LMFDB - Embedded newform 1856.4.a.bl.1.2

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			1.2
Root			$-6.97229$ 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)

$f(q)$	$=$	$q-5.97229 q^{3} +11.6005 q^{5} -8.29658 q^{7} +8.66827 q^{9} +O(q^{10})$	$q-5.97229 q^{3} +11.6005 q^{5} -8.29658 q^{7} +8.66827 q^{9} -49.0756 q^{11} +24.9952 q^{13} -69.2817 q^{15} -42.7599 q^{17} -146.687 q^{19} +49.5496 q^{21} -12.9594 q^{23} +9.57201 q^{25} +109.482 q^{27} -29.0000 q^{29} -133.918 q^{31} +293.094 q^{33} -96.2446 q^{35} +381.762 q^{37} -149.279 q^{39} -416.595 q^{41} -447.227 q^{43} +100.556 q^{45} +555.202 q^{47} -274.167 q^{49} +255.375 q^{51} +308.757 q^{53} -569.302 q^{55} +876.055 q^{57} +647.141 q^{59} +394.081 q^{61} -71.9170 q^{63} +289.958 q^{65} +216.463 q^{67} +77.3971 q^{69} -866.026 q^{71} -408.481 q^{73} -57.1668 q^{75} +407.159 q^{77} -218.022 q^{79} -887.904 q^{81} +928.140 q^{83} -496.037 q^{85} +173.196 q^{87} -1124.93 q^{89} -207.375 q^{91} +799.800 q^{93} -1701.64 q^{95} +1475.58 q^{97} -425.400 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−5.97229	−1.14937	−0.574684	−	0.818375i	$-0.694875\pi$
$3$	−5.97229	−1.14937	−0.574684	+	0.818375i	$0.694875\pi$
$4$	0	0
$5$	11.6005	1.03758	0.518791	−	0.854901i	$-0.326382\pi$
$5$	11.6005	1.03758	0.518791	+	0.854901i	$0.326382\pi$
$6$	0	0
$7$	−8.29658	−0.447973	−0.223987	−	0.974592i	$-0.571907\pi$
$7$	−8.29658	−0.447973	−0.223987	+	0.974592i	$0.571907\pi$
$8$	0	0
$9$	8.66827	0.321047
$10$	0	0
$11$	−49.0756	−1.34517	−0.672584	−	0.740021i	$-0.734815\pi$
$11$	−49.0756	−1.34517	−0.672584	+	0.740021i	$0.734815\pi$
$12$	0	0
$13$	24.9952	0.533264	0.266632	−	0.963798i	$-0.414089\pi$
$13$	24.9952	0.533264	0.266632	+	0.963798i	$0.414089\pi$
$14$	0	0
$15$	−69.2817	−1.19256
$16$	0	0
$17$	−42.7599	−0.610048	−0.305024	−	0.952345i	$-0.598664\pi$
$17$	−42.7599	−0.610048	−0.305024	+	0.952345i	$0.598664\pi$
$18$	0	0
$19$	−146.687	−1.77117	−0.885584	−	0.464479i	$-0.846242\pi$
$19$	−146.687	−1.77117	−0.885584	+	0.464479i	$0.846242\pi$
$20$	0	0
$21$	49.5496	0.514886
$22$	0	0
$23$	−12.9594	−0.117487	−0.0587437	−	0.998273i	$-0.518709\pi$
$23$	−12.9594	−0.117487	−0.0587437	+	0.998273i	$0.518709\pi$
$24$	0	0
$25$	9.57201	0.0765761
$26$	0	0
$27$	109.482	0.780367
$28$	0	0
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	0	0
$31$	−133.918	−0.775886	−0.387943	−	0.921683i	$-0.626814\pi$
$31$	−133.918	−0.775886	−0.387943	+	0.921683i	$0.626814\pi$
$32$	0	0
$33$	293.094	1.54609
$34$	0	0
$35$	−96.2446	−0.464809
$36$	0	0
$37$	381.762	1.69625	0.848125	−	0.529797i	$-0.177732\pi$
$37$	381.762	1.69625	0.848125	+	0.529797i	$0.177732\pi$
$38$	0	0
$39$	−149.279	−0.612917
$40$	0	0
$41$	−416.595	−1.58686	−0.793428	−	0.608663i	$-0.791706\pi$
$41$	−416.595	−1.58686	−0.793428	+	0.608663i	$0.791706\pi$
$42$	0	0
$43$	−447.227	−1.58608	−0.793041	−	0.609168i	$-0.791503\pi$
$43$	−447.227	−1.58608	−0.793041	+	0.609168i	$0.791503\pi$
$44$	0	0
$45$	100.556	0.333113
$46$	0	0
$47$	555.202	1.72307	0.861537	−	0.507694i	$-0.169502\pi$
$47$	555.202	1.72307	0.861537	+	0.507694i	$0.169502\pi$
$48$	0	0
$49$	−274.167	−0.799320
$50$	0	0
$51$	255.375	0.701169
$52$	0	0
$53$	308.757	0.800208	0.400104	−	0.916470i	$-0.368974\pi$
$53$	308.757	0.800208	0.400104	+	0.916470i	$0.368974\pi$
$54$	0	0
$55$	−569.302	−1.39572
$56$	0	0
$57$	876.055	2.03572
$58$	0	0
$59$	647.141	1.42798	0.713988	−	0.700158i	$-0.246887\pi$
$59$	647.141	1.42798	0.713988	+	0.700158i	$0.246887\pi$
$60$	0	0
$61$	394.081	0.827163	0.413581	−	0.910467i	$-0.364278\pi$
$61$	394.081	0.827163	0.413581	+	0.910467i	$0.364278\pi$
$62$	0	0
$63$	−71.9170	−0.143821
$64$	0	0
$65$	289.958	0.553305
$66$	0	0
$67$	216.463	0.394703	0.197352	−	0.980333i	$-0.436766\pi$
$67$	216.463	0.394703	0.197352	+	0.980333i	$0.436766\pi$
$68$	0	0
$69$	77.3971	0.135036
$70$	0	0
$71$	−866.026	−1.44758	−0.723791	−	0.690019i	$-0.757602\pi$
$71$	−866.026	−1.44758	−0.723791	+	0.690019i	$0.757602\pi$
$72$	0	0
$73$	−408.481	−0.654920	−0.327460	−	0.944865i	$-0.606193\pi$
$73$	−408.481	−0.654920	−0.327460	+	0.944865i	$0.606193\pi$
$74$	0	0
$75$	−57.1668	−0.0880141
$76$	0	0
$77$	407.159	0.602599
$78$	0	0
$79$	−218.022	−0.310499	−0.155249	−	0.987875i	$-0.549618\pi$
$79$	−218.022	−0.310499	−0.155249	+	0.987875i	$0.549618\pi$
$80$	0	0
$81$	−887.904	−1.21798
$82$	0	0
$83$	928.140	1.22743	0.613714	−	0.789528i	$-0.289675\pi$
$83$	928.140	1.22743	0.613714	+	0.789528i	$0.289675\pi$
$84$	0	0
$85$	−496.037	−0.632974
$86$	0	0
$87$	173.196	0.213432
$88$	0	0
$89$	−1124.93	−1.33980	−0.669899	−	0.742452i	$-0.733663\pi$
$89$	−1124.93	−1.33980	−0.669899	+	0.742452i	$0.733663\pi$
$90$	0	0
$91$	−207.375	−0.238888
$92$	0	0
$93$	799.800	0.891779
$94$	0	0
$95$	−1701.64	−1.83773
$96$	0	0
$97$	1475.58	1.54456	0.772280	−	0.635282i	$-0.219116\pi$
$97$	1475.58	1.54456	0.772280	+	0.635282i	$0.219116\pi$
$98$	0	0
$99$	−425.400	−0.431862
$100$	0	0
$101$	−1183.76	−1.16622	−0.583110	−	0.812393i	$-0.698164\pi$
$101$	−1183.76	−1.16622	−0.583110	+	0.812393i	$0.698164\pi$
$102$	0	0
$103$	−1758.51	−1.68224	−0.841120	−	0.540849i	$-0.818103\pi$
$103$	−1758.51	−1.68224	−0.841120	+	0.540849i	$0.818103\pi$
$104$	0	0
$105$	574.801	0.534237
$106$	0	0
$107$	−552.772	−0.499425	−0.249712	−	0.968320i	$-0.580336\pi$
$107$	−552.772	−0.499425	−0.249712	+	0.968320i	$0.580336\pi$
$108$	0	0
$109$	796.981	0.700338	0.350169	−	0.936686i	$-0.386124\pi$
$109$	796.981	0.700338	0.350169	+	0.936686i	$0.386124\pi$
$110$	0	0
$111$	−2279.99	−1.94961
$112$	0	0
$113$	1197.45	0.996875	0.498438	−	0.866926i	$-0.333907\pi$
$113$	1197.45	0.996875	0.498438	+	0.866926i	$0.333907\pi$
$114$	0	0
$115$	−150.335	−0.121903
$116$	0	0
$117$	216.666	0.171203
$118$	0	0
$119$	354.761	0.273285
$120$	0	0
$121$	1077.41	0.809475
$122$	0	0
$123$	2488.02	1.82388
$124$	0	0
$125$	−1339.02	−0.958128
$126$	0	0
$127$	866.601	0.605499	0.302750	−	0.953070i	$-0.402095\pi$
$127$	866.601	0.605499	0.302750	+	0.953070i	$0.402095\pi$
$128$	0	0
$129$	2670.97	1.82299
$130$	0	0
$131$	−783.494	−0.522551	−0.261275	−	0.965264i	$-0.584143\pi$
$131$	−783.494	−0.522551	−0.261275	+	0.965264i	$0.584143\pi$
$132$	0	0
$133$	1217.00	0.793436
$134$	0	0
$135$	1270.05	0.809694
$136$	0	0
$137$	1002.80	0.625368	0.312684	−	0.949857i	$-0.398772\pi$
$137$	1002.80	0.625368	0.312684	+	0.949857i	$0.398772\pi$
$138$	0	0
$139$	−117.217	−0.0715269	−0.0357634	−	0.999360i	$-0.511386\pi$
$139$	−117.217	−0.0715269	−0.0357634	+	0.999360i	$0.511386\pi$
$140$	0	0
$141$	−3315.83	−1.98045
$142$	0	0
$143$	−1226.66	−0.717330
$144$	0	0
$145$	−336.415	−0.192674
$146$	0	0
$147$	1637.40	0.918713
$148$	0	0
$149$	−3155.40	−1.73490	−0.867451	−	0.497523i	$-0.834243\pi$
$149$	−3155.40	−1.73490	−0.867451	+	0.497523i	$0.834243\pi$
$150$	0	0
$151$	1758.15	0.947524	0.473762	−	0.880653i	$-0.342896\pi$
$151$	1758.15	0.947524	0.473762	+	0.880653i	$0.342896\pi$
$152$	0	0
$153$	−370.655	−0.195854
$154$	0	0
$155$	−1553.52	−0.805045
$156$	0	0
$157$	83.8007	0.0425989	0.0212994	−	0.999773i	$-0.493220\pi$
$157$	83.8007	0.0425989	0.0212994	+	0.999773i	$0.493220\pi$
$158$	0	0
$159$	−1843.99	−0.919734
$160$	0	0
$161$	107.518	0.0526313
$162$	0	0
$163$	−176.309	−0.0847214	−0.0423607	−	0.999102i	$-0.513488\pi$
$163$	−176.309	−0.0847214	−0.0423607	+	0.999102i	$0.513488\pi$
$164$	0	0
$165$	3400.04	1.60420
$166$	0	0
$167$	36.5061	0.0169157	0.00845786	−	0.999964i	$-0.497308\pi$
$167$	36.5061	0.0169157	0.00845786	+	0.999964i	$0.497308\pi$
$168$	0	0
$169$	−1572.24	−0.715629
$170$	0	0
$171$	−1271.52	−0.568628
$172$	0	0
$173$	2737.81	1.20319	0.601594	−	0.798802i	$-0.294533\pi$
$173$	2737.81	1.20319	0.601594	+	0.798802i	$0.294533\pi$
$174$	0	0
$175$	−79.4150	−0.0343040
$176$	0	0
$177$	−3864.92	−1.64127
$178$	0	0
$179$	3647.96	1.52325	0.761624	−	0.648020i	$-0.224403\pi$
$179$	3647.96	1.52325	0.761624	+	0.648020i	$0.224403\pi$
$180$	0	0
$181$	3553.77	1.45939	0.729696	−	0.683772i	$-0.239662\pi$
$181$	3553.77	1.45939	0.729696	+	0.683772i	$0.239662\pi$
$182$	0	0
$183$	−2353.57	−0.950715
$184$	0	0
$185$	4428.63	1.76000
$186$	0	0
$187$	2098.47	0.820616
$188$	0	0
$189$	−908.330	−0.349584
$190$	0	0
$191$	−969.180	−0.367159	−0.183580	−	0.983005i	$-0.558768\pi$
$191$	−969.180	−0.367159	−0.183580	+	0.983005i	$0.558768\pi$
$192$	0	0
$193$	−2380.64	−0.887888	−0.443944	−	0.896055i	$-0.646421\pi$
$193$	−2380.64	−0.887888	−0.443944	+	0.896055i	$0.646421\pi$
$194$	0	0
$195$	−1731.71	−0.635951
$196$	0	0
$197$	4252.83	1.53808	0.769039	−	0.639202i	$-0.220735\pi$
$197$	4252.83	1.53808	0.769039	+	0.639202i	$0.220735\pi$
$198$	0	0
$199$	3174.09	1.13068	0.565340	−	0.824858i	$-0.308745\pi$
$199$	3174.09	1.13068	0.565340	+	0.824858i	$0.308745\pi$
$200$	0	0
$201$	−1292.78	−0.453659
$202$	0	0
$203$	240.601	0.0831866
$204$	0	0
$205$	−4832.71	−1.64649
$206$	0	0
$207$	−112.335	−0.0377190
$208$	0	0
$209$	7198.72	2.38252
$210$	0	0
$211$	−3100.50	−1.01160	−0.505800	−	0.862651i	$-0.668803\pi$
$211$	−3100.50	−1.01160	−0.505800	+	0.862651i	$0.668803\pi$
$212$	0	0
$213$	5172.16	1.66381
$214$	0	0
$215$	−5188.07	−1.64569
$216$	0	0
$217$	1111.07	0.347576
$218$	0	0
$219$	2439.57	0.752744
$220$	0	0
$221$	−1068.80	−0.325317
$222$	0	0
$223$	2684.13	0.806022	0.403011	−	0.915195i	$-0.367964\pi$
$223$	2684.13	0.806022	0.403011	+	0.915195i	$0.367964\pi$
$224$	0	0
$225$	82.9728	0.0245845
$226$	0	0
$227$	−1196.40	−0.349815	−0.174908	−	0.984585i	$-0.555963\pi$
$227$	−1196.40	−0.349815	−0.174908	+	0.984585i	$0.555963\pi$
$228$	0	0
$229$	−1220.14	−0.352093	−0.176047	−	0.984382i	$-0.556331\pi$
$229$	−1220.14	−0.352093	−0.176047	+	0.984382i	$0.556331\pi$
$230$	0	0
$231$	−2431.68	−0.692608
$232$	0	0
$233$	−1779.72	−0.500402	−0.250201	−	0.968194i	$-0.580497\pi$
$233$	−1779.72	−0.500402	−0.250201	+	0.968194i	$0.580497\pi$
$234$	0	0
$235$	6440.63	1.78783
$236$	0	0
$237$	1302.09	0.356878
$238$	0	0
$239$	4592.51	1.24295	0.621474	−	0.783435i	$-0.286534\pi$
$239$	4592.51	1.24295	0.621474	+	0.783435i	$0.286534\pi$
$240$	0	0
$241$	4689.42	1.25341	0.626706	−	0.779256i	$-0.284403\pi$
$241$	4689.42	1.25341	0.626706	+	0.779256i	$0.284403\pi$
$242$	0	0
$243$	2346.80	0.619536
$244$	0	0
$245$	−3180.48	−0.829360
$246$	0	0
$247$	−3666.47	−0.944500
$248$	0	0
$249$	−5543.12	−1.41077
$250$	0	0
$251$	5916.59	1.48786	0.743929	−	0.668259i	$-0.232960\pi$
$251$	5916.59	1.48786	0.743929	+	0.668259i	$0.232960\pi$
$252$	0	0
$253$	635.988	0.158040
$254$	0	0
$255$	2962.48	0.727521
$256$	0	0
$257$	−1935.87	−0.469869	−0.234934	−	0.972011i	$-0.575488\pi$
$257$	−1935.87	−0.469869	−0.234934	+	0.972011i	$0.575488\pi$
$258$	0	0
$259$	−3167.32	−0.759874
$260$	0	0
$261$	−251.380	−0.0596170
$262$	0	0
$263$	2237.79	0.524670	0.262335	−	0.964977i	$-0.415507\pi$
$263$	2237.79	0.524670	0.262335	+	0.964977i	$0.415507\pi$
$264$	0	0
$265$	3581.74	0.830281
$266$	0	0
$267$	6718.39	1.53992
$268$	0	0
$269$	−541.272	−0.122684	−0.0613418	−	0.998117i	$-0.519538\pi$
$269$	−541.272	−0.122684	−0.0613418	+	0.998117i	$0.519538\pi$
$270$	0	0
$271$	−262.011	−0.0587308	−0.0293654	−	0.999569i	$-0.509349\pi$
$271$	−262.011	−0.0587308	−0.0293654	+	0.999569i	$0.509349\pi$
$272$	0	0
$273$	1238.50	0.274570
$274$	0	0
$275$	−469.752	−0.103008
$276$	0	0
$277$	1408.32	0.305479	0.152740	−	0.988266i	$-0.451190\pi$
$277$	1408.32	0.305479	0.152740	+	0.988266i	$0.451190\pi$
$278$	0	0
$279$	−1160.84	−0.249096
$280$	0	0
$281$	7108.45	1.50909	0.754545	−	0.656248i	$-0.227858\pi$
$281$	7108.45	1.50909	0.754545	+	0.656248i	$0.227858\pi$
$282$	0	0
$283$	−401.964	−0.0844321	−0.0422161	−	0.999109i	$-0.513442\pi$
$283$	−401.964	−0.0844321	−0.0422161	+	0.999109i	$0.513442\pi$
$284$	0	0
$285$	10162.7	2.11223
$286$	0	0
$287$	3456.31	0.710870
$288$	0	0
$289$	−3084.59	−0.627842
$290$	0	0
$291$	−8812.59	−1.77527
$292$	0	0
$293$	6999.20	1.39555	0.697777	−	0.716315i	$-0.254172\pi$
$293$	6999.20	1.39555	0.697777	+	0.716315i	$0.254172\pi$
$294$	0	0
$295$	7507.17	1.48164
$296$	0	0
$297$	−5372.91	−1.04972
$298$	0	0
$299$	−323.922	−0.0626519
$300$	0	0
$301$	3710.46	0.710523
$302$	0	0
$303$	7069.74	1.34042
$304$	0	0
$305$	4571.55	0.858249
$306$	0	0
$307$	491.992	0.0914641	0.0457321	−	0.998954i	$-0.485438\pi$
$307$	491.992	0.0914641	0.0457321	+	0.998954i	$0.485438\pi$
$308$	0	0
$309$	10502.3	1.93351
$310$	0	0
$311$	−7497.31	−1.36699	−0.683494	−	0.729956i	$-0.739541\pi$
$311$	−7497.31	−1.36699	−0.683494	+	0.729956i	$0.739541\pi$
$312$	0	0
$313$	260.548	0.0470512	0.0235256	−	0.999723i	$-0.492511\pi$
$313$	260.548	0.0470512	0.0235256	+	0.999723i	$0.492511\pi$
$314$	0	0
$315$	−834.275	−0.149226
$316$	0	0
$317$	4675.58	0.828412	0.414206	−	0.910183i	$-0.364059\pi$
$317$	4675.58	0.828412	0.414206	+	0.910183i	$0.364059\pi$
$318$	0	0
$319$	1423.19	0.249791
$320$	0	0
$321$	3301.31	0.574023
$322$	0	0
$323$	6272.31	1.08050
$324$	0	0
$325$	239.255	0.0408353
$326$	0	0
$327$	−4759.80	−0.804947
$328$	0	0
$329$	−4606.28	−0.771892
$330$	0	0
$331$	2097.24	0.348263	0.174131	−	0.984722i	$-0.444288\pi$
$331$	2097.24	0.348263	0.174131	+	0.984722i	$0.444288\pi$
$332$	0	0
$333$	3309.21	0.544576
$334$	0	0
$335$	2511.08	0.409537
$336$	0	0
$337$	7156.62	1.15681	0.578407	−	0.815749i	$-0.303675\pi$
$337$	7156.62	1.15681	0.578407	+	0.815749i	$0.303675\pi$
$338$	0	0
$339$	−7151.54	−1.14578
$340$	0	0
$341$	6572.13	1.04370
$342$	0	0
$343$	5120.37	0.806047
$344$	0	0
$345$	897.846	0.140111
$346$	0	0
$347$	10051.3	1.55499	0.777495	−	0.628890i	$-0.216490\pi$
$347$	10051.3	1.55499	0.777495	+	0.628890i	$0.216490\pi$
$348$	0	0
$349$	−4421.41	−0.678145	−0.339072	−	0.940760i	$-0.610113\pi$
$349$	−4421.41	−0.678145	−0.339072	+	0.940760i	$0.610113\pi$
$350$	0	0
$351$	2736.54	0.416142
$352$	0	0
$353$	3639.14	0.548702	0.274351	−	0.961630i	$-0.411537\pi$
$353$	3639.14	0.548702	0.274351	+	0.961630i	$0.411537\pi$
$354$	0	0
$355$	−10046.3	−1.50199
$356$	0	0
$357$	−2118.74	−0.314105
$358$	0	0
$359$	1969.60	0.289558	0.144779	−	0.989464i	$-0.453753\pi$
$359$	1969.60	0.289558	0.144779	+	0.989464i	$0.453753\pi$
$360$	0	0
$361$	14657.9	2.13704
$362$	0	0
$363$	−6434.62	−0.930385
$364$	0	0
$365$	−4738.60	−0.679533
$366$	0	0
$367$	5654.21	0.804216	0.402108	−	0.915592i	$-0.368278\pi$
$367$	5654.21	0.804216	0.402108	+	0.915592i	$0.368278\pi$
$368$	0	0
$369$	−3611.15	−0.509456
$370$	0	0
$371$	−2561.63	−0.358472
$372$	0	0
$373$	−3292.87	−0.457100	−0.228550	−	0.973532i	$-0.573398\pi$
$373$	−3292.87	−0.457100	−0.228550	+	0.973532i	$0.573398\pi$
$374$	0	0
$375$	7997.05	1.10124
$376$	0	0
$377$	−724.862	−0.0990247
$378$	0	0
$379$	−8871.90	−1.20242	−0.601212	−	0.799089i	$-0.705315\pi$
$379$	−8871.90	−1.20242	−0.601212	+	0.799089i	$0.705315\pi$
$380$	0	0
$381$	−5175.60	−0.695942
$382$	0	0
$383$	2677.25	0.357183	0.178591	−	0.983923i	$-0.442846\pi$
$383$	2677.25	0.357183	0.178591	+	0.983923i	$0.442846\pi$
$384$	0	0
$385$	4723.26	0.625246
$386$	0	0
$387$	−3876.69	−0.509207
$388$	0	0
$389$	7084.53	0.923393	0.461697	−	0.887038i	$-0.347241\pi$
$389$	7084.53	0.923393	0.461697	+	0.887038i	$0.347241\pi$
$390$	0	0
$391$	554.141	0.0716730
$392$	0	0
$393$	4679.25	0.600603
$394$	0	0
$395$	−2529.17	−0.322168
$396$	0	0
$397$	−6074.78	−0.767971	−0.383986	−	0.923339i	$-0.625449\pi$
$397$	−6074.78	−0.767971	−0.383986	+	0.923339i	$0.625449\pi$
$398$	0	0
$399$	−7268.26	−0.911950
$400$	0	0
$401$	−6925.49	−0.862450	−0.431225	−	0.902244i	$-0.641918\pi$
$401$	−6925.49	−0.862450	−0.431225	+	0.902244i	$0.641918\pi$
$402$	0	0
$403$	−3347.33	−0.413752
$404$	0	0
$405$	−10300.2	−1.26375
$406$	0	0
$407$	−18735.2	−2.28174
$408$	0	0
$409$	9424.78	1.13943	0.569713	−	0.821843i	$-0.307054\pi$
$409$	9424.78	1.13943	0.569713	+	0.821843i	$0.307054\pi$
$410$	0	0
$411$	−5989.04	−0.718778
$412$	0	0
$413$	−5369.06	−0.639695
$414$	0	0
$415$	10766.9	1.27356
$416$	0	0
$417$	700.055	0.0822107
$418$	0	0
$419$	−8750.08	−1.02021	−0.510106	−	0.860111i	$-0.670394\pi$
$419$	−8750.08	−1.02021	−0.510106	+	0.860111i	$0.670394\pi$
$420$	0	0
$421$	13107.2	1.51736	0.758678	−	0.651466i	$-0.225845\pi$
$421$	13107.2	1.51736	0.758678	+	0.651466i	$0.225845\pi$
$422$	0	0
$423$	4812.64	0.553188
$424$	0	0
$425$	−409.299	−0.0467150
$426$	0	0
$427$	−3269.53	−0.370547
$428$	0	0
$429$	7325.95	0.824476
$430$	0	0
$431$	12662.7	1.41517	0.707586	−	0.706628i	$-0.249784\pi$
$431$	12662.7	1.41517	0.707586	+	0.706628i	$0.249784\pi$
$432$	0	0
$433$	−9718.95	−1.07867	−0.539333	−	0.842092i	$-0.681324\pi$
$433$	−9718.95	−1.07867	−0.539333	+	0.842092i	$0.681324\pi$
$434$	0	0
$435$	2009.17	0.221453
$436$	0	0
$437$	1900.96	0.208090
$438$	0	0
$439$	−1112.31	−0.120928	−0.0604641	−	0.998170i	$-0.519258\pi$
$439$	−1112.31	−0.120928	−0.0604641	+	0.998170i	$0.519258\pi$
$440$	0	0
$441$	−2376.55	−0.256619
$442$	0	0
$443$	7121.64	0.763791	0.381895	−	0.924206i	$-0.375271\pi$
$443$	7121.64	0.763791	0.381895	+	0.924206i	$0.375271\pi$
$444$	0	0
$445$	−13049.7	−1.39015
$446$	0	0
$447$	18845.0	1.99404
$448$	0	0
$449$	−15242.6	−1.60210	−0.801051	−	0.598596i	$-0.795726\pi$
$449$	−15242.6	−1.60210	−0.801051	+	0.598596i	$0.795726\pi$
$450$	0	0
$451$	20444.6	2.13459
$452$	0	0
$453$	−10500.2	−1.08905
$454$	0	0
$455$	−2405.66	−0.247866
$456$	0	0
$457$	3360.45	0.343972	0.171986	−	0.985099i	$-0.444982\pi$
$457$	3360.45	0.343972	0.171986	+	0.985099i	$0.444982\pi$
$458$	0	0
$459$	−4681.46	−0.476061
$460$	0	0
$461$	−4961.99	−0.501308	−0.250654	−	0.968077i	$-0.580646\pi$
$461$	−4961.99	−0.501308	−0.250654	+	0.968077i	$0.580646\pi$
$462$	0	0
$463$	6420.15	0.644427	0.322214	−	0.946667i	$-0.395573\pi$
$463$	6420.15	0.644427	0.322214	+	0.946667i	$0.395573\pi$
$464$	0	0
$465$	9278.10	0.925293
$466$	0	0
$467$	−10238.4	−1.01451	−0.507255	−	0.861796i	$-0.669340\pi$
$467$	−10238.4	−1.01451	−0.507255	+	0.861796i	$0.669340\pi$
$468$	0	0
$469$	−1795.90	−0.176817
$470$	0	0
$471$	−500.482	−0.0489618
$472$	0	0
$473$	21947.9	2.13355
$474$	0	0
$475$	−1404.08	−0.135629
$476$	0	0
$477$	2676.39	0.256904
$478$	0	0
$479$	−4345.01	−0.414465	−0.207232	−	0.978292i	$-0.566446\pi$
$479$	−4345.01	−0.414465	−0.207232	+	0.978292i	$0.566446\pi$
$480$	0	0
$481$	9542.22	0.904549
$482$	0	0
$483$	−642.131	−0.0604927
$484$	0	0
$485$	17117.5	1.60261
$486$	0	0
$487$	4103.05	0.381780	0.190890	−	0.981611i	$-0.438863\pi$
$487$	4103.05	0.381780	0.190890	+	0.981611i	$0.438863\pi$
$488$	0	0
$489$	1052.97	0.0973760
$490$	0	0
$491$	−15593.8	−1.43328	−0.716640	−	0.697443i	$-0.754321\pi$
$491$	−15593.8	−1.43328	−0.716640	+	0.697443i	$0.754321\pi$
$492$	0	0
$493$	1240.04	0.113283
$494$	0	0
$495$	−4934.86	−0.448092
$496$	0	0
$497$	7185.06	0.648478
$498$	0	0
$499$	971.172	0.0871255	0.0435627	−	0.999051i	$-0.486129\pi$
$499$	971.172	0.0871255	0.0435627	+	0.999051i	$0.486129\pi$
$500$	0	0
$501$	−218.025	−0.0194424
$502$	0	0
$503$	−7922.27	−0.702259	−0.351130	−	0.936327i	$-0.614202\pi$
$503$	−7922.27	−0.702259	−0.351130	+	0.936327i	$0.614202\pi$
$504$	0	0
$505$	−13732.2	−1.21005
$506$	0	0
$507$	9389.86	0.822522
$508$	0	0
$509$	13818.0	1.20328	0.601641	−	0.798767i	$-0.294514\pi$
$509$	13818.0	1.20328	0.601641	+	0.798767i	$0.294514\pi$
$510$	0	0
$511$	3389.00	0.293386
$512$	0	0
$513$	−16059.6	−1.38216
$514$	0	0
$515$	−20399.6	−1.74546
$516$	0	0
$517$	−27246.8	−2.31782
$518$	0	0
$519$	−16351.0	−1.38291
$520$	0	0
$521$	19389.5	1.63046	0.815229	−	0.579139i	$-0.196611\pi$
$521$	19389.5	1.63046	0.815229	+	0.579139i	$0.196611\pi$
$522$	0	0
$523$	−12593.2	−1.05289	−0.526444	−	0.850210i	$-0.676475\pi$
$523$	−12593.2	−1.05289	−0.526444	+	0.850210i	$0.676475\pi$
$524$	0	0
$525$	474.289	0.0394280
$526$	0	0
$527$	5726.35	0.473327
$528$	0	0
$529$	−11999.1	−0.986197
$530$	0	0
$531$	5609.59	0.458448
$532$	0	0
$533$	−10412.9	−0.846214
$534$	0	0
$535$	−6412.44	−0.518194
$536$	0	0
$537$	−21786.7	−1.75077
$538$	0	0
$539$	13454.9	1.07522
$540$	0	0
$541$	−9870.15	−0.784382	−0.392191	−	0.919884i	$-0.628283\pi$
$541$	−9870.15	−0.784382	−0.392191	+	0.919884i	$0.628283\pi$
$542$	0	0
$543$	−21224.2	−1.67738
$544$	0	0
$545$	9245.39	0.726658
$546$	0	0
$547$	320.923	0.0250853	0.0125427	−	0.999921i	$-0.496007\pi$
$547$	320.923	0.0250853	0.0125427	+	0.999921i	$0.496007\pi$
$548$	0	0
$549$	3416.00	0.265558
$550$	0	0
$551$	4253.91	0.328898
$552$	0	0
$553$	1808.84	0.139095
$554$	0	0
$555$	−26449.1	−2.02288
$556$	0	0
$557$	−18256.4	−1.38878	−0.694388	−	0.719601i	$-0.744325\pi$
$557$	−18256.4	−1.38878	−0.694388	+	0.719601i	$0.744325\pi$
$558$	0	0
$559$	−11178.6	−0.845801
$560$	0	0
$561$	−12532.7	−0.943190
$562$	0	0
$563$	10854.0	0.812504	0.406252	−	0.913761i	$-0.366835\pi$
$563$	10854.0	0.812504	0.406252	+	0.913761i	$0.366835\pi$
$564$	0	0
$565$	13891.1	1.03434
$566$	0	0
$567$	7366.57	0.545621
$568$	0	0
$569$	6293.86	0.463712	0.231856	−	0.972750i	$-0.425520\pi$
$569$	6293.86	0.463712	0.231856	+	0.972750i	$0.425520\pi$
$570$	0	0
$571$	−13699.0	−1.00401	−0.502003	−	0.864866i	$-0.667403\pi$
$571$	−13699.0	−1.00401	−0.502003	+	0.864866i	$0.667403\pi$
$572$	0	0
$573$	5788.23	0.422001
$574$	0	0
$575$	−124.047	−0.00899673
$576$	0	0
$577$	−4452.52	−0.321249	−0.160625	−	0.987016i	$-0.551351\pi$
$577$	−4452.52	−0.321249	−0.160625	+	0.987016i	$0.551351\pi$
$578$	0	0
$579$	14217.9	1.02051
$580$	0	0
$581$	−7700.39	−0.549855
$582$	0	0
$583$	−15152.4	−1.07641
$584$	0	0
$585$	2513.43	0.177637
$586$	0	0
$587$	20668.1	1.45326	0.726630	−	0.687029i	$-0.241085\pi$
$587$	20668.1	1.45326	0.726630	+	0.687029i	$0.241085\pi$
$588$	0	0
$589$	19644.0	1.37422
$590$	0	0
$591$	−25399.1	−1.76782
$592$	0	0
$593$	−24666.9	−1.70817	−0.854086	−	0.520131i	$-0.825883\pi$
$593$	−24666.9	−1.70817	−0.854086	+	0.520131i	$0.825883\pi$
$594$	0	0
$595$	4115.42	0.283556
$596$	0	0
$597$	−18956.6	−1.29957
$598$	0	0
$599$	7409.35	0.505405	0.252703	−	0.967544i	$-0.418681\pi$
$599$	7409.35	0.505405	0.252703	+	0.967544i	$0.418681\pi$
$600$	0	0
$601$	−7134.72	−0.484245	−0.242122	−	0.970246i	$-0.577844\pi$
$601$	−7134.72	−0.484245	−0.242122	+	0.970246i	$0.577844\pi$
$602$	0	0
$603$	1876.36	0.126718
$604$	0	0
$605$	12498.5	0.839897
$606$	0	0
$607$	−12820.2	−0.857256	−0.428628	−	0.903481i	$-0.641003\pi$
$607$	−12820.2	−0.857256	−0.428628	+	0.903481i	$0.641003\pi$
$608$	0	0
$609$	−1436.94	−0.0956120
$610$	0	0
$611$	13877.4	0.918854
$612$	0	0
$613$	8748.71	0.576439	0.288220	−	0.957564i	$-0.406937\pi$
$613$	8748.71	0.576439	0.288220	+	0.957564i	$0.406937\pi$
$614$	0	0
$615$	28862.4	1.89243
$616$	0	0
$617$	3192.61	0.208314	0.104157	−	0.994561i	$-0.466786\pi$
$617$	3192.61	0.208314	0.104157	+	0.994561i	$0.466786\pi$
$618$	0	0
$619$	−22520.6	−1.46232	−0.731162	−	0.682204i	$-0.761022\pi$
$619$	−22520.6	−1.46232	−0.731162	+	0.682204i	$0.761022\pi$
$620$	0	0
$621$	−1418.82	−0.0916833
$622$	0	0
$623$	9333.05	0.600194
$624$	0	0
$625$	−16729.9	−1.07071
$626$	0	0
$627$	−42992.9	−2.73839
$628$	0	0
$629$	−16324.1	−1.03479
$630$	0	0
$631$	26068.4	1.64464	0.822318	−	0.569029i	$-0.192681\pi$
$631$	26068.4	1.64464	0.822318	+	0.569029i	$0.192681\pi$
$632$	0	0
$633$	18517.1	1.16270
$634$	0	0
$635$	10053.0	0.628255
$636$	0	0
$637$	−6852.86	−0.426249
$638$	0	0
$639$	−7506.95	−0.464742
$640$	0	0
$641$	−8778.42	−0.540915	−0.270458	−	0.962732i	$-0.587175\pi$
$641$	−8778.42	−0.540915	−0.270458	+	0.962732i	$0.587175\pi$
$642$	0	0
$643$	29378.7	1.80184	0.900920	−	0.433985i	$-0.142893\pi$
$643$	29378.7	1.80184	0.900920	+	0.433985i	$0.142893\pi$
$644$	0	0
$645$	30984.7	1.89150
$646$	0	0
$647$	−15681.1	−0.952841	−0.476420	−	0.879218i	$-0.658066\pi$
$647$	−15681.1	−0.952841	−0.476420	+	0.879218i	$0.658066\pi$
$648$	0	0
$649$	−31758.8	−1.92087
$650$	0	0
$651$	−6635.61	−0.399493
$652$	0	0
$653$	−2047.34	−0.122693	−0.0613466	−	0.998117i	$-0.519540\pi$
$653$	−2047.34	−0.122693	−0.0613466	+	0.998117i	$0.519540\pi$
$654$	0	0
$655$	−9088.93	−0.542189
$656$	0	0
$657$	−3540.83	−0.210260
$658$	0	0
$659$	−24563.6	−1.45199	−0.725994	−	0.687701i	$-0.758620\pi$
$659$	−24563.6	−1.45199	−0.725994	+	0.687701i	$0.758620\pi$
$660$	0	0
$661$	−10428.5	−0.613649	−0.306825	−	0.951766i	$-0.599267\pi$
$661$	−10428.5	−0.613649	−0.306825	+	0.951766i	$0.599267\pi$
$662$	0	0
$663$	6383.16	0.373908
$664$	0	0
$665$	14117.8	0.823255
$666$	0	0
$667$	375.821	0.0218169
$668$	0	0
$669$	−16030.4	−0.926416
$670$	0	0
$671$	−19339.8	−1.11267
$672$	0	0
$673$	−26865.3	−1.53875	−0.769377	−	0.638796i	$-0.779433\pi$
$673$	−26865.3	−1.53875	−0.769377	+	0.638796i	$0.779433\pi$
$674$	0	0
$675$	1047.97	0.0597574
$676$	0	0
$677$	7562.94	0.429346	0.214673	−	0.976686i	$-0.431131\pi$
$677$	7562.94	0.429346	0.214673	+	0.976686i	$0.431131\pi$
$678$	0	0
$679$	−12242.3	−0.691922
$680$	0	0
$681$	7145.27	0.402066
$682$	0	0
$683$	9886.11	0.553853	0.276926	−	0.960891i	$-0.410684\pi$
$683$	9886.11	0.553853	0.276926	+	0.960891i	$0.410684\pi$
$684$	0	0
$685$	11633.1	0.648870
$686$	0	0
$687$	7287.05	0.404685
$688$	0	0
$689$	7717.46	0.426722
$690$	0	0
$691$	−7023.52	−0.386668	−0.193334	−	0.981133i	$-0.561930\pi$
$691$	−7023.52	−0.386668	−0.193334	+	0.981133i	$0.561930\pi$
$692$	0	0
$693$	3529.37	0.193463
$694$	0	0
$695$	−1359.78	−0.0742150
$696$	0	0
$697$	17813.6	0.968058
$698$	0	0
$699$	10629.0	0.575146
$700$	0	0
$701$	−17108.5	−0.921798	−0.460899	−	0.887453i	$-0.652473\pi$
$701$	−17108.5	−0.921798	−0.460899	+	0.887453i	$0.652473\pi$
$702$	0	0
$703$	−55999.3	−3.00434
$704$	0	0
$705$	−38465.3	−2.05488
$706$	0	0
$707$	9821.14	0.522436
$708$	0	0
$709$	26717.3	1.41522	0.707610	−	0.706604i	$-0.249774\pi$
$709$	26717.3	1.41522	0.707610	+	0.706604i	$0.249774\pi$
$710$	0	0
$711$	−1889.88	−0.0996848
$712$	0	0
$713$	1735.50	0.0911569
$714$	0	0
$715$	−14229.8	−0.744288
$716$	0	0
$717$	−27427.8	−1.42860
$718$	0	0
$719$	36534.5	1.89500	0.947501	−	0.319754i	$-0.103600\pi$
$719$	36534.5	1.89500	0.947501	+	0.319754i	$0.103600\pi$
$720$	0	0
$721$	14589.6	0.753598
$722$	0	0
$723$	−28006.6	−1.44063
$724$	0	0
$725$	−277.588	−0.0142198
$726$	0	0
$727$	−1857.29	−0.0947497	−0.0473748	−	0.998877i	$-0.515086\pi$
$727$	−1857.29	−0.0947497	−0.0473748	+	0.998877i	$0.515086\pi$
$728$	0	0
$729$	9957.65	0.505901
$730$	0	0
$731$	19123.4	0.967586
$732$	0	0
$733$	−32915.6	−1.65861	−0.829307	−	0.558793i	$-0.811264\pi$
$733$	−32915.6	−1.65861	−0.829307	+	0.558793i	$0.811264\pi$
$734$	0	0
$735$	18994.7	0.953240
$736$	0	0
$737$	−10623.0	−0.530942
$738$	0	0
$739$	21980.4	1.09413	0.547066	−	0.837090i	$-0.315745\pi$
$739$	21980.4	1.09413	0.547066	+	0.837090i	$0.315745\pi$
$740$	0	0
$741$	21897.2	1.08558
$742$	0	0
$743$	−2611.39	−0.128940	−0.0644702	−	0.997920i	$-0.520536\pi$
$743$	−2611.39	−0.128940	−0.0644702	+	0.997920i	$0.520536\pi$
$744$	0	0
$745$	−36604.3	−1.80010
$746$	0	0
$747$	8045.37	0.394062
$748$	0	0
$749$	4586.11	0.223729
$750$	0	0
$751$	30771.3	1.49515	0.747577	−	0.664175i	$-0.231217\pi$
$751$	30771.3	1.49515	0.747577	+	0.664175i	$0.231217\pi$
$752$	0	0
$753$	−35335.6	−1.71010
$754$	0	0
$755$	20395.4	0.983134
$756$	0	0
$757$	−33898.6	−1.62756	−0.813781	−	0.581171i	$-0.802595\pi$
$757$	−33898.6	−1.62756	−0.813781	+	0.581171i	$0.802595\pi$
$758$	0	0
$759$	−3798.30	−0.181647
$760$	0	0
$761$	−9099.21	−0.433438	−0.216719	−	0.976234i	$-0.569535\pi$
$761$	−9099.21	−0.433438	−0.216719	+	0.976234i	$0.569535\pi$
$762$	0	0
$763$	−6612.21	−0.313733
$764$	0	0
$765$	−4299.79	−0.203215
$766$	0	0
$767$	16175.5	0.761489
$768$	0	0
$769$	26319.0	1.23418	0.617092	−	0.786891i	$-0.288310\pi$
$769$	26319.0	1.23418	0.617092	+	0.786891i	$0.288310\pi$
$770$	0	0
$771$	11561.6	0.540052
$772$	0	0
$773$	18733.0	0.871641	0.435821	−	0.900034i	$-0.356458\pi$
$773$	18733.0	0.871641	0.435821	+	0.900034i	$0.356458\pi$
$774$	0	0
$775$	−1281.87	−0.0594143
$776$	0	0
$777$	18916.1	0.873375
$778$	0	0
$779$	61108.8	2.81059
$780$	0	0
$781$	42500.7	1.94724
$782$	0	0
$783$	−3174.99	−0.144910
$784$	0	0
$785$	972.132	0.0441998
$786$	0	0
$787$	6056.20	0.274308	0.137154	−	0.990550i	$-0.456204\pi$
$787$	6056.20	0.274308	0.137154	+	0.990550i	$0.456204\pi$
$788$	0	0
$789$	−13364.7	−0.603039
$790$	0	0
$791$	−9934.77	−0.446573
$792$	0	0
$793$	9850.16	0.441096
$794$	0	0
$795$	−21391.2	−0.954299
$796$	0	0
$797$	41416.5	1.84071	0.920356	−	0.391083i	$-0.127899\pi$
$797$	41416.5	1.84071	0.920356	+	0.391083i	$0.127899\pi$
$798$	0	0
$799$	−23740.4	−1.05116
$800$	0	0
$801$	−9751.17	−0.430138
$802$	0	0
$803$	20046.5	0.880976
$804$	0	0
$805$	1247.27	0.0546092
$806$	0	0
$807$	3232.63	0.141009
$808$	0	0
$809$	−9219.21	−0.400655	−0.200328	−	0.979729i	$-0.564201\pi$
$809$	−9219.21	−0.400655	−0.200328	+	0.979729i	$0.564201\pi$
$810$	0	0
$811$	23382.5	1.01242	0.506209	−	0.862411i	$-0.331046\pi$
$811$	23382.5	1.01242	0.506209	+	0.862411i	$0.331046\pi$
$812$	0	0
$813$	1564.81	0.0675034
$814$	0	0
$815$	−2045.28	−0.0879053
$816$	0	0
$817$	65602.2	2.80922
$818$	0	0
$819$	−1797.58	−0.0766943
$820$	0	0
$821$	−17550.8	−0.746076	−0.373038	−	0.927816i	$-0.621684\pi$
$821$	−17550.8	−0.746076	−0.373038	+	0.927816i	$0.621684\pi$
$822$	0	0
$823$	28325.0	1.19969	0.599846	−	0.800115i	$-0.295228\pi$
$823$	28325.0	1.19969	0.599846	+	0.800115i	$0.295228\pi$
$824$	0	0
$825$	2805.49	0.118394
$826$	0	0
$827$	40292.3	1.69419	0.847097	−	0.531438i	$-0.178348\pi$
$827$	40292.3	1.69419	0.847097	+	0.531438i	$0.178348\pi$
$828$	0	0
$829$	−34935.5	−1.46364	−0.731820	−	0.681497i	$-0.761329\pi$
$829$	−34935.5	−1.46364	−0.731820	+	0.681497i	$0.761329\pi$
$830$	0	0
$831$	−8410.89	−0.351108
$832$	0	0
$833$	11723.4	0.487623
$834$	0	0
$835$	423.489	0.0175514
$836$	0	0
$837$	−14661.7	−0.605476
$838$	0	0
$839$	−224.685	−0.00924551	−0.00462275	−	0.999989i	$-0.501471\pi$
$839$	−224.685	−0.00924551	−0.00462275	+	0.999989i	$0.501471\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	−42453.7	−1.73450
$844$	0	0
$845$	−18238.8	−0.742524
$846$	0	0
$847$	−8938.83	−0.362623
$848$	0	0
$849$	2400.65	0.0970436
$850$	0	0
$851$	−4947.38	−0.199288
$852$	0	0
$853$	35002.5	1.40500	0.702499	−	0.711685i	$-0.252068\pi$
$853$	35002.5	1.40500	0.702499	+	0.711685i	$0.252068\pi$
$854$	0	0
$855$	−14750.3	−0.589999
$856$	0	0
$857$	8836.80	0.352228	0.176114	−	0.984370i	$-0.443647\pi$
$857$	8836.80	0.352228	0.176114	+	0.984370i	$0.443647\pi$
$858$	0	0
$859$	24870.1	0.987842	0.493921	−	0.869507i	$-0.335563\pi$
$859$	24870.1	0.987842	0.493921	+	0.869507i	$0.335563\pi$
$860$	0	0
$861$	−20642.1	−0.817051
$862$	0	0
$863$	11212.1	0.442255	0.221127	−	0.975245i	$-0.429026\pi$
$863$	11212.1	0.442255	0.221127	+	0.975245i	$0.429026\pi$
$864$	0	0
$865$	31760.0	1.24841
$866$	0	0
$867$	18422.1	0.721621
$868$	0	0
$869$	10699.6	0.417673
$870$	0	0
$871$	5410.54	0.210481
$872$	0	0
$873$	12790.7	0.495877
$874$	0	0
$875$	11109.3	0.429216
$876$	0	0
$877$	−19213.6	−0.739790	−0.369895	−	0.929073i	$-0.620606\pi$
$877$	−19213.6	−0.739790	−0.369895	+	0.929073i	$0.620606\pi$
$878$	0	0
$879$	−41801.3	−1.60401
$880$	0	0
$881$	−39679.5	−1.51741	−0.758704	−	0.651435i	$-0.774167\pi$
$881$	−39679.5	−1.51741	−0.758704	+	0.651435i	$0.774167\pi$
$882$	0	0
$883$	−18154.2	−0.691889	−0.345945	−	0.938255i	$-0.612441\pi$
$883$	−18154.2	−0.691889	−0.345945	+	0.938255i	$0.612441\pi$
$884$	0	0
$885$	−44835.0	−1.70295
$886$	0	0
$887$	−24793.3	−0.938530	−0.469265	−	0.883057i	$-0.655481\pi$
$887$	−24793.3	−0.938530	−0.469265	+	0.883057i	$0.655481\pi$
$888$	0	0
$889$	−7189.83	−0.271248
$890$	0	0
$891$	43574.4	1.63838
$892$	0	0
$893$	−81440.6	−3.05185
$894$	0	0
$895$	42318.2	1.58049
$896$	0	0
$897$	1934.56	0.0720101
$898$	0	0
$899$	3883.64	0.144078
$900$	0	0
$901$	−13202.4	−0.488165
$902$	0	0
$903$	−22159.9	−0.816652
$904$	0	0
$905$	41225.6	1.51424
$906$	0	0
$907$	39258.3	1.43721	0.718605	−	0.695418i	$-0.244781\pi$
$907$	39258.3	1.43721	0.718605	+	0.695418i	$0.244781\pi$
$908$	0	0
$909$	−10261.1	−0.374412
$910$	0	0
$911$	11435.6	0.415894	0.207947	−	0.978140i	$-0.433322\pi$
$911$	11435.6	0.415894	0.207947	+	0.978140i	$0.433322\pi$
$912$	0	0
$913$	−45549.0	−1.65110
$914$	0	0
$915$	−27302.6	−0.986444
$916$	0	0
$917$	6500.32	0.234089
$918$	0	0
$919$	20498.3	0.735774	0.367887	−	0.929871i	$-0.380081\pi$
$919$	20498.3	0.735774	0.367887	+	0.929871i	$0.380081\pi$
$920$	0	0
$921$	−2938.32	−0.105126
$922$	0	0
$923$	−21646.5	−0.771944
$924$	0	0
$925$	3654.22	0.129892
$926$	0	0
$927$	−15243.2	−0.540078
$928$	0	0
$929$	−47185.9	−1.66644	−0.833218	−	0.552945i	$-0.813504\pi$
$929$	−47185.9	−1.66644	−0.833218	+	0.552945i	$0.813504\pi$
$930$	0	0
$931$	40216.6	1.41573
$932$	0	0
$933$	44776.1	1.57117
$934$	0	0
$935$	24343.3	0.851456
$936$	0	0
$937$	7622.01	0.265742	0.132871	−	0.991133i	$-0.457580\pi$
$937$	7622.01	0.265742	0.132871	+	0.991133i	$0.457580\pi$
$938$	0	0
$939$	−1556.07	−0.0540792
$940$	0	0
$941$	−3666.12	−0.127005	−0.0635026	−	0.997982i	$-0.520227\pi$
$941$	−3666.12	−0.127005	−0.0635026	+	0.997982i	$0.520227\pi$
$942$	0	0
$943$	5398.80	0.186436
$944$	0	0
$945$	−10537.1	−0.362722
$946$	0	0
$947$	38276.8	1.31344	0.656720	−	0.754134i	$-0.271943\pi$
$947$	38276.8	1.31344	0.656720	+	0.754134i	$0.271943\pi$
$948$	0	0
$949$	−10210.1	−0.349245
$950$	0	0
$951$	−27923.9	−0.952151
$952$	0	0
$953$	−12882.3	−0.437877	−0.218939	−	0.975739i	$-0.570259\pi$
$953$	−12882.3	−0.437877	−0.218939	+	0.975739i	$0.570259\pi$
$954$	0	0
$955$	−11243.0	−0.380958
$956$	0	0
$957$	−8499.72	−0.287102
$958$	0	0
$959$	−8319.85	−0.280148
$960$	0	0
$961$	−11856.8	−0.398001
$962$	0	0
$963$	−4791.57	−0.160339
$964$	0	0
$965$	−27616.7	−0.921256
$966$	0	0
$967$	−19422.3	−0.645894	−0.322947	−	0.946417i	$-0.604674\pi$
$967$	−19422.3	−0.645894	−0.322947	+	0.946417i	$0.604674\pi$
$968$	0	0
$969$	−37460.0	−1.24189
$970$	0	0
$971$	−13473.3	−0.445291	−0.222645	−	0.974899i	$-0.571469\pi$
$971$	−13473.3	−0.445291	−0.222645	+	0.974899i	$0.571469\pi$
$972$	0	0
$973$	972.502	0.0320421
$974$	0	0
$975$	−1428.90	−0.0469348
$976$	0	0
$977$	−32180.8	−1.05379	−0.526895	−	0.849930i	$-0.676644\pi$
$977$	−32180.8	−1.05379	−0.526895	+	0.849930i	$0.676644\pi$
$978$	0	0
$979$	55206.4	1.80225
$980$	0	0
$981$	6908.44	0.224842
$982$	0	0
$983$	−22162.2	−0.719090	−0.359545	−	0.933128i	$-0.617068\pi$
$983$	−22162.2	−0.719090	−0.359545	+	0.933128i	$0.617068\pi$
$984$	0	0
$985$	49335.0	1.59588
$986$	0	0
$987$	27510.0	0.887187
$988$	0	0
$989$	5795.78	0.186345
$990$	0	0
$991$	18481.7	0.592423	0.296212	−	0.955122i	$-0.404277\pi$
$991$	18481.7	0.592423	0.296212	+	0.955122i	$0.404277\pi$
$992$	0	0
$993$	−12525.3	−0.400282
$994$	0	0
$995$	36821.1	1.17317
$996$	0	0
$997$	−4413.36	−0.140193	−0.0700965	−	0.997540i	$-0.522331\pi$
$997$	−4413.36	−0.140193	−0.0700965	+	0.997540i	$0.522331\pi$
$998$	0	0
$999$	41796.2	1.32370

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1856.4.a.bl.1.2		12
4.3	odd	2		1856.4.a.bj.1.11		12
8.3	odd	2		928.4.a.j.1.2	yes	12
8.5	even	2		928.4.a.h.1.11	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
928.4.a.h.1.11	✓	12	8.5	even	2
928.4.a.j.1.2	yes	12	8.3	odd	2
1856.4.a.bj.1.11		12	4.3	odd	2
1856.4.a.bl.1.2		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)

\(f(q)\)	\(=\)	\(q-5.97229 q^{3} +11.6005 q^{5} -8.29658 q^{7} +8.66827 q^{9} +O(q^{10})\)	\(q-5.97229 q^{3} +11.6005 q^{5} -8.29658 q^{7} +8.66827 q^{9} -49.0756 q^{11} +24.9952 q^{13} -69.2817 q^{15} -42.7599 q^{17} -146.687 q^{19} +49.5496 q^{21} -12.9594 q^{23} +9.57201 q^{25} +109.482 q^{27} -29.0000 q^{29} -133.918 q^{31} +293.094 q^{33} -96.2446 q^{35} +381.762 q^{37} -149.279 q^{39} -416.595 q^{41} -447.227 q^{43} +100.556 q^{45} +555.202 q^{47} -274.167 q^{49} +255.375 q^{51} +308.757 q^{53} -569.302 q^{55} +876.055 q^{57} +647.141 q^{59} +394.081 q^{61} -71.9170 q^{63} +289.958 q^{65} +216.463 q^{67} +77.3971 q^{69} -866.026 q^{71} -408.481 q^{73} -57.1668 q^{75} +407.159 q^{77} -218.022 q^{79} -887.904 q^{81} +928.140 q^{83} -496.037 q^{85} +173.196 q^{87} -1124.93 q^{89} -207.375 q^{91} +799.800 q^{93} -1701.64 q^{95} +1475.58 q^{97} -425.400 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 14 q^{3} + 10 q^{5} - 44 q^{7} + 130 q^{9}+O(q^{10}) \) 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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