LMFDB - Embedded newform 928.4.a.j.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [928,4,Mod(1,928)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(928, 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("928.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 928 = 2^{5} \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	928.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:	$54.7537724853$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 2 x^{11} - 217 x^{10} + 520 x^{9} + 16022 x^{8} - 37368 x^{7} - 509640 x^{6} + 989168 x^{5} + \cdots + 14751072 $ x^12 - 2x^11 - 217x^10 + 520x^9 + 16022x^8 - 37368x^7 - 509640x^6 + 989168x^5 + 7106592x^4 - 7979328x^3 - 38912400x^2 + 1083456*x + 14751072
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{17} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-0.676012$ of defining polynomial
Character	$\chi$	$=$	928.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.323988 q^{3} -5.78413 q^{5} -34.7780 q^{7} -26.8950 q^{9} +O(q^{10})$	$q+0.323988 q^{3} -5.78413 q^{5} -34.7780 q^{7} -26.8950 q^{9} +18.6550 q^{11} +38.6717 q^{13} -1.87399 q^{15} -101.230 q^{17} +1.73516 q^{19} -11.2676 q^{21} -24.4055 q^{23} -91.5438 q^{25} -17.4613 q^{27} +29.0000 q^{29} -186.050 q^{31} +6.04400 q^{33} +201.160 q^{35} -339.618 q^{37} +12.5292 q^{39} +34.8517 q^{41} +483.340 q^{43} +155.564 q^{45} -266.551 q^{47} +866.508 q^{49} -32.7972 q^{51} +148.821 q^{53} -107.903 q^{55} +0.562169 q^{57} +6.98434 q^{59} +38.0806 q^{61} +935.355 q^{63} -223.682 q^{65} +544.060 q^{67} -7.90710 q^{69} +1064.67 q^{71} +688.581 q^{73} -29.6591 q^{75} -648.784 q^{77} -424.529 q^{79} +720.509 q^{81} +1247.54 q^{83} +585.526 q^{85} +9.39565 q^{87} -174.320 q^{89} -1344.92 q^{91} -60.2781 q^{93} -10.0364 q^{95} +33.7772 q^{97} -501.727 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 14 q^{3} - 10 q^{5} + 44 q^{7} + 130 q^{9}+O(q^{10}) $ 12 * q + 14 * q^3 - 10 * q^5 + 44 * q^7 + 130 * q^9	$ 12 q + 14 q^{3} - 10 q^{5} + 44 q^{7} + 130 q^{9} + 46 q^{11} - 34 q^{13} - 50 q^{15} + 36 q^{17} + 148 q^{19} - 92 q^{21} + 328 q^{23} + 486 q^{25} + 326 q^{27} + 348 q^{29} + 374 q^{31} + 710 q^{33} + 204 q^{35} - 340 q^{37} - 122 q^{39} + 32 q^{41} + 462 q^{43} - 1132 q^{45} + 434 q^{47} + 1508 q^{49} + 440 q^{51} + 610 q^{53} + 46 q^{55} - 932 q^{57} + 1240 q^{59} - 1228 q^{61} + 4240 q^{63} + 730 q^{65} + 1672 q^{67} - 528 q^{69} + 3220 q^{71} + 564 q^{73} + 6032 q^{75} + 644 q^{77} + 1862 q^{79} + 3040 q^{81} + 3736 q^{83} - 808 q^{85} + 406 q^{87} + 584 q^{89} + 4844 q^{91} - 3226 q^{93} + 2844 q^{95} + 904 q^{97} + 6832 q^{99}+O(q^{100}) $ 12 * q + 14 * q^3 - 10 * q^5 + 44 * q^7 + 130 * q^9 + 46 * q^11 - 34 * q^13 - 50 * q^15 + 36 * q^17 + 148 * q^19 - 92 * q^21 + 328 * q^23 + 486 * q^25 + 326 * q^27 + 348 * q^29 + 374 * q^31 + 710 * q^33 + 204 * q^35 - 340 * q^37 - 122 * q^39 + 32 * q^41 + 462 * q^43 - 1132 * q^45 + 434 * q^47 + 1508 * q^49 + 440 * q^51 + 610 * q^53 + 46 * q^55 - 932 * q^57 + 1240 * q^59 - 1228 * q^61 + 4240 * q^63 + 730 * q^65 + 1672 * q^67 - 528 * q^69 + 3220 * q^71 + 564 * q^73 + 6032 * q^75 + 644 * q^77 + 1862 * q^79 + 3040 * q^81 + 3736 * q^83 - 808 * q^85 + 406 * q^87 + 584 * q^89 + 4844 * q^91 - 3226 * q^93 + 2844 * q^95 + 904 * q^97 + 6832 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	0.323988	0.0623515	0.0311757	−	0.999514i	$-0.490075\pi$
$3$	0.323988	0.0623515	0.0311757	+	0.999514i	$0.490075\pi$
$4$	0	0
$5$	−5.78413	−0.517348	−0.258674	−	0.965965i	$-0.583286\pi$
$5$	−5.78413	−0.517348	−0.258674	+	0.965965i	$0.583286\pi$
$6$	0	0
$7$	−34.7780	−1.87783	−0.938917	−	0.344143i	$-0.888169\pi$
$7$	−34.7780	−1.87783	−0.938917	+	0.344143i	$0.888169\pi$
$8$	0	0
$9$	−26.8950	−0.996112
$10$	0	0
$11$	18.6550	0.511336	0.255668	−	0.966765i	$-0.417705\pi$
$11$	18.6550	0.511336	0.255668	+	0.966765i	$0.417705\pi$
$12$	0	0
$13$	38.6717	0.825046	0.412523	−	0.910947i	$-0.364648\pi$
$13$	38.6717	0.825046	0.412523	+	0.910947i	$0.364648\pi$
$14$	0	0
$15$	−1.87399	−0.0322574
$16$	0	0
$17$	−101.230	−1.44422	−0.722112	−	0.691776i	$-0.756828\pi$
$17$	−101.230	−1.44422	−0.722112	+	0.691776i	$0.756828\pi$
$18$	0	0
$19$	1.73516	0.0209512	0.0104756	−	0.999945i	$-0.496665\pi$
$19$	1.73516	0.0209512	0.0104756	+	0.999945i	$0.496665\pi$
$20$	0	0
$21$	−11.2676	−0.117086
$22$	0	0
$23$	−24.4055	−0.221257	−0.110628	−	0.993862i	$-0.535286\pi$
$23$	−24.4055	−0.221257	−0.110628	+	0.993862i	$0.535286\pi$
$24$	0	0
$25$	−91.5438	−0.732351
$26$	0	0
$27$	−17.4613	−0.124461
$28$	0	0
$29$	29.0000	0.185695
$29$	29.0000	0.185695
$30$	0	0
$31$	−186.050	−1.07792	−0.538962	−	0.842330i	$-0.681183\pi$
$31$	−186.050	−1.07792	−0.538962	+	0.842330i	$0.681183\pi$
$32$	0	0
$33$	6.04400	0.0318826
$34$	0	0
$35$	201.160	0.971495
$36$	0	0
$37$	−339.618	−1.50900	−0.754498	−	0.656303i	$-0.772119\pi$
$37$	−339.618	−1.50900	−0.754498	+	0.656303i	$0.772119\pi$
$38$	0	0
$39$	12.5292	0.0514429
$40$	0	0
$41$	34.8517	0.132754	0.0663770	−	0.997795i	$-0.478856\pi$
$41$	34.8517	0.132754	0.0663770	+	0.997795i	$0.478856\pi$
$42$	0	0
$43$	483.340	1.71416	0.857078	−	0.515187i	$-0.172277\pi$
$43$	483.340	1.71416	0.857078	+	0.515187i	$0.172277\pi$
$44$	0	0
$45$	155.564	0.515337
$46$	0	0
$47$	−266.551	−0.827244	−0.413622	−	0.910449i	$-0.635736\pi$
$47$	−266.551	−0.827244	−0.413622	+	0.910449i	$0.635736\pi$
$48$	0	0
$49$	866.508	2.52626
$50$	0	0
$51$	−32.7972	−0.0900495
$52$	0	0
$53$	148.821	0.385701	0.192850	−	0.981228i	$-0.438227\pi$
$53$	148.821	0.385701	0.192850	+	0.981228i	$0.438227\pi$
$54$	0	0
$55$	−107.903	−0.264539
$56$	0	0
$57$	0.562169	0.00130634
$58$	0	0
$59$	6.98434	0.0154116	0.00770579	−	0.999970i	$-0.497547\pi$
$59$	6.98434	0.0154116	0.00770579	+	0.999970i	$0.497547\pi$
$60$	0	0
$61$	38.0806	0.0799298	0.0399649	−	0.999201i	$-0.487275\pi$
$61$	38.0806	0.0799298	0.0399649	+	0.999201i	$0.487275\pi$
$62$	0	0
$63$	935.355	1.87053
$64$	0	0
$65$	−223.682	−0.426836
$66$	0	0
$67$	544.060	0.992053	0.496026	−	0.868307i	$-0.334792\pi$
$67$	544.060	0.992053	0.496026	+	0.868307i	$0.334792\pi$
$68$	0	0
$69$	−7.90710	−0.0137957
$70$	0	0
$71$	1064.67	1.77962	0.889812	−	0.456327i	$-0.150835\pi$
$71$	1064.67	1.77962	0.889812	+	0.456327i	$0.150835\pi$
$72$	0	0
$73$	688.581	1.10400	0.552002	−	0.833843i	$-0.313864\pi$
$73$	688.581	1.10400	0.552002	+	0.833843i	$0.313864\pi$
$74$	0	0
$75$	−29.6591	−0.0456632
$76$	0	0
$77$	−648.784	−0.960205
$78$	0	0
$79$	−424.529	−0.604597	−0.302299	−	0.953213i	$-0.597754\pi$
$79$	−424.529	−0.604597	−0.302299	+	0.953213i	$0.597754\pi$
$80$	0	0
$81$	720.509	0.988352
$82$	0	0
$83$	1247.54	1.64983	0.824913	−	0.565260i	$-0.191224\pi$
$83$	1247.54	1.64983	0.824913	+	0.565260i	$0.191224\pi$
$84$	0	0
$85$	585.526	0.747167
$86$	0	0
$87$	9.39565	0.0115784
$88$	0	0
$89$	−174.320	−0.207616	−0.103808	−	0.994597i	$-0.533103\pi$
$89$	−174.320	−0.207616	−0.103808	+	0.994597i	$0.533103\pi$
$90$	0	0
$91$	−1344.92	−1.54930
$92$	0	0
$93$	−60.2781	−0.0672102
$94$	0	0
$95$	−10.0364	−0.0108390
$96$	0	0
$97$	33.7772	0.0353563	0.0176781	−	0.999844i	$-0.494373\pi$
$97$	33.7772	0.0353563	0.0176781	+	0.999844i	$0.494373\pi$
$98$	0	0
$99$	−501.727	−0.509348
$100$	0	0
$101$	−1665.04	−1.64037	−0.820184	−	0.572099i	$-0.806129\pi$
$101$	−1665.04	−1.64037	−0.820184	+	0.572099i	$0.806129\pi$
$102$	0	0
$103$	1040.43	0.995309	0.497655	−	0.867375i	$-0.334195\pi$
$103$	1040.43	0.995309	0.497655	+	0.867375i	$0.334195\pi$
$104$	0	0
$105$	65.1735	0.0605741
$106$	0	0
$107$	377.579	0.341139	0.170570	−	0.985346i	$-0.445439\pi$
$107$	377.579	0.341139	0.170570	+	0.985346i	$0.445439\pi$
$108$	0	0
$109$	−859.283	−0.755086	−0.377543	−	0.925992i	$-0.623231\pi$
$109$	−859.283	−0.755086	−0.377543	+	0.925992i	$0.623231\pi$
$110$	0	0
$111$	−110.032	−0.0940881
$112$	0	0
$113$	−1386.98	−1.15465	−0.577326	−	0.816514i	$-0.695904\pi$
$113$	−1386.98	−1.15465	−0.577326	+	0.816514i	$0.695904\pi$
$114$	0	0
$115$	141.165	0.114467
$116$	0	0
$117$	−1040.08	−0.821839
$118$	0	0
$119$	3520.56	2.71201
$120$	0	0
$121$	−982.991	−0.738535
$122$	0	0
$123$	11.2915	0.00827742
$124$	0	0
$125$	1252.52	0.896229
$126$	0	0
$127$	−981.731	−0.685941	−0.342970	−	0.939346i	$-0.611433\pi$
$127$	−981.731	−0.685941	−0.342970	+	0.939346i	$0.611433\pi$
$128$	0	0
$129$	156.596	0.106880
$130$	0	0
$131$	668.964	0.446166	0.223083	−	0.974800i	$-0.428388\pi$
$131$	668.964	0.446166	0.223083	+	0.974800i	$0.428388\pi$
$132$	0	0
$133$	−60.3452	−0.0393428
$134$	0	0
$135$	100.999	0.0643895
$136$	0	0
$137$	138.077	0.0861072	0.0430536	−	0.999073i	$-0.486291\pi$
$137$	138.077	0.0861072	0.0430536	+	0.999073i	$0.486291\pi$
$138$	0	0
$139$	1004.97	0.613241	0.306621	−	0.951832i	$-0.400802\pi$
$139$	1004.97	0.613241	0.306621	+	0.951832i	$0.400802\pi$
$140$	0	0
$141$	−86.3593	−0.0515799
$142$	0	0
$143$	721.421	0.421876
$144$	0	0
$145$	−167.740	−0.0960692
$146$	0	0
$147$	280.738	0.157516
$148$	0	0
$149$	−3548.03	−1.95078	−0.975388	−	0.220495i	$-0.929233\pi$
$149$	−3548.03	−1.95078	−0.975388	+	0.220495i	$0.929233\pi$
$150$	0	0
$151$	1569.44	0.845823	0.422912	−	0.906171i	$-0.361008\pi$
$151$	1569.44	0.845823	0.422912	+	0.906171i	$0.361008\pi$
$152$	0	0
$153$	2722.58	1.43861
$154$	0	0
$155$	1076.14	0.557662
$156$	0	0
$157$	2180.42	1.10839	0.554194	−	0.832388i	$-0.313027\pi$
$157$	2180.42	1.10839	0.554194	+	0.832388i	$0.313027\pi$
$158$	0	0
$159$	48.2162	0.0240490
$160$	0	0
$161$	848.775	0.415484
$162$	0	0
$163$	1450.58	0.697045	0.348523	−	0.937300i	$-0.386683\pi$
$163$	1450.58	0.697045	0.348523	+	0.937300i	$0.386683\pi$
$164$	0	0
$165$	−34.9593	−0.0164944
$166$	0	0
$167$	1600.50	0.741618	0.370809	−	0.928709i	$-0.379081\pi$
$167$	1600.50	0.741618	0.370809	+	0.928709i	$0.379081\pi$
$168$	0	0
$169$	−701.499	−0.319299
$170$	0	0
$171$	−46.6671	−0.0208697
$172$	0	0
$173$	1909.39	0.839124	0.419562	−	0.907727i	$-0.362184\pi$
$173$	1909.39	0.839124	0.419562	+	0.907727i	$0.362184\pi$
$174$	0	0
$175$	3183.71	1.37523
$176$	0	0
$177$	2.26284	0.000960936 0
$178$	0	0
$179$	2562.06	1.06982	0.534908	−	0.844910i	$-0.320346\pi$
$179$	2562.06	1.06982	0.534908	+	0.844910i	$0.320346\pi$
$180$	0	0
$181$	1803.16	0.740485	0.370242	−	0.928935i	$-0.379275\pi$
$181$	1803.16	0.740485	0.370242	+	0.928935i	$0.379275\pi$
$182$	0	0
$183$	12.3376	0.00498374
$184$	0	0
$185$	1964.39	0.780676
$186$	0	0
$187$	−1888.44	−0.738484
$188$	0	0
$189$	607.270	0.233716
$190$	0	0
$191$	−4097.08	−1.55212	−0.776059	−	0.630661i	$-0.782784\pi$
$191$	−4097.08	−1.55212	−0.776059	+	0.630661i	$0.782784\pi$
$192$	0	0
$193$	2395.56	0.893452	0.446726	−	0.894671i	$-0.352590\pi$
$193$	2395.56	0.893452	0.446726	+	0.894671i	$0.352590\pi$
$194$	0	0
$195$	−72.4703	−0.0266139
$196$	0	0
$197$	−2927.37	−1.05871	−0.529357	−	0.848399i	$-0.677567\pi$
$197$	−2927.37	−1.05871	−0.529357	+	0.848399i	$0.677567\pi$
$198$	0	0
$199$	−2894.37	−1.03104	−0.515519	−	0.856878i	$-0.672401\pi$
$199$	−2894.37	−1.03104	−0.515519	+	0.856878i	$0.672401\pi$
$200$	0	0
$201$	176.269	0.0618560
$202$	0	0
$203$	−1008.56	−0.348705
$204$	0	0
$205$	−201.587	−0.0686801
$206$	0	0
$207$	656.388	0.220397
$208$	0	0
$209$	32.3693	0.0107131
$210$	0	0
$211$	2576.49	0.840630	0.420315	−	0.907378i	$-0.361920\pi$
$211$	2576.49	0.840630	0.420315	+	0.907378i	$0.361920\pi$
$212$	0	0
$213$	344.941	0.110962
$214$	0	0
$215$	−2795.70	−0.886816
$216$	0	0
$217$	6470.46	2.02416
$218$	0	0
$219$	223.092	0.0688363
$220$	0	0
$221$	−3914.73	−1.19155
$222$	0	0
$223$	−3521.80	−1.05756	−0.528782	−	0.848758i	$-0.677351\pi$
$223$	−3521.80	−1.05756	−0.528782	+	0.848758i	$0.677351\pi$
$224$	0	0
$225$	2462.07	0.729503
$226$	0	0
$227$	−2334.62	−0.682617	−0.341308	−	0.939951i	$-0.610870\pi$
$227$	−2334.62	−0.682617	−0.341308	+	0.939951i	$0.610870\pi$
$228$	0	0
$229$	2095.27	0.604626	0.302313	−	0.953209i	$-0.402241\pi$
$229$	2095.27	0.604626	0.302313	+	0.953209i	$0.402241\pi$
$230$	0	0
$231$	−210.198	−0.0598702
$232$	0	0
$233$	−4185.52	−1.17683	−0.588417	−	0.808557i	$-0.700249\pi$
$233$	−4185.52	−1.17683	−0.588417	+	0.808557i	$0.700249\pi$
$234$	0	0
$235$	1541.77	0.427973
$236$	0	0
$237$	−137.542	−0.0376975
$238$	0	0
$239$	4505.51	1.21940	0.609702	−	0.792631i	$-0.291289\pi$
$239$	4505.51	1.21940	0.609702	+	0.792631i	$0.291289\pi$
$240$	0	0
$241$	−2957.99	−0.790627	−0.395313	−	0.918546i	$-0.629364\pi$
$241$	−2957.99	−0.790627	−0.395313	+	0.918546i	$0.629364\pi$
$242$	0	0
$243$	704.892	0.186086
$244$	0	0
$245$	−5011.99	−1.30696
$246$	0	0
$247$	67.1014	0.0172857
$248$	0	0
$249$	404.189	0.102869
$250$	0	0
$251$	−2524.50	−0.634842	−0.317421	−	0.948285i	$-0.602817\pi$
$251$	−2524.50	−0.634842	−0.317421	+	0.948285i	$0.602817\pi$
$252$	0	0
$253$	−455.286	−0.113137
$254$	0	0
$255$	189.703	0.0465870
$256$	0	0
$257$	−2763.13	−0.670658	−0.335329	−	0.942101i	$-0.608847\pi$
$257$	−2763.13	−0.670658	−0.335329	+	0.942101i	$0.608847\pi$
$258$	0	0
$259$	11811.2	2.83364
$260$	0	0
$261$	−779.956	−0.184973
$262$	0	0
$263$	1969.77	0.461829	0.230914	−	0.972974i	$-0.425828\pi$
$263$	1969.77	0.461829	0.230914	+	0.972974i	$0.425828\pi$
$264$	0	0
$265$	−860.800	−0.199542
$266$	0	0
$267$	−56.4775	−0.0129452
$268$	0	0
$269$	−843.547	−0.191197	−0.0955984	−	0.995420i	$-0.530476\pi$
$269$	−843.547	−0.191197	−0.0955984	+	0.995420i	$0.530476\pi$
$270$	0	0
$271$	−2221.93	−0.498053	−0.249027	−	0.968497i	$-0.580111\pi$
$271$	−2221.93	−0.498053	−0.249027	+	0.968497i	$0.580111\pi$
$272$	0	0
$273$	−435.739	−0.0966012
$274$	0	0
$275$	−1707.75	−0.374477
$276$	0	0
$277$	−2165.46	−0.469710	−0.234855	−	0.972030i	$-0.575462\pi$
$277$	−2165.46	−0.469710	−0.234855	+	0.972030i	$0.575462\pi$
$278$	0	0
$279$	5003.83	1.07373
$280$	0	0
$281$	2378.13	0.504867	0.252434	−	0.967614i	$-0.418769\pi$
$281$	2378.13	0.504867	0.252434	+	0.967614i	$0.418769\pi$
$282$	0	0
$283$	4690.25	0.985181	0.492591	−	0.870261i	$-0.336050\pi$
$283$	4690.25	0.985181	0.492591	+	0.870261i	$0.336050\pi$
$284$	0	0
$285$	−3.25166	−0.000675831 0
$286$	0	0
$287$	−1212.07	−0.249290
$288$	0	0
$289$	5334.45	1.08578
$290$	0	0
$291$	10.9434	0.00220452
$292$	0	0
$293$	−6610.75	−1.31810	−0.659051	−	0.752098i	$-0.729042\pi$
$293$	−6610.75	−1.31810	−0.659051	+	0.752098i	$0.729042\pi$
$294$	0	0
$295$	−40.3983	−0.00797316
$296$	0	0
$297$	−325.741	−0.0636412
$298$	0	0
$299$	−943.804	−0.182547
$300$	0	0
$301$	−16809.6	−3.21890
$302$	0	0
$303$	−539.451	−0.102279
$304$	0	0
$305$	−220.263	−0.0413516
$306$	0	0
$307$	−8217.54	−1.52769	−0.763843	−	0.645402i	$-0.776690\pi$
$307$	−8217.54	−1.52769	−0.763843	+	0.645402i	$0.776690\pi$
$308$	0	0
$309$	337.088	0.0620590
$310$	0	0
$311$	3610.62	0.658325	0.329163	−	0.944273i	$-0.393234\pi$
$311$	3610.62	0.658325	0.329163	+	0.944273i	$0.393234\pi$
$312$	0	0
$313$	10201.5	1.84224	0.921119	−	0.389280i	$-0.127276\pi$
$313$	10201.5	1.84224	0.921119	+	0.389280i	$0.127276\pi$
$314$	0	0
$315$	−5410.22	−0.967718
$316$	0	0
$317$	6875.78	1.21824	0.609120	−	0.793078i	$-0.291523\pi$
$317$	6875.78	1.21824	0.609120	+	0.793078i	$0.291523\pi$
$318$	0	0
$319$	540.995	0.0949527
$320$	0	0
$321$	122.331	0.0212705
$322$	0	0
$323$	−175.649	−0.0302582
$324$	0	0
$325$	−3540.16	−0.604223
$326$	0	0
$327$	−278.397	−0.0470807
$328$	0	0
$329$	9270.10	1.55343
$330$	0	0
$331$	−9037.73	−1.50078	−0.750391	−	0.660995i	$-0.770135\pi$
$331$	−9037.73	−1.50078	−0.750391	+	0.660995i	$0.770135\pi$
$332$	0	0
$333$	9134.03	1.50313
$334$	0	0
$335$	−3146.92	−0.513237
$336$	0	0
$337$	−884.371	−0.142952	−0.0714759	−	0.997442i	$-0.522771\pi$
$337$	−884.371	−0.142952	−0.0714759	+	0.997442i	$0.522771\pi$
$338$	0	0
$339$	−449.363	−0.0719943
$340$	0	0
$341$	−3470.77	−0.551181
$342$	0	0
$343$	−18206.5	−2.86607
$344$	0	0
$345$	45.7357	0.00713718
$346$	0	0
$347$	6455.66	0.998726	0.499363	−	0.866393i	$-0.333567\pi$
$347$	6455.66	0.998726	0.499363	+	0.866393i	$0.333567\pi$
$348$	0	0
$349$	−9893.81	−1.51749	−0.758744	−	0.651389i	$-0.774187\pi$
$349$	−9893.81	−1.51749	−0.758744	+	0.651389i	$0.774187\pi$
$350$	0	0
$351$	−675.260	−0.102686
$352$	0	0
$353$	11997.3	1.80893	0.904463	−	0.426551i	$-0.140272\pi$
$353$	11997.3	1.80893	0.904463	+	0.426551i	$0.140272\pi$
$354$	0	0
$355$	−6158.20	−0.920686
$356$	0	0
$357$	1140.62	0.169098
$358$	0	0
$359$	−3692.23	−0.542810	−0.271405	−	0.962465i	$-0.587488\pi$
$359$	−3692.23	−0.542810	−0.271405	+	0.962465i	$0.587488\pi$
$360$	0	0
$361$	−6855.99	−0.999561
$362$	0	0
$363$	−318.477	−0.0460488
$364$	0	0
$365$	−3982.84	−0.571155
$366$	0	0
$367$	3510.51	0.499310	0.249655	−	0.968335i	$-0.419683\pi$
$367$	3510.51	0.499310	0.249655	+	0.968335i	$0.419683\pi$
$368$	0	0
$369$	−937.337	−0.132238
$370$	0	0
$371$	−5175.69	−0.724282
$372$	0	0
$373$	6581.99	0.913680	0.456840	−	0.889549i	$-0.348981\pi$
$373$	6581.99	0.913680	0.456840	+	0.889549i	$0.348981\pi$
$374$	0	0
$375$	405.801	0.0558812
$376$	0	0
$377$	1121.48	0.153207
$378$	0	0
$379$	−2946.59	−0.399356	−0.199678	−	0.979862i	$-0.563990\pi$
$379$	−2946.59	−0.399356	−0.199678	+	0.979862i	$0.563990\pi$
$380$	0	0
$381$	−318.069	−0.0427694
$382$	0	0
$383$	5187.37	0.692068	0.346034	−	0.938222i	$-0.387528\pi$
$383$	5187.37	0.692068	0.346034	+	0.938222i	$0.387528\pi$
$384$	0	0
$385$	3752.65	0.496760
$386$	0	0
$387$	−12999.5	−1.70749
$388$	0	0
$389$	6926.04	0.902736	0.451368	−	0.892338i	$-0.350936\pi$
$389$	6926.04	0.902736	0.451368	+	0.892338i	$0.350936\pi$
$390$	0	0
$391$	2470.56	0.319544
$392$	0	0
$393$	216.736	0.0278191
$394$	0	0
$395$	2455.53	0.312787
$396$	0	0
$397$	4750.53	0.600559	0.300280	−	0.953851i	$-0.402920\pi$
$397$	4750.53	0.600559	0.300280	+	0.953851i	$0.402920\pi$
$398$	0	0
$399$	−19.5511	−0.00245308
$400$	0	0
$401$	2766.90	0.344569	0.172285	−	0.985047i	$-0.444885\pi$
$401$	2766.90	0.344569	0.172285	+	0.985047i	$0.444885\pi$
$402$	0	0
$403$	−7194.89	−0.889337
$404$	0	0
$405$	−4167.52	−0.511322
$406$	0	0
$407$	−6335.58	−0.771604
$408$	0	0
$409$	−11381.2	−1.37595	−0.687975	−	0.725734i	$-0.741500\pi$
$409$	−11381.2	−1.37595	−0.687975	+	0.725734i	$0.741500\pi$
$410$	0	0
$411$	44.7352	0.00536891
$412$	0	0
$413$	−242.901	−0.0289404
$414$	0	0
$415$	−7215.95	−0.853535
$416$	0	0
$417$	325.599	0.0382365
$418$	0	0
$419$	−8513.34	−0.992611	−0.496306	−	0.868148i	$-0.665310\pi$
$419$	−8513.34	−0.992611	−0.496306	+	0.868148i	$0.665310\pi$
$420$	0	0
$421$	10096.2	1.16878	0.584391	−	0.811472i	$-0.301333\pi$
$421$	10096.2	1.16878	0.584391	+	0.811472i	$0.301333\pi$
$422$	0	0
$423$	7168.90	0.824027
$424$	0	0
$425$	9266.95	1.05768
$426$	0	0
$427$	−1324.37	−0.150095
$428$	0	0
$429$	233.732	0.0263046
$430$	0	0
$431$	8704.01	0.972755	0.486377	−	0.873749i	$-0.338318\pi$
$431$	8704.01	0.972755	0.486377	+	0.873749i	$0.338318\pi$
$432$	0	0
$433$	14332.2	1.59067	0.795337	−	0.606168i	$-0.207294\pi$
$433$	14332.2	1.59067	0.795337	+	0.606168i	$0.207294\pi$
$434$	0	0
$435$	−54.3457	−0.00599006
$436$	0	0
$437$	−42.3474	−0.00463558
$438$	0	0
$439$	−17693.7	−1.92363	−0.961813	−	0.273707i	$-0.911750\pi$
$439$	−17693.7	−1.92363	−0.961813	+	0.273707i	$0.911750\pi$
$440$	0	0
$441$	−23304.8	−2.51644
$442$	0	0
$443$	−11328.3	−1.21495	−0.607474	−	0.794339i	$-0.707817\pi$
$443$	−11328.3	−1.21495	−0.607474	+	0.794339i	$0.707817\pi$
$444$	0	0
$445$	1008.29	0.107410
$446$	0	0
$447$	−1149.52	−0.121634
$448$	0	0
$449$	14065.7	1.47840	0.739198	−	0.673488i	$-0.235205\pi$
$449$	14065.7	1.47840	0.739198	+	0.673488i	$0.235205\pi$
$450$	0	0
$451$	650.158	0.0678820
$452$	0	0
$453$	508.480	0.0527384
$454$	0	0
$455$	7779.22	0.801528
$456$	0	0
$457$	−13414.1	−1.37305	−0.686524	−	0.727107i	$-0.740864\pi$
$457$	−13414.1	−1.37305	−0.686524	+	0.727107i	$0.740864\pi$
$458$	0	0
$459$	1767.61	0.179749
$460$	0	0
$461$	−4676.87	−0.472502	−0.236251	−	0.971692i	$-0.575919\pi$
$461$	−4676.87	−0.472502	−0.236251	+	0.971692i	$0.575919\pi$
$462$	0	0
$463$	1297.23	0.130211	0.0651054	−	0.997878i	$-0.479262\pi$
$463$	1297.23	0.130211	0.0651054	+	0.997878i	$0.479262\pi$
$464$	0	0
$465$	348.656	0.0347711
$466$	0	0
$467$	9912.42	0.982210	0.491105	−	0.871100i	$-0.336593\pi$
$467$	9912.42	0.982210	0.491105	+	0.871100i	$0.336593\pi$
$468$	0	0
$469$	−18921.3	−1.86291
$470$	0	0
$471$	706.431	0.0691096
$472$	0	0
$473$	9016.72	0.876510
$474$	0	0
$475$	−158.843	−0.0153436
$476$	0	0
$477$	−4002.55	−0.384201
$478$	0	0
$479$	1373.12	0.130980	0.0654900	−	0.997853i	$-0.479139\pi$
$479$	1373.12	0.130980	0.0654900	+	0.997853i	$0.479139\pi$
$480$	0	0
$481$	−13133.6	−1.24499
$482$	0	0
$483$	274.993	0.0259060
$484$	0	0
$485$	−195.372	−0.0182915
$486$	0	0
$487$	−13565.0	−1.26220	−0.631099	−	0.775703i	$-0.717396\pi$
$487$	−13565.0	−1.26220	−0.631099	+	0.775703i	$0.717396\pi$
$488$	0	0
$489$	469.971	0.0434618
$490$	0	0
$491$	−5322.07	−0.489168	−0.244584	−	0.969628i	$-0.578651\pi$
$491$	−5322.07	−0.489168	−0.244584	+	0.969628i	$0.578651\pi$
$492$	0	0
$493$	−2935.66	−0.268186
$494$	0	0
$495$	2902.06	0.263511
$496$	0	0
$497$	−37027.1	−3.34184
$498$	0	0
$499$	−1151.61	−0.103313	−0.0516566	−	0.998665i	$-0.516450\pi$
$499$	−1151.61	−0.103313	−0.0516566	+	0.998665i	$0.516450\pi$
$500$	0	0
$501$	518.541	0.0462410
$502$	0	0
$503$	−15046.7	−1.33380	−0.666900	−	0.745147i	$-0.732379\pi$
$503$	−15046.7	−1.33380	−0.666900	+	0.745147i	$0.732379\pi$
$504$	0	0
$505$	9630.78	0.848642
$506$	0	0
$507$	−227.277	−0.0199087
$508$	0	0
$509$	17124.7	1.49124	0.745619	−	0.666373i	$-0.232154\pi$
$509$	17124.7	1.49124	0.745619	+	0.666373i	$0.232154\pi$
$510$	0	0
$511$	−23947.5	−2.07314
$512$	0	0
$513$	−30.2981	−0.00260759
$514$	0	0
$515$	−6018.00	−0.514922
$516$	0	0
$517$	−4972.51	−0.423000
$518$	0	0
$519$	618.620	0.0523206
$520$	0	0
$521$	15336.5	1.28964	0.644821	−	0.764334i	$-0.276932\pi$
$521$	15336.5	1.28964	0.644821	+	0.764334i	$0.276932\pi$
$522$	0	0
$523$	16190.8	1.35368	0.676839	−	0.736131i	$-0.263349\pi$
$523$	16190.8	1.35368	0.676839	+	0.736131i	$0.263349\pi$
$524$	0	0
$525$	1031.48	0.0857478
$526$	0	0
$527$	18833.8	1.55676
$528$	0	0
$529$	−11571.4	−0.951045
$530$	0	0
$531$	−187.844	−0.0153517
$532$	0	0
$533$	1347.77	0.109528
$534$	0	0
$535$	−2183.96	−0.176488
$536$	0	0
$537$	830.075	0.0667046
$538$	0	0
$539$	16164.7	1.29177
$540$	0	0
$541$	13157.9	1.04566	0.522831	−	0.852436i	$-0.324876\pi$
$541$	13157.9	1.04566	0.522831	+	0.852436i	$0.324876\pi$
$542$	0	0
$543$	584.202	0.0461703
$544$	0	0
$545$	4970.20	0.390643
$546$	0	0
$547$	19078.0	1.49126	0.745628	−	0.666363i	$-0.232150\pi$
$547$	19078.0	1.49126	0.745628	+	0.666363i	$0.232150\pi$
$548$	0	0
$549$	−1024.18	−0.0796191
$550$	0	0
$551$	50.3195	0.00389053
$552$	0	0
$553$	14764.2	1.13533
$554$	0	0
$555$	636.440	0.0486763
$556$	0	0
$557$	−708.764	−0.0539161	−0.0269581	−	0.999637i	$-0.508582\pi$
$557$	−708.764	−0.0539161	−0.0269581	+	0.999637i	$0.508582\pi$
$558$	0	0
$559$	18691.6	1.41426
$560$	0	0
$561$	−611.832	−0.0460456
$562$	0	0
$563$	−14875.1	−1.11352	−0.556759	−	0.830674i	$-0.687955\pi$
$563$	−14875.1	−1.11352	−0.556759	+	0.830674i	$0.687955\pi$
$564$	0	0
$565$	8022.45	0.597357
$566$	0	0
$567$	−25057.8	−1.85596
$568$	0	0
$569$	−10910.6	−0.803863	−0.401932	−	0.915670i	$-0.631661\pi$
$569$	−10910.6	−0.803863	−0.401932	+	0.915670i	$0.631661\pi$
$570$	0	0
$571$	6711.77	0.491907	0.245953	−	0.969282i	$-0.420899\pi$
$571$	6711.77	0.491907	0.245953	+	0.969282i	$0.420899\pi$
$572$	0	0
$573$	−1327.40	−0.0967768
$574$	0	0
$575$	2234.18	0.162038
$576$	0	0
$577$	−18292.6	−1.31981	−0.659906	−	0.751348i	$-0.729404\pi$
$577$	−18292.6	−1.31981	−0.659906	+	0.751348i	$0.729404\pi$
$578$	0	0
$579$	776.133	0.0557081
$580$	0	0
$581$	−43387.0	−3.09810
$582$	0	0
$583$	2776.26	0.197223
$584$	0	0
$585$	6015.94	0.425177
$586$	0	0
$587$	23230.2	1.63341	0.816705	−	0.577056i	$-0.195799\pi$
$587$	23230.2	1.63341	0.816705	+	0.577056i	$0.195799\pi$
$588$	0	0
$589$	−322.826	−0.0225837
$590$	0	0
$591$	−948.433	−0.0660124
$592$	0	0
$593$	−10789.9	−0.747200	−0.373600	−	0.927590i	$-0.621877\pi$
$593$	−10789.9	−0.747200	−0.373600	+	0.927590i	$0.621877\pi$
$594$	0	0
$595$	−20363.4	−1.40306
$596$	0	0
$597$	−937.742	−0.0642868
$598$	0	0
$599$	8421.66	0.574457	0.287228	−	0.957862i	$-0.407266\pi$
$599$	8421.66	0.574457	0.287228	+	0.957862i	$0.407266\pi$
$600$	0	0
$601$	17485.5	1.18677	0.593385	−	0.804919i	$-0.297791\pi$
$601$	17485.5	1.18677	0.593385	+	0.804919i	$0.297791\pi$
$602$	0	0
$603$	−14632.5	−0.988196
$604$	0	0
$605$	5685.75	0.382080
$606$	0	0
$607$	−17492.2	−1.16966	−0.584831	−	0.811155i	$-0.698839\pi$
$607$	−17492.2	−1.16966	−0.584831	+	0.811155i	$0.698839\pi$
$608$	0	0
$609$	−326.762	−0.0217423
$610$	0	0
$611$	−10308.0	−0.682514
$612$	0	0
$613$	−5443.64	−0.358673	−0.179337	−	0.983788i	$-0.557395\pi$
$613$	−5443.64	−0.358673	−0.179337	+	0.983788i	$0.557395\pi$
$614$	0	0
$615$	−65.3116	−0.00428231
$616$	0	0
$617$	26323.3	1.71756	0.858781	−	0.512343i	$-0.171222\pi$
$617$	26323.3	1.71756	0.858781	+	0.512343i	$0.171222\pi$
$618$	0	0
$619$	15401.0	1.00003	0.500013	−	0.866018i	$-0.333328\pi$
$619$	15401.0	1.00003	0.500013	+	0.866018i	$0.333328\pi$
$620$	0	0
$621$	426.153	0.0275377
$622$	0	0
$623$	6062.49	0.389869
$624$	0	0
$625$	4198.25	0.268688
$626$	0	0
$627$	10.4873	0.000667977 0
$628$	0	0
$629$	34379.4	2.17933
$630$	0	0
$631$	20419.0	1.28822	0.644110	−	0.764933i	$-0.277228\pi$
$631$	20419.0	1.28822	0.644110	+	0.764933i	$0.277228\pi$
$632$	0	0
$633$	834.752	0.0524145
$634$	0	0
$635$	5678.46	0.354870
$636$	0	0
$637$	33509.3	2.08428
$638$	0	0
$639$	−28634.4	−1.77271
$640$	0	0
$641$	−11936.3	−0.735497	−0.367749	−	0.929925i	$-0.619871\pi$
$641$	−11936.3	−0.735497	−0.367749	+	0.929925i	$0.619871\pi$
$642$	0	0
$643$	−19181.0	−1.17640	−0.588199	−	0.808716i	$-0.700163\pi$
$643$	−19181.0	−1.17640	−0.588199	+	0.808716i	$0.700163\pi$
$644$	0	0
$645$	−905.774	−0.0552943
$646$	0	0
$647$	4250.18	0.258256	0.129128	−	0.991628i	$-0.458782\pi$
$647$	4250.18	0.258256	0.129128	+	0.991628i	$0.458782\pi$
$648$	0	0
$649$	130.293	0.00788050
$650$	0	0
$651$	2096.35	0.126210
$652$	0	0
$653$	28165.6	1.68791	0.843956	−	0.536413i	$-0.180221\pi$
$653$	28165.6	1.68791	0.843956	+	0.536413i	$0.180221\pi$
$654$	0	0
$655$	−3869.38	−0.230823
$656$	0	0
$657$	−18519.4	−1.09971
$658$	0	0
$659$	32600.6	1.92707	0.963536	−	0.267579i	$-0.0862237\pi$
$659$	32600.6	1.92707	0.963536	+	0.267579i	$0.0862237\pi$
$660$	0	0
$661$	−10591.5	−0.623242	−0.311621	−	0.950207i	$-0.600872\pi$
$661$	−10591.5	−0.623242	−0.311621	+	0.950207i	$0.600872\pi$
$662$	0	0
$663$	−1268.32	−0.0742950
$664$	0	0
$665$	349.044	0.0203539
$666$	0	0
$667$	−707.761	−0.0410864
$668$	0	0
$669$	−1141.02	−0.0659407
$670$	0	0
$671$	710.394	0.0408710
$672$	0	0
$673$	−5065.78	−0.290151	−0.145075	−	0.989421i	$-0.546342\pi$
$673$	−5065.78	−0.290151	−0.145075	+	0.989421i	$0.546342\pi$
$674$	0	0
$675$	1598.48	0.0911488
$676$	0	0
$677$	4021.25	0.228286	0.114143	−	0.993464i	$-0.463588\pi$
$677$	4021.25	0.228286	0.114143	+	0.993464i	$0.463588\pi$
$678$	0	0
$679$	−1174.70	−0.0663932
$680$	0	0
$681$	−756.388	−0.0425622
$682$	0	0
$683$	4846.20	0.271500	0.135750	−	0.990743i	$-0.456656\pi$
$683$	4846.20	0.271500	0.135750	+	0.990743i	$0.456656\pi$
$684$	0	0
$685$	−798.654	−0.0445474
$686$	0	0
$687$	678.842	0.0376993
$688$	0	0
$689$	5755.16	0.318221
$690$	0	0
$691$	−13400.8	−0.737758	−0.368879	−	0.929477i	$-0.620258\pi$
$691$	−13400.8	−0.737758	−0.368879	+	0.929477i	$0.620258\pi$
$692$	0	0
$693$	17449.1	0.956472
$694$	0	0
$695$	−5812.89	−0.317259
$696$	0	0
$697$	−3528.02	−0.191727
$698$	0	0
$699$	−1356.06	−0.0733774
$700$	0	0
$701$	22862.4	1.23182	0.615908	−	0.787818i	$-0.288789\pi$
$701$	22862.4	1.23182	0.615908	+	0.787818i	$0.288789\pi$
$702$	0	0
$703$	−589.290	−0.0316152
$704$	0	0
$705$	499.513	0.0266848
$706$	0	0
$707$	57906.6	3.08034
$708$	0	0
$709$	−13106.1	−0.694229	−0.347115	−	0.937823i	$-0.612839\pi$
$709$	−13106.1	−0.694229	−0.347115	+	0.937823i	$0.612839\pi$
$710$	0	0
$711$	11417.7	0.602247
$712$	0	0
$713$	4540.66	0.238498
$714$	0	0
$715$	−4172.80	−0.218257
$716$	0	0
$717$	1459.73	0.0760316
$718$	0	0
$719$	7109.97	0.368786	0.184393	−	0.982853i	$-0.440968\pi$
$719$	7109.97	0.368786	0.184393	+	0.982853i	$0.440968\pi$
$720$	0	0
$721$	−36184.1	−1.86903
$722$	0	0
$723$	−958.354	−0.0492968
$724$	0	0
$725$	−2654.77	−0.135994
$726$	0	0
$727$	32017.5	1.63338	0.816689	−	0.577079i	$-0.195807\pi$
$727$	32017.5	1.63338	0.816689	+	0.577079i	$0.195807\pi$
$728$	0	0
$729$	−19225.4	−0.976749
$730$	0	0
$731$	−48928.4	−2.47563
$732$	0	0
$733$	1715.94	0.0864662	0.0432331	−	0.999065i	$-0.486234\pi$
$733$	1715.94	0.0864662	0.0432331	+	0.999065i	$0.486234\pi$
$734$	0	0
$735$	−1623.83	−0.0814908
$736$	0	0
$737$	10149.5	0.507273
$738$	0	0
$739$	−14545.0	−0.724016	−0.362008	−	0.932175i	$-0.617909\pi$
$739$	−14545.0	−0.724016	−0.362008	+	0.932175i	$0.617909\pi$
$740$	0	0
$741$	21.7400	0.00107779
$742$	0	0
$743$	15502.4	0.765447	0.382723	−	0.923863i	$-0.374986\pi$
$743$	15502.4	0.765447	0.382723	+	0.923863i	$0.374986\pi$
$744$	0	0
$745$	20522.3	1.00923
$746$	0	0
$747$	−33552.7	−1.64341
$748$	0	0
$749$	−13131.4	−0.640603
$750$	0	0
$751$	32985.0	1.60271	0.801357	−	0.598186i	$-0.204112\pi$
$751$	32985.0	1.60271	0.801357	+	0.598186i	$0.204112\pi$
$752$	0	0
$753$	−817.909	−0.0395833
$754$	0	0
$755$	−9077.86	−0.437585
$756$	0	0
$757$	−38303.7	−1.83906	−0.919532	−	0.393015i	$-0.871432\pi$
$757$	−38303.7	−1.83906	−0.919532	+	0.393015i	$0.871432\pi$
$758$	0	0
$759$	−147.507	−0.00705424
$760$	0	0
$761$	15993.5	0.761843	0.380921	−	0.924607i	$-0.375607\pi$
$761$	15993.5	0.761843	0.380921	+	0.924607i	$0.375607\pi$
$762$	0	0
$763$	29884.1	1.41793
$764$	0	0
$765$	−15747.7	−0.744262
$766$	0	0
$767$	270.096	0.0127153
$768$	0	0
$769$	−5628.40	−0.263934	−0.131967	−	0.991254i	$-0.542129\pi$
$769$	−5628.40	−0.263934	−0.131967	+	0.991254i	$0.542129\pi$
$770$	0	0
$771$	−895.220	−0.0418165
$772$	0	0
$773$	4477.85	0.208353	0.104177	−	0.994559i	$-0.466779\pi$
$773$	4477.85	0.208353	0.104177	+	0.994559i	$0.466779\pi$
$774$	0	0
$775$	17031.8	0.789418
$776$	0	0
$777$	3826.69	0.176682
$778$	0	0
$779$	60.4731	0.00278135
$780$	0	0
$781$	19861.5	0.909986
$782$	0	0
$783$	−506.379	−0.0231118
$784$	0	0
$785$	−12611.9	−0.573422
$786$	0	0
$787$	26974.8	1.22179	0.610894	−	0.791712i	$-0.290810\pi$
$787$	26974.8	1.22179	0.610894	+	0.791712i	$0.290810\pi$
$788$	0	0
$789$	638.180	0.0287957
$790$	0	0
$791$	48236.2	2.16824
$792$	0	0
$793$	1472.64	0.0659458
$794$	0	0
$795$	−278.889	−0.0124417
$796$	0	0
$797$	14755.0	0.655772	0.327886	−	0.944717i	$-0.393664\pi$
$797$	14755.0	0.655772	0.327886	+	0.944717i	$0.393664\pi$
$798$	0	0
$799$	26982.9	1.19472
$800$	0	0
$801$	4688.33	0.206809
$802$	0	0
$803$	12845.5	0.564517
$804$	0	0
$805$	−4909.43	−0.214950
$806$	0	0
$807$	−273.299	−0.0119214
$808$	0	0
$809$	−24539.2	−1.06644	−0.533222	−	0.845975i	$-0.679019\pi$
$809$	−24539.2	−1.06644	−0.533222	+	0.845975i	$0.679019\pi$
$810$	0	0
$811$	14700.6	0.636506	0.318253	−	0.948006i	$-0.396904\pi$
$811$	14700.6	0.636506	0.318253	+	0.948006i	$0.396904\pi$
$812$	0	0
$813$	−719.878	−0.0310544
$814$	0	0
$815$	−8390.36	−0.360615
$816$	0	0
$817$	838.670	0.0359135
$818$	0	0
$819$	36171.8	1.54328
$820$	0	0
$821$	20938.7	0.890092	0.445046	−	0.895508i	$-0.353187\pi$
$821$	20938.7	0.890092	0.445046	+	0.895508i	$0.353187\pi$
$822$	0	0
$823$	−17131.1	−0.725582	−0.362791	−	0.931870i	$-0.618176\pi$
$823$	−17131.1	−0.725582	−0.362791	+	0.931870i	$0.618176\pi$
$824$	0	0
$825$	−553.291	−0.0233492
$826$	0	0
$827$	−1793.47	−0.0754111	−0.0377055	−	0.999289i	$-0.512005\pi$
$827$	−1793.47	−0.0754111	−0.0377055	+	0.999289i	$0.512005\pi$
$828$	0	0
$829$	−41036.5	−1.71925	−0.859624	−	0.510928i	$-0.829302\pi$
$829$	−41036.5	−1.71925	−0.859624	+	0.510928i	$0.829302\pi$
$830$	0	0
$831$	−701.582	−0.0292872
$832$	0	0
$833$	−87716.3	−3.64849
$834$	0	0
$835$	−9257.48	−0.383675
$836$	0	0
$837$	3248.69	0.134159
$838$	0	0
$839$	−13507.8	−0.555830	−0.277915	−	0.960606i	$-0.589643\pi$
$839$	−13507.8	−0.555830	−0.277915	+	0.960606i	$0.589643\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	770.487	0.0314792
$844$	0	0
$845$	4057.56	0.165189
$846$	0	0
$847$	34186.4	1.38685
$848$	0	0
$849$	1519.58	0.0614275
$850$	0	0
$851$	8288.56	0.333875
$852$	0	0
$853$	45261.9	1.81681	0.908405	−	0.418092i	$-0.137301\pi$
$853$	45261.9	1.81681	0.908405	+	0.418092i	$0.137301\pi$
$854$	0	0
$855$	269.928	0.0107969
$856$	0	0
$857$	−19373.8	−0.772224	−0.386112	−	0.922452i	$-0.626182\pi$
$857$	−19373.8	−0.772224	−0.386112	+	0.922452i	$0.626182\pi$
$858$	0	0
$859$	−16119.9	−0.640285	−0.320142	−	0.947369i	$-0.603731\pi$
$859$	−16119.9	−0.640285	−0.320142	+	0.947369i	$0.603731\pi$
$860$	0	0
$861$	−392.696	−0.0155436
$862$	0	0
$863$	−17377.3	−0.685434	−0.342717	−	0.939439i	$-0.611347\pi$
$863$	−17377.3	−0.685434	−0.342717	+	0.939439i	$0.611347\pi$
$864$	0	0
$865$	−11044.2	−0.434119
$866$	0	0
$867$	1728.30	0.0677002
$868$	0	0
$869$	−7919.58	−0.309152
$870$	0	0
$871$	21039.7	0.818490
$872$	0	0
$873$	−908.439	−0.0352188
$874$	0	0
$875$	−43560.0	−1.68297
$876$	0	0
$877$	19608.7	0.755005	0.377503	−	0.926009i	$-0.376783\pi$
$877$	19608.7	0.755005	0.377503	+	0.926009i	$0.376783\pi$
$878$	0	0
$879$	−2141.80	−0.0821856
$880$	0	0
$881$	−40317.9	−1.54182	−0.770911	−	0.636943i	$-0.780199\pi$
$881$	−40317.9	−1.54182	−0.770911	+	0.636943i	$0.780199\pi$
$882$	0	0
$883$	−17384.9	−0.662567	−0.331284	−	0.943531i	$-0.607482\pi$
$883$	−17384.9	−0.662567	−0.331284	+	0.943531i	$0.607482\pi$
$884$	0	0
$885$	−13.0886	−0.000497139 0
$886$	0	0
$887$	32214.4	1.21945	0.609726	−	0.792612i	$-0.291279\pi$
$887$	32214.4	1.21945	0.609726	+	0.792612i	$0.291279\pi$
$888$	0	0
$889$	34142.6	1.28808
$890$	0	0
$891$	13441.1	0.505380
$892$	0	0
$893$	−462.507	−0.0173317
$894$	0	0
$895$	−14819.3	−0.553467
$896$	0	0
$897$	−305.781	−0.0113821
$898$	0	0
$899$	−5395.46	−0.200165
$900$	0	0
$901$	−15065.1	−0.557038
$902$	0	0
$903$	−5446.11	−0.200703
$904$	0	0
$905$	−10429.7	−0.383089
$906$	0	0
$907$	13733.5	0.502771	0.251385	−	0.967887i	$-0.419114\pi$
$907$	13733.5	0.502771	0.251385	+	0.967887i	$0.419114\pi$
$908$	0	0
$909$	44781.2	1.63399
$910$	0	0
$911$	46583.7	1.69417	0.847085	−	0.531458i	$-0.178356\pi$
$911$	46583.7	1.69417	0.847085	+	0.531458i	$0.178356\pi$
$912$	0	0
$913$	23272.9	0.843616
$914$	0	0
$915$	−71.3626	−0.00257833
$916$	0	0
$917$	−23265.2	−0.837825
$918$	0	0
$919$	−50389.1	−1.80869	−0.904343	−	0.426807i	$-0.859638\pi$
$919$	−50389.1	−1.80869	−0.904343	+	0.426807i	$0.859638\pi$
$920$	0	0
$921$	−2662.38	−0.0952535
$922$	0	0
$923$	41172.7	1.46827
$924$	0	0
$925$	31089.9	1.10511
$926$	0	0
$927$	−27982.5	−0.991440
$928$	0	0
$929$	16132.1	0.569728	0.284864	−	0.958568i	$-0.408052\pi$
$929$	16132.1	0.569728	0.284864	+	0.958568i	$0.408052\pi$
$930$	0	0
$931$	1503.53	0.0529281
$932$	0	0
$933$	1169.80	0.0410476
$934$	0	0
$935$	10923.0	0.382054
$936$	0	0
$937$	−14997.5	−0.522888	−0.261444	−	0.965219i	$-0.584199\pi$
$937$	−14997.5	−0.522888	−0.261444	+	0.965219i	$0.584199\pi$
$938$	0	0
$939$	3305.15	0.114866
$940$	0	0
$941$	47704.3	1.65262	0.826310	−	0.563216i	$-0.190436\pi$
$941$	47704.3	1.65262	0.826310	+	0.563216i	$0.190436\pi$
$942$	0	0
$943$	−850.574	−0.0293727
$944$	0	0
$945$	−3512.53	−0.120913
$946$	0	0
$947$	55295.6	1.89743	0.948715	−	0.316133i	$-0.102385\pi$
$947$	55295.6	1.89743	0.948715	+	0.316133i	$0.102385\pi$
$948$	0	0
$949$	26628.6	0.910855
$950$	0	0
$951$	2227.67	0.0759591
$952$	0	0
$953$	25225.6	0.857438	0.428719	−	0.903438i	$-0.358965\pi$
$953$	25225.6	0.857438	0.428719	+	0.903438i	$0.358965\pi$
$954$	0	0
$955$	23698.1	0.802985
$956$	0	0
$957$	175.276	0.00592045
$958$	0	0
$959$	−4802.03	−0.161695
$960$	0	0
$961$	4823.75	0.161920
$962$	0	0
$963$	−10155.0	−0.339813
$964$	0	0
$965$	−13856.2	−0.462226
$966$	0	0
$967$	−8870.00	−0.294974	−0.147487	−	0.989064i	$-0.547118\pi$
$967$	−8870.00	−0.294974	−0.147487	+	0.989064i	$0.547118\pi$
$968$	0	0
$969$	−56.9082	−0.00188664
$970$	0	0
$971$	4362.02	0.144165	0.0720824	−	0.997399i	$-0.477036\pi$
$971$	4362.02	0.144165	0.0720824	+	0.997399i	$0.477036\pi$
$972$	0	0
$973$	−34950.9	−1.15157
$974$	0	0
$975$	−1146.97	−0.0376742
$976$	0	0
$977$	−2432.32	−0.0796486	−0.0398243	−	0.999207i	$-0.512680\pi$
$977$	−2432.32	−0.0796486	−0.0398243	+	0.999207i	$0.512680\pi$
$978$	0	0
$979$	−3251.94	−0.106162
$980$	0	0
$981$	23110.4	0.752150
$982$	0	0
$983$	−5907.53	−0.191680	−0.0958398	−	0.995397i	$-0.530554\pi$
$983$	−5907.53	−0.191680	−0.0958398	+	0.995397i	$0.530554\pi$
$984$	0	0
$985$	16932.3	0.547724
$986$	0	0
$987$	3003.40	0.0968585
$988$	0	0
$989$	−11796.2	−0.379269
$990$	0	0
$991$	−485.384	−0.0155588	−0.00777938	−	0.999970i	$-0.502476\pi$
$991$	−485.384	−0.0155588	−0.00777938	+	0.999970i	$0.502476\pi$
$992$	0	0
$993$	−2928.11	−0.0935760
$994$	0	0
$995$	16741.4	0.533406
$996$	0	0
$997$	32345.7	1.02748	0.513741	−	0.857945i	$-0.328259\pi$
$997$	32345.7	1.02748	0.513741	+	0.857945i	$0.328259\pi$
$998$	0	0
$999$	5930.18	0.187810

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	928.4.a.j.1.6	yes	12
4.3	odd	2		928.4.a.h.1.7	✓	12
8.3	odd	2		1856.4.a.bl.1.6		12
8.5	even	2		1856.4.a.bj.1.7		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
928.4.a.h.1.7	✓	12	4.3	odd	2
928.4.a.j.1.6	yes	12	1.1	even	1	trivial
1856.4.a.bj.1.7		12	8.5	even	2
1856.4.a.bl.1.6		12	8.3	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:	\( N \)	\(=\)	\( 928 = 2^{5} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	928.a (trivial)

\(f(q)\)	\(=\)	\(q+0.323988 q^{3} -5.78413 q^{5} -34.7780 q^{7} -26.8950 q^{9} +O(q^{10})\)	\(q+0.323988 q^{3} -5.78413 q^{5} -34.7780 q^{7} -26.8950 q^{9} +18.6550 q^{11} +38.6717 q^{13} -1.87399 q^{15} -101.230 q^{17} +1.73516 q^{19} -11.2676 q^{21} -24.4055 q^{23} -91.5438 q^{25} -17.4613 q^{27} +29.0000 q^{29} -186.050 q^{31} +6.04400 q^{33} +201.160 q^{35} -339.618 q^{37} +12.5292 q^{39} +34.8517 q^{41} +483.340 q^{43} +155.564 q^{45} -266.551 q^{47} +866.508 q^{49} -32.7972 q^{51} +148.821 q^{53} -107.903 q^{55} +0.562169 q^{57} +6.98434 q^{59} +38.0806 q^{61} +935.355 q^{63} -223.682 q^{65} +544.060 q^{67} -7.90710 q^{69} +1064.67 q^{71} +688.581 q^{73} -29.6591 q^{75} -648.784 q^{77} -424.529 q^{79} +720.509 q^{81} +1247.54 q^{83} +585.526 q^{85} +9.39565 q^{87} -174.320 q^{89} -1344.92 q^{91} -60.2781 q^{93} -10.0364 q^{95} +33.7772 q^{97} -501.727 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 14 q^{3} - 10 q^{5} + 44 q^{7} + 130 q^{9}+O(q^{10}) \) 12 * q + 14 * q^3 - 10 * q^5 + 44 * q^7 + 130 * q^9	\( 12 q + 14 q^{3} - 10 q^{5} + 44 q^{7} + 130 q^{9} + 46 q^{11} - 34 q^{13} - 50 q^{15} + 36 q^{17} + 148 q^{19} - 92 q^{21} + 328 q^{23} + 486 q^{25} + 326 q^{27} + 348 q^{29} + 374 q^{31} + 710 q^{33} + 204 q^{35} - 340 q^{37} - 122 q^{39} + 32 q^{41} + 462 q^{43} - 1132 q^{45} + 434 q^{47} + 1508 q^{49} + 440 q^{51} + 610 q^{53} + 46 q^{55} - 932 q^{57} + 1240 q^{59} - 1228 q^{61} + 4240 q^{63} + 730 q^{65} + 1672 q^{67} - 528 q^{69} + 3220 q^{71} + 564 q^{73} + 6032 q^{75} + 644 q^{77} + 1862 q^{79} + 3040 q^{81} + 3736 q^{83} - 808 q^{85} + 406 q^{87} + 584 q^{89} + 4844 q^{91} - 3226 q^{93} + 2844 q^{95} + 904 q^{97} + 6832 q^{99}+O(q^{100}) \) 12 * q + 14 * q^3 - 10 * q^5 + 44 * q^7 + 130 * q^9 + 46 * q^11 - 34 * q^13 - 50 * q^15 + 36 * q^17 + 148 * q^19 - 92 * q^21 + 328 * q^23 + 486 * q^25 + 326 * q^27 + 348 * q^29 + 374 * q^31 + 710 * q^33 + 204 * q^35 - 340 * q^37 - 122 * q^39 + 32 * q^41 + 462 * q^43 - 1132 * q^45 + 434 * q^47 + 1508 * q^49 + 440 * q^51 + 610 * q^53 + 46 * q^55 - 932 * q^57 + 1240 * q^59 - 1228 * q^61 + 4240 * q^63 + 730 * q^65 + 1672 * q^67 - 528 * q^69 + 3220 * q^71 + 564 * q^73 + 6032 * q^75 + 644 * q^77 + 1862 * q^79 + 3040 * q^81 + 3736 * q^83 - 808 * q^85 + 406 * q^87 + 584 * q^89 + 4844 * q^91 - 3226 * q^93 + 2844 * q^95 + 904 * q^97 + 6832 * q^99

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