LMFDB - Embedded newform 1521.4.a.bn.1.18

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			1.18
Root			$4.85886$ 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)

$f(q)$	$=$	$q+4.85886 q^{2} +15.6085 q^{4} +14.8714 q^{5} +29.6598 q^{7} +36.9687 q^{8} +O(q^{10})$	$q+4.85886 q^{2} +15.6085 q^{4} +14.8714 q^{5} +29.6598 q^{7} +36.9687 q^{8} +72.2582 q^{10} -60.0283 q^{11} +144.113 q^{14} +54.7574 q^{16} +49.0713 q^{17} +96.7335 q^{19} +232.121 q^{20} -291.669 q^{22} -42.4461 q^{23} +96.1597 q^{25} +462.946 q^{28} -176.826 q^{29} +269.559 q^{31} -29.6906 q^{32} +238.431 q^{34} +441.085 q^{35} +186.663 q^{37} +470.014 q^{38} +549.777 q^{40} -52.7630 q^{41} +527.456 q^{43} -936.952 q^{44} -206.240 q^{46} -61.3270 q^{47} +536.706 q^{49} +467.226 q^{50} +340.468 q^{53} -892.707 q^{55} +1096.48 q^{56} -859.172 q^{58} -595.111 q^{59} -415.671 q^{61} +1309.75 q^{62} -582.322 q^{64} -640.901 q^{67} +765.930 q^{68} +2143.17 q^{70} -551.454 q^{71} -431.954 q^{73} +906.968 q^{74} +1509.86 q^{76} -1780.43 q^{77} +998.743 q^{79} +814.322 q^{80} -256.368 q^{82} -461.462 q^{83} +729.762 q^{85} +2562.84 q^{86} -2219.17 q^{88} +1025.25 q^{89} -662.520 q^{92} -297.979 q^{94} +1438.57 q^{95} -261.585 q^{97} +2607.78 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	4.85886	1.71787	0.858933	−	0.512088i	$-0.171128\pi$
$2$	4.85886	1.71787	0.858933	+	0.512088i	$0.171128\pi$
$3$	0	0
$3$	0	0
$4$	15.6085	1.95106
$5$	14.8714	1.33014	0.665071	−	0.746780i	$-0.268401\pi$
$5$	14.8714	1.33014	0.665071	+	0.746780i	$0.268401\pi$
$6$	0	0
$7$	29.6598	1.60148	0.800741	−	0.599011i	$-0.204440\pi$
$7$	29.6598	1.60148	0.800741	+	0.599011i	$0.204440\pi$
$8$	36.9687	1.63380
$9$	0	0
$10$	72.2582	2.28501
$11$	−60.0283	−1.64538	−0.822691	−	0.568488i	$-0.807529\pi$
$11$	−60.0283	−1.64538	−0.822691	+	0.568488i	$0.807529\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	144.113	2.75113
$15$	0	0
$16$	54.7574	0.855585
$17$	49.0713	0.700091	0.350046	−	0.936733i	$-0.386166\pi$
$17$	49.0713	0.700091	0.350046	+	0.936733i	$0.386166\pi$
$18$	0	0
$19$	96.7335	1.16801	0.584005	−	0.811750i	$-0.301485\pi$
$19$	96.7335	1.16801	0.584005	+	0.811750i	$0.301485\pi$
$20$	232.121	2.59519
$21$	0	0
$22$	−291.669	−2.82655
$23$	−42.4461	−0.384810	−0.192405	−	0.981316i	$-0.561629\pi$
$23$	−42.4461	−0.384810	−0.192405	+	0.981316i	$0.561629\pi$
$24$	0	0
$25$	96.1597	0.769278
$26$	0	0
$27$	0	0
$28$	462.946	3.12459
$29$	−176.826	−1.13227	−0.566133	−	0.824314i	$-0.691561\pi$
$29$	−176.826	−1.13227	−0.566133	+	0.824314i	$0.691561\pi$
$30$	0	0
$31$	269.559	1.56175	0.780875	−	0.624688i	$-0.214774\pi$
$31$	269.559	1.56175	0.780875	+	0.624688i	$0.214774\pi$
$32$	−29.6906	−0.164019
$33$	0	0
$34$	238.431	1.20266
$35$	441.085	2.13020
$36$	0	0
$37$	186.663	0.829383	0.414691	−	0.909962i	$-0.363890\pi$
$37$	186.663	0.829383	0.414691	+	0.909962i	$0.363890\pi$
$38$	470.014	2.00648
$39$	0	0
$40$	549.777	2.17319
$41$	−52.7630	−0.200981	−0.100490	−	0.994938i	$-0.532041\pi$
$41$	−52.7630	−0.200981	−0.100490	+	0.994938i	$0.532041\pi$
$42$	0	0
$43$	527.456	1.87061	0.935306	−	0.353839i	$-0.115124\pi$
$43$	527.456	1.87061	0.935306	+	0.353839i	$0.115124\pi$
$44$	−936.952	−3.21025
$45$	0	0
$46$	−206.240	−0.661052
$47$	−61.3270	−0.190329	−0.0951644	−	0.995462i	$-0.530338\pi$
$47$	−61.3270	−0.190329	−0.0951644	+	0.995462i	$0.530338\pi$
$48$	0	0
$49$	536.706	1.56474
$50$	467.226	1.32152
$51$	0	0
$52$	0	0
$53$	340.468	0.882394	0.441197	−	0.897410i	$-0.354554\pi$
$53$	340.468	0.882394	0.441197	+	0.897410i	$0.354554\pi$
$54$	0	0
$55$	−892.707	−2.18859
$56$	1096.48	2.61650
$57$	0	0
$58$	−859.172	−1.94508
$59$	−595.111	−1.31317	−0.656583	−	0.754254i	$-0.727999\pi$
$59$	−595.111	−1.31317	−0.656583	+	0.754254i	$0.727999\pi$
$60$	0	0
$61$	−415.671	−0.872480	−0.436240	−	0.899830i	$-0.643690\pi$
$61$	−415.671	−0.872480	−0.436240	+	0.899830i	$0.643690\pi$
$62$	1309.75	2.68288
$63$	0	0
$64$	−582.322	−1.13735
$65$	0	0
$66$	0	0
$67$	−640.901	−1.16864	−0.584318	−	0.811525i	$-0.698638\pi$
$67$	−640.901	−1.16864	−0.584318	+	0.811525i	$0.698638\pi$
$68$	765.930	1.36592
$69$	0	0
$70$	2143.17	3.65939
$71$	−551.454	−0.921768	−0.460884	−	0.887461i	$-0.652467\pi$
$71$	−551.454	−0.921768	−0.460884	+	0.887461i	$0.652467\pi$
$72$	0	0
$73$	−431.954	−0.692553	−0.346276	−	0.938133i	$-0.612554\pi$
$73$	−431.954	−0.692553	−0.346276	+	0.938133i	$0.612554\pi$
$74$	906.968	1.42477
$75$	0	0
$76$	1509.86	2.27886
$77$	−1780.43	−2.63505
$78$	0	0
$79$	998.743	1.42237	0.711186	−	0.703004i	$-0.248159\pi$
$79$	998.743	1.42237	0.711186	+	0.703004i	$0.248159\pi$
$80$	814.322	1.13805
$81$	0	0
$82$	−256.368	−0.345258
$83$	−461.462	−0.610265	−0.305133	−	0.952310i	$-0.598701\pi$
$83$	−461.462	−0.610265	−0.305133	+	0.952310i	$0.598701\pi$
$84$	0	0
$85$	729.762	0.931221
$86$	2562.84	3.21346
$87$	0	0
$88$	−2219.17	−2.68823
$89$	1025.25	1.22108	0.610541	−	0.791985i	$-0.290952\pi$
$89$	1025.25	1.22108	0.610541	+	0.791985i	$0.290952\pi$
$90$	0	0
$91$	0	0
$92$	−662.520	−0.750788
$93$	0	0
$94$	−297.979	−0.326959
$95$	1438.57	1.55362
$96$	0	0
$97$	−261.585	−0.273813	−0.136907	−	0.990584i	$-0.543716\pi$
$97$	−261.585	−0.273813	−0.136907	+	0.990584i	$0.543716\pi$
$98$	2607.78	2.68802
$99$	0	0
$100$	1500.91	1.50091
$101$	−290.237	−0.285937	−0.142969	−	0.989727i	$-0.545665\pi$
$101$	−290.237	−0.285937	−0.142969	+	0.989727i	$0.545665\pi$
$102$	0	0
$103$	−198.640	−0.190025	−0.0950123	−	0.995476i	$-0.530289\pi$
$103$	−198.640	−0.190025	−0.0950123	+	0.995476i	$0.530289\pi$
$104$	0	0
$105$	0	0
$106$	1654.29	1.51583
$107$	−684.251	−0.618216	−0.309108	−	0.951027i	$-0.600030\pi$
$107$	−684.251	−0.618216	−0.309108	+	0.951027i	$0.600030\pi$
$108$	0	0
$109$	−1725.33	−1.51612	−0.758059	−	0.652186i	$-0.773852\pi$
$109$	−1725.33	−1.51612	−0.758059	+	0.652186i	$0.773852\pi$
$110$	−4337.54	−3.75971
$111$	0	0
$112$	1624.10	1.37020
$113$	2032.55	1.69209	0.846045	−	0.533111i	$-0.178977\pi$
$113$	2032.55	1.69209	0.846045	+	0.533111i	$0.178977\pi$
$114$	0	0
$115$	−631.235	−0.511852
$116$	−2759.99	−2.20912
$117$	0	0
$118$	−2891.56	−2.25584
$119$	1455.45	1.12118
$120$	0	0
$121$	2272.40	1.70728
$122$	−2019.69	−1.49880
$123$	0	0
$124$	4207.41	3.04707
$125$	−428.897	−0.306894
$126$	0	0
$127$	−1074.21	−0.750559	−0.375279	−	0.926912i	$-0.622453\pi$
$127$	−1074.21	−0.750559	−0.375279	+	0.926912i	$0.622453\pi$
$128$	−2591.90	−1.78979
$129$	0	0
$130$	0	0
$131$	−1969.09	−1.31328	−0.656641	−	0.754203i	$-0.728023\pi$
$131$	−1969.09	−1.31328	−0.656641	+	0.754203i	$0.728023\pi$
$132$	0	0
$133$	2869.10	1.87054
$134$	−3114.05	−2.00756
$135$	0	0
$136$	1814.10	1.14381
$137$	−0.346542	−0.000216110 0	−0.000108055	−	1.00000i	$-0.500034\pi$
$137$	−0.346542	−0.000216110 0	−0.000108055		1.00000i	$0.500034\pi$
$138$	0	0
$139$	−1863.25	−1.13697	−0.568486	−	0.822693i	$-0.692471\pi$
$139$	−1863.25	−1.13697	−0.568486	+	0.822693i	$0.692471\pi$
$140$	6884.67	4.15615
$141$	0	0
$142$	−2679.43	−1.58347
$143$	0	0
$144$	0	0
$145$	−2629.65	−1.50608
$146$	−2098.80	−1.18971
$147$	0	0
$148$	2913.53	1.61818
$149$	24.6640	0.0135608	0.00678038	−	0.999977i	$-0.497842\pi$
$149$	24.6640	0.0135608	0.00678038	+	0.999977i	$0.497842\pi$
$150$	0	0
$151$	−447.901	−0.241388	−0.120694	−	0.992690i	$-0.538512\pi$
$151$	−447.901	−0.241388	−0.120694	+	0.992690i	$0.538512\pi$
$152$	3576.11	1.90829
$153$	0	0
$154$	−8650.86	−4.52666
$155$	4008.73	2.07735
$156$	0	0
$157$	−547.454	−0.278290	−0.139145	−	0.990272i	$-0.544435\pi$
$157$	−547.454	−0.278290	−0.139145	+	0.990272i	$0.544435\pi$
$158$	4852.75	2.44344
$159$	0	0
$160$	−441.542	−0.218169
$161$	−1258.94	−0.616266
$162$	0	0
$163$	190.029	0.0913143	0.0456572	−	0.998957i	$-0.485462\pi$
$163$	190.029	0.0913143	0.0456572	+	0.998957i	$0.485462\pi$
$164$	−823.552	−0.392126
$165$	0	0
$166$	−2242.18	−1.04835
$167$	851.170	0.394404	0.197202	−	0.980363i	$-0.436814\pi$
$167$	851.170	0.394404	0.197202	+	0.980363i	$0.436814\pi$
$168$	0	0
$169$	0	0
$170$	3545.81	1.59971
$171$	0	0
$172$	8232.81	3.64968
$173$	2688.72	1.18162	0.590809	−	0.806811i	$-0.298809\pi$
$173$	2688.72	1.18162	0.590809	+	0.806811i	$0.298809\pi$
$174$	0	0
$175$	2852.08	1.23198
$176$	−3287.00	−1.40776
$177$	0	0
$178$	4981.54	2.09765
$179$	991.338	0.413944	0.206972	−	0.978347i	$-0.433639\pi$
$179$	991.338	0.413944	0.206972	+	0.978347i	$0.433639\pi$
$180$	0	0
$181$	225.981	0.0928011	0.0464006	−	0.998923i	$-0.485225\pi$
$181$	225.981	0.0928011	0.0464006	+	0.998923i	$0.485225\pi$
$182$	0	0
$183$	0	0
$184$	−1569.18	−0.628702
$185$	2775.94	1.10320
$186$	0	0
$187$	−2945.67	−1.15192
$188$	−957.222	−0.371344
$189$	0	0
$190$	6989.79	2.66891
$191$	2265.08	0.858090	0.429045	−	0.903283i	$-0.358850\pi$
$191$	2265.08	0.858090	0.429045	+	0.903283i	$0.358850\pi$
$192$	0	0
$193$	20.6018	0.00768369	0.00384184	−	0.999993i	$-0.498777\pi$
$193$	20.6018	0.00768369	0.00384184	+	0.999993i	$0.498777\pi$
$194$	−1271.00	−0.470375
$195$	0	0
$196$	8377.19	3.05291
$197$	72.6929	0.0262901	0.0131451	−	0.999914i	$-0.495816\pi$
$197$	72.6929	0.0262901	0.0131451	+	0.999914i	$0.495816\pi$
$198$	0	0
$199$	−3599.20	−1.28211	−0.641057	−	0.767494i	$-0.721504\pi$
$199$	−3599.20	−1.28211	−0.641057	+	0.767494i	$0.721504\pi$
$200$	3554.90	1.25685
$201$	0	0
$202$	−1410.22	−0.491202
$203$	−5244.63	−1.81330
$204$	0	0
$205$	−784.662	−0.267333
$206$	−965.161	−0.326437
$207$	0	0
$208$	0	0
$209$	−5806.74	−1.92182
$210$	0	0
$211$	−340.466	−0.111084	−0.0555419	−	0.998456i	$-0.517689\pi$
$211$	−340.466	−0.111084	−0.0555419	+	0.998456i	$0.517689\pi$
$212$	5314.20	1.72161
$213$	0	0
$214$	−3324.68	−1.06201
$215$	7844.04	2.48818
$216$	0	0
$217$	7995.08	2.50111
$218$	−8383.14	−2.60449
$219$	0	0
$220$	−13933.8	−4.27008
$221$	0	0
$222$	0	0
$223$	1963.17	0.589522	0.294761	−	0.955571i	$-0.404760\pi$
$223$	1963.17	0.589522	0.294761	+	0.955571i	$0.404760\pi$
$224$	−880.620	−0.262674
$225$	0	0
$226$	9875.87	2.90678
$227$	610.759	0.178579	0.0892897	−	0.996006i	$-0.471540\pi$
$227$	610.759	0.178579	0.0892897	+	0.996006i	$0.471540\pi$
$228$	0	0
$229$	4357.13	1.25732	0.628662	−	0.777679i	$-0.283603\pi$
$229$	4357.13	1.25732	0.628662	+	0.777679i	$0.283603\pi$
$230$	−3067.08	−0.879292
$231$	0	0
$232$	−6537.01	−1.84990
$233$	−2776.38	−0.780630	−0.390315	−	0.920681i	$-0.627634\pi$
$233$	−2776.38	−0.780630	−0.390315	+	0.920681i	$0.627634\pi$
$234$	0	0
$235$	−912.020	−0.253164
$236$	−9288.79	−2.56207
$237$	0	0
$238$	7071.82	1.92604
$239$	2740.18	0.741622	0.370811	−	0.928708i	$-0.379080\pi$
$239$	2740.18	0.741622	0.370811	+	0.928708i	$0.379080\pi$
$240$	0	0
$241$	−1734.06	−0.463489	−0.231745	−	0.972777i	$-0.574443\pi$
$241$	−1734.06	−0.463489	−0.231745	+	0.972777i	$0.574443\pi$
$242$	11041.3	2.93289
$243$	0	0
$244$	−6488.01	−1.70226
$245$	7981.60	2.08133
$246$	0	0
$247$	0	0
$248$	9965.24	2.55159
$249$	0	0
$250$	−2083.95	−0.527202
$251$	−1183.57	−0.297635	−0.148817	−	0.988865i	$-0.547547\pi$
$251$	−1183.57	−0.297635	−0.148817	+	0.988865i	$0.547547\pi$
$252$	0	0
$253$	2547.97	0.633159
$254$	−5219.45	−1.28936
$255$	0	0
$256$	−7935.08	−1.93727
$257$	−7585.34	−1.84109	−0.920546	−	0.390635i	$-0.872255\pi$
$257$	−7585.34	−1.84109	−0.920546	+	0.390635i	$0.872255\pi$
$258$	0	0
$259$	5536.39	1.32824
$260$	0	0
$261$	0	0
$262$	−9567.52	−2.25604
$263$	4698.49	1.10160	0.550801	−	0.834637i	$-0.314322\pi$
$263$	4698.49	1.10160	0.550801	+	0.834637i	$0.314322\pi$
$264$	0	0
$265$	5063.25	1.17371
$266$	13940.5	3.21335
$267$	0	0
$268$	−10003.5	−2.28008
$269$	−7395.94	−1.67635	−0.838175	−	0.545401i	$-0.816378\pi$
$269$	−7395.94	−1.67635	−0.838175	+	0.545401i	$0.816378\pi$
$270$	0	0
$271$	1931.83	0.433028	0.216514	−	0.976280i	$-0.430531\pi$
$271$	1931.83	0.433028	0.216514	+	0.976280i	$0.430531\pi$
$272$	2687.02	0.598987
$273$	0	0
$274$	−1.68380	−0.000371248 0
$275$	−5772.30	−1.26576
$276$	0	0
$277$	1830.40	0.397032	0.198516	−	0.980098i	$-0.436388\pi$
$277$	1830.40	0.397032	0.198516	+	0.980098i	$0.436388\pi$
$278$	−9053.29	−1.95317
$279$	0	0
$280$	16306.3	3.48031
$281$	273.145	0.0579875	0.0289938	−	0.999580i	$-0.490770\pi$
$281$	273.145	0.0579875	0.0289938	+	0.999580i	$0.490770\pi$
$282$	0	0
$283$	−1147.91	−0.241116	−0.120558	−	0.992706i	$-0.538468\pi$
$283$	−1147.91	−0.241116	−0.120558	+	0.992706i	$0.538468\pi$
$284$	−8607.37	−1.79843
$285$	0	0
$286$	0	0
$287$	−1564.94	−0.321867
$288$	0	0
$289$	−2505.00	−0.509872
$290$	−12777.1	−2.58724
$291$	0	0
$292$	−6742.15	−1.35121
$293$	−7106.25	−1.41690	−0.708450	−	0.705761i	$-0.750605\pi$
$293$	−7106.25	−1.41690	−0.708450	+	0.705761i	$0.750605\pi$
$294$	0	0
$295$	−8850.15	−1.74670
$296$	6900.67	1.35504
$297$	0	0
$298$	119.839	0.0232956
$299$	0	0
$300$	0	0
$301$	15644.3	2.99575
$302$	−2176.29	−0.414673
$303$	0	0
$304$	5296.88	0.999331
$305$	−6181.63	−1.16052
$306$	0	0
$307$	7964.11	1.48057	0.740287	−	0.672291i	$-0.234690\pi$
$307$	7964.11	1.48057	0.740287	+	0.672291i	$0.234690\pi$
$308$	−27789.9	−5.14115
$309$	0	0
$310$	19477.9	3.56861
$311$	7003.74	1.27700	0.638498	−	0.769623i	$-0.279556\pi$
$311$	7003.74	1.27700	0.638498	+	0.769623i	$0.279556\pi$
$312$	0	0
$313$	−1848.21	−0.333761	−0.166881	−	0.985977i	$-0.553369\pi$
$313$	−1848.21	−0.333761	−0.166881	+	0.985977i	$0.553369\pi$
$314$	−2660.00	−0.478065
$315$	0	0
$316$	15588.9	2.77514
$317$	2266.75	0.401619	0.200810	−	0.979630i	$-0.435643\pi$
$317$	2266.75	0.401619	0.200810	+	0.979630i	$0.435643\pi$
$318$	0	0
$319$	10614.6	1.86301
$320$	−8659.97	−1.51283
$321$	0	0
$322$	−6117.03	−1.05866
$323$	4746.84	0.817713
$324$	0	0
$325$	0	0
$326$	923.325	0.156866
$327$	0	0
$328$	−1950.58	−0.328362
$329$	−1818.95	−0.304808
$330$	0	0
$331$	−8471.48	−1.40675	−0.703376	−	0.710818i	$-0.748325\pi$
$331$	−8471.48	−1.40675	−0.703376	+	0.710818i	$0.748325\pi$
$332$	−7202.73	−1.19067
$333$	0	0
$334$	4135.71	0.677533
$335$	−9531.13	−1.55445
$336$	0	0
$337$	642.992	0.103935	0.0519674	−	0.998649i	$-0.483451\pi$
$337$	642.992	0.103935	0.0519674	+	0.998649i	$0.483451\pi$
$338$	0	0
$339$	0	0
$340$	11390.5	1.81687
$341$	−16181.2	−2.56968
$342$	0	0
$343$	5745.30	0.904423
$344$	19499.4	3.05621
$345$	0	0
$346$	13064.1	2.02986
$347$	11357.5	1.75706	0.878531	−	0.477686i	$-0.158524\pi$
$347$	11357.5	1.75706	0.878531	+	0.477686i	$0.158524\pi$
$348$	0	0
$349$	7539.82	1.15644	0.578219	−	0.815881i	$-0.303748\pi$
$349$	7539.82	1.15644	0.578219	+	0.815881i	$0.303748\pi$
$350$	13857.9	2.11638
$351$	0	0
$352$	1782.28	0.269874
$353$	−11978.3	−1.80607	−0.903033	−	0.429572i	$-0.858664\pi$
$353$	−11978.3	−1.80607	−0.903033	+	0.429572i	$0.858664\pi$
$354$	0	0
$355$	−8200.91	−1.22608
$356$	16002.6	2.38241
$357$	0	0
$358$	4816.77	0.711101
$359$	−8585.68	−1.26221	−0.631107	−	0.775696i	$-0.717399\pi$
$359$	−8585.68	−1.26221	−0.631107	+	0.775696i	$0.717399\pi$
$360$	0	0
$361$	2498.36	0.364246
$362$	1098.01	0.159420
$363$	0	0
$364$	0	0
$365$	−6423.77	−0.921193
$366$	0	0
$367$	2302.65	0.327514	0.163757	−	0.986501i	$-0.447639\pi$
$367$	2302.65	0.327514	0.163757	+	0.986501i	$0.447639\pi$
$368$	−2324.24	−0.329237
$369$	0	0
$370$	13487.9	1.89514
$371$	10098.2	1.41314
$372$	0	0
$373$	−8961.58	−1.24400	−0.622002	−	0.783016i	$-0.713680\pi$
$373$	−8961.58	−1.24400	−0.622002	+	0.783016i	$0.713680\pi$
$374$	−14312.6	−1.97884
$375$	0	0
$376$	−2267.18	−0.310959
$377$	0	0
$378$	0	0
$379$	4627.11	0.627120	0.313560	−	0.949568i	$-0.398478\pi$
$379$	4627.11	0.627120	0.313560	+	0.949568i	$0.398478\pi$
$380$	22453.9	3.03121
$381$	0	0
$382$	11005.7	1.47408
$383$	−2130.00	−0.284172	−0.142086	−	0.989854i	$-0.545381\pi$
$383$	−2130.00	−0.284172	−0.142086	+	0.989854i	$0.545381\pi$
$384$	0	0
$385$	−26477.6	−3.50499
$386$	100.101	0.0131995
$387$	0	0
$388$	−4082.94	−0.534227
$389$	−5935.17	−0.773586	−0.386793	−	0.922167i	$-0.626417\pi$
$389$	−5935.17	−0.773586	−0.386793	+	0.922167i	$0.626417\pi$
$390$	0	0
$391$	−2082.89	−0.269402
$392$	19841.3	2.55647
$393$	0	0
$394$	353.204	0.0451629
$395$	14852.7	1.89196
$396$	0	0
$397$	−11708.6	−1.48019	−0.740095	−	0.672502i	$-0.765219\pi$
$397$	−11708.6	−1.48019	−0.740095	+	0.672502i	$0.765219\pi$
$398$	−17488.0	−2.20250
$399$	0	0
$400$	5265.46	0.658182
$401$	−4600.01	−0.572852	−0.286426	−	0.958102i	$-0.592467\pi$
$401$	−4600.01	−0.572852	−0.286426	+	0.958102i	$0.592467\pi$
$402$	0	0
$403$	0	0
$404$	−4530.16	−0.557881
$405$	0	0
$406$	−25482.9	−3.11501
$407$	−11205.0	−1.36465
$408$	0	0
$409$	544.220	0.0657945	0.0328972	−	0.999459i	$-0.489527\pi$
$409$	544.220	0.0657945	0.0328972	+	0.999459i	$0.489527\pi$
$410$	−3812.56	−0.459242
$411$	0	0
$412$	−3100.47	−0.370750
$413$	−17650.9	−2.10301
$414$	0	0
$415$	−6862.60	−0.811739
$416$	0	0
$417$	0	0
$418$	−28214.2	−3.30143
$419$	−10617.4	−1.23793	−0.618965	−	0.785418i	$-0.712448\pi$
$419$	−10617.4	−1.23793	−0.618965	+	0.785418i	$0.712448\pi$
$420$	0	0
$421$	−1404.23	−0.162560	−0.0812801	−	0.996691i	$-0.525901\pi$
$421$	−1404.23	−0.162560	−0.0812801	+	0.996691i	$0.525901\pi$
$422$	−1654.28	−0.190827
$423$	0	0
$424$	12586.6	1.44165
$425$	4718.69	0.538564
$426$	0	0
$427$	−12328.8	−1.39726
$428$	−10680.1	−1.20618
$429$	0	0
$430$	38113.1	4.27436
$431$	−1352.99	−0.151209	−0.0756046	−	0.997138i	$-0.524089\pi$
$431$	−1352.99	−0.151209	−0.0756046	+	0.997138i	$0.524089\pi$
$432$	0	0
$433$	12465.1	1.38346	0.691728	−	0.722158i	$-0.256850\pi$
$433$	12465.1	1.38346	0.691728	+	0.722158i	$0.256850\pi$
$434$	38847.0	4.29658
$435$	0	0
$436$	−26929.8	−2.95804
$437$	−4105.96	−0.449461
$438$	0	0
$439$	4202.96	0.456939	0.228470	−	0.973551i	$-0.426628\pi$
$439$	4202.96	0.456939	0.228470	+	0.973551i	$0.426628\pi$
$440$	−33002.2	−3.57572
$441$	0	0
$442$	0	0
$443$	8198.85	0.879321	0.439661	−	0.898164i	$-0.355099\pi$
$443$	8198.85	0.879321	0.439661	+	0.898164i	$0.355099\pi$
$444$	0	0
$445$	15246.9	1.62421
$446$	9538.75	1.01272
$447$	0	0
$448$	−17271.6	−1.82144
$449$	642.438	0.0675245	0.0337623	−	0.999430i	$-0.489251\pi$
$449$	642.438	0.0675245	0.0337623	+	0.999430i	$0.489251\pi$
$450$	0	0
$451$	3167.27	0.330690
$452$	31725.1	3.30138
$453$	0	0
$454$	2967.59	0.306775
$455$	0	0
$456$	0	0
$457$	−1925.81	−0.197124	−0.0985620	−	0.995131i	$-0.531424\pi$
$457$	−1925.81	−0.197124	−0.0985620	+	0.995131i	$0.531424\pi$
$458$	21170.7	2.15991
$459$	0	0
$460$	−9852.63	−0.998655
$461$	9658.81	0.975826	0.487913	−	0.872892i	$-0.337758\pi$
$461$	9658.81	0.975826	0.487913	+	0.872892i	$0.337758\pi$
$462$	0	0
$463$	12025.9	1.20711	0.603556	−	0.797320i	$-0.293750\pi$
$463$	12025.9	1.20711	0.603556	+	0.797320i	$0.293750\pi$
$464$	−9682.53	−0.968750
$465$	0	0
$466$	−13490.0	−1.34102
$467$	2470.93	0.244842	0.122421	−	0.992478i	$-0.460934\pi$
$467$	2470.93	0.244842	0.122421	+	0.992478i	$0.460934\pi$
$468$	0	0
$469$	−19009.0	−1.87155
$470$	−4431.38	−0.434903
$471$	0	0
$472$	−22000.4	−2.14545
$473$	−31662.3	−3.07787
$474$	0	0
$475$	9301.86	0.898523
$476$	22717.4	2.18750
$477$	0	0
$478$	13314.2	1.27401
$479$	5606.05	0.534754	0.267377	−	0.963592i	$-0.413843\pi$
$479$	5606.05	0.534754	0.267377	+	0.963592i	$0.413843\pi$
$480$	0	0
$481$	0	0
$482$	−8425.58	−0.796213
$483$	0	0
$484$	35468.7	3.33102
$485$	−3890.14	−0.364211
$486$	0	0
$487$	778.566	0.0724440	0.0362220	−	0.999344i	$-0.488468\pi$
$487$	778.566	0.0724440	0.0362220	+	0.999344i	$0.488468\pi$
$488$	−15366.8	−1.42546
$489$	0	0
$490$	38781.5	3.57544
$491$	−9336.64	−0.858160	−0.429080	−	0.903266i	$-0.641162\pi$
$491$	−9336.64	−0.858160	−0.429080	+	0.903266i	$0.641162\pi$
$492$	0	0
$493$	−8677.08	−0.792690
$494$	0	0
$495$	0	0
$496$	14760.4	1.33621
$497$	−16356.0	−1.47619
$498$	0	0
$499$	−16615.2	−1.49057	−0.745287	−	0.666744i	$-0.767688\pi$
$499$	−16615.2	−1.49057	−0.745287	+	0.666744i	$0.767688\pi$
$500$	−6694.44	−0.598769
$501$	0	0
$502$	−5750.80	−0.511296
$503$	−7494.12	−0.664307	−0.332153	−	0.943225i	$-0.607775\pi$
$503$	−7494.12	−0.664307	−0.332153	+	0.943225i	$0.607775\pi$
$504$	0	0
$505$	−4316.24	−0.380337
$506$	12380.2	1.08768
$507$	0	0
$508$	−16766.9	−1.46439
$509$	11882.9	1.03478	0.517388	−	0.855751i	$-0.326904\pi$
$509$	11882.9	1.03478	0.517388	+	0.855751i	$0.326904\pi$
$510$	0	0
$511$	−12811.7	−1.10911
$512$	−17820.3	−1.53819
$513$	0	0
$514$	−36856.1	−3.16275
$515$	−2954.06	−0.252760
$516$	0	0
$517$	3681.35	0.313164
$518$	26900.5	2.28174
$519$	0	0
$520$	0	0
$521$	−20244.7	−1.70237	−0.851186	−	0.524864i	$-0.824116\pi$
$521$	−20244.7	−1.70237	−0.851186	+	0.524864i	$0.824116\pi$
$522$	0	0
$523$	−15231.8	−1.27350	−0.636748	−	0.771072i	$-0.719721\pi$
$523$	−15231.8	−1.27350	−0.636748	+	0.771072i	$0.719721\pi$
$524$	−30734.5	−2.56230
$525$	0	0
$526$	22829.3	1.89240
$527$	13227.6	1.09337
$528$	0	0
$529$	−10365.3	−0.851921
$530$	24601.6	2.01628
$531$	0	0
$532$	44782.4	3.64955
$533$	0	0
$534$	0	0
$535$	−10175.8	−0.822315
$536$	−23693.3	−1.90932
$537$	0	0
$538$	−35935.8	−2.87975
$539$	−32217.6	−2.57460
$540$	0	0
$541$	3599.15	0.286025	0.143013	−	0.989721i	$-0.454321\pi$
$541$	3599.15	0.286025	0.143013	+	0.989721i	$0.454321\pi$
$542$	9386.51	0.743884
$543$	0	0
$544$	−1456.96	−0.114828
$545$	−25658.2	−2.01665
$546$	0	0
$547$	−9212.26	−0.720087	−0.360044	−	0.932936i	$-0.617238\pi$
$547$	−9212.26	−0.720087	−0.360044	+	0.932936i	$0.617238\pi$
$548$	−5.40900	−0.000421644 0
$549$	0	0
$550$	−28046.8	−2.17440
$551$	−17105.0	−1.32250
$552$	0	0
$553$	29622.6	2.27790
$554$	8893.64	0.682048
$555$	0	0
$556$	−29082.6	−2.21830
$557$	25245.0	1.92040	0.960202	−	0.279306i	$-0.0901043\pi$
$557$	25245.0	1.92040	0.960202	+	0.279306i	$0.0901043\pi$
$558$	0	0
$559$	0	0
$560$	24152.7	1.82256
$561$	0	0
$562$	1327.18	0.0996148
$563$	1694.84	0.126872	0.0634359	−	0.997986i	$-0.479794\pi$
$563$	1694.84	0.126872	0.0634359	+	0.997986i	$0.479794\pi$
$564$	0	0
$565$	30226.9	2.25072
$566$	−5577.51	−0.414206
$567$	0	0
$568$	−20386.5	−1.50598
$569$	−494.893	−0.0364622	−0.0182311	−	0.999834i	$-0.505803\pi$
$569$	−494.893	−0.0364622	−0.0182311	+	0.999834i	$0.505803\pi$
$570$	0	0
$571$	−1750.91	−0.128325	−0.0641623	−	0.997939i	$-0.520438\pi$
$571$	−1750.91	−0.128325	−0.0641623	+	0.997939i	$0.520438\pi$
$572$	0	0
$573$	0	0
$574$	−7603.84	−0.552924
$575$	−4081.60	−0.296025
$576$	0	0
$577$	8535.56	0.615841	0.307920	−	0.951412i	$-0.400367\pi$
$577$	8535.56	0.615841	0.307920	+	0.951412i	$0.400367\pi$
$578$	−12171.5	−0.875892
$579$	0	0
$580$	−41045.0	−2.93845
$581$	−13686.9	−0.977328
$582$	0	0
$583$	−20437.7	−1.45188
$584$	−15968.7	−1.13149
$585$	0	0
$586$	−34528.3	−2.43404
$587$	−19499.0	−1.37105	−0.685527	−	0.728047i	$-0.740428\pi$
$587$	−19499.0	−1.37105	−0.685527	+	0.728047i	$0.740428\pi$
$588$	0	0
$589$	26075.4	1.82414
$590$	−43001.6	−3.00059
$591$	0	0
$592$	10221.2	0.709607
$593$	7599.31	0.526250	0.263125	−	0.964762i	$-0.415247\pi$
$593$	7599.31	0.526250	0.263125	+	0.964762i	$0.415247\pi$
$594$	0	0
$595$	21644.6	1.49133
$596$	384.968	0.0264579
$597$	0	0
$598$	0	0
$599$	−26240.9	−1.78994	−0.894971	−	0.446124i	$-0.852804\pi$
$599$	−26240.9	−1.78994	−0.894971	+	0.446124i	$0.852804\pi$
$600$	0	0
$601$	20319.6	1.37912	0.689561	−	0.724227i	$-0.257803\pi$
$601$	20319.6	1.37912	0.689561	+	0.724227i	$0.257803\pi$
$602$	76013.3	5.14630
$603$	0	0
$604$	−6991.06	−0.470964
$605$	33793.8	2.27093
$606$	0	0
$607$	1361.39	0.0910333	0.0455166	−	0.998964i	$-0.485507\pi$
$607$	1361.39	0.0910333	0.0455166	+	0.998964i	$0.485507\pi$
$608$	−2872.08	−0.191576
$609$	0	0
$610$	−30035.7	−1.99362
$611$	0	0
$612$	0	0
$613$	−13883.5	−0.914764	−0.457382	−	0.889270i	$-0.651213\pi$
$613$	−13883.5	−0.914764	−0.457382	+	0.889270i	$0.651213\pi$
$614$	38696.5	2.54343
$615$	0	0
$616$	−65820.1	−4.30514
$617$	15316.8	0.999403	0.499701	−	0.866198i	$-0.333443\pi$
$617$	15316.8	0.999403	0.499701	+	0.866198i	$0.333443\pi$
$618$	0	0
$619$	−10325.0	−0.670432	−0.335216	−	0.942141i	$-0.608809\pi$
$619$	−10325.0	−0.670432	−0.335216	+	0.942141i	$0.608809\pi$
$620$	62570.3	4.05304
$621$	0	0
$622$	34030.2	2.19371
$623$	30408.7	1.95554
$624$	0	0
$625$	−18398.3	−1.17749
$626$	−8980.21	−0.573357
$627$	0	0
$628$	−8544.93	−0.542962
$629$	9159.79	0.580643
$630$	0	0
$631$	20967.7	1.32284	0.661420	−	0.750016i	$-0.269954\pi$
$631$	20967.7	1.32284	0.661420	+	0.750016i	$0.269954\pi$
$632$	36922.2	2.32387
$633$	0	0
$634$	11013.8	0.689928
$635$	−15975.1	−0.998349
$636$	0	0
$637$	0	0
$638$	51574.6	3.20041
$639$	0	0
$640$	−38545.2	−2.38068
$641$	20292.1	1.25037	0.625187	−	0.780475i	$-0.285023\pi$
$641$	20292.1	1.25037	0.625187	+	0.780475i	$0.285023\pi$
$642$	0	0
$643$	9106.14	0.558493	0.279247	−	0.960219i	$-0.409915\pi$
$643$	9106.14	0.558493	0.279247	+	0.960219i	$0.409915\pi$
$644$	−19650.2	−1.20237
$645$	0	0
$646$	23064.2	1.40472
$647$	15251.4	0.926732	0.463366	−	0.886167i	$-0.346641\pi$
$647$	15251.4	0.926732	0.463366	+	0.886167i	$0.346641\pi$
$648$	0	0
$649$	35723.5	2.16066
$650$	0	0
$651$	0	0
$652$	2966.07	0.178160
$653$	−14421.0	−0.864220	−0.432110	−	0.901821i	$-0.642231\pi$
$653$	−14421.0	−0.864220	−0.432110	+	0.901821i	$0.642231\pi$
$654$	0	0
$655$	−29283.2	−1.74685
$656$	−2889.17	−0.171956
$657$	0	0
$658$	−8838.01	−0.523619
$659$	11890.9	0.702892	0.351446	−	0.936208i	$-0.385690\pi$
$659$	11890.9	0.702892	0.351446	+	0.936208i	$0.385690\pi$
$660$	0	0
$661$	−3696.05	−0.217488	−0.108744	−	0.994070i	$-0.534683\pi$
$661$	−3696.05	−0.217488	−0.108744	+	0.994070i	$0.534683\pi$
$662$	−41161.7	−2.41661
$663$	0	0
$664$	−17059.6	−0.997051
$665$	42667.6	2.48809
$666$	0	0
$667$	7505.57	0.435707
$668$	13285.5	0.769507
$669$	0	0
$670$	−46310.4	−2.67034
$671$	24952.0	1.43556
$672$	0	0
$673$	−9587.03	−0.549112	−0.274556	−	0.961571i	$-0.588531\pi$
$673$	−9587.03	−0.549112	−0.274556	+	0.961571i	$0.588531\pi$
$674$	3124.21	0.178546
$675$	0	0
$676$	0	0
$677$	12413.6	0.704718	0.352359	−	0.935865i	$-0.385380\pi$
$677$	12413.6	0.704718	0.352359	+	0.935865i	$0.385380\pi$
$678$	0	0
$679$	−7758.56	−0.438507
$680$	26978.3	1.52143
$681$	0	0
$682$	−78622.0	−4.41436
$683$	23012.6	1.28924	0.644622	−	0.764502i	$-0.277015\pi$
$683$	23012.6	1.28924	0.644622	+	0.764502i	$0.277015\pi$
$684$	0	0
$685$	−5.15357	−0.000287457 0
$686$	27915.6	1.55368
$687$	0	0
$688$	28882.2	1.60047
$689$	0	0
$690$	0	0
$691$	30604.2	1.68486	0.842431	−	0.538804i	$-0.181124\pi$
$691$	30604.2	1.68486	0.842431	+	0.538804i	$0.181124\pi$
$692$	41967.0	2.30541
$693$	0	0
$694$	55184.3	3.01840
$695$	−27709.3	−1.51233
$696$	0	0
$697$	−2589.15	−0.140705
$698$	36634.9	1.98661
$699$	0	0
$700$	44516.7	2.40368
$701$	4042.16	0.217789	0.108895	−	0.994053i	$-0.465269\pi$
$701$	4042.16	0.217789	0.108895	+	0.994053i	$0.465269\pi$
$702$	0	0
$703$	18056.5	0.968727
$704$	34955.8	1.87137
$705$	0	0
$706$	−58200.9	−3.10258
$707$	−8608.38	−0.457923
$708$	0	0
$709$	−33867.3	−1.79395	−0.896977	−	0.442077i	$-0.854242\pi$
$709$	−33867.3	−1.79395	−0.896977	+	0.442077i	$0.854242\pi$
$710$	−39847.1	−2.10624
$711$	0	0
$712$	37902.1	1.99500
$713$	−11441.7	−0.600976
$714$	0	0
$715$	0	0
$716$	15473.3	0.807632
$717$	0	0
$718$	−41716.6	−2.16831
$719$	−7082.45	−0.367358	−0.183679	−	0.982986i	$-0.558801\pi$
$719$	−7082.45	−0.367358	−0.183679	+	0.982986i	$0.558801\pi$
$720$	0	0
$721$	−5891.62	−0.304321
$722$	12139.2	0.625725
$723$	0	0
$724$	3527.22	0.181061
$725$	−17003.5	−0.871027
$726$	0	0
$727$	29511.0	1.50551	0.752754	−	0.658302i	$-0.228725\pi$
$727$	29511.0	1.50551	0.752754	+	0.658302i	$0.228725\pi$
$728$	0	0
$729$	0	0
$730$	−31212.2	−1.58249
$731$	25883.0	1.30960
$732$	0	0
$733$	19353.4	0.975217	0.487608	−	0.873062i	$-0.337869\pi$
$733$	19353.4	0.975217	0.487608	+	0.873062i	$0.337869\pi$
$734$	11188.3	0.562625
$735$	0	0
$736$	1260.25	0.0631162
$737$	38472.2	1.92285
$738$	0	0
$739$	35578.3	1.77100	0.885500	−	0.464639i	$-0.153816\pi$
$739$	35578.3	1.77100	0.885500	+	0.464639i	$0.153816\pi$
$740$	43328.3	2.15241
$741$	0	0
$742$	49065.9	2.42758
$743$	−12875.1	−0.635722	−0.317861	−	0.948137i	$-0.602964\pi$
$743$	−12875.1	−0.635722	−0.317861	+	0.948137i	$0.602964\pi$
$744$	0	0
$745$	366.789	0.0180377
$746$	−43543.1	−2.13703
$747$	0	0
$748$	−45977.5	−2.24747
$749$	−20294.8	−0.990061
$750$	0	0
$751$	−37817.4	−1.83752	−0.918758	−	0.394821i	$-0.870807\pi$
$751$	−37817.4	−1.83752	−0.918758	+	0.394821i	$0.870807\pi$
$752$	−3358.11	−0.162843
$753$	0	0
$754$	0	0
$755$	−6660.93	−0.321081
$756$	0	0
$757$	6185.73	0.296994	0.148497	−	0.988913i	$-0.452557\pi$
$757$	6185.73	0.296994	0.148497	+	0.988913i	$0.452557\pi$
$758$	22482.4	1.07731
$759$	0	0
$760$	53181.8	2.53830
$761$	15728.3	0.749213	0.374607	−	0.927184i	$-0.377778\pi$
$761$	15728.3	0.749213	0.374607	+	0.927184i	$0.377778\pi$
$762$	0	0
$763$	−51173.1	−2.42803
$764$	35354.5	1.67419
$765$	0	0
$766$	−10349.4	−0.488169
$767$	0	0
$768$	0	0
$769$	8203.97	0.384711	0.192356	−	0.981325i	$-0.438387\pi$
$769$	8203.97	0.384711	0.192356	+	0.981325i	$0.438387\pi$
$770$	−128651.	−6.02110
$771$	0	0
$772$	321.564	0.0149914
$773$	3418.73	0.159073	0.0795363	−	0.996832i	$-0.474656\pi$
$773$	3418.73	0.159073	0.0795363	+	0.996832i	$0.474656\pi$
$774$	0	0
$775$	25920.7	1.20142
$776$	−9670.43	−0.447356
$777$	0	0
$778$	−28838.1	−1.32892
$779$	−5103.95	−0.234747
$780$	0	0
$781$	33102.8	1.51666
$782$	−10120.5	−0.462796
$783$	0	0
$784$	29388.7	1.33877
$785$	−8141.42	−0.370165
$786$	0	0
$787$	26092.8	1.18184	0.590921	−	0.806730i	$-0.298765\pi$
$787$	26092.8	1.18184	0.590921	+	0.806730i	$0.298765\pi$
$788$	1134.63	0.0512937
$789$	0	0
$790$	72167.4	3.25013
$791$	60285.1	2.70985
$792$	0	0
$793$	0	0
$794$	−56890.2	−2.54277
$795$	0	0
$796$	−56178.1	−2.50148
$797$	−18636.4	−0.828275	−0.414138	−	0.910214i	$-0.635917\pi$
$797$	−18636.4	−0.828275	−0.414138	+	0.910214i	$0.635917\pi$
$798$	0	0
$799$	−3009.40	−0.133248
$800$	−2855.04	−0.126176
$801$	0	0
$802$	−22350.8	−0.984083
$803$	25929.4	1.13951
$804$	0	0
$805$	−18722.3	−0.819721
$806$	0	0
$807$	0	0
$808$	−10729.7	−0.467164
$809$	−9074.08	−0.394348	−0.197174	−	0.980368i	$-0.563176\pi$
$809$	−9074.08	−0.394348	−0.197174	+	0.980368i	$0.563176\pi$
$810$	0	0
$811$	−20112.2	−0.870818	−0.435409	−	0.900233i	$-0.643396\pi$
$811$	−20112.2	−0.870818	−0.435409	+	0.900233i	$0.643396\pi$
$812$	−81860.8	−3.53787
$813$	0	0
$814$	−54443.7	−2.34429
$815$	2826.01	0.121461
$816$	0	0
$817$	51022.7	2.18489
$818$	2644.29	0.113026
$819$	0	0
$820$	−12247.4	−0.521583
$821$	−29190.8	−1.24089	−0.620443	−	0.784252i	$-0.713047\pi$
$821$	−29190.8	−1.24089	−0.620443	+	0.784252i	$0.713047\pi$
$822$	0	0
$823$	27556.8	1.16716	0.583579	−	0.812056i	$-0.301652\pi$
$823$	27556.8	1.16716	0.583579	+	0.812056i	$0.301652\pi$
$824$	−7343.44	−0.310462
$825$	0	0
$826$	−85763.2	−3.61269
$827$	−16596.0	−0.697822	−0.348911	−	0.937156i	$-0.613448\pi$
$827$	−16596.0	−0.697822	−0.348911	+	0.937156i	$0.613448\pi$
$828$	0	0
$829$	20877.6	0.874681	0.437340	−	0.899296i	$-0.355920\pi$
$829$	20877.6	0.874681	0.437340	+	0.899296i	$0.355920\pi$
$830$	−33344.4	−1.39446
$831$	0	0
$832$	0	0
$833$	26336.9	1.09546
$834$	0	0
$835$	12658.1	0.524613
$836$	−90634.6	−3.74960
$837$	0	0
$838$	−51588.3	−2.12660
$839$	−31026.1	−1.27669	−0.638343	−	0.769752i	$-0.720380\pi$
$839$	−31026.1	−1.27669	−0.638343	+	0.769752i	$0.720380\pi$
$840$	0	0
$841$	6878.38	0.282028
$842$	−6822.94	−0.279257
$843$	0	0
$844$	−5314.17	−0.216731
$845$	0	0
$846$	0	0
$847$	67398.9	2.73418
$848$	18643.2	0.754963
$849$	0	0
$850$	22927.4	0.925181
$851$	−7923.10	−0.319154
$852$	0	0
$853$	−24134.3	−0.968748	−0.484374	−	0.874861i	$-0.660953\pi$
$853$	−24134.3	−0.968748	−0.484374	+	0.874861i	$0.660953\pi$
$854$	−59903.7	−2.40031
$855$	0	0
$856$	−25295.9	−1.01004
$857$	4872.72	0.194223	0.0971114	−	0.995274i	$-0.469040\pi$
$857$	4872.72	0.194223	0.0971114	+	0.995274i	$0.469040\pi$
$858$	0	0
$859$	17690.2	0.702658	0.351329	−	0.936252i	$-0.385730\pi$
$859$	17690.2	0.702658	0.351329	+	0.936252i	$0.385730\pi$
$860$	122434.	4.85460
$861$	0	0
$862$	−6573.98	−0.259757
$863$	32855.1	1.29595	0.647973	−	0.761664i	$-0.275617\pi$
$863$	32855.1	1.29595	0.647973	+	0.761664i	$0.275617\pi$
$864$	0	0
$865$	39985.2	1.57172
$866$	60566.4	2.37659
$867$	0	0
$868$	124791.	4.87983
$869$	−59952.8	−2.34035
$870$	0	0
$871$	0	0
$872$	−63783.2	−2.47703
$873$	0	0
$874$	−19950.3	−0.772114
$875$	−12721.0	−0.491484
$876$	0	0
$877$	10874.1	0.418691	0.209346	−	0.977842i	$-0.432867\pi$
$877$	10874.1	0.418691	0.209346	+	0.977842i	$0.432867\pi$
$878$	20421.6	0.784960
$879$	0	0
$880$	−48882.4	−1.87253
$881$	−7130.10	−0.272667	−0.136333	−	0.990663i	$-0.543532\pi$
$881$	−7130.10	−0.272667	−0.136333	+	0.990663i	$0.543532\pi$
$882$	0	0
$883$	12029.0	0.458446	0.229223	−	0.973374i	$-0.426381\pi$
$883$	12029.0	0.458446	0.229223	+	0.973374i	$0.426381\pi$
$884$	0	0
$885$	0	0
$886$	39837.1	1.51056
$887$	40837.9	1.54589	0.772944	−	0.634474i	$-0.218783\pi$
$887$	40837.9	1.54589	0.772944	+	0.634474i	$0.218783\pi$
$888$	0	0
$889$	−31861.0	−1.20201
$890$	74082.7	2.79018
$891$	0	0
$892$	30642.1	1.15019
$893$	−5932.37	−0.222306
$894$	0	0
$895$	14742.6	0.550605
$896$	−76875.2	−2.86632
$897$	0	0
$898$	3121.52	0.115998
$899$	−47665.0	−1.76832
$900$	0	0
$901$	16707.2	0.617756
$902$	15389.3	0.568081
$903$	0	0
$904$	75140.7	2.76454
$905$	3360.66	0.123439
$906$	0	0
$907$	−19247.0	−0.704616	−0.352308	−	0.935884i	$-0.614603\pi$
$907$	−19247.0	−0.704616	−0.352308	+	0.935884i	$0.614603\pi$
$908$	9533.04	0.348420
$909$	0	0
$910$	0	0
$911$	11860.9	0.431359	0.215680	−	0.976464i	$-0.430803\pi$
$911$	11860.9	0.431359	0.215680	+	0.976464i	$0.430803\pi$
$912$	0	0
$913$	27700.8	1.00412
$914$	−9357.25	−0.338633
$915$	0	0
$916$	68008.3	2.45312
$917$	−58402.8	−2.10320
$918$	0	0
$919$	−21211.9	−0.761390	−0.380695	−	0.924701i	$-0.624315\pi$
$919$	−21211.9	−0.761390	−0.380695	+	0.924701i	$0.624315\pi$
$920$	−23335.9	−0.836263
$921$	0	0
$922$	46930.8	1.67634
$923$	0	0
$924$	0	0
$925$	17949.4	0.638025
$926$	58432.4	2.07366
$927$	0	0
$928$	5250.07	0.185713
$929$	−37450.9	−1.32263	−0.661316	−	0.750108i	$-0.730002\pi$
$929$	−37450.9	−1.32263	−0.661316	+	0.750108i	$0.730002\pi$
$930$	0	0
$931$	51917.5	1.82763
$932$	−43335.2	−1.52306
$933$	0	0
$934$	12005.9	0.420605
$935$	−43806.3	−1.53221
$936$	0	0
$937$	−45853.2	−1.59868	−0.799338	−	0.600882i	$-0.794816\pi$
$937$	−45853.2	−1.59868	−0.799338	+	0.600882i	$0.794816\pi$
$938$	−92362.2	−3.21507
$939$	0	0
$940$	−14235.3	−0.493940
$941$	14680.3	0.508569	0.254285	−	0.967129i	$-0.418160\pi$
$941$	14680.3	0.508569	0.254285	+	0.967129i	$0.418160\pi$
$942$	0	0
$943$	2239.59	0.0773393
$944$	−32586.7	−1.12353
$945$	0	0
$946$	−153843.	−5.28738
$947$	34542.6	1.18531	0.592653	−	0.805458i	$-0.298081\pi$
$947$	34542.6	1.18531	0.592653	+	0.805458i	$0.298081\pi$
$948$	0	0
$949$	0	0
$950$	45196.4	1.54354
$951$	0	0
$952$	53806.0	1.83179
$953$	44993.2	1.52935	0.764677	−	0.644414i	$-0.222899\pi$
$953$	44993.2	1.52935	0.764677	+	0.644414i	$0.222899\pi$
$954$	0	0
$955$	33684.9	1.14138
$956$	42770.1	1.44695
$957$	0	0
$958$	27239.0	0.918636
$959$	−10.2784	−0.000346096 0
$960$	0	0
$961$	42871.1	1.43906
$962$	0	0
$963$	0	0
$964$	−27066.2	−0.904297
$965$	306.379	0.0102204
$966$	0	0
$967$	38006.9	1.26393	0.631964	−	0.774997i	$-0.282249\pi$
$967$	38006.9	1.26393	0.631964	+	0.774997i	$0.282249\pi$
$968$	84007.4	2.78936
$969$	0	0
$970$	−18901.6	−0.625665
$971$	−29954.2	−0.989987	−0.494993	−	0.868897i	$-0.664829\pi$
$971$	−29954.2	−0.989987	−0.494993	+	0.868897i	$0.664829\pi$
$972$	0	0
$973$	−55263.8	−1.82084
$974$	3782.94	0.124449
$975$	0	0
$976$	−22761.1	−0.746481
$977$	43605.7	1.42791	0.713957	−	0.700190i	$-0.246901\pi$
$977$	43605.7	1.42791	0.713957	+	0.700190i	$0.246901\pi$
$978$	0	0
$979$	−61544.0	−2.00915
$980$	124581.	4.06080
$981$	0	0
$982$	−45365.4	−1.47420
$983$	45624.8	1.48037	0.740185	−	0.672403i	$-0.234738\pi$
$983$	45624.8	1.48037	0.740185	+	0.672403i	$0.234738\pi$
$984$	0	0
$985$	1081.05	0.0349696
$986$	−42160.7	−1.36173
$987$	0	0
$988$	0	0
$989$	−22388.5	−0.719830
$990$	0	0
$991$	34882.4	1.11814	0.559069	−	0.829121i	$-0.311159\pi$
$991$	34882.4	1.11814	0.559069	+	0.829121i	$0.311159\pi$
$992$	−8003.38	−0.256157
$993$	0	0
$994$	−79471.6	−2.53590
$995$	−53525.3	−1.70539
$996$	0	0
$997$	−18448.4	−0.586024	−0.293012	−	0.956109i	$-0.594658\pi$
$997$	−18448.4	−0.586024	−0.293012	+	0.956109i	$0.594658\pi$
$998$	−80730.7	−2.56061
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.bn.1.18	yes	18
3.2	odd	2	inner	1521.4.a.bn.1.1	yes	18
13.12	even	2		1521.4.a.bm.1.1	✓	18
39.38	odd	2		1521.4.a.bm.1.18	yes	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1521.4.a.bm.1.1	✓	18	13.12	even	2
1521.4.a.bm.1.18	yes	18	39.38	odd	2
1521.4.a.bn.1.1	yes	18	3.2	odd	2	inner
1521.4.a.bn.1.18	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)

\(f(q)\)	\(=\)	\(q+4.85886 q^{2} +15.6085 q^{4} +14.8714 q^{5} +29.6598 q^{7} +36.9687 q^{8} +O(q^{10})\)	\(q+4.85886 q^{2} +15.6085 q^{4} +14.8714 q^{5} +29.6598 q^{7} +36.9687 q^{8} +72.2582 q^{10} -60.0283 q^{11} +144.113 q^{14} +54.7574 q^{16} +49.0713 q^{17} +96.7335 q^{19} +232.121 q^{20} -291.669 q^{22} -42.4461 q^{23} +96.1597 q^{25} +462.946 q^{28} -176.826 q^{29} +269.559 q^{31} -29.6906 q^{32} +238.431 q^{34} +441.085 q^{35} +186.663 q^{37} +470.014 q^{38} +549.777 q^{40} -52.7630 q^{41} +527.456 q^{43} -936.952 q^{44} -206.240 q^{46} -61.3270 q^{47} +536.706 q^{49} +467.226 q^{50} +340.468 q^{53} -892.707 q^{55} +1096.48 q^{56} -859.172 q^{58} -595.111 q^{59} -415.671 q^{61} +1309.75 q^{62} -582.322 q^{64} -640.901 q^{67} +765.930 q^{68} +2143.17 q^{70} -551.454 q^{71} -431.954 q^{73} +906.968 q^{74} +1509.86 q^{76} -1780.43 q^{77} +998.743 q^{79} +814.322 q^{80} -256.368 q^{82} -461.462 q^{83} +729.762 q^{85} +2562.84 q^{86} -2219.17 q^{88} +1025.25 q^{89} -662.520 q^{92} -297.979 q^{94} +1438.57 q^{95} -261.585 q^{97} +2607.78 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 56 q^{4} + 94 q^{7}+O(q^{10}) \) 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

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