LMFDB - Embedded newform 1856.4.a.bl.1.12

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			1.12
Root			$8.74783$ 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)

$f(q)$	$=$	$q+9.74783 q^{3} +20.5968 q^{5} -29.4259 q^{7} +68.0201 q^{9} +O(q^{10})$	$q+9.74783 q^{3} +20.5968 q^{5} -29.4259 q^{7} +68.0201 q^{9} +61.9578 q^{11} -11.1763 q^{13} +200.774 q^{15} -54.9444 q^{17} -78.4587 q^{19} -286.838 q^{21} -56.6544 q^{23} +299.229 q^{25} +399.857 q^{27} -29.0000 q^{29} +151.103 q^{31} +603.954 q^{33} -606.079 q^{35} +272.217 q^{37} -108.945 q^{39} +382.766 q^{41} +29.9746 q^{43} +1401.00 q^{45} -68.4004 q^{47} +522.881 q^{49} -535.588 q^{51} -108.793 q^{53} +1276.13 q^{55} -764.802 q^{57} +617.270 q^{59} +27.0352 q^{61} -2001.55 q^{63} -230.197 q^{65} -621.800 q^{67} -552.257 q^{69} -225.766 q^{71} +265.131 q^{73} +2916.84 q^{75} -1823.16 q^{77} +114.305 q^{79} +2061.20 q^{81} +700.367 q^{83} -1131.68 q^{85} -282.687 q^{87} +80.4339 q^{89} +328.873 q^{91} +1472.92 q^{93} -1616.00 q^{95} +549.014 q^{97} +4214.38 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	9.74783	1.87597	0.937985	−	0.346675i	$-0.112689\pi$
$3$	9.74783	1.87597	0.937985	+	0.346675i	$0.112689\pi$
$4$	0	0
$5$	20.5968	1.84224	0.921118	−	0.389283i	$-0.127277\pi$
$5$	20.5968	1.84224	0.921118	+	0.389283i	$0.127277\pi$
$6$	0	0
$7$	−29.4259	−1.58885	−0.794423	−	0.607364i	$-0.792227\pi$
$7$	−29.4259	−1.58885	−0.794423	+	0.607364i	$0.792227\pi$
$8$	0	0
$9$	68.0201	2.51926
$10$	0	0
$11$	61.9578	1.69827	0.849135	−	0.528175i	$-0.177124\pi$
$11$	61.9578	1.69827	0.849135	+	0.528175i	$0.177124\pi$
$12$	0	0
$13$	−11.1763	−0.238443	−0.119221	−	0.992868i	$-0.538040\pi$
$13$	−11.1763	−0.238443	−0.119221	+	0.992868i	$0.538040\pi$
$14$	0	0
$15$	200.774	3.45598
$16$	0	0
$17$	−54.9444	−0.783880	−0.391940	−	0.919991i	$-0.628196\pi$
$17$	−54.9444	−0.783880	−0.391940	+	0.919991i	$0.628196\pi$
$18$	0	0
$19$	−78.4587	−0.947351	−0.473675	−	0.880699i	$-0.657073\pi$
$19$	−78.4587	−0.947351	−0.473675	+	0.880699i	$0.657073\pi$
$20$	0	0
$21$	−286.838	−2.98063
$22$	0	0
$23$	−56.6544	−0.513620	−0.256810	−	0.966462i	$-0.582671\pi$
$23$	−56.6544	−0.513620	−0.256810	+	0.966462i	$0.582671\pi$
$24$	0	0
$25$	299.229	2.39383
$26$	0	0
$27$	399.857	2.85010
$28$	0	0
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	0	0
$31$	151.103	0.875447	0.437723	−	0.899110i	$-0.355785\pi$
$31$	151.103	0.875447	0.437723	+	0.899110i	$0.355785\pi$
$32$	0	0
$33$	603.954	3.18591
$34$	0	0
$35$	−606.079	−2.92703
$36$	0	0
$37$	272.217	1.20952	0.604759	−	0.796409i	$-0.293269\pi$
$37$	272.217	1.20952	0.604759	+	0.796409i	$0.293269\pi$
$38$	0	0
$39$	−108.945	−0.447311
$40$	0	0
$41$	382.766	1.45800	0.728999	−	0.684515i	$-0.239986\pi$
$41$	382.766	1.45800	0.728999	+	0.684515i	$0.239986\pi$
$42$	0	0
$43$	29.9746	0.106304	0.0531522	−	0.998586i	$-0.483073\pi$
$43$	29.9746	0.106304	0.0531522	+	0.998586i	$0.483073\pi$
$44$	0	0
$45$	1401.00	4.64108
$46$	0	0
$47$	−68.4004	−0.212281	−0.106141	−	0.994351i	$-0.533849\pi$
$47$	−68.4004	−0.212281	−0.106141	+	0.994351i	$0.533849\pi$
$48$	0	0
$49$	522.881	1.52443
$50$	0	0
$51$	−535.588	−1.47054
$52$	0	0
$53$	−108.793	−0.281959	−0.140980	−	0.990012i	$-0.545025\pi$
$53$	−108.793	−0.281959	−0.140980	+	0.990012i	$0.545025\pi$
$54$	0	0
$55$	1276.13	3.12862
$56$	0	0
$57$	−764.802	−1.77720
$58$	0	0
$59$	617.270	1.36206	0.681031	−	0.732255i	$-0.261532\pi$
$59$	617.270	1.36206	0.681031	+	0.732255i	$0.261532\pi$
$60$	0	0
$61$	27.0352	0.0567460	0.0283730	−	0.999597i	$-0.490967\pi$
$61$	27.0352	0.0567460	0.0283730	+	0.999597i	$0.490967\pi$
$62$	0	0
$63$	−2001.55	−4.00273
$64$	0	0
$65$	−230.197	−0.439268
$66$	0	0
$67$	−621.800	−1.13381	−0.566903	−	0.823785i	$-0.691858\pi$
$67$	−621.800	−1.13381	−0.566903	+	0.823785i	$0.691858\pi$
$68$	0	0
$69$	−552.257	−0.963535
$70$	0	0
$71$	−225.766	−0.377374	−0.188687	−	0.982037i	$-0.560423\pi$
$71$	−225.766	−0.377374	−0.188687	+	0.982037i	$0.560423\pi$
$72$	0	0
$73$	265.131	0.425085	0.212543	−	0.977152i	$-0.431826\pi$
$73$	265.131	0.425085	0.212543	+	0.977152i	$0.431826\pi$
$74$	0	0
$75$	2916.84	4.49076
$76$	0	0
$77$	−1823.16	−2.69829
$78$	0	0
$79$	114.305	0.162788	0.0813941	−	0.996682i	$-0.474063\pi$
$79$	114.305	0.162788	0.0813941	+	0.996682i	$0.474063\pi$
$80$	0	0
$81$	2061.20	2.82743
$82$	0	0
$83$	700.367	0.926208	0.463104	−	0.886304i	$-0.346736\pi$
$83$	700.367	0.926208	0.463104	+	0.886304i	$0.346736\pi$
$84$	0	0
$85$	−1131.68	−1.44409
$86$	0	0
$87$	−282.687	−0.348359
$88$	0	0
$89$	80.4339	0.0957975	0.0478988	−	0.998852i	$-0.484748\pi$
$89$	80.4339	0.0957975	0.0478988	+	0.998852i	$0.484748\pi$
$90$	0	0
$91$	328.873	0.378849
$92$	0	0
$93$	1472.92	1.64231
$94$	0	0
$95$	−1616.00	−1.74524
$96$	0	0
$97$	549.014	0.574679	0.287340	−	0.957829i	$-0.407229\pi$
$97$	549.014	0.574679	0.287340	+	0.957829i	$0.407229\pi$
$98$	0	0
$99$	4214.38	4.27839
$100$	0	0
$101$	1033.40	1.01809	0.509044	−	0.860741i	$-0.329999\pi$
$101$	1033.40	1.01809	0.509044	+	0.860741i	$0.329999\pi$
$102$	0	0
$103$	−1994.16	−1.90767	−0.953837	−	0.300326i	$-0.902904\pi$
$103$	−1994.16	−1.90767	−0.953837	+	0.300326i	$0.902904\pi$
$104$	0	0
$105$	−5907.96	−5.49102
$106$	0	0
$107$	−519.290	−0.469175	−0.234587	−	0.972095i	$-0.575374\pi$
$107$	−519.290	−0.469175	−0.234587	+	0.972095i	$0.575374\pi$
$108$	0	0
$109$	−791.977	−0.695942	−0.347971	−	0.937505i	$-0.613129\pi$
$109$	−791.977	−0.695942	−0.347971	+	0.937505i	$0.613129\pi$
$110$	0	0
$111$	2653.52	2.26902
$112$	0	0
$113$	−1125.51	−0.936980	−0.468490	−	0.883469i	$-0.655202\pi$
$113$	−1125.51	−0.936980	−0.468490	+	0.883469i	$0.655202\pi$
$114$	0	0
$115$	−1166.90	−0.946209
$116$	0	0
$117$	−760.215	−0.600700
$118$	0	0
$119$	1616.78	1.24547
$120$	0	0
$121$	2507.77	1.88412
$122$	0	0
$123$	3731.13	2.73516
$124$	0	0
$125$	3588.57	2.56777
$126$	0	0
$127$	1507.96	1.05362	0.526809	−	0.849984i	$-0.323388\pi$
$127$	1507.96	1.05362	0.526809	+	0.849984i	$0.323388\pi$
$128$	0	0
$129$	292.187	0.199424
$130$	0	0
$131$	1005.27	0.670464	0.335232	−	0.942136i	$-0.391185\pi$
$131$	1005.27	0.670464	0.335232	+	0.942136i	$0.391185\pi$
$132$	0	0
$133$	2308.72	1.50520
$134$	0	0
$135$	8235.79	5.25055
$136$	0	0
$137$	1355.15	0.845100	0.422550	−	0.906340i	$-0.361135\pi$
$137$	1355.15	0.845100	0.422550	+	0.906340i	$0.361135\pi$
$138$	0	0
$139$	−1874.71	−1.14396	−0.571981	−	0.820267i	$-0.693825\pi$
$139$	−1874.71	−1.14396	−0.571981	+	0.820267i	$0.693825\pi$
$140$	0	0
$141$	−666.756	−0.398234
$142$	0	0
$143$	−692.460	−0.404940
$144$	0	0
$145$	−597.308	−0.342095
$146$	0	0
$147$	5096.95	2.85979
$148$	0	0
$149$	−2198.14	−1.20858	−0.604290	−	0.796765i	$-0.706543\pi$
$149$	−2198.14	−1.20858	−0.604290	+	0.796765i	$0.706543\pi$
$150$	0	0
$151$	30.1284	0.0162372	0.00811859	−	0.999967i	$-0.497416\pi$
$151$	30.1284	0.0162372	0.00811859	+	0.999967i	$0.497416\pi$
$152$	0	0
$153$	−3737.32	−1.97480
$154$	0	0
$155$	3112.24	1.61278
$156$	0	0
$157$	747.880	0.380174	0.190087	−	0.981767i	$-0.439123\pi$
$157$	747.880	0.380174	0.190087	+	0.981767i	$0.439123\pi$
$158$	0	0
$159$	−1060.49	−0.528947
$160$	0	0
$161$	1667.10	0.816063
$162$	0	0
$163$	−3018.90	−1.45067	−0.725334	−	0.688397i	$-0.758315\pi$
$163$	−3018.90	−1.45067	−0.725334	+	0.688397i	$0.758315\pi$
$164$	0	0
$165$	12439.5	5.86919
$166$	0	0
$167$	−1112.20	−0.515356	−0.257678	−	0.966231i	$-0.582957\pi$
$167$	−1112.20	−0.515356	−0.257678	+	0.966231i	$0.582957\pi$
$168$	0	0
$169$	−2072.09	−0.943145
$170$	0	0
$171$	−5336.77	−2.38663
$172$	0	0
$173$	−1795.92	−0.789257	−0.394628	−	0.918841i	$-0.629127\pi$
$173$	−1795.92	−0.789257	−0.394628	+	0.918841i	$0.629127\pi$
$174$	0	0
$175$	−8805.08	−3.80344
$176$	0	0
$177$	6017.04	2.55519
$178$	0	0
$179$	−2323.30	−0.970121	−0.485061	−	0.874481i	$-0.661202\pi$
$179$	−2323.30	−0.970121	−0.485061	+	0.874481i	$0.661202\pi$
$180$	0	0
$181$	−754.107	−0.309681	−0.154841	−	0.987939i	$-0.549486\pi$
$181$	−754.107	−0.309681	−0.154841	+	0.987939i	$0.549486\pi$
$182$	0	0
$183$	263.535	0.106454
$184$	0	0
$185$	5606.80	2.22822
$186$	0	0
$187$	−3404.23	−1.33124
$188$	0	0
$189$	−11766.1	−4.52837
$190$	0	0
$191$	511.294	0.193696	0.0968480	−	0.995299i	$-0.469124\pi$
$191$	511.294	0.193696	0.0968480	+	0.995299i	$0.469124\pi$
$192$	0	0
$193$	−3198.38	−1.19287	−0.596436	−	0.802660i	$-0.703417\pi$
$193$	−3198.38	−1.19287	−0.596436	+	0.802660i	$0.703417\pi$
$194$	0	0
$195$	−2243.92	−0.824053
$196$	0	0
$197$	−1510.59	−0.546321	−0.273161	−	0.961968i	$-0.588069\pi$
$197$	−1510.59	−0.546321	−0.273161	+	0.961968i	$0.588069\pi$
$198$	0	0
$199$	−4707.50	−1.67691	−0.838457	−	0.544968i	$-0.816542\pi$
$199$	−4707.50	−1.67691	−0.838457	+	0.544968i	$0.816542\pi$
$200$	0	0
$201$	−6061.20	−2.12698
$202$	0	0
$203$	853.350	0.295041
$204$	0	0
$205$	7883.76	2.68598
$206$	0	0
$207$	−3853.64	−1.29394
$208$	0	0
$209$	−4861.13	−1.60886
$210$	0	0
$211$	−4482.02	−1.46235	−0.731173	−	0.682192i	$-0.761027\pi$
$211$	−4482.02	−1.46235	−0.731173	+	0.682192i	$0.761027\pi$
$212$	0	0
$213$	−2200.73	−0.707942
$214$	0	0
$215$	617.382	0.195838
$216$	0	0
$217$	−4446.33	−1.39095
$218$	0	0
$219$	2584.45	0.797447
$220$	0	0
$221$	614.076	0.186910
$222$	0	0
$223$	1320.02	0.396392	0.198196	−	0.980162i	$-0.436492\pi$
$223$	1320.02	0.396392	0.198196	+	0.980162i	$0.436492\pi$
$224$	0	0
$225$	20353.6	6.03070
$226$	0	0
$227$	4007.58	1.17177	0.585886	−	0.810393i	$-0.300747\pi$
$227$	4007.58	1.17177	0.585886	+	0.810393i	$0.300747\pi$
$228$	0	0
$229$	4270.66	1.23237	0.616186	−	0.787600i	$-0.288677\pi$
$229$	4270.66	1.23237	0.616186	+	0.787600i	$0.288677\pi$
$230$	0	0
$231$	−17771.9	−5.06192
$232$	0	0
$233$	−1311.96	−0.368882	−0.184441	−	0.982844i	$-0.559047\pi$
$233$	−1311.96	−0.368882	−0.184441	+	0.982844i	$0.559047\pi$
$234$	0	0
$235$	−1408.83	−0.391073
$236$	0	0
$237$	1114.22	0.305386
$238$	0	0
$239$	6527.86	1.76674	0.883372	−	0.468672i	$-0.155267\pi$
$239$	6527.86	1.76674	0.883372	+	0.468672i	$0.155267\pi$
$240$	0	0
$241$	−5532.77	−1.47883	−0.739413	−	0.673252i	$-0.764897\pi$
$241$	−5532.77	−1.47883	−0.739413	+	0.673252i	$0.764897\pi$
$242$	0	0
$243$	9296.04	2.45408
$244$	0	0
$245$	10769.7	2.80837
$246$	0	0
$247$	876.880	0.225889
$248$	0	0
$249$	6827.06	1.73754
$250$	0	0
$251$	−7055.49	−1.77426	−0.887129	−	0.461522i	$-0.847303\pi$
$251$	−7055.49	−1.77426	−0.887129	+	0.461522i	$0.847303\pi$
$252$	0	0
$253$	−3510.18	−0.872265
$254$	0	0
$255$	−11031.4	−2.70907
$256$	0	0
$257$	3010.05	0.730590	0.365295	−	0.930892i	$-0.380968\pi$
$257$	3010.05	0.730590	0.365295	+	0.930892i	$0.380968\pi$
$258$	0	0
$259$	−8010.21	−1.92174
$260$	0	0
$261$	−1972.58	−0.467816
$262$	0	0
$263$	1206.91	0.282970	0.141485	−	0.989940i	$-0.454812\pi$
$263$	1206.91	0.282970	0.141485	+	0.989940i	$0.454812\pi$
$264$	0	0
$265$	−2240.79	−0.519436
$266$	0	0
$267$	784.056	0.179713
$268$	0	0
$269$	603.567	0.136803	0.0684017	−	0.997658i	$-0.478210\pi$
$269$	603.567	0.136803	0.0684017	+	0.997658i	$0.478210\pi$
$270$	0	0
$271$	−113.696	−0.0254853	−0.0127426	−	0.999919i	$-0.504056\pi$
$271$	−113.696	−0.0254853	−0.0127426	+	0.999919i	$0.504056\pi$
$272$	0	0
$273$	3205.80	0.710709
$274$	0	0
$275$	18539.6	4.06538
$276$	0	0
$277$	2241.91	0.486294	0.243147	−	0.969989i	$-0.421820\pi$
$277$	2241.91	0.486294	0.243147	+	0.969989i	$0.421820\pi$
$278$	0	0
$279$	10278.0	2.20548
$280$	0	0
$281$	7146.06	1.51708	0.758538	−	0.651629i	$-0.225914\pi$
$281$	7146.06	1.51708	0.758538	+	0.651629i	$0.225914\pi$
$282$	0	0
$283$	−3603.35	−0.756879	−0.378439	−	0.925626i	$-0.623539\pi$
$283$	−3603.35	−0.756879	−0.378439	+	0.925626i	$0.623539\pi$
$284$	0	0
$285$	−15752.5	−3.27403
$286$	0	0
$287$	−11263.2	−2.31654
$288$	0	0
$289$	−1894.12	−0.385532
$290$	0	0
$291$	5351.69	1.07808
$292$	0	0
$293$	−3720.10	−0.741743	−0.370872	−	0.928684i	$-0.620941\pi$
$293$	−3720.10	−0.741743	−0.370872	+	0.928684i	$0.620941\pi$
$294$	0	0
$295$	12713.8	2.50924
$296$	0	0
$297$	24774.3	4.84023
$298$	0	0
$299$	633.187	0.122469
$300$	0	0
$301$	−882.029	−0.168901
$302$	0	0
$303$	10073.4	1.90990
$304$	0	0
$305$	556.840	0.104540
$306$	0	0
$307$	−5780.96	−1.07471	−0.537357	−	0.843355i	$-0.680577\pi$
$307$	−5780.96	−1.07471	−0.537357	+	0.843355i	$0.680577\pi$
$308$	0	0
$309$	−19438.7	−3.57874
$310$	0	0
$311$	3569.02	0.650741	0.325371	−	0.945587i	$-0.394511\pi$
$311$	3569.02	0.650741	0.325371	+	0.945587i	$0.394511\pi$
$312$	0	0
$313$	7087.11	1.27983	0.639915	−	0.768445i	$-0.278969\pi$
$313$	7087.11	1.27983	0.639915	+	0.768445i	$0.278969\pi$
$314$	0	0
$315$	−41225.6	−7.37397
$316$	0	0
$317$	7977.84	1.41350	0.706751	−	0.707462i	$-0.250160\pi$
$317$	7977.84	1.41350	0.706751	+	0.707462i	$0.250160\pi$
$318$	0	0
$319$	−1796.78	−0.315361
$320$	0	0
$321$	−5061.95	−0.880158
$322$	0	0
$323$	4310.86	0.742610
$324$	0	0
$325$	−3344.28	−0.570792
$326$	0	0
$327$	−7720.06	−1.30557
$328$	0	0
$329$	2012.74	0.337283
$330$	0	0
$331$	−4890.63	−0.812126	−0.406063	−	0.913845i	$-0.633099\pi$
$331$	−4890.63	−0.812126	−0.406063	+	0.913845i	$0.633099\pi$
$332$	0	0
$333$	18516.2	3.04710
$334$	0	0
$335$	−12807.1	−2.08874
$336$	0	0
$337$	1788.10	0.289033	0.144517	−	0.989502i	$-0.453837\pi$
$337$	1788.10	0.289033	0.144517	+	0.989502i	$0.453837\pi$
$338$	0	0
$339$	−10971.2	−1.75775
$340$	0	0
$341$	9361.99	1.48675
$342$	0	0
$343$	−5293.15	−0.833246
$344$	0	0
$345$	−11374.7	−1.77506
$346$	0	0
$347$	−7586.55	−1.17368	−0.586841	−	0.809702i	$-0.699629\pi$
$347$	−7586.55	−1.17368	−0.586841	+	0.809702i	$0.699629\pi$
$348$	0	0
$349$	−5871.25	−0.900518	−0.450259	−	0.892898i	$-0.648668\pi$
$349$	−5871.25	−0.900518	−0.450259	+	0.892898i	$0.648668\pi$
$350$	0	0
$351$	−4468.93	−0.679584
$352$	0	0
$353$	−8959.32	−1.35087	−0.675434	−	0.737420i	$-0.736044\pi$
$353$	−8959.32	−1.35087	−0.675434	+	0.737420i	$0.736044\pi$
$354$	0	0
$355$	−4650.07	−0.695211
$356$	0	0
$357$	15760.1	2.33646
$358$	0	0
$359$	3887.18	0.571469	0.285734	−	0.958309i	$-0.407762\pi$
$359$	3887.18	0.571469	0.285734	+	0.958309i	$0.407762\pi$
$360$	0	0
$361$	−703.228	−0.102526
$362$	0	0
$363$	24445.3	3.53456
$364$	0	0
$365$	5460.85	0.783107
$366$	0	0
$367$	−876.255	−0.124633	−0.0623163	−	0.998056i	$-0.519849\pi$
$367$	−876.255	−0.124633	−0.0623163	+	0.998056i	$0.519849\pi$
$368$	0	0
$369$	26035.8	3.67308
$370$	0	0
$371$	3201.32	0.447990
$372$	0	0
$373$	−13518.8	−1.87662	−0.938310	−	0.345795i	$-0.887609\pi$
$373$	−13518.8	−1.87662	−0.938310	+	0.345795i	$0.887609\pi$
$374$	0	0
$375$	34980.8	4.81706
$376$	0	0
$377$	324.113	0.0442777
$378$	0	0
$379$	914.215	0.123905	0.0619526	−	0.998079i	$-0.480267\pi$
$379$	914.215	0.123905	0.0619526	+	0.998079i	$0.480267\pi$
$380$	0	0
$381$	14699.3	1.97656
$382$	0	0
$383$	−11101.1	−1.48105	−0.740523	−	0.672031i	$-0.765422\pi$
$383$	−11101.1	−1.48105	−0.740523	+	0.672031i	$0.765422\pi$
$384$	0	0
$385$	−37551.3	−4.97089
$386$	0	0
$387$	2038.88	0.267809
$388$	0	0
$389$	12552.2	1.63604	0.818021	−	0.575189i	$-0.195071\pi$
$389$	12552.2	1.63604	0.818021	+	0.575189i	$0.195071\pi$
$390$	0	0
$391$	3112.84	0.402616
$392$	0	0
$393$	9799.19	1.25777
$394$	0	0
$395$	2354.31	0.299894
$396$	0	0
$397$	−909.419	−0.114968	−0.0574842	−	0.998346i	$-0.518308\pi$
$397$	−909.419	−0.114968	−0.0574842	+	0.998346i	$0.518308\pi$
$398$	0	0
$399$	22505.0	2.82370
$400$	0	0
$401$	9299.03	1.15803	0.579017	−	0.815316i	$-0.303437\pi$
$401$	9299.03	1.15803	0.579017	+	0.815316i	$0.303437\pi$
$402$	0	0
$403$	−1688.77	−0.208744
$404$	0	0
$405$	42454.1	5.20879
$406$	0	0
$407$	16866.0	2.05409
$408$	0	0
$409$	−12374.1	−1.49600	−0.747998	−	0.663702i	$-0.768985\pi$
$409$	−12374.1	−1.49600	−0.747998	+	0.663702i	$0.768985\pi$
$410$	0	0
$411$	13209.8	1.58538
$412$	0	0
$413$	−18163.7	−2.16411
$414$	0	0
$415$	14425.3	1.70629
$416$	0	0
$417$	−18274.3	−2.14604
$418$	0	0
$419$	−2301.22	−0.268310	−0.134155	−	0.990960i	$-0.542832\pi$
$419$	−2301.22	−0.268310	−0.134155	+	0.990960i	$0.542832\pi$
$420$	0	0
$421$	12150.3	1.40658	0.703291	−	0.710902i	$-0.251713\pi$
$421$	12150.3	1.40658	0.703291	+	0.710902i	$0.251713\pi$
$422$	0	0
$423$	−4652.61	−0.534793
$424$	0	0
$425$	−16441.0	−1.87648
$426$	0	0
$427$	−795.534	−0.0901607
$428$	0	0
$429$	−6749.98	−0.759656
$430$	0	0
$431$	−8614.70	−0.962774	−0.481387	−	0.876508i	$-0.659867\pi$
$431$	−8614.70	−0.962774	−0.481387	+	0.876508i	$0.659867\pi$
$432$	0	0
$433$	−1358.43	−0.150767	−0.0753833	−	0.997155i	$-0.524018\pi$
$433$	−1358.43	−0.150767	−0.0753833	+	0.997155i	$0.524018\pi$
$434$	0	0
$435$	−5822.46	−0.641759
$436$	0	0
$437$	4445.03	0.486578
$438$	0	0
$439$	−4.35089	−0.000473023 0	−0.000236511	−	1.00000i	$-0.500075\pi$
$439$	−4.35089	−0.000473023 0	−0.000236511		1.00000i	$0.500075\pi$
$440$	0	0
$441$	35566.4	3.84045
$442$	0	0
$443$	−8144.24	−0.873464	−0.436732	−	0.899592i	$-0.643864\pi$
$443$	−8144.24	−0.873464	−0.436732	+	0.899592i	$0.643864\pi$
$444$	0	0
$445$	1656.68	0.176482
$446$	0	0
$447$	−21427.0	−2.26726
$448$	0	0
$449$	321.807	0.0338241	0.0169120	−	0.999857i	$-0.494616\pi$
$449$	321.807	0.0338241	0.0169120	+	0.999857i	$0.494616\pi$
$450$	0	0
$451$	23715.3	2.47608
$452$	0	0
$453$	293.686	0.0304605
$454$	0	0
$455$	6773.74	0.697929
$456$	0	0
$457$	17921.7	1.83444	0.917222	−	0.398377i	$-0.130426\pi$
$457$	17921.7	1.83444	0.917222	+	0.398377i	$0.130426\pi$
$458$	0	0
$459$	−21969.9	−2.23413
$460$	0	0
$461$	1475.58	0.149078	0.0745388	−	0.997218i	$-0.476252\pi$
$461$	1475.58	0.149078	0.0745388	+	0.997218i	$0.476252\pi$
$462$	0	0
$463$	−15103.3	−1.51600	−0.758000	−	0.652254i	$-0.773823\pi$
$463$	−15103.3	−1.51600	−0.758000	+	0.652254i	$0.773823\pi$
$464$	0	0
$465$	30337.6	3.02553
$466$	0	0
$467$	1755.45	0.173946	0.0869729	−	0.996211i	$-0.472281\pi$
$467$	1755.45	0.173946	0.0869729	+	0.996211i	$0.472281\pi$
$468$	0	0
$469$	18297.0	1.80144
$470$	0	0
$471$	7290.20	0.713195
$472$	0	0
$473$	1857.16	0.180534
$474$	0	0
$475$	−23477.2	−2.26780
$476$	0	0
$477$	−7400.11	−0.710330
$478$	0	0
$479$	−683.538	−0.0652018	−0.0326009	−	0.999468i	$-0.510379\pi$
$479$	−683.538	−0.0652018	−0.0326009	+	0.999468i	$0.510379\pi$
$480$	0	0
$481$	−3042.38	−0.288401
$482$	0	0
$483$	16250.6	1.53091
$484$	0	0
$485$	11307.9	1.05870
$486$	0	0
$487$	11775.5	1.09569	0.547844	−	0.836581i	$-0.315449\pi$
$487$	11775.5	1.09569	0.547844	+	0.836581i	$0.315449\pi$
$488$	0	0
$489$	−29427.8	−2.72141
$490$	0	0
$491$	−1490.36	−0.136984	−0.0684921	−	0.997652i	$-0.521819\pi$
$491$	−1490.36	−0.136984	−0.0684921	+	0.997652i	$0.521819\pi$
$492$	0	0
$493$	1593.39	0.145563
$494$	0	0
$495$	86802.8	7.88181
$496$	0	0
$497$	6643.37	0.599589
$498$	0	0
$499$	6051.89	0.542926	0.271463	−	0.962449i	$-0.412493\pi$
$499$	6051.89	0.542926	0.271463	+	0.962449i	$0.412493\pi$
$500$	0	0
$501$	−10841.5	−0.966793
$502$	0	0
$503$	8442.96	0.748416	0.374208	−	0.927345i	$-0.377915\pi$
$503$	8442.96	0.748416	0.374208	+	0.927345i	$0.377915\pi$
$504$	0	0
$505$	21284.7	1.87556
$506$	0	0
$507$	−20198.4	−1.76931
$508$	0	0
$509$	−1151.07	−0.100237	−0.0501183	−	0.998743i	$-0.515960\pi$
$509$	−1151.07	−0.100237	−0.0501183	+	0.998743i	$0.515960\pi$
$510$	0	0
$511$	−7801.70	−0.675395
$512$	0	0
$513$	−31372.3	−2.70004
$514$	0	0
$515$	−41073.3	−3.51438
$516$	0	0
$517$	−4237.94	−0.360511
$518$	0	0
$519$	−17506.3	−1.48062
$520$	0	0
$521$	2071.19	0.174166	0.0870829	−	0.996201i	$-0.472245\pi$
$521$	2071.19	0.174166	0.0870829	+	0.996201i	$0.472245\pi$
$522$	0	0
$523$	11250.3	0.940613	0.470306	−	0.882503i	$-0.344143\pi$
$523$	11250.3	0.940613	0.470306	+	0.882503i	$0.344143\pi$
$524$	0	0
$525$	−85830.4	−7.13513
$526$	0	0
$527$	−8302.24	−0.686246
$528$	0	0
$529$	−8957.28	−0.736195
$530$	0	0
$531$	41986.8	3.43139
$532$	0	0
$533$	−4277.91	−0.347649
$534$	0	0
$535$	−10695.7	−0.864331
$536$	0	0
$537$	−22647.1	−1.81992
$538$	0	0
$539$	32396.6	2.58890
$540$	0	0
$541$	−2757.73	−0.219158	−0.109579	−	0.993978i	$-0.534950\pi$
$541$	−2757.73	−0.219158	−0.109579	+	0.993978i	$0.534950\pi$
$542$	0	0
$543$	−7350.91	−0.580953
$544$	0	0
$545$	−16312.2	−1.28209
$546$	0	0
$547$	−23518.5	−1.83835	−0.919174	−	0.393851i	$-0.871143\pi$
$547$	−23518.5	−1.83835	−0.919174	+	0.393851i	$0.871143\pi$
$548$	0	0
$549$	1838.94	0.142958
$550$	0	0
$551$	2275.30	0.175919
$552$	0	0
$553$	−3363.51	−0.258646
$554$	0	0
$555$	54654.1	4.18007
$556$	0	0
$557$	20884.4	1.58869	0.794346	−	0.607466i	$-0.207814\pi$
$557$	20884.4	1.58869	0.794346	+	0.607466i	$0.207814\pi$
$558$	0	0
$559$	−335.006	−0.0253475
$560$	0	0
$561$	−33183.9	−2.49737
$562$	0	0
$563$	7596.28	0.568641	0.284321	−	0.958729i	$-0.408232\pi$
$563$	7596.28	0.568641	0.284321	+	0.958729i	$0.408232\pi$
$564$	0	0
$565$	−23181.9	−1.72614
$566$	0	0
$567$	−60652.5	−4.49235
$568$	0	0
$569$	9129.38	0.672625	0.336313	−	0.941750i	$-0.390820\pi$
$569$	9129.38	0.672625	0.336313	+	0.941750i	$0.390820\pi$
$570$	0	0
$571$	−14309.7	−1.04876	−0.524379	−	0.851485i	$-0.675702\pi$
$571$	−14309.7	−1.04876	−0.524379	+	0.851485i	$0.675702\pi$
$572$	0	0
$573$	4984.01	0.363368
$574$	0	0
$575$	−16952.6	−1.22952
$576$	0	0
$577$	−4202.76	−0.303229	−0.151614	−	0.988440i	$-0.548447\pi$
$577$	−4202.76	−0.303229	−0.151614	+	0.988440i	$0.548447\pi$
$578$	0	0
$579$	−31177.3	−2.23779
$580$	0	0
$581$	−20608.9	−1.47160
$582$	0	0
$583$	−6740.57	−0.478843
$584$	0	0
$585$	−15658.0	−1.10663
$586$	0	0
$587$	11189.6	0.786786	0.393393	−	0.919370i	$-0.371301\pi$
$587$	11189.6	0.786786	0.393393	+	0.919370i	$0.371301\pi$
$588$	0	0
$589$	−11855.3	−0.829356
$590$	0	0
$591$	−14725.0	−1.02488
$592$	0	0
$593$	−21100.7	−1.46122	−0.730608	−	0.682798i	$-0.760763\pi$
$593$	−21100.7	−1.46122	−0.730608	+	0.682798i	$0.760763\pi$
$594$	0	0
$595$	33300.6	2.29444
$596$	0	0
$597$	−45887.9	−3.14584
$598$	0	0
$599$	326.913	0.0222993	0.0111497	−	0.999938i	$-0.496451\pi$
$599$	326.913	0.0222993	0.0111497	+	0.999938i	$0.496451\pi$
$600$	0	0
$601$	8171.63	0.554622	0.277311	−	0.960780i	$-0.410557\pi$
$601$	8171.63	0.554622	0.277311	+	0.960780i	$0.410557\pi$
$602$	0	0
$603$	−42294.9	−2.85636
$604$	0	0
$605$	51652.1	3.47100
$606$	0	0
$607$	12295.5	0.822175	0.411087	−	0.911596i	$-0.365149\pi$
$607$	12295.5	0.822175	0.411087	+	0.911596i	$0.365149\pi$
$608$	0	0
$609$	8318.31	0.553489
$610$	0	0
$611$	764.465	0.0506169
$612$	0	0
$613$	−678.776	−0.0447235	−0.0223617	−	0.999750i	$-0.507119\pi$
$613$	−678.776	−0.0447235	−0.0223617	+	0.999750i	$0.507119\pi$
$614$	0	0
$615$	76849.5	5.03881
$616$	0	0
$617$	−7924.46	−0.517061	−0.258530	−	0.966003i	$-0.583238\pi$
$617$	−7924.46	−0.517061	−0.258530	+	0.966003i	$0.583238\pi$
$618$	0	0
$619$	7166.94	0.465369	0.232685	−	0.972552i	$-0.425249\pi$
$619$	7166.94	0.465369	0.232685	+	0.972552i	$0.425249\pi$
$620$	0	0
$621$	−22653.7	−1.46387
$622$	0	0
$623$	−2366.84	−0.152208
$624$	0	0
$625$	36509.5	2.33661
$626$	0	0
$627$	−47385.5	−3.01817
$628$	0	0
$629$	−14956.8	−0.948117
$630$	0	0
$631$	−5408.21	−0.341200	−0.170600	−	0.985340i	$-0.554571\pi$
$631$	−5408.21	−0.341200	−0.170600	+	0.985340i	$0.554571\pi$
$632$	0	0
$633$	−43690.0	−2.74332
$634$	0	0
$635$	31059.1	1.94101
$636$	0	0
$637$	−5843.89	−0.363490
$638$	0	0
$639$	−15356.7	−0.950704
$640$	0	0
$641$	1566.31	0.0965143	0.0482571	−	0.998835i	$-0.484633\pi$
$641$	1566.31	0.0965143	0.0482571	+	0.998835i	$0.484633\pi$
$642$	0	0
$643$	14290.9	0.876481	0.438241	−	0.898858i	$-0.355602\pi$
$643$	14290.9	0.876481	0.438241	+	0.898858i	$0.355602\pi$
$644$	0	0
$645$	6018.13	0.367386
$646$	0	0
$647$	−24303.5	−1.47677	−0.738384	−	0.674380i	$-0.764411\pi$
$647$	−24303.5	−1.47677	−0.738384	+	0.674380i	$0.764411\pi$
$648$	0	0
$649$	38244.7	2.31315
$650$	0	0
$651$	−43342.0	−2.60938
$652$	0	0
$653$	28275.5	1.69450	0.847248	−	0.531197i	$-0.178258\pi$
$653$	28275.5	1.69450	0.847248	+	0.531197i	$0.178258\pi$
$654$	0	0
$655$	20705.3	1.23515
$656$	0	0
$657$	18034.2	1.07090
$658$	0	0
$659$	16179.5	0.956391	0.478196	−	0.878253i	$-0.341291\pi$
$659$	16179.5	0.956391	0.478196	+	0.878253i	$0.341291\pi$
$660$	0	0
$661$	32415.8	1.90745	0.953727	−	0.300673i	$-0.0972113\pi$
$661$	32415.8	1.90745	0.953727	+	0.300673i	$0.0972113\pi$
$662$	0	0
$663$	5985.91	0.350638
$664$	0	0
$665$	47552.2	2.77293
$666$	0	0
$667$	1642.98	0.0953768
$668$	0	0
$669$	12867.4	0.743619
$670$	0	0
$671$	1675.04	0.0963700
$672$	0	0
$673$	−12237.1	−0.700899	−0.350449	−	0.936582i	$-0.613971\pi$
$673$	−12237.1	−0.700899	−0.350449	+	0.936582i	$0.613971\pi$
$674$	0	0
$675$	119649.	6.82266
$676$	0	0
$677$	−12849.2	−0.729448	−0.364724	−	0.931116i	$-0.618837\pi$
$677$	−12849.2	−0.729448	−0.364724	+	0.931116i	$0.618837\pi$
$678$	0	0
$679$	−16155.2	−0.913077
$680$	0	0
$681$	39065.2	2.19821
$682$	0	0
$683$	−7311.10	−0.409592	−0.204796	−	0.978805i	$-0.565653\pi$
$683$	−7311.10	−0.409592	−0.204796	+	0.978805i	$0.565653\pi$
$684$	0	0
$685$	27911.9	1.55687
$686$	0	0
$687$	41629.7	2.31189
$688$	0	0
$689$	1215.90	0.0672311
$690$	0	0
$691$	31990.9	1.76120	0.880601	−	0.473858i	$-0.157139\pi$
$691$	31990.9	1.76120	0.880601	+	0.473858i	$0.157139\pi$
$692$	0	0
$693$	−124012.	−6.79771
$694$	0	0
$695$	−38613.0	−2.10745
$696$	0	0
$697$	−21030.8	−1.14290
$698$	0	0
$699$	−12788.8	−0.692012
$700$	0	0
$701$	18537.3	0.998777	0.499388	−	0.866378i	$-0.333558\pi$
$701$	18537.3	0.998777	0.499388	+	0.866378i	$0.333558\pi$
$702$	0	0
$703$	−21357.8	−1.14584
$704$	0	0
$705$	−13733.1	−0.733641
$706$	0	0
$707$	−30408.6	−1.61759
$708$	0	0
$709$	−34171.2	−1.81005	−0.905026	−	0.425357i	$-0.860149\pi$
$709$	−34171.2	−1.81005	−0.905026	+	0.425357i	$0.860149\pi$
$710$	0	0
$711$	7775.02	0.410107
$712$	0	0
$713$	−8560.63	−0.449647
$714$	0	0
$715$	−14262.5	−0.745995
$716$	0	0
$717$	63632.4	3.31436
$718$	0	0
$719$	−5542.46	−0.287481	−0.143741	−	0.989615i	$-0.545913\pi$
$719$	−5542.46	−0.287481	−0.143741	+	0.989615i	$0.545913\pi$
$720$	0	0
$721$	58679.8	3.03100
$722$	0	0
$723$	−53932.5	−2.77423
$724$	0	0
$725$	−8677.65	−0.444524
$726$	0	0
$727$	−18293.4	−0.933236	−0.466618	−	0.884459i	$-0.654528\pi$
$727$	−18293.4	−0.933236	−0.466618	+	0.884459i	$0.654528\pi$
$728$	0	0
$729$	34963.9	1.77635
$730$	0	0
$731$	−1646.94	−0.0833299
$732$	0	0
$733$	4221.75	0.212734	0.106367	−	0.994327i	$-0.466078\pi$
$733$	4221.75	0.212734	0.106367	+	0.994327i	$0.466078\pi$
$734$	0	0
$735$	104981.	5.26842
$736$	0	0
$737$	−38525.4	−1.92551
$738$	0	0
$739$	−20916.2	−1.04116	−0.520579	−	0.853813i	$-0.674284\pi$
$739$	−20916.2	−1.04116	−0.520579	+	0.853813i	$0.674284\pi$
$740$	0	0
$741$	8547.68	0.423761
$742$	0	0
$743$	29318.2	1.44762	0.723810	−	0.690000i	$-0.242389\pi$
$743$	29318.2	1.44762	0.723810	+	0.690000i	$0.242389\pi$
$744$	0	0
$745$	−45274.6	−2.22649
$746$	0	0
$747$	47639.1	2.33336
$748$	0	0
$749$	15280.6	0.745447
$750$	0	0
$751$	−2258.98	−0.109762	−0.0548811	−	0.998493i	$-0.517478\pi$
$751$	−2258.98	−0.109762	−0.0548811	+	0.998493i	$0.517478\pi$
$752$	0	0
$753$	−68775.7	−3.32845
$754$	0	0
$755$	620.549	0.0299127
$756$	0	0
$757$	29352.2	1.40928	0.704639	−	0.709566i	$-0.251109\pi$
$757$	29352.2	1.40928	0.704639	+	0.709566i	$0.251109\pi$
$758$	0	0
$759$	−34216.6	−1.63634
$760$	0	0
$761$	18420.7	0.877462	0.438731	−	0.898618i	$-0.355428\pi$
$761$	18420.7	0.877462	0.438731	+	0.898618i	$0.355428\pi$
$762$	0	0
$763$	23304.6	1.10574
$764$	0	0
$765$	−76977.0	−3.63805
$766$	0	0
$767$	−6898.80	−0.324774
$768$	0	0
$769$	19280.4	0.904120	0.452060	−	0.891987i	$-0.350689\pi$
$769$	19280.4	0.904120	0.452060	+	0.891987i	$0.350689\pi$
$770$	0	0
$771$	29341.4	1.37057
$772$	0	0
$773$	−8109.24	−0.377321	−0.188661	−	0.982042i	$-0.560415\pi$
$773$	−8109.24	−0.377321	−0.188661	+	0.982042i	$0.560415\pi$
$774$	0	0
$775$	45214.4	2.09568
$776$	0	0
$777$	−78082.2	−3.60513
$778$	0	0
$779$	−30031.3	−1.38124
$780$	0	0
$781$	−13988.0	−0.640883
$782$	0	0
$783$	−11595.9	−0.529249
$784$	0	0
$785$	15403.9	0.700370
$786$	0	0
$787$	−32384.7	−1.46682	−0.733412	−	0.679784i	$-0.762073\pi$
$787$	−32384.7	−1.46682	−0.733412	+	0.679784i	$0.762073\pi$
$788$	0	0
$789$	11764.7	0.530844
$790$	0	0
$791$	33119.0	1.48872
$792$	0	0
$793$	−302.154	−0.0135307
$794$	0	0
$795$	−21842.8	−0.974446
$796$	0	0
$797$	23032.3	1.02365	0.511823	−	0.859091i	$-0.328970\pi$
$797$	23032.3	1.02365	0.511823	+	0.859091i	$0.328970\pi$
$798$	0	0
$799$	3758.22	0.166403
$800$	0	0
$801$	5471.13	0.241339
$802$	0	0
$803$	16426.9	0.721910
$804$	0	0
$805$	34337.0	1.50338
$806$	0	0
$807$	5883.47	0.256639
$808$	0	0
$809$	30006.9	1.30406	0.652031	−	0.758192i	$-0.273917\pi$
$809$	30006.9	1.30406	0.652031	+	0.758192i	$0.273917\pi$
$810$	0	0
$811$	12221.9	0.529185	0.264593	−	0.964360i	$-0.414763\pi$
$811$	12221.9	0.529185	0.264593	+	0.964360i	$0.414763\pi$
$812$	0	0
$813$	−1108.28	−0.0478096
$814$	0	0
$815$	−62179.8	−2.67247
$816$	0	0
$817$	−2351.77	−0.100707
$818$	0	0
$819$	22370.0	0.954420
$820$	0	0
$821$	38742.1	1.64690	0.823452	−	0.567386i	$-0.192045\pi$
$821$	38742.1	1.64690	0.823452	+	0.567386i	$0.192045\pi$
$822$	0	0
$823$	−22341.9	−0.946283	−0.473142	−	0.880986i	$-0.656880\pi$
$823$	−22341.9	−0.946283	−0.473142	+	0.880986i	$0.656880\pi$
$824$	0	0
$825$	180721.	7.62653
$826$	0	0
$827$	11103.7	0.466884	0.233442	−	0.972371i	$-0.425001\pi$
$827$	11103.7	0.466884	0.233442	+	0.972371i	$0.425001\pi$
$828$	0	0
$829$	−24249.0	−1.01593	−0.507963	−	0.861379i	$-0.669601\pi$
$829$	−24249.0	−1.01593	−0.507963	+	0.861379i	$0.669601\pi$
$830$	0	0
$831$	21853.8	0.912273
$832$	0	0
$833$	−28729.4	−1.19497
$834$	0	0
$835$	−22907.8	−0.949408
$836$	0	0
$837$	60419.5	2.49511
$838$	0	0
$839$	−43189.3	−1.77719	−0.888595	−	0.458693i	$-0.848318\pi$
$839$	−43189.3	−1.77719	−0.888595	+	0.458693i	$0.848318\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	69658.6	2.84599
$844$	0	0
$845$	−42678.5	−1.73750
$846$	0	0
$847$	−73793.2	−2.99358
$848$	0	0
$849$	−35124.8	−1.41988
$850$	0	0
$851$	−15422.3	−0.621232
$852$	0	0
$853$	−12591.7	−0.505430	−0.252715	−	0.967541i	$-0.581324\pi$
$853$	−12591.7	−0.505430	−0.252715	+	0.967541i	$0.581324\pi$
$854$	0	0
$855$	−109921.	−4.39673
$856$	0	0
$857$	−32926.7	−1.31243	−0.656216	−	0.754573i	$-0.727844\pi$
$857$	−32926.7	−1.31243	−0.656216	+	0.754573i	$0.727844\pi$
$858$	0	0
$859$	−27777.3	−1.10332	−0.551659	−	0.834070i	$-0.686005\pi$
$859$	−27777.3	−1.10332	−0.551659	+	0.834070i	$0.686005\pi$
$860$	0	0
$861$	−109792.	−4.34575
$862$	0	0
$863$	12717.9	0.501649	0.250825	−	0.968033i	$-0.419298\pi$
$863$	12717.9	0.501649	0.250825	+	0.968033i	$0.419298\pi$
$864$	0	0
$865$	−36990.3	−1.45400
$866$	0	0
$867$	−18463.5	−0.723246
$868$	0	0
$869$	7082.06	0.276459
$870$	0	0
$871$	6949.44	0.270347
$872$	0	0
$873$	37344.0	1.44777
$874$	0	0
$875$	−105597.	−4.07980
$876$	0	0
$877$	−11042.4	−0.425171	−0.212585	−	0.977142i	$-0.568188\pi$
$877$	−11042.4	−0.425171	−0.212585	+	0.977142i	$0.568188\pi$
$878$	0	0
$879$	−36262.9	−1.39149
$880$	0	0
$881$	36901.8	1.41119	0.705593	−	0.708618i	$-0.250681\pi$
$881$	36901.8	1.41119	0.705593	+	0.708618i	$0.250681\pi$
$882$	0	0
$883$	−21471.0	−0.818298	−0.409149	−	0.912468i	$-0.634174\pi$
$883$	−21471.0	−0.818298	−0.409149	+	0.912468i	$0.634174\pi$
$884$	0	0
$885$	123932.	4.70726
$886$	0	0
$887$	−13867.8	−0.524953	−0.262477	−	0.964938i	$-0.584539\pi$
$887$	−13867.8	−0.524953	−0.262477	+	0.964938i	$0.584539\pi$
$888$	0	0
$889$	−44372.9	−1.67404
$890$	0	0
$891$	127707.	4.80174
$892$	0	0
$893$	5366.61	0.201105
$894$	0	0
$895$	−47852.6	−1.78719
$896$	0	0
$897$	6172.20	0.229748
$898$	0	0
$899$	−4381.98	−0.162566
$900$	0	0
$901$	5977.55	0.221022
$902$	0	0
$903$	−8597.86	−0.316854
$904$	0	0
$905$	−15532.2	−0.570506
$906$	0	0
$907$	16554.7	0.606053	0.303026	−	0.952982i	$-0.402003\pi$
$907$	16554.7	0.606053	0.303026	+	0.952982i	$0.402003\pi$
$908$	0	0
$909$	70291.8	2.56483
$910$	0	0
$911$	28515.8	1.03707	0.518535	−	0.855056i	$-0.326478\pi$
$911$	28515.8	1.03707	0.518535	+	0.855056i	$0.326478\pi$
$912$	0	0
$913$	43393.2	1.57295
$914$	0	0
$915$	5427.98	0.196113
$916$	0	0
$917$	−29580.9	−1.06526
$918$	0	0
$919$	−49951.6	−1.79298	−0.896492	−	0.443061i	$-0.853893\pi$
$919$	−49951.6	−1.79298	−0.896492	+	0.443061i	$0.853893\pi$
$920$	0	0
$921$	−56351.8	−2.01613
$922$	0	0
$923$	2523.24	0.0899820
$924$	0	0
$925$	81455.2	2.89539
$926$	0	0
$927$	−135643.	−4.80593
$928$	0	0
$929$	15111.1	0.533668	0.266834	−	0.963742i	$-0.414022\pi$
$929$	15111.1	0.533668	0.266834	+	0.963742i	$0.414022\pi$
$930$	0	0
$931$	−41024.6	−1.44417
$932$	0	0
$933$	34790.2	1.22077
$934$	0	0
$935$	−70116.4	−2.45246
$936$	0	0
$937$	−5975.89	−0.208350	−0.104175	−	0.994559i	$-0.533220\pi$
$937$	−5975.89	−0.208350	−0.104175	+	0.994559i	$0.533220\pi$
$938$	0	0
$939$	69083.9	2.40092
$940$	0	0
$941$	9500.09	0.329112	0.164556	−	0.986368i	$-0.447381\pi$
$941$	9500.09	0.329112	0.164556	+	0.986368i	$0.447381\pi$
$942$	0	0
$943$	−21685.3	−0.748857
$944$	0	0
$945$	−242345.	−8.34232
$946$	0	0
$947$	−10249.4	−0.351702	−0.175851	−	0.984417i	$-0.556268\pi$
$947$	−10249.4	−0.351702	−0.175851	+	0.984417i	$0.556268\pi$
$948$	0	0
$949$	−2963.19	−0.101358
$950$	0	0
$951$	77766.7	2.65169
$952$	0	0
$953$	−42222.9	−1.43519	−0.717594	−	0.696462i	$-0.754756\pi$
$953$	−42222.9	−1.43519	−0.717594	+	0.696462i	$0.754756\pi$
$954$	0	0
$955$	10531.0	0.356834
$956$	0	0
$957$	−17514.7	−0.591608
$958$	0	0
$959$	−39876.6	−1.34273
$960$	0	0
$961$	−6958.96	−0.233593
$962$	0	0
$963$	−35322.2	−1.18198
$964$	0	0
$965$	−65876.5	−2.19755
$966$	0	0
$967$	−1541.73	−0.0512706	−0.0256353	−	0.999671i	$-0.508161\pi$
$967$	−1541.73	−0.0512706	−0.0256353	+	0.999671i	$0.508161\pi$
$968$	0	0
$969$	42021.6	1.39311
$970$	0	0
$971$	−5603.31	−0.185189	−0.0925947	−	0.995704i	$-0.529516\pi$
$971$	−5603.31	−0.185189	−0.0925947	+	0.995704i	$0.529516\pi$
$972$	0	0
$973$	55164.9	1.81758
$974$	0	0
$975$	−32599.5	−1.07079
$976$	0	0
$977$	−46326.9	−1.51702	−0.758511	−	0.651660i	$-0.774073\pi$
$977$	−46326.9	−1.51702	−0.758511	+	0.651660i	$0.774073\pi$
$978$	0	0
$979$	4983.51	0.162690
$980$	0	0
$981$	−53870.4	−1.75326
$982$	0	0
$983$	−111.878	−0.00363006	−0.00181503	−	0.999998i	$-0.500578\pi$
$983$	−111.878	−0.00363006	−0.00181503	+	0.999998i	$0.500578\pi$
$984$	0	0
$985$	−31113.4	−1.00645
$986$	0	0
$987$	19619.9	0.632732
$988$	0	0
$989$	−1698.19	−0.0546000
$990$	0	0
$991$	57563.4	1.84517	0.922584	−	0.385796i	$-0.126073\pi$
$991$	57563.4	1.84517	0.922584	+	0.385796i	$0.126073\pi$
$992$	0	0
$993$	−47673.1	−1.52352
$994$	0	0
$995$	−96959.6	−3.08927
$996$	0	0
$997$	11300.4	0.358963	0.179481	−	0.983761i	$-0.442558\pi$
$997$	11300.4	0.358963	0.179481	+	0.983761i	$0.442558\pi$
$998$	0	0
$999$	108848.	3.44724

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1856.4.a.bl.1.12		12
4.3	odd	2		1856.4.a.bj.1.1		12
8.3	odd	2		928.4.a.j.1.12	yes	12
8.5	even	2		928.4.a.h.1.1	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
928.4.a.h.1.1	✓	12	8.5	even	2
928.4.a.j.1.12	yes	12	8.3	odd	2
1856.4.a.bj.1.1		12	4.3	odd	2
1856.4.a.bl.1.12		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)

\(f(q)\)	\(=\)	\(q+9.74783 q^{3} +20.5968 q^{5} -29.4259 q^{7} +68.0201 q^{9} +O(q^{10})\)	\(q+9.74783 q^{3} +20.5968 q^{5} -29.4259 q^{7} +68.0201 q^{9} +61.9578 q^{11} -11.1763 q^{13} +200.774 q^{15} -54.9444 q^{17} -78.4587 q^{19} -286.838 q^{21} -56.6544 q^{23} +299.229 q^{25} +399.857 q^{27} -29.0000 q^{29} +151.103 q^{31} +603.954 q^{33} -606.079 q^{35} +272.217 q^{37} -108.945 q^{39} +382.766 q^{41} +29.9746 q^{43} +1401.00 q^{45} -68.4004 q^{47} +522.881 q^{49} -535.588 q^{51} -108.793 q^{53} +1276.13 q^{55} -764.802 q^{57} +617.270 q^{59} +27.0352 q^{61} -2001.55 q^{63} -230.197 q^{65} -621.800 q^{67} -552.257 q^{69} -225.766 q^{71} +265.131 q^{73} +2916.84 q^{75} -1823.16 q^{77} +114.305 q^{79} +2061.20 q^{81} +700.367 q^{83} -1131.68 q^{85} -282.687 q^{87} +80.4339 q^{89} +328.873 q^{91} +1472.92 q^{93} -1616.00 q^{95} +549.014 q^{97} +4214.38 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 14 q^{3} + 10 q^{5} - 44 q^{7} + 130 q^{9}+O(q^{10}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more