LMFDB - Embedded newform 1856.4.a.bk.1.6

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} - 245x^{10} + 21294x^{8} - 755514x^{6} + 8955005x^{4} - 27099393x^{2} + 23710340 $ x^12 - 245x^10 + 21294x^8 - 755514x^6 + 8955005x^4 - 27099393*x^2 + 23710340
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{19} $
Twist minimal:	no (minimal twist has level 928)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-1.30165$ 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)

$f(q)$	$=$	$q-1.30165 q^{3} +15.1013 q^{5} -22.0989 q^{7} -25.3057 q^{9} +O(q^{10})$	$q-1.30165 q^{3} +15.1013 q^{5} -22.0989 q^{7} -25.3057 q^{9} +46.1006 q^{11} +57.4585 q^{13} -19.6567 q^{15} -52.7069 q^{17} +45.2101 q^{19} +28.7651 q^{21} +80.1240 q^{23} +103.050 q^{25} +68.0838 q^{27} +29.0000 q^{29} -47.9675 q^{31} -60.0070 q^{33} -333.722 q^{35} +91.2348 q^{37} -74.7910 q^{39} +252.437 q^{41} -480.084 q^{43} -382.150 q^{45} -551.935 q^{47} +145.360 q^{49} +68.6061 q^{51} +263.472 q^{53} +696.180 q^{55} -58.8478 q^{57} -315.638 q^{59} +628.577 q^{61} +559.228 q^{63} +867.700 q^{65} +693.637 q^{67} -104.294 q^{69} -785.772 q^{71} +807.667 q^{73} -134.135 q^{75} -1018.77 q^{77} -299.051 q^{79} +594.632 q^{81} +333.152 q^{83} -795.944 q^{85} -37.7479 q^{87} +34.5437 q^{89} -1269.77 q^{91} +62.4370 q^{93} +682.732 q^{95} +929.042 q^{97} -1166.61 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 10 q^{5} + 166 q^{9}+O(q^{10}) $ 12 * q - 10 * q^5 + 166 * q^9	$ 12 q - 10 q^{5} + 166 q^{9} - 30 q^{13} + 336 q^{17} - 56 q^{21} + 294 q^{25} + 348 q^{29} + 214 q^{33} - 196 q^{37} + 940 q^{41} - 1672 q^{45} + 972 q^{49} + 406 q^{53} + 1224 q^{57} - 400 q^{61} + 2998 q^{65} - 1004 q^{69} + 2252 q^{73} - 3032 q^{77} + 3932 q^{81} - 5564 q^{85} + 5212 q^{89} - 3722 q^{93} + 6164 q^{97}+O(q^{100}) $ 12 * q - 10 * q^5 + 166 * q^9 - 30 * q^13 + 336 * q^17 - 56 * q^21 + 294 * q^25 + 348 * q^29 + 214 * q^33 - 196 * q^37 + 940 * q^41 - 1672 * q^45 + 972 * q^49 + 406 * q^53 + 1224 * q^57 - 400 * q^61 + 2998 * q^65 - 1004 * q^69 + 2252 * q^73 - 3032 * q^77 + 3932 * q^81 - 5564 * q^85 + 5212 * q^89 - 3722 * q^93 + 6164 * q^97

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
$3$	−1.30165	−0.250503	−0.125252	−	0.992125i	$-0.539974\pi$
$3$	−1.30165	−0.250503	−0.125252	+	0.992125i	$0.539974\pi$
$4$	0	0
$5$	15.1013	1.35070	0.675352	−	0.737496i	$-0.263992\pi$
$5$	15.1013	1.35070	0.675352	+	0.737496i	$0.263992\pi$
$6$	0	0
$7$	−22.0989	−1.19323	−0.596614	−	0.802529i	$-0.703487\pi$
$7$	−22.0989	−1.19323	−0.596614	+	0.802529i	$0.703487\pi$
$8$	0	0
$9$	−25.3057	−0.937248
$10$	0	0
$11$	46.1006	1.26362	0.631812	−	0.775122i	$-0.282312\pi$
$11$	46.1006	1.26362	0.631812	+	0.775122i	$0.282312\pi$
$12$	0	0
$13$	57.4585	1.22586	0.612928	−	0.790139i	$-0.289992\pi$
$13$	57.4585	1.22586	0.612928	+	0.790139i	$0.289992\pi$
$14$	0	0
$15$	−19.6567	−0.338355
$16$	0	0
$17$	−52.7069	−0.751959	−0.375979	−	0.926628i	$-0.622694\pi$
$17$	−52.7069	−0.751959	−0.375979	+	0.926628i	$0.622694\pi$
$18$	0	0
$19$	45.2101	0.545889	0.272945	−	0.962030i	$-0.412002\pi$
$19$	45.2101	0.545889	0.272945	+	0.962030i	$0.412002\pi$
$20$	0	0
$21$	28.7651	0.298907
$22$	0	0
$23$	80.1240	0.726392	0.363196	−	0.931713i	$-0.381686\pi$
$23$	80.1240	0.726392	0.363196	+	0.931713i	$0.381686\pi$
$24$	0	0
$25$	103.050	0.824399
$26$	0	0
$27$	68.0838	0.485287
$28$	0	0
$29$	29.0000	0.185695
$29$	29.0000	0.185695
$30$	0	0
$31$	−47.9675	−0.277910	−0.138955	−	0.990299i	$-0.544374\pi$
$31$	−47.9675	−0.277910	−0.138955	+	0.990299i	$0.544374\pi$
$32$	0	0
$33$	−60.0070	−0.316542
$34$	0	0
$35$	−333.722	−1.61170
$36$	0	0
$37$	91.2348	0.405376	0.202688	−	0.979243i	$-0.435032\pi$
$37$	91.2348	0.405376	0.202688	+	0.979243i	$0.435032\pi$
$38$	0	0
$39$	−74.7910	−0.307081
$40$	0	0
$41$	252.437	0.961561	0.480781	−	0.876841i	$-0.340353\pi$
$41$	252.437	0.961561	0.480781	+	0.876841i	$0.340353\pi$
$42$	0	0
$43$	−480.084	−1.70261	−0.851303	−	0.524674i	$-0.824187\pi$
$43$	−480.084	−1.70261	−0.851303	+	0.524674i	$0.824187\pi$
$44$	0	0
$45$	−382.150	−1.26594
$46$	0	0
$47$	−551.935	−1.71294	−0.856468	−	0.516200i	$-0.827346\pi$
$47$	−551.935	−1.71294	−0.856468	+	0.516200i	$0.827346\pi$
$48$	0	0
$49$	145.360	0.423791
$50$	0	0
$51$	68.6061	0.188368
$52$	0	0
$53$	263.472	0.682842	0.341421	−	0.939910i	$-0.389092\pi$
$53$	263.472	0.682842	0.341421	+	0.939910i	$0.389092\pi$
$54$	0	0
$55$	696.180	1.70678
$56$	0	0
$57$	−58.8478	−0.136747
$58$	0	0
$59$	−315.638	−0.696485	−0.348242	−	0.937405i	$-0.613221\pi$
$59$	−315.638	−0.696485	−0.348242	+	0.937405i	$0.613221\pi$
$60$	0	0
$61$	628.577	1.31936	0.659680	−	0.751546i	$-0.270692\pi$
$61$	628.577	1.31936	0.659680	+	0.751546i	$0.270692\pi$
$62$	0	0
$63$	559.228	1.11835
$64$	0	0
$65$	867.700	1.65577
$66$	0	0
$67$	693.637	1.26479	0.632397	−	0.774644i	$-0.282071\pi$
$67$	693.637	1.26479	0.632397	+	0.774644i	$0.282071\pi$
$68$	0	0
$69$	−104.294	−0.181963
$70$	0	0
$71$	−785.772	−1.31344	−0.656718	−	0.754136i	$-0.728056\pi$
$71$	−785.772	−1.31344	−0.656718	+	0.754136i	$0.728056\pi$
$72$	0	0
$73$	807.667	1.29493	0.647467	−	0.762093i	$-0.275828\pi$
$73$	807.667	1.29493	0.647467	+	0.762093i	$0.275828\pi$
$74$	0	0
$75$	−134.135	−0.206515
$76$	0	0
$77$	−1018.77	−1.50779
$78$	0	0
$79$	−299.051	−0.425897	−0.212948	−	0.977063i	$-0.568307\pi$
$79$	−299.051	−0.425897	−0.212948	+	0.977063i	$0.568307\pi$
$80$	0	0
$81$	594.632	0.815682
$82$	0	0
$83$	333.152	0.440580	0.220290	−	0.975434i	$-0.429300\pi$
$83$	333.152	0.440580	0.220290	+	0.975434i	$0.429300\pi$
$84$	0	0
$85$	−795.944	−1.01567
$86$	0	0
$87$	−37.7479	−0.0465173
$88$	0	0
$89$	34.5437	0.0411419	0.0205709	−	0.999788i	$-0.493452\pi$
$89$	34.5437	0.0411419	0.0205709	+	0.999788i	$0.493452\pi$
$90$	0	0
$91$	−1269.77	−1.46272
$92$	0	0
$93$	62.4370	0.0696174
$94$	0	0
$95$	682.732	0.737334
$96$	0	0
$97$	929.042	0.972474	0.486237	−	0.873827i	$-0.338369\pi$
$97$	929.042	0.972474	0.486237	+	0.873827i	$0.338369\pi$
$98$	0	0
$99$	−1166.61	−1.18433
$100$	0	0
$101$	294.519	0.290156	0.145078	−	0.989420i	$-0.453657\pi$
$101$	294.519	0.290156	0.145078	+	0.989420i	$0.453657\pi$
$102$	0	0
$103$	−1509.54	−1.44407	−0.722035	−	0.691856i	$-0.756793\pi$
$103$	−1509.54	−1.44407	−0.722035	+	0.691856i	$0.756793\pi$
$104$	0	0
$105$	434.390	0.403735
$106$	0	0
$107$	2036.36	1.83984	0.919918	−	0.392110i	$-0.128255\pi$
$107$	2036.36	1.83984	0.919918	+	0.392110i	$0.128255\pi$
$108$	0	0
$109$	1324.89	1.16423	0.582116	−	0.813106i	$-0.302225\pi$
$109$	1324.89	1.16423	0.582116	+	0.813106i	$0.302225\pi$
$110$	0	0
$111$	−118.756	−0.101548
$112$	0	0
$113$	1466.80	1.22110	0.610552	−	0.791976i	$-0.290948\pi$
$113$	1466.80	1.22110	0.610552	+	0.791976i	$0.290948\pi$
$114$	0	0
$115$	1209.98	0.981140
$116$	0	0
$117$	−1454.03	−1.14893
$118$	0	0
$119$	1164.76	0.897258
$120$	0	0
$121$	794.266	0.596744
$122$	0	0
$123$	−328.585	−0.240874
$124$	0	0
$125$	−331.475	−0.237184
$126$	0	0
$127$	−1413.23	−0.987430	−0.493715	−	0.869624i	$-0.664361\pi$
$127$	−1413.23	−0.987430	−0.493715	+	0.869624i	$0.664361\pi$
$128$	0	0
$129$	624.902	0.426508
$130$	0	0
$131$	1705.88	1.13774	0.568870	−	0.822428i	$-0.307381\pi$
$131$	1705.88	1.13774	0.568870	+	0.822428i	$0.307381\pi$
$132$	0	0
$133$	−999.091	−0.651370
$134$	0	0
$135$	1028.16	0.655478
$136$	0	0
$137$	−656.331	−0.409301	−0.204650	−	0.978835i	$-0.565606\pi$
$137$	−656.331	−0.409301	−0.204650	+	0.978835i	$0.565606\pi$
$138$	0	0
$139$	−74.2148	−0.0452864	−0.0226432	−	0.999744i	$-0.507208\pi$
$139$	−74.2148	−0.0452864	−0.0226432	+	0.999744i	$0.507208\pi$
$140$	0	0
$141$	718.428	0.429096
$142$	0	0
$143$	2648.87	1.54902
$144$	0	0
$145$	437.938	0.250819
$146$	0	0
$147$	−189.209	−0.106161
$148$	0	0
$149$	−1155.09	−0.635092	−0.317546	−	0.948243i	$-0.602859\pi$
$149$	−1155.09	−0.635092	−0.317546	+	0.948243i	$0.602859\pi$
$150$	0	0
$151$	−76.8840	−0.0414353	−0.0207176	−	0.999785i	$-0.506595\pi$
$151$	−76.8840	−0.0414353	−0.0207176	+	0.999785i	$0.506595\pi$
$152$	0	0
$153$	1333.79	0.704772
$154$	0	0
$155$	−724.373	−0.375374
$156$	0	0
$157$	1462.75	0.743568	0.371784	−	0.928319i	$-0.378746\pi$
$157$	1462.75	0.743568	0.371784	+	0.928319i	$0.378746\pi$
$158$	0	0
$159$	−342.949	−0.171054
$160$	0	0
$161$	−1770.65	−0.866751
$162$	0	0
$163$	1205.28	0.579170	0.289585	−	0.957152i	$-0.406483\pi$
$163$	1205.28	0.579170	0.289585	+	0.957152i	$0.406483\pi$
$164$	0	0
$165$	−906.185	−0.427554
$166$	0	0
$167$	147.650	0.0684162	0.0342081	−	0.999415i	$-0.489109\pi$
$167$	147.650	0.0684162	0.0342081	+	0.999415i	$0.489109\pi$
$168$	0	0
$169$	1104.48	0.502723
$170$	0	0
$171$	−1144.07	−0.511634
$172$	0	0
$173$	1413.80	0.621326	0.310663	−	0.950520i	$-0.399449\pi$
$173$	1413.80	0.621326	0.310663	+	0.950520i	$0.399449\pi$
$174$	0	0
$175$	−2277.29	−0.983696
$176$	0	0
$177$	410.851	0.174472
$178$	0	0
$179$	3919.57	1.63666	0.818331	−	0.574748i	$-0.194900\pi$
$179$	3919.57	1.63666	0.818331	+	0.574748i	$0.194900\pi$
$180$	0	0
$181$	336.944	0.138369	0.0691847	−	0.997604i	$-0.477960\pi$
$181$	336.944	0.138369	0.0691847	+	0.997604i	$0.477960\pi$
$182$	0	0
$183$	−818.188	−0.330504
$184$	0	0
$185$	1377.77	0.547542
$186$	0	0
$187$	−2429.82	−0.950193
$188$	0	0
$189$	−1504.58	−0.579057
$190$	0	0
$191$	4908.91	1.85966	0.929832	−	0.367984i	$-0.119952\pi$
$191$	4908.91	1.85966	0.929832	+	0.367984i	$0.119952\pi$
$192$	0	0
$193$	−3014.21	−1.12418	−0.562092	−	0.827075i	$-0.690003\pi$
$193$	−3014.21	−1.12418	−0.562092	+	0.827075i	$0.690003\pi$
$194$	0	0
$195$	−1129.44	−0.414775
$196$	0	0
$197$	−5241.86	−1.89577	−0.947886	−	0.318609i	$-0.896784\pi$
$197$	−5241.86	−1.89577	−0.947886	+	0.318609i	$0.896784\pi$
$198$	0	0
$199$	2001.36	0.712929	0.356464	−	0.934309i	$-0.383982\pi$
$199$	2001.36	0.712929	0.356464	+	0.934309i	$0.383982\pi$
$200$	0	0
$201$	−902.874	−0.316835
$202$	0	0
$203$	−640.867	−0.221577
$204$	0	0
$205$	3812.13	1.29878
$206$	0	0
$207$	−2027.59	−0.680810
$208$	0	0
$209$	2084.21	0.689798
$210$	0	0
$211$	2125.08	0.693349	0.346675	−	0.937985i	$-0.387311\pi$
$211$	2125.08	0.693349	0.346675	+	0.937985i	$0.387311\pi$
$212$	0	0
$213$	1022.80	0.329020
$214$	0	0
$215$	−7249.90	−2.29972
$216$	0	0
$217$	1060.03	0.331610
$218$	0	0
$219$	−1051.30	−0.324385
$220$	0	0
$221$	−3028.46	−0.921793
$222$	0	0
$223$	993.788	0.298426	0.149213	−	0.988805i	$-0.452326\pi$
$223$	993.788	0.298426	0.149213	+	0.988805i	$0.452326\pi$
$224$	0	0
$225$	−2607.75	−0.772667
$226$	0	0
$227$	−3650.98	−1.06751	−0.533753	−	0.845640i	$-0.679219\pi$
$227$	−3650.98	−1.06751	−0.533753	+	0.845640i	$0.679219\pi$
$228$	0	0
$229$	−3709.46	−1.07043	−0.535214	−	0.844717i	$-0.679769\pi$
$229$	−3709.46	−1.07043	−0.535214	+	0.844717i	$0.679769\pi$
$230$	0	0
$231$	1326.09	0.377706
$232$	0	0
$233$	2473.21	0.695387	0.347693	−	0.937608i	$-0.386965\pi$
$233$	2473.21	0.695387	0.347693	+	0.937608i	$0.386965\pi$
$234$	0	0
$235$	−8334.95	−2.31367
$236$	0	0
$237$	389.260	0.106689
$238$	0	0
$239$	−2443.26	−0.661261	−0.330630	−	0.943760i	$-0.607261\pi$
$239$	−2443.26	−0.661261	−0.330630	+	0.943760i	$0.607261\pi$
$240$	0	0
$241$	2783.17	0.743900	0.371950	−	0.928253i	$-0.378689\pi$
$241$	2783.17	0.743900	0.371950	+	0.928253i	$0.378689\pi$
$242$	0	0
$243$	−2612.27	−0.689618
$244$	0	0
$245$	2195.13	0.572416
$246$	0	0
$247$	2597.70	0.669182
$248$	0	0
$249$	−433.648	−0.110367
$250$	0	0
$251$	3661.49	0.920762	0.460381	−	0.887721i	$-0.347713\pi$
$251$	3661.49	0.920762	0.460381	+	0.887721i	$0.347713\pi$
$252$	0	0
$253$	3693.77	0.917886
$254$	0	0
$255$	1036.04	0.254429
$256$	0	0
$257$	5369.76	1.30333	0.651667	−	0.758505i	$-0.274070\pi$
$257$	5369.76	1.30333	0.651667	+	0.758505i	$0.274070\pi$
$258$	0	0
$259$	−2016.19	−0.483705
$260$	0	0
$261$	−733.865	−0.174043
$262$	0	0
$263$	−3302.62	−0.774329	−0.387164	−	0.922011i	$-0.626545\pi$
$263$	−3302.62	−0.774329	−0.387164	+	0.922011i	$0.626545\pi$
$264$	0	0
$265$	3978.77	0.922317
$266$	0	0
$267$	−44.9639	−0.0103062
$268$	0	0
$269$	6890.49	1.56179	0.780894	−	0.624664i	$-0.214764\pi$
$269$	6890.49	1.56179	0.780894	+	0.624664i	$0.214764\pi$
$270$	0	0
$271$	1259.62	0.282350	0.141175	−	0.989985i	$-0.454912\pi$
$271$	1259.62	0.282350	0.141175	+	0.989985i	$0.454912\pi$
$272$	0	0
$273$	1652.80	0.366417
$274$	0	0
$275$	4750.66	1.04173
$276$	0	0
$277$	4323.60	0.937834	0.468917	−	0.883242i	$-0.344644\pi$
$277$	4323.60	0.937834	0.468917	+	0.883242i	$0.344644\pi$
$278$	0	0
$279$	1213.85	0.260471
$280$	0	0
$281$	8721.69	1.85157	0.925787	−	0.378045i	$-0.123404\pi$
$281$	8721.69	1.85157	0.925787	+	0.378045i	$0.123404\pi$
$282$	0	0
$283$	2497.86	0.524672	0.262336	−	0.964977i	$-0.415507\pi$
$283$	2497.86	0.524672	0.262336	+	0.964977i	$0.415507\pi$
$284$	0	0
$285$	−888.679	−0.184705
$286$	0	0
$287$	−5578.57	−1.14736
$288$	0	0
$289$	−2134.98	−0.434558
$290$	0	0
$291$	−1209.29	−0.243608
$292$	0	0
$293$	1248.76	0.248988	0.124494	−	0.992220i	$-0.460269\pi$
$293$	1248.76	0.248988	0.124494	+	0.992220i	$0.460269\pi$
$294$	0	0
$295$	−4766.55	−0.940744
$296$	0	0
$297$	3138.71	0.613220
$298$	0	0
$299$	4603.81	0.890452
$300$	0	0
$301$	10609.3	2.03160
$302$	0	0
$303$	−383.361	−0.0726849
$304$	0	0
$305$	9492.34	1.78207
$306$	0	0
$307$	−221.109	−0.0411054	−0.0205527	−	0.999789i	$-0.506543\pi$
$307$	−221.109	−0.0411054	−0.0205527	+	0.999789i	$0.506543\pi$
$308$	0	0
$309$	1964.89	0.361744
$310$	0	0
$311$	4174.82	0.761198	0.380599	−	0.924740i	$-0.375718\pi$
$311$	4174.82	0.761198	0.380599	+	0.924740i	$0.375718\pi$
$312$	0	0
$313$	1852.46	0.334529	0.167264	−	0.985912i	$-0.446507\pi$
$313$	1852.46	0.334529	0.167264	+	0.985912i	$0.446507\pi$
$314$	0	0
$315$	8445.08	1.51056
$316$	0	0
$317$	7417.23	1.31417	0.657087	−	0.753814i	$-0.271788\pi$
$317$	7417.23	1.31417	0.657087	+	0.753814i	$0.271788\pi$
$318$	0	0
$319$	1336.92	0.234649
$320$	0	0
$321$	−2650.64	−0.460885
$322$	0	0
$323$	−2382.88	−0.410486
$324$	0	0
$325$	5921.10	1.01059
$326$	0	0
$327$	−1724.54	−0.291644
$328$	0	0
$329$	12197.1	2.04392
$330$	0	0
$331$	20.5890	0.00341896	0.00170948	−	0.999999i	$-0.499456\pi$
$331$	20.5890	0.00341896	0.00170948	+	0.999999i	$0.499456\pi$
$332$	0	0
$333$	−2308.76	−0.379938
$334$	0	0
$335$	10474.8	1.70836
$336$	0	0
$337$	−5486.22	−0.886806	−0.443403	−	0.896322i	$-0.646229\pi$
$337$	−5486.22	−0.886806	−0.443403	+	0.896322i	$0.646229\pi$
$338$	0	0
$339$	−1909.26	−0.305890
$340$	0	0
$341$	−2211.33	−0.351174
$342$	0	0
$343$	4367.61	0.687548
$344$	0	0
$345$	−1574.97	−0.245779
$346$	0	0
$347$	11764.3	1.82001	0.910003	−	0.414603i	$-0.136079\pi$
$347$	11764.3	1.82001	0.910003	+	0.414603i	$0.136079\pi$
$348$	0	0
$349$	3916.16	0.600651	0.300326	−	0.953837i	$-0.402905\pi$
$349$	3916.16	0.600651	0.300326	+	0.953837i	$0.402905\pi$
$350$	0	0
$351$	3912.00	0.594892
$352$	0	0
$353$	−308.427	−0.0465039	−0.0232520	−	0.999730i	$-0.507402\pi$
$353$	−308.427	−0.0465039	−0.0232520	+	0.999730i	$0.507402\pi$
$354$	0	0
$355$	−11866.2	−1.77406
$356$	0	0
$357$	−1516.12	−0.224766
$358$	0	0
$359$	4540.73	0.667551	0.333775	−	0.942653i	$-0.391677\pi$
$359$	4540.73	0.667551	0.333775	+	0.942653i	$0.391677\pi$
$360$	0	0
$361$	−4815.05	−0.702005
$362$	0	0
$363$	−1033.86	−0.149486
$364$	0	0
$365$	12196.8	1.74907
$366$	0	0
$367$	−12573.3	−1.78834	−0.894169	−	0.447730i	$-0.852233\pi$
$367$	−12573.3	−1.78834	−0.894169	+	0.447730i	$0.852233\pi$
$368$	0	0
$369$	−6388.09	−0.901222
$370$	0	0
$371$	−5822.43	−0.814786
$372$	0	0
$373$	12962.4	1.79937	0.899687	−	0.436535i	$-0.143795\pi$
$373$	12962.4	1.79937	0.899687	+	0.436535i	$0.143795\pi$
$374$	0	0
$375$	431.466	0.0594154
$376$	0	0
$377$	1666.30	0.227636
$378$	0	0
$379$	8731.04	1.18333	0.591667	−	0.806182i	$-0.298470\pi$
$379$	8731.04	1.18333	0.591667	+	0.806182i	$0.298470\pi$
$380$	0	0
$381$	1839.53	0.247354
$382$	0	0
$383$	13357.9	1.78213	0.891067	−	0.453872i	$-0.149958\pi$
$383$	13357.9	1.78213	0.891067	+	0.453872i	$0.149958\pi$
$384$	0	0
$385$	−15384.8	−2.03658
$386$	0	0
$387$	12148.9	1.59576
$388$	0	0
$389$	−2592.44	−0.337897	−0.168949	−	0.985625i	$-0.554037\pi$
$389$	−2592.44	−0.337897	−0.168949	+	0.985625i	$0.554037\pi$
$390$	0	0
$391$	−4223.09	−0.546217
$392$	0	0
$393$	−2220.47	−0.285007
$394$	0	0
$395$	−4516.06	−0.575260
$396$	0	0
$397$	−7765.25	−0.981679	−0.490840	−	0.871250i	$-0.663310\pi$
$397$	−7765.25	−0.981679	−0.490840	+	0.871250i	$0.663310\pi$
$398$	0	0
$399$	1300.47	0.163170
$400$	0	0
$401$	−3597.04	−0.447949	−0.223974	−	0.974595i	$-0.571903\pi$
$401$	−3597.04	−0.447949	−0.223974	+	0.974595i	$0.571903\pi$
$402$	0	0
$403$	−2756.14	−0.340678
$404$	0	0
$405$	8979.74	1.10174
$406$	0	0
$407$	4205.98	0.512242
$408$	0	0
$409$	−15100.2	−1.82556	−0.912780	−	0.408451i	$-0.866069\pi$
$409$	−15100.2	−1.82556	−0.912780	+	0.408451i	$0.866069\pi$
$410$	0	0
$411$	854.315	0.102531
$412$	0	0
$413$	6975.25	0.831064
$414$	0	0
$415$	5031.03	0.595093
$416$	0	0
$417$	96.6018	0.0113444
$418$	0	0
$419$	11571.7	1.34920	0.674602	−	0.738182i	$-0.264315\pi$
$419$	11571.7	1.34920	0.674602	+	0.738182i	$0.264315\pi$
$420$	0	0
$421$	8890.20	1.02917	0.514587	−	0.857438i	$-0.327945\pi$
$421$	8890.20	1.02917	0.514587	+	0.857438i	$0.327945\pi$
$422$	0	0
$423$	13967.1	1.60545
$424$	0	0
$425$	−5431.44	−0.619914
$426$	0	0
$427$	−13890.8	−1.57430
$428$	0	0
$429$	−3447.91	−0.388034
$430$	0	0
$431$	−773.030	−0.0863934	−0.0431967	−	0.999067i	$-0.513754\pi$
$431$	−773.030	−0.0863934	−0.0431967	+	0.999067i	$0.513754\pi$
$432$	0	0
$433$	10979.3	1.21855	0.609275	−	0.792959i	$-0.291461\pi$
$433$	10979.3	1.21855	0.609275	+	0.792959i	$0.291461\pi$
$434$	0	0
$435$	−570.043	−0.0628310
$436$	0	0
$437$	3622.41	0.396530
$438$	0	0
$439$	1273.70	0.138475	0.0692373	−	0.997600i	$-0.477943\pi$
$439$	1273.70	0.138475	0.0692373	+	0.997600i	$0.477943\pi$
$440$	0	0
$441$	−3678.45	−0.397197
$442$	0	0
$443$	−15411.4	−1.65287	−0.826433	−	0.563035i	$-0.809634\pi$
$443$	−15411.4	−1.65287	−0.826433	+	0.563035i	$0.809634\pi$
$444$	0	0
$445$	521.656	0.0555704
$446$	0	0
$447$	1503.53	0.159093
$448$	0	0
$449$	−2282.01	−0.239855	−0.119927	−	0.992783i	$-0.538266\pi$
$449$	−2282.01	−0.239855	−0.119927	+	0.992783i	$0.538266\pi$
$450$	0	0
$451$	11637.5	1.21505
$452$	0	0
$453$	100.076	0.0103797
$454$	0	0
$455$	−19175.2	−1.97571
$456$	0	0
$457$	−10792.9	−1.10475	−0.552377	−	0.833595i	$-0.686279\pi$
$457$	−10792.9	−1.10475	−0.552377	+	0.833595i	$0.686279\pi$
$458$	0	0
$459$	−3588.49	−0.364916
$460$	0	0
$461$	461.180	0.0465928	0.0232964	−	0.999729i	$-0.492584\pi$
$461$	461.180	0.0465928	0.0232964	+	0.999729i	$0.492584\pi$
$462$	0	0
$463$	−11833.2	−1.18776	−0.593881	−	0.804553i	$-0.702405\pi$
$463$	−11833.2	−1.18776	−0.593881	+	0.804553i	$0.702405\pi$
$464$	0	0
$465$	942.881	0.0940324
$466$	0	0
$467$	4049.01	0.401212	0.200606	−	0.979672i	$-0.435709\pi$
$467$	4049.01	0.401212	0.200606	+	0.979672i	$0.435709\pi$
$468$	0	0
$469$	−15328.6	−1.50919
$470$	0	0
$471$	−1903.99	−0.186266
$472$	0	0
$473$	−22132.1	−2.15145
$474$	0	0
$475$	4658.89	0.450031
$476$	0	0
$477$	−6667.34	−0.639992
$478$	0	0
$479$	−15144.5	−1.44462	−0.722308	−	0.691571i	$-0.756919\pi$
$479$	−15144.5	−1.44462	−0.722308	+	0.691571i	$0.756919\pi$
$480$	0	0
$481$	5242.22	0.496932
$482$	0	0
$483$	2304.77	0.217124
$484$	0	0
$485$	14029.8	1.31352
$486$	0	0
$487$	2164.85	0.201435	0.100717	−	0.994915i	$-0.467886\pi$
$487$	2164.85	0.201435	0.100717	+	0.994915i	$0.467886\pi$
$488$	0	0
$489$	−1568.85	−0.145084
$490$	0	0
$491$	3518.52	0.323398	0.161699	−	0.986840i	$-0.448303\pi$
$491$	3518.52	0.323398	0.161699	+	0.986840i	$0.448303\pi$
$492$	0	0
$493$	−1528.50	−0.139635
$494$	0	0
$495$	−17617.3	−1.59968
$496$	0	0
$497$	17364.7	1.56723
$498$	0	0
$499$	−19259.8	−1.72783	−0.863914	−	0.503639i	$-0.831994\pi$
$499$	−19259.8	−1.72783	−0.863914	+	0.503639i	$0.831994\pi$
$500$	0	0
$501$	−192.189	−0.0171385
$502$	0	0
$503$	−4171.40	−0.369768	−0.184884	−	0.982760i	$-0.559191\pi$
$503$	−4171.40	−0.369768	−0.184884	+	0.982760i	$0.559191\pi$
$504$	0	0
$505$	4447.62	0.391914
$506$	0	0
$507$	−1437.65	−0.125934
$508$	0	0
$509$	−14191.8	−1.23584	−0.617919	−	0.786242i	$-0.712024\pi$
$509$	−14191.8	−1.23584	−0.617919	+	0.786242i	$0.712024\pi$
$510$	0	0
$511$	−17848.5	−1.54515
$512$	0	0
$513$	3078.07	0.264913
$514$	0	0
$515$	−22796.0	−1.95051
$516$	0	0
$517$	−25444.5	−2.16451
$518$	0	0
$519$	−1840.28	−0.155644
$520$	0	0
$521$	−8401.44	−0.706476	−0.353238	−	0.935534i	$-0.614919\pi$
$521$	−8401.44	−0.706476	−0.353238	+	0.935534i	$0.614919\pi$
$522$	0	0
$523$	7612.18	0.636439	0.318219	−	0.948017i	$-0.396915\pi$
$523$	7612.18	0.636439	0.318219	+	0.948017i	$0.396915\pi$
$524$	0	0
$525$	2964.24	0.246419
$526$	0	0
$527$	2528.22	0.208977
$528$	0	0
$529$	−5747.14	−0.472355
$530$	0	0
$531$	7987.45	0.652779
$532$	0	0
$533$	14504.6	1.17874
$534$	0	0
$535$	30751.8	2.48507
$536$	0	0
$537$	−5101.92	−0.409989
$538$	0	0
$539$	6701.20	0.535512
$540$	0	0
$541$	2661.53	0.211512	0.105756	−	0.994392i	$-0.466274\pi$
$541$	2661.53	0.211512	0.105756	+	0.994392i	$0.466274\pi$
$542$	0	0
$543$	−438.584	−0.0346620
$544$	0	0
$545$	20007.6	1.57253
$546$	0	0
$547$	−4554.56	−0.356013	−0.178006	−	0.984029i	$-0.556965\pi$
$547$	−4554.56	−0.356013	−0.178006	+	0.984029i	$0.556965\pi$
$548$	0	0
$549$	−15906.6	−1.23657
$550$	0	0
$551$	1311.09	0.101369
$552$	0	0
$553$	6608.69	0.508192
$554$	0	0
$555$	−1793.37	−0.137161
$556$	0	0
$557$	−4752.54	−0.361529	−0.180765	−	0.983526i	$-0.557857\pi$
$557$	−4752.54	−0.361529	−0.180765	+	0.983526i	$0.557857\pi$
$558$	0	0
$559$	−27584.9	−2.08715
$560$	0	0
$561$	3162.78	0.238026
$562$	0	0
$563$	14781.0	1.10647	0.553237	−	0.833024i	$-0.313392\pi$
$563$	14781.0	1.10647	0.553237	+	0.833024i	$0.313392\pi$
$564$	0	0
$565$	22150.6	1.64935
$566$	0	0
$567$	−13140.7	−0.973294
$568$	0	0
$569$	14584.0	1.07450	0.537252	−	0.843422i	$-0.319462\pi$
$569$	14584.0	1.07450	0.537252	+	0.843422i	$0.319462\pi$
$570$	0	0
$571$	18739.7	1.37343	0.686717	−	0.726925i	$-0.259051\pi$
$571$	18739.7	1.37343	0.686717	+	0.726925i	$0.259051\pi$
$572$	0	0
$573$	−6389.69	−0.465852
$574$	0	0
$575$	8256.78	0.598837
$576$	0	0
$577$	6024.34	0.434656	0.217328	−	0.976099i	$-0.430266\pi$
$577$	6024.34	0.434656	0.217328	+	0.976099i	$0.430266\pi$
$578$	0	0
$579$	3923.45	0.281612
$580$	0	0
$581$	−7362.28	−0.525712
$582$	0	0
$583$	12146.2	0.862855
$584$	0	0
$585$	−21957.7	−1.55187
$586$	0	0
$587$	−7334.77	−0.515739	−0.257869	−	0.966180i	$-0.583020\pi$
$587$	−7334.77	−0.515739	−0.257869	+	0.966180i	$0.583020\pi$
$588$	0	0
$589$	−2168.61	−0.151708
$590$	0	0
$591$	6823.08	0.474897
$592$	0	0
$593$	24716.4	1.71161	0.855803	−	0.517302i	$-0.173064\pi$
$593$	24716.4	1.71161	0.855803	+	0.517302i	$0.173064\pi$
$594$	0	0
$595$	17589.5	1.21193
$596$	0	0
$597$	−2605.08	−0.178591
$598$	0	0
$599$	24229.0	1.65271	0.826354	−	0.563152i	$-0.190411\pi$
$599$	24229.0	1.65271	0.826354	+	0.563152i	$0.190411\pi$
$600$	0	0
$601$	−3426.86	−0.232587	−0.116293	−	0.993215i	$-0.537101\pi$
$601$	−3426.86	−0.232587	−0.116293	+	0.993215i	$0.537101\pi$
$602$	0	0
$603$	−17553.0	−1.18543
$604$	0	0
$605$	11994.5	0.806024
$606$	0	0
$607$	−17056.6	−1.14054	−0.570269	−	0.821458i	$-0.693161\pi$
$607$	−17056.6	−1.14054	−0.570269	+	0.821458i	$0.693161\pi$
$608$	0	0
$609$	834.187	0.0555057
$610$	0	0
$611$	−31713.4	−2.09981
$612$	0	0
$613$	−11897.9	−0.783935	−0.391967	−	0.919979i	$-0.628205\pi$
$613$	−11897.9	−0.783935	−0.391967	+	0.919979i	$0.628205\pi$
$614$	0	0
$615$	−4962.07	−0.325349
$616$	0	0
$617$	−19490.8	−1.27175	−0.635874	−	0.771793i	$-0.719360\pi$
$617$	−19490.8	−1.27175	−0.635874	+	0.771793i	$0.719360\pi$
$618$	0	0
$619$	−17797.8	−1.15566	−0.577829	−	0.816158i	$-0.696100\pi$
$619$	−17797.8	−1.15566	−0.577829	+	0.816158i	$0.696100\pi$
$620$	0	0
$621$	5455.15	0.352508
$622$	0	0
$623$	−763.377	−0.0490916
$624$	0	0
$625$	−17887.0	−1.14477
$626$	0	0
$627$	−2712.92	−0.172797
$628$	0	0
$629$	−4808.70	−0.304826
$630$	0	0
$631$	8332.83	0.525712	0.262856	−	0.964835i	$-0.415336\pi$
$631$	8332.83	0.525712	0.262856	+	0.964835i	$0.415336\pi$
$632$	0	0
$633$	−2766.12	−0.173686
$634$	0	0
$635$	−21341.6	−1.33373
$636$	0	0
$637$	8352.19	0.519507
$638$	0	0
$639$	19884.5	1.23102
$640$	0	0
$641$	−20666.6	−1.27345	−0.636724	−	0.771092i	$-0.719711\pi$
$641$	−20666.6	−1.27345	−0.636724	+	0.771092i	$0.719711\pi$
$642$	0	0
$643$	−3852.70	−0.236292	−0.118146	−	0.992996i	$-0.537695\pi$
$643$	−3852.70	−0.236292	−0.118146	+	0.992996i	$0.537695\pi$
$644$	0	0
$645$	9436.84	0.576086
$646$	0	0
$647$	−7994.37	−0.485767	−0.242884	−	0.970055i	$-0.578093\pi$
$647$	−7994.37	−0.485767	−0.242884	+	0.970055i	$0.578093\pi$
$648$	0	0
$649$	−14551.1	−0.880094
$650$	0	0
$651$	−1379.79	−0.0830693
$652$	0	0
$653$	1626.28	0.0974598	0.0487299	−	0.998812i	$-0.484483\pi$
$653$	1626.28	0.0974598	0.0487299	+	0.998812i	$0.484483\pi$
$654$	0	0
$655$	25761.1	1.53675
$656$	0	0
$657$	−20438.6	−1.21368
$658$	0	0
$659$	−11582.8	−0.684676	−0.342338	−	0.939577i	$-0.611219\pi$
$659$	−11582.8	−0.684676	−0.342338	+	0.939577i	$0.611219\pi$
$660$	0	0
$661$	−21487.1	−1.26438	−0.632188	−	0.774815i	$-0.717843\pi$
$661$	−21487.1	−1.26438	−0.632188	+	0.774815i	$0.717843\pi$
$662$	0	0
$663$	3942.00	0.230912
$664$	0	0
$665$	−15087.6	−0.879808
$666$	0	0
$667$	2323.60	0.134888
$668$	0	0
$669$	−1293.57	−0.0747566
$670$	0	0
$671$	28977.8	1.66718
$672$	0	0
$673$	12392.1	0.709775	0.354888	−	0.934909i	$-0.384519\pi$
$673$	12392.1	0.709775	0.354888	+	0.934909i	$0.384519\pi$
$674$	0	0
$675$	7016.03	0.400070
$676$	0	0
$677$	−19645.9	−1.11529	−0.557646	−	0.830079i	$-0.688295\pi$
$677$	−19645.9	−1.11529	−0.557646	+	0.830079i	$0.688295\pi$
$678$	0	0
$679$	−20530.8	−1.16038
$680$	0	0
$681$	4752.31	0.267414
$682$	0	0
$683$	10245.3	0.573978	0.286989	−	0.957934i	$-0.407346\pi$
$683$	10245.3	0.573978	0.286989	+	0.957934i	$0.407346\pi$
$684$	0	0
$685$	−9911.47	−0.552844
$686$	0	0
$687$	4828.43	0.268145
$688$	0	0
$689$	15138.7	0.837066
$690$	0	0
$691$	32913.1	1.81197	0.905986	−	0.423307i	$-0.139131\pi$
$691$	32913.1	1.81197	0.905986	+	0.423307i	$0.139131\pi$
$692$	0	0
$693$	25780.7	1.41317
$694$	0	0
$695$	−1120.74	−0.0611685
$696$	0	0
$697$	−13305.2	−0.723055
$698$	0	0
$699$	−3219.25	−0.174196
$700$	0	0
$701$	−8561.99	−0.461315	−0.230658	−	0.973035i	$-0.574088\pi$
$701$	−8561.99	−0.461315	−0.230658	+	0.973035i	$0.574088\pi$
$702$	0	0
$703$	4124.73	0.221290
$704$	0	0
$705$	10849.2	0.579581
$706$	0	0
$707$	−6508.53	−0.346221
$708$	0	0
$709$	−37078.0	−1.96402	−0.982012	−	0.188817i	$-0.939535\pi$
$709$	−37078.0	−1.96402	−0.982012	+	0.188817i	$0.939535\pi$
$710$	0	0
$711$	7567.69	0.399171
$712$	0	0
$713$	−3843.35	−0.201872
$714$	0	0
$715$	40001.5	2.09227
$716$	0	0
$717$	3180.27	0.165648
$718$	0	0
$719$	−12120.0	−0.628650	−0.314325	−	0.949315i	$-0.601778\pi$
$719$	−12120.0	−0.628650	−0.314325	+	0.949315i	$0.601778\pi$
$720$	0	0
$721$	33359.1	1.72310
$722$	0	0
$723$	−3622.72	−0.186349
$724$	0	0
$725$	2988.45	0.153087
$726$	0	0
$727$	−30947.7	−1.57880	−0.789400	−	0.613879i	$-0.789608\pi$
$727$	−30947.7	−1.57880	−0.789400	+	0.613879i	$0.789608\pi$
$728$	0	0
$729$	−12654.8	−0.642931
$730$	0	0
$731$	25303.7	1.28029
$732$	0	0
$733$	25598.9	1.28993	0.644963	−	0.764213i	$-0.276873\pi$
$733$	25598.9	1.28993	0.644963	+	0.764213i	$0.276873\pi$
$734$	0	0
$735$	−2857.30	−0.143392
$736$	0	0
$737$	31977.1	1.59822
$738$	0	0
$739$	21931.6	1.09170	0.545851	−	0.837882i	$-0.316207\pi$
$739$	21931.6	1.09170	0.545851	+	0.837882i	$0.316207\pi$
$740$	0	0
$741$	−3381.31	−0.167632
$742$	0	0
$743$	27774.4	1.37139	0.685695	−	0.727889i	$-0.259498\pi$
$743$	27774.4	1.37139	0.685695	+	0.727889i	$0.259498\pi$
$744$	0	0
$745$	−17443.4	−0.857821
$746$	0	0
$747$	−8430.64	−0.412933
$748$	0	0
$749$	−45001.3	−2.19534
$750$	0	0
$751$	−11111.6	−0.539906	−0.269953	−	0.962873i	$-0.587008\pi$
$751$	−11111.6	−0.539906	−0.269953	+	0.962873i	$0.587008\pi$
$752$	0	0
$753$	−4765.99	−0.230654
$754$	0	0
$755$	−1161.05	−0.0559668
$756$	0	0
$757$	−12323.1	−0.591666	−0.295833	−	0.955240i	$-0.595597\pi$
$757$	−12323.1	−0.591666	−0.295833	+	0.955240i	$0.595597\pi$
$758$	0	0
$759$	−4808.00	−0.229933
$760$	0	0
$761$	−7634.91	−0.363686	−0.181843	−	0.983328i	$-0.558206\pi$
$761$	−7634.91	−0.363686	−0.181843	+	0.983328i	$0.558206\pi$
$762$	0	0
$763$	−29278.5	−1.38919
$764$	0	0
$765$	20141.9	0.951938
$766$	0	0
$767$	−18136.1	−0.853790
$768$	0	0
$769$	9070.83	0.425361	0.212680	−	0.977122i	$-0.431781\pi$
$769$	9070.83	0.425361	0.212680	+	0.977122i	$0.431781\pi$
$770$	0	0
$771$	−6989.57	−0.326489
$772$	0	0
$773$	−27518.9	−1.28045	−0.640224	−	0.768188i	$-0.721159\pi$
$773$	−27518.9	−1.28045	−0.640224	+	0.768188i	$0.721159\pi$
$774$	0	0
$775$	−4943.05	−0.229109
$776$	0	0
$777$	2624.37	0.121170
$778$	0	0
$779$	11412.7	0.524906
$780$	0	0
$781$	−36224.5	−1.65969
$782$	0	0
$783$	1974.43	0.0901155
$784$	0	0
$785$	22089.4	1.00434
$786$	0	0
$787$	−34518.8	−1.56348	−0.781742	−	0.623602i	$-0.785669\pi$
$787$	−34518.8	−1.56348	−0.781742	+	0.623602i	$0.785669\pi$
$788$	0	0
$789$	4298.87	0.193972
$790$	0	0
$791$	−32414.6	−1.45706
$792$	0	0
$793$	36117.1	1.61735
$794$	0	0
$795$	−5178.98	−0.231043
$796$	0	0
$797$	−3193.61	−0.141937	−0.0709684	−	0.997479i	$-0.522609\pi$
$797$	−3193.61	−0.141937	−0.0709684	+	0.997479i	$0.522609\pi$
$798$	0	0
$799$	29090.8	1.28806
$800$	0	0
$801$	−874.153	−0.0385601
$802$	0	0
$803$	37233.9	1.63631
$804$	0	0
$805$	−26739.2	−1.17072
$806$	0	0
$807$	−8969.03	−0.391233
$808$	0	0
$809$	3218.15	0.139857	0.0699284	−	0.997552i	$-0.477723\pi$
$809$	3218.15	0.139857	0.0699284	+	0.997552i	$0.477723\pi$
$810$	0	0
$811$	6376.50	0.276090	0.138045	−	0.990426i	$-0.455918\pi$
$811$	6376.50	0.276090	0.138045	+	0.990426i	$0.455918\pi$
$812$	0	0
$813$	−1639.59	−0.0707294
$814$	0	0
$815$	18201.3	0.782287
$816$	0	0
$817$	−21704.6	−0.929434
$818$	0	0
$819$	32132.4	1.37094
$820$	0	0
$821$	8106.11	0.344586	0.172293	−	0.985046i	$-0.444882\pi$
$821$	8106.11	0.344586	0.172293	+	0.985046i	$0.444882\pi$
$822$	0	0
$823$	7634.99	0.323377	0.161688	−	0.986842i	$-0.448306\pi$
$823$	7634.99	0.323377	0.161688	+	0.986842i	$0.448306\pi$
$824$	0	0
$825$	−6183.71	−0.260957
$826$	0	0
$827$	−30618.6	−1.28744	−0.643721	−	0.765260i	$-0.722610\pi$
$827$	−30618.6	−1.28744	−0.643721	+	0.765260i	$0.722610\pi$
$828$	0	0
$829$	28104.9	1.17747	0.588736	−	0.808325i	$-0.299626\pi$
$829$	28104.9	1.17747	0.588736	+	0.808325i	$0.299626\pi$
$830$	0	0
$831$	−5627.82	−0.234930
$832$	0	0
$833$	−7661.49	−0.318674
$834$	0	0
$835$	2229.71	0.0924100
$836$	0	0
$837$	−3265.81	−0.134866
$838$	0	0
$839$	27574.4	1.13465	0.567326	−	0.823493i	$-0.307978\pi$
$839$	27574.4	1.13465	0.567326	+	0.823493i	$0.307978\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	−11352.6	−0.463825
$844$	0	0
$845$	16679.1	0.679029
$846$	0	0
$847$	−17552.4	−0.712051
$848$	0	0
$849$	−3251.34	−0.131432
$850$	0	0
$851$	7310.10	0.294462
$852$	0	0
$853$	−48368.8	−1.94152	−0.970760	−	0.240052i	$-0.922836\pi$
$853$	−48368.8	−1.94152	−0.970760	+	0.240052i	$0.922836\pi$
$854$	0	0
$855$	−17277.0	−0.691065
$856$	0	0
$857$	−27292.3	−1.08785	−0.543924	−	0.839134i	$-0.683062\pi$
$857$	−27292.3	−1.08785	−0.543924	+	0.839134i	$0.683062\pi$
$858$	0	0
$859$	14609.4	0.580286	0.290143	−	0.956983i	$-0.406297\pi$
$859$	14609.4	0.580286	0.290143	+	0.956983i	$0.406297\pi$
$860$	0	0
$861$	7261.36	0.287418
$862$	0	0
$863$	−46715.8	−1.84267	−0.921334	−	0.388771i	$-0.872900\pi$
$863$	−46715.8	−1.84267	−0.921334	+	0.388771i	$0.872900\pi$
$864$	0	0
$865$	21350.3	0.839227
$866$	0	0
$867$	2779.01	0.108858
$868$	0	0
$869$	−13786.4	−0.538173
$870$	0	0
$871$	39855.4	1.55046
$872$	0	0
$873$	−23510.1	−0.911449
$874$	0	0
$875$	7325.23	0.283015
$876$	0	0
$877$	37316.5	1.43682	0.718408	−	0.695622i	$-0.244871\pi$
$877$	37316.5	1.43682	0.718408	+	0.695622i	$0.244871\pi$
$878$	0	0
$879$	−1625.45	−0.0623721
$880$	0	0
$881$	40989.9	1.56752	0.783759	−	0.621065i	$-0.213300\pi$
$881$	40989.9	1.56752	0.783759	+	0.621065i	$0.213300\pi$
$882$	0	0
$883$	−16505.0	−0.629035	−0.314517	−	0.949252i	$-0.601843\pi$
$883$	−16505.0	−0.629035	−0.314517	+	0.949252i	$0.601843\pi$
$884$	0	0
$885$	6204.40	0.235659
$886$	0	0
$887$	−23388.7	−0.885362	−0.442681	−	0.896679i	$-0.645973\pi$
$887$	−23388.7	−0.885362	−0.442681	+	0.896679i	$0.645973\pi$
$888$	0	0
$889$	31230.7	1.17823
$890$	0	0
$891$	27412.9	1.03072
$892$	0	0
$893$	−24953.0	−0.935074
$894$	0	0
$895$	59190.7	2.21064
$896$	0	0
$897$	−5992.56	−0.223061
$898$	0	0
$899$	−1391.06	−0.0516066
$900$	0	0
$901$	−13886.8	−0.513469
$902$	0	0
$903$	−13809.6	−0.508921
$904$	0	0
$905$	5088.30	0.186896
$906$	0	0
$907$	−39865.7	−1.45945	−0.729723	−	0.683743i	$-0.760351\pi$
$907$	−39865.7	−1.45945	−0.729723	+	0.683743i	$0.760351\pi$
$908$	0	0
$909$	−7453.00	−0.271948
$910$	0	0
$911$	−16794.3	−0.610781	−0.305390	−	0.952227i	$-0.598787\pi$
$911$	−16794.3	−0.610781	−0.305390	+	0.952227i	$0.598787\pi$
$912$	0	0
$913$	15358.5	0.556728
$914$	0	0
$915$	−12355.7	−0.446413
$916$	0	0
$917$	−37698.1	−1.35758
$918$	0	0
$919$	39775.6	1.42772	0.713861	−	0.700287i	$-0.246945\pi$
$919$	39775.6	1.42772	0.713861	+	0.700287i	$0.246945\pi$
$920$	0	0
$921$	287.807	0.0102970
$922$	0	0
$923$	−45149.3	−1.61008
$924$	0	0
$925$	9401.74	0.334192
$926$	0	0
$927$	38199.9	1.35345
$928$	0	0
$929$	−19272.0	−0.680616	−0.340308	−	0.940314i	$-0.610531\pi$
$929$	−19272.0	−0.680616	−0.340308	+	0.940314i	$0.610531\pi$
$930$	0	0
$931$	6571.75	0.231343
$932$	0	0
$933$	−5434.17	−0.190682
$934$	0	0
$935$	−36693.5	−1.28343
$936$	0	0
$937$	38117.4	1.32897	0.664483	−	0.747303i	$-0.268652\pi$
$937$	38117.4	1.32897	0.664483	+	0.747303i	$0.268652\pi$
$938$	0	0
$939$	−2411.26	−0.0838005
$940$	0	0
$941$	−41006.4	−1.42058	−0.710292	−	0.703907i	$-0.751437\pi$
$941$	−41006.4	−1.42058	−0.710292	+	0.703907i	$0.751437\pi$
$942$	0	0
$943$	20226.3	0.698470
$944$	0	0
$945$	−22721.1	−0.782135
$946$	0	0
$947$	−10726.2	−0.368063	−0.184031	−	0.982920i	$-0.558915\pi$
$947$	−10726.2	−0.368063	−0.184031	+	0.982920i	$0.558915\pi$
$948$	0	0
$949$	46407.3	1.58740
$950$	0	0
$951$	−9654.66	−0.329205
$952$	0	0
$953$	43028.4	1.46257	0.731283	−	0.682074i	$-0.238922\pi$
$953$	43028.4	1.46257	0.731283	+	0.682074i	$0.238922\pi$
$954$	0	0
$955$	74131.0	2.51185
$956$	0	0
$957$	−1740.20	−0.0587803
$958$	0	0
$959$	14504.2	0.488389
$960$	0	0
$961$	−27490.1	−0.922766
$962$	0	0
$963$	−51531.6	−1.72438
$964$	0	0
$965$	−45518.5	−1.51844
$966$	0	0
$967$	−47920.1	−1.59359	−0.796797	−	0.604247i	$-0.793474\pi$
$967$	−47920.1	−1.59359	−0.796797	+	0.604247i	$0.793474\pi$
$968$	0	0
$969$	3101.68	0.102828
$970$	0	0
$971$	47501.5	1.56992	0.784961	−	0.619545i	$-0.212683\pi$
$971$	47501.5	1.56992	0.784961	+	0.619545i	$0.212683\pi$
$972$	0	0
$973$	1640.06	0.0540370
$974$	0	0
$975$	−7707.21	−0.253157
$976$	0	0
$977$	46081.7	1.50899	0.754496	−	0.656305i	$-0.227882\pi$
$977$	46081.7	1.50899	0.754496	+	0.656305i	$0.227882\pi$
$978$	0	0
$979$	1592.49	0.0519878
$980$	0	0
$981$	−33527.2	−1.09117
$982$	0	0
$983$	−22330.4	−0.724545	−0.362273	−	0.932072i	$-0.617999\pi$
$983$	−22330.4	−0.724545	−0.362273	+	0.932072i	$0.617999\pi$
$984$	0	0
$985$	−79159.1	−2.56063
$986$	0	0
$987$	−15876.4	−0.512009
$988$	0	0
$989$	−38466.2	−1.23676
$990$	0	0
$991$	37242.5	1.19379	0.596895	−	0.802319i	$-0.296401\pi$
$991$	37242.5	1.19379	0.596895	+	0.802319i	$0.296401\pi$
$992$	0	0
$993$	−26.7997	−0.000856460 0
$994$	0	0
$995$	30223.2	0.962955
$996$	0	0
$997$	−27326.5	−0.868043	−0.434021	−	0.900903i	$-0.642906\pi$
$997$	−27326.5	−0.868043	−0.434021	+	0.900903i	$0.642906\pi$
$998$	0	0
$999$	6211.61	0.196723

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
928.4.a.i.1.6	✓	12	8.3	odd	2
928.4.a.i.1.7	yes	12	8.5	even	2
1856.4.a.bk.1.6		12	1.1	even	1	trivial
1856.4.a.bk.1.7		12	4.3	odd	2	inner

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)

\(f(q)\)	\(=\)	\(q-1.30165 q^{3} +15.1013 q^{5} -22.0989 q^{7} -25.3057 q^{9} +O(q^{10})\)	\(q-1.30165 q^{3} +15.1013 q^{5} -22.0989 q^{7} -25.3057 q^{9} +46.1006 q^{11} +57.4585 q^{13} -19.6567 q^{15} -52.7069 q^{17} +45.2101 q^{19} +28.7651 q^{21} +80.1240 q^{23} +103.050 q^{25} +68.0838 q^{27} +29.0000 q^{29} -47.9675 q^{31} -60.0070 q^{33} -333.722 q^{35} +91.2348 q^{37} -74.7910 q^{39} +252.437 q^{41} -480.084 q^{43} -382.150 q^{45} -551.935 q^{47} +145.360 q^{49} +68.6061 q^{51} +263.472 q^{53} +696.180 q^{55} -58.8478 q^{57} -315.638 q^{59} +628.577 q^{61} +559.228 q^{63} +867.700 q^{65} +693.637 q^{67} -104.294 q^{69} -785.772 q^{71} +807.667 q^{73} -134.135 q^{75} -1018.77 q^{77} -299.051 q^{79} +594.632 q^{81} +333.152 q^{83} -795.944 q^{85} -37.7479 q^{87} +34.5437 q^{89} -1269.77 q^{91} +62.4370 q^{93} +682.732 q^{95} +929.042 q^{97} -1166.61 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 10 q^{5} + 166 q^{9}+O(q^{10}) \) 12 * q - 10 * q^5 + 166 * q^9	\( 12 q - 10 q^{5} + 166 q^{9} - 30 q^{13} + 336 q^{17} - 56 q^{21} + 294 q^{25} + 348 q^{29} + 214 q^{33} - 196 q^{37} + 940 q^{41} - 1672 q^{45} + 972 q^{49} + 406 q^{53} + 1224 q^{57} - 400 q^{61} + 2998 q^{65} - 1004 q^{69} + 2252 q^{73} - 3032 q^{77} + 3932 q^{81} - 5564 q^{85} + 5212 q^{89} - 3722 q^{93} + 6164 q^{97}+O(q^{100}) \) 12 * q - 10 * q^5 + 166 * q^9 - 30 * q^13 + 336 * q^17 - 56 * q^21 + 294 * q^25 + 348 * q^29 + 214 * q^33 - 196 * q^37 + 940 * q^41 - 1672 * q^45 + 972 * q^49 + 406 * q^53 + 1224 * q^57 - 400 * q^61 + 2998 * q^65 - 1004 * q^69 + 2252 * q^73 - 3032 * q^77 + 3932 * q^81 - 5564 * q^85 + 5212 * q^89 - 3722 * q^93 + 6164 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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