LMFDB - Embedded newform 1521.4.a.bm.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,4,Mod(1,1521)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1521, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1521.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1521 = 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1521.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:	$89.7419051187$
Analytic rank:	$1$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 100 x^{16} + 4066 x^{14} - 86197 x^{12} + 1016064 x^{10} - 6594119 x^{8} + 22251817 x^{6} + \cdots - 6889792 $ x^18 - 100x^16 + 4066x^14 - 86197x^12 + 1016064x^10 - 6594119x^8 + 22251817x^6 - 37096332x^4 + 27824160x^2 - 6889792
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3}\cdot 7\cdot 13^{4} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-3.45813$ of defining polynomial
Character	$\chi$	$=$	1521.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.45813 q^{2} +3.95865 q^{4} -9.27918 q^{5} +27.2299 q^{7} +13.9755 q^{8} +O(q^{10})$	$q-3.45813 q^{2} +3.95865 q^{4} -9.27918 q^{5} +27.2299 q^{7} +13.9755 q^{8} +32.0886 q^{10} +38.3557 q^{11} -94.1645 q^{14} -79.9983 q^{16} +85.0844 q^{17} -114.360 q^{19} -36.7331 q^{20} -132.639 q^{22} +49.4232 q^{23} -38.8968 q^{25} +107.794 q^{28} -11.5644 q^{29} -220.904 q^{31} +164.840 q^{32} -294.233 q^{34} -252.671 q^{35} -24.7406 q^{37} +395.470 q^{38} -129.681 q^{40} -30.8365 q^{41} -409.346 q^{43} +151.837 q^{44} -170.912 q^{46} -434.519 q^{47} +398.468 q^{49} +134.510 q^{50} +716.136 q^{53} -355.909 q^{55} +380.552 q^{56} +39.9913 q^{58} -618.055 q^{59} +213.058 q^{61} +763.915 q^{62} +69.9470 q^{64} -578.504 q^{67} +336.820 q^{68} +873.770 q^{70} -238.263 q^{71} +748.837 q^{73} +85.5562 q^{74} -452.710 q^{76} +1044.42 q^{77} +883.090 q^{79} +742.319 q^{80} +106.637 q^{82} -1401.67 q^{83} -789.513 q^{85} +1415.57 q^{86} +536.040 q^{88} +1301.10 q^{89} +195.649 q^{92} +1502.62 q^{94} +1061.16 q^{95} +258.030 q^{97} -1377.95 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 56 q^{4} - 94 q^{7}+O(q^{10}) $ 18 * q + 56 * q^4 - 94 * q^7	$ 18 q + 56 q^{4} - 94 q^{7} + 156 q^{10} + 88 q^{16} - 448 q^{19} - 256 q^{22} - 16 q^{25} - 1300 q^{28} - 818 q^{31} - 900 q^{34} - 524 q^{37} + 800 q^{40} + 752 q^{43} - 1650 q^{46} + 2948 q^{49} - 464 q^{55} - 1626 q^{58} - 1784 q^{61} - 4274 q^{64} - 2872 q^{67} - 5772 q^{70} - 3078 q^{73} - 5726 q^{76} + 3326 q^{79} - 1038 q^{82} - 2340 q^{85} + 780 q^{88} + 1220 q^{94} - 4630 q^{97}+O(q^{100}) $ 18 * q + 56 * q^4 - 94 * q^7 + 156 * q^10 + 88 * q^16 - 448 * q^19 - 256 * q^22 - 16 * q^25 - 1300 * q^28 - 818 * q^31 - 900 * q^34 - 524 * q^37 + 800 * q^40 + 752 * q^43 - 1650 * q^46 + 2948 * q^49 - 464 * q^55 - 1626 * q^58 - 1784 * q^61 - 4274 * q^64 - 2872 * q^67 - 5772 * q^70 - 3078 * q^73 - 5726 * q^76 + 3326 * q^79 - 1038 * q^82 - 2340 * q^85 + 780 * q^88 + 1220 * q^94 - 4630 * 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 $

$2$	−3.45813	−1.22263	−0.611317	−	0.791386i	$-0.709360\pi$
$2$	−3.45813	−1.22263	−0.611317	+	0.791386i	$0.709360\pi$
$3$	0	0
$3$	0	0
$4$	3.95865	0.494832
$5$	−9.27918	−0.829955	−0.414978	−	0.909832i	$-0.636211\pi$
$5$	−9.27918	−0.829955	−0.414978	+	0.909832i	$0.636211\pi$
$6$	0	0
$7$	27.2299	1.47028	0.735139	−	0.677917i	$-0.237117\pi$
$7$	27.2299	1.47028	0.735139	+	0.677917i	$0.237117\pi$
$8$	13.9755	0.617635
$9$	0	0
$10$	32.0886	1.01473
$11$	38.3557	1.05133	0.525667	−	0.850691i	$-0.323816\pi$
$11$	38.3557	1.05133	0.525667	+	0.850691i	$0.323816\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−94.1645	−1.79761
$15$	0	0
$16$	−79.9983	−1.24997
$17$	85.0844	1.21388	0.606941	−	0.794747i	$-0.292396\pi$
$17$	85.0844	1.21388	0.606941	+	0.794747i	$0.292396\pi$
$18$	0	0
$19$	−114.360	−1.38084	−0.690419	−	0.723410i	$-0.742574\pi$
$19$	−114.360	−1.38084	−0.690419	+	0.723410i	$0.742574\pi$
$20$	−36.7331	−0.410688
$21$	0	0
$22$	−132.639	−1.28540
$23$	49.4232	0.448063	0.224032	−	0.974582i	$-0.428078\pi$
$23$	49.4232	0.448063	0.224032	+	0.974582i	$0.428078\pi$
$24$	0	0
$25$	−38.8968	−0.311174
$26$	0	0
$27$	0	0
$28$	107.794	0.727540
$29$	−11.5644	−0.0740504	−0.0370252	−	0.999314i	$-0.511788\pi$
$29$	−11.5644	−0.0740504	−0.0370252	+	0.999314i	$0.511788\pi$
$30$	0	0
$31$	−220.904	−1.27986	−0.639929	−	0.768434i	$-0.721036\pi$
$31$	−220.904	−1.27986	−0.639929	+	0.768434i	$0.721036\pi$
$32$	164.840	0.910623
$33$	0	0
$34$	−294.233	−1.48413
$35$	−252.671	−1.22026
$36$	0	0
$37$	−24.7406	−0.109928	−0.0549639	−	0.998488i	$-0.517504\pi$
$37$	−24.7406	−0.109928	−0.0549639	+	0.998488i	$0.517504\pi$
$38$	395.470	1.68826
$39$	0	0
$40$	−129.681	−0.512610
$41$	−30.8365	−0.117460	−0.0587300	−	0.998274i	$-0.518705\pi$
$41$	−30.8365	−0.117460	−0.0587300	+	0.998274i	$0.518705\pi$
$42$	0	0
$43$	−409.346	−1.45174	−0.725868	−	0.687834i	$-0.758562\pi$
$43$	−409.346	−1.45174	−0.725868	+	0.687834i	$0.758562\pi$
$44$	151.837	0.520233
$45$	0	0
$46$	−170.912	−0.547817
$47$	−434.519	−1.34854	−0.674268	−	0.738487i	$-0.735541\pi$
$47$	−434.519	−1.34854	−0.674268	+	0.738487i	$0.735541\pi$
$48$	0	0
$49$	398.468	1.16171
$50$	134.510	0.380452
$51$	0	0
$52$	0	0
$53$	716.136	1.85602	0.928008	−	0.372561i	$-0.121520\pi$
$53$	716.136	1.85602	0.928008	+	0.372561i	$0.121520\pi$
$54$	0	0
$55$	−355.909	−0.872560
$56$	380.552	0.908095
$57$	0	0
$58$	39.9913	0.0905365
$59$	−618.055	−1.36380	−0.681898	−	0.731448i	$-0.738845\pi$
$59$	−618.055	−1.36380	−0.681898	+	0.731448i	$0.738845\pi$
$60$	0	0
$61$	213.058	0.447202	0.223601	−	0.974681i	$-0.428219\pi$
$61$	213.058	0.447202	0.223601	+	0.974681i	$0.428219\pi$
$62$	763.915	1.56480
$63$	0	0
$64$	69.9470	0.136615
$65$	0	0
$66$	0	0
$67$	−578.504	−1.05486	−0.527429	−	0.849599i	$-0.676844\pi$
$67$	−578.504	−1.05486	−0.527429	+	0.849599i	$0.676844\pi$
$68$	336.820	0.600667
$69$	0	0
$70$	873.770	1.49194
$71$	−238.263	−0.398262	−0.199131	−	0.979973i	$-0.563812\pi$
$71$	−238.263	−0.398262	−0.199131	+	0.979973i	$0.563812\pi$
$72$	0	0
$73$	748.837	1.20061	0.600307	−	0.799770i	$-0.295045\pi$
$73$	748.837	1.20061	0.600307	+	0.799770i	$0.295045\pi$
$74$	85.5562	0.134401
$75$	0	0
$76$	−452.710	−0.683282
$77$	1044.42	1.54575
$78$	0	0
$79$	883.090	1.25766	0.628832	−	0.777541i	$-0.283533\pi$
$79$	883.090	1.25766	0.628832	+	0.777541i	$0.283533\pi$
$80$	742.319	1.03742
$81$	0	0
$82$	106.637	0.143610
$83$	−1401.67	−1.85365	−0.926827	−	0.375488i	$-0.877475\pi$
$83$	−1401.67	−1.85365	−0.926827	+	0.375488i	$0.877475\pi$
$84$	0	0
$85$	−789.513	−1.00747
$86$	1415.57	1.77494
$87$	0	0
$88$	536.040	0.649341
$89$	1301.10	1.54962	0.774810	−	0.632194i	$-0.217846\pi$
$89$	1301.10	1.54962	0.774810	+	0.632194i	$0.217846\pi$
$90$	0	0
$91$	0	0
$92$	195.649	0.221716
$93$	0	0
$94$	1502.62	1.64876
$95$	1061.16	1.14603
$96$	0	0
$97$	258.030	0.270093	0.135046	−	0.990839i	$-0.456882\pi$
$97$	258.030	0.270093	0.135046	+	0.990839i	$0.456882\pi$
$98$	−1377.95	−1.42035
$99$	0	0
$100$	−153.979	−0.153979
$101$	261.469	0.257595	0.128798	−	0.991671i	$-0.458888\pi$
$101$	261.469	0.257595	0.128798	+	0.991671i	$0.458888\pi$
$102$	0	0
$103$	−1088.23	−1.04104	−0.520518	−	0.853851i	$-0.674261\pi$
$103$	−1088.23	−1.04104	−0.520518	+	0.853851i	$0.674261\pi$
$104$	0	0
$105$	0	0
$106$	−2476.49	−2.26923
$107$	−618.466	−0.558779	−0.279390	−	0.960178i	$-0.590132\pi$
$107$	−618.466	−0.558779	−0.279390	+	0.960178i	$0.590132\pi$
$108$	0	0
$109$	474.867	0.417285	0.208642	−	0.977992i	$-0.433096\pi$
$109$	474.867	0.417285	0.208642	+	0.977992i	$0.433096\pi$
$110$	1230.78	1.06682
$111$	0	0
$112$	−2178.35	−1.83781
$113$	−1590.05	−1.32371	−0.661854	−	0.749633i	$-0.730230\pi$
$113$	−1590.05	−1.32371	−0.661854	+	0.749633i	$0.730230\pi$
$114$	0	0
$115$	−458.607	−0.371873
$116$	−45.7796	−0.0366425
$117$	0	0
$118$	2137.31	1.66742
$119$	2316.84	1.78474
$120$	0	0
$121$	140.158	0.105303
$122$	−736.783	−0.546764
$123$	0	0
$124$	−874.483	−0.633314
$125$	1520.83	1.08822
$126$	0	0
$127$	−2211.91	−1.54548	−0.772739	−	0.634724i	$-0.781114\pi$
$127$	−2211.91	−1.54548	−0.772739	+	0.634724i	$0.781114\pi$
$128$	−1560.61	−1.07765
$129$	0	0
$130$	0	0
$131$	2348.00	1.56600	0.782999	−	0.622023i	$-0.213689\pi$
$131$	2348.00	1.56600	0.782999	+	0.622023i	$0.213689\pi$
$132$	0	0
$133$	−3114.00	−2.03021
$134$	2000.54	1.28970
$135$	0	0
$136$	1189.10	0.749736
$137$	−2656.99	−1.65695	−0.828473	−	0.560029i	$-0.810790\pi$
$137$	−2656.99	−1.65695	−0.828473	+	0.560029i	$0.810790\pi$
$138$	0	0
$139$	−143.480	−0.0875525	−0.0437762	−	0.999041i	$-0.513939\pi$
$139$	−143.480	−0.0875525	−0.0437762	+	0.999041i	$0.513939\pi$
$140$	−1000.24	−0.603825
$141$	0	0
$142$	823.944	0.486928
$143$	0	0
$144$	0	0
$145$	107.309	0.0614585
$146$	−2589.58	−1.46791
$147$	0	0
$148$	−97.9395	−0.0543958
$149$	−566.289	−0.311357	−0.155679	−	0.987808i	$-0.549756\pi$
$149$	−566.289	−0.311357	−0.155679	+	0.987808i	$0.549756\pi$
$150$	0	0
$151$	−1781.62	−0.960175	−0.480087	−	0.877221i	$-0.659395\pi$
$151$	−1781.62	−0.960175	−0.480087	+	0.877221i	$0.659395\pi$
$152$	−1598.23	−0.852854
$153$	0	0
$154$	−3611.74	−1.88989
$155$	2049.81	1.06222
$156$	0	0
$157$	3203.84	1.62863	0.814314	−	0.580425i	$-0.197114\pi$
$157$	3203.84	1.62863	0.814314	+	0.580425i	$0.197114\pi$
$158$	−3053.84	−1.53766
$159$	0	0
$160$	−1529.58	−0.755777
$161$	1345.79	0.658777
$162$	0	0
$163$	1298.71	0.624069	0.312034	−	0.950071i	$-0.398990\pi$
$163$	1298.71	0.624069	0.312034	+	0.950071i	$0.398990\pi$
$164$	−122.071	−0.0581229
$165$	0	0
$166$	4847.16	2.26634
$167$	1058.26	0.490361	0.245181	−	0.969477i	$-0.421153\pi$
$167$	1058.26	0.490361	0.245181	+	0.969477i	$0.421153\pi$
$168$	0	0
$169$	0	0
$170$	2730.24	1.23176
$171$	0	0
$172$	−1620.46	−0.718365
$173$	−2757.94	−1.21204	−0.606019	−	0.795450i	$-0.707234\pi$
$173$	−2757.94	−1.21204	−0.606019	+	0.795450i	$0.707234\pi$
$174$	0	0
$175$	−1059.16	−0.457512
$176$	−3068.39	−1.31414
$177$	0	0
$178$	−4499.37	−1.89462
$179$	1989.50	0.830740	0.415370	−	0.909653i	$-0.363652\pi$
$179$	1989.50	0.830740	0.415370	+	0.909653i	$0.363652\pi$
$180$	0	0
$181$	129.804	0.0533053	0.0266527	−	0.999645i	$-0.491515\pi$
$181$	129.804	0.0533053	0.0266527	+	0.999645i	$0.491515\pi$
$182$	0	0
$183$	0	0
$184$	690.714	0.276740
$185$	229.573	0.0912352
$186$	0	0
$187$	3263.47	1.27619
$188$	−1720.11	−0.667298
$189$	0	0
$190$	−3669.64	−1.40118
$191$	2086.63	0.790487	0.395243	−	0.918576i	$-0.370660\pi$
$191$	2086.63	0.790487	0.395243	+	0.918576i	$0.370660\pi$
$192$	0	0
$193$	−4678.77	−1.74500	−0.872500	−	0.488614i	$-0.837503\pi$
$193$	−4678.77	−1.74500	−0.872500	+	0.488614i	$0.837503\pi$
$194$	−892.302	−0.330225
$195$	0	0
$196$	1577.40	0.574853
$197$	4448.12	1.60871	0.804354	−	0.594151i	$-0.202512\pi$
$197$	4448.12	1.60871	0.804354	+	0.594151i	$0.202512\pi$
$198$	0	0
$199$	−3707.75	−1.32078	−0.660391	−	0.750922i	$-0.729609\pi$
$199$	−3707.75	−1.32078	−0.660391	+	0.750922i	$0.729609\pi$
$200$	−543.602	−0.192192
$201$	0	0
$202$	−904.193	−0.314945
$203$	−314.899	−0.108875
$204$	0	0
$205$	286.138	0.0974865
$206$	3763.24	1.27280
$207$	0	0
$208$	0	0
$209$	−4386.34	−1.45172
$210$	0	0
$211$	886.156	0.289126	0.144563	−	0.989496i	$-0.453822\pi$
$211$	886.156	0.289126	0.144563	+	0.989496i	$0.453822\pi$
$212$	2834.93	0.918415
$213$	0	0
$214$	2138.73	0.683182
$215$	3798.40	1.20488
$216$	0	0
$217$	−6015.20	−1.88174
$218$	−1642.15	−0.510186
$219$	0	0
$220$	−1408.92	−0.431770
$221$	0	0
$222$	0	0
$223$	130.541	0.0392002	0.0196001	−	0.999808i	$-0.493761\pi$
$223$	130.541	0.0392002	0.0196001	+	0.999808i	$0.493761\pi$
$224$	4488.59	1.33887
$225$	0	0
$226$	5498.59	1.61841
$227$	−2573.36	−0.752424	−0.376212	−	0.926534i	$-0.622774\pi$
$227$	−2573.36	−0.752424	−0.376212	+	0.926534i	$0.622774\pi$
$228$	0	0
$229$	−1370.40	−0.395453	−0.197726	−	0.980257i	$-0.563356\pi$
$229$	−1370.40	−0.395453	−0.197726	+	0.980257i	$0.563356\pi$
$230$	1585.92	0.454664
$231$	0	0
$232$	−161.619	−0.0457362
$233$	3855.81	1.08413	0.542066	−	0.840336i	$-0.317642\pi$
$233$	3855.81	1.08413	0.542066	+	0.840336i	$0.317642\pi$
$234$	0	0
$235$	4031.98	1.11922
$236$	−2446.67	−0.674849
$237$	0	0
$238$	−8011.93	−2.18209
$239$	−713.811	−0.193191	−0.0965954	−	0.995324i	$-0.530795\pi$
$239$	−713.811	−0.193191	−0.0965954	+	0.995324i	$0.530795\pi$
$240$	0	0
$241$	−4456.92	−1.19127	−0.595634	−	0.803256i	$-0.703099\pi$
$241$	−4456.92	−1.19127	−0.595634	+	0.803256i	$0.703099\pi$
$242$	−484.683	−0.128746
$243$	0	0
$244$	843.424	0.221290
$245$	−3697.46	−0.964171
$246$	0	0
$247$	0	0
$248$	−3087.25	−0.790485
$249$	0	0
$250$	−5259.22	−1.33049
$251$	−1612.02	−0.405378	−0.202689	−	0.979243i	$-0.564968\pi$
$251$	−1612.02	−0.405378	−0.202689	+	0.979243i	$0.564968\pi$
$252$	0	0
$253$	1895.66	0.471064
$254$	7649.08	1.88955
$255$	0	0
$256$	4837.21	1.18096
$257$	649.846	0.157729	0.0788644	−	0.996885i	$-0.474871\pi$
$257$	649.846	0.157729	0.0788644	+	0.996885i	$0.474871\pi$
$258$	0	0
$259$	−673.684	−0.161624
$260$	0	0
$261$	0	0
$262$	−8119.69	−1.91464
$263$	−299.125	−0.0701325	−0.0350663	−	0.999385i	$-0.511164\pi$
$263$	−299.125	−0.0701325	−0.0350663	+	0.999385i	$0.511164\pi$
$264$	0	0
$265$	−6645.16	−1.54041
$266$	10768.6	2.48221
$267$	0	0
$268$	−2290.10	−0.521977
$269$	−1475.49	−0.334432	−0.167216	−	0.985920i	$-0.553478\pi$
$269$	−1475.49	−0.334432	−0.167216	+	0.985920i	$0.553478\pi$
$270$	0	0
$271$	−3385.03	−0.758767	−0.379383	−	0.925240i	$-0.623864\pi$
$271$	−3385.03	−0.758767	−0.379383	+	0.925240i	$0.623864\pi$
$272$	−6806.60	−1.51732
$273$	0	0
$274$	9188.20	2.02584
$275$	−1491.91	−0.327148
$276$	0	0
$277$	6899.32	1.49653	0.748267	−	0.663398i	$-0.230886\pi$
$277$	6899.32	1.49653	0.748267	+	0.663398i	$0.230886\pi$
$278$	496.171	0.107045
$279$	0	0
$280$	−3531.21	−0.753679
$281$	−7728.70	−1.64077	−0.820384	−	0.571813i	$-0.806240\pi$
$281$	−7728.70	−1.64077	−0.820384	+	0.571813i	$0.806240\pi$
$282$	0	0
$283$	4538.48	0.953303	0.476652	−	0.879092i	$-0.341850\pi$
$283$	4538.48	0.953303	0.476652	+	0.879092i	$0.341850\pi$
$284$	−943.200	−0.197073
$285$	0	0
$286$	0	0
$287$	−839.676	−0.172699
$288$	0	0
$289$	2326.35	0.473509
$290$	−371.087	−0.0751412
$291$	0	0
$292$	2964.39	0.594102
$293$	5575.74	1.11174	0.555868	−	0.831271i	$-0.312386\pi$
$293$	5575.74	1.11174	0.555868	+	0.831271i	$0.312386\pi$
$294$	0	0
$295$	5735.05	1.13189
$296$	−345.762	−0.0678953
$297$	0	0
$298$	1958.30	0.380676
$299$	0	0
$300$	0	0
$301$	−11146.5	−2.13446
$302$	6161.08	1.17394
$303$	0	0
$304$	9148.58	1.72601
$305$	−1977.01	−0.371157
$306$	0	0
$307$	−7026.63	−1.30629	−0.653145	−	0.757233i	$-0.726551\pi$
$307$	−7026.63	−1.30629	−0.653145	+	0.757233i	$0.726551\pi$
$308$	4134.50	0.764887
$309$	0	0
$310$	−7088.51	−1.29871
$311$	−10343.8	−1.88600	−0.942998	−	0.332799i	$-0.892007\pi$
$311$	−10343.8	−1.88600	−0.942998	+	0.332799i	$0.892007\pi$
$312$	0	0
$313$	−1491.82	−0.269402	−0.134701	−	0.990886i	$-0.543007\pi$
$313$	−1491.82	−0.269402	−0.134701	+	0.990886i	$0.543007\pi$
$314$	−11079.3	−1.99121
$315$	0	0
$316$	3495.85	0.622332
$317$	6071.14	1.07568	0.537838	−	0.843048i	$-0.319241\pi$
$317$	6071.14	1.07568	0.537838	+	0.843048i	$0.319241\pi$
$318$	0	0
$319$	−443.562	−0.0778517
$320$	−649.051	−0.113385
$321$	0	0
$322$	−4653.92	−0.805443
$323$	−9730.22	−1.67617
$324$	0	0
$325$	0	0
$326$	−4491.12	−0.763007
$327$	0	0
$328$	−430.956	−0.0725474
$329$	−11831.9	−1.98272
$330$	0	0
$331$	−10627.8	−1.76483	−0.882414	−	0.470474i	$-0.844083\pi$
$331$	−10627.8	−1.76483	−0.882414	+	0.470474i	$0.844083\pi$
$332$	−5548.73	−0.917247
$333$	0	0
$334$	−3659.58	−0.599532
$335$	5368.04	0.875485
$336$	0	0
$337$	−8774.58	−1.41834	−0.709172	−	0.705035i	$-0.750931\pi$
$337$	−8774.58	−1.41834	−0.709172	+	0.705035i	$0.750931\pi$
$338$	0	0
$339$	0	0
$340$	−3125.41	−0.498527
$341$	−8472.93	−1.34556
$342$	0	0
$343$	1510.39	0.237766
$344$	−5720.81	−0.896644
$345$	0	0
$346$	9537.32	1.48188
$347$	−2291.37	−0.354488	−0.177244	−	0.984167i	$-0.556718\pi$
$347$	−2291.37	−0.354488	−0.177244	+	0.984167i	$0.556718\pi$
$348$	0	0
$349$	5902.75	0.905350	0.452675	−	0.891676i	$-0.350470\pi$
$349$	5902.75	0.905350	0.452675	+	0.891676i	$0.350470\pi$
$350$	3662.70	0.559370
$351$	0	0
$352$	6322.57	0.957369
$353$	−3478.03	−0.524410	−0.262205	−	0.965012i	$-0.584450\pi$
$353$	−3478.03	−0.524410	−0.262205	+	0.965012i	$0.584450\pi$
$354$	0	0
$355$	2210.88	0.330540
$356$	5150.60	0.766801
$357$	0	0
$358$	−6879.96	−1.01569
$359$	−6464.23	−0.950332	−0.475166	−	0.879896i	$-0.657612\pi$
$359$	−6464.23	−0.950332	−0.475166	+	0.879896i	$0.657612\pi$
$360$	0	0
$361$	6219.13	0.906711
$362$	−448.879	−0.0651728
$363$	0	0
$364$	0	0
$365$	−6948.60	−0.996456
$366$	0	0
$367$	1351.60	0.192242	0.0961210	−	0.995370i	$-0.469356\pi$
$367$	1351.60	0.192242	0.0961210	+	0.995370i	$0.469356\pi$
$368$	−3953.77	−0.560067
$369$	0	0
$370$	−793.891	−0.111547
$371$	19500.3	2.72886
$372$	0	0
$373$	−3306.27	−0.458961	−0.229480	−	0.973313i	$-0.573703\pi$
$373$	−3306.27	−0.458961	−0.229480	+	0.973313i	$0.573703\pi$
$374$	−11285.5	−1.56032
$375$	0	0
$376$	−6072.62	−0.832903
$377$	0	0
$378$	0	0
$379$	−3978.58	−0.539225	−0.269612	−	0.962969i	$-0.586896\pi$
$379$	−3978.58	−0.539225	−0.269612	+	0.962969i	$0.586896\pi$
$380$	4200.78	0.567093
$381$	0	0
$382$	−7215.82	−0.966476
$383$	−8149.13	−1.08721	−0.543605	−	0.839341i	$-0.682941\pi$
$383$	−8149.13	−1.08721	−0.543605	+	0.839341i	$0.682941\pi$
$384$	0	0
$385$	−9691.38	−1.28290
$386$	16179.8	2.13349
$387$	0	0
$388$	1021.45	0.133651
$389$	−8791.24	−1.14584	−0.572922	−	0.819610i	$-0.694190\pi$
$389$	−8791.24	−1.14584	−0.572922	+	0.819610i	$0.694190\pi$
$390$	0	0
$391$	4205.14	0.543896
$392$	5568.79	0.717516
$393$	0	0
$394$	−15382.2	−1.96686
$395$	−8194.36	−1.04380
$396$	0	0
$397$	5764.94	0.728801	0.364401	−	0.931242i	$-0.381274\pi$
$397$	5764.94	0.728801	0.364401	+	0.931242i	$0.381274\pi$
$398$	12821.9	1.61483
$399$	0	0
$400$	3111.68	0.388960
$401$	13107.5	1.63232	0.816158	−	0.577829i	$-0.196100\pi$
$401$	13107.5	1.63232	0.816158	+	0.577829i	$0.196100\pi$
$402$	0	0
$403$	0	0
$404$	1035.07	0.127466
$405$	0	0
$406$	1088.96	0.133114
$407$	−948.942	−0.115571
$408$	0	0
$409$	11160.2	1.34923	0.674614	−	0.738171i	$-0.264310\pi$
$409$	11160.2	1.34923	0.674614	+	0.738171i	$0.264310\pi$
$410$	−989.501	−0.119190
$411$	0	0
$412$	−4307.93	−0.515137
$413$	−16829.6	−2.00516
$414$	0	0
$415$	13006.4	1.53845
$416$	0	0
$417$	0	0
$418$	15168.5	1.77492
$419$	−4322.04	−0.503927	−0.251963	−	0.967737i	$-0.581076\pi$
$419$	−4322.04	−0.503927	−0.251963	+	0.967737i	$0.581076\pi$
$420$	0	0
$421$	3832.29	0.443645	0.221822	−	0.975087i	$-0.428799\pi$
$421$	3832.29	0.443645	0.221822	+	0.975087i	$0.428799\pi$
$422$	−3064.44	−0.353494
$423$	0	0
$424$	10008.4	1.14634
$425$	−3309.51	−0.377729
$426$	0	0
$427$	5801.56	0.657511
$428$	−2448.29	−0.276502
$429$	0	0
$430$	−13135.3	−1.47312
$431$	14043.1	1.56945	0.784725	−	0.619844i	$-0.212804\pi$
$431$	14043.1	1.56945	0.784725	+	0.619844i	$0.212804\pi$
$432$	0	0
$433$	−1687.12	−0.187247	−0.0936234	−	0.995608i	$-0.529845\pi$
$433$	−1687.12	−0.187247	−0.0936234	+	0.995608i	$0.529845\pi$
$434$	20801.3	2.30068
$435$	0	0
$436$	1879.84	0.206486
$437$	−5652.02	−0.618702
$438$	0	0
$439$	306.227	0.0332926	0.0166463	−	0.999861i	$-0.494701\pi$
$439$	306.227	0.0332926	0.0166463	+	0.999861i	$0.494701\pi$
$440$	−4974.01	−0.538924
$441$	0	0
$442$	0	0
$443$	−9036.50	−0.969158	−0.484579	−	0.874747i	$-0.661027\pi$
$443$	−9036.50	−0.969158	−0.484579	+	0.874747i	$0.661027\pi$
$444$	0	0
$445$	−12073.1	−1.28611
$446$	−451.427	−0.0479275
$447$	0	0
$448$	1904.65	0.200862
$449$	−2142.20	−0.225160	−0.112580	−	0.993643i	$-0.535911\pi$
$449$	−2142.20	−0.225160	−0.112580	+	0.993643i	$0.535911\pi$
$450$	0	0
$451$	−1182.76	−0.123490
$452$	−6294.44	−0.655013
$453$	0	0
$454$	8899.03	0.919938
$455$	0	0
$456$	0	0
$457$	8727.96	0.893385	0.446692	−	0.894688i	$-0.352602\pi$
$457$	8727.96	0.893385	0.446692	+	0.894688i	$0.352602\pi$
$458$	4739.02	0.483493
$459$	0	0
$460$	−1815.47	−0.184014
$461$	118.371	0.0119590	0.00597948	−	0.999982i	$-0.498097\pi$
$461$	118.371	0.0119590	0.00597948	+	0.999982i	$0.498097\pi$
$462$	0	0
$463$	−11738.4	−1.17825	−0.589127	−	0.808041i	$-0.700528\pi$
$463$	−11738.4	−1.17825	−0.589127	+	0.808041i	$0.700528\pi$
$464$	925.135	0.0925611
$465$	0	0
$466$	−13333.9	−1.32550
$467$	4327.25	0.428782	0.214391	−	0.976748i	$-0.431223\pi$
$467$	4327.25	0.428782	0.214391	+	0.976748i	$0.431223\pi$
$468$	0	0
$469$	−15752.6	−1.55093
$470$	−13943.1	−1.36840
$471$	0	0
$472$	−8637.63	−0.842328
$473$	−15700.7	−1.52626
$474$	0	0
$475$	4448.22	0.429681
$476$	9171.57	0.883147
$477$	0	0
$478$	2468.45	0.236201
$479$	−16087.9	−1.53460	−0.767302	−	0.641286i	$-0.778401\pi$
$479$	−16087.9	−1.53460	−0.767302	+	0.641286i	$0.778401\pi$
$480$	0	0
$481$	0	0
$482$	15412.6	1.45648
$483$	0	0
$484$	554.836	0.0521071
$485$	−2394.31	−0.224165
$486$	0	0
$487$	−15331.7	−1.42658	−0.713291	−	0.700868i	$-0.752796\pi$
$487$	−15331.7	−1.42658	−0.713291	+	0.700868i	$0.752796\pi$
$488$	2977.59	0.276208
$489$	0	0
$490$	12786.3	1.17883
$491$	8060.00	0.740820	0.370410	−	0.928868i	$-0.379217\pi$
$491$	8060.00	0.740820	0.370410	+	0.928868i	$0.379217\pi$
$492$	0	0
$493$	−983.953	−0.0898885
$494$	0	0
$495$	0	0
$496$	17672.0	1.59979
$497$	−6487.88	−0.585555
$498$	0	0
$499$	−8448.26	−0.757908	−0.378954	−	0.925416i	$-0.623716\pi$
$499$	−8448.26	−0.757908	−0.378954	+	0.925416i	$0.623716\pi$
$500$	6020.43	0.538484
$501$	0	0
$502$	5574.58	0.495629
$503$	−11093.9	−0.983406	−0.491703	−	0.870763i	$-0.663625\pi$
$503$	−11093.9	−0.983406	−0.491703	+	0.870763i	$0.663625\pi$
$504$	0	0
$505$	−2426.22	−0.213793
$506$	−6555.44	−0.575939
$507$	0	0
$508$	−8756.20	−0.764751
$509$	1935.86	0.168576	0.0842882	−	0.996441i	$-0.473138\pi$
$509$	1935.86	0.168576	0.0842882	+	0.996441i	$0.473138\pi$
$510$	0	0
$511$	20390.8	1.76523
$512$	−4242.82	−0.366227
$513$	0	0
$514$	−2247.25	−0.192844
$515$	10097.9	0.864013
$516$	0	0
$517$	−16666.3	−1.41776
$518$	2329.69	0.197607
$519$	0	0
$520$	0	0
$521$	−1520.44	−0.127853	−0.0639266	−	0.997955i	$-0.520362\pi$
$521$	−1520.44	−0.127853	−0.0639266	+	0.997955i	$0.520362\pi$
$522$	0	0
$523$	3592.38	0.300351	0.150176	−	0.988659i	$-0.452016\pi$
$523$	3592.38	0.300351	0.150176	+	0.988659i	$0.452016\pi$
$524$	9294.92	0.774906
$525$	0	0
$526$	1034.41	0.0857464
$527$	−18795.5	−1.55360
$528$	0	0
$529$	−9724.34	−0.799239
$530$	22979.8	1.88336
$531$	0	0
$532$	−12327.3	−1.00461
$533$	0	0
$534$	0	0
$535$	5738.86	0.463762
$536$	−8084.88	−0.651518
$537$	0	0
$538$	5102.43	0.408888
$539$	15283.5	1.22135
$540$	0	0
$541$	18029.5	1.43281	0.716405	−	0.697685i	$-0.245786\pi$
$541$	18029.5	1.43281	0.716405	+	0.697685i	$0.245786\pi$
$542$	11705.9	0.927693
$543$	0	0
$544$	14025.3	1.10539
$545$	−4406.38	−0.346328
$546$	0	0
$547$	19024.1	1.48704	0.743522	−	0.668712i	$-0.233154\pi$
$547$	19024.1	1.48704	0.743522	+	0.668712i	$0.233154\pi$
$548$	−10518.1	−0.819909
$549$	0	0
$550$	5159.22	0.399982
$551$	1322.51	0.102252
$552$	0	0
$553$	24046.5	1.84911
$554$	−23858.7	−1.82971
$555$	0	0
$556$	−567.987	−0.0433237
$557$	8681.36	0.660397	0.330198	−	0.943912i	$-0.392884\pi$
$557$	8681.36	0.660397	0.330198	+	0.943912i	$0.392884\pi$
$558$	0	0
$559$	0	0
$560$	20213.3	1.52530
$561$	0	0
$562$	26726.9	2.00606
$563$	−14357.6	−1.07478	−0.537389	−	0.843335i	$-0.680589\pi$
$563$	−14357.6	−1.07478	−0.537389	+	0.843335i	$0.680589\pi$
$564$	0	0
$565$	14754.3	1.09862
$566$	−15694.7	−1.16554
$567$	0	0
$568$	−3329.84	−0.245981
$569$	20395.7	1.50269	0.751346	−	0.659909i	$-0.229405\pi$
$569$	20395.7	1.50269	0.751346	+	0.659909i	$0.229405\pi$
$570$	0	0
$571$	11216.8	0.822079	0.411039	−	0.911618i	$-0.365166\pi$
$571$	11216.8	0.822079	0.411039	+	0.911618i	$0.365166\pi$
$572$	0	0
$573$	0	0
$574$	2903.71	0.211147
$575$	−1922.40	−0.139426
$576$	0	0
$577$	−21270.1	−1.53464	−0.767318	−	0.641267i	$-0.778409\pi$
$577$	−21270.1	−1.53464	−0.767318	+	0.641267i	$0.778409\pi$
$578$	−8044.81	−0.578927
$579$	0	0
$580$	424.797	0.0304116
$581$	−38167.4	−2.72539
$582$	0	0
$583$	27467.9	1.95129
$584$	10465.4	0.741542
$585$	0	0
$586$	−19281.6	−1.35924
$587$	15987.3	1.12414	0.562068	−	0.827091i	$-0.310006\pi$
$587$	15987.3	1.12414	0.562068	+	0.827091i	$0.310006\pi$
$588$	0	0
$589$	25262.5	1.76727
$590$	−19832.5	−1.38388
$591$	0	0
$592$	1979.21	0.137407
$593$	3936.60	0.272608	0.136304	−	0.990667i	$-0.456478\pi$
$593$	3936.60	0.272608	0.136304	+	0.990667i	$0.456478\pi$
$594$	0	0
$595$	−21498.4	−1.48126
$596$	−2241.74	−0.154069
$597$	0	0
$598$	0	0
$599$	−6235.60	−0.425341	−0.212671	−	0.977124i	$-0.568216\pi$
$599$	−6235.60	−0.425341	−0.212671	+	0.977124i	$0.568216\pi$
$600$	0	0
$601$	−8763.86	−0.594817	−0.297409	−	0.954750i	$-0.596122\pi$
$601$	−8763.86	−0.594817	−0.297409	+	0.954750i	$0.596122\pi$
$602$	38545.9	2.60966
$603$	0	0
$604$	−7052.83	−0.475125
$605$	−1300.55	−0.0873964
$606$	0	0
$607$	6395.57	0.427658	0.213829	−	0.976871i	$-0.431407\pi$
$607$	6395.57	0.427658	0.213829	+	0.976871i	$0.431407\pi$
$608$	−18851.1	−1.25742
$609$	0	0
$610$	6836.74	0.453789
$611$	0	0
$612$	0	0
$613$	−28911.8	−1.90495	−0.952476	−	0.304614i	$-0.901472\pi$
$613$	−28911.8	−1.90495	−0.952476	+	0.304614i	$0.901472\pi$
$614$	24299.0	1.59711
$615$	0	0
$616$	14596.3	0.954711
$617$	7201.88	0.469913	0.234957	−	0.972006i	$-0.424505\pi$
$617$	7201.88	0.469913	0.234957	+	0.972006i	$0.424505\pi$
$618$	0	0
$619$	5577.21	0.362144	0.181072	−	0.983470i	$-0.442043\pi$
$619$	5577.21	0.362144	0.181072	+	0.983470i	$0.442043\pi$
$620$	8114.49	0.525622
$621$	0	0
$622$	35770.3	2.30588
$623$	35428.8	2.27837
$624$	0	0
$625$	−9249.94	−0.591996
$626$	5158.92	0.329380
$627$	0	0
$628$	12682.9	0.805896
$629$	−2105.04	−0.133439
$630$	0	0
$631$	12199.0	0.769625	0.384812	−	0.922995i	$-0.374266\pi$
$631$	12199.0	0.769625	0.384812	+	0.922995i	$0.374266\pi$
$632$	12341.6	0.776778
$633$	0	0
$634$	−20994.8	−1.31516
$635$	20524.8	1.28268
$636$	0	0
$637$	0	0
$638$	1533.89	0.0951841
$639$	0	0
$640$	14481.2	0.894404
$641$	23744.1	1.46308	0.731541	−	0.681797i	$-0.238801\pi$
$641$	23744.1	1.46308	0.731541	+	0.681797i	$0.238801\pi$
$642$	0	0
$643$	−12818.4	−0.786174	−0.393087	−	0.919501i	$-0.628593\pi$
$643$	−12818.4	−0.786174	−0.393087	+	0.919501i	$0.628593\pi$
$644$	5327.52	0.325984
$645$	0	0
$646$	33648.3	2.04934
$647$	17693.3	1.07511	0.537554	−	0.843229i	$-0.319348\pi$
$647$	17693.3	1.07511	0.537554	+	0.843229i	$0.319348\pi$
$648$	0	0
$649$	−23705.9	−1.43380
$650$	0	0
$651$	0	0
$652$	5141.16	0.308809
$653$	13264.9	0.794938	0.397469	−	0.917616i	$-0.369889\pi$
$653$	13264.9	0.794938	0.397469	+	0.917616i	$0.369889\pi$
$654$	0	0
$655$	−21787.5	−1.29971
$656$	2466.87	0.146822
$657$	0	0
$658$	40916.3	2.42414
$659$	21758.7	1.28619	0.643094	−	0.765787i	$-0.277651\pi$
$659$	21758.7	1.28619	0.643094	+	0.765787i	$0.277651\pi$
$660$	0	0
$661$	−24372.3	−1.43415	−0.717073	−	0.696998i	$-0.754519\pi$
$661$	−24372.3	−1.43415	−0.717073	+	0.696998i	$0.754519\pi$
$662$	36752.4	2.15774
$663$	0	0
$664$	−19589.0	−1.14488
$665$	28895.4	1.68499
$666$	0	0
$667$	−571.552	−0.0331793
$668$	4189.27	0.242646
$669$	0	0
$670$	−18563.4	−1.07040
$671$	8171.99	0.470158
$672$	0	0
$673$	−3842.50	−0.220085	−0.110043	−	0.993927i	$-0.535099\pi$
$673$	−3842.50	−0.220085	−0.110043	+	0.993927i	$0.535099\pi$
$674$	30343.6	1.73411
$675$	0	0
$676$	0	0
$677$	9373.85	0.532151	0.266076	−	0.963952i	$-0.414273\pi$
$677$	9373.85	0.532151	0.266076	+	0.963952i	$0.414273\pi$
$678$	0	0
$679$	7026.14	0.397111
$680$	−11033.8	−0.622248
$681$	0	0
$682$	29300.5	1.64512
$683$	−12382.7	−0.693718	−0.346859	−	0.937917i	$-0.612752\pi$
$683$	−12382.7	−0.693718	−0.346859	+	0.937917i	$0.612752\pi$
$684$	0	0
$685$	24654.6	1.37519
$686$	−5223.14	−0.290700
$687$	0	0
$688$	32747.0	1.81463
$689$	0	0
$690$	0	0
$691$	32829.2	1.80736	0.903678	−	0.428213i	$-0.140857\pi$
$691$	32829.2	1.80736	0.903678	+	0.428213i	$0.140857\pi$
$692$	−10917.7	−0.599755
$693$	0	0
$694$	7923.86	0.433409
$695$	1331.37	0.0726646
$696$	0	0
$697$	−2623.71	−0.142582
$698$	−20412.5	−1.10691
$699$	0	0
$700$	−4192.83	−0.226392
$701$	5575.51	0.300405	0.150203	−	0.988655i	$-0.452007\pi$
$701$	5575.51	0.300405	0.150203	+	0.988655i	$0.452007\pi$
$702$	0	0
$703$	2829.33	0.151792
$704$	2682.86	0.143628
$705$	0	0
$706$	12027.5	0.641161
$707$	7119.78	0.378737
$708$	0	0
$709$	−972.214	−0.0514982	−0.0257491	−	0.999668i	$-0.508197\pi$
$709$	−972.214	−0.0514982	−0.0257491	+	0.999668i	$0.508197\pi$
$710$	−7645.52	−0.404129
$711$	0	0
$712$	18183.5	0.957100
$713$	−10917.8	−0.573457
$714$	0	0
$715$	0	0
$716$	7875.75	0.411076
$717$	0	0
$718$	22354.2	1.16191
$719$	18512.8	0.960238	0.480119	−	0.877203i	$-0.340593\pi$
$719$	18512.8	0.960238	0.480119	+	0.877203i	$0.340593\pi$
$720$	0	0
$721$	−29632.4	−1.53061
$722$	−21506.6	−1.10858
$723$	0	0
$724$	513.849	0.0263772
$725$	449.819	0.0230426
$726$	0	0
$727$	−17690.7	−0.902491	−0.451246	−	0.892400i	$-0.649020\pi$
$727$	−17690.7	−0.902491	−0.451246	+	0.892400i	$0.649020\pi$
$728$	0	0
$729$	0	0
$730$	24029.1	1.21830
$731$	−34828.9	−1.76224
$732$	0	0
$733$	−9237.54	−0.465479	−0.232740	−	0.972539i	$-0.574769\pi$
$733$	−9237.54	−0.465479	−0.232740	+	0.972539i	$0.574769\pi$
$734$	−4674.00	−0.235042
$735$	0	0
$736$	8146.95	0.408017
$737$	−22188.9	−1.10901
$738$	0	0
$739$	12911.2	0.642686	0.321343	−	0.946963i	$-0.395866\pi$
$739$	12911.2	0.642686	0.321343	+	0.946963i	$0.395866\pi$
$740$	908.798	0.0451461
$741$	0	0
$742$	−67434.6	−3.33639
$743$	−8257.27	−0.407712	−0.203856	−	0.979001i	$-0.565347\pi$
$743$	−8257.27	−0.407712	−0.203856	+	0.979001i	$0.565347\pi$
$744$	0	0
$745$	5254.70	0.258412
$746$	11433.5	0.561140
$747$	0	0
$748$	12918.9	0.631502
$749$	−16840.8	−0.821560
$750$	0	0
$751$	−8283.88	−0.402508	−0.201254	−	0.979539i	$-0.564502\pi$
$751$	−8283.88	−0.402508	−0.201254	+	0.979539i	$0.564502\pi$
$752$	34760.8	1.68563
$753$	0	0
$754$	0	0
$755$	16532.0	0.796902
$756$	0	0
$757$	−8293.52	−0.398194	−0.199097	−	0.979980i	$-0.563801\pi$
$757$	−8293.52	−0.398194	−0.199097	+	0.979980i	$0.563801\pi$
$758$	13758.5	0.659274
$759$	0	0
$760$	14830.3	0.707831
$761$	−33681.1	−1.60439	−0.802194	−	0.597063i	$-0.796334\pi$
$761$	−33681.1	−1.60439	−0.802194	+	0.597063i	$0.796334\pi$
$762$	0	0
$763$	12930.6	0.613524
$764$	8260.23	0.391158
$765$	0	0
$766$	28180.7	1.32926
$767$	0	0
$768$	0	0
$769$	7179.40	0.336666	0.168333	−	0.985730i	$-0.446162\pi$
$769$	7179.40	0.336666	0.168333	+	0.985730i	$0.446162\pi$
$770$	33514.0	1.56852
$771$	0	0
$772$	−18521.6	−0.863481
$773$	−6454.17	−0.300311	−0.150156	−	0.988662i	$-0.547977\pi$
$773$	−6454.17	−0.300311	−0.150156	+	0.988662i	$0.547977\pi$
$774$	0	0
$775$	8592.46	0.398259
$776$	3606.10	0.166819
$777$	0	0
$778$	30401.2	1.40095
$779$	3526.45	0.162193
$780$	0	0
$781$	−9138.73	−0.418706
$782$	−14541.9	−0.664985
$783$	0	0
$784$	−31876.8	−1.45211
$785$	−29729.0	−1.35169
$786$	0	0
$787$	−30111.7	−1.36387	−0.681936	−	0.731412i	$-0.738862\pi$
$787$	−30111.7	−1.36387	−0.681936	+	0.731412i	$0.738862\pi$
$788$	17608.6	0.796039
$789$	0	0
$790$	28337.1	1.27619
$791$	−43296.8	−1.94622
$792$	0	0
$793$	0	0
$794$	−19935.9	−0.891057
$795$	0	0
$796$	−14677.7	−0.653565
$797$	−30078.3	−1.33680	−0.668398	−	0.743803i	$-0.733020\pi$
$797$	−30078.3	−1.33680	−0.668398	+	0.743803i	$0.733020\pi$
$798$	0	0
$799$	−36970.8	−1.63696
$800$	−6411.76	−0.283363
$801$	0	0
$802$	−45327.5	−1.99572
$803$	28722.2	1.26225
$804$	0	0
$805$	−12487.8	−0.546756
$806$	0	0
$807$	0	0
$808$	3654.16	0.159100
$809$	−22965.9	−0.998071	−0.499036	−	0.866581i	$-0.666312\pi$
$809$	−22965.9	−0.998071	−0.499036	+	0.866581i	$0.666312\pi$
$810$	0	0
$811$	−5344.98	−0.231428	−0.115714	−	0.993283i	$-0.536916\pi$
$811$	−5344.98	−0.231428	−0.115714	+	0.993283i	$0.536916\pi$
$812$	−1246.57	−0.0538746
$813$	0	0
$814$	3281.56	0.141301
$815$	−12051.0	−0.517949
$816$	0	0
$817$	46812.7	2.00461
$818$	−38593.3	−1.64961
$819$	0	0
$820$	1132.72	0.0482394
$821$	−18197.0	−0.773546	−0.386773	−	0.922175i	$-0.626410\pi$
$821$	−18197.0	−0.773546	−0.386773	+	0.922175i	$0.626410\pi$
$822$	0	0
$823$	−1437.68	−0.0608921	−0.0304461	−	0.999536i	$-0.509693\pi$
$823$	−1437.68	−0.0608921	−0.0304461	+	0.999536i	$0.509693\pi$
$824$	−15208.6	−0.642980
$825$	0	0
$826$	58198.9	2.45157
$827$	16163.8	0.679651	0.339826	−	0.940488i	$-0.389632\pi$
$827$	16163.8	0.679651	0.339826	+	0.940488i	$0.389632\pi$
$828$	0	0
$829$	8071.82	0.338174	0.169087	−	0.985601i	$-0.445918\pi$
$829$	8071.82	0.338174	0.169087	+	0.985601i	$0.445918\pi$
$830$	−44977.6	−1.88096
$831$	0	0
$832$	0	0
$833$	33903.4	1.41018
$834$	0	0
$835$	−9819.75	−0.406978
$836$	−17364.0	−0.718357
$837$	0	0
$838$	14946.2	0.616117
$839$	4068.42	0.167410	0.0837052	−	0.996491i	$-0.473325\pi$
$839$	4068.42	0.167410	0.0837052	+	0.996491i	$0.473325\pi$
$840$	0	0
$841$	−24255.3	−0.994517
$842$	−13252.6	−0.542415
$843$	0	0
$844$	3507.98	0.143068
$845$	0	0
$846$	0	0
$847$	3816.48	0.154824
$848$	−57289.7	−2.31997
$849$	0	0
$850$	11444.7	0.461824
$851$	−1222.76	−0.0492546
$852$	0	0
$853$	13940.8	0.559582	0.279791	−	0.960061i	$-0.409735\pi$
$853$	13940.8	0.559582	0.279791	+	0.960061i	$0.409735\pi$
$854$	−20062.5	−0.803894
$855$	0	0
$856$	−8643.37	−0.345122
$857$	10966.7	0.437123	0.218561	−	0.975823i	$-0.429864\pi$
$857$	10966.7	0.437123	0.218561	+	0.975823i	$0.429864\pi$
$858$	0	0
$859$	−15139.8	−0.601355	−0.300678	−	0.953726i	$-0.597213\pi$
$859$	−15139.8	−0.601355	−0.300678	+	0.953726i	$0.597213\pi$
$860$	15036.5	0.596211
$861$	0	0
$862$	−48562.9	−1.91886
$863$	32822.5	1.29466	0.647330	−	0.762210i	$-0.275886\pi$
$863$	32822.5	1.29466	0.647330	+	0.762210i	$0.275886\pi$
$864$	0	0
$865$	25591.5	1.00594
$866$	5834.28	0.228934
$867$	0	0
$868$	−23812.1	−0.931147
$869$	33871.5	1.32222
$870$	0	0
$871$	0	0
$872$	6636.51	0.257730
$873$	0	0
$874$	19545.4	0.756446
$875$	41412.0	1.59998
$876$	0	0
$877$	−28497.8	−1.09727	−0.548634	−	0.836063i	$-0.684852\pi$
$877$	−28497.8	−1.09727	−0.548634	+	0.836063i	$0.684852\pi$
$878$	−1058.97	−0.0407046
$879$	0	0
$880$	28472.1	1.09068
$881$	−29945.2	−1.14515	−0.572576	−	0.819852i	$-0.694056\pi$
$881$	−29945.2	−1.14515	−0.572576	+	0.819852i	$0.694056\pi$
$882$	0	0
$883$	−28435.0	−1.08371	−0.541855	−	0.840472i	$-0.682278\pi$
$883$	−28435.0	−1.08371	−0.541855	+	0.840472i	$0.682278\pi$
$884$	0	0
$885$	0	0
$886$	31249.4	1.18493
$887$	19869.2	0.752133	0.376067	−	0.926593i	$-0.377276\pi$
$887$	19869.2	0.752133	0.376067	+	0.926593i	$0.377276\pi$
$888$	0	0
$889$	−60230.2	−2.27228
$890$	41750.4	1.57245
$891$	0	0
$892$	516.765	0.0193975
$893$	49691.5	1.86211
$894$	0	0
$895$	−18461.0	−0.689477
$896$	−42495.2	−1.58445
$897$	0	0
$898$	7408.02	0.275288
$899$	2554.63	0.0947740
$900$	0	0
$901$	60932.0	2.25298
$902$	4090.12	0.150982
$903$	0	0
$904$	−22221.7	−0.817569
$905$	−1204.48	−0.0442410
$906$	0	0
$907$	−12973.9	−0.474963	−0.237481	−	0.971392i	$-0.576322\pi$
$907$	−12973.9	−0.474963	−0.237481	+	0.971392i	$0.576322\pi$
$908$	−10187.1	−0.372323
$909$	0	0
$910$	0	0
$911$	−34595.4	−1.25817	−0.629087	−	0.777335i	$-0.716571\pi$
$911$	−34595.4	−1.25817	−0.629087	+	0.777335i	$0.716571\pi$
$912$	0	0
$913$	−53762.0	−1.94881
$914$	−30182.4	−1.09228
$915$	0	0
$916$	−5424.94	−0.195682
$917$	63935.9	2.30245
$918$	0	0
$919$	−28458.7	−1.02151	−0.510754	−	0.859727i	$-0.670634\pi$
$919$	−28458.7	−1.02151	−0.510754	+	0.859727i	$0.670634\pi$
$920$	−6409.26	−0.229682
$921$	0	0
$922$	−409.342	−0.0146214
$923$	0	0
$924$	0	0
$925$	962.330	0.0342067
$926$	40593.0	1.44057
$927$	0	0
$928$	−1906.29	−0.0674320
$929$	−18360.5	−0.648426	−0.324213	−	0.945984i	$-0.605100\pi$
$929$	−18360.5	−0.648426	−0.324213	+	0.945984i	$0.605100\pi$
$930$	0	0
$931$	−45568.7	−1.60414
$932$	15263.8	0.536463
$933$	0	0
$934$	−14964.2	−0.524243
$935$	−30282.3	−1.05918
$936$	0	0
$937$	11470.9	0.399933	0.199966	−	0.979803i	$-0.435917\pi$
$937$	11470.9	0.399933	0.199966	+	0.979803i	$0.435917\pi$
$938$	54474.6	1.89622
$939$	0	0
$940$	15961.2	0.553827
$941$	−32739.5	−1.13419	−0.567097	−	0.823651i	$-0.691934\pi$
$941$	−32739.5	−1.13419	−0.567097	+	0.823651i	$0.691934\pi$
$942$	0	0
$943$	−1524.04	−0.0526295
$944$	49443.3	1.70471
$945$	0	0
$946$	54295.2	1.86606
$947$	−5369.11	−0.184237	−0.0921186	−	0.995748i	$-0.529364\pi$
$947$	−5369.11	−0.184237	−0.0921186	+	0.995748i	$0.529364\pi$
$948$	0	0
$949$	0	0
$950$	−15382.5	−0.525342
$951$	0	0
$952$	32379.0	1.10232
$953$	−11553.1	−0.392698	−0.196349	−	0.980534i	$-0.562908\pi$
$953$	−11553.1	−0.392698	−0.196349	+	0.980534i	$0.562908\pi$
$954$	0	0
$955$	−19362.2	−0.656069
$956$	−2825.73	−0.0955969
$957$	0	0
$958$	55634.1	1.87626
$959$	−72349.5	−2.43617
$960$	0	0
$961$	19007.7	0.638034
$962$	0	0
$963$	0	0
$964$	−17643.4	−0.589477
$965$	43415.1	1.44827
$966$	0	0
$967$	16826.0	0.559554	0.279777	−	0.960065i	$-0.409739\pi$
$967$	16826.0	0.559554	0.279777	+	0.960065i	$0.409739\pi$
$968$	1958.77	0.0650386
$969$	0	0
$970$	8279.83	0.274072
$971$	12031.9	0.397653	0.198827	−	0.980035i	$-0.436287\pi$
$971$	12031.9	0.397653	0.198827	+	0.980035i	$0.436287\pi$
$972$	0	0
$973$	−3906.94	−0.128726
$974$	53019.0	1.74419
$975$	0	0
$976$	−17044.3	−0.558990
$977$	−19509.0	−0.638842	−0.319421	−	0.947613i	$-0.603488\pi$
$977$	−19509.0	−0.638842	−0.319421	+	0.947613i	$0.603488\pi$
$978$	0	0
$979$	49904.5	1.62917
$980$	−14637.0	−0.477103
$981$	0	0
$982$	−27872.5	−0.905752
$983$	−12696.7	−0.411964	−0.205982	−	0.978556i	$-0.566039\pi$
$983$	−12696.7	−0.411964	−0.205982	+	0.978556i	$0.566039\pi$
$984$	0	0
$985$	−41274.9	−1.33515
$986$	3402.64	0.109901
$987$	0	0
$988$	0	0
$989$	−20231.2	−0.650470
$990$	0	0
$991$	26580.3	0.852020	0.426010	−	0.904718i	$-0.359919\pi$
$991$	26580.3	0.852020	0.426010	+	0.904718i	$0.359919\pi$
$992$	−36413.9	−1.16547
$993$	0	0
$994$	22435.9	0.715920
$995$	34404.9	1.09619
$996$	0	0
$997$	7492.31	0.237998	0.118999	−	0.992894i	$-0.462031\pi$
$997$	7492.31	0.237998	0.118999	+	0.992894i	$0.462031\pi$
$998$	29215.2	0.926643
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.bm.1.5	✓	18
3.2	odd	2	inner	1521.4.a.bm.1.14	yes	18
13.12	even	2		1521.4.a.bn.1.14	yes	18
39.38	odd	2		1521.4.a.bn.1.5	yes	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1521.4.a.bm.1.5	✓	18	1.1	even	1	trivial
1521.4.a.bm.1.14	yes	18	3.2	odd	2	inner
1521.4.a.bn.1.5	yes	18	39.38	odd	2
1521.4.a.bn.1.14	yes	18	13.12	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:	\( N \)	\(=\)	\( 1521 = 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1521.a (trivial)

\(f(q)\)	\(=\)	\(q-3.45813 q^{2} +3.95865 q^{4} -9.27918 q^{5} +27.2299 q^{7} +13.9755 q^{8} +O(q^{10})\)	\(q-3.45813 q^{2} +3.95865 q^{4} -9.27918 q^{5} +27.2299 q^{7} +13.9755 q^{8} +32.0886 q^{10} +38.3557 q^{11} -94.1645 q^{14} -79.9983 q^{16} +85.0844 q^{17} -114.360 q^{19} -36.7331 q^{20} -132.639 q^{22} +49.4232 q^{23} -38.8968 q^{25} +107.794 q^{28} -11.5644 q^{29} -220.904 q^{31} +164.840 q^{32} -294.233 q^{34} -252.671 q^{35} -24.7406 q^{37} +395.470 q^{38} -129.681 q^{40} -30.8365 q^{41} -409.346 q^{43} +151.837 q^{44} -170.912 q^{46} -434.519 q^{47} +398.468 q^{49} +134.510 q^{50} +716.136 q^{53} -355.909 q^{55} +380.552 q^{56} +39.9913 q^{58} -618.055 q^{59} +213.058 q^{61} +763.915 q^{62} +69.9470 q^{64} -578.504 q^{67} +336.820 q^{68} +873.770 q^{70} -238.263 q^{71} +748.837 q^{73} +85.5562 q^{74} -452.710 q^{76} +1044.42 q^{77} +883.090 q^{79} +742.319 q^{80} +106.637 q^{82} -1401.67 q^{83} -789.513 q^{85} +1415.57 q^{86} +536.040 q^{88} +1301.10 q^{89} +195.649 q^{92} +1502.62 q^{94} +1061.16 q^{95} +258.030 q^{97} -1377.95 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 56 q^{4} - 94 q^{7}+O(q^{10}) \) 18 * q + 56 * q^4 - 94 * q^7	\( 18 q + 56 q^{4} - 94 q^{7} + 156 q^{10} + 88 q^{16} - 448 q^{19} - 256 q^{22} - 16 q^{25} - 1300 q^{28} - 818 q^{31} - 900 q^{34} - 524 q^{37} + 800 q^{40} + 752 q^{43} - 1650 q^{46} + 2948 q^{49} - 464 q^{55} - 1626 q^{58} - 1784 q^{61} - 4274 q^{64} - 2872 q^{67} - 5772 q^{70} - 3078 q^{73} - 5726 q^{76} + 3326 q^{79} - 1038 q^{82} - 2340 q^{85} + 780 q^{88} + 1220 q^{94} - 4630 q^{97}+O(q^{100}) \) 18 * q + 56 * q^4 - 94 * q^7 + 156 * q^10 + 88 * q^16 - 448 * q^19 - 256 * q^22 - 16 * q^25 - 1300 * q^28 - 818 * q^31 - 900 * q^34 - 524 * q^37 + 800 * q^40 + 752 * q^43 - 1650 * q^46 + 2948 * q^49 - 464 * q^55 - 1626 * q^58 - 1784 * q^61 - 4274 * q^64 - 2872 * q^67 - 5772 * q^70 - 3078 * q^73 - 5726 * q^76 + 3326 * q^79 - 1038 * q^82 - 2340 * q^85 + 780 * q^88 + 1220 * q^94 - 4630 * q^97

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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