LMFDB - Embedded newform 1856.4.a.bl.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1856,4,Mod(1,1856)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1856, 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("1856.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1856 = 2^{6} \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1856.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:	$109.507544971$
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:	no (minimal twist has level 928)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$5.24549$ of defining polynomial
Character	$\chi$	$=$	1856.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+6.24549 q^{3} -5.77776 q^{5} -10.3273 q^{7} +12.0061 q^{9} +O(q^{10})$	$q+6.24549 q^{3} -5.77776 q^{5} -10.3273 q^{7} +12.0061 q^{9} +21.9555 q^{11} -61.7074 q^{13} -36.0850 q^{15} +41.9208 q^{17} +141.827 q^{19} -64.4992 q^{21} +193.579 q^{23} -91.6175 q^{25} -93.6440 q^{27} -29.0000 q^{29} -115.138 q^{31} +137.123 q^{33} +59.6688 q^{35} +99.1634 q^{37} -385.393 q^{39} +244.674 q^{41} -385.333 q^{43} -69.3686 q^{45} +108.288 q^{47} -236.346 q^{49} +261.816 q^{51} -515.263 q^{53} -126.853 q^{55} +885.778 q^{57} +856.269 q^{59} -402.678 q^{61} -123.991 q^{63} +356.530 q^{65} +758.717 q^{67} +1209.00 q^{69} +696.409 q^{71} -1103.52 q^{73} -572.196 q^{75} -226.741 q^{77} +817.740 q^{79} -909.018 q^{81} +1083.67 q^{83} -242.209 q^{85} -181.119 q^{87} +119.417 q^{89} +637.272 q^{91} -719.091 q^{93} -819.442 q^{95} +1609.79 q^{97} +263.600 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	6.24549	1.20195	0.600973	−	0.799270i	$-0.294780\pi$
$3$	6.24549	1.20195	0.600973	+	0.799270i	$0.294780\pi$
$4$	0	0
$5$	−5.77776	−0.516779	−0.258389	−	0.966041i	$-0.583192\pi$
$5$	−5.77776	−0.516779	−0.258389	+	0.966041i	$0.583192\pi$
$6$	0	0
$7$	−10.3273	−0.557623	−0.278811	−	0.960346i	$-0.589940\pi$
$7$	−10.3273	−0.557623	−0.278811	+	0.960346i	$0.589940\pi$
$8$	0	0
$9$	12.0061	0.444672
$10$	0	0
$11$	21.9555	0.601802	0.300901	−	0.953655i	$-0.402713\pi$
$11$	21.9555	0.601802	0.300901	+	0.953655i	$0.402713\pi$
$12$	0	0
$13$	−61.7074	−1.31650	−0.658252	−	0.752798i	$-0.728704\pi$
$13$	−61.7074	−1.31650	−0.658252	+	0.752798i	$0.728704\pi$
$14$	0	0
$15$	−36.0850	−0.621140
$16$	0	0
$17$	41.9208	0.598076	0.299038	−	0.954241i	$-0.403334\pi$
$17$	41.9208	0.598076	0.299038	+	0.954241i	$0.403334\pi$
$18$	0	0
$19$	141.827	1.71249	0.856245	−	0.516570i	$-0.172791\pi$
$19$	141.827	1.71249	0.856245	+	0.516570i	$0.172791\pi$
$20$	0	0
$21$	−64.4992	−0.670232
$22$	0	0
$23$	193.579	1.75496	0.877479	−	0.479616i	$-0.159224\pi$
$23$	193.579	1.75496	0.877479	+	0.479616i	$0.159224\pi$
$24$	0	0
$25$	−91.6175	−0.732940
$26$	0	0
$27$	−93.6440	−0.667474
$28$	0	0
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	0	0
$31$	−115.138	−0.667075	−0.333537	−	0.942737i	$-0.608242\pi$
$31$	−115.138	−0.667075	−0.333537	+	0.942737i	$0.608242\pi$
$32$	0	0
$33$	137.123	0.723333
$34$	0	0
$35$	59.6688	0.288168
$36$	0	0
$37$	99.1634	0.440604	0.220302	−	0.975432i	$-0.429296\pi$
$37$	99.1634	0.440604	0.220302	+	0.975432i	$0.429296\pi$
$38$	0	0
$39$	−385.393	−1.58236
$40$	0	0
$41$	244.674	0.931993	0.465997	−	0.884786i	$-0.345696\pi$
$41$	244.674	0.931993	0.465997	+	0.884786i	$0.345696\pi$
$42$	0	0
$43$	−385.333	−1.36657	−0.683287	−	0.730150i	$-0.739450\pi$
$43$	−385.333	−1.36657	−0.683287	+	0.730150i	$0.739450\pi$
$44$	0	0
$45$	−69.3686	−0.229797
$46$	0	0
$47$	108.288	0.336073	0.168036	−	0.985781i	$-0.446257\pi$
$47$	108.288	0.336073	0.168036	+	0.985781i	$0.446257\pi$
$48$	0	0
$49$	−236.346	−0.689057
$50$	0	0
$51$	261.816	0.718855
$52$	0	0
$53$	−515.263	−1.33541	−0.667705	−	0.744426i	$-0.732723\pi$
$53$	−515.263	−1.33541	−0.667705	+	0.744426i	$0.732723\pi$
$54$	0	0
$55$	−126.853	−0.310998
$56$	0	0
$57$	885.778	2.05832
$58$	0	0
$59$	856.269	1.88944	0.944718	−	0.327884i	$-0.106336\pi$
$59$	856.269	1.88944	0.944718	+	0.327884i	$0.106336\pi$
$60$	0	0
$61$	−402.678	−0.845206	−0.422603	−	0.906315i	$-0.638884\pi$
$61$	−402.678	−0.845206	−0.422603	+	0.906315i	$0.638884\pi$
$62$	0	0
$63$	−123.991	−0.247959
$64$	0	0
$65$	356.530	0.680341
$66$	0	0
$67$	758.717	1.38346	0.691731	−	0.722155i	$-0.256848\pi$
$67$	758.717	1.38346	0.691731	+	0.722155i	$0.256848\pi$
$68$	0	0
$69$	1209.00	2.10936
$70$	0	0
$71$	696.409	1.16406	0.582032	−	0.813166i	$-0.302258\pi$
$71$	696.409	1.16406	0.582032	+	0.813166i	$0.302258\pi$
$72$	0	0
$73$	−1103.52	−1.76928	−0.884639	−	0.466277i	$-0.845595\pi$
$73$	−1103.52	−1.76928	−0.884639	+	0.466277i	$0.845595\pi$
$74$	0	0
$75$	−572.196	−0.880953
$76$	0	0
$77$	−226.741	−0.335579
$78$	0	0
$79$	817.740	1.16459	0.582297	−	0.812976i	$-0.302154\pi$
$79$	817.740	1.16459	0.582297	+	0.812976i	$0.302154\pi$
$80$	0	0
$81$	−909.018	−1.24694
$82$	0	0
$83$	1083.67	1.43311	0.716555	−	0.697531i	$-0.245718\pi$
$83$	1083.67	1.43311	0.716555	+	0.697531i	$0.245718\pi$
$84$	0	0
$85$	−242.209	−0.309073
$86$	0	0
$87$	−181.119	−0.223196
$88$	0	0
$89$	119.417	0.142226	0.0711132	−	0.997468i	$-0.477345\pi$
$89$	119.417	0.142226	0.0711132	+	0.997468i	$0.477345\pi$
$90$	0	0
$91$	637.272	0.734112
$92$	0	0
$93$	−719.091	−0.801787
$94$	0	0
$95$	−819.442	−0.884979
$96$	0	0
$97$	1609.79	1.68505	0.842525	−	0.538658i	$-0.181068\pi$
$97$	1609.79	1.68505	0.842525	+	0.538658i	$0.181068\pi$
$98$	0	0
$99$	263.600	0.267604
$100$	0	0
$101$	660.447	0.650663	0.325331	−	0.945600i	$-0.394524\pi$
$101$	660.447	0.650663	0.325331	+	0.945600i	$0.394524\pi$
$102$	0	0
$103$	−486.244	−0.465156	−0.232578	−	0.972578i	$-0.574716\pi$
$103$	−486.244	−0.465156	−0.232578	+	0.972578i	$0.574716\pi$
$104$	0	0
$105$	372.661	0.346362
$106$	0	0
$107$	791.169	0.714815	0.357407	−	0.933949i	$-0.383661\pi$
$107$	791.169	0.714815	0.357407	+	0.933949i	$0.383661\pi$
$108$	0	0
$109$	2200.75	1.93389	0.966944	−	0.254989i	$-0.0820719\pi$
$109$	2200.75	1.93389	0.966944	+	0.254989i	$0.0820719\pi$
$110$	0	0
$111$	619.324	0.529582
$112$	0	0
$113$	1877.78	1.56324	0.781622	−	0.623753i	$-0.214393\pi$
$113$	1877.78	1.56324	0.781622	+	0.623753i	$0.214393\pi$
$114$	0	0
$115$	−1118.45	−0.906925
$116$	0	0
$117$	−740.867	−0.585412
$118$	0	0
$119$	−432.930	−0.333501
$120$	0	0
$121$	−848.958	−0.637834
$122$	0	0
$123$	1528.11	1.12021
$124$	0	0
$125$	1251.56	0.895546
$126$	0	0
$127$	−1024.98	−0.716163	−0.358081	−	0.933690i	$-0.616569\pi$
$127$	−1024.98	−0.716163	−0.358081	+	0.933690i	$0.616569\pi$
$128$	0	0
$129$	−2406.59	−1.64255
$130$	0	0
$131$	705.014	0.470209	0.235104	−	0.971970i	$-0.424457\pi$
$131$	705.014	0.470209	0.235104	+	0.971970i	$0.424457\pi$
$132$	0	0
$133$	−1464.69	−0.954924
$134$	0	0
$135$	541.053	0.344936
$136$	0	0
$137$	−1394.21	−0.869456	−0.434728	−	0.900562i	$-0.643156\pi$
$137$	−1394.21	−0.869456	−0.434728	+	0.900562i	$0.643156\pi$
$138$	0	0
$139$	3006.12	1.83436	0.917179	−	0.398475i	$-0.130461\pi$
$139$	3006.12	1.83436	0.917179	+	0.398475i	$0.130461\pi$
$140$	0	0
$141$	676.311	0.403941
$142$	0	0
$143$	−1354.81	−0.792274
$144$	0	0
$145$	167.555	0.0959634
$146$	0	0
$147$	−1476.10	−0.828208
$148$	0	0
$149$	3184.10	1.75068	0.875340	−	0.483508i	$-0.160637\pi$
$149$	3184.10	1.75068	0.875340	+	0.483508i	$0.160637\pi$
$150$	0	0
$151$	237.510	0.128002	0.0640010	−	0.997950i	$-0.479614\pi$
$151$	237.510	0.128002	0.0640010	+	0.997950i	$0.479614\pi$
$152$	0	0
$153$	503.307	0.265948
$154$	0	0
$155$	665.237	0.344730
$156$	0	0
$157$	905.423	0.460259	0.230129	−	0.973160i	$-0.426085\pi$
$157$	905.423	0.460259	0.230129	+	0.973160i	$0.426085\pi$
$158$	0	0
$159$	−3218.07	−1.60509
$160$	0	0
$161$	−1999.15	−0.978605
$162$	0	0
$163$	1565.78	0.752402	0.376201	−	0.926538i	$-0.377230\pi$
$163$	1565.78	0.752402	0.376201	+	0.926538i	$0.377230\pi$
$164$	0	0
$165$	−792.262	−0.373803
$166$	0	0
$167$	2765.87	1.28161	0.640806	−	0.767703i	$-0.278600\pi$
$167$	2765.87	1.28161	0.640806	+	0.767703i	$0.278600\pi$
$168$	0	0
$169$	1610.80	0.733181
$170$	0	0
$171$	1702.79	0.761496
$172$	0	0
$173$	2849.37	1.25222	0.626110	−	0.779735i	$-0.284646\pi$
$173$	2849.37	1.25222	0.626110	+	0.779735i	$0.284646\pi$
$174$	0	0
$175$	946.163	0.408704
$176$	0	0
$177$	5347.82	2.27100
$178$	0	0
$179$	946.923	0.395399	0.197699	−	0.980263i	$-0.436653\pi$
$179$	946.923	0.395399	0.197699	+	0.980263i	$0.436653\pi$
$180$	0	0
$181$	707.512	0.290547	0.145273	−	0.989392i	$-0.453594\pi$
$181$	707.512	0.290547	0.145273	+	0.989392i	$0.453594\pi$
$182$	0	0
$183$	−2514.92	−1.01589
$184$	0	0
$185$	−572.943	−0.227695
$186$	0	0
$187$	920.391	0.359923
$188$	0	0
$189$	967.092	0.372199
$190$	0	0
$191$	−4300.24	−1.62908	−0.814540	−	0.580107i	$-0.803011\pi$
$191$	−4300.24	−1.62908	−0.814540	+	0.580107i	$0.803011\pi$
$192$	0	0
$193$	−195.060	−0.0727500	−0.0363750	−	0.999338i	$-0.511581\pi$
$193$	−195.060	−0.0727500	−0.0363750	+	0.999338i	$0.511581\pi$
$194$	0	0
$195$	2226.71	0.817732
$196$	0	0
$197$	2635.98	0.953328	0.476664	−	0.879086i	$-0.341846\pi$
$197$	2635.98	0.953328	0.476664	+	0.879086i	$0.341846\pi$
$198$	0	0
$199$	1495.24	0.532637	0.266319	−	0.963885i	$-0.414193\pi$
$199$	1495.24	0.532637	0.266319	+	0.963885i	$0.414193\pi$
$200$	0	0
$201$	4738.56	1.66285
$202$	0	0
$203$	299.492	0.103548
$204$	0	0
$205$	−1413.67	−0.481634
$206$	0	0
$207$	2324.14	0.780380
$208$	0	0
$209$	3113.87	1.03058
$210$	0	0
$211$	−195.204	−0.0636892	−0.0318446	−	0.999493i	$-0.510138\pi$
$211$	−195.204	−0.0636892	−0.0318446	+	0.999493i	$0.510138\pi$
$212$	0	0
$213$	4349.42	1.39914
$214$	0	0
$215$	2226.36	0.706217
$216$	0	0
$217$	1189.06	0.371976
$218$	0	0
$219$	−6892.03	−2.12657
$220$	0	0
$221$	−2586.82	−0.787369
$222$	0	0
$223$	−1004.00	−0.301493	−0.150746	−	0.988572i	$-0.548168\pi$
$223$	−1004.00	−0.301493	−0.150746	+	0.988572i	$0.548168\pi$
$224$	0	0
$225$	−1099.97	−0.325918
$226$	0	0
$227$	−207.164	−0.0605726	−0.0302863	−	0.999541i	$-0.509642\pi$
$227$	−207.164	−0.0605726	−0.0302863	+	0.999541i	$0.509642\pi$
$228$	0	0
$229$	−1397.42	−0.403248	−0.201624	−	0.979463i	$-0.564622\pi$
$229$	−1397.42	−0.403248	−0.201624	+	0.979463i	$0.564622\pi$
$230$	0	0
$231$	−1416.11	−0.403347
$232$	0	0
$233$	1996.12	0.561246	0.280623	−	0.959818i	$-0.409459\pi$
$233$	1996.12	0.561246	0.280623	+	0.959818i	$0.409459\pi$
$234$	0	0
$235$	−625.662	−0.173675
$236$	0	0
$237$	5107.19	1.39978
$238$	0	0
$239$	−3953.90	−1.07011	−0.535056	−	0.844817i	$-0.679710\pi$
$239$	−3953.90	−1.07011	−0.535056	+	0.844817i	$0.679710\pi$
$240$	0	0
$241$	−4964.66	−1.32698	−0.663489	−	0.748186i	$-0.730925\pi$
$241$	−4964.66	−1.32698	−0.663489	+	0.748186i	$0.730925\pi$
$242$	0	0
$243$	−3148.88	−0.831278
$244$	0	0
$245$	1365.55	0.356090
$246$	0	0
$247$	−8751.76	−2.25450
$248$	0	0
$249$	6768.04	1.72252
$250$	0	0
$251$	−4470.84	−1.12429	−0.562145	−	0.827039i	$-0.690024\pi$
$251$	−4470.84	−1.12429	−0.562145	+	0.827039i	$0.690024\pi$
$252$	0	0
$253$	4250.12	1.05614
$254$	0	0
$255$	−1512.71	−0.371489
$256$	0	0
$257$	−1157.45	−0.280934	−0.140467	−	0.990085i	$-0.544860\pi$
$257$	−1157.45	−0.280934	−0.140467	+	0.990085i	$0.544860\pi$
$258$	0	0
$259$	−1024.09	−0.245691
$260$	0	0
$261$	−348.178	−0.0825735
$262$	0	0
$263$	1390.87	0.326102	0.163051	−	0.986618i	$-0.447866\pi$
$263$	1390.87	0.326102	0.163051	+	0.986618i	$0.447866\pi$
$264$	0	0
$265$	2977.06	0.690112
$266$	0	0
$267$	745.817	0.170948
$268$	0	0
$269$	−3984.78	−0.903184	−0.451592	−	0.892225i	$-0.649144\pi$
$269$	−3984.78	−0.903184	−0.451592	+	0.892225i	$0.649144\pi$
$270$	0	0
$271$	−1291.22	−0.289431	−0.144716	−	0.989473i	$-0.546227\pi$
$271$	−1291.22	−0.289431	−0.144716	+	0.989473i	$0.546227\pi$
$272$	0	0
$273$	3980.07	0.882363
$274$	0	0
$275$	−2011.50	−0.441085
$276$	0	0
$277$	−6409.29	−1.39024	−0.695121	−	0.718893i	$-0.744649\pi$
$277$	−6409.29	−1.39024	−0.695121	+	0.718893i	$0.744649\pi$
$278$	0	0
$279$	−1382.36	−0.296629
$280$	0	0
$281$	5964.42	1.26622	0.633110	−	0.774062i	$-0.281778\pi$
$281$	5964.42	1.26622	0.633110	+	0.774062i	$0.281778\pi$
$282$	0	0
$283$	−6950.38	−1.45992	−0.729960	−	0.683490i	$-0.760461\pi$
$283$	−6950.38	−1.45992	−0.729960	+	0.683490i	$0.760461\pi$
$284$	0	0
$285$	−5117.82	−1.06370
$286$	0	0
$287$	−2526.83	−0.519701
$288$	0	0
$289$	−3155.64	−0.642305
$290$	0	0
$291$	10054.0	2.02534
$292$	0	0
$293$	−7403.08	−1.47608	−0.738042	−	0.674755i	$-0.764249\pi$
$293$	−7403.08	−1.47608	−0.738042	+	0.674755i	$0.764249\pi$
$294$	0	0
$295$	−4947.32	−0.976420
$296$	0	0
$297$	−2056.00	−0.401687
$298$	0	0
$299$	−11945.3	−2.31041
$300$	0	0
$301$	3979.46	0.762034
$302$	0	0
$303$	4124.82	0.782061
$304$	0	0
$305$	2326.58	0.436785
$306$	0	0
$307$	−9442.07	−1.75533	−0.877667	−	0.479272i	$-0.840901\pi$
$307$	−9442.07	−1.75533	−0.877667	+	0.479272i	$0.840901\pi$
$308$	0	0
$309$	−3036.83	−0.559092
$310$	0	0
$311$	10536.5	1.92113	0.960565	−	0.278054i	$-0.0896896\pi$
$311$	10536.5	1.92113	0.960565	+	0.278054i	$0.0896896\pi$
$312$	0	0
$313$	3096.03	0.559099	0.279550	−	0.960131i	$-0.409815\pi$
$313$	3096.03	0.559099	0.279550	+	0.960131i	$0.409815\pi$
$314$	0	0
$315$	716.392	0.128140
$316$	0	0
$317$	−863.049	−0.152914	−0.0764569	−	0.997073i	$-0.524361\pi$
$317$	−863.049	−0.152914	−0.0764569	+	0.997073i	$0.524361\pi$
$318$	0	0
$319$	−636.708	−0.111752
$320$	0	0
$321$	4941.24	0.859168
$322$	0	0
$323$	5945.50	1.02420
$324$	0	0
$325$	5653.47	0.964917
$326$	0	0
$327$	13744.8	2.32443
$328$	0	0
$329$	−1118.32	−0.187402
$330$	0	0
$331$	833.660	0.138435	0.0692177	−	0.997602i	$-0.477950\pi$
$331$	833.660	0.138435	0.0692177	+	0.997602i	$0.477950\pi$
$332$	0	0
$333$	1190.57	0.195924
$334$	0	0
$335$	−4383.68	−0.714944
$336$	0	0
$337$	−9333.09	−1.50862	−0.754311	−	0.656517i	$-0.772029\pi$
$337$	−9333.09	−1.50862	−0.754311	+	0.656517i	$0.772029\pi$
$338$	0	0
$339$	11727.6	1.87893
$340$	0	0
$341$	−2527.90	−0.401447
$342$	0	0
$343$	5983.10	0.941857
$344$	0	0
$345$	−6985.29	−1.09007
$346$	0	0
$347$	−1010.85	−0.156384	−0.0781919	−	0.996938i	$-0.524915\pi$
$347$	−1010.85	−0.156384	−0.0781919	+	0.996938i	$0.524915\pi$
$348$	0	0
$349$	4815.64	0.738612	0.369306	−	0.929308i	$-0.379595\pi$
$349$	4815.64	0.738612	0.369306	+	0.929308i	$0.379595\pi$
$350$	0	0
$351$	5778.52	0.878731
$352$	0	0
$353$	−6923.20	−1.04387	−0.521933	−	0.852986i	$-0.674789\pi$
$353$	−6923.20	−1.04387	−0.521933	+	0.852986i	$0.674789\pi$
$354$	0	0
$355$	−4023.69	−0.601564
$356$	0	0
$357$	−2703.86	−0.400850
$358$	0	0
$359$	136.845	0.0201181	0.0100590	−	0.999949i	$-0.496798\pi$
$359$	136.845	0.0201181	0.0100590	+	0.999949i	$0.496798\pi$
$360$	0	0
$361$	13255.9	1.93262
$362$	0	0
$363$	−5302.16	−0.766642
$364$	0	0
$365$	6375.88	0.914325
$366$	0	0
$367$	−8897.66	−1.26554	−0.632772	−	0.774339i	$-0.718083\pi$
$367$	−8897.66	−1.26554	−0.632772	+	0.774339i	$0.718083\pi$
$368$	0	0
$369$	2937.60	0.414431
$370$	0	0
$371$	5321.28	0.744656
$372$	0	0
$373$	13459.6	1.86839	0.934196	−	0.356761i	$-0.116119\pi$
$373$	13459.6	1.86839	0.934196	+	0.356761i	$0.116119\pi$
$374$	0	0
$375$	7816.63	1.07640
$376$	0	0
$377$	1789.51	0.244469
$378$	0	0
$379$	−4584.69	−0.621371	−0.310686	−	0.950513i	$-0.600559\pi$
$379$	−4584.69	−0.621371	−0.310686	+	0.950513i	$0.600559\pi$
$380$	0	0
$381$	−6401.53	−0.860788
$382$	0	0
$383$	−295.534	−0.0394284	−0.0197142	−	0.999806i	$-0.506276\pi$
$383$	−295.534	−0.0394284	−0.0197142	+	0.999806i	$0.506276\pi$
$384$	0	0
$385$	1310.06	0.173420
$386$	0	0
$387$	−4626.36	−0.607678
$388$	0	0
$389$	5802.99	0.756358	0.378179	−	0.925733i	$-0.376550\pi$
$389$	5802.99	0.756358	0.378179	+	0.925733i	$0.376550\pi$
$390$	0	0
$391$	8114.99	1.04960
$392$	0	0
$393$	4403.16	0.565165
$394$	0	0
$395$	−4724.71	−0.601838
$396$	0	0
$397$	8614.61	1.08905	0.544527	−	0.838743i	$-0.316709\pi$
$397$	8614.61	1.08905	0.544527	+	0.838743i	$0.316709\pi$
$398$	0	0
$399$	−9147.72	−1.14777
$400$	0	0
$401$	−10952.4	−1.36393	−0.681964	−	0.731385i	$-0.738874\pi$
$401$	−10952.4	−1.36393	−0.681964	+	0.731385i	$0.738874\pi$
$402$	0	0
$403$	7104.84	0.878206
$404$	0	0
$405$	5252.09	0.644391
$406$	0	0
$407$	2177.18	0.265157
$408$	0	0
$409$	−332.788	−0.0402330	−0.0201165	−	0.999798i	$-0.506404\pi$
$409$	−332.788	−0.0402330	−0.0201165	+	0.999798i	$0.506404\pi$
$410$	0	0
$411$	−8707.53	−1.04504
$412$	0	0
$413$	−8842.96	−1.05359
$414$	0	0
$415$	−6261.18	−0.740601
$416$	0	0
$417$	18774.7	2.20480
$418$	0	0
$419$	−3824.08	−0.445868	−0.222934	−	0.974834i	$-0.571563\pi$
$419$	−3824.08	−0.445868	−0.222934	+	0.974834i	$0.571563\pi$
$420$	0	0
$421$	16249.7	1.88115	0.940574	−	0.339588i	$-0.110288\pi$
$421$	16249.7	1.88115	0.940574	+	0.339588i	$0.110288\pi$
$422$	0	0
$423$	1300.12	0.149442
$424$	0	0
$425$	−3840.68	−0.438354
$426$	0	0
$427$	4158.58	0.471306
$428$	0	0
$429$	−8461.47	−0.952270
$430$	0	0
$431$	4479.60	0.500638	0.250319	−	0.968163i	$-0.419465\pi$
$431$	4479.60	0.500638	0.250319	+	0.968163i	$0.419465\pi$
$432$	0	0
$433$	−5959.36	−0.661405	−0.330702	−	0.943735i	$-0.607286\pi$
$433$	−5959.36	−0.661405	−0.330702	+	0.943735i	$0.607286\pi$
$434$	0	0
$435$	1046.46	0.115343
$436$	0	0
$437$	27454.7	3.00535
$438$	0	0
$439$	−9708.95	−1.05554	−0.527771	−	0.849387i	$-0.676972\pi$
$439$	−9708.95	−1.05554	−0.527771	+	0.849387i	$0.676972\pi$
$440$	0	0
$441$	−2837.61	−0.306404
$442$	0	0
$443$	9447.46	1.01323	0.506616	−	0.862172i	$-0.330896\pi$
$443$	9447.46	1.01323	0.506616	+	0.862172i	$0.330896\pi$
$444$	0	0
$445$	−689.962	−0.0734996
$446$	0	0
$447$	19886.2	2.10422
$448$	0	0
$449$	6246.29	0.656527	0.328263	−	0.944586i	$-0.393537\pi$
$449$	6246.29	0.656527	0.328263	+	0.944586i	$0.393537\pi$
$450$	0	0
$451$	5371.94	0.560876
$452$	0	0
$453$	1483.37	0.153851
$454$	0	0
$455$	−3682.00	−0.379374
$456$	0	0
$457$	−2396.84	−0.245337	−0.122669	−	0.992448i	$-0.539145\pi$
$457$	−2396.84	−0.245337	−0.122669	+	0.992448i	$0.539145\pi$
$458$	0	0
$459$	−3925.63	−0.399200
$460$	0	0
$461$	−3796.18	−0.383527	−0.191763	−	0.981441i	$-0.561421\pi$
$461$	−3796.18	−0.383527	−0.191763	+	0.981441i	$0.561421\pi$
$462$	0	0
$463$	8531.69	0.856374	0.428187	−	0.903690i	$-0.359152\pi$
$463$	8531.69	0.856374	0.428187	+	0.903690i	$0.359152\pi$
$464$	0	0
$465$	4154.73	0.414347
$466$	0	0
$467$	−6660.92	−0.660023	−0.330011	−	0.943977i	$-0.607053\pi$
$467$	−6660.92	−0.660023	−0.330011	+	0.943977i	$0.607053\pi$
$468$	0	0
$469$	−7835.51	−0.771451
$470$	0	0
$471$	5654.81	0.553206
$472$	0	0
$473$	−8460.16	−0.822407
$474$	0	0
$475$	−12993.8	−1.25515
$476$	0	0
$477$	−6186.32	−0.593820
$478$	0	0
$479$	4730.39	0.451225	0.225613	−	0.974217i	$-0.427562\pi$
$479$	4730.39	0.451225	0.225613	+	0.974217i	$0.427562\pi$
$480$	0	0
$481$	−6119.11	−0.580057
$482$	0	0
$483$	−12485.7	−1.17623
$484$	0	0
$485$	−9301.01	−0.870798
$486$	0	0
$487$	8122.73	0.755803	0.377901	−	0.925846i	$-0.376646\pi$
$487$	8122.73	0.755803	0.377901	+	0.925846i	$0.376646\pi$
$488$	0	0
$489$	9779.08	0.904346
$490$	0	0
$491$	11777.8	1.08254	0.541268	−	0.840850i	$-0.317944\pi$
$491$	11777.8	1.08254	0.541268	+	0.840850i	$0.317944\pi$
$492$	0	0
$493$	−1215.70	−0.111060
$494$	0	0
$495$	−1523.02	−0.138292
$496$	0	0
$497$	−7192.04	−0.649109
$498$	0	0
$499$	−6631.25	−0.594901	−0.297450	−	0.954737i	$-0.596136\pi$
$499$	−6631.25	−0.594901	−0.297450	+	0.954737i	$0.596136\pi$
$500$	0	0
$501$	17274.2	1.54043
$502$	0	0
$503$	−12700.4	−1.12581	−0.562905	−	0.826521i	$-0.690316\pi$
$503$	−12700.4	−1.12581	−0.562905	+	0.826521i	$0.690316\pi$
$504$	0	0
$505$	−3815.91	−0.336249
$506$	0	0
$507$	10060.2	0.881243
$508$	0	0
$509$	16365.4	1.42512	0.712559	−	0.701612i	$-0.247536\pi$
$509$	16365.4	1.42512	0.712559	+	0.701612i	$0.247536\pi$
$510$	0	0
$511$	11396.4	0.986590
$512$	0	0
$513$	−13281.2	−1.14304
$514$	0	0
$515$	2809.40	0.240383
$516$	0	0
$517$	2377.51	0.202249
$518$	0	0
$519$	17795.7	1.50510
$520$	0	0
$521$	−2603.78	−0.218952	−0.109476	−	0.993989i	$-0.534917\pi$
$521$	−2603.78	−0.218952	−0.109476	+	0.993989i	$0.534917\pi$
$522$	0	0
$523$	−14885.7	−1.24456	−0.622281	−	0.782794i	$-0.713794\pi$
$523$	−14885.7	−1.24456	−0.622281	+	0.782794i	$0.713794\pi$
$524$	0	0
$525$	5909.25	0.491240
$526$	0	0
$527$	−4826.66	−0.398961
$528$	0	0
$529$	25305.8	2.07988
$530$	0	0
$531$	10280.5	0.840179
$532$	0	0
$533$	−15098.2	−1.22697
$534$	0	0
$535$	−4571.19	−0.369401
$536$	0	0
$537$	5914.00	0.475247
$538$	0	0
$539$	−5189.10	−0.414676
$540$	0	0
$541$	−11538.3	−0.916949	−0.458474	−	0.888708i	$-0.651604\pi$
$541$	−11538.3	−0.916949	−0.458474	+	0.888708i	$0.651604\pi$
$542$	0	0
$543$	4418.76	0.349221
$544$	0	0
$545$	−12715.4	−0.999392
$546$	0	0
$547$	−22259.5	−1.73994	−0.869971	−	0.493102i	$-0.835863\pi$
$547$	−22259.5	−1.73994	−0.869971	+	0.493102i	$0.835863\pi$
$548$	0	0
$549$	−4834.60	−0.375840
$550$	0	0
$551$	−4112.98	−0.318001
$552$	0	0
$553$	−8445.06	−0.649404
$554$	0	0
$555$	−3578.31	−0.273677
$556$	0	0
$557$	−4804.26	−0.365463	−0.182732	−	0.983163i	$-0.558494\pi$
$557$	−4804.26	−0.365463	−0.182732	+	0.983163i	$0.558494\pi$
$558$	0	0
$559$	23777.9	1.79910
$560$	0	0
$561$	5748.29	0.432608
$562$	0	0
$563$	22389.3	1.67602	0.838009	−	0.545657i	$-0.183720\pi$
$563$	22389.3	1.67602	0.838009	+	0.545657i	$0.183720\pi$
$564$	0	0
$565$	−10849.4	−0.807851
$566$	0	0
$567$	9387.73	0.695322
$568$	0	0
$569$	6381.15	0.470144	0.235072	−	0.971978i	$-0.424467\pi$
$569$	6381.15	0.470144	0.235072	+	0.971978i	$0.424467\pi$
$570$	0	0
$571$	−7423.90	−0.544099	−0.272050	−	0.962283i	$-0.587702\pi$
$571$	−7423.90	−0.544099	−0.272050	+	0.962283i	$0.587702\pi$
$572$	0	0
$573$	−26857.1	−1.95806
$574$	0	0
$575$	−17735.2	−1.28628
$576$	0	0
$577$	6829.23	0.492729	0.246364	−	0.969177i	$-0.420764\pi$
$577$	6829.23	0.492729	0.246364	+	0.969177i	$0.420764\pi$
$578$	0	0
$579$	−1218.25	−0.0874415
$580$	0	0
$581$	−11191.4	−0.799135
$582$	0	0
$583$	−11312.8	−0.803653
$584$	0	0
$585$	4280.56	0.302529
$586$	0	0
$587$	13882.6	0.976146	0.488073	−	0.872803i	$-0.337700\pi$
$587$	13882.6	0.976146	0.488073	+	0.872803i	$0.337700\pi$
$588$	0	0
$589$	−16329.6	−1.14236
$590$	0	0
$591$	16463.0	1.14585
$592$	0	0
$593$	−14144.5	−0.979503	−0.489751	−	0.871862i	$-0.662912\pi$
$593$	−14144.5	−0.979503	−0.489751	+	0.871862i	$0.662912\pi$
$594$	0	0
$595$	2501.37	0.172346
$596$	0	0
$597$	9338.51	0.640201
$598$	0	0
$599$	11966.6	0.816263	0.408132	−	0.912923i	$-0.366180\pi$
$599$	11966.6	0.816263	0.408132	+	0.912923i	$0.366180\pi$
$600$	0	0
$601$	12554.2	0.852075	0.426037	−	0.904706i	$-0.359909\pi$
$601$	12554.2	0.852075	0.426037	+	0.904706i	$0.359909\pi$
$602$	0	0
$603$	9109.26	0.615187
$604$	0	0
$605$	4905.08	0.329619
$606$	0	0
$607$	−17486.4	−1.16928	−0.584639	−	0.811294i	$-0.698764\pi$
$607$	−17486.4	−1.16928	−0.584639	+	0.811294i	$0.698764\pi$
$608$	0	0
$609$	1870.48	0.124459
$610$	0	0
$611$	−6682.16	−0.442441
$612$	0	0
$613$	5283.10	0.348095	0.174047	−	0.984737i	$-0.444315\pi$
$613$	5283.10	0.348095	0.174047	+	0.984737i	$0.444315\pi$
$614$	0	0
$615$	−8829.07	−0.578898
$616$	0	0
$617$	−27318.1	−1.78248	−0.891238	−	0.453537i	$-0.850162\pi$
$617$	−27318.1	−1.78248	−0.891238	+	0.453537i	$0.850162\pi$
$618$	0	0
$619$	2978.98	0.193433	0.0967166	−	0.995312i	$-0.469166\pi$
$619$	2978.98	0.193433	0.0967166	+	0.995312i	$0.469166\pi$
$620$	0	0
$621$	−18127.5	−1.17139
$622$	0	0
$623$	−1233.26	−0.0793088
$624$	0	0
$625$	4220.94	0.270140
$626$	0	0
$627$	19447.7	1.23870
$628$	0	0
$629$	4157.01	0.263515
$630$	0	0
$631$	18585.8	1.17257	0.586283	−	0.810106i	$-0.300591\pi$
$631$	18585.8	1.17257	0.586283	+	0.810106i	$0.300591\pi$
$632$	0	0
$633$	−1219.15	−0.0765509
$634$	0	0
$635$	5922.12	0.370098
$636$	0	0
$637$	14584.3	0.907145
$638$	0	0
$639$	8361.19	0.517627
$640$	0	0
$641$	−25979.8	−1.60084	−0.800420	−	0.599439i	$-0.795390\pi$
$641$	−25979.8	−1.60084	−0.800420	+	0.599439i	$0.795390\pi$
$642$	0	0
$643$	21899.5	1.34313	0.671565	−	0.740946i	$-0.265622\pi$
$643$	21899.5	1.34313	0.671565	+	0.740946i	$0.265622\pi$
$644$	0	0
$645$	13904.7	0.848834
$646$	0	0
$647$	1017.24	0.0618114	0.0309057	−	0.999522i	$-0.490161\pi$
$647$	1017.24	0.0618114	0.0309057	+	0.999522i	$0.490161\pi$
$648$	0	0
$649$	18799.8	1.13707
$650$	0	0
$651$	7426.28	0.447095
$652$	0	0
$653$	−13166.3	−0.789032	−0.394516	−	0.918889i	$-0.629088\pi$
$653$	−13166.3	−0.789032	−0.394516	+	0.918889i	$0.629088\pi$
$654$	0	0
$655$	−4073.40	−0.242994
$656$	0	0
$657$	−13249.0	−0.786748
$658$	0	0
$659$	14519.2	0.858252	0.429126	−	0.903245i	$-0.358822\pi$
$659$	14519.2	0.858252	0.429126	+	0.903245i	$0.358822\pi$
$660$	0	0
$661$	24867.3	1.46328	0.731639	−	0.681692i	$-0.238756\pi$
$661$	24867.3	1.46328	0.731639	+	0.681692i	$0.238756\pi$
$662$	0	0
$663$	−16156.0	−0.946374
$664$	0	0
$665$	8462.64	0.493484
$666$	0	0
$667$	−5613.79	−0.325887
$668$	0	0
$669$	−6270.48	−0.362378
$670$	0	0
$671$	−8840.97	−0.508647
$672$	0	0
$673$	2718.48	0.155705	0.0778526	−	0.996965i	$-0.475194\pi$
$673$	2718.48	0.155705	0.0778526	+	0.996965i	$0.475194\pi$
$674$	0	0
$675$	8579.42	0.489218
$676$	0	0
$677$	19569.7	1.11097	0.555484	−	0.831527i	$-0.312533\pi$
$677$	19569.7	1.11097	0.555484	+	0.831527i	$0.312533\pi$
$678$	0	0
$679$	−16624.9	−0.939622
$680$	0	0
$681$	−1293.84	−0.0728049
$682$	0	0
$683$	−978.536	−0.0548209	−0.0274104	−	0.999624i	$-0.508726\pi$
$683$	−978.536	−0.0548209	−0.0274104	+	0.999624i	$0.508726\pi$
$684$	0	0
$685$	8055.42	0.449317
$686$	0	0
$687$	−8727.54	−0.484682
$688$	0	0
$689$	31795.5	1.75807
$690$	0	0
$691$	16567.7	0.912103	0.456051	−	0.889953i	$-0.349263\pi$
$691$	16567.7	0.912103	0.456051	+	0.889953i	$0.349263\pi$
$692$	0	0
$693$	−2722.29	−0.149222
$694$	0	0
$695$	−17368.6	−0.947957
$696$	0	0
$697$	10257.0	0.557403
$698$	0	0
$699$	12466.8	0.674587
$700$	0	0
$701$	−24735.9	−1.33275	−0.666377	−	0.745615i	$-0.732156\pi$
$701$	−24735.9	−1.33275	−0.666377	+	0.745615i	$0.732156\pi$
$702$	0	0
$703$	14064.0	0.754531
$704$	0	0
$705$	−3907.56	−0.208748
$706$	0	0
$707$	−6820.65	−0.362825
$708$	0	0
$709$	−29565.8	−1.56610	−0.783051	−	0.621958i	$-0.786338\pi$
$709$	−29565.8	−1.56610	−0.783051	+	0.621958i	$0.786338\pi$
$710$	0	0
$711$	9817.90	0.517862
$712$	0	0
$713$	−22288.2	−1.17069
$714$	0	0
$715$	7827.79	0.409430
$716$	0	0
$717$	−24694.1	−1.28622
$718$	0	0
$719$	−23673.4	−1.22791	−0.613956	−	0.789341i	$-0.710423\pi$
$719$	−23673.4	−1.22791	−0.613956	+	0.789341i	$0.710423\pi$
$720$	0	0
$721$	5021.60	0.259382
$722$	0	0
$723$	−31006.7	−1.59495
$724$	0	0
$725$	2656.91	0.136103
$726$	0	0
$727$	−13493.6	−0.688376	−0.344188	−	0.938901i	$-0.611846\pi$
$727$	−13493.6	−0.688376	−0.344188	+	0.938901i	$0.611846\pi$
$728$	0	0
$729$	4877.21	0.247788
$730$	0	0
$731$	−16153.5	−0.817316
$732$	0	0
$733$	15739.3	0.793101	0.396551	−	0.918013i	$-0.370207\pi$
$733$	15739.3	0.793101	0.396551	+	0.918013i	$0.370207\pi$
$734$	0	0
$735$	8528.55	0.428000
$736$	0	0
$737$	16658.0	0.832570
$738$	0	0
$739$	−25229.6	−1.25587	−0.627934	−	0.778266i	$-0.716099\pi$
$739$	−25229.6	−1.25587	−0.627934	+	0.778266i	$0.716099\pi$
$740$	0	0
$741$	−54659.0	−2.70978
$742$	0	0
$743$	−11063.4	−0.546267	−0.273134	−	0.961976i	$-0.588060\pi$
$743$	−11063.4	−0.546267	−0.273134	+	0.961976i	$0.588060\pi$
$744$	0	0
$745$	−18397.0	−0.904714
$746$	0	0
$747$	13010.7	0.637264
$748$	0	0
$749$	−8170.66	−0.398597
$750$	0	0
$751$	20969.5	1.01889	0.509447	−	0.860502i	$-0.329850\pi$
$751$	20969.5	1.01889	0.509447	+	0.860502i	$0.329850\pi$
$752$	0	0
$753$	−27922.6	−1.35133
$754$	0	0
$755$	−1372.28	−0.0661487
$756$	0	0
$757$	−16610.9	−0.797532	−0.398766	−	0.917053i	$-0.630562\pi$
$757$	−16610.9	−0.797532	−0.398766	+	0.917053i	$0.630562\pi$
$758$	0	0
$759$	26544.1	1.26942
$760$	0	0
$761$	26618.0	1.26794	0.633971	−	0.773357i	$-0.281424\pi$
$761$	26618.0	1.26794	0.633971	+	0.773357i	$0.281424\pi$
$762$	0	0
$763$	−22727.9	−1.07838
$764$	0	0
$765$	−2907.99	−0.137436
$766$	0	0
$767$	−52838.1	−2.48745
$768$	0	0
$769$	−17272.7	−0.809973	−0.404987	−	0.914323i	$-0.632724\pi$
$769$	−17272.7	−0.809973	−0.404987	+	0.914323i	$0.632724\pi$
$770$	0	0
$771$	−7228.86	−0.337667
$772$	0	0
$773$	−30188.1	−1.40465	−0.702323	−	0.711859i	$-0.747854\pi$
$773$	−30188.1	−1.40465	−0.702323	+	0.711859i	$0.747854\pi$
$774$	0	0
$775$	10548.6	0.488926
$776$	0	0
$777$	−6395.96	−0.295307
$778$	0	0
$779$	34701.4	1.59603
$780$	0	0
$781$	15290.0	0.700536
$782$	0	0
$783$	2715.68	0.123947
$784$	0	0
$785$	−5231.32	−0.237852
$786$	0	0
$787$	14440.7	0.654073	0.327036	−	0.945012i	$-0.393950\pi$
$787$	14440.7	0.654073	0.327036	+	0.945012i	$0.393950\pi$
$788$	0	0
$789$	8686.68	0.391957
$790$	0	0
$791$	−19392.4	−0.871700
$792$	0	0
$793$	24848.2	1.11272
$794$	0	0
$795$	18593.2	0.829476
$796$	0	0
$797$	3602.11	0.160092	0.0800460	−	0.996791i	$-0.474493\pi$
$797$	3602.11	0.160092	0.0800460	+	0.996791i	$0.474493\pi$
$798$	0	0
$799$	4539.52	0.200997
$800$	0	0
$801$	1433.74	0.0632441
$802$	0	0
$803$	−24228.3	−1.06475
$804$	0	0
$805$	11550.6	0.505722
$806$	0	0
$807$	−24886.9	−1.08558
$808$	0	0
$809$	−15459.1	−0.671831	−0.335916	−	0.941892i	$-0.609046\pi$
$809$	−15459.1	−0.671831	−0.335916	+	0.941892i	$0.609046\pi$
$810$	0	0
$811$	2609.86	0.113002	0.0565010	−	0.998403i	$-0.482006\pi$
$811$	2609.86	0.113002	0.0565010	+	0.998403i	$0.482006\pi$
$812$	0	0
$813$	−8064.29	−0.347881
$814$	0	0
$815$	−9046.72	−0.388826
$816$	0	0
$817$	−54650.6	−2.34025
$818$	0	0
$819$	7651.18	0.326439
$820$	0	0
$821$	29330.7	1.24683	0.623417	−	0.781890i	$-0.285744\pi$
$821$	29330.7	1.24683	0.623417	+	0.781890i	$0.285744\pi$
$822$	0	0
$823$	−1154.78	−0.0489101	−0.0244551	−	0.999701i	$-0.507785\pi$
$823$	−1154.78	−0.0489101	−0.0244551	+	0.999701i	$0.507785\pi$
$824$	0	0
$825$	−12562.8	−0.530159
$826$	0	0
$827$	9260.30	0.389374	0.194687	−	0.980865i	$-0.437631\pi$
$827$	9260.30	0.389374	0.194687	+	0.980865i	$0.437631\pi$
$828$	0	0
$829$	12223.9	0.512127	0.256063	−	0.966660i	$-0.417574\pi$
$829$	12223.9	0.512127	0.256063	+	0.966660i	$0.417574\pi$
$830$	0	0
$831$	−40029.2	−1.67099
$832$	0	0
$833$	−9907.84	−0.412108
$834$	0	0
$835$	−15980.5	−0.662310
$836$	0	0
$837$	10781.9	0.445255
$838$	0	0
$839$	7484.05	0.307959	0.153980	−	0.988074i	$-0.450791\pi$
$839$	7484.05	0.307959	0.153980	+	0.988074i	$0.450791\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	37250.8	1.52193
$844$	0	0
$845$	−9306.81	−0.378892
$846$	0	0
$847$	8767.46	0.355671
$848$	0	0
$849$	−43408.5	−1.75474
$850$	0	0
$851$	19196.0	0.773242
$852$	0	0
$853$	−16877.0	−0.677442	−0.338721	−	0.940887i	$-0.609994\pi$
$853$	−16877.0	−0.677442	−0.338721	+	0.940887i	$0.609994\pi$
$854$	0	0
$855$	−9838.34	−0.393525
$856$	0	0
$857$	−12829.0	−0.511355	−0.255677	−	0.966762i	$-0.582299\pi$
$857$	−12829.0	−0.511355	−0.255677	+	0.966762i	$0.582299\pi$
$858$	0	0
$859$	−436.566	−0.0173404	−0.00867021	−	0.999962i	$-0.502760\pi$
$859$	−436.566	−0.0173404	−0.00867021	+	0.999962i	$0.502760\pi$
$860$	0	0
$861$	−15781.3	−0.624652
$862$	0	0
$863$	−9587.19	−0.378160	−0.189080	−	0.981962i	$-0.560551\pi$
$863$	−9587.19	−0.378160	−0.189080	+	0.981962i	$0.560551\pi$
$864$	0	0
$865$	−16463.0	−0.647120
$866$	0	0
$867$	−19708.5	−0.772015
$868$	0	0
$869$	17953.9	0.700855
$870$	0	0
$871$	−46818.4	−1.82133
$872$	0	0
$873$	19327.4	0.749294
$874$	0	0
$875$	−12925.3	−0.499377
$876$	0	0
$877$	−28153.2	−1.08400	−0.541999	−	0.840379i	$-0.682332\pi$
$877$	−28153.2	−1.08400	−0.541999	+	0.840379i	$0.682332\pi$
$878$	0	0
$879$	−46235.9	−1.77417
$880$	0	0
$881$	−21712.0	−0.830301	−0.415150	−	0.909753i	$-0.636271\pi$
$881$	−21712.0	−0.830301	−0.415150	+	0.909753i	$0.636271\pi$
$882$	0	0
$883$	38443.8	1.46516	0.732581	−	0.680680i	$-0.238315\pi$
$883$	38443.8	1.46516	0.732581	+	0.680680i	$0.238315\pi$
$884$	0	0
$885$	−30898.4	−1.17360
$886$	0	0
$887$	9216.28	0.348875	0.174438	−	0.984668i	$-0.444189\pi$
$887$	9216.28	0.348875	0.174438	+	0.984668i	$0.444189\pi$
$888$	0	0
$889$	10585.3	0.399349
$890$	0	0
$891$	−19957.9	−0.750410
$892$	0	0
$893$	15358.1	0.575521
$894$	0	0
$895$	−5471.10	−0.204334
$896$	0	0
$897$	−74603.9	−2.77698
$898$	0	0
$899$	3338.99	0.123873
$900$	0	0
$901$	−21600.2	−0.798677
$902$	0	0
$903$	24853.7	0.915922
$904$	0	0
$905$	−4087.84	−0.150148
$906$	0	0
$907$	−20093.0	−0.735586	−0.367793	−	0.929908i	$-0.619887\pi$
$907$	−20093.0	−0.735586	−0.367793	+	0.929908i	$0.619887\pi$
$908$	0	0
$909$	7929.42	0.289332
$910$	0	0
$911$	−4076.94	−0.148271	−0.0741356	−	0.997248i	$-0.523620\pi$
$911$	−4076.94	−0.148271	−0.0741356	+	0.997248i	$0.523620\pi$
$912$	0	0
$913$	23792.5	0.862448
$914$	0	0
$915$	14530.6	0.524991
$916$	0	0
$917$	−7280.91	−0.262199
$918$	0	0
$919$	11917.0	0.427755	0.213878	−	0.976860i	$-0.431391\pi$
$919$	11917.0	0.427755	0.213878	+	0.976860i	$0.431391\pi$
$920$	0	0
$921$	−58970.3	−2.10981
$922$	0	0
$923$	−42973.6	−1.53249
$924$	0	0
$925$	−9085.10	−0.322936
$926$	0	0
$927$	−5837.92	−0.206842
$928$	0	0
$929$	32110.0	1.13401	0.567004	−	0.823715i	$-0.308102\pi$
$929$	32110.0	1.13401	0.567004	+	0.823715i	$0.308102\pi$
$930$	0	0
$931$	−33520.3	−1.18000
$932$	0	0
$933$	65805.8	2.30909
$934$	0	0
$935$	−5317.80	−0.186001
$936$	0	0
$937$	5112.86	0.178260	0.0891301	−	0.996020i	$-0.471591\pi$
$937$	5112.86	0.178260	0.0891301	+	0.996020i	$0.471591\pi$
$938$	0	0
$939$	19336.2	0.672006
$940$	0	0
$941$	−16176.1	−0.560390	−0.280195	−	0.959943i	$-0.590399\pi$
$941$	−16176.1	−0.560390	−0.280195	+	0.959943i	$0.590399\pi$
$942$	0	0
$943$	47363.8	1.63561
$944$	0	0
$945$	−5587.63	−0.192344
$946$	0	0
$947$	43229.1	1.48337	0.741687	−	0.670746i	$-0.234026\pi$
$947$	43229.1	1.48337	0.741687	+	0.670746i	$0.234026\pi$
$948$	0	0
$949$	68095.3	2.32926
$950$	0	0
$951$	−5390.16	−0.183794
$952$	0	0
$953$	31964.2	1.08649	0.543243	−	0.839575i	$-0.317196\pi$
$953$	31964.2	1.08649	0.543243	+	0.839575i	$0.317196\pi$
$954$	0	0
$955$	24845.8	0.841874
$956$	0	0
$957$	−3976.56	−0.134320
$958$	0	0
$959$	14398.5	0.484829
$960$	0	0
$961$	−16534.3	−0.555011
$962$	0	0
$963$	9498.89	0.317858
$964$	0	0
$965$	1127.01	0.0375956
$966$	0	0
$967$	−20710.8	−0.688743	−0.344372	−	0.938833i	$-0.611908\pi$
$967$	−20710.8	−0.688743	−0.344372	+	0.938833i	$0.611908\pi$
$968$	0	0
$969$	37132.6	1.23103
$970$	0	0
$971$	−21766.4	−0.719378	−0.359689	−	0.933072i	$-0.617117\pi$
$971$	−21766.4	−0.719378	−0.359689	+	0.933072i	$0.617117\pi$
$972$	0	0
$973$	−31045.2	−1.02288
$974$	0	0
$975$	35308.7	1.15978
$976$	0	0
$977$	24558.3	0.804187	0.402093	−	0.915599i	$-0.368283\pi$
$977$	24558.3	0.804187	0.402093	+	0.915599i	$0.368283\pi$
$978$	0	0
$979$	2621.85	0.0855922
$980$	0	0
$981$	26422.5	0.859946
$982$	0	0
$983$	−10521.7	−0.341395	−0.170697	−	0.985324i	$-0.554602\pi$
$983$	−10521.7	−0.341395	−0.170697	+	0.985324i	$0.554602\pi$
$984$	0	0
$985$	−15230.1	−0.492660
$986$	0	0
$987$	−6984.48	−0.225247
$988$	0	0
$989$	−74592.4	−2.39828
$990$	0	0
$991$	30258.6	0.969927	0.484963	−	0.874535i	$-0.338833\pi$
$991$	30258.6	0.969927	0.484963	+	0.874535i	$0.338833\pi$
$992$	0	0
$993$	5206.61	0.166392
$994$	0	0
$995$	−8639.14	−0.275256
$996$	0	0
$997$	27589.2	0.876388	0.438194	−	0.898881i	$-0.355618\pi$
$997$	27589.2	0.876388	0.438194	+	0.898881i	$0.355618\pi$
$998$	0	0
$999$	−9286.06	−0.294092

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1856.4.a.bl.1.9		12
4.3	odd	2		1856.4.a.bj.1.4		12
8.3	odd	2		928.4.a.j.1.9	yes	12
8.5	even	2		928.4.a.h.1.4	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
928.4.a.h.1.4	✓	12	8.5	even	2
928.4.a.j.1.9	yes	12	8.3	odd	2
1856.4.a.bj.1.4		12	4.3	odd	2
1856.4.a.bl.1.9		12	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:	\( N \)	\(=\)	\( 1856 = 2^{6} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1856.a (trivial)

\(f(q)\)	\(=\)	\(q+6.24549 q^{3} -5.77776 q^{5} -10.3273 q^{7} +12.0061 q^{9} +O(q^{10})\)	\(q+6.24549 q^{3} -5.77776 q^{5} -10.3273 q^{7} +12.0061 q^{9} +21.9555 q^{11} -61.7074 q^{13} -36.0850 q^{15} +41.9208 q^{17} +141.827 q^{19} -64.4992 q^{21} +193.579 q^{23} -91.6175 q^{25} -93.6440 q^{27} -29.0000 q^{29} -115.138 q^{31} +137.123 q^{33} +59.6688 q^{35} +99.1634 q^{37} -385.393 q^{39} +244.674 q^{41} -385.333 q^{43} -69.3686 q^{45} +108.288 q^{47} -236.346 q^{49} +261.816 q^{51} -515.263 q^{53} -126.853 q^{55} +885.778 q^{57} +856.269 q^{59} -402.678 q^{61} -123.991 q^{63} +356.530 q^{65} +758.717 q^{67} +1209.00 q^{69} +696.409 q^{71} -1103.52 q^{73} -572.196 q^{75} -226.741 q^{77} +817.740 q^{79} -909.018 q^{81} +1083.67 q^{83} -242.209 q^{85} -181.119 q^{87} +119.417 q^{89} +637.272 q^{91} -719.091 q^{93} -819.442 q^{95} +1609.79 q^{97} +263.600 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 14 q^{3} + 10 q^{5} - 44 q^{7} + 130 q^{9}+O(q^{10}) \) 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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