LMFDB - Embedded newform 1521.4.a.bn.1.13

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			1.13
Root			$1.86163$ 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)

$f(q)$	$=$	$q+1.86163 q^{2} -4.53433 q^{4} -3.46613 q^{5} -21.2285 q^{7} -23.3343 q^{8} +O(q^{10})$	$q+1.86163 q^{2} -4.53433 q^{4} -3.46613 q^{5} -21.2285 q^{7} -23.3343 q^{8} -6.45265 q^{10} -54.1559 q^{11} -39.5196 q^{14} -7.16523 q^{16} -99.0529 q^{17} -64.1160 q^{19} +15.7166 q^{20} -100.818 q^{22} -153.983 q^{23} -112.986 q^{25} +96.2569 q^{28} +22.5438 q^{29} +241.319 q^{31} +173.335 q^{32} -184.400 q^{34} +73.5806 q^{35} -34.5346 q^{37} -119.360 q^{38} +80.8796 q^{40} -117.778 q^{41} -101.468 q^{43} +245.561 q^{44} -286.660 q^{46} +451.273 q^{47} +107.649 q^{49} -210.338 q^{50} -6.41346 q^{53} +187.711 q^{55} +495.352 q^{56} +41.9682 q^{58} -303.502 q^{59} -622.411 q^{61} +449.248 q^{62} +380.008 q^{64} -289.575 q^{67} +449.138 q^{68} +136.980 q^{70} -949.340 q^{71} +56.4739 q^{73} -64.2906 q^{74} +290.723 q^{76} +1149.65 q^{77} -968.723 q^{79} +24.8356 q^{80} -219.259 q^{82} +480.985 q^{83} +343.330 q^{85} -188.897 q^{86} +1263.69 q^{88} -240.037 q^{89} +698.210 q^{92} +840.104 q^{94} +222.234 q^{95} +1871.81 q^{97} +200.402 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.86163	0.658186	0.329093	−	0.944298i	$-0.393257\pi$
$2$	1.86163	0.658186	0.329093	+	0.944298i	$0.393257\pi$
$3$	0	0
$3$	0	0
$4$	−4.53433	−0.566791
$5$	−3.46613	−0.310020	−0.155010	−	0.987913i	$-0.549541\pi$
$5$	−3.46613	−0.310020	−0.155010	+	0.987913i	$0.549541\pi$
$6$	0	0
$7$	−21.2285	−1.14623	−0.573115	−	0.819475i	$-0.694265\pi$
$7$	−21.2285	−1.14623	−0.573115	+	0.819475i	$0.694265\pi$
$8$	−23.3343	−1.03124
$9$	0	0
$10$	−6.45265	−0.204051
$11$	−54.1559	−1.48442	−0.742210	−	0.670168i	$-0.766222\pi$
$11$	−54.1559	−1.48442	−0.742210	+	0.670168i	$0.766222\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−39.5196	−0.754433
$15$	0	0
$16$	−7.16523	−0.111957
$17$	−99.0529	−1.41317	−0.706584	−	0.707629i	$-0.749765\pi$
$17$	−99.0529	−1.41317	−0.706584	+	0.707629i	$0.749765\pi$
$18$	0	0
$19$	−64.1160	−0.774169	−0.387085	−	0.922044i	$-0.626518\pi$
$19$	−64.1160	−0.774169	−0.387085	+	0.922044i	$0.626518\pi$
$20$	15.7166	0.175716
$21$	0	0
$22$	−100.818	−0.977024
$23$	−153.983	−1.39599	−0.697994	−	0.716104i	$-0.745924\pi$
$23$	−153.983	−1.39599	−0.697994	+	0.716104i	$0.745924\pi$
$24$	0	0
$25$	−112.986	−0.903888
$26$	0	0
$27$	0	0
$28$	96.2569	0.649673
$29$	22.5438	0.144354	0.0721771	−	0.997392i	$-0.477005\pi$
$29$	22.5438	0.144354	0.0721771	+	0.997392i	$0.477005\pi$
$30$	0	0
$31$	241.319	1.39814	0.699068	−	0.715055i	$-0.253598\pi$
$31$	241.319	1.39814	0.699068	+	0.715055i	$0.253598\pi$
$32$	173.335	0.957552
$33$	0	0
$34$	−184.400	−0.930128
$35$	73.5806	0.355354
$36$	0	0
$37$	−34.5346	−0.153444	−0.0767222	−	0.997053i	$-0.524445\pi$
$37$	−34.5346	−0.153444	−0.0767222	+	0.997053i	$0.524445\pi$
$38$	−119.360	−0.509547
$39$	0	0
$40$	80.8796	0.319705
$41$	−117.778	−0.448630	−0.224315	−	0.974517i	$-0.572014\pi$
$41$	−117.778	−0.448630	−0.224315	+	0.974517i	$0.572014\pi$
$42$	0	0
$43$	−101.468	−0.359856	−0.179928	−	0.983680i	$-0.557586\pi$
$43$	−101.468	−0.359856	−0.179928	+	0.983680i	$0.557586\pi$
$44$	245.561	0.841356
$45$	0	0
$46$	−286.660	−0.918819
$47$	451.273	1.40053	0.700265	−	0.713883i	$-0.253065\pi$
$47$	451.273	1.40053	0.700265	+	0.713883i	$0.253065\pi$
$48$	0	0
$49$	107.649	0.313844
$50$	−210.338	−0.594926
$51$	0	0
$52$	0	0
$53$	−6.41346	−0.0166218	−0.00831091	−	0.999965i	$-0.502645\pi$
$53$	−6.41346	−0.0166218	−0.00831091	+	0.999965i	$0.502645\pi$
$54$	0	0
$55$	187.711	0.460199
$56$	495.352	1.18204
$57$	0	0
$58$	41.9682	0.0950119
$59$	−303.502	−0.669704	−0.334852	−	0.942271i	$-0.608686\pi$
$59$	−303.502	−0.669704	−0.334852	+	0.942271i	$0.608686\pi$
$60$	0	0
$61$	−622.411	−1.30642	−0.653210	−	0.757177i	$-0.726578\pi$
$61$	−622.411	−1.30642	−0.653210	+	0.757177i	$0.726578\pi$
$62$	449.248	0.920234
$63$	0	0
$64$	380.008	0.742204
$65$	0	0
$66$	0	0
$67$	−289.575	−0.528018	−0.264009	−	0.964520i	$-0.585045\pi$
$67$	−289.575	−0.528018	−0.264009	+	0.964520i	$0.585045\pi$
$68$	449.138	0.800971
$69$	0	0
$70$	136.980	0.233889
$71$	−949.340	−1.58684	−0.793422	−	0.608671i	$-0.791703\pi$
$71$	−949.340	−1.58684	−0.793422	+	0.608671i	$0.791703\pi$
$72$	0	0
$73$	56.4739	0.0905448	0.0452724	−	0.998975i	$-0.485584\pi$
$73$	56.4739	0.0905448	0.0452724	+	0.998975i	$0.485584\pi$
$74$	−64.2906	−0.100995
$75$	0	0
$76$	290.723	0.438792
$77$	1149.65	1.70149
$78$	0	0
$79$	−968.723	−1.37962	−0.689809	−	0.723991i	$-0.742306\pi$
$79$	−968.723	−1.37962	−0.689809	+	0.723991i	$0.742306\pi$
$80$	24.8356	0.0347088
$81$	0	0
$82$	−219.259	−0.295282
$83$	480.985	0.636084	0.318042	−	0.948077i	$-0.396975\pi$
$83$	480.985	0.636084	0.318042	+	0.948077i	$0.396975\pi$
$84$	0	0
$85$	343.330	0.438110
$86$	−188.897	−0.236852
$87$	0	0
$88$	1263.69	1.53079
$89$	−240.037	−0.285887	−0.142943	−	0.989731i	$-0.545657\pi$
$89$	−240.037	−0.285887	−0.142943	+	0.989731i	$0.545657\pi$
$90$	0	0
$91$	0	0
$92$	698.210	0.791233
$93$	0	0
$94$	840.104	0.921810
$95$	222.234	0.240008
$96$	0	0
$97$	1871.81	1.95931	0.979657	−	0.200680i	$-0.0643152\pi$
$97$	1871.81	1.95931	0.979657	+	0.200680i	$0.0643152\pi$
$98$	200.402	0.206568
$99$	0	0
$100$	512.316	0.512316
$101$	−1798.53	−1.77189	−0.885944	−	0.463792i	$-0.846489\pi$
$101$	−1798.53	−1.77189	−0.885944	+	0.463792i	$0.846489\pi$
$102$	0	0
$103$	723.429	0.692054	0.346027	−	0.938225i	$-0.387531\pi$
$103$	723.429	0.692054	0.346027	+	0.938225i	$0.387531\pi$
$104$	0	0
$105$	0	0
$106$	−11.9395	−0.0109403
$107$	1767.01	1.59648	0.798241	−	0.602338i	$-0.205764\pi$
$107$	1767.01	1.59648	0.798241	+	0.602338i	$0.205764\pi$
$108$	0	0
$109$	401.486	0.352801	0.176401	−	0.984318i	$-0.443555\pi$
$109$	401.486	0.352801	0.176401	+	0.984318i	$0.443555\pi$
$110$	349.449	0.302897
$111$	0	0
$112$	152.107	0.128328
$113$	1042.23	0.867649	0.433824	−	0.900997i	$-0.357164\pi$
$113$	1042.23	0.867649	0.433824	+	0.900997i	$0.357164\pi$
$114$	0	0
$115$	533.725	0.432784
$116$	−102.221	−0.0818187
$117$	0	0
$118$	−565.008	−0.440790
$119$	2102.74	1.61982
$120$	0	0
$121$	1601.86	1.20350
$122$	−1158.70	−0.859867
$123$	0	0
$124$	−1094.22	−0.792452
$125$	824.889	0.590243
$126$	0	0
$127$	818.340	0.571779	0.285889	−	0.958263i	$-0.407711\pi$
$127$	818.340	0.571779	0.285889	+	0.958263i	$0.407711\pi$
$128$	−679.248	−0.469044
$129$	0	0
$130$	0	0
$131$	−1359.91	−0.906993	−0.453496	−	0.891258i	$-0.649824\pi$
$131$	−1359.91	−0.906993	−0.453496	+	0.891258i	$0.649824\pi$
$132$	0	0
$133$	1361.09	0.887376
$134$	−539.082	−0.347534
$135$	0	0
$136$	2311.33	1.45732
$137$	−2233.38	−1.39278	−0.696390	−	0.717664i	$-0.745211\pi$
$137$	−2233.38	−1.39278	−0.696390	+	0.717664i	$0.745211\pi$
$138$	0	0
$139$	−3119.54	−1.90357	−0.951784	−	0.306768i	$-0.900752\pi$
$139$	−3119.54	−1.90357	−0.951784	+	0.306768i	$0.900752\pi$
$140$	−333.639	−0.201412
$141$	0	0
$142$	−1767.32	−1.04444
$143$	0	0
$144$	0	0
$145$	−78.1395	−0.0447527
$146$	105.134	0.0595953
$147$	0	0
$148$	156.591	0.0869710
$149$	2157.72	1.18636	0.593180	−	0.805070i	$-0.297872\pi$
$149$	2157.72	1.18636	0.593180	+	0.805070i	$0.297872\pi$
$150$	0	0
$151$	2588.04	1.39478	0.697391	−	0.716691i	$-0.254344\pi$
$151$	2588.04	1.39478	0.697391	+	0.716691i	$0.254344\pi$
$152$	1496.10	0.798354
$153$	0	0
$154$	2140.22	1.11989
$155$	−836.443	−0.433450
$156$	0	0
$157$	−1535.63	−0.780614	−0.390307	−	0.920685i	$-0.627631\pi$
$157$	−1535.63	−0.780614	−0.390307	+	0.920685i	$0.627631\pi$
$158$	−1803.40	−0.908046
$159$	0	0
$160$	−600.802	−0.296860
$161$	3268.83	1.60012
$162$	0	0
$163$	−2318.00	−1.11386	−0.556931	−	0.830559i	$-0.688021\pi$
$163$	−2318.00	−1.11386	−0.556931	+	0.830559i	$0.688021\pi$
$164$	534.044	0.254280
$165$	0	0
$166$	895.417	0.418662
$167$	−1684.04	−0.780327	−0.390163	−	0.920746i	$-0.627581\pi$
$167$	−1684.04	−0.780327	−0.390163	+	0.920746i	$0.627581\pi$
$168$	0	0
$169$	0	0
$170$	639.154	0.288358
$171$	0	0
$172$	460.091	0.203963
$173$	−1269.76	−0.558021	−0.279011	−	0.960288i	$-0.590007\pi$
$173$	−1269.76	−0.558021	−0.279011	+	0.960288i	$0.590007\pi$
$174$	0	0
$175$	2398.52	1.03606
$176$	388.039	0.166191
$177$	0	0
$178$	−446.861	−0.188167
$179$	2236.72	0.933967	0.466984	−	0.884266i	$-0.345341\pi$
$179$	2236.72	0.933967	0.466984	+	0.884266i	$0.345341\pi$
$180$	0	0
$181$	−3560.87	−1.46230	−0.731152	−	0.682214i	$-0.761017\pi$
$181$	−3560.87	−1.46230	−0.731152	+	0.682214i	$0.761017\pi$
$182$	0	0
$183$	0	0
$184$	3593.09	1.43960
$185$	119.701	0.0475708
$186$	0	0
$187$	5364.30	2.09773
$188$	−2046.22	−0.793808
$189$	0	0
$190$	413.718	0.157970
$191$	−859.826	−0.325732	−0.162866	−	0.986648i	$-0.552074\pi$
$191$	−859.826	−0.325732	−0.162866	+	0.986648i	$0.552074\pi$
$192$	0	0
$193$	4824.95	1.79952	0.899761	−	0.436383i	$-0.143741\pi$
$193$	4824.95	1.79952	0.899761	+	0.436383i	$0.143741\pi$
$194$	3484.62	1.28959
$195$	0	0
$196$	−488.114	−0.177884
$197$	−2722.78	−0.984721	−0.492360	−	0.870392i	$-0.663866\pi$
$197$	−2722.78	−0.984721	−0.492360	+	0.870392i	$0.663866\pi$
$198$	0	0
$199$	1752.86	0.624405	0.312203	−	0.950016i	$-0.398933\pi$
$199$	1752.86	0.624405	0.312203	+	0.950016i	$0.398933\pi$
$200$	2636.45	0.932125
$201$	0	0
$202$	−3348.21	−1.16623
$203$	−478.570	−0.165463
$204$	0	0
$205$	408.233	0.139084
$206$	1346.76	0.455500
$207$	0	0
$208$	0	0
$209$	3472.26	1.14919
$210$	0	0
$211$	3839.73	1.25279	0.626393	−	0.779508i	$-0.284531\pi$
$211$	3839.73	1.25279	0.626393	+	0.779508i	$0.284531\pi$
$212$	29.0807	0.00942110
$213$	0	0
$214$	3289.53	1.05078
$215$	351.703	0.111562
$216$	0	0
$217$	−5122.84	−1.60259
$218$	747.418	0.232209
$219$	0	0
$220$	−851.144	−0.260837
$221$	0	0
$222$	0	0
$223$	−1070.50	−0.321461	−0.160730	−	0.986998i	$-0.551385\pi$
$223$	−1070.50	−0.321461	−0.160730	+	0.986998i	$0.551385\pi$
$224$	−3679.65	−1.09757
$225$	0	0
$226$	1940.24	0.571074
$227$	−2762.09	−0.807606	−0.403803	−	0.914846i	$-0.632312\pi$
$227$	−2762.09	−0.807606	−0.403803	+	0.914846i	$0.632312\pi$
$228$	0	0
$229$	443.465	0.127969	0.0639847	−	0.997951i	$-0.479619\pi$
$229$	443.465	0.127969	0.0639847	+	0.997951i	$0.479619\pi$
$230$	993.599	0.284852
$231$	0	0
$232$	−526.043	−0.148864
$233$	−4535.09	−1.27512	−0.637561	−	0.770399i	$-0.720057\pi$
$233$	−4535.09	−1.27512	−0.637561	+	0.770399i	$0.720057\pi$
$234$	0	0
$235$	−1564.17	−0.434192
$236$	1376.18	0.379582
$237$	0	0
$238$	3914.53	1.06614
$239$	−6158.94	−1.66690	−0.833449	−	0.552596i	$-0.813637\pi$
$239$	−6158.94	−1.66690	−0.833449	+	0.552596i	$0.813637\pi$
$240$	0	0
$241$	−794.221	−0.212283	−0.106142	−	0.994351i	$-0.533850\pi$
$241$	−794.221	−0.212283	−0.106142	+	0.994351i	$0.533850\pi$
$242$	2982.07	0.792128
$243$	0	0
$244$	2822.22	0.740467
$245$	−373.123	−0.0972978
$246$	0	0
$247$	0	0
$248$	−5631.02	−1.44181
$249$	0	0
$250$	1535.64	0.388490
$251$	5736.65	1.44261	0.721303	−	0.692620i	$-0.243544\pi$
$251$	5736.65	1.44261	0.721303	+	0.692620i	$0.243544\pi$
$252$	0	0
$253$	8339.09	2.07223
$254$	1523.45	0.376337
$255$	0	0
$256$	−4304.58	−1.05092
$257$	−3815.47	−0.926080	−0.463040	−	0.886337i	$-0.653241\pi$
$257$	−3815.47	−0.926080	−0.463040	+	0.886337i	$0.653241\pi$
$258$	0	0
$259$	733.116	0.175883
$260$	0	0
$261$	0	0
$262$	−2531.65	−0.596970
$263$	1307.88	0.306644	0.153322	−	0.988176i	$-0.451003\pi$
$263$	1307.88	0.306644	0.153322	+	0.988176i	$0.451003\pi$
$264$	0	0
$265$	22.2299	0.00515309
$266$	2533.84	0.584059
$267$	0	0
$268$	1313.03	0.299276
$269$	−4631.27	−1.04972	−0.524858	−	0.851190i	$-0.675882\pi$
$269$	−4631.27	−1.04972	−0.524858	+	0.851190i	$0.675882\pi$
$270$	0	0
$271$	3912.21	0.876937	0.438468	−	0.898747i	$-0.355521\pi$
$271$	3912.21	0.876937	0.438468	+	0.898747i	$0.355521\pi$
$272$	709.736	0.158214
$273$	0	0
$274$	−4157.73	−0.916708
$275$	6118.86	1.34175
$276$	0	0
$277$	−7619.61	−1.65277	−0.826387	−	0.563103i	$-0.809607\pi$
$277$	−7619.61	−1.65277	−0.826387	+	0.563103i	$0.809607\pi$
$278$	−5807.43	−1.25290
$279$	0	0
$280$	−1716.95	−0.366455
$281$	−7136.58	−1.51506	−0.757531	−	0.652799i	$-0.773595\pi$
$281$	−7136.58	−1.51506	−0.757531	+	0.652799i	$0.773595\pi$
$282$	0	0
$283$	8223.69	1.72738	0.863689	−	0.504025i	$-0.168148\pi$
$283$	8223.69	1.72738	0.863689	+	0.504025i	$0.168148\pi$
$284$	4304.62	0.899410
$285$	0	0
$286$	0	0
$287$	2500.25	0.514233
$288$	0	0
$289$	4898.48	0.997044
$290$	−145.467	−0.0294556
$291$	0	0
$292$	−256.071	−0.0513200
$293$	6094.65	1.21520	0.607600	−	0.794243i	$-0.292133\pi$
$293$	6094.65	1.21520	0.607600	+	0.794243i	$0.292133\pi$
$294$	0	0
$295$	1051.97	0.207621
$296$	805.840	0.158238
$297$	0	0
$298$	4016.89	0.780846
$299$	0	0
$300$	0	0
$301$	2154.02	0.412478
$302$	4817.99	0.918026
$303$	0	0
$304$	459.406	0.0866734
$305$	2157.36	0.405016
$306$	0	0
$307$	−189.227	−0.0351783	−0.0175892	−	0.999845i	$-0.505599\pi$
$307$	−189.227	−0.0351783	−0.0175892	+	0.999845i	$0.505599\pi$
$308$	−5212.88	−0.964388
$309$	0	0
$310$	−1557.15	−0.285291
$311$	−2282.65	−0.416196	−0.208098	−	0.978108i	$-0.566727\pi$
$311$	−2282.65	−0.416196	−0.208098	+	0.978108i	$0.566727\pi$
$312$	0	0
$313$	−5417.71	−0.978361	−0.489180	−	0.872183i	$-0.662704\pi$
$313$	−5417.71	−0.978361	−0.489180	+	0.872183i	$0.662704\pi$
$314$	−2858.77	−0.513789
$315$	0	0
$316$	4392.51	0.781955
$317$	2033.88	0.360360	0.180180	−	0.983634i	$-0.442332\pi$
$317$	2033.88	0.360360	0.180180	+	0.983634i	$0.442332\pi$
$318$	0	0
$319$	−1220.88	−0.214282
$320$	−1317.16	−0.230098
$321$	0	0
$322$	6085.35	1.05318
$323$	6350.88	1.09403
$324$	0	0
$325$	0	0
$326$	−4315.26	−0.733128
$327$	0	0
$328$	2748.27	0.462645
$329$	−9579.84	−1.60533
$330$	0	0
$331$	6413.08	1.06494	0.532469	−	0.846449i	$-0.321264\pi$
$331$	6413.08	1.06494	0.532469	+	0.846449i	$0.321264\pi$
$332$	−2180.95	−0.360527
$333$	0	0
$334$	−3135.05	−0.513600
$335$	1003.70	0.163696
$336$	0	0
$337$	10577.9	1.70984	0.854922	−	0.518756i	$-0.173605\pi$
$337$	10577.9	1.70984	0.854922	+	0.518756i	$0.173605\pi$
$338$	0	0
$339$	0	0
$340$	−1556.77	−0.248317
$341$	−13068.9	−2.07542
$342$	0	0
$343$	4996.16	0.786493
$344$	2367.70	0.371098
$345$	0	0
$346$	−2363.82	−0.367282
$347$	−5339.55	−0.826058	−0.413029	−	0.910718i	$-0.635529\pi$
$347$	−5339.55	−0.826058	−0.413029	+	0.910718i	$0.635529\pi$
$348$	0	0
$349$	2188.63	0.335687	0.167843	−	0.985814i	$-0.446320\pi$
$349$	2188.63	0.335687	0.167843	+	0.985814i	$0.446320\pi$
$350$	4465.16	0.681923
$351$	0	0
$352$	−9387.13	−1.42141
$353$	−5534.31	−0.834452	−0.417226	−	0.908803i	$-0.636998\pi$
$353$	−5534.31	−0.834452	−0.417226	+	0.908803i	$0.636998\pi$
$354$	0	0
$355$	3290.53	0.491953
$356$	1088.41	0.162038
$357$	0	0
$358$	4163.94	0.614724
$359$	−4937.93	−0.725945	−0.362972	−	0.931800i	$-0.618238\pi$
$359$	−4937.93	−0.725945	−0.362972	+	0.931800i	$0.618238\pi$
$360$	0	0
$361$	−2748.14	−0.400662
$362$	−6629.02	−0.962468
$363$	0	0
$364$	0	0
$365$	−195.746	−0.0280707
$366$	0	0
$367$	−451.014	−0.0641492	−0.0320746	−	0.999485i	$-0.510211\pi$
$367$	−451.014	−0.0641492	−0.0320746	+	0.999485i	$0.510211\pi$
$368$	1103.32	0.156290
$369$	0	0
$370$	222.839	0.0313104
$371$	136.148	0.0190524
$372$	0	0
$373$	−3759.18	−0.521831	−0.260915	−	0.965362i	$-0.584024\pi$
$373$	−3759.18	−0.521831	−0.260915	+	0.965362i	$0.584024\pi$
$374$	9986.35	1.38070
$375$	0	0
$376$	−10530.1	−1.44428
$377$	0	0
$378$	0	0
$379$	−6247.82	−0.846778	−0.423389	−	0.905948i	$-0.639160\pi$
$379$	−6247.82	−0.846778	−0.423389	+	0.905948i	$0.639160\pi$
$380$	−1007.68	−0.136034
$381$	0	0
$382$	−1600.68	−0.214392
$383$	−7950.94	−1.06077	−0.530384	−	0.847757i	$-0.677952\pi$
$383$	−7950.94	−1.06077	−0.530384	+	0.847757i	$0.677952\pi$
$384$	0	0
$385$	−3984.82	−0.527495
$386$	8982.28	1.18442
$387$	0	0
$388$	−8487.40	−1.11052
$389$	−11176.0	−1.45668	−0.728338	−	0.685218i	$-0.759707\pi$
$389$	−11176.0	−1.45668	−0.728338	+	0.685218i	$0.759707\pi$
$390$	0	0
$391$	15252.5	1.97276
$392$	−2511.90	−0.323649
$393$	0	0
$394$	−5068.81	−0.648129
$395$	3357.72	0.427709
$396$	0	0
$397$	−7407.12	−0.936404	−0.468202	−	0.883621i	$-0.655098\pi$
$397$	−7407.12	−0.936404	−0.468202	+	0.883621i	$0.655098\pi$
$398$	3263.17	0.410975
$399$	0	0
$400$	809.570	0.101196
$401$	4464.87	0.556022	0.278011	−	0.960578i	$-0.410325\pi$
$401$	4464.87	0.556022	0.278011	+	0.960578i	$0.410325\pi$
$402$	0	0
$403$	0	0
$404$	8155.14	1.00429
$405$	0	0
$406$	−890.921	−0.108906
$407$	1870.25	0.227776
$408$	0	0
$409$	−2362.97	−0.285676	−0.142838	−	0.989746i	$-0.545623\pi$
$409$	−2362.97	−0.285676	−0.142838	+	0.989746i	$0.545623\pi$
$410$	759.980	0.0915433
$411$	0	0
$412$	−3280.26	−0.392250
$413$	6442.88	0.767635
$414$	0	0
$415$	−1667.16	−0.197199
$416$	0	0
$417$	0	0
$418$	6464.07	0.756382
$419$	−15637.3	−1.82323	−0.911614	−	0.411048i	$-0.865163\pi$
$419$	−15637.3	−1.82323	−0.911614	+	0.411048i	$0.865163\pi$
$420$	0	0
$421$	4087.68	0.473210	0.236605	−	0.971606i	$-0.423965\pi$
$421$	4087.68	0.473210	0.236605	+	0.971606i	$0.423965\pi$
$422$	7148.16	0.824566
$423$	0	0
$424$	149.654	0.0171411
$425$	11191.6	1.27735
$426$	0	0
$427$	13212.8	1.49746
$428$	−8012.22	−0.904872
$429$	0	0
$430$	654.740	0.0734288
$431$	−1822.93	−0.203730	−0.101865	−	0.994798i	$-0.532481\pi$
$431$	−1822.93	−0.203730	−0.101865	+	0.994798i	$0.532481\pi$
$432$	0	0
$433$	−6100.85	−0.677109	−0.338555	−	0.940947i	$-0.609938\pi$
$433$	−6100.85	−0.677109	−0.338555	+	0.940947i	$0.609938\pi$
$434$	−9536.85	−1.05480
$435$	0	0
$436$	−1820.47	−0.199965
$437$	9872.78	1.08073
$438$	0	0
$439$	−2269.61	−0.246749	−0.123374	−	0.992360i	$-0.539372\pi$
$439$	−2269.61	−0.246749	−0.123374	+	0.992360i	$0.539372\pi$
$440$	−4380.11	−0.474576
$441$	0	0
$442$	0	0
$443$	1508.03	0.161735	0.0808674	−	0.996725i	$-0.474231\pi$
$443$	1508.03	0.161735	0.0808674	+	0.996725i	$0.474231\pi$
$444$	0	0
$445$	832.000	0.0886305
$446$	−1992.87	−0.211581
$447$	0	0
$448$	−8067.00	−0.850737
$449$	−4573.69	−0.480725	−0.240363	−	0.970683i	$-0.577266\pi$
$449$	−4573.69	−0.480725	−0.240363	+	0.970683i	$0.577266\pi$
$450$	0	0
$451$	6378.37	0.665955
$452$	−4725.79	−0.491776
$453$	0	0
$454$	−5142.00	−0.531555
$455$	0	0
$456$	0	0
$457$	8999.57	0.921186	0.460593	−	0.887611i	$-0.347637\pi$
$457$	8999.57	0.921186	0.460593	+	0.887611i	$0.347637\pi$
$458$	825.569	0.0842277
$459$	0	0
$460$	−2420.08	−0.245298
$461$	4945.51	0.499643	0.249821	−	0.968292i	$-0.419628\pi$
$461$	4945.51	0.499643	0.249821	+	0.968292i	$0.419628\pi$
$462$	0	0
$463$	3371.96	0.338463	0.169231	−	0.985576i	$-0.445872\pi$
$463$	3371.96	0.338463	0.169231	+	0.985576i	$0.445872\pi$
$464$	−161.531	−0.0161614
$465$	0	0
$466$	−8442.67	−0.839268
$467$	6411.64	0.635322	0.317661	−	0.948204i	$-0.397103\pi$
$467$	6411.64	0.635322	0.317661	+	0.948204i	$0.397103\pi$
$468$	0	0
$469$	6147.24	0.605230
$470$	−2911.91	−0.285779
$471$	0	0
$472$	7082.00	0.690626
$473$	5495.12	0.534177
$474$	0	0
$475$	7244.21	0.699762
$476$	−9534.53	−0.918098
$477$	0	0
$478$	−11465.7	−1.09713
$479$	13472.4	1.28512	0.642559	−	0.766237i	$-0.277873\pi$
$479$	13472.4	1.28512	0.642559	+	0.766237i	$0.277873\pi$
$480$	0	0
$481$	0	0
$482$	−1478.55	−0.139722
$483$	0	0
$484$	−7263.37	−0.682134
$485$	−6487.93	−0.607426
$486$	0	0
$487$	−14439.2	−1.34354	−0.671770	−	0.740760i	$-0.734466\pi$
$487$	−14439.2	−1.34354	−0.671770	+	0.740760i	$0.734466\pi$
$488$	14523.5	1.34723
$489$	0	0
$490$	−694.618	−0.0640401
$491$	−12856.5	−1.18168	−0.590842	−	0.806787i	$-0.701204\pi$
$491$	−12856.5	−1.18168	−0.590842	+	0.806787i	$0.701204\pi$
$492$	0	0
$493$	−2233.03	−0.203997
$494$	0	0
$495$	0	0
$496$	−1729.11	−0.156531
$497$	20153.1	1.81889
$498$	0	0
$499$	2996.45	0.268817	0.134409	−	0.990926i	$-0.457087\pi$
$499$	2996.45	0.268817	0.134409	+	0.990926i	$0.457087\pi$
$500$	−3740.32	−0.334544
$501$	0	0
$502$	10679.5	0.949503
$503$	−20295.0	−1.79902	−0.899511	−	0.436899i	$-0.856077\pi$
$503$	−20295.0	−1.79902	−0.899511	+	0.436899i	$0.856077\pi$
$504$	0	0
$505$	6233.94	0.549320
$506$	15524.3	1.36391
$507$	0	0
$508$	−3710.62	−0.324079
$509$	16270.4	1.41684	0.708421	−	0.705790i	$-0.249408\pi$
$509$	16270.4	1.41684	0.708421	+	0.705790i	$0.249408\pi$
$510$	0	0
$511$	−1198.86	−0.103785
$512$	−2579.55	−0.222658
$513$	0	0
$514$	−7103.00	−0.609533
$515$	−2507.50	−0.214550
$516$	0	0
$517$	−24439.1	−2.07897
$518$	1364.79	0.115764
$519$	0	0
$520$	0	0
$521$	−10390.5	−0.873734	−0.436867	−	0.899526i	$-0.643912\pi$
$521$	−10390.5	−0.873734	−0.436867	+	0.899526i	$0.643912\pi$
$522$	0	0
$523$	−14355.4	−1.20023	−0.600113	−	0.799916i	$-0.704878\pi$
$523$	−14355.4	−1.20023	−0.600113	+	0.799916i	$0.704878\pi$
$524$	6166.29	0.514076
$525$	0	0
$526$	2434.79	0.201829
$527$	−23903.4	−1.97580
$528$	0	0
$529$	11543.8	0.948780
$530$	41.3838	0.00339169
$531$	0	0
$532$	−6171.61	−0.502957
$533$	0	0
$534$	0	0
$535$	−6124.69	−0.494941
$536$	6757.03	0.544513
$537$	0	0
$538$	−8621.72	−0.690909
$539$	−5829.80	−0.465876
$540$	0	0
$541$	−14187.5	−1.12748	−0.563741	−	0.825951i	$-0.690638\pi$
$541$	−14187.5	−1.12748	−0.563741	+	0.825951i	$0.690638\pi$
$542$	7283.09	0.577188
$543$	0	0
$544$	−17169.4	−1.35318
$545$	−1391.60	−0.109375
$546$	0	0
$547$	8864.63	0.692915	0.346457	−	0.938066i	$-0.387385\pi$
$547$	8864.63	0.692915	0.346457	+	0.938066i	$0.387385\pi$
$548$	10126.9	0.789415
$549$	0	0
$550$	11391.1	0.883120
$551$	−1445.42	−0.111755
$552$	0	0
$553$	20564.5	1.58136
$554$	−14184.9	−1.08783
$555$	0	0
$556$	14145.0	1.07893
$557$	−16846.3	−1.28151	−0.640755	−	0.767746i	$-0.721378\pi$
$557$	−16846.3	−1.28151	−0.640755	+	0.767746i	$0.721378\pi$
$558$	0	0
$559$	0	0
$560$	−527.222	−0.0397842
$561$	0	0
$562$	−13285.7	−0.997193
$563$	4754.01	0.355875	0.177938	−	0.984042i	$-0.443057\pi$
$563$	4754.01	0.355875	0.177938	+	0.984042i	$0.443057\pi$
$564$	0	0
$565$	−3612.49	−0.268988
$566$	15309.5	1.13694
$567$	0	0
$568$	22152.2	1.63642
$569$	8299.82	0.611506	0.305753	−	0.952111i	$-0.401092\pi$
$569$	8299.82	0.611506	0.305753	+	0.952111i	$0.401092\pi$
$570$	0	0
$571$	9383.57	0.687723	0.343862	−	0.939020i	$-0.388265\pi$
$571$	9383.57	0.687723	0.343862	+	0.939020i	$0.388265\pi$
$572$	0	0
$573$	0	0
$574$	4654.54	0.338461
$575$	17397.9	1.26182
$576$	0	0
$577$	−24230.2	−1.74821	−0.874105	−	0.485736i	$-0.838551\pi$
$577$	−24230.2	−1.74821	−0.874105	+	0.485736i	$0.838551\pi$
$578$	9119.16	0.656241
$579$	0	0
$580$	354.310	0.0253654
$581$	−10210.6	−0.729099
$582$	0	0
$583$	347.327	0.0246738
$584$	−1317.78	−0.0933735
$585$	0	0
$586$	11346.0	0.799827
$587$	−446.729	−0.0314113	−0.0157057	−	0.999877i	$-0.504999\pi$
$587$	−446.729	−0.0314113	−0.0157057	+	0.999877i	$0.504999\pi$
$588$	0	0
$589$	−15472.4	−1.08239
$590$	1958.39	0.136654
$591$	0	0
$592$	247.448	0.0171791
$593$	9449.99	0.654409	0.327205	−	0.944954i	$-0.393893\pi$
$593$	9449.99	0.654409	0.327205	+	0.944954i	$0.393893\pi$
$594$	0	0
$595$	−7288.37	−0.502175
$596$	−9783.83	−0.672418
$597$	0	0
$598$	0	0
$599$	3273.52	0.223293	0.111646	−	0.993748i	$-0.464388\pi$
$599$	3273.52	0.223293	0.111646	+	0.993748i	$0.464388\pi$
$600$	0	0
$601$	15211.4	1.03242	0.516211	−	0.856462i	$-0.327342\pi$
$601$	15211.4	1.03242	0.516211	+	0.856462i	$0.327342\pi$
$602$	4009.99	0.271487
$603$	0	0
$604$	−11735.0	−0.790550
$605$	−5552.25	−0.373109
$606$	0	0
$607$	14322.4	0.957705	0.478852	−	0.877895i	$-0.341053\pi$
$607$	14322.4	0.957705	0.478852	+	0.877895i	$0.341053\pi$
$608$	−11113.6	−0.741307
$609$	0	0
$610$	4016.20	0.266576
$611$	0	0
$612$	0	0
$613$	−17695.5	−1.16593	−0.582966	−	0.812497i	$-0.698108\pi$
$613$	−17695.5	−1.16593	−0.582966	+	0.812497i	$0.698108\pi$
$614$	−352.271	−0.0231539
$615$	0	0
$616$	−26826.2	−1.75464
$617$	−24230.3	−1.58100	−0.790499	−	0.612463i	$-0.790179\pi$
$617$	−24230.3	−1.58100	−0.790499	+	0.612463i	$0.790179\pi$
$618$	0	0
$619$	16761.3	1.08836	0.544178	−	0.838970i	$-0.316842\pi$
$619$	16761.3	1.08836	0.544178	+	0.838970i	$0.316842\pi$
$620$	3792.71	0.245676
$621$	0	0
$622$	−4249.45	−0.273935
$623$	5095.63	0.327692
$624$	0	0
$625$	11264.1	0.720901
$626$	−10085.8	−0.643943
$627$	0	0
$628$	6963.04	0.442445
$629$	3420.75	0.216843
$630$	0	0
$631$	26812.2	1.69156	0.845780	−	0.533531i	$-0.179135\pi$
$631$	26812.2	1.69156	0.845780	+	0.533531i	$0.179135\pi$
$632$	22604.5	1.42272
$633$	0	0
$634$	3786.33	0.237184
$635$	−2836.47	−0.177263
$636$	0	0
$637$	0	0
$638$	−2272.82	−0.141038
$639$	0	0
$640$	2354.36	0.145413
$641$	23803.2	1.46673	0.733363	−	0.679838i	$-0.237950\pi$
$641$	23803.2	1.46673	0.733363	+	0.679838i	$0.237950\pi$
$642$	0	0
$643$	−9688.90	−0.594235	−0.297117	−	0.954841i	$-0.596025\pi$
$643$	−9688.90	−0.594235	−0.297117	+	0.954841i	$0.596025\pi$
$644$	−14821.9	−0.906935
$645$	0	0
$646$	11823.0	0.720076
$647$	−14666.7	−0.891204	−0.445602	−	0.895231i	$-0.647010\pi$
$647$	−14666.7	−0.891204	−0.445602	+	0.895231i	$0.647010\pi$
$648$	0	0
$649$	16436.4	0.994122
$650$	0	0
$651$	0	0
$652$	10510.6	0.631327
$653$	−16774.9	−1.00528	−0.502642	−	0.864494i	$-0.667639\pi$
$653$	−16774.9	−1.00528	−0.502642	+	0.864494i	$0.667639\pi$
$654$	0	0
$655$	4713.63	0.281186
$656$	843.906	0.0502271
$657$	0	0
$658$	−17834.1	−1.05661
$659$	33153.6	1.95976	0.979880	−	0.199587i	$-0.0639599\pi$
$659$	33153.6	1.95976	0.979880	+	0.199587i	$0.0639599\pi$
$660$	0	0
$661$	−26998.6	−1.58869	−0.794343	−	0.607469i	$-0.792185\pi$
$661$	−26998.6	−1.58869	−0.794343	+	0.607469i	$0.792185\pi$
$662$	11938.8	0.700928
$663$	0	0
$664$	−11223.5	−0.655956
$665$	−4717.69	−0.275104
$666$	0	0
$667$	−3471.36	−0.201517
$668$	7635.97	0.442282
$669$	0	0
$670$	1868.52	0.107742
$671$	33707.2	1.93928
$672$	0	0
$673$	−31725.4	−1.81713	−0.908563	−	0.417749i	$-0.862819\pi$
$673$	−31725.4	−1.81713	−0.908563	+	0.417749i	$0.862819\pi$
$674$	19692.2	1.12540
$675$	0	0
$676$	0	0
$677$	28609.6	1.62416	0.812081	−	0.583544i	$-0.198335\pi$
$677$	28609.6	1.62416	0.812081	+	0.583544i	$0.198335\pi$
$678$	0	0
$679$	−39735.7	−2.24582
$680$	−8011.36	−0.451797
$681$	0	0
$682$	−24329.4	−1.36601
$683$	16876.6	0.945482	0.472741	−	0.881201i	$-0.343265\pi$
$683$	16876.6	0.945482	0.472741	+	0.881201i	$0.343265\pi$
$684$	0	0
$685$	7741.18	0.431789
$686$	9301.00	0.517659
$687$	0	0
$688$	727.045	0.0402882
$689$	0	0
$690$	0	0
$691$	−14682.7	−0.808329	−0.404165	−	0.914686i	$-0.632438\pi$
$691$	−14682.7	−0.808329	−0.404165	+	0.914686i	$0.632438\pi$
$692$	5757.49	0.316282
$693$	0	0
$694$	−9940.27	−0.543700
$695$	10812.7	0.590144
$696$	0	0
$697$	11666.3	0.633990
$698$	4074.42	0.220944
$699$	0	0
$700$	−10875.7	−0.587232
$701$	−12377.6	−0.666898	−0.333449	−	0.942768i	$-0.608213\pi$
$701$	−12377.6	−0.666898	−0.333449	+	0.942768i	$0.608213\pi$
$702$	0	0
$703$	2214.22	0.118792
$704$	−20579.7	−1.10174
$705$	0	0
$706$	−10302.8	−0.549224
$707$	38180.1	2.03099
$708$	0	0
$709$	25093.4	1.32920	0.664599	−	0.747200i	$-0.268602\pi$
$709$	25093.4	1.32920	0.664599	+	0.747200i	$0.268602\pi$
$710$	6125.76	0.323797
$711$	0	0
$712$	5601.10	0.294818
$713$	−37159.1	−1.95178
$714$	0	0
$715$	0	0
$716$	−10142.0	−0.529364
$717$	0	0
$718$	−9192.61	−0.477807
$719$	−19834.1	−1.02877	−0.514385	−	0.857559i	$-0.671980\pi$
$719$	−19834.1	−1.02877	−0.514385	+	0.857559i	$0.671980\pi$
$720$	0	0
$721$	−15357.3	−0.793253
$722$	−5116.02	−0.263710
$723$	0	0
$724$	16146.1	0.828821
$725$	−2547.13	−0.130480
$726$	0	0
$727$	26992.7	1.37703	0.688517	−	0.725221i	$-0.258262\pi$
$727$	26992.7	1.37703	0.688517	+	0.725221i	$0.258262\pi$
$728$	0	0
$729$	0	0
$730$	−364.406	−0.0184757
$731$	10050.7	0.508537
$732$	0	0
$733$	12248.4	0.617195	0.308597	−	0.951193i	$-0.400140\pi$
$733$	12248.4	0.617195	0.308597	+	0.951193i	$0.400140\pi$
$734$	−839.622	−0.0422221
$735$	0	0
$736$	−26690.7	−1.33673
$737$	15682.2	0.783800
$738$	0	0
$739$	13980.0	0.695889	0.347945	−	0.937515i	$-0.386880\pi$
$739$	13980.0	0.695889	0.347945	+	0.937515i	$0.386880\pi$
$740$	−542.764	−0.0269627
$741$	0	0
$742$	253.458	0.0125401
$743$	−13730.3	−0.677949	−0.338975	−	0.940795i	$-0.610080\pi$
$743$	−13730.3	−0.677949	−0.338975	+	0.940795i	$0.610080\pi$
$744$	0	0
$745$	−7478.94	−0.367795
$746$	−6998.20	−0.343462
$747$	0	0
$748$	−24323.5	−1.18898
$749$	−37511.0	−1.82994
$750$	0	0
$751$	−16795.7	−0.816092	−0.408046	−	0.912961i	$-0.633790\pi$
$751$	−16795.7	−0.816092	−0.408046	+	0.912961i	$0.633790\pi$
$752$	−3233.47	−0.156799
$753$	0	0
$754$	0	0
$755$	−8970.49	−0.432410
$756$	0	0
$757$	−3680.42	−0.176707	−0.0883533	−	0.996089i	$-0.528160\pi$
$757$	−3680.42	−0.176707	−0.0883533	+	0.996089i	$0.528160\pi$
$758$	−11631.1	−0.557338
$759$	0	0
$760$	−5185.68	−0.247506
$761$	−17732.8	−0.844695	−0.422347	−	0.906434i	$-0.638794\pi$
$761$	−17732.8	−0.844695	−0.422347	+	0.906434i	$0.638794\pi$
$762$	0	0
$763$	−8522.93	−0.404392
$764$	3898.74	0.184622
$765$	0	0
$766$	−14801.7	−0.698183
$767$	0	0
$768$	0	0
$769$	−5198.88	−0.243792	−0.121896	−	0.992543i	$-0.538897\pi$
$769$	−5198.88	−0.243792	−0.121896	+	0.992543i	$0.538897\pi$
$770$	−7418.27	−0.347190
$771$	0	0
$772$	−21877.9	−1.01995
$773$	10049.6	0.467605	0.233802	−	0.972284i	$-0.424883\pi$
$773$	10049.6	0.467605	0.233802	+	0.972284i	$0.424883\pi$
$774$	0	0
$775$	−27265.7	−1.26376
$776$	−43677.4	−2.02052
$777$	0	0
$778$	−20805.6	−0.958764
$779$	7551.45	0.347316
$780$	0	0
$781$	51412.4	2.35554
$782$	28394.5	1.29845
$783$	0	0
$784$	−771.326	−0.0351369
$785$	5322.68	0.242006
$786$	0	0
$787$	2283.57	0.103431	0.0517157	−	0.998662i	$-0.483531\pi$
$787$	2283.57	0.103431	0.0517157	+	0.998662i	$0.483531\pi$
$788$	12346.0	0.558131
$789$	0	0
$790$	6250.83	0.281512
$791$	−22124.9	−0.994526
$792$	0	0
$793$	0	0
$794$	−13789.3	−0.616328
$795$	0	0
$796$	−7948.02	−0.353907
$797$	15767.9	0.700787	0.350394	−	0.936603i	$-0.386048\pi$
$797$	15767.9	0.700787	0.350394	+	0.936603i	$0.386048\pi$
$798$	0	0
$799$	−44699.9	−1.97918
$800$	−19584.5	−0.865519
$801$	0	0
$802$	8311.94	0.365966
$803$	−3058.40	−0.134407
$804$	0	0
$805$	−11330.2	−0.496070
$806$	0	0
$807$	0	0
$808$	41967.5	1.82724
$809$	35014.4	1.52168	0.760842	−	0.648937i	$-0.224786\pi$
$809$	35014.4	1.52168	0.760842	+	0.648937i	$0.224786\pi$
$810$	0	0
$811$	15741.1	0.681561	0.340780	−	0.940143i	$-0.389309\pi$
$811$	15741.1	0.681561	0.340780	+	0.940143i	$0.389309\pi$
$812$	2169.99	0.0937831
$813$	0	0
$814$	3481.72	0.149919
$815$	8034.47	0.345319
$816$	0	0
$817$	6505.75	0.278589
$818$	−4398.98	−0.188028
$819$	0	0
$820$	−1851.06	−0.0788317
$821$	28093.1	1.19422	0.597111	−	0.802158i	$-0.296315\pi$
$821$	28093.1	1.19422	0.597111	+	0.802158i	$0.296315\pi$
$822$	0	0
$823$	−17076.7	−0.723276	−0.361638	−	0.932319i	$-0.617782\pi$
$823$	−17076.7	−0.723276	−0.361638	+	0.932319i	$0.617782\pi$
$824$	−16880.7	−0.713674
$825$	0	0
$826$	11994.3	0.505247
$827$	−21398.3	−0.899748	−0.449874	−	0.893092i	$-0.648531\pi$
$827$	−21398.3	−0.899748	−0.449874	+	0.893092i	$0.648531\pi$
$828$	0	0
$829$	−34951.3	−1.46430	−0.732152	−	0.681141i	$-0.761484\pi$
$829$	−34951.3	−1.46430	−0.732152	+	0.681141i	$0.761484\pi$
$830$	−3103.63	−0.129793
$831$	0	0
$832$	0	0
$833$	−10662.9	−0.443514
$834$	0	0
$835$	5837.08	0.241917
$836$	−15744.4	−0.651352
$837$	0	0
$838$	−29110.9	−1.20002
$839$	14168.0	0.582996	0.291498	−	0.956571i	$-0.405846\pi$
$839$	14168.0	0.582996	0.291498	+	0.956571i	$0.405846\pi$
$840$	0	0
$841$	−23880.8	−0.979162
$842$	7609.76	0.311460
$843$	0	0
$844$	−17410.6	−0.710068
$845$	0	0
$846$	0	0
$847$	−34005.1	−1.37949
$848$	45.9539	0.00186092
$849$	0	0
$850$	20834.6	0.840731
$851$	5317.74	0.214206
$852$	0	0
$853$	−2604.09	−0.104528	−0.0522639	−	0.998633i	$-0.516644\pi$
$853$	−2604.09	−0.104528	−0.0522639	+	0.998633i	$0.516644\pi$
$854$	24597.5	0.985606
$855$	0	0
$856$	−41232.0	−1.64636
$857$	7726.51	0.307973	0.153986	−	0.988073i	$-0.450789\pi$
$857$	7726.51	0.307973	0.153986	+	0.988073i	$0.450789\pi$
$858$	0	0
$859$	16487.4	0.654883	0.327441	−	0.944872i	$-0.393814\pi$
$859$	16487.4	0.654883	0.327441	+	0.944872i	$0.393814\pi$
$860$	−1594.74	−0.0632326
$861$	0	0
$862$	−3393.63	−0.134092
$863$	−19510.7	−0.769585	−0.384792	−	0.923003i	$-0.625727\pi$
$863$	−19510.7	−0.769585	−0.384792	+	0.923003i	$0.625727\pi$
$864$	0	0
$865$	4401.13	0.172998
$866$	−11357.5	−0.445664
$867$	0	0
$868$	23228.7	0.908332
$869$	52462.1	2.04793
$870$	0	0
$871$	0	0
$872$	−9368.39	−0.363823
$873$	0	0
$874$	18379.5	0.711322
$875$	−17511.1	−0.676554
$876$	0	0
$877$	8989.31	0.346120	0.173060	−	0.984911i	$-0.444635\pi$
$877$	8989.31	0.346120	0.173060	+	0.984911i	$0.444635\pi$
$878$	−4225.18	−0.162407
$879$	0	0
$880$	−1344.99	−0.0515224
$881$	−36597.1	−1.39953	−0.699766	−	0.714372i	$-0.746713\pi$
$881$	−36597.1	−1.39953	−0.699766	+	0.714372i	$0.746713\pi$
$882$	0	0
$883$	−5749.48	−0.219123	−0.109561	−	0.993980i	$-0.534945\pi$
$883$	−5749.48	−0.219123	−0.109561	+	0.993980i	$0.534945\pi$
$884$	0	0
$885$	0	0
$886$	2807.39	0.106452
$887$	−14331.4	−0.542503	−0.271252	−	0.962508i	$-0.587438\pi$
$887$	−14331.4	−0.542503	−0.271252	+	0.962508i	$0.587438\pi$
$888$	0	0
$889$	−17372.1	−0.655390
$890$	1548.88	0.0583353
$891$	0	0
$892$	4853.98	0.182201
$893$	−28933.8	−1.08425
$894$	0	0
$895$	−7752.75	−0.289548
$896$	14419.4	0.537632
$897$	0	0
$898$	−8514.52	−0.316407
$899$	5440.25	0.201827
$900$	0	0
$901$	635.272	0.0234894
$902$	11874.2	0.438323
$903$	0	0
$904$	−24319.6	−0.894754
$905$	12342.4	0.453343
$906$	0	0
$907$	−25035.2	−0.916516	−0.458258	−	0.888819i	$-0.651526\pi$
$907$	−25035.2	−0.916516	−0.458258	+	0.888819i	$0.651526\pi$
$908$	12524.2	0.457744
$909$	0	0
$910$	0	0
$911$	41464.5	1.50799	0.753996	−	0.656879i	$-0.228124\pi$
$911$	41464.5	1.50799	0.753996	+	0.656879i	$0.228124\pi$
$912$	0	0
$913$	−26048.2	−0.944216
$914$	16753.9	0.606312
$915$	0	0
$916$	−2010.82	−0.0725319
$917$	28868.9	1.03962
$918$	0	0
$919$	6761.65	0.242705	0.121353	−	0.992609i	$-0.461277\pi$
$919$	6761.65	0.242705	0.121353	+	0.992609i	$0.461277\pi$
$920$	−12454.1	−0.446304
$921$	0	0
$922$	9206.71	0.328858
$923$	0	0
$924$	0	0
$925$	3901.92	0.138697
$926$	6277.34	0.222771
$927$	0	0
$928$	3907.63	0.138227
$929$	−1309.62	−0.0462512	−0.0231256	−	0.999733i	$-0.507362\pi$
$929$	−1309.62	−0.0462512	−0.0231256	+	0.999733i	$0.507362\pi$
$930$	0	0
$931$	−6901.99	−0.242968
$932$	20563.6	0.722728
$933$	0	0
$934$	11936.1	0.418160
$935$	−18593.3	−0.650339
$936$	0	0
$937$	−31660.7	−1.10385	−0.551926	−	0.833893i	$-0.686107\pi$
$937$	−31660.7	−1.10385	−0.551926	+	0.833893i	$0.686107\pi$
$938$	11443.9	0.398354
$939$	0	0
$940$	7092.46	0.246096
$941$	−54545.0	−1.88960	−0.944802	−	0.327643i	$-0.893746\pi$
$941$	−54545.0	−1.88960	−0.944802	+	0.327643i	$0.893746\pi$
$942$	0	0
$943$	18135.8	0.626282
$944$	2174.66	0.0749778
$945$	0	0
$946$	10229.9	0.351588
$947$	25624.5	0.879286	0.439643	−	0.898173i	$-0.355105\pi$
$947$	25624.5	0.879286	0.439643	+	0.898173i	$0.355105\pi$
$948$	0	0
$949$	0	0
$950$	13486.0	0.460574
$951$	0	0
$952$	−49066.0	−1.67042
$953$	44620.8	1.51670	0.758348	−	0.651850i	$-0.226007\pi$
$953$	44620.8	1.51670	0.758348	+	0.651850i	$0.226007\pi$
$954$	0	0
$955$	2980.27	0.100983
$956$	27926.7	0.944783
$957$	0	0
$958$	25080.7	0.845846
$959$	47411.3	1.59645
$960$	0	0
$961$	28444.0	0.954786
$962$	0	0
$963$	0	0
$964$	3601.26	0.120320
$965$	−16723.9	−0.557887
$966$	0	0
$967$	13882.9	0.461679	0.230839	−	0.972992i	$-0.425853\pi$
$967$	13882.9	0.461679	0.230839	+	0.972992i	$0.425853\pi$
$968$	−37378.3	−1.24110
$969$	0	0
$970$	−12078.1	−0.399799
$971$	−13687.2	−0.452363	−0.226181	−	0.974085i	$-0.572624\pi$
$971$	−13687.2	−0.452363	−0.226181	+	0.974085i	$0.572624\pi$
$972$	0	0
$973$	66223.1	2.18193
$974$	−26880.5	−0.884299
$975$	0	0
$976$	4459.72	0.146262
$977$	−23474.5	−0.768694	−0.384347	−	0.923189i	$-0.625573\pi$
$977$	−23474.5	−0.768694	−0.384347	+	0.923189i	$0.625573\pi$
$978$	0	0
$979$	12999.4	0.424376
$980$	1691.86	0.0551476
$981$	0	0
$982$	−23934.1	−0.777768
$983$	−528.996	−0.0171641	−0.00858207	−	0.999963i	$-0.502732\pi$
$983$	−528.996	−0.0171641	−0.00858207	+	0.999963i	$0.502732\pi$
$984$	0	0
$985$	9437.49	0.305283
$986$	−4157.07	−0.134268
$987$	0	0
$988$	0	0
$989$	15624.4	0.502354
$990$	0	0
$991$	52472.4	1.68198	0.840989	−	0.541052i	$-0.181974\pi$
$991$	52472.4	1.68198	0.840989	+	0.541052i	$0.181974\pi$
$992$	41829.2	1.33879
$993$	0	0
$994$	37517.6	1.19717
$995$	−6075.62	−0.193578
$996$	0	0
$997$	−5163.27	−0.164015	−0.0820073	−	0.996632i	$-0.526133\pi$
$997$	−5163.27	−0.164015	−0.0820073	+	0.996632i	$0.526133\pi$
$998$	5578.29	0.176932
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.bn.1.13	yes	18
3.2	odd	2	inner	1521.4.a.bn.1.6	yes	18
13.12	even	2		1521.4.a.bm.1.6	✓	18
39.38	odd	2		1521.4.a.bm.1.13	yes	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1521.4.a.bm.1.6	✓	18	13.12	even	2
1521.4.a.bm.1.13	yes	18	39.38	odd	2
1521.4.a.bn.1.6	yes	18	3.2	odd	2	inner
1521.4.a.bn.1.13	yes	18	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+1.86163 q^{2} -4.53433 q^{4} -3.46613 q^{5} -21.2285 q^{7} -23.3343 q^{8} +O(q^{10})\)	\(q+1.86163 q^{2} -4.53433 q^{4} -3.46613 q^{5} -21.2285 q^{7} -23.3343 q^{8} -6.45265 q^{10} -54.1559 q^{11} -39.5196 q^{14} -7.16523 q^{16} -99.0529 q^{17} -64.1160 q^{19} +15.7166 q^{20} -100.818 q^{22} -153.983 q^{23} -112.986 q^{25} +96.2569 q^{28} +22.5438 q^{29} +241.319 q^{31} +173.335 q^{32} -184.400 q^{34} +73.5806 q^{35} -34.5346 q^{37} -119.360 q^{38} +80.8796 q^{40} -117.778 q^{41} -101.468 q^{43} +245.561 q^{44} -286.660 q^{46} +451.273 q^{47} +107.649 q^{49} -210.338 q^{50} -6.41346 q^{53} +187.711 q^{55} +495.352 q^{56} +41.9682 q^{58} -303.502 q^{59} -622.411 q^{61} +449.248 q^{62} +380.008 q^{64} -289.575 q^{67} +449.138 q^{68} +136.980 q^{70} -949.340 q^{71} +56.4739 q^{73} -64.2906 q^{74} +290.723 q^{76} +1149.65 q^{77} -968.723 q^{79} +24.8356 q^{80} -219.259 q^{82} +480.985 q^{83} +343.330 q^{85} -188.897 q^{86} +1263.69 q^{88} -240.037 q^{89} +698.210 q^{92} +840.104 q^{94} +222.234 q^{95} +1871.81 q^{97} +200.402 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 56 q^{4} + 94 q^{7}+O(q^{10}) \) 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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