LMFDB - Embedded newform 480.4.a.q.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [480,4,Mod(1,480)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(480, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("480.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 480 = 2^{5} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	480.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:	$28.3209168028$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{41}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 10 $ x^2 - x - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.70156$ of defining polynomial
Character	$\chi$	$=$	480.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} -5.00000 q^{5} +18.8062 q^{7} +9.00000 q^{9} -63.2250 q^{11} +1.58125 q^{13} -15.0000 q^{15} +135.256 q^{17} +97.2562 q^{19} +56.4187 q^{21} +41.1938 q^{23} +25.0000 q^{25} +27.0000 q^{27} -207.675 q^{29} +193.969 q^{31} -189.675 q^{33} -94.0312 q^{35} +339.256 q^{37} +4.74376 q^{39} -490.187 q^{41} +74.3250 q^{43} -45.0000 q^{45} +544.481 q^{47} +10.6750 q^{49} +405.769 q^{51} +663.862 q^{53} +316.125 q^{55} +291.769 q^{57} +344.775 q^{59} +5.16251 q^{61} +169.256 q^{63} -7.90627 q^{65} -671.475 q^{67} +123.581 q^{69} +425.550 q^{71} -94.8375 q^{73} +75.0000 q^{75} -1189.02 q^{77} +770.681 q^{79} +81.0000 q^{81} +589.800 q^{83} -676.281 q^{85} -623.025 q^{87} -409.813 q^{89} +29.7375 q^{91} +581.906 q^{93} -486.281 q^{95} -152.050 q^{97} -569.025 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{3} - 10 q^{5} + 12 q^{7} + 18 q^{9} - 24 q^{11} + 80 q^{13} - 30 q^{15} + 40 q^{17} - 36 q^{19} + 36 q^{21} + 108 q^{23} + 50 q^{25} + 54 q^{27} - 108 q^{29} + 516 q^{31} - 72 q^{33} - 60 q^{35}+ \cdots - 216 q^{99}+O(q^{100}) $ 2 * q + 6 * q^3 - 10 * q^5 + 12 * q^7 + 18 * q^9 - 24 * q^11 + 80 * q^13 - 30 * q^15 + 40 * q^17 - 36 * q^19 + 36 * q^21 + 108 * q^23 + 50 * q^25 + 54 * q^27 - 108 * q^29 + 516 * q^31 - 72 * q^33 - 60 * q^35 + 448 * q^37 + 240 * q^39 - 212 * q^41 + 456 * q^43 - 90 * q^45 + 756 * q^47 - 286 * q^49 + 120 * q^51 + 252 * q^53 + 120 * q^55 - 108 * q^57 + 792 * q^59 + 164 * q^61 + 108 * q^63 - 400 * q^65 - 216 * q^67 + 324 * q^69 + 1056 * q^71 - 36 * q^73 + 150 * q^75 - 1456 * q^77 + 2028 * q^79 + 162 * q^81 + 360 * q^83 - 200 * q^85 - 324 * q^87 - 1588 * q^89 - 504 * q^91 + 1548 * q^93 + 180 * q^95 + 1540 * q^97 - 216 * q^99

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
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	18.8062	1.01544	0.507721	−	0.861522i	$-0.330488\pi$
$7$	18.8062	1.01544	0.507721	+	0.861522i	$0.330488\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−63.2250	−1.73300	−0.866502	−	0.499173i	$-0.833637\pi$
$11$	−63.2250	−1.73300	−0.866502	+	0.499173i	$0.833637\pi$
$12$	0	0
$13$	1.58125	0.0337355	0.0168677	−	0.999858i	$-0.494631\pi$
$13$	1.58125	0.0337355	0.0168677	+	0.999858i	$0.494631\pi$
$14$	0	0
$15$	−15.0000	−0.258199
$16$	0	0
$17$	135.256	1.92967	0.964837	−	0.262849i	$-0.0846622\pi$
$17$	135.256	1.92967	0.964837	+	0.262849i	$0.0846622\pi$
$18$	0	0
$19$	97.2562	1.17432	0.587161	−	0.809470i	$-0.300246\pi$
$19$	97.2562	1.17432	0.587161	+	0.809470i	$0.300246\pi$
$20$	0	0
$21$	56.4187	0.586266
$22$	0	0
$23$	41.1938	0.373456	0.186728	−	0.982412i	$-0.440212\pi$
$23$	41.1938	0.373456	0.186728	+	0.982412i	$0.440212\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−207.675	−1.32980	−0.664901	−	0.746931i	$-0.731526\pi$
$29$	−207.675	−1.32980	−0.664901	+	0.746931i	$0.731526\pi$
$30$	0	0
$31$	193.969	1.12380	0.561900	−	0.827205i	$-0.310070\pi$
$31$	193.969	1.12380	0.561900	+	0.827205i	$0.310070\pi$
$32$	0	0
$33$	−189.675	−1.00055
$34$	0	0
$35$	−94.0312	−0.454119
$36$	0	0
$37$	339.256	1.50739	0.753694	−	0.657225i	$-0.228270\pi$
$37$	339.256	1.50739	0.753694	+	0.657225i	$0.228270\pi$
$38$	0	0
$39$	4.74376	0.0194772
$40$	0	0
$41$	−490.187	−1.86718	−0.933590	−	0.358342i	$-0.883342\pi$
$41$	−490.187	−1.86718	−0.933590	+	0.358342i	$0.883342\pi$
$42$	0	0
$43$	74.3250	0.263592	0.131796	−	0.991277i	$-0.457926\pi$
$43$	74.3250	0.263592	0.131796	+	0.991277i	$0.457926\pi$
$44$	0	0
$45$	−45.0000	−0.149071
$46$	0	0
$47$	544.481	1.68980	0.844902	−	0.534922i	$-0.179659\pi$
$47$	544.481	1.68980	0.844902	+	0.534922i	$0.179659\pi$
$48$	0	0
$49$	10.6750	0.0311224
$50$	0	0
$51$	405.769	1.11410
$52$	0	0
$53$	663.862	1.72054	0.860269	−	0.509840i	$-0.170296\pi$
$53$	663.862	1.72054	0.860269	+	0.509840i	$0.170296\pi$
$54$	0	0
$55$	316.125	0.775023
$56$	0	0
$57$	291.769	0.677995
$58$	0	0
$59$	344.775	0.760778	0.380389	−	0.924827i	$-0.375790\pi$
$59$	344.775	0.760778	0.380389	+	0.924827i	$0.375790\pi$
$60$	0	0
$61$	5.16251	0.0108359	0.00541796	−	0.999985i	$-0.498275\pi$
$61$	5.16251	0.0108359	0.00541796	+	0.999985i	$0.498275\pi$
$62$	0	0
$63$	169.256	0.338481
$64$	0	0
$65$	−7.90627	−0.0150870
$66$	0	0
$67$	−671.475	−1.22438	−0.612192	−	0.790709i	$-0.709712\pi$
$67$	−671.475	−1.22438	−0.612192	+	0.790709i	$0.709712\pi$
$68$	0	0
$69$	123.581	0.215615
$70$	0	0
$71$	425.550	0.711317	0.355658	−	0.934616i	$-0.384257\pi$
$71$	425.550	0.711317	0.355658	+	0.934616i	$0.384257\pi$
$72$	0	0
$73$	−94.8375	−0.152053	−0.0760266	−	0.997106i	$-0.524223\pi$
$73$	−94.8375	−0.152053	−0.0760266	+	0.997106i	$0.524223\pi$
$74$	0	0
$75$	75.0000	0.115470
$76$	0	0
$77$	−1189.02	−1.75977
$78$	0	0
$79$	770.681	1.09757	0.548787	−	0.835962i	$-0.315090\pi$
$79$	770.681	1.09757	0.548787	+	0.835962i	$0.315090\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	589.800	0.779987	0.389994	−	0.920818i	$-0.372477\pi$
$83$	589.800	0.779987	0.389994	+	0.920818i	$0.372477\pi$
$84$	0	0
$85$	−676.281	−0.862976
$86$	0	0
$87$	−623.025	−0.767762
$88$	0	0
$89$	−409.813	−0.488090	−0.244045	−	0.969764i	$-0.578475\pi$
$89$	−409.813	−0.488090	−0.244045	+	0.969764i	$0.578475\pi$
$90$	0	0
$91$	29.7375	0.0342564
$92$	0	0
$93$	581.906	0.648827
$94$	0	0
$95$	−486.281	−0.525173
$96$	0	0
$97$	−152.050	−0.159158	−0.0795790	−	0.996829i	$-0.525358\pi$
$97$	−152.050	−0.159158	−0.0795790	+	0.996829i	$0.525358\pi$
$98$	0	0
$99$	−569.025	−0.577668
$100$	0	0
$101$	−484.325	−0.477150	−0.238575	−	0.971124i	$-0.576680\pi$
$101$	−484.325	−0.477150	−0.238575	+	0.971124i	$0.576680\pi$
$102$	0	0
$103$	1617.51	1.54736	0.773678	−	0.633579i	$-0.218415\pi$
$103$	1617.51	1.54736	0.773678	+	0.633579i	$0.218415\pi$
$104$	0	0
$105$	−282.094	−0.262186
$106$	0	0
$107$	1010.36	0.912854	0.456427	−	0.889761i	$-0.349129\pi$
$107$	1010.36	0.912854	0.456427	+	0.889761i	$0.349129\pi$
$108$	0	0
$109$	1.72487	0.00151571	0.000757857	−	1.00000i	$-0.499759\pi$
$109$	1.72487	0.00151571	0.000757857		1.00000i	$0.499759\pi$
$110$	0	0
$111$	1017.77	0.870291
$112$	0	0
$113$	−729.119	−0.606989	−0.303494	−	0.952833i	$-0.598153\pi$
$113$	−729.119	−0.606989	−0.303494	+	0.952833i	$0.598153\pi$
$114$	0	0
$115$	−205.969	−0.167015
$116$	0	0
$117$	14.2313	0.0112452
$118$	0	0
$119$	2543.66	1.95947
$120$	0	0
$121$	2666.40	2.00331
$122$	0	0
$123$	−1470.56	−1.07802
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	388.481	0.271434	0.135717	−	0.990748i	$-0.456666\pi$
$127$	388.481	0.271434	0.135717	+	0.990748i	$0.456666\pi$
$128$	0	0
$129$	222.975	0.152185
$130$	0	0
$131$	−2777.06	−1.85216	−0.926080	−	0.377326i	$-0.876844\pi$
$131$	−2777.06	−1.85216	−0.926080	+	0.377326i	$0.876844\pi$
$132$	0	0
$133$	1829.02	1.19246
$134$	0	0
$135$	−135.000	−0.0860663
$136$	0	0
$137$	173.119	0.107960	0.0539800	−	0.998542i	$-0.482809\pi$
$137$	173.119	0.107960	0.0539800	+	0.998542i	$0.482809\pi$
$138$	0	0
$139$	−450.806	−0.275086	−0.137543	−	0.990496i	$-0.543920\pi$
$139$	−450.806	−0.275086	−0.137543	+	0.990496i	$0.543920\pi$
$140$	0	0
$141$	1633.44	0.975608
$142$	0	0
$143$	−99.9748	−0.0584637
$144$	0	0
$145$	1038.37	0.594706
$146$	0	0
$147$	32.0249	0.0179685
$148$	0	0
$149$	−1059.02	−0.582273	−0.291137	−	0.956681i	$-0.594033\pi$
$149$	−1059.02	−0.582273	−0.291137	+	0.956681i	$0.594033\pi$
$150$	0	0
$151$	−1273.63	−0.686402	−0.343201	−	0.939262i	$-0.611511\pi$
$151$	−1273.63	−0.686402	−0.343201	+	0.939262i	$0.611511\pi$
$152$	0	0
$153$	1217.31	0.643225
$154$	0	0
$155$	−969.844	−0.502579
$156$	0	0
$157$	−1442.61	−0.733328	−0.366664	−	0.930353i	$-0.619500\pi$
$157$	−1442.61	−0.733328	−0.366664	+	0.930353i	$0.619500\pi$
$158$	0	0
$159$	1991.59	0.993353
$160$	0	0
$161$	774.700	0.379223
$162$	0	0
$163$	2440.35	1.17266	0.586328	−	0.810074i	$-0.300573\pi$
$163$	2440.35	1.17266	0.586328	+	0.810074i	$0.300573\pi$
$164$	0	0
$165$	948.375	0.447460
$166$	0	0
$167$	−3655.18	−1.69369	−0.846846	−	0.531839i	$-0.821501\pi$
$167$	−3655.18	−1.69369	−0.846846	+	0.531839i	$0.821501\pi$
$168$	0	0
$169$	−2194.50	−0.998862
$170$	0	0
$171$	875.306	0.391441
$172$	0	0
$173$	−3014.75	−1.32490	−0.662449	−	0.749107i	$-0.730483\pi$
$173$	−3014.75	−1.32490	−0.662449	+	0.749107i	$0.730483\pi$
$174$	0	0
$175$	470.156	0.203088
$176$	0	0
$177$	1034.33	0.439235
$178$	0	0
$179$	−1683.04	−0.702772	−0.351386	−	0.936231i	$-0.614289\pi$
$179$	−1683.04	−0.702772	−0.351386	+	0.936231i	$0.614289\pi$
$180$	0	0
$181$	−2507.12	−1.02958	−0.514788	−	0.857318i	$-0.672129\pi$
$181$	−2507.12	−1.02958	−0.514788	+	0.857318i	$0.672129\pi$
$182$	0	0
$183$	15.4875	0.00625613
$184$	0	0
$185$	−1696.28	−0.674125
$186$	0	0
$187$	−8551.57	−3.34413
$188$	0	0
$189$	507.769	0.195422
$190$	0	0
$191$	3251.17	1.23166	0.615829	−	0.787880i	$-0.288821\pi$
$191$	3251.17	1.23166	0.615829	+	0.787880i	$0.288821\pi$
$192$	0	0
$193$	2468.89	0.920800	0.460400	−	0.887712i	$-0.347706\pi$
$193$	2468.89	0.920800	0.460400	+	0.887712i	$0.347706\pi$
$194$	0	0
$195$	−23.7188	−0.00871046
$196$	0	0
$197$	155.212	0.0561341	0.0280671	−	0.999606i	$-0.491065\pi$
$197$	155.212	0.0561341	0.0280671	+	0.999606i	$0.491065\pi$
$198$	0	0
$199$	−2117.98	−0.754471	−0.377235	−	0.926117i	$-0.623125\pi$
$199$	−2117.98	−0.754471	−0.377235	+	0.926117i	$0.623125\pi$
$200$	0	0
$201$	−2014.42	−0.706898
$202$	0	0
$203$	−3905.59	−1.35034
$204$	0	0
$205$	2450.94	0.835029
$206$	0	0
$207$	370.744	0.124485
$208$	0	0
$209$	−6149.02	−2.03511
$210$	0	0
$211$	613.444	0.200148	0.100074	−	0.994980i	$-0.468092\pi$
$211$	613.444	0.200148	0.100074	+	0.994980i	$0.468092\pi$
$212$	0	0
$213$	1276.65	0.410679
$214$	0	0
$215$	−371.625	−0.117882
$216$	0	0
$217$	3647.82	1.14115
$218$	0	0
$219$	−284.512	−0.0877880
$220$	0	0
$221$	213.875	0.0650985
$222$	0	0
$223$	−1153.26	−0.346313	−0.173156	−	0.984894i	$-0.555397\pi$
$223$	−1153.26	−0.346313	−0.173156	+	0.984894i	$0.555397\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	−4873.65	−1.42500	−0.712501	−	0.701671i	$-0.752438\pi$
$227$	−4873.65	−1.42500	−0.712501	+	0.701671i	$0.752438\pi$
$228$	0	0
$229$	597.163	0.172321	0.0861607	−	0.996281i	$-0.472540\pi$
$229$	597.163	0.172321	0.0861607	+	0.996281i	$0.472540\pi$
$230$	0	0
$231$	−3567.07	−1.01600
$232$	0	0
$233$	−5302.61	−1.49092	−0.745462	−	0.666548i	$-0.767771\pi$
$233$	−5302.61	−1.49092	−0.745462	+	0.666548i	$0.767771\pi$
$234$	0	0
$235$	−2722.41	−0.755703
$236$	0	0
$237$	2312.04	0.633685
$238$	0	0
$239$	3571.31	0.966565	0.483282	−	0.875464i	$-0.339444\pi$
$239$	3571.31	0.966565	0.483282	+	0.875464i	$0.339444\pi$
$240$	0	0
$241$	4796.98	1.28216	0.641080	−	0.767474i	$-0.278487\pi$
$241$	4796.98	1.28216	0.641080	+	0.767474i	$0.278487\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	−53.3749	−0.0139184
$246$	0	0
$247$	153.787	0.0396163
$248$	0	0
$249$	1769.40	0.450326
$250$	0	0
$251$	2441.44	0.613953	0.306976	−	0.951717i	$-0.400683\pi$
$251$	2441.44	0.613953	0.306976	+	0.951717i	$0.400683\pi$
$252$	0	0
$253$	−2604.47	−0.647201
$254$	0	0
$255$	−2028.84	−0.498240
$256$	0	0
$257$	−4163.17	−1.01047	−0.505236	−	0.862981i	$-0.668595\pi$
$257$	−4163.17	−1.01047	−0.505236	+	0.862981i	$0.668595\pi$
$258$	0	0
$259$	6380.14	1.53067
$260$	0	0
$261$	−1869.07	−0.443268
$262$	0	0
$263$	−543.694	−0.127474	−0.0637369	−	0.997967i	$-0.520302\pi$
$263$	−543.694	−0.127474	−0.0637369	+	0.997967i	$0.520302\pi$
$264$	0	0
$265$	−3319.31	−0.769448
$266$	0	0
$267$	−1229.44	−0.281799
$268$	0	0
$269$	992.325	0.224919	0.112459	−	0.993656i	$-0.464127\pi$
$269$	992.325	0.224919	0.112459	+	0.993656i	$0.464127\pi$
$270$	0	0
$271$	−5302.11	−1.18849	−0.594244	−	0.804285i	$-0.702548\pi$
$271$	−5302.11	−1.18849	−0.594244	+	0.804285i	$0.702548\pi$
$272$	0	0
$273$	89.2124	0.0197779
$274$	0	0
$275$	−1580.62	−0.346601
$276$	0	0
$277$	−929.107	−0.201533	−0.100766	−	0.994910i	$-0.532129\pi$
$277$	−929.107	−0.201533	−0.100766	+	0.994910i	$0.532129\pi$
$278$	0	0
$279$	1745.72	0.374600
$280$	0	0
$281$	−7372.24	−1.56509	−0.782546	−	0.622593i	$-0.786079\pi$
$281$	−7372.24	−1.56509	−0.782546	+	0.622593i	$0.786079\pi$
$282$	0	0
$283$	4510.05	0.947331	0.473665	−	0.880705i	$-0.342931\pi$
$283$	4510.05	0.947331	0.473665	+	0.880705i	$0.342931\pi$
$284$	0	0
$285$	−1458.84	−0.303209
$286$	0	0
$287$	−9218.59	−1.89601
$288$	0	0
$289$	13381.2	2.72364
$290$	0	0
$291$	−456.150	−0.0918899
$292$	0	0
$293$	4358.94	0.869119	0.434559	−	0.900643i	$-0.356904\pi$
$293$	4358.94	0.869119	0.434559	+	0.900643i	$0.356904\pi$
$294$	0	0
$295$	−1723.88	−0.340230
$296$	0	0
$297$	−1707.07	−0.333517
$298$	0	0
$299$	65.1378	0.0125987
$300$	0	0
$301$	1397.77	0.267662
$302$	0	0
$303$	−1452.98	−0.275483
$304$	0	0
$305$	−25.8125	−0.00484597
$306$	0	0
$307$	−788.962	−0.146673	−0.0733363	−	0.997307i	$-0.523365\pi$
$307$	−788.962	−0.146673	−0.0733363	+	0.997307i	$0.523365\pi$
$308$	0	0
$309$	4852.52	0.893366
$310$	0	0
$311$	956.062	0.174319	0.0871597	−	0.996194i	$-0.472221\pi$
$311$	956.062	0.174319	0.0871597	+	0.996194i	$0.472221\pi$
$312$	0	0
$313$	−728.887	−0.131627	−0.0658133	−	0.997832i	$-0.520964\pi$
$313$	−728.887	−0.131627	−0.0658133	+	0.997832i	$0.520964\pi$
$314$	0	0
$315$	−846.281	−0.151373
$316$	0	0
$317$	816.788	0.144717	0.0723586	−	0.997379i	$-0.476947\pi$
$317$	816.788	0.144717	0.0723586	+	0.997379i	$0.476947\pi$
$318$	0	0
$319$	13130.2	2.30455
$320$	0	0
$321$	3031.09	0.527037
$322$	0	0
$323$	13154.5	2.26606
$324$	0	0
$325$	39.5314	0.00674709
$326$	0	0
$327$	5.17462	0.000875098 0
$328$	0	0
$329$	10239.6	1.71590
$330$	0	0
$331$	−3787.26	−0.628902	−0.314451	−	0.949274i	$-0.601820\pi$
$331$	−3787.26	−0.628902	−0.314451	+	0.949274i	$0.601820\pi$
$332$	0	0
$333$	3053.31	0.502463
$334$	0	0
$335$	3357.37	0.547561
$336$	0	0
$337$	−4959.79	−0.801712	−0.400856	−	0.916141i	$-0.631287\pi$
$337$	−4959.79	−0.801712	−0.400856	+	0.916141i	$0.631287\pi$
$338$	0	0
$339$	−2187.36	−0.350445
$340$	0	0
$341$	−12263.7	−1.94755
$342$	0	0
$343$	−6249.79	−0.983839
$344$	0	0
$345$	−617.906	−0.0964260
$346$	0	0
$347$	1001.51	0.154939	0.0774697	−	0.996995i	$-0.475316\pi$
$347$	1001.51	0.154939	0.0774697	+	0.996995i	$0.475316\pi$
$348$	0	0
$349$	−11048.8	−1.69464	−0.847320	−	0.531083i	$-0.821785\pi$
$349$	−11048.8	−1.69464	−0.847320	+	0.531083i	$0.821785\pi$
$350$	0	0
$351$	42.6939	0.00649239
$352$	0	0
$353$	−1216.63	−0.183441	−0.0917207	−	0.995785i	$-0.529237\pi$
$353$	−1216.63	−0.183441	−0.0917207	+	0.995785i	$0.529237\pi$
$354$	0	0
$355$	−2127.75	−0.318111
$356$	0	0
$357$	7630.99	1.13130
$358$	0	0
$359$	967.125	0.142181	0.0710904	−	0.997470i	$-0.477352\pi$
$359$	967.125	0.142181	0.0710904	+	0.997470i	$0.477352\pi$
$360$	0	0
$361$	2599.78	0.379031
$362$	0	0
$363$	7999.20	1.15661
$364$	0	0
$365$	474.187	0.0680003
$366$	0	0
$367$	−4356.06	−0.619576	−0.309788	−	0.950806i	$-0.600258\pi$
$367$	−4356.06	−0.619576	−0.309788	+	0.950806i	$0.600258\pi$
$368$	0	0
$369$	−4411.69	−0.622394
$370$	0	0
$371$	12484.8	1.74711
$372$	0	0
$373$	−319.982	−0.0444183	−0.0222092	−	0.999753i	$-0.507070\pi$
$373$	−319.982	−0.0444183	−0.0222092	+	0.999753i	$0.507070\pi$
$374$	0	0
$375$	−375.000	−0.0516398
$376$	0	0
$377$	−328.387	−0.0448615
$378$	0	0
$379$	−1188.39	−0.161065	−0.0805326	−	0.996752i	$-0.525662\pi$
$379$	−1188.39	−0.161065	−0.0805326	+	0.996752i	$0.525662\pi$
$380$	0	0
$381$	1165.44	0.156713
$382$	0	0
$383$	3150.51	0.420322	0.210161	−	0.977667i	$-0.432601\pi$
$383$	3150.51	0.420322	0.210161	+	0.977667i	$0.432601\pi$
$384$	0	0
$385$	5945.12	0.786991
$386$	0	0
$387$	668.925	0.0878640
$388$	0	0
$389$	6201.45	0.808293	0.404147	−	0.914694i	$-0.367569\pi$
$389$	6201.45	0.808293	0.404147	+	0.914694i	$0.367569\pi$
$390$	0	0
$391$	5571.71	0.720649
$392$	0	0
$393$	−8331.19	−1.06935
$394$	0	0
$395$	−3853.41	−0.490850
$396$	0	0
$397$	−4225.39	−0.534172	−0.267086	−	0.963673i	$-0.586061\pi$
$397$	−4225.39	−0.534172	−0.267086	+	0.963673i	$0.586061\pi$
$398$	0	0
$399$	5487.07	0.688464
$400$	0	0
$401$	−5691.02	−0.708719	−0.354359	−	0.935109i	$-0.615301\pi$
$401$	−5691.02	−0.708719	−0.354359	+	0.935109i	$0.615301\pi$
$402$	0	0
$403$	306.714	0.0379119
$404$	0	0
$405$	−405.000	−0.0496904
$406$	0	0
$407$	−21449.5	−2.61231
$408$	0	0
$409$	2831.60	0.342331	0.171166	−	0.985242i	$-0.445247\pi$
$409$	2831.60	0.342331	0.171166	+	0.985242i	$0.445247\pi$
$410$	0	0
$411$	519.356	0.0623308
$412$	0	0
$413$	6483.92	0.772526
$414$	0	0
$415$	−2949.00	−0.348821
$416$	0	0
$417$	−1352.42	−0.158821
$418$	0	0
$419$	13660.9	1.59279	0.796395	−	0.604776i	$-0.206738\pi$
$419$	13660.9	1.59279	0.796395	+	0.604776i	$0.206738\pi$
$420$	0	0
$421$	9252.46	1.07111	0.535555	−	0.844500i	$-0.320102\pi$
$421$	9252.46	1.07111	0.535555	+	0.844500i	$0.320102\pi$
$422$	0	0
$423$	4900.33	0.563268
$424$	0	0
$425$	3381.41	0.385935
$426$	0	0
$427$	97.0874	0.0110033
$428$	0	0
$429$	−299.924	−0.0337541
$430$	0	0
$431$	−397.275	−0.0443992	−0.0221996	−	0.999754i	$-0.507067\pi$
$431$	−397.275	−0.0443992	−0.0221996	+	0.999754i	$0.507067\pi$
$432$	0	0
$433$	−1122.39	−0.124569	−0.0622846	−	0.998058i	$-0.519839\pi$
$433$	−1122.39	−0.124569	−0.0622846	+	0.998058i	$0.519839\pi$
$434$	0	0
$435$	3115.12	0.343354
$436$	0	0
$437$	4006.35	0.438558
$438$	0	0
$439$	17424.7	1.89439	0.947193	−	0.320665i	$-0.103906\pi$
$439$	17424.7	1.89439	0.947193	+	0.320665i	$0.103906\pi$
$440$	0	0
$441$	96.0748	0.0103741
$442$	0	0
$443$	6448.20	0.691565	0.345782	−	0.938315i	$-0.387614\pi$
$443$	6448.20	0.691565	0.345782	+	0.938315i	$0.387614\pi$
$444$	0	0
$445$	2049.06	0.218281
$446$	0	0
$447$	−3177.07	−0.336176
$448$	0	0
$449$	−9622.97	−1.01144	−0.505720	−	0.862698i	$-0.668773\pi$
$449$	−9622.97	−1.01144	−0.505720	+	0.862698i	$0.668773\pi$
$450$	0	0
$451$	30992.1	3.23583
$452$	0	0
$453$	−3820.89	−0.396294
$454$	0	0
$455$	−148.687	−0.0153199
$456$	0	0
$457$	6424.55	0.657610	0.328805	−	0.944398i	$-0.393354\pi$
$457$	6424.55	0.657610	0.328805	+	0.944398i	$0.393354\pi$
$458$	0	0
$459$	3651.92	0.371366
$460$	0	0
$461$	7773.75	0.785379	0.392689	−	0.919671i	$-0.371545\pi$
$461$	7773.75	0.785379	0.392689	+	0.919671i	$0.371545\pi$
$462$	0	0
$463$	7122.96	0.714972	0.357486	−	0.933919i	$-0.383634\pi$
$463$	7122.96	0.714972	0.357486	+	0.933919i	$0.383634\pi$
$464$	0	0
$465$	−2909.53	−0.290164
$466$	0	0
$467$	4292.21	0.425310	0.212655	−	0.977127i	$-0.431789\pi$
$467$	4292.21	0.425310	0.212655	+	0.977127i	$0.431789\pi$
$468$	0	0
$469$	−12627.9	−1.24329
$470$	0	0
$471$	−4327.82	−0.423387
$472$	0	0
$473$	−4699.20	−0.456806
$474$	0	0
$475$	2431.41	0.234864
$476$	0	0
$477$	5974.76	0.573513
$478$	0	0
$479$	12607.8	1.20264	0.601321	−	0.799008i	$-0.294641\pi$
$479$	12607.8	1.20264	0.601321	+	0.799008i	$0.294641\pi$
$480$	0	0
$481$	536.450	0.0508525
$482$	0	0
$483$	2324.10	0.218945
$484$	0	0
$485$	760.249	0.0711776
$486$	0	0
$487$	−14242.9	−1.32527	−0.662634	−	0.748943i	$-0.730561\pi$
$487$	−14242.9	−1.32527	−0.662634	+	0.748943i	$0.730561\pi$
$488$	0	0
$489$	7321.05	0.677033
$490$	0	0
$491$	−19491.8	−1.79156	−0.895778	−	0.444502i	$-0.853381\pi$
$491$	−19491.8	−1.79156	−0.895778	+	0.444502i	$0.853381\pi$
$492$	0	0
$493$	−28089.3	−2.56609
$494$	0	0
$495$	2845.12	0.258341
$496$	0	0
$497$	8003.00	0.722301
$498$	0	0
$499$	−8728.93	−0.783087	−0.391544	−	0.920160i	$-0.628059\pi$
$499$	−8728.93	−0.783087	−0.391544	+	0.920160i	$0.628059\pi$
$500$	0	0
$501$	−10965.5	−0.977853
$502$	0	0
$503$	−10470.8	−0.928175	−0.464087	−	0.885789i	$-0.653618\pi$
$503$	−10470.8	−0.928175	−0.464087	+	0.885789i	$0.653618\pi$
$504$	0	0
$505$	2421.63	0.213388
$506$	0	0
$507$	−6583.50	−0.576693
$508$	0	0
$509$	−4779.80	−0.416230	−0.208115	−	0.978104i	$-0.566733\pi$
$509$	−4779.80	−0.416230	−0.208115	+	0.978104i	$0.566733\pi$
$510$	0	0
$511$	−1783.54	−0.154401
$512$	0	0
$513$	2625.92	0.225998
$514$	0	0
$515$	−8087.53	−0.691998
$516$	0	0
$517$	−34424.8	−2.92844
$518$	0	0
$519$	−9044.25	−0.764930
$520$	0	0
$521$	20647.1	1.73621	0.868104	−	0.496382i	$-0.165338\pi$
$521$	20647.1	1.73621	0.868104	+	0.496382i	$0.165338\pi$
$522$	0	0
$523$	18853.5	1.57630	0.788149	−	0.615484i	$-0.211039\pi$
$523$	18853.5	1.57630	0.788149	+	0.615484i	$0.211039\pi$
$524$	0	0
$525$	1410.47	0.117253
$526$	0	0
$527$	26235.5	2.16857
$528$	0	0
$529$	−10470.1	−0.860531
$530$	0	0
$531$	3102.98	0.253593
$532$	0	0
$533$	−775.111	−0.0629902
$534$	0	0
$535$	−5051.81	−0.408241
$536$	0	0
$537$	−5049.11	−0.405745
$538$	0	0
$539$	−674.926	−0.0539353
$540$	0	0
$541$	8575.48	0.681494	0.340747	−	0.940155i	$-0.389320\pi$
$541$	8575.48	0.681494	0.340747	+	0.940155i	$0.389320\pi$
$542$	0	0
$543$	−7521.37	−0.594426
$544$	0	0
$545$	−8.62436	−0.000677848 0
$546$	0	0
$547$	14884.0	1.16343	0.581715	−	0.813393i	$-0.302382\pi$
$547$	14884.0	1.16343	0.581715	+	0.813393i	$0.302382\pi$
$548$	0	0
$549$	46.4626	0.00361198
$550$	0	0
$551$	−20197.7	−1.56162
$552$	0	0
$553$	14493.6	1.11452
$554$	0	0
$555$	−5088.84	−0.389206
$556$	0	0
$557$	9780.58	0.744015	0.372007	−	0.928230i	$-0.378670\pi$
$557$	9780.58	0.744015	0.372007	+	0.928230i	$0.378670\pi$
$558$	0	0
$559$	117.527	0.00889240
$560$	0	0
$561$	−25654.7	−1.93074
$562$	0	0
$563$	25413.3	1.90239	0.951194	−	0.308594i	$-0.0998583\pi$
$563$	25413.3	1.90239	0.951194	+	0.308594i	$0.0998583\pi$
$564$	0	0
$565$	3645.59	0.271454
$566$	0	0
$567$	1523.31	0.112827
$568$	0	0
$569$	−766.399	−0.0564659	−0.0282330	−	0.999601i	$-0.508988\pi$
$569$	−766.399	−0.0564659	−0.0282330	+	0.999601i	$0.508988\pi$
$570$	0	0
$571$	−2066.98	−0.151489	−0.0757447	−	0.997127i	$-0.524133\pi$
$571$	−2066.98	−0.151489	−0.0757447	+	0.997127i	$0.524133\pi$
$572$	0	0
$573$	9753.52	0.711098
$574$	0	0
$575$	1029.84	0.0746912
$576$	0	0
$577$	2235.48	0.161290	0.0806448	−	0.996743i	$-0.474302\pi$
$577$	2235.48	0.161290	0.0806448	+	0.996743i	$0.474302\pi$
$578$	0	0
$579$	7406.66	0.531624
$580$	0	0
$581$	11091.9	0.792032
$582$	0	0
$583$	−41972.7	−2.98170
$584$	0	0
$585$	−71.1565	−0.00502899
$586$	0	0
$587$	−14951.7	−1.05131	−0.525657	−	0.850697i	$-0.676180\pi$
$587$	−14951.7	−1.05131	−0.525657	+	0.850697i	$0.676180\pi$
$588$	0	0
$589$	18864.7	1.31970
$590$	0	0
$591$	465.637	0.0324091
$592$	0	0
$593$	14265.4	0.987872	0.493936	−	0.869498i	$-0.335558\pi$
$593$	14265.4	0.987872	0.493936	+	0.869498i	$0.335558\pi$
$594$	0	0
$595$	−12718.3	−0.876302
$596$	0	0
$597$	−6353.94	−0.435594
$598$	0	0
$599$	16069.6	1.09614	0.548068	−	0.836434i	$-0.315364\pi$
$599$	16069.6	1.09614	0.548068	+	0.836434i	$0.315364\pi$
$600$	0	0
$601$	19227.5	1.30500	0.652501	−	0.757788i	$-0.273720\pi$
$601$	19227.5	1.30500	0.652501	+	0.757788i	$0.273720\pi$
$602$	0	0
$603$	−6043.27	−0.408128
$604$	0	0
$605$	−13332.0	−0.895906
$606$	0	0
$607$	−22962.2	−1.53543	−0.767716	−	0.640790i	$-0.778607\pi$
$607$	−22962.2	−1.53543	−0.767716	+	0.640790i	$0.778607\pi$
$608$	0	0
$609$	−11716.8	−0.779618
$610$	0	0
$611$	860.963	0.0570063
$612$	0	0
$613$	23035.5	1.51777	0.758887	−	0.651223i	$-0.225744\pi$
$613$	23035.5	1.51777	0.758887	+	0.651223i	$0.225744\pi$
$614$	0	0
$615$	7352.81	0.482104
$616$	0	0
$617$	17460.2	1.13926	0.569628	−	0.821903i	$-0.307087\pi$
$617$	17460.2	1.13926	0.569628	+	0.821903i	$0.307087\pi$
$618$	0	0
$619$	−14611.4	−0.948759	−0.474380	−	0.880320i	$-0.657328\pi$
$619$	−14611.4	−0.948759	−0.474380	+	0.880320i	$0.657328\pi$
$620$	0	0
$621$	1112.23	0.0718717
$622$	0	0
$623$	−7707.04	−0.495627
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−18447.1	−1.17497
$628$	0	0
$629$	45886.5	2.90877
$630$	0	0
$631$	4126.33	0.260327	0.130164	−	0.991493i	$-0.458450\pi$
$631$	4126.33	0.260327	0.130164	+	0.991493i	$0.458450\pi$
$632$	0	0
$633$	1840.33	0.115555
$634$	0	0
$635$	−1942.41	−0.121389
$636$	0	0
$637$	16.8799	0.00104993
$638$	0	0
$639$	3829.95	0.237106
$640$	0	0
$641$	−21550.1	−1.32789	−0.663946	−	0.747780i	$-0.731120\pi$
$641$	−21550.1	−1.32789	−0.663946	+	0.747780i	$0.731120\pi$
$642$	0	0
$643$	−13528.4	−0.829718	−0.414859	−	0.909886i	$-0.636169\pi$
$643$	−13528.4	−0.829718	−0.414859	+	0.909886i	$0.636169\pi$
$644$	0	0
$645$	−1114.88	−0.0680592
$646$	0	0
$647$	3157.14	0.191839	0.0959197	−	0.995389i	$-0.469421\pi$
$647$	3157.14	0.191839	0.0959197	+	0.995389i	$0.469421\pi$
$648$	0	0
$649$	−21798.4	−1.31843
$650$	0	0
$651$	10943.5	0.658846
$652$	0	0
$653$	9644.38	0.577969	0.288984	−	0.957334i	$-0.406682\pi$
$653$	9644.38	0.577969	0.288984	+	0.957334i	$0.406682\pi$
$654$	0	0
$655$	13885.3	0.828311
$656$	0	0
$657$	−853.537	−0.0506844
$658$	0	0
$659$	−19276.7	−1.13948	−0.569738	−	0.821826i	$-0.692955\pi$
$659$	−19276.7	−1.13948	−0.569738	+	0.821826i	$0.692955\pi$
$660$	0	0
$661$	−7366.06	−0.433444	−0.216722	−	0.976233i	$-0.569537\pi$
$661$	−7366.06	−0.433444	−0.216722	+	0.976233i	$0.569537\pi$
$662$	0	0
$663$	641.624	0.0375846
$664$	0	0
$665$	−9145.12	−0.533282
$666$	0	0
$667$	−8554.91	−0.496623
$668$	0	0
$669$	−3459.77	−0.199944
$670$	0	0
$671$	−326.400	−0.0187787
$672$	0	0
$673$	3283.74	0.188081	0.0940407	−	0.995568i	$-0.470022\pi$
$673$	3283.74	0.188081	0.0940407	+	0.995568i	$0.470022\pi$
$674$	0	0
$675$	675.000	0.0384900
$676$	0	0
$677$	−8895.32	−0.504985	−0.252493	−	0.967599i	$-0.581250\pi$
$677$	−8895.32	−0.504985	−0.252493	+	0.967599i	$0.581250\pi$
$678$	0	0
$679$	−2859.49	−0.161616
$680$	0	0
$681$	−14620.9	−0.822725
$682$	0	0
$683$	−14632.3	−0.819751	−0.409875	−	0.912141i	$-0.634428\pi$
$683$	−14632.3	−0.819751	−0.409875	+	0.912141i	$0.634428\pi$
$684$	0	0
$685$	−865.593	−0.0482812
$686$	0	0
$687$	1791.49	0.0994898
$688$	0	0
$689$	1049.74	0.0580432
$690$	0	0
$691$	−7677.28	−0.422659	−0.211330	−	0.977415i	$-0.567779\pi$
$691$	−7677.28	−0.422659	−0.211330	+	0.977415i	$0.567779\pi$
$692$	0	0
$693$	−10701.2	−0.586589
$694$	0	0
$695$	2254.03	0.123022
$696$	0	0
$697$	−66300.9	−3.60305
$698$	0	0
$699$	−15907.8	−0.860785
$700$	0	0
$701$	2348.58	0.126540	0.0632700	−	0.997996i	$-0.479847\pi$
$701$	2348.58	0.126540	0.0632700	+	0.997996i	$0.479847\pi$
$702$	0	0
$703$	32994.8	1.77016
$704$	0	0
$705$	−8167.22	−0.436305
$706$	0	0
$707$	−9108.34	−0.484518
$708$	0	0
$709$	−30897.1	−1.63662	−0.818310	−	0.574777i	$-0.805089\pi$
$709$	−30897.1	−1.63662	−0.818310	+	0.574777i	$0.805089\pi$
$710$	0	0
$711$	6936.13	0.365858
$712$	0	0
$713$	7990.30	0.419690
$714$	0	0
$715$	499.874	0.0261458
$716$	0	0
$717$	10713.9	0.558047
$718$	0	0
$719$	−7448.66	−0.386353	−0.193177	−	0.981164i	$-0.561879\pi$
$719$	−7448.66	−0.386353	−0.193177	+	0.981164i	$0.561879\pi$
$720$	0	0
$721$	30419.2	1.57125
$722$	0	0
$723$	14390.9	0.740255
$724$	0	0
$725$	−5191.87	−0.265961
$726$	0	0
$727$	−5301.58	−0.270460	−0.135230	−	0.990814i	$-0.543177\pi$
$727$	−5301.58	−0.270460	−0.135230	+	0.990814i	$0.543177\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	10052.9	0.508647
$732$	0	0
$733$	−18909.0	−0.952825	−0.476413	−	0.879222i	$-0.658063\pi$
$733$	−18909.0	−0.952825	−0.476413	+	0.879222i	$0.658063\pi$
$734$	0	0
$735$	−160.125	−0.00803577
$736$	0	0
$737$	42454.0	2.12186
$738$	0	0
$739$	−30354.5	−1.51097	−0.755486	−	0.655165i	$-0.772599\pi$
$739$	−30354.5	−1.51097	−0.755486	+	0.655165i	$0.772599\pi$
$740$	0	0
$741$	461.361	0.0228725
$742$	0	0
$743$	−26539.9	−1.31044	−0.655219	−	0.755439i	$-0.727424\pi$
$743$	−26539.9	−1.31044	−0.655219	+	0.755439i	$0.727424\pi$
$744$	0	0
$745$	5295.12	0.260400
$746$	0	0
$747$	5308.20	0.259996
$748$	0	0
$749$	19001.1	0.926951
$750$	0	0
$751$	−23097.1	−1.12227	−0.561136	−	0.827724i	$-0.689636\pi$
$751$	−23097.1	−1.12227	−0.561136	+	0.827724i	$0.689636\pi$
$752$	0	0
$753$	7324.31	0.354466
$754$	0	0
$755$	6368.16	0.306968
$756$	0	0
$757$	25979.5	1.24734	0.623672	−	0.781686i	$-0.285640\pi$
$757$	25979.5	1.24734	0.623672	+	0.781686i	$0.285640\pi$
$758$	0	0
$759$	−7813.42	−0.373662
$760$	0	0
$761$	−21314.3	−1.01530	−0.507649	−	0.861564i	$-0.669485\pi$
$761$	−21314.3	−1.01530	−0.507649	+	0.861564i	$0.669485\pi$
$762$	0	0
$763$	32.4384	0.00153912
$764$	0	0
$765$	−6086.53	−0.287659
$766$	0	0
$767$	545.177	0.0256652
$768$	0	0
$769$	−2318.70	−0.108731	−0.0543657	−	0.998521i	$-0.517314\pi$
$769$	−2318.70	−0.108731	−0.0543657	+	0.998521i	$0.517314\pi$
$770$	0	0
$771$	−12489.5	−0.583396
$772$	0	0
$773$	−13988.2	−0.650868	−0.325434	−	0.945565i	$-0.605510\pi$
$773$	−13988.2	−0.650868	−0.325434	+	0.945565i	$0.605510\pi$
$774$	0	0
$775$	4849.22	0.224760
$776$	0	0
$777$	19140.4	0.883730
$778$	0	0
$779$	−47673.8	−2.19267
$780$	0	0
$781$	−26905.4	−1.23272
$782$	0	0
$783$	−5607.22	−0.255921
$784$	0	0
$785$	7213.03	0.327954
$786$	0	0
$787$	9663.71	0.437705	0.218853	−	0.975758i	$-0.429769\pi$
$787$	9663.71	0.437705	0.218853	+	0.975758i	$0.429769\pi$
$788$	0	0
$789$	−1631.08	−0.0735970
$790$	0	0
$791$	−13712.0	−0.616362
$792$	0	0
$793$	8.16324	0.000365555 0
$794$	0	0
$795$	−9957.94	−0.444241
$796$	0	0
$797$	35878.9	1.59460	0.797299	−	0.603584i	$-0.206261\pi$
$797$	35878.9	1.59460	0.797299	+	0.603584i	$0.206261\pi$
$798$	0	0
$799$	73644.5	3.26077
$800$	0	0
$801$	−3688.31	−0.162697
$802$	0	0
$803$	5996.10	0.263509
$804$	0	0
$805$	−3873.50	−0.169594
$806$	0	0
$807$	2976.98	0.129857
$808$	0	0
$809$	−18021.0	−0.783172	−0.391586	−	0.920142i	$-0.628073\pi$
$809$	−18021.0	−0.783172	−0.391586	+	0.920142i	$0.628073\pi$
$810$	0	0
$811$	−31290.5	−1.35482	−0.677409	−	0.735606i	$-0.736897\pi$
$811$	−31290.5	−1.35482	−0.677409	+	0.735606i	$0.736897\pi$
$812$	0	0
$813$	−15906.3	−0.686173
$814$	0	0
$815$	−12201.7	−0.524428
$816$	0	0
$817$	7228.57	0.309542
$818$	0	0
$819$	267.637	0.0114188
$820$	0	0
$821$	17983.2	0.764458	0.382229	−	0.924068i	$-0.375157\pi$
$821$	17983.2	0.764458	0.382229	+	0.924068i	$0.375157\pi$
$822$	0	0
$823$	−2663.27	−0.112802	−0.0564008	−	0.998408i	$-0.517962\pi$
$823$	−2663.27	−0.112802	−0.0564008	+	0.998408i	$0.517962\pi$
$824$	0	0
$825$	−4741.87	−0.200110
$826$	0	0
$827$	21334.0	0.897043	0.448522	−	0.893772i	$-0.351951\pi$
$827$	21334.0	0.897043	0.448522	+	0.893772i	$0.351951\pi$
$828$	0	0
$829$	−34240.2	−1.43451	−0.717256	−	0.696810i	$-0.754602\pi$
$829$	−34240.2	−1.43451	−0.717256	+	0.696810i	$0.754602\pi$
$830$	0	0
$831$	−2787.32	−0.116355
$832$	0	0
$833$	1443.86	0.0600561
$834$	0	0
$835$	18275.9	0.757442
$836$	0	0
$837$	5237.16	0.216276
$838$	0	0
$839$	23674.2	0.974165	0.487082	−	0.873356i	$-0.338061\pi$
$839$	23674.2	0.974165	0.487082	+	0.873356i	$0.338061\pi$
$840$	0	0
$841$	18739.9	0.768375
$842$	0	0
$843$	−22116.7	−0.903606
$844$	0	0
$845$	10972.5	0.446705
$846$	0	0
$847$	50145.0	2.03424
$848$	0	0
$849$	13530.1	0.546942
$850$	0	0
$851$	13975.2	0.562944
$852$	0	0
$853$	−6935.77	−0.278401	−0.139201	−	0.990264i	$-0.544453\pi$
$853$	−6935.77	−0.278401	−0.139201	+	0.990264i	$0.544453\pi$
$854$	0	0
$855$	−4376.53	−0.175058
$856$	0	0
$857$	8482.56	0.338108	0.169054	−	0.985607i	$-0.445929\pi$
$857$	8482.56	0.338108	0.169054	+	0.985607i	$0.445929\pi$
$858$	0	0
$859$	−37384.8	−1.48493	−0.742464	−	0.669886i	$-0.766343\pi$
$859$	−37384.8	−1.48493	−0.742464	+	0.669886i	$0.766343\pi$
$860$	0	0
$861$	−27655.8	−1.09466
$862$	0	0
$863$	−16182.0	−0.638286	−0.319143	−	0.947707i	$-0.603395\pi$
$863$	−16182.0	−0.638286	−0.319143	+	0.947707i	$0.603395\pi$
$864$	0	0
$865$	15073.7	0.592512
$866$	0	0
$867$	40143.7	1.57250
$868$	0	0
$869$	−48726.3	−1.90210
$870$	0	0
$871$	−1061.77	−0.0413052
$872$	0	0
$873$	−1368.45	−0.0530527
$874$	0	0
$875$	−2350.78	−0.0908239
$876$	0	0
$877$	−15908.8	−0.612545	−0.306273	−	0.951944i	$-0.599082\pi$
$877$	−15908.8	−0.612545	−0.306273	+	0.951944i	$0.599082\pi$
$878$	0	0
$879$	13076.8	0.501786
$880$	0	0
$881$	−32957.3	−1.26034	−0.630170	−	0.776457i	$-0.717015\pi$
$881$	−32957.3	−1.26034	−0.630170	+	0.776457i	$0.717015\pi$
$882$	0	0
$883$	2932.72	0.111771	0.0558856	−	0.998437i	$-0.482202\pi$
$883$	2932.72	0.111771	0.0558856	+	0.998437i	$0.482202\pi$
$884$	0	0
$885$	−5171.63	−0.196432
$886$	0	0
$887$	10991.8	0.416086	0.208043	−	0.978120i	$-0.433291\pi$
$887$	10991.8	0.416086	0.208043	+	0.978120i	$0.433291\pi$
$888$	0	0
$889$	7305.87	0.275626
$890$	0	0
$891$	−5121.22	−0.192556
$892$	0	0
$893$	52954.2	1.98437
$894$	0	0
$895$	8415.19	0.314289
$896$	0	0
$897$	195.413	0.00727387
$898$	0	0
$899$	−40282.5	−1.49443
$900$	0	0
$901$	89791.5	3.32008
$902$	0	0
$903$	4193.32	0.154535
$904$	0	0
$905$	12535.6	0.460440
$906$	0	0
$907$	15117.6	0.553440	0.276720	−	0.960951i	$-0.410752\pi$
$907$	15117.6	0.553440	0.276720	+	0.960951i	$0.410752\pi$
$908$	0	0
$909$	−4358.93	−0.159050
$910$	0	0
$911$	25259.2	0.918633	0.459317	−	0.888273i	$-0.348094\pi$
$911$	25259.2	0.918633	0.459317	+	0.888273i	$0.348094\pi$
$912$	0	0
$913$	−37290.1	−1.35172
$914$	0	0
$915$	−77.4376	−0.00279782
$916$	0	0
$917$	−52226.1	−1.88076
$918$	0	0
$919$	−43485.1	−1.56087	−0.780436	−	0.625235i	$-0.785003\pi$
$919$	−43485.1	−1.56087	−0.780436	+	0.625235i	$0.785003\pi$
$920$	0	0
$921$	−2366.89	−0.0846814
$922$	0	0
$923$	672.903	0.0239966
$924$	0	0
$925$	8481.41	0.301478
$926$	0	0
$927$	14557.6	0.515785
$928$	0	0
$929$	22312.2	0.787988	0.393994	−	0.919113i	$-0.371093\pi$
$929$	22312.2	0.787988	0.393994	+	0.919113i	$0.371093\pi$
$930$	0	0
$931$	1038.21	0.0365477
$932$	0	0
$933$	2868.19	0.100643
$934$	0	0
$935$	42757.9	1.49554
$936$	0	0
$937$	−50965.7	−1.77692	−0.888461	−	0.458951i	$-0.848225\pi$
$937$	−50965.7	−1.77692	−0.888461	+	0.458951i	$0.848225\pi$
$938$	0	0
$939$	−2186.66	−0.0759947
$940$	0	0
$941$	−29611.4	−1.02583	−0.512914	−	0.858440i	$-0.671434\pi$
$941$	−29611.4	−1.02583	−0.512914	+	0.858440i	$0.671434\pi$
$942$	0	0
$943$	−20192.7	−0.697310
$944$	0	0
$945$	−2538.84	−0.0873953
$946$	0	0
$947$	−16026.1	−0.549925	−0.274962	−	0.961455i	$-0.588665\pi$
$947$	−16026.1	−0.549925	−0.274962	+	0.961455i	$0.588665\pi$
$948$	0	0
$949$	−149.962	−0.00512959
$950$	0	0
$951$	2450.36	0.0835525
$952$	0	0
$953$	41491.2	1.41032	0.705159	−	0.709050i	$-0.250876\pi$
$953$	41491.2	1.41032	0.705159	+	0.709050i	$0.250876\pi$
$954$	0	0
$955$	−16255.9	−0.550814
$956$	0	0
$957$	39390.7	1.33054
$958$	0	0
$959$	3255.71	0.109627
$960$	0	0
$961$	7832.88	0.262928
$962$	0	0
$963$	9093.26	0.304285
$964$	0	0
$965$	−12344.4	−0.411794
$966$	0	0
$967$	16698.6	0.555317	0.277658	−	0.960680i	$-0.410442\pi$
$967$	16698.6	0.555317	0.277658	+	0.960680i	$0.410442\pi$
$968$	0	0
$969$	39463.5	1.30831
$970$	0	0
$971$	44370.0	1.46643	0.733213	−	0.679999i	$-0.238020\pi$
$971$	44370.0	1.46643	0.733213	+	0.679999i	$0.238020\pi$
$972$	0	0
$973$	−8477.97	−0.279333
$974$	0	0
$975$	118.594	0.00389544
$976$	0	0
$977$	−12443.8	−0.407484	−0.203742	−	0.979025i	$-0.565310\pi$
$977$	−12443.8	−0.407484	−0.203742	+	0.979025i	$0.565310\pi$
$978$	0	0
$979$	25910.4	0.845863
$980$	0	0
$981$	15.5238	0.000505238 0
$982$	0	0
$983$	45382.4	1.47251	0.736253	−	0.676707i	$-0.236594\pi$
$983$	45382.4	1.47251	0.736253	+	0.676707i	$0.236594\pi$
$984$	0	0
$985$	−776.062	−0.0251040
$986$	0	0
$987$	30718.9	0.990674
$988$	0	0
$989$	3061.73	0.0984401
$990$	0	0
$991$	2468.08	0.0791132	0.0395566	−	0.999217i	$-0.487405\pi$
$991$	2468.08	0.0791132	0.0395566	+	0.999217i	$0.487405\pi$
$992$	0	0
$993$	−11361.8	−0.363096
$994$	0	0
$995$	10589.9	0.337410
$996$	0	0
$997$	−8805.68	−0.279718	−0.139859	−	0.990171i	$-0.544665\pi$
$997$	−8805.68	−0.279718	−0.139859	+	0.990171i	$0.544665\pi$
$998$	0	0
$999$	9159.92	0.290097

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	480.4.a.q.1.2	yes	2
3.2	odd	2		1440.4.a.bg.1.2		2
4.3	odd	2		480.4.a.m.1.1	✓	2
5.4	even	2		2400.4.a.x.1.1		2
8.3	odd	2		960.4.a.bo.1.1		2
8.5	even	2		960.4.a.bm.1.2		2
12.11	even	2		1440.4.a.z.1.1		2
20.19	odd	2		2400.4.a.bc.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
480.4.a.m.1.1	✓	2	4.3	odd	2
480.4.a.q.1.2	yes	2	1.1	even	1	trivial
960.4.a.bm.1.2		2	8.5	even	2
960.4.a.bo.1.1		2	8.3	odd	2
1440.4.a.z.1.1		2	12.11	even	2
1440.4.a.bg.1.2		2	3.2	odd	2
2400.4.a.x.1.1		2	5.4	even	2
2400.4.a.bc.1.2		2	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 480 = 2^{5} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	480.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} -5.00000 q^{5} +18.8062 q^{7} +9.00000 q^{9} -63.2250 q^{11} +1.58125 q^{13} -15.0000 q^{15} +135.256 q^{17} +97.2562 q^{19} +56.4187 q^{21} +41.1938 q^{23} +25.0000 q^{25} +27.0000 q^{27} -207.675 q^{29} +193.969 q^{31} -189.675 q^{33} -94.0312 q^{35} +339.256 q^{37} +4.74376 q^{39} -490.187 q^{41} +74.3250 q^{43} -45.0000 q^{45} +544.481 q^{47} +10.6750 q^{49} +405.769 q^{51} +663.862 q^{53} +316.125 q^{55} +291.769 q^{57} +344.775 q^{59} +5.16251 q^{61} +169.256 q^{63} -7.90627 q^{65} -671.475 q^{67} +123.581 q^{69} +425.550 q^{71} -94.8375 q^{73} +75.0000 q^{75} -1189.02 q^{77} +770.681 q^{79} +81.0000 q^{81} +589.800 q^{83} -676.281 q^{85} -623.025 q^{87} -409.813 q^{89} +29.7375 q^{91} +581.906 q^{93} -486.281 q^{95} -152.050 q^{97} -569.025 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{3} - 10 q^{5} + 12 q^{7} + 18 q^{9} - 24 q^{11} + 80 q^{13} - 30 q^{15} + 40 q^{17} - 36 q^{19} + 36 q^{21} + 108 q^{23} + 50 q^{25} + 54 q^{27} - 108 q^{29} + 516 q^{31} - 72 q^{33} - 60 q^{35}+ \cdots - 216 q^{99}+O(q^{100}) \) 2 * q + 6 * q^3 - 10 * q^5 + 12 * q^7 + 18 * q^9 - 24 * q^11 + 80 * q^13 - 30 * q^15 + 40 * q^17 - 36 * q^19 + 36 * q^21 + 108 * q^23 + 50 * q^25 + 54 * q^27 - 108 * q^29 + 516 * q^31 - 72 * q^33 - 60 * q^35 + 448 * q^37 + 240 * q^39 - 212 * q^41 + 456 * q^43 - 90 * q^45 + 756 * q^47 - 286 * q^49 + 120 * q^51 + 252 * q^53 + 120 * q^55 - 108 * q^57 + 792 * q^59 + 164 * q^61 + 108 * q^63 - 400 * q^65 - 216 * q^67 + 324 * q^69 + 1056 * q^71 - 36 * q^73 + 150 * q^75 - 1456 * q^77 + 2028 * q^79 + 162 * q^81 + 360 * q^83 - 200 * q^85 - 324 * q^87 - 1588 * q^89 - 504 * q^91 + 1548 * q^93 + 180 * q^95 + 1540 * q^97 - 216 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more