LMFDB - Embedded newform 1521.4.a.bn.1.7

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:	$0$
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			1.7
Root			$-1.32218$ 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)

$f(q)$	$=$	$q-1.32218 q^{2} -6.25184 q^{4} +12.9610 q^{5} +5.54391 q^{7} +18.8435 q^{8} +O(q^{10})$	$q-1.32218 q^{2} -6.25184 q^{4} +12.9610 q^{5} +5.54391 q^{7} +18.8435 q^{8} -17.1367 q^{10} -0.841909 q^{11} -7.33006 q^{14} +25.1002 q^{16} +10.1858 q^{17} -73.0894 q^{19} -81.0298 q^{20} +1.11316 q^{22} +139.073 q^{23} +42.9863 q^{25} -34.6596 q^{28} +250.377 q^{29} -161.445 q^{31} -183.935 q^{32} -13.4674 q^{34} +71.8544 q^{35} +195.826 q^{37} +96.6374 q^{38} +244.230 q^{40} +183.760 q^{41} +115.071 q^{43} +5.26348 q^{44} -183.880 q^{46} +551.780 q^{47} -312.265 q^{49} -56.8357 q^{50} -324.717 q^{53} -10.9119 q^{55} +104.467 q^{56} -331.043 q^{58} -521.559 q^{59} -444.586 q^{61} +213.459 q^{62} +42.3942 q^{64} +55.0046 q^{67} -63.6797 q^{68} -95.0045 q^{70} +279.871 q^{71} +908.097 q^{73} -258.918 q^{74} +456.943 q^{76} -4.66747 q^{77} +941.507 q^{79} +325.322 q^{80} -242.964 q^{82} -946.563 q^{83} +132.017 q^{85} -152.145 q^{86} -15.8645 q^{88} -32.9886 q^{89} -869.461 q^{92} -729.553 q^{94} -947.308 q^{95} -68.2675 q^{97} +412.871 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.32218	−0.467462	−0.233731	−	0.972301i	$-0.575093\pi$
$2$	−1.32218	−0.467462	−0.233731	+	0.972301i	$0.575093\pi$
$3$	0	0
$3$	0	0
$4$	−6.25184	−0.781480
$5$	12.9610	1.15926	0.579631	−	0.814879i	$-0.303197\pi$
$5$	12.9610	1.15926	0.579631	+	0.814879i	$0.303197\pi$
$6$	0	0
$7$	5.54391	0.299343	0.149672	−	0.988736i	$-0.452178\pi$
$7$	5.54391	0.299343	0.149672	+	0.988736i	$0.452178\pi$
$8$	18.8435	0.832773
$9$	0	0
$10$	−17.1367	−0.541911
$11$	−0.841909	−0.0230768	−0.0115384	−	0.999933i	$-0.503673\pi$
$11$	−0.841909	−0.0230768	−0.0115384	+	0.999933i	$0.503673\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−7.33006	−0.139931
$15$	0	0
$16$	25.1002	0.392190
$17$	10.1858	0.145318	0.0726591	−	0.997357i	$-0.476851\pi$
$17$	10.1858	0.145318	0.0726591	+	0.997357i	$0.476851\pi$
$18$	0	0
$19$	−73.0894	−0.882519	−0.441260	−	0.897380i	$-0.645468\pi$
$19$	−73.0894	−0.882519	−0.441260	+	0.897380i	$0.645468\pi$
$20$	−81.0298	−0.905940
$21$	0	0
$22$	1.11316	0.0107875
$23$	139.073	1.26081	0.630407	−	0.776265i	$-0.282888\pi$
$23$	139.073	1.26081	0.630407	+	0.776265i	$0.282888\pi$
$24$	0	0
$25$	42.9863	0.343890
$26$	0	0
$27$	0	0
$28$	−34.6596	−0.233931
$29$	250.377	1.60323	0.801617	−	0.597838i	$-0.203973\pi$
$29$	250.377	1.60323	0.801617	+	0.597838i	$0.203973\pi$
$30$	0	0
$31$	−161.445	−0.935366	−0.467683	−	0.883896i	$-0.654911\pi$
$31$	−161.445	−0.935366	−0.467683	+	0.883896i	$0.654911\pi$
$32$	−183.935	−1.01611
$33$	0	0
$34$	−13.4674	−0.0679307
$35$	71.8544	0.347017
$36$	0	0
$37$	195.826	0.870099	0.435049	−	0.900407i	$-0.356731\pi$
$37$	195.826	0.870099	0.435049	+	0.900407i	$0.356731\pi$
$38$	96.6374	0.412544
$39$	0	0
$40$	244.230	0.965403
$41$	183.760	0.699962	0.349981	−	0.936757i	$-0.386188\pi$
$41$	183.760	0.699962	0.349981	+	0.936757i	$0.386188\pi$
$42$	0	0
$43$	115.071	0.408097	0.204049	−	0.978961i	$-0.434590\pi$
$43$	115.071	0.408097	0.204049	+	0.978961i	$0.434590\pi$
$44$	5.26348	0.0180341
$45$	0	0
$46$	−183.880	−0.589382
$47$	551.780	1.71245	0.856227	−	0.516600i	$-0.172802\pi$
$47$	551.780	1.71245	0.856227	+	0.516600i	$0.172802\pi$
$48$	0	0
$49$	−312.265	−0.910394
$50$	−56.8357	−0.160755
$51$	0	0
$52$	0	0
$53$	−324.717	−0.841572	−0.420786	−	0.907160i	$-0.638246\pi$
$53$	−324.717	−0.841572	−0.420786	+	0.907160i	$0.638246\pi$
$54$	0	0
$55$	−10.9119	−0.0267521
$56$	104.467	0.249285
$57$	0	0
$58$	−331.043	−0.749450
$59$	−521.559	−1.15087	−0.575434	−	0.817848i	$-0.695167\pi$
$59$	−521.559	−1.15087	−0.575434	+	0.817848i	$0.695167\pi$
$60$	0	0
$61$	−444.586	−0.933170	−0.466585	−	0.884476i	$-0.654516\pi$
$61$	−444.586	−0.933170	−0.466585	+	0.884476i	$0.654516\pi$
$62$	213.459	0.437248
$63$	0	0
$64$	42.3942	0.0828011
$65$	0	0
$66$	0	0
$67$	55.0046	0.100297	0.0501484	−	0.998742i	$-0.484031\pi$
$67$	55.0046	0.100297	0.0501484	+	0.998742i	$0.484031\pi$
$68$	−63.6797	−0.113563
$69$	0	0
$70$	−95.0045	−0.162217
$71$	279.871	0.467811	0.233906	−	0.972259i	$-0.424849\pi$
$71$	279.871	0.467811	0.233906	+	0.972259i	$0.424849\pi$
$72$	0	0
$73$	908.097	1.45596	0.727978	−	0.685601i	$-0.240460\pi$
$73$	908.097	1.45596	0.727978	+	0.685601i	$0.240460\pi$
$74$	−258.918	−0.406738
$75$	0	0
$76$	456.943	0.689671
$77$	−4.66747	−0.00690789
$78$	0	0
$79$	941.507	1.34086	0.670429	−	0.741974i	$-0.266110\pi$
$79$	941.507	1.34086	0.670429	+	0.741974i	$0.266110\pi$
$80$	325.322	0.454651
$81$	0	0
$82$	−242.964	−0.327206
$83$	−946.563	−1.25179	−0.625896	−	0.779906i	$-0.715267\pi$
$83$	−946.563	−1.25179	−0.625896	+	0.779906i	$0.715267\pi$
$84$	0	0
$85$	132.017	0.168462
$86$	−152.145	−0.190770
$87$	0	0
$88$	−15.8645	−0.0192178
$89$	−32.9886	−0.0392897	−0.0196448	−	0.999807i	$-0.506254\pi$
$89$	−32.9886	−0.0392897	−0.0196448	+	0.999807i	$0.506254\pi$
$90$	0	0
$91$	0	0
$92$	−869.461	−0.985300
$93$	0	0
$94$	−729.553	−0.800507
$95$	−947.308	−1.02307
$96$	0	0
$97$	−68.2675	−0.0714589	−0.0357294	−	0.999361i	$-0.511375\pi$
$97$	−68.2675	−0.0714589	−0.0357294	+	0.999361i	$0.511375\pi$
$98$	412.871	0.425574
$99$	0	0
$100$	−268.743	−0.268743
$101$	−1643.68	−1.61933	−0.809667	−	0.586890i	$-0.800352\pi$
$101$	−1643.68	−1.61933	−0.809667	+	0.586890i	$0.800352\pi$
$102$	0	0
$103$	−166.289	−0.159078	−0.0795388	−	0.996832i	$-0.525345\pi$
$103$	−166.289	−0.159078	−0.0795388	+	0.996832i	$0.525345\pi$
$104$	0	0
$105$	0	0
$106$	429.335	0.393403
$107$	−445.544	−0.402546	−0.201273	−	0.979535i	$-0.564508\pi$
$107$	−445.544	−0.402546	−0.201273	+	0.979535i	$0.564508\pi$
$108$	0	0
$109$	−884.955	−0.777645	−0.388822	−	0.921313i	$-0.627118\pi$
$109$	−884.955	−0.777645	−0.388822	+	0.921313i	$0.627118\pi$
$110$	14.4276	0.0125056
$111$	0	0
$112$	139.153	0.117399
$113$	1513.58	1.26005	0.630023	−	0.776576i	$-0.283045\pi$
$113$	1513.58	1.26005	0.630023	+	0.776576i	$0.283045\pi$
$114$	0	0
$115$	1802.52	1.46161
$116$	−1565.31	−1.25289
$117$	0	0
$118$	689.596	0.537987
$119$	56.4690	0.0435000
$120$	0	0
$121$	−1330.29	−0.999467
$122$	587.823	0.436221
$123$	0	0
$124$	1009.33	0.730970
$125$	−1062.98	−0.760604
$126$	0	0
$127$	2584.41	1.80574	0.902871	−	0.429912i	$-0.141455\pi$
$127$	2584.41	1.80574	0.902871	+	0.429912i	$0.141455\pi$
$128$	1415.43	0.977401
$129$	0	0
$130$	0	0
$131$	1486.33	0.991311	0.495655	−	0.868519i	$-0.334928\pi$
$131$	1486.33	0.991311	0.495655	+	0.868519i	$0.334928\pi$
$132$	0	0
$133$	−405.201	−0.264176
$134$	−72.7261	−0.0468849
$135$	0	0
$136$	191.935	0.121017
$137$	−1481.51	−0.923895	−0.461947	−	0.886907i	$-0.652849\pi$
$137$	−1481.51	−0.923895	−0.461947	+	0.886907i	$0.652849\pi$
$138$	0	0
$139$	−917.381	−0.559793	−0.279897	−	0.960030i	$-0.590300\pi$
$139$	−917.381	−0.559793	−0.279897	+	0.960030i	$0.590300\pi$
$140$	−449.222	−0.271187
$141$	0	0
$142$	−370.040	−0.218684
$143$	0	0
$144$	0	0
$145$	3245.12	1.85857
$146$	−1200.67	−0.680603
$147$	0	0
$148$	−1224.27	−0.679965
$149$	2278.14	1.25257	0.626283	−	0.779596i	$-0.284576\pi$
$149$	2278.14	1.25257	0.626283	+	0.779596i	$0.284576\pi$
$150$	0	0
$151$	2196.92	1.18399	0.591996	−	0.805941i	$-0.298340\pi$
$151$	2196.92	1.18399	0.591996	+	0.805941i	$0.298340\pi$
$152$	−1377.26	−0.734938
$153$	0	0
$154$	6.17124	0.00322917
$155$	−2092.48	−1.08434
$156$	0	0
$157$	3197.69	1.62550	0.812750	−	0.582612i	$-0.197969\pi$
$157$	3197.69	1.62550	0.812750	+	0.582612i	$0.197969\pi$
$158$	−1244.84	−0.626800
$159$	0	0
$160$	−2383.97	−1.17794
$161$	771.008	0.377416
$162$	0	0
$163$	3083.17	1.48155	0.740774	−	0.671754i	$-0.234459\pi$
$163$	3083.17	1.48155	0.740774	+	0.671754i	$0.234459\pi$
$164$	−1148.84	−0.547006
$165$	0	0
$166$	1251.53	0.585165
$167$	524.082	0.242842	0.121421	−	0.992601i	$-0.461255\pi$
$167$	524.082	0.242842	0.121421	+	0.992601i	$0.461255\pi$
$168$	0	0
$169$	0	0
$170$	−174.551	−0.0787495
$171$	0	0
$172$	−719.405	−0.318919
$173$	−2981.68	−1.31036	−0.655182	−	0.755471i	$-0.727408\pi$
$173$	−2981.68	−1.31036	−0.655182	+	0.755471i	$0.727408\pi$
$174$	0	0
$175$	238.312	0.102941
$176$	−21.1321	−0.00905050
$177$	0	0
$178$	43.6168	0.0183664
$179$	332.862	0.138990	0.0694952	−	0.997582i	$-0.477861\pi$
$179$	332.862	0.138990	0.0694952	+	0.997582i	$0.477861\pi$
$180$	0	0
$181$	−2546.13	−1.04559	−0.522796	−	0.852458i	$-0.675111\pi$
$181$	−2546.13	−1.04559	−0.522796	+	0.852458i	$0.675111\pi$
$182$	0	0
$183$	0	0
$184$	2620.62	1.04997
$185$	2538.10	1.00867
$186$	0	0
$187$	−8.57548	−0.00335348
$188$	−3449.64	−1.33825
$189$	0	0
$190$	1252.51	0.478247
$191$	3424.25	1.29723	0.648613	−	0.761118i	$-0.275349\pi$
$191$	3424.25	1.29723	0.648613	+	0.761118i	$0.275349\pi$
$192$	0	0
$193$	2912.13	1.08611	0.543055	−	0.839697i	$-0.317267\pi$
$193$	2912.13	1.08611	0.543055	+	0.839697i	$0.317267\pi$
$194$	90.2620	0.0334043
$195$	0	0
$196$	1952.23	0.711454
$197$	80.9189	0.0292651	0.0146326	−	0.999893i	$-0.495342\pi$
$197$	80.9189	0.0292651	0.0146326	+	0.999893i	$0.495342\pi$
$198$	0	0
$199$	−875.929	−0.312025	−0.156013	−	0.987755i	$-0.549864\pi$
$199$	−875.929	−0.312025	−0.156013	+	0.987755i	$0.549864\pi$
$200$	810.012	0.286383
$201$	0	0
$202$	2173.25	0.756977
$203$	1388.07	0.479917
$204$	0	0
$205$	2381.70	0.811440
$206$	219.865	0.0743626
$207$	0	0
$208$	0	0
$209$	61.5347	0.0203657
$210$	0	0
$211$	−3862.86	−1.26033	−0.630167	−	0.776460i	$-0.717014\pi$
$211$	−3862.86	−1.26033	−0.630167	+	0.776460i	$0.717014\pi$
$212$	2030.08	0.657672
$213$	0	0
$214$	589.090	0.188175
$215$	1491.43	0.473092
$216$	0	0
$217$	−895.036	−0.279996
$218$	1170.07	0.363519
$219$	0	0
$220$	68.2197	0.0209062
$221$	0	0
$222$	0	0
$223$	6391.24	1.91923	0.959617	−	0.281311i	$-0.0907693\pi$
$223$	6391.24	1.91923	0.959617	+	0.281311i	$0.0907693\pi$
$224$	−1019.72	−0.304165
$225$	0	0
$226$	−2001.22	−0.589023
$227$	1271.34	0.371727	0.185864	−	0.982576i	$-0.440492\pi$
$227$	1271.34	0.371727	0.185864	+	0.982576i	$0.440492\pi$
$228$	0	0
$229$	5964.71	1.72122	0.860610	−	0.509265i	$-0.170083\pi$
$229$	5964.71	1.72122	0.860610	+	0.509265i	$0.170083\pi$
$230$	−2383.25	−0.683249
$231$	0	0
$232$	4717.98	1.33513
$233$	5160.98	1.45110	0.725551	−	0.688168i	$-0.241585\pi$
$233$	5160.98	1.45110	0.725551	+	0.688168i	$0.241585\pi$
$234$	0	0
$235$	7151.59	1.98518
$236$	3260.70	0.899380
$237$	0	0
$238$	−74.6622	−0.0203346
$239$	−3799.33	−1.02828	−0.514139	−	0.857707i	$-0.671888\pi$
$239$	−3799.33	−1.02828	−0.514139	+	0.857707i	$0.671888\pi$
$240$	0	0
$241$	−120.502	−0.0322083	−0.0161042	−	0.999870i	$-0.505126\pi$
$241$	−120.502	−0.0322083	−0.0161042	+	0.999870i	$0.505126\pi$
$242$	1758.89	0.467213
$243$	0	0
$244$	2779.48	0.729253
$245$	−4047.25	−1.05539
$246$	0	0
$247$	0	0
$248$	−3042.19	−0.778948
$249$	0	0
$250$	1405.45	0.355553
$251$	6961.35	1.75058	0.875292	−	0.483595i	$-0.160669\pi$
$251$	6961.35	1.75058	0.875292	+	0.483595i	$0.160669\pi$
$252$	0	0
$253$	−117.087	−0.0290956
$254$	−3417.06	−0.844115
$255$	0	0
$256$	−2210.60	−0.539698
$257$	1154.03	0.280103	0.140052	−	0.990144i	$-0.455273\pi$
$257$	1154.03	0.280103	0.140052	+	0.990144i	$0.455273\pi$
$258$	0	0
$259$	1085.64	0.260458
$260$	0	0
$261$	0	0
$262$	−1965.20	−0.463400
$263$	2627.40	0.616018	0.308009	−	0.951383i	$-0.400337\pi$
$263$	2627.40	0.616018	0.308009	+	0.951383i	$0.400337\pi$
$264$	0	0
$265$	−4208.64	−0.975604
$266$	535.750	0.123492
$267$	0	0
$268$	−343.880	−0.0783799
$269$	−5162.02	−1.17001	−0.585007	−	0.811028i	$-0.698908\pi$
$269$	−5162.02	−1.17001	−0.585007	+	0.811028i	$0.698908\pi$
$270$	0	0
$271$	3053.24	0.684394	0.342197	−	0.939628i	$-0.388829\pi$
$271$	3053.24	0.684394	0.342197	+	0.939628i	$0.388829\pi$
$272$	255.664	0.0569924
$273$	0	0
$274$	1958.82	0.431885
$275$	−36.1905	−0.00793590
$276$	0	0
$277$	−3425.81	−0.743094	−0.371547	−	0.928414i	$-0.621173\pi$
$277$	−3425.81	−0.743094	−0.371547	+	0.928414i	$0.621173\pi$
$278$	1212.94	0.261682
$279$	0	0
$280$	1353.99	0.288987
$281$	5619.38	1.19297	0.596484	−	0.802625i	$-0.296564\pi$
$281$	5619.38	1.19297	0.596484	+	0.802625i	$0.296564\pi$
$282$	0	0
$283$	−2181.00	−0.458116	−0.229058	−	0.973413i	$-0.573565\pi$
$283$	−2181.00	−0.458116	−0.229058	+	0.973413i	$0.573565\pi$
$284$	−1749.71	−0.365585
$285$	0	0
$286$	0	0
$287$	1018.75	0.209529
$288$	0	0
$289$	−4809.25	−0.978883
$290$	−4290.64	−0.868810
$291$	0	0
$292$	−5677.28	−1.13780
$293$	4610.74	0.919325	0.459662	−	0.888094i	$-0.347970\pi$
$293$	4610.74	0.919325	0.459662	+	0.888094i	$0.347970\pi$
$294$	0	0
$295$	−6759.91	−1.33416
$296$	3690.06	0.724595
$297$	0	0
$298$	−3012.11	−0.585526
$299$	0	0
$300$	0	0
$301$	637.944	0.122161
$302$	−2904.73	−0.553471
$303$	0	0
$304$	−1834.56	−0.346115
$305$	−5762.25	−1.08179
$306$	0	0
$307$	4432.85	0.824091	0.412045	−	0.911163i	$-0.364814\pi$
$307$	4432.85	0.824091	0.412045	+	0.911163i	$0.364814\pi$
$308$	29.1803	0.00539838
$309$	0	0
$310$	2766.64	0.506885
$311$	8942.00	1.63040	0.815200	−	0.579180i	$-0.196627\pi$
$311$	8942.00	1.63040	0.815200	+	0.579180i	$0.196627\pi$
$312$	0	0
$313$	7883.46	1.42364	0.711820	−	0.702362i	$-0.247871\pi$
$313$	7883.46	1.42364	0.711820	+	0.702362i	$0.247871\pi$
$314$	−4227.93	−0.759859
$315$	0	0
$316$	−5886.15	−1.04785
$317$	−670.707	−0.118835	−0.0594174	−	0.998233i	$-0.518924\pi$
$317$	−670.707	−0.118835	−0.0594174	+	0.998233i	$0.518924\pi$
$318$	0	0
$319$	−210.794	−0.0369976
$320$	549.469	0.0959882
$321$	0	0
$322$	−1019.41	−0.176427
$323$	−744.471	−0.128246
$324$	0	0
$325$	0	0
$326$	−4076.51	−0.692567
$327$	0	0
$328$	3462.68	0.582910
$329$	3059.02	0.512611
$330$	0	0
$331$	−2842.70	−0.472051	−0.236026	−	0.971747i	$-0.575845\pi$
$331$	−2842.70	−0.472051	−0.236026	+	0.971747i	$0.575845\pi$
$332$	5917.76	0.978251
$333$	0	0
$334$	−692.931	−0.113520
$335$	712.912	0.116270
$336$	0	0
$337$	1886.14	0.304880	0.152440	−	0.988313i	$-0.451287\pi$
$337$	1886.14	0.304880	0.152440	+	0.988313i	$0.451287\pi$
$338$	0	0
$339$	0	0
$340$	−825.350	−0.131650
$341$	135.922	0.0215853
$342$	0	0
$343$	−3632.73	−0.571863
$344$	2168.34	0.339852
$345$	0	0
$346$	3942.32	0.612545
$347$	4059.65	0.628050	0.314025	−	0.949415i	$-0.398322\pi$
$347$	4059.65	0.628050	0.314025	+	0.949415i	$0.398322\pi$
$348$	0	0
$349$	7316.53	1.12219	0.561096	−	0.827751i	$-0.310380\pi$
$349$	7316.53	1.12219	0.561096	+	0.827751i	$0.310380\pi$
$350$	−315.092	−0.0481211
$351$	0	0
$352$	154.857	0.0234485
$353$	3390.13	0.511157	0.255579	−	0.966788i	$-0.417734\pi$
$353$	3390.13	0.511157	0.255579	+	0.966788i	$0.417734\pi$
$354$	0	0
$355$	3627.40	0.542316
$356$	206.239	0.0307041
$357$	0	0
$358$	−440.104	−0.0649727
$359$	−12580.6	−1.84952	−0.924759	−	0.380552i	$-0.875734\pi$
$359$	−12580.6	−1.84952	−0.924759	+	0.380552i	$0.875734\pi$
$360$	0	0
$361$	−1516.94	−0.221160
$362$	3366.44	0.488774
$363$	0	0
$364$	0	0
$365$	11769.8	1.68783
$366$	0	0
$367$	−1833.08	−0.260725	−0.130362	−	0.991466i	$-0.541614\pi$
$367$	−1833.08	−0.260725	−0.130362	+	0.991466i	$0.541614\pi$
$368$	3490.75	0.494478
$369$	0	0
$370$	−3355.82	−0.471516
$371$	−1800.20	−0.251919
$372$	0	0
$373$	1383.58	0.192062	0.0960312	−	0.995378i	$-0.469385\pi$
$373$	1383.58	0.192062	0.0960312	+	0.995378i	$0.469385\pi$
$374$	11.3383	0.00156763
$375$	0	0
$376$	10397.5	1.42609
$377$	0	0
$378$	0	0
$379$	−1495.54	−0.202693	−0.101346	−	0.994851i	$-0.532315\pi$
$379$	−1495.54	−0.202693	−0.101346	+	0.994851i	$0.532315\pi$
$380$	5922.42	0.799510
$381$	0	0
$382$	−4527.48	−0.606404
$383$	1454.92	0.194107	0.0970537	−	0.995279i	$-0.469058\pi$
$383$	1454.92	0.194107	0.0970537	+	0.995279i	$0.469058\pi$
$384$	0	0
$385$	−60.4949	−0.00800806
$386$	−3850.36	−0.507715
$387$	0	0
$388$	426.797	0.0558437
$389$	−14163.0	−1.84600	−0.922999	−	0.384801i	$-0.874270\pi$
$389$	−14163.0	−1.84600	−0.922999	+	0.384801i	$0.874270\pi$
$390$	0	0
$391$	1416.56	0.183219
$392$	−5884.17	−0.758152
$393$	0	0
$394$	−106.989	−0.0136803
$395$	12202.8	1.55441
$396$	0	0
$397$	−4877.38	−0.616596	−0.308298	−	0.951290i	$-0.599759\pi$
$397$	−4877.38	−0.616596	−0.308298	+	0.951290i	$0.599759\pi$
$398$	1158.14	0.145860
$399$	0	0
$400$	1078.96	0.134870
$401$	−1233.69	−0.153635	−0.0768176	−	0.997045i	$-0.524476\pi$
$401$	−1233.69	−0.153635	−0.0768176	+	0.997045i	$0.524476\pi$
$402$	0	0
$403$	0	0
$404$	10276.0	1.26548
$405$	0	0
$406$	−1835.28	−0.224343
$407$	−164.868	−0.0200791
$408$	0	0
$409$	5113.24	0.618174	0.309087	−	0.951034i	$-0.399977\pi$
$409$	5113.24	0.618174	0.309087	+	0.951034i	$0.399977\pi$
$410$	−3149.04	−0.379317
$411$	0	0
$412$	1039.61	0.124316
$413$	−2891.48	−0.344505
$414$	0	0
$415$	−12268.4	−1.45116
$416$	0	0
$417$	0	0
$418$	−81.3600	−0.00952021
$419$	−8630.26	−1.00624	−0.503121	−	0.864216i	$-0.667815\pi$
$419$	−8630.26	−1.00624	−0.503121	+	0.864216i	$0.667815\pi$
$420$	0	0
$421$	13265.8	1.53571	0.767854	−	0.640624i	$-0.221324\pi$
$421$	13265.8	1.53571	0.767854	+	0.640624i	$0.221324\pi$
$422$	5107.40	0.589158
$423$	0	0
$424$	−6118.81	−0.700839
$425$	437.848	0.0499735
$426$	0	0
$427$	−2464.74	−0.279338
$428$	2785.47	0.314581
$429$	0	0
$430$	−1971.94	−0.221152
$431$	−4617.20	−0.516015	−0.258008	−	0.966143i	$-0.583066\pi$
$431$	−4617.20	−0.516015	−0.258008	+	0.966143i	$0.583066\pi$
$432$	0	0
$433$	12484.6	1.38561	0.692805	−	0.721125i	$-0.256374\pi$
$433$	12484.6	1.38561	0.692805	+	0.721125i	$0.256374\pi$
$434$	1183.40	0.130887
$435$	0	0
$436$	5532.59	0.607714
$437$	−10164.8	−1.11269
$438$	0	0
$439$	−5668.93	−0.616317	−0.308159	−	0.951335i	$-0.599713\pi$
$439$	−5668.93	−0.616317	−0.308159	+	0.951335i	$0.599713\pi$
$440$	−205.619	−0.0222785
$441$	0	0
$442$	0	0
$443$	−4801.47	−0.514954	−0.257477	−	0.966284i	$-0.582891\pi$
$443$	−4801.47	−0.514954	−0.257477	+	0.966284i	$0.582891\pi$
$444$	0	0
$445$	−427.563	−0.0455470
$446$	−8450.38	−0.897168
$447$	0	0
$448$	235.029	0.0247859
$449$	16254.1	1.70841	0.854207	−	0.519934i	$-0.174043\pi$
$449$	16254.1	1.70841	0.854207	+	0.519934i	$0.174043\pi$
$450$	0	0
$451$	−154.709	−0.0161529
$452$	−9462.63	−0.984700
$453$	0	0
$454$	−1680.95	−0.173768
$455$	0	0
$456$	0	0
$457$	7646.07	0.782643	0.391322	−	0.920254i	$-0.372018\pi$
$457$	7646.07	0.782643	0.391322	+	0.920254i	$0.372018\pi$
$458$	−7886.43	−0.804604
$459$	0	0
$460$	−11269.0	−1.14222
$461$	−5645.67	−0.570380	−0.285190	−	0.958471i	$-0.592057\pi$
$461$	−5645.67	−0.570380	−0.285190	+	0.958471i	$0.592057\pi$
$462$	0	0
$463$	3141.05	0.315285	0.157642	−	0.987496i	$-0.449611\pi$
$463$	3141.05	0.315285	0.157642	+	0.987496i	$0.449611\pi$
$464$	6284.49	0.628772
$465$	0	0
$466$	−6823.75	−0.678335
$467$	1234.77	0.122352	0.0611759	−	0.998127i	$-0.480515\pi$
$467$	1234.77	0.122352	0.0611759	+	0.998127i	$0.480515\pi$
$468$	0	0
$469$	304.941	0.0300231
$470$	−9455.70	−0.927998
$471$	0	0
$472$	−9828.01	−0.958413
$473$	−96.8794	−0.00941759
$474$	0	0
$475$	−3141.84	−0.303490
$476$	−353.035	−0.0339944
$477$	0	0
$478$	5023.41	0.480680
$479$	11758.1	1.12159	0.560795	−	0.827955i	$-0.310495\pi$
$479$	11758.1	1.12159	0.560795	+	0.827955i	$0.310495\pi$
$480$	0	0
$481$	0	0
$482$	159.325	0.0150562
$483$	0	0
$484$	8316.76	0.781063
$485$	−884.811	−0.0828396
$486$	0	0
$487$	8650.25	0.804888	0.402444	−	0.915445i	$-0.368161\pi$
$487$	8650.25	0.804888	0.402444	+	0.915445i	$0.368161\pi$
$488$	−8377.56	−0.777119
$489$	0	0
$490$	5351.20	0.493352
$491$	3819.18	0.351033	0.175516	−	0.984476i	$-0.443840\pi$
$491$	3819.18	0.351033	0.175516	+	0.984476i	$0.443840\pi$
$492$	0	0
$493$	2550.28	0.232979
$494$	0	0
$495$	0	0
$496$	−4052.29	−0.366841
$497$	1551.58	0.140036
$498$	0	0
$499$	17077.9	1.53208	0.766042	−	0.642791i	$-0.222224\pi$
$499$	17077.9	1.53208	0.766042	+	0.642791i	$0.222224\pi$
$500$	6645.55	0.594396
$501$	0	0
$502$	−9204.17	−0.818331
$503$	−2996.47	−0.265618	−0.132809	−	0.991142i	$-0.542400\pi$
$503$	−2996.47	−0.265618	−0.132809	+	0.991142i	$0.542400\pi$
$504$	0	0
$505$	−21303.7	−1.87723
$506$	154.810	0.0136011
$507$	0	0
$508$	−16157.3	−1.41115
$509$	8256.37	0.718973	0.359486	−	0.933150i	$-0.382952\pi$
$509$	8256.37	0.718973	0.359486	+	0.933150i	$0.382952\pi$
$510$	0	0
$511$	5034.41	0.435830
$512$	−8400.60	−0.725113
$513$	0	0
$514$	−1525.84	−0.130938
$515$	−2155.27	−0.184413
$516$	0	0
$517$	−464.548	−0.0395180
$518$	−1435.42	−0.121754
$519$	0	0
$520$	0	0
$521$	−6348.47	−0.533842	−0.266921	−	0.963718i	$-0.586006\pi$
$521$	−6348.47	−0.533842	−0.266921	+	0.963718i	$0.586006\pi$
$522$	0	0
$523$	7444.93	0.622455	0.311228	−	0.950335i	$-0.399260\pi$
$523$	7444.93	0.622455	0.311228	+	0.950335i	$0.399260\pi$
$524$	−9292.32	−0.774689
$525$	0	0
$526$	−3473.91	−0.287965
$527$	−1644.44	−0.135926
$528$	0	0
$529$	7174.28	0.589651
$530$	5564.59	0.456057
$531$	0	0
$532$	2533.25	0.206448
$533$	0	0
$534$	0	0
$535$	−5774.68	−0.466656
$536$	1036.48	0.0835245
$537$	0	0
$538$	6825.12	0.546937
$539$	262.899	0.0210090
$540$	0	0
$541$	−6247.50	−0.496490	−0.248245	−	0.968697i	$-0.579854\pi$
$541$	−6247.50	−0.496490	−0.248245	+	0.968697i	$0.579854\pi$
$542$	−4036.93	−0.319928
$543$	0	0
$544$	−1873.52	−0.147659
$545$	−11469.9	−0.901495
$546$	0	0
$547$	−23338.4	−1.82427	−0.912137	−	0.409884i	$-0.865569\pi$
$547$	−23338.4	−1.82427	−0.912137	+	0.409884i	$0.865569\pi$
$548$	9262.13	0.722005
$549$	0	0
$550$	47.8505	0.00370973
$551$	−18299.9	−1.41488
$552$	0	0
$553$	5219.63	0.401377
$554$	4529.54	0.347368
$555$	0	0
$556$	5735.32	0.437467
$557$	−14975.3	−1.13918	−0.569592	−	0.821927i	$-0.692899\pi$
$557$	−14975.3	−1.13918	−0.569592	+	0.821927i	$0.692899\pi$
$558$	0	0
$559$	0	0
$560$	1803.56	0.136097
$561$	0	0
$562$	−7429.84	−0.557667
$563$	23463.5	1.75642	0.878212	−	0.478271i	$-0.158736\pi$
$563$	23463.5	1.75642	0.878212	+	0.478271i	$0.158736\pi$
$564$	0	0
$565$	19617.4	1.46072
$566$	2883.67	0.214152
$567$	0	0
$568$	5273.75	0.389581
$569$	−22359.6	−1.64739	−0.823694	−	0.567035i	$-0.808091\pi$
$569$	−22359.6	−1.64739	−0.823694	+	0.567035i	$0.808091\pi$
$570$	0	0
$571$	−2998.13	−0.219733	−0.109867	−	0.993946i	$-0.535042\pi$
$571$	−2998.13	−0.219733	−0.109867	+	0.993946i	$0.535042\pi$
$572$	0	0
$573$	0	0
$574$	−1346.97	−0.0979467
$575$	5978.23	0.433581
$576$	0	0
$577$	3370.06	0.243150	0.121575	−	0.992582i	$-0.461205\pi$
$577$	3370.06	0.243150	0.121575	+	0.992582i	$0.461205\pi$
$578$	6358.70	0.457590
$579$	0	0
$580$	−20288.0	−1.45243
$581$	−5247.66	−0.374716
$582$	0	0
$583$	273.382	0.0194208
$584$	17111.7	1.21248
$585$	0	0
$586$	−6096.23	−0.429749
$587$	21645.2	1.52197	0.760983	−	0.648772i	$-0.224717\pi$
$587$	21645.2	1.52197	0.760983	+	0.648772i	$0.224717\pi$
$588$	0	0
$589$	11799.9	0.825479
$590$	8937.82	0.623668
$591$	0	0
$592$	4915.27	0.341244
$593$	22232.6	1.53960	0.769801	−	0.638284i	$-0.220355\pi$
$593$	22232.6	1.53960	0.769801	+	0.638284i	$0.220355\pi$
$594$	0	0
$595$	731.891	0.0504279
$596$	−14242.5	−0.978854
$597$	0	0
$598$	0	0
$599$	−21337.0	−1.45544	−0.727718	−	0.685877i	$-0.759419\pi$
$599$	−21337.0	−1.45544	−0.727718	+	0.685877i	$0.759419\pi$
$600$	0	0
$601$	−1592.47	−0.108084	−0.0540419	−	0.998539i	$-0.517210\pi$
$601$	−1592.47	−0.108084	−0.0540419	+	0.998539i	$0.517210\pi$
$602$	−843.477	−0.0571056
$603$	0	0
$604$	−13734.8	−0.925266
$605$	−17241.8	−1.15865
$606$	0	0
$607$	18462.3	1.23454	0.617268	−	0.786753i	$-0.288239\pi$
$607$	18462.3	1.23454	0.617268	+	0.786753i	$0.288239\pi$
$608$	13443.7	0.896734
$609$	0	0
$610$	7618.74	0.505695
$611$	0	0
$612$	0	0
$613$	−20041.2	−1.32048	−0.660241	−	0.751054i	$-0.729546\pi$
$613$	−20041.2	−1.32048	−0.660241	+	0.751054i	$0.729546\pi$
$614$	−5861.03	−0.385231
$615$	0	0
$616$	−87.9515	−0.00575271
$617$	−18752.7	−1.22359	−0.611796	−	0.791016i	$-0.709553\pi$
$617$	−18752.7	−1.22359	−0.611796	+	0.791016i	$0.709553\pi$
$618$	0	0
$619$	−7276.59	−0.472489	−0.236245	−	0.971694i	$-0.575917\pi$
$619$	−7276.59	−0.472489	−0.236245	+	0.971694i	$0.575917\pi$
$620$	13081.8	0.847386
$621$	0	0
$622$	−11822.9	−0.762149
$623$	−182.886	−0.0117611
$624$	0	0
$625$	−19150.5	−1.22563
$626$	−10423.4	−0.665497
$627$	0	0
$628$	−19991.5	−1.27030
$629$	1994.64	0.126441
$630$	0	0
$631$	−22411.5	−1.41393	−0.706963	−	0.707251i	$-0.749935\pi$
$631$	−22411.5	−1.41393	−0.706963	+	0.707251i	$0.749935\pi$
$632$	17741.3	1.11663
$633$	0	0
$634$	886.796	0.0555507
$635$	33496.4	2.09333
$636$	0	0
$637$	0	0
$638$	278.708	0.0172949
$639$	0	0
$640$	18345.3	1.13306
$641$	−18756.0	−1.15572	−0.577861	−	0.816135i	$-0.696113\pi$
$641$	−18756.0	−1.15572	−0.577861	+	0.816135i	$0.696113\pi$
$642$	0	0
$643$	14712.9	0.902366	0.451183	−	0.892431i	$-0.351002\pi$
$643$	14712.9	0.902366	0.451183	+	0.892431i	$0.351002\pi$
$644$	−4820.22	−0.294943
$645$	0	0
$646$	984.326	0.0599501
$647$	17409.3	1.05785	0.528925	−	0.848668i	$-0.322595\pi$
$647$	17409.3	1.05785	0.528925	+	0.848668i	$0.322595\pi$
$648$	0	0
$649$	439.106	0.0265584
$650$	0	0
$651$	0	0
$652$	−19275.5	−1.15780
$653$	17232.4	1.03270	0.516351	−	0.856377i	$-0.327290\pi$
$653$	17232.4	1.03270	0.516351	+	0.856377i	$0.327290\pi$
$654$	0	0
$655$	19264.3	1.14919
$656$	4612.40	0.274518
$657$	0	0
$658$	−4044.58	−0.239626
$659$	7718.22	0.456236	0.228118	−	0.973634i	$-0.426743\pi$
$659$	7718.22	0.456236	0.228118	+	0.973634i	$0.426743\pi$
$660$	0	0
$661$	−31375.2	−1.84623	−0.923113	−	0.384530i	$-0.874364\pi$
$661$	−31375.2	−1.84623	−0.923113	+	0.384530i	$0.874364\pi$
$662$	3758.57	0.220666
$663$	0	0
$664$	−17836.6	−1.04246
$665$	−5251.79	−0.306249
$666$	0	0
$667$	34820.6	2.02138
$668$	−3276.48	−0.189776
$669$	0	0
$670$	−942.599	−0.0543519
$671$	374.301	0.0215346
$672$	0	0
$673$	−24768.7	−1.41867	−0.709333	−	0.704873i	$-0.751004\pi$
$673$	−24768.7	−1.41867	−0.709333	+	0.704873i	$0.751004\pi$
$674$	−2493.82	−0.142520
$675$	0	0
$676$	0	0
$677$	−17050.3	−0.967944	−0.483972	−	0.875084i	$-0.660806\pi$
$677$	−17050.3	−0.967944	−0.483972	+	0.875084i	$0.660806\pi$
$678$	0	0
$679$	−378.469	−0.0213907
$680$	2487.67	0.140291
$681$	0	0
$682$	−179.713	−0.0100903
$683$	−23209.9	−1.30029	−0.650147	−	0.759808i	$-0.725293\pi$
$683$	−23209.9	−1.30029	−0.650147	+	0.759808i	$0.725293\pi$
$684$	0	0
$685$	−19201.7	−1.07104
$686$	4803.13	0.267324
$687$	0	0
$688$	2888.30	0.160052
$689$	0	0
$690$	0	0
$691$	1501.07	0.0826389	0.0413195	−	0.999146i	$-0.486844\pi$
$691$	1501.07	0.0826389	0.0413195	+	0.999146i	$0.486844\pi$
$692$	18641.0	1.02402
$693$	0	0
$694$	−5367.59	−0.293589
$695$	−11890.1	−0.648948
$696$	0	0
$697$	1871.73	0.101717
$698$	−9673.78	−0.524581
$699$	0	0
$700$	−1489.89	−0.0804464
$701$	−3944.82	−0.212544	−0.106272	−	0.994337i	$-0.533891\pi$
$701$	−3944.82	−0.212544	−0.106272	+	0.994337i	$0.533891\pi$
$702$	0	0
$703$	−14312.8	−0.767879
$704$	−35.6920	−0.00191079
$705$	0	0
$706$	−4482.37	−0.238946
$707$	−9112.44	−0.484736
$708$	0	0
$709$	−24251.8	−1.28462	−0.642311	−	0.766444i	$-0.722024\pi$
$709$	−24251.8	−1.28462	−0.642311	+	0.766444i	$0.722024\pi$
$710$	−4796.07	−0.253512
$711$	0	0
$712$	−621.620	−0.0327194
$713$	−22452.6	−1.17932
$714$	0	0
$715$	0	0
$716$	−2081.00	−0.108618
$717$	0	0
$718$	16633.8	0.864579
$719$	−6941.00	−0.360022	−0.180011	−	0.983665i	$-0.557613\pi$
$719$	−6941.00	−0.360022	−0.180011	+	0.983665i	$0.557613\pi$
$720$	0	0
$721$	−921.894	−0.0476188
$722$	2005.67	0.103384
$723$	0	0
$724$	15918.0	0.817108
$725$	10762.8	0.551336
$726$	0	0
$727$	3587.40	0.183011	0.0915056	−	0.995805i	$-0.470832\pi$
$727$	3587.40	0.183011	0.0915056	+	0.995805i	$0.470832\pi$
$728$	0	0
$729$	0	0
$730$	−15561.8	−0.788998
$731$	1172.09	0.0593039
$732$	0	0
$733$	32867.4	1.65619	0.828093	−	0.560591i	$-0.189426\pi$
$733$	32867.4	1.65619	0.828093	+	0.560591i	$0.189426\pi$
$734$	2423.66	0.121879
$735$	0	0
$736$	−25580.4	−1.28112
$737$	−46.3089	−0.00231453
$738$	0	0
$739$	−19818.9	−0.986534	−0.493267	−	0.869878i	$-0.664197\pi$
$739$	−19818.9	−0.986534	−0.493267	+	0.869878i	$0.664197\pi$
$740$	−15867.8	−0.788258
$741$	0	0
$742$	2380.20	0.117762
$743$	13341.0	0.658727	0.329364	−	0.944203i	$-0.393166\pi$
$743$	13341.0	0.658727	0.329364	+	0.944203i	$0.393166\pi$
$744$	0	0
$745$	29526.8	1.45205
$746$	−1829.35	−0.0897818
$747$	0	0
$748$	53.6125	0.00262068
$749$	−2470.06	−0.120499
$750$	0	0
$751$	−32392.8	−1.57394	−0.786971	−	0.616990i	$-0.788352\pi$
$751$	−32392.8	−1.57394	−0.786971	+	0.616990i	$0.788352\pi$
$752$	13849.8	0.671607
$753$	0	0
$754$	0	0
$755$	28474.2	1.37256
$756$	0	0
$757$	−6177.77	−0.296612	−0.148306	−	0.988942i	$-0.547382\pi$
$757$	−6177.77	−0.296612	−0.148306	+	0.988942i	$0.547382\pi$
$758$	1977.37	0.0947512
$759$	0	0
$760$	−17850.6	−0.851987
$761$	−4215.28	−0.200793	−0.100397	−	0.994947i	$-0.532011\pi$
$761$	−4215.28	−0.200793	−0.100397	+	0.994947i	$0.532011\pi$
$762$	0	0
$763$	−4906.11	−0.232783
$764$	−21407.9	−1.01376
$765$	0	0
$766$	−1923.67	−0.0907378
$767$	0	0
$768$	0	0
$769$	−6536.90	−0.306536	−0.153268	−	0.988185i	$-0.548980\pi$
$769$	−6536.90	−0.306536	−0.153268	+	0.988185i	$0.548980\pi$
$770$	79.9852	0.00374346
$771$	0	0
$772$	−18206.1	−0.848773
$773$	9953.50	0.463134	0.231567	−	0.972819i	$-0.425615\pi$
$773$	9953.50	0.463134	0.231567	+	0.972819i	$0.425615\pi$
$774$	0	0
$775$	−6939.92	−0.321663
$776$	−1286.40	−0.0595091
$777$	0	0
$778$	18726.1	0.862934
$779$	−13430.9	−0.617730
$780$	0	0
$781$	−235.626	−0.0107956
$782$	−1872.95	−0.0856479
$783$	0	0
$784$	−7837.90	−0.357047
$785$	41445.1	1.88438
$786$	0	0
$787$	−17827.4	−0.807467	−0.403733	−	0.914877i	$-0.632288\pi$
$787$	−17827.4	−0.807467	−0.403733	+	0.914877i	$0.632288\pi$
$788$	−505.892	−0.0228701
$789$	0	0
$790$	−16134.3	−0.726626
$791$	8391.13	0.377186
$792$	0	0
$793$	0	0
$794$	6448.78	0.288235
$795$	0	0
$796$	5476.17	0.243841
$797$	249.882	0.0111057	0.00555287	−	0.999985i	$-0.498232\pi$
$797$	249.882	0.0111057	0.00555287	+	0.999985i	$0.498232\pi$
$798$	0	0
$799$	5620.30	0.248851
$800$	−7906.68	−0.349429
$801$	0	0
$802$	1631.17	0.0718185
$803$	−764.535	−0.0335988
$804$	0	0
$805$	9993.00	0.437524
$806$	0	0
$807$	0	0
$808$	−30972.8	−1.34854
$809$	−13019.9	−0.565829	−0.282914	−	0.959145i	$-0.591301\pi$
$809$	−13019.9	−0.565829	−0.282914	+	0.959145i	$0.591301\pi$
$810$	0	0
$811$	−956.251	−0.0414038	−0.0207019	−	0.999786i	$-0.506590\pi$
$811$	−956.251	−0.0414038	−0.0207019	+	0.999786i	$0.506590\pi$
$812$	−8677.96	−0.375045
$813$	0	0
$814$	217.985	0.00938622
$815$	39960.8	1.71750
$816$	0	0
$817$	−8410.48	−0.360153
$818$	−6760.62	−0.288973
$819$	0	0
$820$	−14890.0	−0.634124
$821$	7853.26	0.333837	0.166919	−	0.985971i	$-0.446618\pi$
$821$	7853.26	0.333837	0.166919	+	0.985971i	$0.446618\pi$
$822$	0	0
$823$	33820.9	1.43247	0.716236	−	0.697858i	$-0.245863\pi$
$823$	33820.9	1.43247	0.716236	+	0.697858i	$0.245863\pi$
$824$	−3133.48	−0.132476
$825$	0	0
$826$	3823.06	0.161043
$827$	−7297.39	−0.306838	−0.153419	−	0.988161i	$-0.549028\pi$
$827$	−7297.39	−0.306838	−0.153419	+	0.988161i	$0.549028\pi$
$828$	0	0
$829$	−8454.13	−0.354191	−0.177095	−	0.984194i	$-0.556670\pi$
$829$	−8454.13	−0.354191	−0.177095	+	0.984194i	$0.556670\pi$
$830$	16221.0	0.678360
$831$	0	0
$832$	0	0
$833$	−3180.66	−0.132297
$834$	0	0
$835$	6792.60	0.281518
$836$	−384.705	−0.0159154
$837$	0	0
$838$	11410.8	0.470380
$839$	32007.9	1.31709	0.658543	−	0.752543i	$-0.271173\pi$
$839$	32007.9	1.31709	0.658543	+	0.752543i	$0.271173\pi$
$840$	0	0
$841$	38299.5	1.57036
$842$	−17539.7	−0.717885
$843$	0	0
$844$	24150.0	0.984925
$845$	0	0
$846$	0	0
$847$	−7375.02	−0.299184
$848$	−8150.45	−0.330056
$849$	0	0
$850$	−578.914	−0.0233607
$851$	27234.1	1.09703
$852$	0	0
$853$	9986.02	0.400838	0.200419	−	0.979710i	$-0.435770\pi$
$853$	9986.02	0.400838	0.200419	+	0.979710i	$0.435770\pi$
$854$	3258.84	0.130580
$855$	0	0
$856$	−8395.62	−0.335229
$857$	−289.524	−0.0115402	−0.00577010	−	0.999983i	$-0.501837\pi$
$857$	−289.524	−0.0115402	−0.00577010	+	0.999983i	$0.501837\pi$
$858$	0	0
$859$	−7833.85	−0.311161	−0.155581	−	0.987823i	$-0.549725\pi$
$859$	−7833.85	−0.311161	−0.155581	+	0.987823i	$0.549725\pi$
$860$	−9324.18	−0.369712
$861$	0	0
$862$	6104.77	0.241217
$863$	−33370.8	−1.31629	−0.658143	−	0.752893i	$-0.728658\pi$
$863$	−33370.8	−1.31629	−0.658143	+	0.752893i	$0.728658\pi$
$864$	0	0
$865$	−38645.4	−1.51906
$866$	−16506.8	−0.647720
$867$	0	0
$868$	5595.62	0.218811
$869$	−792.663	−0.0309428
$870$	0	0
$871$	0	0
$872$	−16675.7	−0.647602
$873$	0	0
$874$	13439.7	0.520141
$875$	−5893.04	−0.227681
$876$	0	0
$877$	−25422.4	−0.978852	−0.489426	−	0.872045i	$-0.662794\pi$
$877$	−25422.4	−0.978852	−0.489426	+	0.872045i	$0.662794\pi$
$878$	7495.35	0.288105
$879$	0	0
$880$	−273.892	−0.0104919
$881$	−40373.6	−1.54395	−0.771976	−	0.635652i	$-0.780731\pi$
$881$	−40373.6	−1.54395	−0.771976	+	0.635652i	$0.780731\pi$
$882$	0	0
$883$	−16898.9	−0.644045	−0.322023	−	0.946732i	$-0.604363\pi$
$883$	−16898.9	−0.644045	−0.322023	+	0.946732i	$0.604363\pi$
$884$	0	0
$885$	0	0
$886$	6348.41	0.240721
$887$	−49889.1	−1.88851	−0.944257	−	0.329210i	$-0.893218\pi$
$887$	−49889.1	−1.88851	−0.944257	+	0.329210i	$0.893218\pi$
$888$	0	0
$889$	14327.7	0.540536
$890$	565.316	0.0212915
$891$	0	0
$892$	−39957.0	−1.49984
$893$	−40329.3	−1.51127
$894$	0	0
$895$	4314.21	0.161126
$896$	7847.01	0.292578
$897$	0	0
$898$	−21490.8	−0.798618
$899$	−40422.0	−1.49961
$900$	0	0
$901$	−3307.49	−0.122296
$902$	204.553	0.00755087
$903$	0	0
$904$	28521.1	1.04933
$905$	−33000.2	−1.21212
$906$	0	0
$907$	−3061.06	−0.112063	−0.0560313	−	0.998429i	$-0.517845\pi$
$907$	−3061.06	−0.112063	−0.0560313	+	0.998429i	$0.517845\pi$
$908$	−7948.24	−0.290497
$909$	0	0
$910$	0	0
$911$	15247.2	0.554514	0.277257	−	0.960796i	$-0.410575\pi$
$911$	15247.2	0.554514	0.277257	+	0.960796i	$0.410575\pi$
$912$	0	0
$913$	796.920	0.0288874
$914$	−10109.5	−0.365856
$915$	0	0
$916$	−37290.4	−1.34510
$917$	8240.11	0.296742
$918$	0	0
$919$	−29419.2	−1.05599	−0.527993	−	0.849249i	$-0.677055\pi$
$919$	−29419.2	−1.05599	−0.527993	+	0.849249i	$0.677055\pi$
$920$	33965.8	1.21719
$921$	0	0
$922$	7464.60	0.266631
$923$	0	0
$924$	0	0
$925$	8417.85	0.299219
$926$	−4153.03	−0.147383
$927$	0	0
$928$	−46053.0	−1.62906
$929$	16910.0	0.597200	0.298600	−	0.954378i	$-0.403480\pi$
$929$	16910.0	0.597200	0.298600	+	0.954378i	$0.403480\pi$
$930$	0	0
$931$	22823.3	0.803440
$932$	−32265.6	−1.13401
$933$	0	0
$934$	−1632.59	−0.0571948
$935$	−111.146	−0.00388757
$936$	0	0
$937$	32828.7	1.14457	0.572287	−	0.820053i	$-0.306056\pi$
$937$	32828.7	1.14457	0.572287	+	0.820053i	$0.306056\pi$
$938$	−403.187	−0.0140347
$939$	0	0
$940$	−44710.6	−1.55138
$941$	−22244.7	−0.770623	−0.385311	−	0.922787i	$-0.625906\pi$
$941$	−22244.7	−0.770623	−0.385311	+	0.922787i	$0.625906\pi$
$942$	0	0
$943$	25556.0	0.882522
$944$	−13091.2	−0.451359
$945$	0	0
$946$	128.092	0.00440236
$947$	−17502.1	−0.600573	−0.300287	−	0.953849i	$-0.597082\pi$
$947$	−17502.1	−0.600573	−0.300287	+	0.953849i	$0.597082\pi$
$948$	0	0
$949$	0	0
$950$	4154.08	0.141870
$951$	0	0
$952$	1064.07	0.0362256
$953$	−31522.6	−1.07148	−0.535739	−	0.844384i	$-0.679967\pi$
$953$	−31522.6	−1.07148	−0.535739	+	0.844384i	$0.679967\pi$
$954$	0	0
$955$	44381.6	1.50383
$956$	23752.8	0.803578
$957$	0	0
$958$	−15546.4	−0.524301
$959$	−8213.34	−0.276562
$960$	0	0
$961$	−3726.54	−0.125090
$962$	0	0
$963$	0	0
$964$	753.358	0.0251701
$965$	37743.9	1.25909
$966$	0	0
$967$	35433.0	1.17833	0.589167	−	0.808011i	$-0.299456\pi$
$967$	35433.0	1.17833	0.589167	+	0.808011i	$0.299456\pi$
$968$	−25067.4	−0.832330
$969$	0	0
$970$	1169.88	0.0387243
$971$	5907.77	0.195252	0.0976258	−	0.995223i	$-0.468875\pi$
$971$	5907.77	0.195252	0.0976258	+	0.995223i	$0.468875\pi$
$972$	0	0
$973$	−5085.88	−0.167570
$974$	−11437.2	−0.376254
$975$	0	0
$976$	−11159.2	−0.365980
$977$	231.490	0.00758035	0.00379018	−	0.999993i	$-0.498794\pi$
$977$	231.490	0.00758035	0.00379018	+	0.999993i	$0.498794\pi$
$978$	0	0
$979$	27.7734	0.000906681 0
$980$	25302.8	0.824762
$981$	0	0
$982$	−5049.65	−0.164094
$983$	7577.90	0.245877	0.122939	−	0.992414i	$-0.460768\pi$
$983$	7577.90	0.245877	0.122939	+	0.992414i	$0.460768\pi$
$984$	0	0
$985$	1048.79	0.0339260
$986$	−3371.93	−0.108909
$987$	0	0
$988$	0	0
$989$	16003.3	0.514534
$990$	0	0
$991$	−6578.71	−0.210878	−0.105439	−	0.994426i	$-0.533625\pi$
$991$	−6578.71	−0.210878	−0.105439	+	0.994426i	$0.533625\pi$
$992$	29695.4	0.950433
$993$	0	0
$994$	−2051.47	−0.0654615
$995$	−11352.9	−0.361719
$996$	0	0
$997$	14907.8	0.473556	0.236778	−	0.971564i	$-0.423909\pi$
$997$	14907.8	0.473556	0.236778	+	0.971564i	$0.423909\pi$
$998$	−22580.0	−0.716190
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.bn.1.7	yes	18
3.2	odd	2	inner	1521.4.a.bn.1.12	yes	18
13.12	even	2		1521.4.a.bm.1.12	yes	18
39.38	odd	2		1521.4.a.bm.1.7	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1521.4.a.bm.1.7	✓	18	39.38	odd	2
1521.4.a.bm.1.12	yes	18	13.12	even	2
1521.4.a.bn.1.7	yes	18	1.1	even	1	trivial
1521.4.a.bn.1.12	yes	18	3.2	odd	2	inner

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)

\(f(q)\)	\(=\)	\(q-1.32218 q^{2} -6.25184 q^{4} +12.9610 q^{5} +5.54391 q^{7} +18.8435 q^{8} +O(q^{10})\)	\(q-1.32218 q^{2} -6.25184 q^{4} +12.9610 q^{5} +5.54391 q^{7} +18.8435 q^{8} -17.1367 q^{10} -0.841909 q^{11} -7.33006 q^{14} +25.1002 q^{16} +10.1858 q^{17} -73.0894 q^{19} -81.0298 q^{20} +1.11316 q^{22} +139.073 q^{23} +42.9863 q^{25} -34.6596 q^{28} +250.377 q^{29} -161.445 q^{31} -183.935 q^{32} -13.4674 q^{34} +71.8544 q^{35} +195.826 q^{37} +96.6374 q^{38} +244.230 q^{40} +183.760 q^{41} +115.071 q^{43} +5.26348 q^{44} -183.880 q^{46} +551.780 q^{47} -312.265 q^{49} -56.8357 q^{50} -324.717 q^{53} -10.9119 q^{55} +104.467 q^{56} -331.043 q^{58} -521.559 q^{59} -444.586 q^{61} +213.459 q^{62} +42.3942 q^{64} +55.0046 q^{67} -63.6797 q^{68} -95.0045 q^{70} +279.871 q^{71} +908.097 q^{73} -258.918 q^{74} +456.943 q^{76} -4.66747 q^{77} +941.507 q^{79} +325.322 q^{80} -242.964 q^{82} -946.563 q^{83} +132.017 q^{85} -152.145 q^{86} -15.8645 q^{88} -32.9886 q^{89} -869.461 q^{92} -729.553 q^{94} -947.308 q^{95} -68.2675 q^{97} +412.871 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 56 q^{4} + 94 q^{7}+O(q^{10}) \) 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

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more