
$273$	0	0
$274$	31.0847	1.87790
$275$	23.2231	1.40041
$276$	0	0
$277$	8.23442	0.494758	0.247379	−	0.968919i	$-0.420431\pi$
$277$	8.23442	0.494758	0.247379	+	0.968919i	$0.420431\pi$
$278$	−30.5625	−1.83302
$279$	0	0
$280$	37.1976	2.22298
$281$	−19.8298	−1.18295	−0.591474	−	0.806324i	$-0.701454\pi$
$281$	−19.8298	−1.18295	−0.591474	+	0.806324i	$0.701454\pi$
$282$	0	0
$283$	−15.6631	−0.931077	−0.465538	−	0.885028i	$-0.654139\pi$
$283$	−15.6631	−0.931077	−0.465538	+	0.885028i	$0.654139\pi$
$284$	31.4699	1.86739
$285$	0	0
$286$	−29.3304	−1.73434
$287$	0.609662	0.0359872
$288$	0	0
$289$	0	0
$290$	30.3043	1.77953
$291$	0	0
$292$	−3.98545	−0.233231
$293$	−26.4282	−1.54395	−0.771975	−	0.635653i	$-0.780731\pi$
$293$	−26.4282	−1.54395	−0.771975	+	0.635653i	$0.780731\pi$
$294$	0	0
$295$	−41.8803	−2.43837
$296$	40.7113	2.36630
$297$	0	0
$298$	−28.6705	−1.66084
$299$	22.1145	1.27892
$300$	0	0
$301$	1.56212	0.0900390
$302$	20.1579	1.15996
$303$	0	0
$304$	−28.6878	−1.64536
$305$	6.22020	0.356168
$306$	0	0
$307$	−1.67230	−0.0954434	−0.0477217	−	0.998861i	$-0.515196\pi$
$307$	−1.67230	−0.0954434	−0.0477217	+	0.998861i	$0.515196\pi$
$308$	−14.9521	−0.851977
$309$	0	0
$310$	78.8548	4.47865
$311$	33.4964	1.89941	0.949704	−	0.313149i	$-0.101384\pi$
$311$	33.4964	1.89941	0.949704	+	0.313149i	$0.101384\pi$
$312$	0	0
$313$	12.6509	0.715074	0.357537	−	0.933899i	$-0.383617\pi$
$313$	12.6509	0.715074	0.357537	+	0.933899i	$0.383617\pi$
$314$	9.58686	0.541018
$315$	0	0
$316$	−17.8375	−1.00344
$317$	5.89215	0.330936	0.165468	−	0.986215i	$-0.447087\pi$
$317$	5.89215	0.330936	0.165468	+	0.986215i	$0.447087\pi$
$318$	0	0
$319$	−6.80571	−0.381047
$320$	−3.13515	−0.175260
$321$	0	0
$322$	16.2486	0.905502
$323$	0	0
$324$	0	0
$325$	−52.7597	−2.92658
$326$	42.7378	2.36703
$327$	0	0
$328$	−2.68779	−0.148408
$329$	14.5542	0.802400
$330$	0	0
$331$	19.6340	1.07918	0.539592	−	0.841927i	$-0.318578\pi$
$331$	19.6340	1.07918	0.539592	+	0.841927i	$0.318578\pi$
$332$	42.1411	2.31279
$333$	0	0
$334$	−14.9641	−0.818799
$335$	41.4349	2.26383
$336$	0	0
$337$	−27.0574	−1.47391	−0.736954	−	0.675942i	$-0.763737\pi$
$337$	−27.0574	−1.47391	−0.736954	+	0.675942i	$0.763737\pi$
$338$	33.4091	1.81722
$339$	0	0
$340$	0	0
$341$	−17.7091	−0.959004
$342$	0	0
$343$	17.3878	0.938851
$344$	−6.88683	−0.371313
$345$	0	0
$346$	40.6732	2.18661
$347$	−6.77977	−0.363957	−0.181978	−	0.983303i	$-0.558250\pi$
$347$	−6.77977	−0.363957	−0.181978	+	0.983303i	$0.558250\pi$
$348$	0	0
$349$	22.3577	1.19678	0.598391	−	0.801204i	$-0.295807\pi$
$349$	22.3577	1.19678	0.598391	+	0.801204i	$0.295807\pi$
$350$	−38.7652	−2.07209
$351$	0	0
$352$	13.8520	0.738316
$353$	20.7372	1.10373	0.551866	−	0.833933i	$-0.313916\pi$
$353$	20.7372	1.10373	0.551866	+	0.833933i	$0.313916\pi$
$354$	0	0
$355$	−27.1898	−1.44309
$356$	−19.8868	−1.05400
$357$	0	0
$358$	47.6245	2.51704
$359$	17.4882	0.922989	0.461495	−	0.887143i	$-0.347313\pi$
$359$	17.4882	0.922989	0.461495	+	0.887143i	$0.347313\pi$
$360$	0	0
$361$	−4.27362	−0.224928
$362$	−31.5701	−1.65929
$363$	0	0
$364$	33.9691	1.78047
$365$	3.44341	0.180237
$366$	0	0
$367$	−27.1506	−1.41725	−0.708626	−	0.705584i	$-0.750685\pi$
$367$	−27.1506	−1.41725	−0.708626	+	0.705584i	$0.750685\pi$
$368$	−32.3773	−1.68778
$369$	0	0
$370$	−62.9573	−3.27299
$371$	−1.54372	−0.0801459
$372$	0	0
$373$	14.6117	0.756568	0.378284	−	0.925690i	$-0.376514\pi$
$373$	14.6117	0.756568	0.378284	+	0.925690i	$0.376514\pi$
$374$	0	0
$375$	0	0
$376$	−64.1644	−3.30902
$377$	15.4616	0.796314
$378$	0	0
$379$	9.68273	0.497369	0.248684	−	0.968585i	$-0.420002\pi$
$379$	9.68273	0.497369	0.248684	+	0.968585i	$0.420002\pi$
$380$	68.1015	3.49353
$381$	0	0
$382$	26.8212	1.37229
$383$	−6.47150	−0.330678	−0.165339	−	0.986237i	$-0.552872\pi$
$383$	−6.47150	−0.330678	−0.165339	+	0.986237i	$0.552872\pi$
$384$	0	0
$385$	12.9186	0.658392
$386$	−59.3081	−3.01870
$387$	0	0
$388$	55.6373	2.82456
$389$	−37.4121	−1.89687	−0.948435	−	0.316971i	$-0.897334\pi$
$389$	−37.4121	−1.89687	−0.948435	+	0.316971i	$0.897334\pi$
$390$	0	0
$391$	0	0
$392$	−31.3560	−1.58371
$393$	0	0
$394$	−46.6911	−2.35226
$395$	15.4115	0.775438
$396$	0	0
$397$	11.6209	0.583237	0.291619	−	0.956535i	$-0.405806\pi$
$397$	11.6209	0.583237	0.291619	+	0.956535i	$0.405806\pi$
$398$	−18.9422	−0.949487
$399$	0	0
$400$	77.2440	3.86220
$401$	−26.9005	−1.34335	−0.671673	−	0.740848i	$-0.734424\pi$
$401$	−26.9005	−1.34335	−0.671673	+	0.740848i	$0.734424\pi$
$402$	0	0
$403$	40.2327	2.00413
$404$	−3.36905	−0.167617
$405$	0	0
$406$	11.3604	0.563808
$407$	14.1389	0.700839
$408$	0	0
$409$	−10.8402	−0.536013	−0.268006	−	0.963417i	$-0.586365\pi$
$409$	−10.8402	−0.536013	−0.268006	+	0.963417i	$0.586365\pi$
$410$	4.15648	0.205274
$411$	0	0
$412$	25.9982	1.28084
$413$	−15.7000	−0.772547
$414$	0	0
$415$	−36.4097	−1.78728
$416$	−31.4699	−1.54294
$417$	0	0
$418$	−22.0434	−1.07818
$419$	−27.6612	−1.35134	−0.675670	−	0.737204i	$-0.736146\pi$
$419$	−27.6612	−1.35134	−0.675670	+	0.737204i	$0.736146\pi$
$420$	0	0
$421$	3.03415	0.147875	0.0739377	−	0.997263i	$-0.476443\pi$
$421$	3.03415	0.147875	0.0739377	+	0.997263i	$0.476443\pi$
$422$	−27.2831	−1.32812
$423$	0	0
$424$	6.80571	0.330515
$425$	0	0
$426$	0	0
$427$	2.33181	0.112844
$428$	−6.16323	−0.297911
$429$	0	0
$430$	10.6500	0.513589
$431$	−28.1836	−1.35756	−0.678779	−	0.734343i	$-0.737490\pi$
$431$	−28.1836	−1.35756	−0.678779	+	0.734343i	$0.737490\pi$
$432$	0	0
$433$	4.59896	0.221012	0.110506	−	0.993875i	$-0.464753\pi$
$433$	4.59896	0.221012	0.110506	+	0.993875i	$0.464753\pi$
$434$	29.5609	1.41897
$435$	0	0
$436$	−30.0188	−1.43764
$437$	16.6203	0.795058
$438$	0	0
$439$	−18.4456	−0.880362	−0.440181	−	0.897909i	$-0.645086\pi$
$439$	−18.4456	−0.880362	−0.440181	+	0.897909i	$0.645086\pi$
$440$	−56.9535	−2.71515
$441$	0	0
$442$	0	0
$443$	−27.6612	−1.31423	−0.657113	−	0.753792i	$-0.728222\pi$
$443$	−27.6612	−1.31423	−0.657113	+	0.753792i	$0.728222\pi$
$444$	0	0
$445$	17.1821	0.814510
$446$	58.8358	2.78595
$447$	0	0
$448$	−1.17530	−0.0555276
$449$	7.68717	0.362780	0.181390	−	0.983411i	$-0.441940\pi$
$449$	7.68717	0.362780	0.181390	+	0.983411i	$0.441940\pi$
$450$	0	0
$451$	−0.933458	−0.0439548
$452$	84.1925	3.96008
$453$	0	0
$454$	−13.4834	−0.632807
$455$	−29.3492	−1.37591
$456$	0	0
$457$	11.1034	0.519394	0.259697	−	0.965690i	$-0.416377\pi$
$457$	11.1034	0.519394	0.259697	+	0.965690i	$0.416377\pi$
$458$	−1.19588	−0.0558800
$459$	0	0
$460$	76.8599	3.58361
$461$	−7.88151	−0.367078	−0.183539	−	0.983012i	$-0.558755\pi$
$461$	−7.88151	−0.367078	−0.183539	+	0.983012i	$0.558755\pi$
$462$	0	0
$463$	−23.5895	−1.09630	−0.548148	−	0.836382i	$-0.684667\pi$
$463$	−23.5895	−1.09630	−0.548148	+	0.836382i	$0.684667\pi$
$464$	−22.6369	−1.05089
$465$	0	0
$466$	42.8138	1.98331
$467$	1.05164	0.0486643	0.0243321	−	0.999704i	$-0.492254\pi$
$467$	1.05164	0.0486643	0.0243321	+	0.999704i	$0.492254\pi$
$468$	0	0
$469$	15.5330	0.717248
$470$	99.2259	4.57695
$471$	0	0
$472$	69.2158	3.18591
$473$	−2.39177	−0.109974
$474$	0	0
$475$	−39.6519	−1.81935
$476$	0	0
$477$	0	0
$478$	60.9623	2.78835
$479$	−40.2192	−1.83766	−0.918832	−	0.394650i	$-0.870866\pi$
$479$	−40.2192	−1.83766	−0.918832	+	0.394650i	$0.870866\pi$
$480$	0	0
$481$	−32.1215	−1.46462
$482$	−64.2997	−2.92877
$483$	0	0
$484$	−26.9597	−1.22544
$485$	−48.0704	−2.18277
$486$	0	0
$487$	−31.2499	−1.41607	−0.708034	−	0.706178i	$-0.750418\pi$
$487$	−31.2499	−1.41607	−0.708034	+	0.706178i	$0.750418\pi$
$488$	−10.2801	−0.465360
$489$	0	0
$490$	48.4899	2.19055
$491$	−16.7646	−0.756574	−0.378287	−	0.925688i	$-0.623487\pi$
$491$	−16.7646	−0.756574	−0.378287	+	0.925688i	$0.623487\pi$
$492$	0	0
$493$	0	0
$494$	50.0796	2.25319
$495$	0	0
$496$	−58.9035	−2.64485
$497$	−10.1929	−0.457213
$498$	0	0
$499$	26.0847	1.16771	0.583856	−	0.811857i	$-0.301543\pi$
$499$	26.0847	1.16771	0.583856	+	0.811857i	$0.301543\pi$
$500$	−94.6367	−4.23228
$501$	0	0
$502$	−51.7425	−2.30938
$503$	8.76912	0.390996	0.195498	−	0.980704i	$-0.437368\pi$
$503$	8.76912	0.390996	0.195498	+	0.980704i	$0.437368\pi$
$504$	0	0
$505$	2.91085	0.129531
$506$	−24.8784	−1.10598
$507$	0	0
$508$	−83.5663	−3.70766
$509$	−26.0500	−1.15465	−0.577324	−	0.816515i	$-0.695903\pi$
$509$	−26.0500	−1.15465	−0.577324	+	0.816515i	$0.695903\pi$
$510$	0	0
$511$	1.29086	0.0571043
$512$	−50.6832	−2.23990
$513$	0	0
$514$	−5.61619	−0.247720
$515$	−22.4624	−0.989810
$516$	0	0
$517$	−22.2841	−0.980051
$518$	−23.6013	−1.03698
$519$	0	0
$520$	129.390	5.67414
$521$	−9.53677	−0.417813	−0.208907	−	0.977936i	$-0.566991\pi$
$521$	−9.53677	−0.417813	−0.208907	+	0.977936i	$0.566991\pi$
$522$	0	0
$523$	−11.5125	−0.503406	−0.251703	−	0.967805i	$-0.580991\pi$
$523$	−11.5125	−0.503406	−0.251703	+	0.967805i	$0.580991\pi$
$524$	46.3324	2.02404
$525$	0	0
$526$	−62.1189	−2.70851
$527$	0	0
$528$	0	0
$529$	−4.24216	−0.184442
$530$	−10.5246	−0.457158
$531$	0	0
$532$	25.5297	1.10685
$533$	2.12068	0.0918571
$534$	0	0
$535$	5.32501	0.230220
$536$	−68.4796	−2.95787
$537$	0	0
$538$	67.6400	2.91617
$539$	−10.8898	−0.469057
$540$	0	0
$541$	−39.3346	−1.69113	−0.845564	−	0.533875i	$-0.820735\pi$
$541$	−39.3346	−1.69113	−0.845564	+	0.533875i	$0.820735\pi$
$542$	15.0091	0.644696
$543$	0	0
$544$	0	0
$545$	25.9361	1.11098
$546$	0	0
$547$	−21.0077	−0.898226	−0.449113	−	0.893475i	$-0.648260\pi$
$547$	−21.0077	−0.898226	−0.449113	+	0.893475i	$0.648260\pi$
$548$	55.1213	2.35466
$549$	0	0
$550$	59.3536	2.53085
$551$	11.6203	0.495040
$552$	0	0
$553$	5.77744	0.245682
$554$	21.0455	0.894137
$555$	0	0
$556$	−54.1952	−2.29839
$557$	1.48437	0.0628947	0.0314473	−	0.999505i	$-0.489988\pi$
$557$	1.48437	0.0628947	0.0314473	+	0.999505i	$0.489988\pi$
$558$	0	0
$559$	5.43376	0.229824
$560$	42.9694	1.81579
$561$	0	0
$562$	−50.6810	−2.13785
$563$	23.2163	0.978449	0.489225	−	0.872158i	$-0.337280\pi$
$563$	23.2163	0.978449	0.489225	+	0.872158i	$0.337280\pi$
$564$	0	0
$565$	−72.7420	−3.06028
$566$	−40.0318	−1.68266
$567$	0	0
$568$	44.9368	1.88550
$569$	−22.7068	−0.951919	−0.475959	−	0.879467i	$-0.657899\pi$
$569$	−22.7068	−0.951919	−0.475959	+	0.879467i	$0.657899\pi$
$570$	0	0
$571$	14.3979	0.602532	0.301266	−	0.953540i	$-0.402591\pi$
$571$	14.3979	0.602532	0.301266	+	0.953540i	$0.402591\pi$
$572$	−52.0104	−2.17466
$573$	0	0
$574$	1.55817	0.0650368
$575$	−44.7515	−1.86626
$576$	0	0
$577$	−9.55707	−0.397866	−0.198933	−	0.980013i	$-0.563748\pi$
$577$	−9.55707	−0.397866	−0.198933	+	0.980013i	$0.563748\pi$
$578$	0	0
$579$	0	0
$580$	53.7374	2.23132
$581$	−13.6492	−0.566264
$582$	0	0
$583$	2.36360	0.0978903
$584$	−5.69094	−0.235493
$585$	0	0
$586$	−67.5450	−2.79026
$587$	43.0833	1.77824	0.889119	−	0.457677i	$-0.151318\pi$
$587$	43.0833	1.77824	0.889119	+	0.457677i	$0.151318\pi$
$588$	0	0
$589$	30.2371	1.24590
$590$	−107.038	−4.40666
$591$	0	0
$592$	47.0283	1.93285
$593$	15.0463	0.617877	0.308939	−	0.951082i	$-0.400026\pi$
$593$	15.0463	0.617877	0.308939	+	0.951082i	$0.400026\pi$
$594$	0	0
$595$	0	0
$596$	−50.8403	−2.08250
$597$	0	0
$598$	56.5202	2.31128
$599$	−14.7752	−0.603698	−0.301849	−	0.953356i	$-0.597604\pi$
$599$	−14.7752	−0.603698	−0.301849	+	0.953356i	$0.597604\pi$
$600$	0	0
$601$	11.3919	0.464684	0.232342	−	0.972634i	$-0.425361\pi$
$601$	11.3919	0.464684	0.232342	+	0.972634i	$0.425361\pi$
$602$	3.99245	0.162720
$603$	0	0
$604$	35.7452	1.45445
$605$	23.2930	0.946996
$606$	0	0
$607$	−12.3645	−0.501861	−0.250930	−	0.968005i	$-0.580737\pi$
$607$	−12.3645	−0.501861	−0.250930	+	0.968005i	$0.580737\pi$
$608$	−23.6514	−0.959190
$609$	0	0
$610$	15.8976	0.643673
$611$	50.6262	2.04812
$612$	0	0
$613$	6.05737	0.244655	0.122327	−	0.992490i	$-0.460964\pi$
$613$	6.05737	0.244655	0.122327	+	0.992490i	$0.460964\pi$
$614$	−4.27407	−0.172487
$615$	0	0
$616$	−21.3506	−0.860240
$617$	−15.1905	−0.611548	−0.305774	−	0.952104i	$-0.598915\pi$
$617$	−15.1905	−0.611548	−0.305774	+	0.952104i	$0.598915\pi$
$618$	0	0
$619$	−28.9077	−1.16190	−0.580948	−	0.813940i	$-0.697318\pi$
$619$	−28.9077	−1.16190	−0.580948	+	0.813940i	$0.697318\pi$
$620$	139.830	5.61571
$621$	0	0
$622$	85.6100	3.43265
$623$	6.44119	0.258061
$624$	0	0
$625$	30.1019	1.20408
$626$	32.3332	1.29230
$627$	0	0
$628$	17.0000	0.678374
$629$	0	0
$630$	0	0
$631$	−18.7820	−0.747699	−0.373850	−	0.927489i	$-0.621962\pi$
$631$	−18.7820	−0.747699	−0.373850	+	0.927489i	$0.621962\pi$
$632$	−25.4707	−1.01317
$633$	0	0
$634$	15.0591	0.598074
$635$	72.2010	2.86521
$636$	0	0
$637$	24.7401	0.980239
$638$	−17.3940	−0.688635
$639$	0	0
$640$	40.2540	1.59118
$641$	−21.1526	−0.835476	−0.417738	−	0.908568i	$-0.637177\pi$
$641$	−21.1526	−0.835476	−0.417738	+	0.908568i	$0.637177\pi$
$642$	0	0
$643$	13.3122	0.524982	0.262491	−	0.964934i	$-0.415456\pi$
$643$	13.3122	0.524982	0.262491	+	0.964934i	$0.415456\pi$
$644$	28.8131	1.13539
$645$	0	0
$646$	0	0
$647$	−3.32948	−0.130895	−0.0654477	−	0.997856i	$-0.520848\pi$
$647$	−3.32948	−0.130895	−0.0654477	+	0.997856i	$0.520848\pi$
$648$	0	0
$649$	24.0384	0.943589
$650$	−134.843	−5.28898
$651$	0	0
$652$	75.7853	2.96798
$653$	3.24908	0.127146	0.0635731	−	0.997977i	$-0.479750\pi$
$653$	3.24908	0.127146	0.0635731	+	0.997977i	$0.479750\pi$
$654$	0	0
$655$	−40.0310	−1.56414
$656$	−3.10484	−0.121223
$657$	0	0
$658$	37.1976	1.45011
$659$	9.19132	0.358043	0.179022	−	0.983845i	$-0.442707\pi$
$659$	9.19132	0.358043	0.179022	+	0.983845i	$0.442707\pi$
$660$	0	0
$661$	25.9736	1.01026	0.505128	−	0.863045i	$-0.331445\pi$
$661$	25.9736	1.01026	0.505128	+	0.863045i	$0.331445\pi$
$662$	50.1806	1.95032
$663$	0	0
$664$	60.1745	2.33522
$665$	−22.0576	−0.855356
$666$	0	0
$667$	13.1147	0.507805
$668$	−26.5352	−1.02668
$669$	0	0
$670$	105.899	4.09124
$671$	−3.57026	−0.137828
$672$	0	0
$673$	36.4861	1.40644	0.703218	−	0.710975i	$-0.251746\pi$
$673$	36.4861	1.40644	0.703218	+	0.710975i	$0.251746\pi$
$674$	−69.1531	−2.66368
$675$	0	0
$676$	59.2431	2.27858
$677$	12.7487	0.489971	0.244986	−	0.969527i	$-0.421217\pi$
$677$	12.7487	0.489971	0.244986	+	0.969527i	$0.421217\pi$
$678$	0	0
$679$	−18.0205	−0.691565
$680$	0	0
$681$	0	0
$682$	−45.2609	−1.73313
$683$	6.48441	0.248119	0.124060	−	0.992275i	$-0.460409\pi$
$683$	6.48441	0.248119	0.124060	+	0.992275i	$0.460409\pi$
$684$	0	0
$685$	−47.6245	−1.81964
$686$	44.4395	1.69671
$687$	0	0
$688$	−7.95542	−0.303298
$689$	−5.36976	−0.204572
$690$	0	0
$691$	−27.5645	−1.04860	−0.524301	−	0.851533i	$-0.675673\pi$
$691$	−27.5645	−1.04860	−0.524301	+	0.851533i	$0.675673\pi$
$692$	72.1242	2.74175
$693$	0	0
$694$	−17.3277	−0.657750
$695$	46.8244	1.77615
$696$	0	0
$697$	0	0
$698$	57.1418	2.16285
$699$	0	0
$700$	−68.7407	−2.59815
$701$	12.8489	0.485295	0.242647	−	0.970115i	$-0.421984\pi$
$701$	12.8489	0.485295	0.242647	+	0.970115i	$0.421984\pi$
$702$	0	0
$703$	−24.1411	−0.910501
$704$	1.79951	0.0678214
$705$	0	0
$706$	53.0001	1.99469
$707$	1.09121	0.0410392
$708$	0	0
$709$	45.4388	1.70649	0.853245	−	0.521510i	$-0.174631\pi$
$709$	45.4388	1.70649	0.853245	+	0.521510i	$0.174631\pi$
$710$	−69.4917	−2.60798
$711$	0	0
$712$	−28.3969	−1.06422
$713$	34.1259	1.27802
$714$	0	0
$715$	44.9368	1.68054
$716$	84.4507	3.15607
$717$	0	0
$718$	44.6961	1.66804
$719$	40.3158	1.50352	0.751762	−	0.659435i	$-0.229204\pi$
$719$	40.3158	1.50352	0.751762	+	0.659435i	$0.229204\pi$
$720$	0	0
$721$	−8.42065	−0.313601
$722$	−10.9225	−0.406494
$723$	0	0
$724$	−55.9819	−2.08055
$725$	−31.2885	−1.16202
$726$	0	0
$727$	−0.386497	−0.0143344	−0.00716719	−	0.999974i	$-0.502281\pi$
$727$	−0.386497	−0.0143344	−0.00716719	+	0.999974i	$0.502281\pi$
$728$	48.5055	1.79773
$729$	0	0
$730$	8.80066	0.325727
$731$	0	0
$732$	0	0
$733$	11.6023	0.428539	0.214269	−	0.976775i	$-0.431263\pi$
$733$	11.6023	0.428539	0.214269	+	0.976775i	$0.431263\pi$
$734$	−69.3915	−2.56129
$735$	0	0
$736$	−26.6932	−0.983923
$737$	−23.7827	−0.876048
$738$	0	0
$739$	−28.1506	−1.03554	−0.517769	−	0.855520i	$-0.673237\pi$
$739$	−28.1506	−1.03554	−0.517769	+	0.855520i	$0.673237\pi$
$740$	−111.640	−4.10395
$741$	0	0
$742$	−3.94543	−0.144841
$743$	−17.9301	−0.657793	−0.328896	−	0.944366i	$-0.606677\pi$
$743$	−17.9301	−0.657793	−0.328896	+	0.944366i	$0.606677\pi$
$744$	0	0
$745$	43.9258	1.60932
$746$	37.3446	1.36728
$747$	0	0
$748$	0	0
$749$	1.99623	0.0729406
$750$	0	0
$751$	−25.6013	−0.934205	−0.467103	−	0.884203i	$-0.654702\pi$
$751$	−25.6013	−0.934205	−0.467103	+	0.884203i	$0.654702\pi$
$752$	−74.1204	−2.70289
$753$	0	0
$754$	39.5167	1.43911
$755$	−30.8837	−1.12397
$756$	0	0
$757$	15.0550	0.547184	0.273592	−	0.961846i	$-0.411788\pi$
$757$	15.0550	0.547184	0.273592	+	0.961846i	$0.411788\pi$
$758$	24.7471	0.898855
$759$	0	0
$760$	97.2440	3.52741
$761$	−8.08958	−0.293247	−0.146624	−	0.989192i	$-0.546841\pi$
$761$	−8.08958	−0.293247	−0.146624	+	0.989192i	$0.546841\pi$
$762$	0	0
$763$	9.72287	0.351991
$764$	47.5609	1.72069
$765$	0	0
$766$	−16.5398	−0.597608
$767$	−54.6118	−1.97192
$768$	0	0
$769$	13.6973	0.493937	0.246968	−	0.969024i	$-0.420566\pi$
$769$	13.6973	0.493937	0.246968	+	0.969024i	$0.420566\pi$
$770$	33.0173	1.18986
$771$	0	0
$772$	−105.169	−3.78510
$773$	15.0766	0.542268	0.271134	−	0.962542i	$-0.412601\pi$
$773$	15.0766	0.542268	0.271134	+	0.962542i	$0.412601\pi$
$774$	0	0
$775$	−81.4157	−2.92454
$776$	79.4462	2.85195
$777$	0	0
$778$	−95.6177	−3.42806
$779$	1.59381	0.0571043
$780$	0	0
$781$	15.6064	0.558440
$782$	0	0
$783$	0	0
$784$	−36.2213	−1.29362
$785$	−14.6879	−0.524235
$786$	0	0
$787$	10.6290	0.378882	0.189441	−	0.981892i	$-0.439332\pi$
$787$	10.6290	0.378882	0.189441	+	0.981892i	$0.439332\pi$
$788$	−82.7954	−2.94946
$789$	0	0
$790$	39.3887	1.40139
$791$	−27.2694	−0.969587
$792$	0	0
$793$	8.11112	0.288034
$794$	29.7007	1.05404
$795$	0	0
$796$	−33.5895	−1.19055
$797$	1.98936	0.0704666	0.0352333	−	0.999379i	$-0.488783\pi$
$797$	1.98936	0.0704666	0.0352333	+	0.999379i	$0.488783\pi$
$798$	0	0
$799$	0	0
$800$	63.6831	2.25154
$801$	0	0
$802$	−68.7521	−2.42772
$803$	−1.97644	−0.0697472
$804$	0	0
$805$	−24.8944	−0.877411
$806$	102.826	3.62191
$807$	0	0
$808$	−4.81076	−0.169242
$809$	−28.5359	−1.00327	−0.501635	−	0.865079i	$-0.667268\pi$
$809$	−28.5359	−1.00327	−0.501635	+	0.865079i	$0.667268\pi$
$810$	0	0
$811$	−38.0223	−1.33514	−0.667572	−	0.744546i	$-0.732666\pi$
$811$	−38.0223	−1.33514	−0.667572	+	0.744546i	$0.732666\pi$
$812$	20.1450	0.706950
$813$	0	0
$814$	36.1361	1.26657
$815$	−65.4782	−2.29360
$816$	0	0
$817$	4.08378	0.142873
$818$	−27.7053	−0.968693
$819$	0	0
$820$	7.37052	0.257390
$821$	18.1245	0.632548	0.316274	−	0.948668i	$-0.397568\pi$
$821$	18.1245	0.632548	0.316274	+	0.948668i	$0.397568\pi$
$822$	0	0
$823$	−17.4023	−0.606606	−0.303303	−	0.952894i	$-0.598089\pi$
$823$	−17.4023	−0.606606	−0.303303	+	0.952894i	$0.598089\pi$
$824$	37.1236	1.29326
$825$	0	0
$826$	−40.1260	−1.39616
$827$	4.71605	0.163993	0.0819965	−	0.996633i	$-0.473870\pi$
$827$	4.71605	0.163993	0.0819965	+	0.996633i	$0.473870\pi$
$828$	0	0
$829$	13.8675	0.481639	0.240820	−	0.970570i	$-0.422584\pi$
$829$	13.8675	0.481639	0.240820	+	0.970570i	$0.422584\pi$
$830$	−93.0558	−3.23001
$831$	0	0
$832$	−4.08822	−0.141734
$833$	0	0
$834$	0	0
$835$	22.9263	0.793398
$836$	−39.0887	−1.35191
$837$	0	0
$838$	−70.6965	−2.44217
$839$	−28.8070	−0.994529	−0.497265	−	0.867599i	$-0.665662\pi$
$839$	−28.8070	−0.994529	−0.497265	+	0.867599i	$0.665662\pi$
$840$	0	0
$841$	−19.8307	−0.683817
$842$	7.75466	0.267243
$843$	0	0
$844$	−48.3800	−1.66531
$845$	−51.1858	−1.76084
$846$	0	0
$847$	8.73204	0.300036
$848$	7.86172	0.269973
$849$	0	0
$850$	0	0
$851$	−27.2459	−0.933978
$852$	0	0
$853$	34.7701	1.19051	0.595253	−	0.803538i	$-0.297052\pi$
$853$	34.7701	1.19051	0.595253	+	0.803538i	$0.297052\pi$
$854$	5.95964	0.203935
$855$	0	0
$856$	−8.80066	−0.300800
$857$	−25.1624	−0.859532	−0.429766	−	0.902940i	$-0.641404\pi$
$857$	−25.1624	−0.859532	−0.429766	+	0.902940i	$0.641404\pi$
$858$	0	0
$859$	44.4005	1.51493	0.757464	−	0.652877i	$-0.226438\pi$
$859$	44.4005	1.51493	0.757464	+	0.652877i	$0.226438\pi$
$860$	18.8852	0.643981
$861$	0	0
$862$	−72.0316	−2.45341
$863$	22.4054	0.762689	0.381344	−	0.924433i	$-0.375461\pi$
$863$	22.4054	0.762689	0.381344	+	0.924433i	$0.375461\pi$
$864$	0	0
$865$	−62.3150	−2.11877
$866$	11.7540	0.399417
$867$	0	0
$868$	52.4192	1.77922
$869$	−8.84587	−0.300076
$870$	0	0
$871$	54.0310	1.83077
$872$	−42.8647	−1.45158
$873$	0	0
$874$	42.4781	1.43684
$875$	30.6521	1.03623
$876$	0	0
$877$	34.9665	1.18073	0.590367	−	0.807135i	$-0.298983\pi$
$877$	34.9665	1.18073	0.590367	+	0.807135i	$0.298983\pi$
$878$	−47.1432	−1.59101
$879$	0	0
$880$	−65.7907	−2.21780
$881$	−22.6567	−0.763324	−0.381662	−	0.924302i	$-0.624648\pi$
$881$	−22.6567	−0.763324	−0.381662	+	0.924302i	$0.624648\pi$
$882$	0	0
$883$	−5.75702	−0.193739	−0.0968695	−	0.995297i	$-0.530883\pi$
$883$	−5.75702	−0.193739	−0.0968695	+	0.995297i	$0.530883\pi$
$884$	0	0
$885$	0	0
$886$	−70.6965	−2.37509
$887$	−50.9914	−1.71213	−0.856063	−	0.516872i	$-0.827096\pi$
$887$	−50.9914	−1.71213	−0.856063	+	0.516872i	$0.827096\pi$
$888$	0	0
$889$	27.0665	0.907783
$890$	43.9139	1.47200
$891$	0	0
$892$	104.331	3.49326
$893$	38.0484	1.27324
$894$	0	0
$895$	−72.9650	−2.43895
$896$	15.0903	0.504133
$897$	0	0
$898$	19.6468	0.655623
$899$	23.8595	0.795758
$900$	0	0
$901$	0	0
$902$	−2.38573	−0.0794360
$903$	0	0
$904$	120.221	3.99849
$905$	48.3682	1.60781
$906$	0	0
$907$	−23.5175	−0.780887	−0.390444	−	0.920627i	$-0.627678\pi$
$907$	−23.5175	−0.780887	−0.390444	+	0.920627i	$0.627678\pi$
$908$	−23.9096	−0.793467
$909$	0	0
$910$	−75.0105	−2.48658
$911$	13.0675	0.432944	0.216472	−	0.976289i	$-0.430545\pi$
$911$	13.0675	0.432944	0.216472	+	0.976289i	$0.430545\pi$
$912$	0	0
$913$	20.8984	0.691635
$914$	28.3780	0.938660
$915$	0	0
$916$	−2.12061	−0.0700671
$917$	−15.0067	−0.495566
$918$	0	0
$919$	−30.1453	−0.994401	−0.497200	−	0.867636i	$-0.665639\pi$
$919$	−30.1453	−0.994401	−0.497200	+	0.867636i	$0.665639\pi$
$920$	109.750	3.61837
$921$	0	0
$922$	−20.1435	−0.663391
$923$	−35.4555	−1.16703
$924$	0	0
$925$	65.0019	2.13725
$926$	−60.2898	−1.98125
$927$	0	0
$928$	−18.6628	−0.612637
$929$	2.88385	0.0946159	0.0473080	−	0.998880i	$-0.484936\pi$
$929$	2.88385	0.0946159	0.0473080	+	0.998880i	$0.484936\pi$
$930$	0	0
$931$	18.5936	0.609380
$932$	75.9200	2.48684
$933$	0	0
$934$	2.68779	0.0879470
$935$	0	0
$936$	0	0
$937$	−3.57903	−0.116922	−0.0584609	−	0.998290i	$-0.518619\pi$
$937$	−3.57903	−0.116922	−0.0584609	+	0.998290i	$0.518619\pi$
$938$	39.6992	1.29623
$939$	0	0
$940$	175.953	5.73896
$941$	43.3067	1.41176	0.705878	−	0.708333i	$-0.250552\pi$
$941$	43.3067	1.41176	0.705878	+	0.708333i	$0.250552\pi$
$942$	0	0
$943$	1.79879	0.0585767
$944$	79.9556	2.60233
$945$	0	0
$946$	−6.11287	−0.198747
$947$	−29.3666	−0.954286	−0.477143	−	0.878826i	$-0.658328\pi$
$947$	−29.3666	−0.954286	−0.477143	+	0.878826i	$0.658328\pi$
$948$	0	0
$949$	4.49020	0.145758
$950$	−101.342	−3.28797
$951$	0	0
$952$	0	0
$953$	−48.8009	−1.58082	−0.790408	−	0.612581i	$-0.790131\pi$
$953$	−48.8009	−1.58082	−0.790408	+	0.612581i	$0.790131\pi$
$954$	0	0
$955$	−41.0925	−1.32972
$956$	108.102	3.49627
$957$	0	0
$958$	−102.792	−3.32106
$959$	−17.8534	−0.576516
$960$	0	0
$961$	31.0847	1.00273
$962$	−82.0961	−2.64688
$963$	0	0
$964$	−114.020	−3.67234
$965$	90.8652	2.92505
$966$	0	0
$967$	11.0537	0.355465	0.177732	−	0.984079i	$-0.443124\pi$
$967$	11.0537	0.355465	0.177732	+	0.984079i	$0.443124\pi$
$968$	−38.4965	−1.23732
$969$	0	0
$970$	−122.858	−3.94474
$971$	20.0007	0.641854	0.320927	−	0.947104i	$-0.396006\pi$
$971$	20.0007	0.641854	0.320927	+	0.947104i	$0.396006\pi$
$972$	0	0
$973$	17.5534	0.562738
$974$	−79.8684	−2.55915
$975$	0	0
$976$	−11.8753	−0.380118
$977$	45.0120	1.44006	0.720031	−	0.693942i	$-0.244128\pi$
$977$	45.0120	1.44006	0.720031	+	0.693942i	$0.244128\pi$
$978$	0	0
$979$	−9.86215	−0.315196
$980$	85.9851	2.74669
$981$	0	0
$982$	−42.8468	−1.36730
$983$	11.9915	0.382471	0.191235	−	0.981544i	$-0.438751\pi$
$983$	11.9915	0.382471	0.191235	+	0.981544i	$0.438751\pi$
$984$	0	0
$985$	71.5349	2.27929
$986$	0	0
$987$	0	0
$988$	88.8041	2.82523
$989$	4.60899	0.146557
$990$	0	0
$991$	26.7802	0.850702	0.425351	−	0.905028i	$-0.360151\pi$
$991$	26.7802	0.850702	0.425351	+	0.905028i	$0.360151\pi$
$992$	−48.5625	−1.54186
$993$	0	0
$994$	−26.0509	−0.826284
$995$	29.0211	0.920032
$996$	0	0
$997$	8.27456	0.262058	0.131029	−	0.991379i	$-0.458172\pi$
$997$	8.27456	0.262058	0.131029	+	0.991379i	$0.458172\pi$
$998$	66.6672	2.11031
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2601.2.a.bj.1.6	yes	6
3.2	odd	2	inner	2601.2.a.bj.1.1	✓	6
17.16	even	2		2601.2.a.bk.1.6	yes	6
51.50	odd	2		2601.2.a.bk.1.1	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2601.2.a.bj.1.1	✓	6	3.2	odd	2	inner
2601.2.a.bj.1.6	yes	6	1.1	even	1	trivial
2601.2.a.bk.1.1	yes	6	51.50	odd	2
2601.2.a.bk.1.6	yes	6	17.16	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:	\( N \)	\(=\)	\( 2601 = 3^{2} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2601.a (trivial)

\(f(q)\)	\(=\)	\(q+2.55580 q^{2} +4.53209 q^{4} -3.91571 q^{5} -1.46791 q^{7} +6.47150 q^{8} +O(q^{10})\)	\(q+2.55580 q^{2} +4.53209 q^{4} -3.91571 q^{5} -1.46791 q^{7} +6.47150 q^{8} -10.0077 q^{10} +2.24753 q^{11} -5.10607 q^{13} -3.75168 q^{14} +7.47565 q^{16} -3.83750 q^{19} -17.7463 q^{20} +5.74422 q^{22} -4.33103 q^{23} +10.3327 q^{25} -13.0501 q^{26} -6.65270 q^{28} -3.02809 q^{29} -7.87939 q^{31} +6.16323 q^{32} +5.74791 q^{35} +6.29086 q^{37} -9.80785 q^{38} -25.3405 q^{40} -0.415326 q^{41} -1.06418 q^{43} +10.1860 q^{44} -11.0692 q^{46} -9.91491 q^{47} -4.84524 q^{49} +26.4084 q^{50} -23.1411 q^{52} +1.05164 q^{53} -8.80066 q^{55} -9.49959 q^{56} -7.73917 q^{58} +10.6955 q^{59} -1.58853 q^{61} -20.1381 q^{62} +0.800660 q^{64} +19.9939 q^{65} -10.5817 q^{67} +14.6905 q^{70} +6.94379 q^{71} -0.879385 q^{73} +16.0781 q^{74} -17.3919 q^{76} -3.29917 q^{77} -3.93582 q^{79} -29.2725 q^{80} -1.06149 q^{82} +9.29838 q^{83} -2.71982 q^{86} +14.5449 q^{88} -4.38800 q^{89} +7.49525 q^{91} -19.6286 q^{92} -25.3405 q^{94} +15.0265 q^{95} +12.2763 q^{97} -12.3834 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 18 q^{4} - 18 q^{7}+O(q^{10}) \) 6 * q + 18 * q^4 - 18 * q^7	\( 6 q + 18 q^{4} - 18 q^{7} - 12 q^{10} - 6 q^{13} + 6 q^{16} - 18 q^{19} - 24 q^{22} + 24 q^{25} - 42 q^{28} - 36 q^{31} + 6 q^{37} - 66 q^{40} + 12 q^{43} - 18 q^{46} + 24 q^{49} - 24 q^{52} - 24 q^{55} - 18 q^{58} - 30 q^{61} - 24 q^{64} - 18 q^{70} + 6 q^{73} - 78 q^{76} - 42 q^{79} - 6 q^{82} + 12 q^{91} - 66 q^{94} + 6 q^{97}+O(q^{100}) \) 6 * q + 18 * q^4 - 18 * q^7 - 12 * q^10 - 6 * q^13 + 6 * q^16 - 18 * q^19 - 24 * q^22 + 24 * q^25 - 42 * q^28 - 36 * q^31 + 6 * q^37 - 66 * q^40 + 12 * q^43 - 18 * q^46 + 24 * q^49 - 24 * q^52 - 24 * q^55 - 18 * q^58 - 30 * q^61 - 24 * q^64 - 18 * q^70 + 6 * q^73 - 78 * q^76 - 42 * q^79 - 6 * q^82 + 12 * q^91 - 66 * q^94 + 6 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more