LMFDB - Embedded newform 1305.4.a.i.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1305,4,Mod(1,1305)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1305, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1305.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1305 = 3^{2} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1305.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:	$76.9974925575$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 31x^{4} + 9x^{3} + 230x^{2} + 32x - 128 $ x^6 - x^5 - 31x^4 + 9x^3 + 230x^2 + 32x - 128
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 435)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$4.98965$ of defining polynomial
Character	$\chi$	$=$	1305.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+4.98965 q^{2} +16.8966 q^{4} -5.00000 q^{5} +22.5076 q^{7} +44.3911 q^{8} -24.9483 q^{10} +32.8384 q^{11} +31.3437 q^{13} +112.305 q^{14} +86.3230 q^{16} +66.5556 q^{17} -103.349 q^{19} -84.4831 q^{20} +163.852 q^{22} -5.86439 q^{23} +25.0000 q^{25} +156.394 q^{26} +380.302 q^{28} +29.0000 q^{29} -26.9324 q^{31} +75.5933 q^{32} +332.089 q^{34} -112.538 q^{35} -18.2548 q^{37} -515.675 q^{38} -221.955 q^{40} +115.564 q^{41} +415.780 q^{43} +554.859 q^{44} -29.2613 q^{46} -11.7223 q^{47} +163.592 q^{49} +124.741 q^{50} +529.604 q^{52} +275.059 q^{53} -164.192 q^{55} +999.136 q^{56} +144.700 q^{58} -212.467 q^{59} -766.954 q^{61} -134.383 q^{62} -313.400 q^{64} -156.719 q^{65} +256.113 q^{67} +1124.57 q^{68} -561.525 q^{70} +711.750 q^{71} +45.4465 q^{73} -91.0850 q^{74} -1746.25 q^{76} +739.114 q^{77} +329.147 q^{79} -431.615 q^{80} +576.623 q^{82} +283.281 q^{83} -332.778 q^{85} +2074.60 q^{86} +1457.73 q^{88} +802.821 q^{89} +705.472 q^{91} -99.0885 q^{92} -58.4903 q^{94} +516.744 q^{95} +389.532 q^{97} +816.265 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + q^{2} + 15 q^{4} - 30 q^{5} + 23 q^{7} + 51 q^{8} - 5 q^{10} + 111 q^{11} - 83 q^{13} + 102 q^{14} - 37 q^{16} + 35 q^{17} - 76 q^{19} - 75 q^{20} + 66 q^{22} - 166 q^{23} + 150 q^{25} + 282 q^{26}+ \cdots + 1903 q^{98}+O(q^{100}) $ 6 * q + q^2 + 15 * q^4 - 30 * q^5 + 23 * q^7 + 51 * q^8 - 5 * q^10 + 111 * q^11 - 83 * q^13 + 102 * q^14 - 37 * q^16 + 35 * q^17 - 76 * q^19 - 75 * q^20 + 66 * q^22 - 166 * q^23 + 150 * q^25 + 282 * q^26 + 368 * q^28 + 174 * q^29 - 164 * q^31 + 259 * q^32 + 512 * q^34 - 115 * q^35 - 538 * q^37 - 6 * q^38 - 255 * q^40 + 1250 * q^41 - 80 * q^43 + 1004 * q^44 - 1190 * q^46 + 491 * q^47 - 969 * q^49 + 25 * q^50 - 336 * q^52 + 356 * q^53 - 555 * q^55 + 1260 * q^56 + 29 * q^58 + 646 * q^59 - 1476 * q^61 + 4 * q^62 - 845 * q^64 + 415 * q^65 + 87 * q^67 + 558 * q^68 - 510 * q^70 + 1312 * q^71 - 1142 * q^73 + 2270 * q^74 - 1594 * q^76 + 2561 * q^77 - 142 * q^79 + 185 * q^80 - 1544 * q^82 - 346 * q^83 - 175 * q^85 + 2056 * q^86 + 1400 * q^88 + 2827 * q^89 + 1379 * q^91 + 1434 * q^92 + 978 * q^94 + 380 * q^95 - 4 * q^97 + 1903 * q^98

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 $

$2$	4.98965	1.76411	0.882054	−	0.471148i	$-0.156160\pi$
$2$	4.98965	1.76411	0.882054	+	0.471148i	$0.156160\pi$
$3$	0	0
$3$	0	0
$4$	16.8966	2.11208
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	22.5076	1.21530	0.607648	−	0.794207i	$-0.292113\pi$
$7$	22.5076	1.21530	0.607648	+	0.794207i	$0.292113\pi$
$8$	44.3911	1.96183
$9$	0	0
$10$	−24.9483	−0.788933
$11$	32.8384	0.900106	0.450053	−	0.893002i	$-0.351405\pi$
$11$	32.8384	0.900106	0.450053	+	0.893002i	$0.351405\pi$
$12$	0	0
$13$	31.3437	0.668707	0.334353	−	0.942448i	$-0.391482\pi$
$13$	31.3437	0.668707	0.334353	+	0.942448i	$0.391482\pi$
$14$	112.305	2.14391
$15$	0	0
$16$	86.3230	1.34880
$17$	66.5556	0.949536	0.474768	−	0.880111i	$-0.342532\pi$
$17$	66.5556	0.949536	0.474768	+	0.880111i	$0.342532\pi$
$18$	0	0
$19$	−103.349	−1.24789	−0.623943	−	0.781470i	$-0.714470\pi$
$19$	−103.349	−1.24789	−0.623943	+	0.781470i	$0.714470\pi$
$20$	−84.4831	−0.944550
$21$	0	0
$22$	163.852	1.58788
$23$	−5.86439	−0.0531657	−0.0265828	−	0.999647i	$-0.508463\pi$
$23$	−5.86439	−0.0531657	−0.0265828	+	0.999647i	$0.508463\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	156.394	1.17967
$27$	0	0
$28$	380.302	2.56680
$29$	29.0000	0.185695
$29$	29.0000	0.185695
$30$	0	0
$31$	−26.9324	−0.156039	−0.0780195	−	0.996952i	$-0.524860\pi$
$31$	−26.9324	−0.156039	−0.0780195	+	0.996952i	$0.524860\pi$
$32$	75.5933	0.417598
$33$	0	0
$34$	332.089	1.67508
$35$	−112.538	−0.543497
$36$	0	0
$37$	−18.2548	−0.0811099	−0.0405550	−	0.999177i	$-0.512913\pi$
$37$	−18.2548	−0.0811099	−0.0405550	+	0.999177i	$0.512913\pi$
$38$	−515.675	−2.20141
$39$	0	0
$40$	−221.955	−0.877356
$41$	115.564	0.440196	0.220098	−	0.975478i	$-0.429362\pi$
$41$	115.564	0.440196	0.220098	+	0.975478i	$0.429362\pi$
$42$	0	0
$43$	415.780	1.47455	0.737277	−	0.675591i	$-0.236111\pi$
$43$	415.780	1.47455	0.737277	+	0.675591i	$0.236111\pi$
$44$	554.859	1.90109
$45$	0	0
$46$	−29.2613	−0.0937900
$47$	−11.7223	−0.0363804	−0.0181902	−	0.999835i	$-0.505790\pi$
$47$	−11.7223	−0.0363804	−0.0181902	+	0.999835i	$0.505790\pi$
$48$	0	0
$49$	163.592	0.476943
$50$	124.741	0.352822
$51$	0	0
$52$	529.604	1.41236
$53$	275.059	0.712872	0.356436	−	0.934320i	$-0.383992\pi$
$53$	275.059	0.712872	0.356436	+	0.934320i	$0.383992\pi$
$54$	0	0
$55$	−164.192	−0.402540
$56$	999.136	2.38420
$57$	0	0
$58$	144.700	0.327587
$59$	−212.467	−0.468829	−0.234414	−	0.972137i	$-0.575317\pi$
$59$	−212.467	−0.468829	−0.234414	+	0.972137i	$0.575317\pi$
$60$	0	0
$61$	−766.954	−1.60981	−0.804905	−	0.593403i	$-0.797784\pi$
$61$	−766.954	−1.60981	−0.804905	+	0.593403i	$0.797784\pi$
$62$	−134.383	−0.275270
$63$	0	0
$64$	−313.400	−0.612110
$65$	−156.719	−0.299055
$66$	0	0
$67$	256.113	0.467003	0.233501	−	0.972356i	$-0.424982\pi$
$67$	256.113	0.467003	0.233501	+	0.972356i	$0.424982\pi$
$68$	1124.57	2.00549
$69$	0	0
$70$	−561.525	−0.958787
$71$	711.750	1.18971	0.594854	−	0.803834i	$-0.297210\pi$
$71$	711.750	1.18971	0.594854	+	0.803834i	$0.297210\pi$
$72$	0	0
$73$	45.4465	0.0728646	0.0364323	−	0.999336i	$-0.488401\pi$
$73$	45.4465	0.0728646	0.0364323	+	0.999336i	$0.488401\pi$
$74$	−91.0850	−0.143087
$75$	0	0
$76$	−1746.25	−2.63563
$77$	739.114	1.09389
$78$	0	0
$79$	329.147	0.468759	0.234380	−	0.972145i	$-0.424694\pi$
$79$	329.147	0.468759	0.234380	+	0.972145i	$0.424694\pi$
$80$	−431.615	−0.603201
$81$	0	0
$82$	576.623	0.776553
$83$	283.281	0.374628	0.187314	−	0.982300i	$-0.440022\pi$
$83$	283.281	0.374628	0.187314	+	0.982300i	$0.440022\pi$
$84$	0	0
$85$	−332.778	−0.424645
$86$	2074.60	2.60127
$87$	0	0
$88$	1457.73	1.76585
$89$	802.821	0.956167	0.478083	−	0.878314i	$-0.341332\pi$
$89$	802.821	0.956167	0.478083	+	0.878314i	$0.341332\pi$
$90$	0	0
$91$	705.472	0.812677
$92$	−99.0885	−0.112290
$93$	0	0
$94$	−58.4903	−0.0641789
$95$	516.744	0.558072
$96$	0	0
$97$	389.532	0.407742	0.203871	−	0.978998i	$-0.434648\pi$
$97$	389.532	0.407742	0.203871	+	0.978998i	$0.434648\pi$
$98$	816.265	0.841379
$99$	0	0
$100$	422.416	0.422416
$101$	−274.559	−0.270492	−0.135246	−	0.990812i	$-0.543182\pi$
$101$	−274.559	−0.270492	−0.135246	+	0.990812i	$0.543182\pi$
$102$	0	0
$103$	−841.113	−0.804635	−0.402317	−	0.915500i	$-0.631795\pi$
$103$	−841.113	−0.804635	−0.402317	+	0.915500i	$0.631795\pi$
$104$	1391.38	1.31189
$105$	0	0
$106$	1372.45	1.25758
$107$	1489.04	1.34534	0.672670	−	0.739943i	$-0.265147\pi$
$107$	1489.04	1.34534	0.672670	+	0.739943i	$0.265147\pi$
$108$	0	0
$109$	−987.120	−0.867422	−0.433711	−	0.901052i	$-0.642796\pi$
$109$	−987.120	−0.867422	−0.433711	+	0.901052i	$0.642796\pi$
$110$	−819.262	−0.710123
$111$	0	0
$112$	1942.92	1.63919
$113$	−135.446	−0.112758	−0.0563791	−	0.998409i	$-0.517956\pi$
$113$	−135.446	−0.112758	−0.0563791	+	0.998409i	$0.517956\pi$
$114$	0	0
$115$	29.3220	0.0237764
$116$	490.002	0.392203
$117$	0	0
$118$	−1060.14	−0.827065
$119$	1498.01	1.15397
$120$	0	0
$121$	−252.636	−0.189809
$122$	−3826.84	−2.83988
$123$	0	0
$124$	−455.067	−0.329567
$125$	−125.000	−0.0894427
$126$	0	0
$127$	1539.42	1.07560	0.537802	−	0.843071i	$-0.319255\pi$
$127$	1539.42	1.07560	0.537802	+	0.843071i	$0.319255\pi$
$128$	−2168.50	−1.49743
$129$	0	0
$130$	−781.972	−0.527565
$131$	−1625.05	−1.08383	−0.541914	−	0.840434i	$-0.682300\pi$
$131$	−1625.05	−1.08383	−0.541914	+	0.840434i	$0.682300\pi$
$132$	0	0
$133$	−2326.13	−1.51655
$134$	1277.91	0.823843
$135$	0	0
$136$	2954.48	1.86282
$137$	925.116	0.576920	0.288460	−	0.957492i	$-0.406857\pi$
$137$	925.116	0.576920	0.288460	+	0.957492i	$0.406857\pi$
$138$	0	0
$139$	−3232.14	−1.97228	−0.986138	−	0.165928i	$-0.946938\pi$
$139$	−3232.14	−1.97228	−0.986138	+	0.165928i	$0.946938\pi$
$140$	−1901.51	−1.14791
$141$	0	0
$142$	3551.39	2.09877
$143$	1029.28	0.601907
$144$	0	0
$145$	−145.000	−0.0830455
$146$	226.762	0.128541
$147$	0	0
$148$	−308.444	−0.171311
$149$	−764.349	−0.420255	−0.210127	−	0.977674i	$-0.567388\pi$
$149$	−764.349	−0.420255	−0.210127	+	0.977674i	$0.567388\pi$
$150$	0	0
$151$	2721.64	1.46678	0.733390	−	0.679808i	$-0.237937\pi$
$151$	2721.64	1.46678	0.733390	+	0.679808i	$0.237937\pi$
$152$	−4587.76	−2.44814
$153$	0	0
$154$	3687.92	1.92975
$155$	134.662	0.0697827
$156$	0	0
$157$	1760.38	0.894862	0.447431	−	0.894319i	$-0.352339\pi$
$157$	1760.38	0.894862	0.447431	+	0.894319i	$0.352339\pi$
$158$	1642.33	0.826942
$159$	0	0
$160$	−377.966	−0.186755
$161$	−131.993	−0.0646120
$162$	0	0
$163$	−3566.14	−1.71363	−0.856816	−	0.515622i	$-0.827561\pi$
$163$	−3566.14	−1.71363	−0.856816	+	0.515622i	$0.827561\pi$
$164$	1952.64	0.929728
$165$	0	0
$166$	1413.47	0.660885
$167$	1976.07	0.915647	0.457824	−	0.889043i	$-0.348629\pi$
$167$	1976.07	0.915647	0.457824	+	0.889043i	$0.348629\pi$
$168$	0	0
$169$	−1214.57	−0.552831
$170$	−1660.45	−0.749120
$171$	0	0
$172$	7025.28	3.11437
$173$	1836.14	0.806934	0.403467	−	0.914994i	$-0.367805\pi$
$173$	1836.14	0.806934	0.403467	+	0.914994i	$0.367805\pi$
$174$	0	0
$175$	562.690	0.243059
$176$	2834.71	1.21406
$177$	0	0
$178$	4005.80	1.68678
$179$	−2241.87	−0.936117	−0.468058	−	0.883698i	$-0.655046\pi$
$179$	−2241.87	−0.936117	−0.468058	+	0.883698i	$0.655046\pi$
$180$	0	0
$181$	2084.75	0.856122	0.428061	−	0.903750i	$-0.359197\pi$
$181$	2084.75	0.856122	0.428061	+	0.903750i	$0.359197\pi$
$182$	3520.06	1.43365
$183$	0	0
$184$	−260.327	−0.104302
$185$	91.2739	0.0362735
$186$	0	0
$187$	2185.58	0.854682
$188$	−198.068	−0.0768382
$189$	0	0
$190$	2578.37	0.984499
$191$	−4206.24	−1.59347	−0.796735	−	0.604328i	$-0.793442\pi$
$191$	−4206.24	−1.59347	−0.796735	+	0.604328i	$0.793442\pi$
$192$	0	0
$193$	−2047.28	−0.763558	−0.381779	−	0.924254i	$-0.624688\pi$
$193$	−2047.28	−0.763558	−0.381779	+	0.924254i	$0.624688\pi$
$194$	1943.63	0.719301
$195$	0	0
$196$	2764.15	1.00734
$197$	−4461.88	−1.61369	−0.806843	−	0.590766i	$-0.798826\pi$
$197$	−4461.88	−1.61369	−0.806843	+	0.590766i	$0.798826\pi$
$198$	0	0
$199$	−1926.80	−0.686367	−0.343183	−	0.939268i	$-0.611505\pi$
$199$	−1926.80	−0.686367	−0.343183	+	0.939268i	$0.611505\pi$
$200$	1109.78	0.392365
$201$	0	0
$202$	−1369.95	−0.477177
$203$	652.720	0.225675
$204$	0	0
$205$	−577.819	−0.196862
$206$	−4196.86	−1.41946
$207$	0	0
$208$	2705.69	0.901950
$209$	−3393.81	−1.12323
$210$	0	0
$211$	−4487.22	−1.46404	−0.732021	−	0.681282i	$-0.761423\pi$
$211$	−4487.22	−1.46404	−0.732021	+	0.681282i	$0.761423\pi$
$212$	4647.57	1.50564
$213$	0	0
$214$	7429.81	2.37332
$215$	−2078.90	−0.659441
$216$	0	0
$217$	−606.184	−0.189633
$218$	−4925.39	−1.53023
$219$	0	0
$220$	−2774.30	−0.850195
$221$	2086.10	0.634961
$222$	0	0
$223$	−5113.28	−1.53547	−0.767737	−	0.640765i	$-0.778617\pi$
$223$	−5113.28	−1.53547	−0.767737	+	0.640765i	$0.778617\pi$
$224$	1701.42	0.507505
$225$	0	0
$226$	−675.827	−0.198918
$227$	5258.42	1.53751	0.768753	−	0.639546i	$-0.220878\pi$
$227$	5258.42	1.53751	0.768753	+	0.639546i	$0.220878\pi$
$228$	0	0
$229$	−5242.30	−1.51276	−0.756378	−	0.654135i	$-0.773033\pi$
$229$	−5242.30	−1.51276	−0.756378	+	0.654135i	$0.773033\pi$
$230$	146.306	0.0419442
$231$	0	0
$232$	1287.34	0.364302
$233$	6046.86	1.70018	0.850092	−	0.526635i	$-0.176547\pi$
$233$	6046.86	1.70018	0.850092	+	0.526635i	$0.176547\pi$
$234$	0	0
$235$	58.6116	0.0162698
$236$	−3589.98	−0.990203
$237$	0	0
$238$	7474.53	2.03572
$239$	212.645	0.0575518	0.0287759	−	0.999586i	$-0.490839\pi$
$239$	212.645	0.0575518	0.0287759	+	0.999586i	$0.490839\pi$
$240$	0	0
$241$	−5857.59	−1.56565	−0.782823	−	0.622245i	$-0.786221\pi$
$241$	−5857.59	−1.56565	−0.782823	+	0.622245i	$0.786221\pi$
$242$	−1260.57	−0.334844
$243$	0	0
$244$	−12958.9	−3.40005
$245$	−817.958	−0.213295
$246$	0	0
$247$	−3239.34	−0.834470
$248$	−1195.56	−0.306121
$249$	0	0
$250$	−623.707	−0.157787
$251$	3510.70	0.882842	0.441421	−	0.897300i	$-0.354475\pi$
$251$	3510.70	0.882842	0.441421	+	0.897300i	$0.354475\pi$
$252$	0	0
$253$	−192.578	−0.0478547
$254$	7681.18	1.89748
$255$	0	0
$256$	−8312.88	−2.02951
$257$	−2682.10	−0.650990	−0.325495	−	0.945544i	$-0.605531\pi$
$257$	−2682.10	−0.650990	−0.325495	+	0.945544i	$0.605531\pi$
$258$	0	0
$259$	−410.871	−0.0985725
$260$	−2648.02	−0.631627
$261$	0	0
$262$	−8108.45	−1.91199
$263$	−4147.94	−0.972521	−0.486261	−	0.873814i	$-0.661639\pi$
$263$	−4147.94	−0.972521	−0.486261	+	0.873814i	$0.661639\pi$
$264$	0	0
$265$	−1375.29	−0.318806
$266$	−11606.6	−2.67536
$267$	0	0
$268$	4327.45	0.986346
$269$	7462.57	1.69145	0.845726	−	0.533617i	$-0.179168\pi$
$269$	7462.57	1.69145	0.845726	+	0.533617i	$0.179168\pi$
$270$	0	0
$271$	6848.81	1.53519	0.767594	−	0.640937i	$-0.221454\pi$
$271$	6848.81	1.53519	0.767594	+	0.640937i	$0.221454\pi$
$272$	5745.28	1.28073
$273$	0	0
$274$	4616.01	1.01775
$275$	820.961	0.180021
$276$	0	0
$277$	−100.396	−0.0217769	−0.0108884	−	0.999941i	$-0.503466\pi$
$277$	−100.396	−0.0217769	−0.0108884	+	0.999941i	$0.503466\pi$
$278$	−16127.2	−3.47931
$279$	0	0
$280$	−4995.68	−1.06625
$281$	−3619.61	−0.768426	−0.384213	−	0.923244i	$-0.625527\pi$
$281$	−3619.61	−0.768426	−0.384213	+	0.923244i	$0.625527\pi$
$282$	0	0
$283$	−3692.84	−0.775677	−0.387839	−	0.921727i	$-0.626778\pi$
$283$	−3692.84	−0.775677	−0.387839	+	0.921727i	$0.626778\pi$
$284$	12026.2	2.51276
$285$	0	0
$286$	5135.75	1.06183
$287$	2601.06	0.534968
$288$	0	0
$289$	−483.352	−0.0983823
$290$	−723.500	−0.146501
$291$	0	0
$292$	767.893	0.153896
$293$	1172.85	0.233851	0.116926	−	0.993141i	$-0.462696\pi$
$293$	1172.85	0.233851	0.116926	+	0.993141i	$0.462696\pi$
$294$	0	0
$295$	1062.34	0.209667
$296$	−810.349	−0.159124
$297$	0	0
$298$	−3813.84	−0.741375
$299$	−183.812	−0.0355523
$300$	0	0
$301$	9358.20	1.79202
$302$	13580.0	2.58756
$303$	0	0
$304$	−8921.38	−1.68315
$305$	3834.77	0.719929
$306$	0	0
$307$	−4625.08	−0.859829	−0.429915	−	0.902870i	$-0.641456\pi$
$307$	−4625.08	−0.859829	−0.429915	+	0.902870i	$0.641456\pi$
$308$	12488.5	2.31039
$309$	0	0
$310$	671.917	0.123104
$311$	−5757.91	−1.04984	−0.524921	−	0.851151i	$-0.675905\pi$
$311$	−5757.91	−1.04984	−0.524921	+	0.851151i	$0.675905\pi$
$312$	0	0
$313$	−7518.69	−1.35777	−0.678884	−	0.734246i	$-0.737536\pi$
$313$	−7518.69	−1.35777	−0.678884	+	0.734246i	$0.737536\pi$
$314$	8783.66	1.57863
$315$	0	0
$316$	5561.48	0.990056
$317$	−695.769	−0.123275	−0.0616376	−	0.998099i	$-0.519632\pi$
$317$	−695.769	−0.123275	−0.0616376	+	0.998099i	$0.519632\pi$
$318$	0	0
$319$	952.315	0.167145
$320$	1567.00	0.273744
$321$	0	0
$322$	−658.601	−0.113983
$323$	−6878.44	−1.18491
$324$	0	0
$325$	783.594	0.133741
$326$	−17793.8	−3.02303
$327$	0	0
$328$	5130.00	0.863588
$329$	−263.841	−0.0442129
$330$	0	0
$331$	−7536.11	−1.25143	−0.625713	−	0.780053i	$-0.715192\pi$
$331$	−7536.11	−1.25143	−0.625713	+	0.780053i	$0.715192\pi$
$332$	4786.50	0.791245
$333$	0	0
$334$	9859.92	1.61530
$335$	−1280.56	−0.208850
$336$	0	0
$337$	6481.09	1.04762	0.523809	−	0.851836i	$-0.324511\pi$
$337$	6481.09	1.04762	0.523809	+	0.851836i	$0.324511\pi$
$338$	−6060.28	−0.975254
$339$	0	0
$340$	−5622.83	−0.896884
$341$	−884.419	−0.140452
$342$	0	0
$343$	−4038.05	−0.635669
$344$	18456.9	2.89282
$345$	0	0
$346$	9161.72	1.42352
$347$	1154.45	0.178600	0.0893000	−	0.996005i	$-0.471537\pi$
$347$	1154.45	0.178600	0.0893000	+	0.996005i	$0.471537\pi$
$348$	0	0
$349$	7656.69	1.17436	0.587182	−	0.809455i	$-0.300237\pi$
$349$	7656.69	1.17436	0.587182	+	0.809455i	$0.300237\pi$
$350$	2807.63	0.428783
$351$	0	0
$352$	2482.37	0.375882
$353$	−1112.90	−0.167801	−0.0839003	−	0.996474i	$-0.526738\pi$
$353$	−1112.90	−0.167801	−0.0839003	+	0.996474i	$0.526738\pi$
$354$	0	0
$355$	−3558.75	−0.532053
$356$	13565.0	2.01950
$357$	0	0
$358$	−11186.1	−1.65141
$359$	−12401.6	−1.82321	−0.911604	−	0.411068i	$-0.865156\pi$
$359$	−12401.6	−1.82321	−0.911604	+	0.411068i	$0.865156\pi$
$360$	0	0
$361$	3821.97	0.557220
$362$	10402.2	1.51029
$363$	0	0
$364$	11920.1	1.71644
$365$	−227.233	−0.0325860
$366$	0	0
$367$	11259.8	1.60152	0.800759	−	0.598987i	$-0.204430\pi$
$367$	11259.8	1.60152	0.800759	+	0.598987i	$0.204430\pi$
$368$	−506.232	−0.0717097
$369$	0	0
$370$	455.425	0.0639903
$371$	6190.91	0.866350
$372$	0	0
$373$	−8049.64	−1.11741	−0.558706	−	0.829366i	$-0.688702\pi$
$373$	−8049.64	−1.11741	−0.558706	+	0.829366i	$0.688702\pi$
$374$	10905.3	1.50775
$375$	0	0
$376$	−520.367	−0.0713720
$377$	908.969	0.124176
$378$	0	0
$379$	9811.51	1.32977	0.664886	−	0.746945i	$-0.268480\pi$
$379$	9811.51	1.32977	0.664886	+	0.746945i	$0.268480\pi$
$380$	8731.23	1.17869
$381$	0	0
$382$	−20987.7	−2.81106
$383$	−6660.10	−0.888551	−0.444276	−	0.895890i	$-0.646539\pi$
$383$	−6660.10	−0.888551	−0.444276	+	0.895890i	$0.646539\pi$
$384$	0	0
$385$	−3695.57	−0.489205
$386$	−10215.2	−1.34700
$387$	0	0
$388$	6581.78	0.861183
$389$	15036.0	1.95979	0.979894	−	0.199521i	$-0.0639386\pi$
$389$	15036.0	1.95979	0.979894	+	0.199521i	$0.0639386\pi$
$390$	0	0
$391$	−390.308	−0.0504827
$392$	7262.00	0.935680
$393$	0	0
$394$	−22263.2	−2.84672
$395$	−1645.74	−0.209636
$396$	0	0
$397$	−2227.56	−0.281607	−0.140804	−	0.990038i	$-0.544969\pi$
$397$	−2227.56	−0.281607	−0.140804	+	0.990038i	$0.544969\pi$
$398$	−9614.05	−1.21083
$399$	0	0
$400$	2158.08	0.269759
$401$	7601.22	0.946600	0.473300	−	0.880901i	$-0.343063\pi$
$401$	7601.22	0.946600	0.473300	+	0.880901i	$0.343063\pi$
$402$	0	0
$403$	−844.163	−0.104344
$404$	−4639.13	−0.571300
$405$	0	0
$406$	3256.85	0.398115
$407$	−599.458	−0.0730075
$408$	0	0
$409$	−13097.7	−1.58347	−0.791733	−	0.610867i	$-0.790821\pi$
$409$	−13097.7	−1.58347	−0.791733	+	0.610867i	$0.790821\pi$
$410$	−2883.12	−0.347285
$411$	0	0
$412$	−14212.0	−1.69945
$413$	−4782.13	−0.569765
$414$	0	0
$415$	−1416.41	−0.167539
$416$	2369.38	0.279251
$417$	0	0
$418$	−16934.0	−1.98150
$419$	−10997.7	−1.28227	−0.641135	−	0.767428i	$-0.721536\pi$
$419$	−10997.7	−1.28227	−0.641135	+	0.767428i	$0.721536\pi$
$420$	0	0
$421$	−13032.2	−1.50867	−0.754333	−	0.656492i	$-0.772040\pi$
$421$	−13032.2	−1.50867	−0.754333	+	0.656492i	$0.772040\pi$
$422$	−22389.7	−2.58273
$423$	0	0
$424$	12210.2	1.39853
$425$	1663.89	0.189907
$426$	0	0
$427$	−17262.3	−1.95640
$428$	25159.8	2.84146
$429$	0	0
$430$	−10373.0	−1.16332
$431$	781.858	0.0873800	0.0436900	−	0.999045i	$-0.486089\pi$
$431$	781.858	0.0873800	0.0436900	+	0.999045i	$0.486089\pi$
$432$	0	0
$433$	9159.83	1.01661	0.508306	−	0.861176i	$-0.330272\pi$
$433$	9159.83	1.01661	0.508306	+	0.861176i	$0.330272\pi$
$434$	−3024.65	−0.334534
$435$	0	0
$436$	−16679.0	−1.83206
$437$	606.078	0.0663447
$438$	0	0
$439$	−4048.62	−0.440160	−0.220080	−	0.975482i	$-0.570632\pi$
$439$	−4048.62	−0.440160	−0.220080	+	0.975482i	$0.570632\pi$
$440$	−7288.67	−0.789713
$441$	0	0
$442$	10408.9	1.12014
$443$	6522.28	0.699510	0.349755	−	0.936841i	$-0.386265\pi$
$443$	6522.28	0.699510	0.349755	+	0.936841i	$0.386265\pi$
$444$	0	0
$445$	−4014.10	−0.427611
$446$	−25513.5	−2.70874
$447$	0	0
$448$	−7053.88	−0.743894
$449$	4946.78	0.519940	0.259970	−	0.965617i	$-0.416287\pi$
$449$	4946.78	0.519940	0.259970	+	0.965617i	$0.416287\pi$
$450$	0	0
$451$	3794.94	0.396223
$452$	−2288.58	−0.238154
$453$	0	0
$454$	26237.7	2.71233
$455$	−3527.36	−0.363440
$456$	0	0
$457$	16111.7	1.64918	0.824590	−	0.565731i	$-0.191406\pi$
$457$	16111.7	1.64918	0.824590	+	0.565731i	$0.191406\pi$
$458$	−26157.3	−2.66866
$459$	0	0
$460$	495.443	0.0502177
$461$	−3232.48	−0.326576	−0.163288	−	0.986578i	$-0.552210\pi$
$461$	−3232.48	−0.326576	−0.163288	+	0.986578i	$0.552210\pi$
$462$	0	0
$463$	6763.53	0.678894	0.339447	−	0.940625i	$-0.389760\pi$
$463$	6763.53	0.678894	0.339447	+	0.940625i	$0.389760\pi$
$464$	2503.37	0.250465
$465$	0	0
$466$	30171.7	2.99931
$467$	−2459.52	−0.243711	−0.121855	−	0.992548i	$-0.538884\pi$
$467$	−2459.52	−0.243711	−0.121855	+	0.992548i	$0.538884\pi$
$468$	0	0
$469$	5764.48	0.567546
$470$	292.452	0.0287017
$471$	0	0
$472$	−9431.66	−0.919761
$473$	13653.6	1.32725
$474$	0	0
$475$	−2583.72	−0.249577
$476$	25311.3	2.43727
$477$	0	0
$478$	1061.03	0.101528
$479$	−3207.84	−0.305991	−0.152996	−	0.988227i	$-0.548892\pi$
$479$	−3207.84	−0.305991	−0.152996	+	0.988227i	$0.548892\pi$
$480$	0	0
$481$	−572.173	−0.0542388
$482$	−29227.3	−2.76197
$483$	0	0
$484$	−4268.70	−0.400892
$485$	−1947.66	−0.182348
$486$	0	0
$487$	12362.0	1.15026	0.575130	−	0.818062i	$-0.304952\pi$
$487$	12362.0	1.15026	0.575130	+	0.818062i	$0.304952\pi$
$488$	−34045.9	−3.15817
$489$	0	0
$490$	−4081.32	−0.376276
$491$	−18430.3	−1.69399	−0.846993	−	0.531604i	$-0.821590\pi$
$491$	−18430.3	−1.69399	−0.846993	+	0.531604i	$0.821590\pi$
$492$	0	0
$493$	1930.11	0.176324
$494$	−16163.2	−1.47210
$495$	0	0
$496$	−2324.89	−0.210465
$497$	16019.8	1.44585
$498$	0	0
$499$	9338.78	0.837798	0.418899	−	0.908033i	$-0.362416\pi$
$499$	9338.78	0.837798	0.418899	+	0.908033i	$0.362416\pi$
$500$	−2112.08	−0.188910
$501$	0	0
$502$	17517.2	1.55743
$503$	−19444.6	−1.72364	−0.861819	−	0.507216i	$-0.830674\pi$
$503$	−19444.6	−1.72364	−0.861819	+	0.507216i	$0.830674\pi$
$504$	0	0
$505$	1372.80	0.120968
$506$	−960.895	−0.0844210
$507$	0	0
$508$	26011.0	2.27176
$509$	−11959.2	−1.04142	−0.520711	−	0.853733i	$-0.674333\pi$
$509$	−11959.2	−1.04142	−0.520711	+	0.853733i	$0.674333\pi$
$510$	0	0
$511$	1022.89	0.0885520
$512$	−24130.3	−2.08285
$513$	0	0
$514$	−13382.7	−1.14842
$515$	4205.57	0.359844
$516$	0	0
$517$	−384.943	−0.0327462
$518$	−2050.10	−0.173893
$519$	0	0
$520$	−6956.91	−0.586694
$521$	2757.67	0.231892	0.115946	−	0.993256i	$-0.463010\pi$
$521$	2757.67	0.231892	0.115946	+	0.993256i	$0.463010\pi$
$522$	0	0
$523$	3886.64	0.324954	0.162477	−	0.986712i	$-0.448052\pi$
$523$	3886.64	0.324954	0.162477	+	0.986712i	$0.448052\pi$
$524$	−27457.9	−2.28913
$525$	0	0
$526$	−20696.8	−1.71563
$527$	−1792.50	−0.148165
$528$	0	0
$529$	−12132.6	−0.997173
$530$	−6862.24	−0.562409
$531$	0	0
$532$	−39303.8	−3.20307
$533$	3622.20	0.294362
$534$	0	0
$535$	−7445.22	−0.601654
$536$	11369.1	0.916178
$537$	0	0
$538$	37235.6	2.98390
$539$	5372.09	0.429299
$540$	0	0
$541$	−11307.5	−0.898610	−0.449305	−	0.893379i	$-0.648328\pi$
$541$	−11307.5	−0.898610	−0.449305	+	0.893379i	$0.648328\pi$
$542$	34173.2	2.70824
$543$	0	0
$544$	5031.15	0.396524
$545$	4935.60	0.387923
$546$	0	0
$547$	9195.12	0.718748	0.359374	−	0.933194i	$-0.382990\pi$
$547$	9195.12	0.718748	0.359374	+	0.933194i	$0.382990\pi$
$548$	15631.3	1.21850
$549$	0	0
$550$	4096.31	0.317577
$551$	−2997.11	−0.231727
$552$	0	0
$553$	7408.32	0.569681
$554$	−500.940	−0.0384168
$555$	0	0
$556$	−54612.2	−4.16560
$557$	−12391.1	−0.942596	−0.471298	−	0.881974i	$-0.656214\pi$
$557$	−12391.1	−0.942596	−0.471298	+	0.881974i	$0.656214\pi$
$558$	0	0
$559$	13032.1	0.986045
$560$	−9714.62	−0.733067
$561$	0	0
$562$	−18060.6	−1.35559
$563$	−9161.81	−0.685834	−0.342917	−	0.939366i	$-0.611415\pi$
$563$	−9161.81	−0.685834	−0.342917	+	0.939366i	$0.611415\pi$
$564$	0	0
$565$	677.229	0.0504270
$566$	−18426.0	−1.36838
$567$	0	0
$568$	31595.4	2.33400
$569$	15614.8	1.15045	0.575226	−	0.817994i	$-0.304914\pi$
$569$	15614.8	1.15045	0.575226	+	0.817994i	$0.304914\pi$
$570$	0	0
$571$	13526.3	0.991345	0.495673	−	0.868509i	$-0.334922\pi$
$571$	13526.3	0.991345	0.495673	+	0.868509i	$0.334922\pi$
$572$	17391.4	1.27128
$573$	0	0
$574$	12978.4	0.943742
$575$	−146.610	−0.0106331
$576$	0	0
$577$	21208.3	1.53018	0.765090	−	0.643923i	$-0.222694\pi$
$577$	21208.3	1.53018	0.765090	+	0.643923i	$0.222694\pi$
$578$	−2411.76	−0.173557
$579$	0	0
$580$	−2450.01	−0.175399
$581$	6375.98	0.455284
$582$	0	0
$583$	9032.50	0.641660
$584$	2017.42	0.142948
$585$	0	0
$586$	5852.09	0.412539
$587$	−7975.12	−0.560764	−0.280382	−	0.959888i	$-0.590461\pi$
$587$	−7975.12	−0.560764	−0.280382	+	0.959888i	$0.590461\pi$
$588$	0	0
$589$	2783.43	0.194719
$590$	5300.69	0.369875
$591$	0	0
$592$	−1575.81	−0.109401
$593$	6169.18	0.427214	0.213607	−	0.976920i	$-0.431479\pi$
$593$	6169.18	0.427214	0.213607	+	0.976920i	$0.431479\pi$
$594$	0	0
$595$	−7490.03	−0.516069
$596$	−12914.9	−0.887611
$597$	0	0
$598$	−917.158	−0.0627181
$599$	−10909.8	−0.744180	−0.372090	−	0.928197i	$-0.621359\pi$
$599$	−10909.8	−0.744180	−0.372090	+	0.928197i	$0.621359\pi$
$600$	0	0
$601$	−11926.7	−0.809483	−0.404741	−	0.914431i	$-0.632638\pi$
$601$	−11926.7	−0.809483	−0.404741	+	0.914431i	$0.632638\pi$
$602$	46694.2	3.16132
$603$	0	0
$604$	45986.5	3.09795
$605$	1263.18	0.0848853
$606$	0	0
$607$	−12043.2	−0.805301	−0.402650	−	0.915354i	$-0.631911\pi$
$607$	−12043.2	−0.805301	−0.402650	+	0.915354i	$0.631911\pi$
$608$	−7812.47	−0.521114
$609$	0	0
$610$	19134.2	1.27003
$611$	−367.422	−0.0243278
$612$	0	0
$613$	11777.4	0.775992	0.387996	−	0.921661i	$-0.373168\pi$
$613$	11777.4	0.775992	0.387996	+	0.921661i	$0.373168\pi$
$614$	−23077.6	−1.51683
$615$	0	0
$616$	32810.1	2.14603
$617$	2907.19	0.189691	0.0948454	−	0.995492i	$-0.469764\pi$
$617$	2907.19	0.189691	0.0948454	+	0.995492i	$0.469764\pi$
$618$	0	0
$619$	−839.647	−0.0545206	−0.0272603	−	0.999628i	$-0.508678\pi$
$619$	−839.647	−0.0545206	−0.0272603	+	0.999628i	$0.508678\pi$
$620$	2275.34	0.147387
$621$	0	0
$622$	−28730.0	−1.85204
$623$	18069.6	1.16203
$624$	0	0
$625$	625.000	0.0400000
$626$	−37515.6	−2.39525
$627$	0	0
$628$	29744.4	1.89002
$629$	−1214.96	−0.0770167
$630$	0	0
$631$	9825.37	0.619876	0.309938	−	0.950757i	$-0.399692\pi$
$631$	9825.37	0.619876	0.309938	+	0.950757i	$0.399692\pi$
$632$	14611.2	0.919625
$633$	0	0
$634$	−3471.64	−0.217471
$635$	−7697.11	−0.481024
$636$	0	0
$637$	5127.57	0.318935
$638$	4751.72	0.294863
$639$	0	0
$640$	10842.5	0.669669
$641$	22040.3	1.35810	0.679049	−	0.734093i	$-0.262392\pi$
$641$	22040.3	1.35810	0.679049	+	0.734093i	$0.262392\pi$
$642$	0	0
$643$	−21366.9	−1.31047	−0.655233	−	0.755427i	$-0.727429\pi$
$643$	−21366.9	−1.31047	−0.655233	+	0.755427i	$0.727429\pi$
$644$	−2230.24	−0.136466
$645$	0	0
$646$	−34321.0	−2.09031
$647$	5826.03	0.354011	0.177005	−	0.984210i	$-0.443359\pi$
$647$	5826.03	0.354011	0.177005	+	0.984210i	$0.443359\pi$
$648$	0	0
$649$	−6977.10	−0.421995
$650$	3909.86	0.235934
$651$	0	0
$652$	−60255.8	−3.61933
$653$	20058.4	1.20206	0.601030	−	0.799227i	$-0.294757\pi$
$653$	20058.4	1.20206	0.601030	+	0.799227i	$0.294757\pi$
$654$	0	0
$655$	8125.26	0.484703
$656$	9975.82	0.593735
$657$	0	0
$658$	−1316.48	−0.0779963
$659$	28535.3	1.68676	0.843382	−	0.537315i	$-0.180561\pi$
$659$	28535.3	1.68676	0.843382	+	0.537315i	$0.180561\pi$
$660$	0	0
$661$	15044.6	0.885274	0.442637	−	0.896701i	$-0.354043\pi$
$661$	15044.6	0.885274	0.442637	+	0.896701i	$0.354043\pi$
$662$	−37602.6	−2.20765
$663$	0	0
$664$	12575.2	0.734956
$665$	11630.7	0.678222
$666$	0	0
$667$	−170.067	−0.00987262
$668$	33389.0	1.93392
$669$	0	0
$670$	−6389.57	−0.368434
$671$	−25185.6	−1.44900
$672$	0	0
$673$	15511.3	0.888434	0.444217	−	0.895919i	$-0.353482\pi$
$673$	15511.3	0.888434	0.444217	+	0.895919i	$0.353482\pi$
$674$	32338.4	1.84811
$675$	0	0
$676$	−20522.1	−1.16762
$677$	10695.3	0.607170	0.303585	−	0.952804i	$-0.401816\pi$
$677$	10695.3	0.607170	0.303585	+	0.952804i	$0.401816\pi$
$678$	0	0
$679$	8767.43	0.495527
$680$	−14772.4	−0.833081
$681$	0	0
$682$	−4412.94	−0.247772
$683$	11144.8	0.624367	0.312184	−	0.950022i	$-0.398940\pi$
$683$	11144.8	0.624367	0.312184	+	0.950022i	$0.398940\pi$
$684$	0	0
$685$	−4625.58	−0.258006
$686$	−20148.5	−1.12139
$687$	0	0
$688$	35891.4	1.98887
$689$	8621.37	0.476703
$690$	0	0
$691$	−5380.30	−0.296203	−0.148102	−	0.988972i	$-0.547316\pi$
$691$	−5380.30	−0.296203	−0.148102	+	0.988972i	$0.547316\pi$
$692$	31024.7	1.70431
$693$	0	0
$694$	5760.31	0.315070
$695$	16160.7	0.882029
$696$	0	0
$697$	7691.42	0.417982
$698$	38204.2	2.07171
$699$	0	0
$700$	9507.56	0.513360
$701$	−21377.2	−1.15179	−0.575896	−	0.817523i	$-0.695347\pi$
$701$	−21377.2	−1.15179	−0.575896	+	0.817523i	$0.695347\pi$
$702$	0	0
$703$	1886.61	0.101216
$704$	−10291.6	−0.550964
$705$	0	0
$706$	−5552.97	−0.296018
$707$	−6179.67	−0.328727
$708$	0	0
$709$	−1384.02	−0.0733116	−0.0366558	−	0.999328i	$-0.511671\pi$
$709$	−1384.02	−0.0733116	−0.0366558	+	0.999328i	$0.511671\pi$
$710$	−17756.9	−0.938600
$711$	0	0
$712$	35638.1	1.87583
$713$	157.942	0.00829592
$714$	0	0
$715$	−5146.40	−0.269181
$716$	−37880.0	−1.97715
$717$	0	0
$718$	−61879.7	−3.21634
$719$	10439.9	0.541504	0.270752	−	0.962649i	$-0.412728\pi$
$719$	10439.9	0.541504	0.270752	+	0.962649i	$0.412728\pi$
$720$	0	0
$721$	−18931.4	−0.977869
$722$	19070.3	0.982996
$723$	0	0
$724$	35225.2	1.80820
$725$	725.000	0.0371391
$726$	0	0
$727$	24684.5	1.25928	0.629640	−	0.776887i	$-0.283203\pi$
$727$	24684.5	1.25928	0.629640	+	0.776887i	$0.283203\pi$
$728$	31316.7	1.59433
$729$	0	0
$730$	−1133.81	−0.0574853
$731$	27672.5	1.40014
$732$	0	0
$733$	−858.035	−0.0432363	−0.0216182	−	0.999766i	$-0.506882\pi$
$733$	−858.035	−0.0432363	−0.0216182	+	0.999766i	$0.506882\pi$
$734$	56182.5	2.82525
$735$	0	0
$736$	−443.309	−0.0222019
$737$	8410.35	0.420352
$738$	0	0
$739$	−10331.9	−0.514299	−0.257149	−	0.966372i	$-0.582783\pi$
$739$	−10331.9	−0.514299	−0.257149	+	0.966372i	$0.582783\pi$
$740$	1542.22	0.0766124
$741$	0	0
$742$	30890.5	1.52834
$743$	27385.1	1.35217	0.676084	−	0.736824i	$-0.263676\pi$
$743$	27385.1	1.35217	0.676084	+	0.736824i	$0.263676\pi$
$744$	0	0
$745$	3821.75	0.187944
$746$	−40164.9	−1.97123
$747$	0	0
$748$	36929.0	1.80516
$749$	33514.8	1.63499
$750$	0	0
$751$	−40267.8	−1.95658	−0.978291	−	0.207234i	$-0.933554\pi$
$751$	−40267.8	−1.95658	−0.978291	+	0.207234i	$0.933554\pi$
$752$	−1011.91	−0.0490697
$753$	0	0
$754$	4535.44	0.219060
$755$	−13608.2	−0.655964
$756$	0	0
$757$	−7047.02	−0.338346	−0.169173	−	0.985586i	$-0.554110\pi$
$757$	−7047.02	−0.338346	−0.169173	+	0.985586i	$0.554110\pi$
$758$	48956.0	2.34586
$759$	0	0
$760$	22938.8	1.09484
$761$	−38699.3	−1.84343	−0.921714	−	0.387871i	$-0.873211\pi$
$761$	−38699.3	−1.84343	−0.921714	+	0.387871i	$0.873211\pi$
$762$	0	0
$763$	−22217.7	−1.05417
$764$	−71071.3	−3.36554
$765$	0	0
$766$	−33231.6	−1.56750
$767$	−6659.52	−0.313509
$768$	0	0
$769$	−2374.27	−0.111337	−0.0556686	−	0.998449i	$-0.517729\pi$
$769$	−2374.27	−0.111337	−0.0556686	+	0.998449i	$0.517729\pi$
$770$	−18439.6	−0.863010
$771$	0	0
$772$	−34592.2	−1.61270
$773$	12947.9	0.602464	0.301232	−	0.953551i	$-0.402602\pi$
$773$	12947.9	0.602464	0.301232	+	0.953551i	$0.402602\pi$
$774$	0	0
$775$	−673.311	−0.0312078
$776$	17291.7	0.799920
$777$	0	0
$778$	75024.6	3.45728
$779$	−11943.4	−0.549314
$780$	0	0
$781$	23372.8	1.07086
$782$	−1947.50	−0.0890570
$783$	0	0
$784$	14121.7	0.643300
$785$	−8801.88	−0.400194
$786$	0	0
$787$	−15659.1	−0.709260	−0.354630	−	0.935007i	$-0.615393\pi$
$787$	−15659.1	−0.709260	−0.354630	+	0.935007i	$0.615393\pi$
$788$	−75390.8	−3.40823
$789$	0	0
$790$	−8211.66	−0.369820
$791$	−3048.56	−0.137034
$792$	0	0
$793$	−24039.2	−1.07649
$794$	−11114.7	−0.496786
$795$	0	0
$796$	−32556.4	−1.44966
$797$	−36033.7	−1.60148	−0.800739	−	0.599013i	$-0.795560\pi$
$797$	−36033.7	−1.60148	−0.800739	+	0.599013i	$0.795560\pi$
$798$	0	0
$799$	−780.186	−0.0345444
$800$	1889.83	0.0835195
$801$	0	0
$802$	37927.4	1.66991
$803$	1492.39	0.0655858
$804$	0	0
$805$	659.967	0.0288954
$806$	−4212.08	−0.184075
$807$	0	0
$808$	−12188.0	−0.530658
$809$	−8941.26	−0.388576	−0.194288	−	0.980945i	$-0.562240\pi$
$809$	−8941.26	−0.388576	−0.194288	+	0.980945i	$0.562240\pi$
$810$	0	0
$811$	31988.8	1.38506	0.692528	−	0.721391i	$-0.256497\pi$
$811$	31988.8	1.38506	0.692528	+	0.721391i	$0.256497\pi$
$812$	11028.8	0.476643
$813$	0	0
$814$	−2991.09	−0.128793
$815$	17830.7	0.766359
$816$	0	0
$817$	−42970.3	−1.84008
$818$	−65352.8	−2.79341
$819$	0	0
$820$	−9763.19	−0.415787
$821$	27919.3	1.18683	0.593417	−	0.804895i	$-0.297778\pi$
$821$	27919.3	1.18683	0.593417	+	0.804895i	$0.297778\pi$
$822$	0	0
$823$	44835.5	1.89899	0.949494	−	0.313784i	$-0.101597\pi$
$823$	44835.5	1.89899	0.949494	+	0.313784i	$0.101597\pi$
$824$	−37337.9	−1.57855
$825$	0	0
$826$	−23861.2	−1.00513
$827$	26829.4	1.12811	0.564056	−	0.825737i	$-0.309240\pi$
$827$	26829.4	1.12811	0.564056	+	0.825737i	$0.309240\pi$
$828$	0	0
$829$	−34790.3	−1.45756	−0.728780	−	0.684748i	$-0.759912\pi$
$829$	−34790.3	−1.45756	−0.728780	+	0.684748i	$0.759912\pi$
$830$	−7067.37	−0.295557
$831$	0	0
$832$	−9823.14	−0.409322
$833$	10887.9	0.452874
$834$	0	0
$835$	−9880.36	−0.409490
$836$	−57344.0	−2.37235
$837$	0	0
$838$	−54874.6	−2.26206
$839$	−23387.9	−0.962383	−0.481192	−	0.876615i	$-0.659796\pi$
$839$	−23387.9	−0.962383	−0.481192	+	0.876615i	$0.659796\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	−65025.9	−2.66145
$843$	0	0
$844$	−75818.9	−3.09217
$845$	6072.85	0.247234
$846$	0	0
$847$	−5686.23	−0.230675
$848$	23743.9	0.961520
$849$	0	0
$850$	8302.23	0.335017
$851$	107.053	0.00431226
$852$	0	0
$853$	3915.34	0.157161	0.0785807	−	0.996908i	$-0.474961\pi$
$853$	3915.34	0.157161	0.0785807	+	0.996908i	$0.474961\pi$
$854$	−86132.8	−3.45129
$855$	0	0
$856$	66100.3	2.63932
$857$	−31094.8	−1.23942	−0.619708	−	0.784833i	$-0.712749\pi$
$857$	−31094.8	−1.23942	−0.619708	+	0.784833i	$0.712749\pi$
$858$	0	0
$859$	9069.15	0.360227	0.180114	−	0.983646i	$-0.442353\pi$
$859$	9069.15	0.360227	0.180114	+	0.983646i	$0.442353\pi$
$860$	−35126.4	−1.39279
$861$	0	0
$862$	3901.20	0.154148
$863$	18282.8	0.721151	0.360576	−	0.932730i	$-0.382580\pi$
$863$	18282.8	0.721151	0.360576	+	0.932730i	$0.382580\pi$
$864$	0	0
$865$	−9180.72	−0.360872
$866$	45704.3	1.79341
$867$	0	0
$868$	−10242.5	−0.400521
$869$	10808.7	0.421933
$870$	0	0
$871$	8027.54	0.312288
$872$	−43819.3	−1.70173
$873$	0	0
$874$	3024.12	0.117039
$875$	−2813.45	−0.108699
$876$	0	0
$877$	4534.49	0.174594	0.0872970	−	0.996182i	$-0.472177\pi$
$877$	4534.49	0.174594	0.0872970	+	0.996182i	$0.472177\pi$
$878$	−20201.2	−0.776490
$879$	0	0
$880$	−14173.6	−0.542944
$881$	−27949.4	−1.06883	−0.534415	−	0.845223i	$-0.679468\pi$
$881$	−27949.4	−1.06883	−0.534415	+	0.845223i	$0.679468\pi$
$882$	0	0
$883$	7755.96	0.295593	0.147797	−	0.989018i	$-0.452782\pi$
$883$	7755.96	0.295593	0.147797	+	0.989018i	$0.452782\pi$
$884$	35248.1	1.34109
$885$	0	0
$886$	32543.9	1.23401
$887$	−28219.5	−1.06823	−0.534115	−	0.845412i	$-0.679355\pi$
$887$	−28219.5	−1.06823	−0.534115	+	0.845412i	$0.679355\pi$
$888$	0	0
$889$	34648.7	1.30718
$890$	−20029.0	−0.754352
$891$	0	0
$892$	−86397.2	−3.24304
$893$	1211.49	0.0453985
$894$	0	0
$895$	11209.3	0.418644
$896$	−48807.8	−1.81981
$897$	0	0
$898$	24682.7	0.917230
$899$	−781.041	−0.0289757
$900$	0	0
$901$	18306.7	0.676897
$902$	18935.4	0.698980
$903$	0	0
$904$	−6012.58	−0.221212
$905$	−10423.7	−0.382869
$906$	0	0
$907$	182.994	0.00669926	0.00334963	−	0.999994i	$-0.498934\pi$
$907$	182.994	0.00669926	0.00334963	+	0.999994i	$0.498934\pi$
$908$	88849.6	3.24733
$909$	0	0
$910$	−17600.3	−0.641148
$911$	15870.4	0.577179	0.288590	−	0.957453i	$-0.406814\pi$
$911$	15870.4	0.577179	0.288590	+	0.957453i	$0.406814\pi$
$912$	0	0
$913$	9302.52	0.337205
$914$	80391.9	2.90933
$915$	0	0
$916$	−88577.2	−3.19506
$917$	−36576.0	−1.31717
$918$	0	0
$919$	−14707.3	−0.527908	−0.263954	−	0.964535i	$-0.585027\pi$
$919$	−14707.3	−0.527908	−0.263954	+	0.964535i	$0.585027\pi$
$920$	1301.63	0.0466452
$921$	0	0
$922$	−16129.0	−0.576116
$923$	22308.9	0.795566
$924$	0	0
$925$	−456.369	−0.0162220
$926$	33747.6	1.19764
$927$	0	0
$928$	2192.20	0.0775459
$929$	−4008.12	−0.141552	−0.0707762	−	0.997492i	$-0.522548\pi$
$929$	−4008.12	−0.141552	−0.0707762	+	0.997492i	$0.522548\pi$
$930$	0	0
$931$	−16907.0	−0.595171
$932$	102171.	3.59092
$933$	0	0
$934$	−12272.1	−0.429932
$935$	−10927.9	−0.382226
$936$	0	0
$937$	17088.4	0.595788	0.297894	−	0.954599i	$-0.403716\pi$
$937$	17088.4	0.595788	0.297894	+	0.954599i	$0.403716\pi$
$938$	28762.8	1.00121
$939$	0	0
$940$	990.339	0.0343631
$941$	−1981.60	−0.0686486	−0.0343243	−	0.999411i	$-0.510928\pi$
$941$	−1981.60	−0.0686486	−0.0343243	+	0.999411i	$0.510928\pi$
$942$	0	0
$943$	−677.712	−0.0234033
$944$	−18340.8	−0.632355
$945$	0	0
$946$	68126.5	2.34142
$947$	19125.0	0.656260	0.328130	−	0.944633i	$-0.393582\pi$
$947$	19125.0	0.656260	0.328130	+	0.944633i	$0.393582\pi$
$948$	0	0
$949$	1424.46	0.0487251
$950$	−12891.9	−0.440281
$951$	0	0
$952$	66498.1	2.26388
$953$	−10338.6	−0.351416	−0.175708	−	0.984442i	$-0.556222\pi$
$953$	−10338.6	−0.351416	−0.175708	+	0.984442i	$0.556222\pi$
$954$	0	0
$955$	21031.2	0.712622
$956$	3592.99	0.121554
$957$	0	0
$958$	−16006.0	−0.539802
$959$	20822.1	0.701128
$960$	0	0
$961$	−29065.6	−0.975652
$962$	−2854.94	−0.0956831
$963$	0	0
$964$	−98973.5	−3.30677
$965$	10236.4	0.341474
$966$	0	0
$967$	−18633.6	−0.619664	−0.309832	−	0.950791i	$-0.600273\pi$
$967$	−18633.6	−0.619664	−0.309832	+	0.950791i	$0.600273\pi$
$968$	−11214.8	−0.372373
$969$	0	0
$970$	−9718.15	−0.321681
$971$	39486.5	1.30503	0.652515	−	0.757776i	$-0.273714\pi$
$971$	39486.5	1.30503	0.652515	+	0.757776i	$0.273714\pi$
$972$	0	0
$973$	−72747.6	−2.39690
$974$	61682.1	2.02918
$975$	0	0
$976$	−66205.8	−2.17131
$977$	−33392.3	−1.09346	−0.546731	−	0.837308i	$-0.684128\pi$
$977$	−33392.3	−1.09346	−0.546731	+	0.837308i	$0.684128\pi$
$978$	0	0
$979$	26363.4	0.860651
$980$	−13820.7	−0.450497
$981$	0	0
$982$	−91960.7	−2.98838
$983$	−20923.0	−0.678880	−0.339440	−	0.940628i	$-0.610238\pi$
$983$	−20923.0	−0.678880	−0.339440	+	0.940628i	$0.610238\pi$
$984$	0	0
$985$	22309.4	0.721662
$986$	9630.59	0.311055
$987$	0	0
$988$	−54733.9	−1.76247
$989$	−2438.30	−0.0783957
$990$	0	0
$991$	44681.6	1.43225	0.716124	−	0.697973i	$-0.245914\pi$
$991$	44681.6	1.43225	0.716124	+	0.697973i	$0.245914\pi$
$992$	−2035.91	−0.0651615
$993$	0	0
$994$	79933.2	2.55063
$995$	9633.98	0.306953
$996$	0	0
$997$	−49844.7	−1.58335	−0.791674	−	0.610944i	$-0.790790\pi$
$997$	−49844.7	−1.58335	−0.791674	+	0.610944i	$0.790790\pi$
$998$	46597.2	1.47797
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1305.4.a.i.1.6		6
3.2	odd	2		435.4.a.g.1.1	✓	6
15.14	odd	2		2175.4.a.l.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
435.4.a.g.1.1	✓	6	3.2	odd	2
1305.4.a.i.1.6		6	1.1	even	1	trivial
2175.4.a.l.1.6		6	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1305.a (trivial)

\(f(q)\)	\(=\)	\(q+4.98965 q^{2} +16.8966 q^{4} -5.00000 q^{5} +22.5076 q^{7} +44.3911 q^{8} -24.9483 q^{10} +32.8384 q^{11} +31.3437 q^{13} +112.305 q^{14} +86.3230 q^{16} +66.5556 q^{17} -103.349 q^{19} -84.4831 q^{20} +163.852 q^{22} -5.86439 q^{23} +25.0000 q^{25} +156.394 q^{26} +380.302 q^{28} +29.0000 q^{29} -26.9324 q^{31} +75.5933 q^{32} +332.089 q^{34} -112.538 q^{35} -18.2548 q^{37} -515.675 q^{38} -221.955 q^{40} +115.564 q^{41} +415.780 q^{43} +554.859 q^{44} -29.2613 q^{46} -11.7223 q^{47} +163.592 q^{49} +124.741 q^{50} +529.604 q^{52} +275.059 q^{53} -164.192 q^{55} +999.136 q^{56} +144.700 q^{58} -212.467 q^{59} -766.954 q^{61} -134.383 q^{62} -313.400 q^{64} -156.719 q^{65} +256.113 q^{67} +1124.57 q^{68} -561.525 q^{70} +711.750 q^{71} +45.4465 q^{73} -91.0850 q^{74} -1746.25 q^{76} +739.114 q^{77} +329.147 q^{79} -431.615 q^{80} +576.623 q^{82} +283.281 q^{83} -332.778 q^{85} +2074.60 q^{86} +1457.73 q^{88} +802.821 q^{89} +705.472 q^{91} -99.0885 q^{92} -58.4903 q^{94} +516.744 q^{95} +389.532 q^{97} +816.265 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + q^{2} + 15 q^{4} - 30 q^{5} + 23 q^{7} + 51 q^{8} - 5 q^{10} + 111 q^{11} - 83 q^{13} + 102 q^{14} - 37 q^{16} + 35 q^{17} - 76 q^{19} - 75 q^{20} + 66 q^{22} - 166 q^{23} + 150 q^{25} + 282 q^{26}+ \cdots + 1903 q^{98}+O(q^{100}) \) 6 * q + q^2 + 15 * q^4 - 30 * q^5 + 23 * q^7 + 51 * q^8 - 5 * q^10 + 111 * q^11 - 83 * q^13 + 102 * q^14 - 37 * q^16 + 35 * q^17 - 76 * q^19 - 75 * q^20 + 66 * q^22 - 166 * q^23 + 150 * q^25 + 282 * q^26 + 368 * q^28 + 174 * q^29 - 164 * q^31 + 259 * q^32 + 512 * q^34 - 115 * q^35 - 538 * q^37 - 6 * q^38 - 255 * q^40 + 1250 * q^41 - 80 * q^43 + 1004 * q^44 - 1190 * q^46 + 491 * q^47 - 969 * q^49 + 25 * q^50 - 336 * q^52 + 356 * q^53 - 555 * q^55 + 1260 * q^56 + 29 * q^58 + 646 * q^59 - 1476 * q^61 + 4 * q^62 - 845 * q^64 + 415 * q^65 + 87 * q^67 + 558 * q^68 - 510 * q^70 + 1312 * q^71 - 1142 * q^73 + 2270 * q^74 - 1594 * q^76 + 2561 * q^77 - 142 * q^79 + 185 * q^80 - 1544 * q^82 - 346 * q^83 - 175 * q^85 + 2056 * q^86 + 1400 * q^88 + 2827 * q^89 + 1379 * q^91 + 1434 * q^92 + 978 * q^94 + 380 * q^95 - 4 * q^97 + 1903 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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