LMFDB - Embedded newform 152.6.a.c.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [152,6,Mod(1,152)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("152.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(152, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 152 = 2^{3} \cdot 19 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	152.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$24.3783406116$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 1012x^{4} + 2938x^{3} + 184643x^{2} - 1214504x + 729856 $ x^6 - 2x^5 - 1012x^4 + 2938x^3 + 184643x^2 - 1214504*x + 729856
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-17.4790$ of defining polynomial
Character	$\chi$	$=$	152.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-18.4790 q^{3} +70.0177 q^{5} -152.088 q^{7} +98.4744 q^{9} +259.063 q^{11} +625.486 q^{13} -1293.86 q^{15} +160.671 q^{17} -361.000 q^{19} +2810.44 q^{21} -2528.59 q^{23} +1777.48 q^{25} +2670.69 q^{27} +1719.07 q^{29} -5144.77 q^{31} -4787.23 q^{33} -10648.9 q^{35} -13394.5 q^{37} -11558.4 q^{39} -19614.0 q^{41} -2498.66 q^{43} +6894.95 q^{45} -17793.8 q^{47} +6323.77 q^{49} -2969.04 q^{51} -13506.5 q^{53} +18139.0 q^{55} +6670.93 q^{57} -27961.8 q^{59} +36418.3 q^{61} -14976.8 q^{63} +43795.1 q^{65} +33655.4 q^{67} +46725.9 q^{69} -5865.51 q^{71} -49109.8 q^{73} -32846.1 q^{75} -39400.3 q^{77} +65185.2 q^{79} -73281.1 q^{81} -38353.1 q^{83} +11249.8 q^{85} -31766.7 q^{87} +10409.3 q^{89} -95129.0 q^{91} +95070.4 q^{93} -25276.4 q^{95} +16402.2 q^{97} +25511.0 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{3} - 25 q^{5} - 161 q^{7} + 572 q^{9} - 205 q^{11} - 382 q^{13} - 950 q^{15} - 65 q^{17} - 2166 q^{19} - 6006 q^{21} - 5584 q^{23} - 3111 q^{25} - 6874 q^{27} - 9972 q^{29} - 7140 q^{31} - 3774 q^{33}+ \cdots + 49107 q^{99}+O(q^{100}) $ 6 * q - 4 * q^3 - 25 * q^5 - 161 * q^7 + 572 * q^9 - 205 * q^11 - 382 * q^13 - 950 * q^15 - 65 * q^17 - 2166 * q^19 - 6006 * q^21 - 5584 * q^23 - 3111 * q^25 - 6874 * q^27 - 9972 * q^29 - 7140 * q^31 - 3774 * q^33 - 14979 * q^35 - 7872 * q^37 - 46448 * q^39 - 12476 * q^41 - 45829 * q^43 - 41933 * q^45 - 61259 * q^47 + 5315 * q^49 - 72572 * q^51 + 13382 * q^53 - 55199 * q^55 + 1444 * q^57 - 56812 * q^59 + 2559 * q^61 - 97953 * q^63 + 37452 * q^65 - 36678 * q^67 - 12888 * q^69 + 14934 * q^71 + 43567 * q^73 - 57842 * q^75 + 36359 * q^77 - 155476 * q^79 + 199946 * q^81 - 149856 * q^83 + 112601 * q^85 - 95180 * q^87 + 176022 * q^89 + 76214 * q^91 + 434620 * q^93 + 9025 * q^95 + 117190 * q^97 + 49107 * 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
$3$	−18.4790	−1.18543	−0.592715	−	0.805412i	$-0.701944\pi$
$3$	−18.4790	−1.18543	−0.592715	+	0.805412i	$0.701944\pi$
$4$	0	0
$5$	70.0177	1.25252	0.626258	−	0.779616i	$-0.284586\pi$
$5$	70.0177	1.25252	0.626258	+	0.779616i	$0.284586\pi$
$6$	0	0
$7$	−152.088	−1.17314	−0.586570	−	0.809898i	$-0.699522\pi$
$7$	−152.088	−1.17314	−0.586570	+	0.809898i	$0.699522\pi$
$8$	0	0
$9$	98.4744	0.405244
$10$	0	0
$11$	259.063	0.645540	0.322770	−	0.946477i	$-0.395386\pi$
$11$	259.063	0.645540	0.322770	+	0.946477i	$0.395386\pi$
$12$	0	0
$13$	625.486	1.02650	0.513251	−	0.858239i	$-0.328441\pi$
$13$	625.486	1.02650	0.513251	+	0.858239i	$0.328441\pi$
$14$	0	0
$15$	−1293.86	−1.48477
$16$	0	0
$17$	160.671	0.134839	0.0674193	−	0.997725i	$-0.478523\pi$
$17$	160.671	0.134839	0.0674193	+	0.997725i	$0.478523\pi$
$18$	0	0
$19$	−361.000	−0.229416
$19$	−361.000	−0.229416
$20$	0	0
$21$	2810.44	1.39068
$22$	0	0
$23$	−2528.59	−0.996688	−0.498344	−	0.866979i	$-0.666058\pi$
$23$	−2528.59	−0.996688	−0.498344	+	0.866979i	$0.666058\pi$
$24$	0	0
$25$	1777.48	0.568794
$26$	0	0
$27$	2670.69	0.705041
$28$	0	0
$29$	1719.07	0.379576	0.189788	−	0.981825i	$-0.439220\pi$
$29$	1719.07	0.379576	0.189788	+	0.981825i	$0.439220\pi$
$30$	0	0
$31$	−5144.77	−0.961528	−0.480764	−	0.876850i	$-0.659641\pi$
$31$	−5144.77	−0.961528	−0.480764	+	0.876850i	$0.659641\pi$
$32$	0	0
$33$	−4787.23	−0.765242
$34$	0	0
$35$	−10648.9	−1.46938
$36$	0	0
$37$	−13394.5	−1.60850	−0.804252	−	0.594289i	$-0.797434\pi$
$37$	−13394.5	−1.60850	−0.804252	+	0.594289i	$0.797434\pi$
$38$	0	0
$39$	−11558.4	−1.21685
$40$	0	0
$41$	−19614.0	−1.82225	−0.911123	−	0.412135i	$-0.864783\pi$
$41$	−19614.0	−1.82225	−0.911123	+	0.412135i	$0.864783\pi$
$42$	0	0
$43$	−2498.66	−0.206080	−0.103040	−	0.994677i	$-0.532857\pi$
$43$	−2498.66	−0.206080	−0.103040	+	0.994677i	$0.532857\pi$
$44$	0	0
$45$	6894.95	0.507575
$46$	0	0
$47$	−17793.8	−1.17496	−0.587481	−	0.809238i	$-0.699880\pi$
$47$	−17793.8	−1.17496	−0.587481	+	0.809238i	$0.699880\pi$
$48$	0	0
$49$	6323.77	0.376258
$50$	0	0
$51$	−2969.04	−0.159842
$52$	0	0
$53$	−13506.5	−0.660470	−0.330235	−	0.943899i	$-0.607128\pi$
$53$	−13506.5	−0.660470	−0.330235	+	0.943899i	$0.607128\pi$
$54$	0	0
$55$	18139.0	0.808549
$56$	0	0
$57$	6670.93	0.271956
$58$	0	0
$59$	−27961.8	−1.04577	−0.522883	−	0.852404i	$-0.675144\pi$
$59$	−27961.8	−1.04577	−0.522883	+	0.852404i	$0.675144\pi$
$60$	0	0
$61$	36418.3	1.25313	0.626564	−	0.779370i	$-0.284461\pi$
$61$	36418.3	1.25313	0.626564	+	0.779370i	$0.284461\pi$
$62$	0	0
$63$	−14976.8	−0.475409
$64$	0	0
$65$	43795.1	1.28571
$66$	0	0
$67$	33655.4	0.915940	0.457970	−	0.888968i	$-0.348577\pi$
$67$	33655.4	0.915940	0.457970	+	0.888968i	$0.348577\pi$
$68$	0	0
$69$	46725.9	1.18150
$70$	0	0
$71$	−5865.51	−0.138089	−0.0690446	−	0.997614i	$-0.521995\pi$
$71$	−5865.51	−0.138089	−0.0690446	+	0.997614i	$0.521995\pi$
$72$	0	0
$73$	−49109.8	−1.07860	−0.539301	−	0.842113i	$-0.681311\pi$
$73$	−49109.8	−1.07860	−0.539301	+	0.842113i	$0.681311\pi$
$74$	0	0
$75$	−32846.1	−0.674266
$76$	0	0
$77$	−39400.3	−0.757309
$78$	0	0
$79$	65185.2	1.17512	0.587558	−	0.809182i	$-0.300089\pi$
$79$	65185.2	1.17512	0.587558	+	0.809182i	$0.300089\pi$
$80$	0	0
$81$	−73281.1	−1.24102
$82$	0	0
$83$	−38353.1	−0.611090	−0.305545	−	0.952178i	$-0.598839\pi$
$83$	−38353.1	−0.611090	−0.305545	+	0.952178i	$0.598839\pi$
$84$	0	0
$85$	11249.8	0.168887
$86$	0	0
$87$	−31766.7	−0.449961
$88$	0	0
$89$	10409.3	0.139298	0.0696492	−	0.997572i	$-0.477812\pi$
$89$	10409.3	0.139298	0.0696492	+	0.997572i	$0.477812\pi$
$90$	0	0
$91$	−95129.0	−1.20423
$92$	0	0
$93$	95070.4	1.13982
$94$	0	0
$95$	−25276.4	−0.287347
$96$	0	0
$97$	16402.2	0.177000	0.0885000	−	0.996076i	$-0.471793\pi$
$97$	16402.2	0.177000	0.0885000	+	0.996076i	$0.471793\pi$
$98$	0	0
$99$	25511.0	0.261602
$100$	0	0
$101$	15177.5	0.148046	0.0740232	−	0.997257i	$-0.476416\pi$
$101$	15177.5	0.148046	0.0740232	+	0.997257i	$0.476416\pi$
$102$	0	0
$103$	76659.9	0.711992	0.355996	−	0.934488i	$-0.384142\pi$
$103$	76659.9	0.711992	0.355996	+	0.934488i	$0.384142\pi$
$104$	0	0
$105$	196781.	1.74184
$106$	0	0
$107$	−156306.	−1.31982	−0.659912	−	0.751343i	$-0.729406\pi$
$107$	−156306.	−1.31982	−0.659912	+	0.751343i	$0.729406\pi$
$108$	0	0
$109$	170983.	1.37843	0.689217	−	0.724555i	$-0.257955\pi$
$109$	170983.	1.37843	0.689217	+	0.724555i	$0.257955\pi$
$110$	0	0
$111$	247517.	1.90677
$112$	0	0
$113$	64792.4	0.477340	0.238670	−	0.971101i	$-0.423289\pi$
$113$	64792.4	0.477340	0.238670	+	0.971101i	$0.423289\pi$
$114$	0	0
$115$	−177046.	−1.24837
$116$	0	0
$117$	61594.4	0.415984
$118$	0	0
$119$	−24436.1	−0.158185
$120$	0	0
$121$	−93937.5	−0.583278
$122$	0	0
$123$	362448.	2.16015
$124$	0	0
$125$	−94350.2	−0.540092
$126$	0	0
$127$	−295505.	−1.62576	−0.812880	−	0.582432i	$-0.802101\pi$
$127$	−295505.	−1.62576	−0.812880	+	0.582432i	$0.802101\pi$
$128$	0	0
$129$	46172.8	0.244293
$130$	0	0
$131$	−59737.2	−0.304135	−0.152067	−	0.988370i	$-0.548593\pi$
$131$	−59737.2	−0.304135	−0.152067	+	0.988370i	$0.548593\pi$
$132$	0	0
$133$	54903.8	0.269137
$134$	0	0
$135$	186996.	0.883075
$136$	0	0
$137$	223726.	1.01839	0.509195	−	0.860651i	$-0.329943\pi$
$137$	223726.	1.01839	0.509195	+	0.860651i	$0.329943\pi$
$138$	0	0
$139$	−154449.	−0.678027	−0.339014	−	0.940781i	$-0.610093\pi$
$139$	−154449.	−0.678027	−0.339014	+	0.940781i	$0.610093\pi$
$140$	0	0
$141$	328812.	1.39283
$142$	0	0
$143$	162040.	0.662648
$144$	0	0
$145$	120365.	0.475424
$146$	0	0
$147$	−116857.	−0.446028
$148$	0	0
$149$	2314.71	0.00854145	0.00427072	−	0.999991i	$-0.498641\pi$
$149$	2314.71	0.00854145	0.00427072	+	0.999991i	$0.498641\pi$
$150$	0	0
$151$	157129.	0.560809	0.280405	−	0.959882i	$-0.409531\pi$
$151$	157129.	0.560809	0.280405	+	0.959882i	$0.409531\pi$
$152$	0	0
$153$	15822.0	0.0546426
$154$	0	0
$155$	−360225.	−1.20433
$156$	0	0
$157$	−515617.	−1.66947	−0.834734	−	0.550654i	$-0.814378\pi$
$157$	−515617.	−1.66947	−0.834734	+	0.550654i	$0.814378\pi$
$158$	0	0
$159$	249587.	0.782941
$160$	0	0
$161$	384568.	1.16925
$162$	0	0
$163$	−603036.	−1.77777	−0.888883	−	0.458135i	$-0.848518\pi$
$163$	−603036.	−1.77777	−0.888883	+	0.458135i	$0.848518\pi$
$164$	0	0
$165$	−335191.	−0.958478
$166$	0	0
$167$	423435.	1.17489	0.587443	−	0.809265i	$-0.300135\pi$
$167$	423435.	1.17489	0.587443	+	0.809265i	$0.300135\pi$
$168$	0	0
$169$	19940.2	0.0537047
$170$	0	0
$171$	−35549.3	−0.0929695
$172$	0	0
$173$	−96321.5	−0.244685	−0.122343	−	0.992488i	$-0.539041\pi$
$173$	−96321.5	−0.244685	−0.122343	+	0.992488i	$0.539041\pi$
$174$	0	0
$175$	−270334.	−0.667275
$176$	0	0
$177$	516706.	1.23968
$178$	0	0
$179$	716878.	1.67229	0.836147	−	0.548506i	$-0.184803\pi$
$179$	716878.	1.67229	0.836147	+	0.548506i	$0.184803\pi$
$180$	0	0
$181$	−345785.	−0.784531	−0.392266	−	0.919852i	$-0.628309\pi$
$181$	−345785.	−0.784531	−0.392266	+	0.919852i	$0.628309\pi$
$182$	0	0
$183$	−672975.	−1.48550
$184$	0	0
$185$	−937852.	−2.01467
$186$	0	0
$187$	41623.8	0.0870437
$188$	0	0
$189$	−406180.	−0.827112
$190$	0	0
$191$	517892.	1.02720	0.513601	−	0.858029i	$-0.328311\pi$
$191$	517892.	1.02720	0.513601	+	0.858029i	$0.328311\pi$
$192$	0	0
$193$	260809.	0.503998	0.251999	−	0.967728i	$-0.418912\pi$
$193$	260809.	0.503998	0.251999	+	0.967728i	$0.418912\pi$
$194$	0	0
$195$	−809291.	−1.52412
$196$	0	0
$197$	−908450.	−1.66777	−0.833884	−	0.551940i	$-0.813888\pi$
$197$	−908450.	−1.66777	−0.833884	+	0.551940i	$0.813888\pi$
$198$	0	0
$199$	−55645.3	−0.0996084	−0.0498042	−	0.998759i	$-0.515860\pi$
$199$	−55645.3	−0.0996084	−0.0498042	+	0.998759i	$0.515860\pi$
$200$	0	0
$201$	−621918.	−1.08578
$202$	0	0
$203$	−261450.	−0.445296
$204$	0	0
$205$	−1.37333e6	−2.28239
$206$	0	0
$207$	−249002.	−0.403902
$208$	0	0
$209$	−93521.6	−0.148097
$210$	0	0
$211$	1.24888e6	1.93114	0.965570	−	0.260145i	$-0.0837702\pi$
$211$	1.24888e6	1.93114	0.965570	+	0.260145i	$0.0837702\pi$
$212$	0	0
$213$	108389.	0.163695
$214$	0	0
$215$	−174950.	−0.258118
$216$	0	0
$217$	782458.	1.12801
$218$	0	0
$219$	907501.	1.27861
$220$	0	0
$221$	100497.	0.138412
$222$	0	0
$223$	−608221.	−0.819029	−0.409514	−	0.912304i	$-0.634302\pi$
$223$	−608221.	−0.819029	−0.409514	+	0.912304i	$0.634302\pi$
$224$	0	0
$225$	175036.	0.230501
$226$	0	0
$227$	1.10525e6	1.42362	0.711810	−	0.702372i	$-0.247875\pi$
$227$	1.10525e6	1.42362	0.711810	+	0.702372i	$0.247875\pi$
$228$	0	0
$229$	−786681.	−0.991311	−0.495656	−	0.868519i	$-0.665072\pi$
$229$	−786681.	−0.991311	−0.495656	+	0.868519i	$0.665072\pi$
$230$	0	0
$231$	728080.	0.897737
$232$	0	0
$233$	−573358.	−0.691888	−0.345944	−	0.938255i	$-0.612441\pi$
$233$	−573358.	−0.691888	−0.345944	+	0.938255i	$0.612441\pi$
$234$	0	0
$235$	−1.24588e6	−1.47166
$236$	0	0
$237$	−1.20456e6	−1.39302
$238$	0	0
$239$	325087.	0.368133	0.184066	−	0.982914i	$-0.441074\pi$
$239$	325087.	0.368133	0.184066	+	0.982914i	$0.441074\pi$
$240$	0	0
$241$	463016.	0.513515	0.256758	−	0.966476i	$-0.417346\pi$
$241$	463016.	0.513515	0.256758	+	0.966476i	$0.417346\pi$
$242$	0	0
$243$	705185.	0.766103
$244$	0	0
$245$	442776.	0.471269
$246$	0	0
$247$	−225801.	−0.235496
$248$	0	0
$249$	708728.	0.724404
$250$	0	0
$251$	982838.	0.984686	0.492343	−	0.870401i	$-0.336141\pi$
$251$	982838.	0.984686	0.492343	+	0.870401i	$0.336141\pi$
$252$	0	0
$253$	−655064.	−0.643402
$254$	0	0
$255$	−207885.	−0.200204
$256$	0	0
$257$	171205.	0.161690	0.0808451	−	0.996727i	$-0.474238\pi$
$257$	171205.	0.161690	0.0808451	+	0.996727i	$0.474238\pi$
$258$	0	0
$259$	2.03714e6	1.88700
$260$	0	0
$261$	169284.	0.153821
$262$	0	0
$263$	−1.24374e6	−1.10877	−0.554384	−	0.832261i	$-0.687046\pi$
$263$	−1.24374e6	−1.10877	−0.554384	+	0.832261i	$0.687046\pi$
$264$	0	0
$265$	−945695.	−0.827249
$266$	0	0
$267$	−192353.	−0.165128
$268$	0	0
$269$	2.13594e6	1.79973	0.899865	−	0.436168i	$-0.143665\pi$
$269$	2.13594e6	1.79973	0.899865	+	0.436168i	$0.143665\pi$
$270$	0	0
$271$	530072.	0.438441	0.219221	−	0.975675i	$-0.429649\pi$
$271$	530072.	0.438441	0.219221	+	0.975675i	$0.429649\pi$
$272$	0	0
$273$	1.75789e6	1.42753
$274$	0	0
$275$	460479.	0.367179
$276$	0	0
$277$	1.22727e6	0.961038	0.480519	−	0.876984i	$-0.340448\pi$
$277$	1.22727e6	0.961038	0.480519	+	0.876984i	$0.340448\pi$
$278$	0	0
$279$	−506628.	−0.389654
$280$	0	0
$281$	−1.77694e6	−1.34248	−0.671239	−	0.741241i	$-0.734238\pi$
$281$	−1.77694e6	−1.34248	−0.671239	+	0.741241i	$0.734238\pi$
$282$	0	0
$283$	−1.23061e6	−0.913383	−0.456692	−	0.889625i	$-0.650966\pi$
$283$	−1.23061e6	−0.913383	−0.456692	+	0.889625i	$0.650966\pi$
$284$	0	0
$285$	467083.	0.340629
$286$	0	0
$287$	2.98306e6	2.13775
$288$	0	0
$289$	−1.39404e6	−0.981819
$290$	0	0
$291$	−303097.	−0.209821
$292$	0	0
$293$	1.95416e6	1.32981	0.664906	−	0.746927i	$-0.268472\pi$
$293$	1.95416e6	1.32981	0.664906	+	0.746927i	$0.268472\pi$
$294$	0	0
$295$	−1.95782e6	−1.30984
$296$	0	0
$297$	691877.	0.455132
$298$	0	0
$299$	−1.58160e6	−1.02310
$300$	0	0
$301$	380016.	0.241761
$302$	0	0
$303$	−280466.	−0.175499
$304$	0	0
$305$	2.54993e6	1.56956
$306$	0	0
$307$	1.71778e6	1.04021	0.520105	−	0.854102i	$-0.325893\pi$
$307$	1.71778e6	1.04021	0.520105	+	0.854102i	$0.325893\pi$
$308$	0	0
$309$	−1.41660e6	−0.844017
$310$	0	0
$311$	−2.56907e6	−1.50617	−0.753087	−	0.657920i	$-0.771436\pi$
$311$	−2.56907e6	−1.50617	−0.753087	+	0.657920i	$0.771436\pi$
$312$	0	0
$313$	−2.49115e6	−1.43727	−0.718636	−	0.695387i	$-0.755233\pi$
$313$	−2.49115e6	−1.43727	−0.718636	+	0.695387i	$0.755233\pi$
$314$	0	0
$315$	−1.04864e6	−0.595456
$316$	0	0
$317$	−2.54780e6	−1.42403	−0.712013	−	0.702166i	$-0.752216\pi$
$317$	−2.54780e6	−1.42403	−0.712013	+	0.702166i	$0.752216\pi$
$318$	0	0
$319$	445347.	0.245031
$320$	0	0
$321$	2.88838e6	1.56456
$322$	0	0
$323$	−58002.1	−0.0309341
$324$	0	0
$325$	1.11179e6	0.583868
$326$	0	0
$327$	−3.15959e6	−1.63404
$328$	0	0
$329$	2.70622e6	1.37839
$330$	0	0
$331$	1.16690e6	0.585417	0.292708	−	0.956202i	$-0.405443\pi$
$331$	1.16690e6	0.585417	0.292708	+	0.956202i	$0.405443\pi$
$332$	0	0
$333$	−1.31901e6	−0.651837
$334$	0	0
$335$	2.35647e6	1.14723
$336$	0	0
$337$	2.29823e6	1.10235	0.551174	−	0.834391i	$-0.314180\pi$
$337$	2.29823e6	1.10235	0.551174	+	0.834391i	$0.314180\pi$
$338$	0	0
$339$	−1.19730e6	−0.565854
$340$	0	0
$341$	−1.33282e6	−0.620705
$342$	0	0
$343$	1.59437e6	0.731737
$344$	0	0
$345$	3.27164e6	1.47985
$346$	0	0
$347$	1.21467e6	0.541546	0.270773	−	0.962643i	$-0.412721\pi$
$347$	1.21467e6	0.541546	0.270773	+	0.962643i	$0.412721\pi$
$348$	0	0
$349$	−517696.	−0.227516	−0.113758	−	0.993509i	$-0.536289\pi$
$349$	−517696.	−0.227516	−0.113758	+	0.993509i	$0.536289\pi$
$350$	0	0
$351$	1.67048e6	0.723726
$352$	0	0
$353$	117465.	0.0501730	0.0250865	−	0.999685i	$-0.492014\pi$
$353$	117465.	0.0501730	0.0250865	+	0.999685i	$0.492014\pi$
$354$	0	0
$355$	−410690.	−0.172959
$356$	0	0
$357$	451555.	0.187517
$358$	0	0
$359$	559009.	0.228920	0.114460	−	0.993428i	$-0.463486\pi$
$359$	559009.	0.228920	0.114460	+	0.993428i	$0.463486\pi$
$360$	0	0
$361$	130321.	0.0526316
$362$	0	0
$363$	1.73587e6	0.691436
$364$	0	0
$365$	−3.43856e6	−1.35096
$366$	0	0
$367$	2.04987e6	0.794441	0.397221	−	0.917723i	$-0.369975\pi$
$367$	2.04987e6	0.794441	0.397221	+	0.917723i	$0.369975\pi$
$368$	0	0
$369$	−1.93148e6	−0.738455
$370$	0	0
$371$	2.05418e6	0.774824
$372$	0	0
$373$	−4.30519e6	−1.60221	−0.801106	−	0.598522i	$-0.795755\pi$
$373$	−4.30519e6	−1.60221	−0.801106	+	0.598522i	$0.795755\pi$
$374$	0	0
$375$	1.74350e6	0.640241
$376$	0	0
$377$	1.07525e6	0.389635
$378$	0	0
$379$	−379620.	−0.135753	−0.0678767	−	0.997694i	$-0.521622\pi$
$379$	−379620.	−0.135753	−0.0678767	+	0.997694i	$0.521622\pi$
$380$	0	0
$381$	5.46065e6	1.92722
$382$	0	0
$383$	−3.32092e6	−1.15681	−0.578405	−	0.815750i	$-0.696325\pi$
$383$	−3.32092e6	−1.15681	−0.578405	+	0.815750i	$0.696325\pi$
$384$	0	0
$385$	−2.75872e6	−0.948541
$386$	0	0
$387$	−246054.	−0.0835128
$388$	0	0
$389$	−2.41572e6	−0.809417	−0.404708	−	0.914446i	$-0.632627\pi$
$389$	−2.41572e6	−0.809417	−0.404708	+	0.914446i	$0.632627\pi$
$390$	0	0
$391$	−406271.	−0.134392
$392$	0	0
$393$	1.10388e6	0.360531
$394$	0	0
$395$	4.56412e6	1.47185
$396$	0	0
$397$	3.56568e6	1.13545	0.567723	−	0.823220i	$-0.307824\pi$
$397$	3.56568e6	1.13545	0.567723	+	0.823220i	$0.307824\pi$
$398$	0	0
$399$	−1.01457e6	−0.319043
$400$	0	0
$401$	2.47294e6	0.767985	0.383992	−	0.923336i	$-0.374549\pi$
$401$	2.47294e6	0.767985	0.383992	+	0.923336i	$0.374549\pi$
$402$	0	0
$403$	−3.21798e6	−0.987009
$404$	0	0
$405$	−5.13097e6	−1.55440
$406$	0	0
$407$	−3.47001e6	−1.03835
$408$	0	0
$409$	−3.42792e6	−1.01326	−0.506631	−	0.862163i	$-0.669109\pi$
$409$	−3.42792e6	−1.01326	−0.506631	+	0.862163i	$0.669109\pi$
$410$	0	0
$411$	−4.13423e6	−1.20723
$412$	0	0
$413$	4.25265e6	1.22683
$414$	0	0
$415$	−2.68540e6	−0.765399
$416$	0	0
$417$	2.85406e6	0.803754
$418$	0	0
$419$	39082.0	0.0108753	0.00543766	−	0.999985i	$-0.498269\pi$
$419$	39082.0	0.0108753	0.00543766	+	0.999985i	$0.498269\pi$
$420$	0	0
$421$	−6.28126e6	−1.72720	−0.863598	−	0.504181i	$-0.831794\pi$
$421$	−6.28126e6	−1.72720	−0.863598	+	0.504181i	$0.831794\pi$
$422$	0	0
$423$	−1.75223e6	−0.476147
$424$	0	0
$425$	285589.	0.0766954
$426$	0	0
$427$	−5.53879e6	−1.47009
$428$	0	0
$429$	−2.99434e6	−0.785522
$430$	0	0
$431$	−6.66064e6	−1.72712	−0.863561	−	0.504244i	$-0.831771\pi$
$431$	−6.66064e6	−1.72712	−0.863561	+	0.504244i	$0.831771\pi$
$432$	0	0
$433$	−5.40926e6	−1.38650	−0.693248	−	0.720699i	$-0.743821\pi$
$433$	−5.40926e6	−1.38650	−0.693248	+	0.720699i	$0.743821\pi$
$434$	0	0
$435$	−2.22423e6	−0.563582
$436$	0	0
$437$	912821.	0.228656
$438$	0	0
$439$	114943.	0.0284657	0.0142328	−	0.999899i	$-0.495469\pi$
$439$	114943.	0.0284657	0.0142328	+	0.999899i	$0.495469\pi$
$440$	0	0
$441$	622729.	0.152476
$442$	0	0
$443$	−5.64940e6	−1.36771	−0.683854	−	0.729619i	$-0.739697\pi$
$443$	−5.64940e6	−1.36771	−0.683854	+	0.729619i	$0.739697\pi$
$444$	0	0
$445$	728834.	0.174473
$446$	0	0
$447$	−42773.6	−0.0101253
$448$	0	0
$449$	8.29484e6	1.94174	0.970872	−	0.239597i	$-0.0770153\pi$
$449$	8.29484e6	1.94174	0.970872	+	0.239597i	$0.0770153\pi$
$450$	0	0
$451$	−5.08126e6	−1.17633
$452$	0	0
$453$	−2.90360e6	−0.664800
$454$	0	0
$455$	−6.66072e6	−1.50832
$456$	0	0
$457$	7.30985e6	1.63726	0.818631	−	0.574321i	$-0.194734\pi$
$457$	7.30985e6	1.63726	0.818631	+	0.574321i	$0.194734\pi$
$458$	0	0
$459$	429102.	0.0950668
$460$	0	0
$461$	6.42356e6	1.40774	0.703872	−	0.710327i	$-0.251453\pi$
$461$	6.42356e6	1.40774	0.703872	+	0.710327i	$0.251453\pi$
$462$	0	0
$463$	−3.76505e6	−0.816240	−0.408120	−	0.912928i	$-0.633816\pi$
$463$	−3.76505e6	−0.816240	−0.408120	+	0.912928i	$0.633816\pi$
$464$	0	0
$465$	6.65661e6	1.42765
$466$	0	0
$467$	−1.26052e6	−0.267459	−0.133729	−	0.991018i	$-0.542695\pi$
$467$	−1.26052e6	−0.267459	−0.133729	+	0.991018i	$0.542695\pi$
$468$	0	0
$469$	−5.11858e6	−1.07453
$470$	0	0
$471$	9.52810e6	1.97904
$472$	0	0
$473$	−647309.	−0.133033
$474$	0	0
$475$	−641671.	−0.130490
$476$	0	0
$477$	−1.33005e6	−0.267652
$478$	0	0
$479$	6.90007e6	1.37409	0.687044	−	0.726615i	$-0.258908\pi$
$479$	6.90007e6	1.37409	0.687044	+	0.726615i	$0.258908\pi$
$480$	0	0
$481$	−8.37807e6	−1.65113
$482$	0	0
$483$	−7.10645e6	−1.38607
$484$	0	0
$485$	1.14845e6	0.221695
$486$	0	0
$487$	9.68962e6	1.85133	0.925666	−	0.378342i	$-0.123506\pi$
$487$	9.68962e6	1.85133	0.925666	+	0.378342i	$0.123506\pi$
$488$	0	0
$489$	1.11435e7	2.10742
$490$	0	0
$491$	7.27751e6	1.36232	0.681160	−	0.732134i	$-0.261476\pi$
$491$	7.27751e6	1.36232	0.681160	+	0.732134i	$0.261476\pi$
$492$	0	0
$493$	276204.	0.0511815
$494$	0	0
$495$	1.78623e6	0.327660
$496$	0	0
$497$	892074.	0.161998
$498$	0	0
$499$	4.89050e6	0.879228	0.439614	−	0.898187i	$-0.355115\pi$
$499$	4.89050e6	0.879228	0.439614	+	0.898187i	$0.355115\pi$
$500$	0	0
$501$	−7.82467e6	−1.39275
$502$	0	0
$503$	−4.85291e6	−0.855228	−0.427614	−	0.903961i	$-0.640646\pi$
$503$	−4.85291e6	−0.855228	−0.427614	+	0.903961i	$0.640646\pi$
$504$	0	0
$505$	1.06270e6	0.185430
$506$	0	0
$507$	−368475.	−0.0636632
$508$	0	0
$509$	1.85291e6	0.317000	0.158500	−	0.987359i	$-0.449334\pi$
$509$	1.85291e6	0.317000	0.158500	+	0.987359i	$0.449334\pi$
$510$	0	0
$511$	7.46901e6	1.26535
$512$	0	0
$513$	−964120.	−0.161748
$514$	0	0
$515$	5.36755e6	0.891781
$516$	0	0
$517$	−4.60970e6	−0.758484
$518$	0	0
$519$	1.77993e6	0.290057
$520$	0	0
$521$	9.95920e6	1.60742	0.803711	−	0.595019i	$-0.202856\pi$
$521$	9.95920e6	1.60742	0.803711	+	0.595019i	$0.202856\pi$
$522$	0	0
$523$	3.15094e6	0.503716	0.251858	−	0.967764i	$-0.418958\pi$
$523$	3.15094e6	0.503716	0.251858	+	0.967764i	$0.418958\pi$
$524$	0	0
$525$	4.99550e6	0.791008
$526$	0	0
$527$	−826614.	−0.129651
$528$	0	0
$529$	−42568.6	−0.00661379
$530$	0	0
$531$	−2.75352e6	−0.423791
$532$	0	0
$533$	−1.22683e7	−1.87054
$534$	0	0
$535$	−1.09442e7	−1.65310
$536$	0	0
$537$	−1.32472e7	−1.98239
$538$	0	0
$539$	1.63825e6	0.242890
$540$	0	0
$541$	−2.29503e6	−0.337128	−0.168564	−	0.985691i	$-0.553913\pi$
$541$	−2.29503e6	−0.337128	−0.168564	+	0.985691i	$0.553913\pi$
$542$	0	0
$543$	6.38978e6	0.930007
$544$	0	0
$545$	1.19718e7	1.72651
$546$	0	0
$547$	−5.53456e6	−0.790888	−0.395444	−	0.918490i	$-0.629409\pi$
$547$	−5.53456e6	−0.790888	−0.395444	+	0.918490i	$0.629409\pi$
$548$	0	0
$549$	3.58627e6	0.507823
$550$	0	0
$551$	−620584.	−0.0870807
$552$	0	0
$553$	−9.91388e6	−1.37858
$554$	0	0
$555$	1.73306e7	2.38826
$556$	0	0
$557$	602885.	0.0823373	0.0411686	−	0.999152i	$-0.486892\pi$
$557$	602885.	0.0823373	0.0411686	+	0.999152i	$0.486892\pi$
$558$	0	0
$559$	−1.56288e6	−0.211541
$560$	0	0
$561$	−769167.	−0.103184
$562$	0	0
$563$	7.59100e6	1.00932	0.504659	−	0.863319i	$-0.331618\pi$
$563$	7.59100e6	1.00932	0.504659	+	0.863319i	$0.331618\pi$
$564$	0	0
$565$	4.53662e6	0.597876
$566$	0	0
$567$	1.11452e7	1.45589
$568$	0	0
$569$	−6.56162e6	−0.849631	−0.424815	−	0.905280i	$-0.639661\pi$
$569$	−6.56162e6	−0.849631	−0.424815	+	0.905280i	$0.639661\pi$
$570$	0	0
$571$	6.75971e6	0.867637	0.433818	−	0.901000i	$-0.357166\pi$
$571$	6.75971e6	0.867637	0.433818	+	0.901000i	$0.357166\pi$
$572$	0	0
$573$	−9.57014e6	−1.21768
$574$	0	0
$575$	−4.49453e6	−0.566910
$576$	0	0
$577$	1.06670e7	1.33383	0.666915	−	0.745133i	$-0.267614\pi$
$577$	1.06670e7	1.33383	0.666915	+	0.745133i	$0.267614\pi$
$578$	0	0
$579$	−4.81949e6	−0.597454
$580$	0	0
$581$	5.83304e6	0.716894
$582$	0	0
$583$	−3.49903e6	−0.426360
$584$	0	0
$585$	4.31270e6	0.521026
$586$	0	0
$587$	−3.29726e6	−0.394964	−0.197482	−	0.980306i	$-0.563276\pi$
$587$	−3.29726e6	−0.394964	−0.197482	+	0.980306i	$0.563276\pi$
$588$	0	0
$589$	1.85726e6	0.220590
$590$	0	0
$591$	1.67873e7	1.97702
$592$	0	0
$593$	5.79620e6	0.676872	0.338436	−	0.940989i	$-0.390102\pi$
$593$	5.79620e6	0.676872	0.338436	+	0.940989i	$0.390102\pi$
$594$	0	0
$595$	−1.71096e6	−0.198129
$596$	0	0
$597$	1.02827e6	0.118079
$598$	0	0
$599$	8.70225e6	0.990979	0.495489	−	0.868614i	$-0.334989\pi$
$599$	8.70225e6	0.990979	0.495489	+	0.868614i	$0.334989\pi$
$600$	0	0
$601$	−6.62613e6	−0.748296	−0.374148	−	0.927369i	$-0.622065\pi$
$601$	−6.62613e6	−0.748296	−0.374148	+	0.927369i	$0.622065\pi$
$602$	0	0
$603$	3.31419e6	0.371180
$604$	0	0
$605$	−6.57729e6	−0.730565
$606$	0	0
$607$	−7.72626e6	−0.851133	−0.425566	−	0.904927i	$-0.639925\pi$
$607$	−7.72626e6	−0.851133	−0.425566	+	0.904927i	$0.639925\pi$
$608$	0	0
$609$	4.83134e6	0.527867
$610$	0	0
$611$	−1.11298e7	−1.20610
$612$	0	0
$613$	1.03025e6	0.110737	0.0553684	−	0.998466i	$-0.482367\pi$
$613$	1.03025e6	0.110737	0.0553684	+	0.998466i	$0.482367\pi$
$614$	0	0
$615$	2.53778e7	2.70561
$616$	0	0
$617$	−2.43072e6	−0.257052	−0.128526	−	0.991706i	$-0.541025\pi$
$617$	−2.43072e6	−0.257052	−0.128526	+	0.991706i	$0.541025\pi$
$618$	0	0
$619$	1.04157e6	0.109260	0.0546300	−	0.998507i	$-0.482602\pi$
$619$	1.04157e6	0.109260	0.0546300	+	0.998507i	$0.482602\pi$
$620$	0	0
$621$	−6.75309e6	−0.702706
$622$	0	0
$623$	−1.58313e6	−0.163416
$624$	0	0
$625$	−1.21608e7	−1.24527
$626$	0	0
$627$	1.72819e6	0.175559
$628$	0	0
$629$	−2.15210e6	−0.216888
$630$	0	0
$631$	8.12769e6	0.812631	0.406316	−	0.913733i	$-0.366813\pi$
$631$	8.12769e6	0.812631	0.406316	+	0.913733i	$0.366813\pi$
$632$	0	0
$633$	−2.30780e7	−2.28923
$634$	0	0
$635$	−2.06906e7	−2.03629
$636$	0	0
$637$	3.95543e6	0.386229
$638$	0	0
$639$	−577603.	−0.0559599
$640$	0	0
$641$	7.32327e6	0.703979	0.351990	−	0.936004i	$-0.385505\pi$
$641$	7.32327e6	0.703979	0.351990	+	0.936004i	$0.385505\pi$
$642$	0	0
$643$	1.15623e7	1.10285	0.551424	−	0.834225i	$-0.314085\pi$
$643$	1.15623e7	1.10285	0.551424	+	0.834225i	$0.314085\pi$
$644$	0	0
$645$	3.23291e6	0.305981
$646$	0	0
$647$	−1.23804e7	−1.16271	−0.581356	−	0.813649i	$-0.697478\pi$
$647$	−1.23804e7	−1.16271	−0.581356	+	0.813649i	$0.697478\pi$
$648$	0	0
$649$	−7.24385e6	−0.675084
$650$	0	0
$651$	−1.44591e7	−1.33717
$652$	0	0
$653$	1.90958e7	1.75249	0.876245	−	0.481865i	$-0.160041\pi$
$653$	1.90958e7	1.75249	0.876245	+	0.481865i	$0.160041\pi$
$654$	0	0
$655$	−4.18266e6	−0.380934
$656$	0	0
$657$	−4.83606e6	−0.437097
$658$	0	0
$659$	−9.31704e6	−0.835727	−0.417863	−	0.908510i	$-0.637221\pi$
$659$	−9.31704e6	−0.835727	−0.417863	+	0.908510i	$0.637221\pi$
$660$	0	0
$661$	−214237.	−0.0190718	−0.00953588	−	0.999955i	$-0.503035\pi$
$661$	−214237.	−0.0190718	−0.00953588	+	0.999955i	$0.503035\pi$
$662$	0	0
$663$	−1.85709e6	−0.164078
$664$	0	0
$665$	3.84424e6	0.337098
$666$	0	0
$667$	−4.34682e6	−0.378318
$668$	0	0
$669$	1.12393e7	0.970901
$670$	0	0
$671$	9.43463e6	0.808944
$672$	0	0
$673$	−1.26137e7	−1.07351	−0.536755	−	0.843738i	$-0.680350\pi$
$673$	−1.26137e7	−1.07351	−0.536755	+	0.843738i	$0.680350\pi$
$674$	0	0
$675$	4.74711e6	0.401023
$676$	0	0
$677$	−7.31311e6	−0.613240	−0.306620	−	0.951832i	$-0.599198\pi$
$677$	−7.31311e6	−0.613240	−0.306620	+	0.951832i	$0.599198\pi$
$678$	0	0
$679$	−2.49458e6	−0.207646
$680$	0	0
$681$	−2.04239e7	−1.68760
$682$	0	0
$683$	1.63995e7	1.34518	0.672589	−	0.740016i	$-0.265182\pi$
$683$	1.63995e7	1.34518	0.672589	+	0.740016i	$0.265182\pi$
$684$	0	0
$685$	1.56648e7	1.27555
$686$	0	0
$687$	1.45371e7	1.17513
$688$	0	0
$689$	−8.44813e6	−0.677974
$690$	0	0
$691$	435527.	0.0346992	0.0173496	−	0.999849i	$-0.494477\pi$
$691$	435527.	0.0346992	0.0173496	+	0.999849i	$0.494477\pi$
$692$	0	0
$693$	−3.87992e6	−0.306895
$694$	0	0
$695$	−1.08141e7	−0.849239
$696$	0	0
$697$	−3.15140e6	−0.245709
$698$	0	0
$699$	1.05951e7	0.820185
$700$	0	0
$701$	−2.43006e7	−1.86776	−0.933882	−	0.357580i	$-0.883602\pi$
$701$	−2.43006e7	−1.86776	−0.933882	+	0.357580i	$0.883602\pi$
$702$	0	0
$703$	4.83541e6	0.369016
$704$	0	0
$705$	2.30226e7	1.74455
$706$	0	0
$707$	−2.30832e6	−0.173679
$708$	0	0
$709$	−1.63023e7	−1.21796	−0.608981	−	0.793185i	$-0.708422\pi$
$709$	−1.63023e7	−1.21796	−0.608981	+	0.793185i	$0.708422\pi$
$710$	0	0
$711$	6.41907e6	0.476210
$712$	0	0
$713$	1.30090e7	0.958343
$714$	0	0
$715$	1.13457e7	0.829976
$716$	0	0
$717$	−6.00728e6	−0.436396
$718$	0	0
$719$	−1.08258e6	−0.0780974	−0.0390487	−	0.999237i	$-0.512433\pi$
$719$	−1.08258e6	−0.0780974	−0.0390487	+	0.999237i	$0.512433\pi$
$720$	0	0
$721$	−1.16590e7	−0.835266
$722$	0	0
$723$	−8.55609e6	−0.608737
$724$	0	0
$725$	3.05562e6	0.215900
$726$	0	0
$727$	−2.21287e7	−1.55282	−0.776409	−	0.630229i	$-0.782961\pi$
$727$	−2.21287e7	−1.55282	−0.776409	+	0.630229i	$0.782961\pi$
$728$	0	0
$729$	4.77617e6	0.332860
$730$	0	0
$731$	−401461.	−0.0277875
$732$	0	0
$733$	1.35432e7	0.931028	0.465514	−	0.885041i	$-0.345869\pi$
$733$	1.35432e7	0.931028	0.465514	+	0.885041i	$0.345869\pi$
$734$	0	0
$735$	−8.18207e6	−0.558656
$736$	0	0
$737$	8.71884e6	0.591276
$738$	0	0
$739$	−3.84195e6	−0.258786	−0.129393	−	0.991593i	$-0.541303\pi$
$739$	−3.84195e6	−0.258786	−0.129393	+	0.991593i	$0.541303\pi$
$740$	0	0
$741$	4.17257e6	0.279163
$742$	0	0
$743$	−1.06878e7	−0.710256	−0.355128	−	0.934818i	$-0.615563\pi$
$743$	−1.06878e7	−0.710256	−0.355128	+	0.934818i	$0.615563\pi$
$744$	0	0
$745$	162071.	0.0106983
$746$	0	0
$747$	−3.77680e6	−0.247641
$748$	0	0
$749$	2.37723e7	1.54834
$750$	0	0
$751$	−2.09801e7	−1.35740	−0.678701	−	0.734414i	$-0.737457\pi$
$751$	−2.09801e7	−1.35740	−0.678701	+	0.734414i	$0.737457\pi$
$752$	0	0
$753$	−1.81619e7	−1.16728
$754$	0	0
$755$	1.10018e7	0.702422
$756$	0	0
$757$	2.90118e7	1.84007	0.920035	−	0.391837i	$-0.128160\pi$
$757$	2.90118e7	1.84007	0.920035	+	0.391837i	$0.128160\pi$
$758$	0	0
$759$	1.21049e7	0.762708
$760$	0	0
$761$	4.43022e6	0.277309	0.138654	−	0.990341i	$-0.455722\pi$
$761$	4.43022e6	0.277309	0.138654	+	0.990341i	$0.455722\pi$
$762$	0	0
$763$	−2.60044e7	−1.61710
$764$	0	0
$765$	1.10782e6	0.0684407
$766$	0	0
$767$	−1.74897e7	−1.07348
$768$	0	0
$769$	1.08879e6	0.0663936	0.0331968	−	0.999449i	$-0.489431\pi$
$769$	1.08879e6	0.0663936	0.0331968	+	0.999449i	$0.489431\pi$
$770$	0	0
$771$	−3.16370e6	−0.191673
$772$	0	0
$773$	−1.43240e7	−0.862212	−0.431106	−	0.902301i	$-0.641877\pi$
$773$	−1.43240e7	−0.862212	−0.431106	+	0.902301i	$0.641877\pi$
$774$	0	0
$775$	−9.14474e6	−0.546911
$776$	0	0
$777$	−3.76444e7	−2.23691
$778$	0	0
$779$	7.08066e6	0.418052
$780$	0	0
$781$	−1.51953e6	−0.0891422
$782$	0	0
$783$	4.59111e6	0.267617
$784$	0	0
$785$	−3.61023e7	−2.09103
$786$	0	0
$787$	−3.25819e7	−1.87517	−0.937583	−	0.347761i	$-0.886942\pi$
$787$	−3.25819e7	−1.87517	−0.937583	+	0.347761i	$0.886942\pi$
$788$	0	0
$789$	2.29831e7	1.31437
$790$	0	0
$791$	−9.85415e6	−0.559987
$792$	0	0
$793$	2.27792e7	1.28634
$794$	0	0
$795$	1.74755e7	0.980646
$796$	0	0
$797$	7.67176e6	0.427808	0.213904	−	0.976855i	$-0.431382\pi$
$797$	7.67176e6	0.427808	0.213904	+	0.976855i	$0.431382\pi$
$798$	0	0
$799$	−2.85894e6	−0.158430
$800$	0	0
$801$	1.02505e6	0.0564499
$802$	0	0
$803$	−1.27225e7	−0.696280
$804$	0	0
$805$	2.69266e7	1.46451
$806$	0	0
$807$	−3.94700e7	−2.13346
$808$	0	0
$809$	1.98169e7	1.06455	0.532273	−	0.846573i	$-0.321338\pi$
$809$	1.98169e7	1.06455	0.532273	+	0.846573i	$0.321338\pi$
$810$	0	0
$811$	2.53549e7	1.35366	0.676829	−	0.736140i	$-0.263354\pi$
$811$	2.53549e7	1.35366	0.676829	+	0.736140i	$0.263354\pi$
$812$	0	0
$813$	−9.79521e6	−0.519742
$814$	0	0
$815$	−4.22232e7	−2.22668
$816$	0	0
$817$	902016.	0.0472780
$818$	0	0
$819$	−9.36777e6	−0.488008
$820$	0	0
$821$	−2.27985e7	−1.18045	−0.590227	−	0.807238i	$-0.700962\pi$
$821$	−2.27985e7	−1.18045	−0.590227	+	0.807238i	$0.700962\pi$
$822$	0	0
$823$	3.51660e6	0.180977	0.0904885	−	0.995898i	$-0.471157\pi$
$823$	3.51660e6	0.180977	0.0904885	+	0.995898i	$0.471157\pi$
$824$	0	0
$825$	−8.50921e6	−0.435265
$826$	0	0
$827$	760144.	0.0386484	0.0193242	−	0.999813i	$-0.493849\pi$
$827$	760144.	0.0386484	0.0193242	+	0.999813i	$0.493849\pi$
$828$	0	0
$829$	2.94228e7	1.48695	0.743476	−	0.668763i	$-0.233176\pi$
$829$	2.94228e7	1.48695	0.743476	+	0.668763i	$0.233176\pi$
$830$	0	0
$831$	−2.26788e7	−1.13924
$832$	0	0
$833$	1.01604e6	0.0507341
$834$	0	0
$835$	2.96480e7	1.47156
$836$	0	0
$837$	−1.37401e7	−0.677917
$838$	0	0
$839$	−1.82408e6	−0.0894620	−0.0447310	−	0.998999i	$-0.514243\pi$
$839$	−1.82408e6	−0.0894620	−0.0447310	+	0.998999i	$0.514243\pi$
$840$	0	0
$841$	−1.75559e7	−0.855922
$842$	0	0
$843$	3.28361e7	1.59141
$844$	0	0
$845$	1.39617e6	0.0672660
$846$	0	0
$847$	1.42868e7	0.684267
$848$	0	0
$849$	2.27404e7	1.08275
$850$	0	0
$851$	3.38692e7	1.60318
$852$	0	0
$853$	7.39342e6	0.347915	0.173957	−	0.984753i	$-0.444345\pi$
$853$	7.39342e6	0.347915	0.173957	+	0.984753i	$0.444345\pi$
$854$	0	0
$855$	−2.48908e6	−0.116446
$856$	0	0
$857$	2.92351e7	1.35973	0.679864	−	0.733338i	$-0.262039\pi$
$857$	2.92351e7	1.35973	0.679864	+	0.733338i	$0.262039\pi$
$858$	0	0
$859$	3.19321e7	1.47654	0.738269	−	0.674506i	$-0.235644\pi$
$859$	3.19321e7	1.47654	0.738269	+	0.674506i	$0.235644\pi$
$860$	0	0
$861$	−5.51240e7	−2.53415
$862$	0	0
$863$	2.41161e7	1.10225	0.551124	−	0.834423i	$-0.314199\pi$
$863$	2.41161e7	1.10225	0.551124	+	0.834423i	$0.314199\pi$
$864$	0	0
$865$	−6.74421e6	−0.306472
$866$	0	0
$867$	2.57605e7	1.16388
$868$	0	0
$869$	1.68870e7	0.758585
$870$	0	0
$871$	2.10510e7	0.940214
$872$	0	0
$873$	1.61520e6	0.0717282
$874$	0	0
$875$	1.43495e7	0.633603
$876$	0	0
$877$	1.75569e7	0.770812	0.385406	−	0.922747i	$-0.374061\pi$
$877$	1.75569e7	0.770812	0.385406	+	0.922747i	$0.374061\pi$
$878$	0	0
$879$	−3.61109e7	−1.57640
$880$	0	0
$881$	−402812.	−0.0174849	−0.00874244	−	0.999962i	$-0.502783\pi$
$881$	−402812.	−0.0174849	−0.00874244	+	0.999962i	$0.502783\pi$
$882$	0	0
$883$	−3.52030e7	−1.51942	−0.759711	−	0.650261i	$-0.774660\pi$
$883$	−3.52030e7	−1.51942	−0.759711	+	0.650261i	$0.774660\pi$
$884$	0	0
$885$	3.61786e7	1.55272
$886$	0	0
$887$	−2.28152e7	−0.973679	−0.486840	−	0.873491i	$-0.661850\pi$
$887$	−2.28152e7	−0.973679	−0.486840	+	0.873491i	$0.661850\pi$
$888$	0	0
$889$	4.49428e7	1.90724
$890$	0	0
$891$	−1.89844e7	−0.801129
$892$	0	0
$893$	6.42355e6	0.269555
$894$	0	0
$895$	5.01942e7	2.09457
$896$	0	0
$897$	2.92264e7	1.21281
$898$	0	0
$899$	−8.84422e6	−0.364973
$900$	0	0
$901$	−2.17010e6	−0.0890569
$902$	0	0
$903$	−7.02233e6	−0.286590
$904$	0	0
$905$	−2.42111e7	−0.982637
$906$	0	0
$907$	2.63621e7	1.06405	0.532025	−	0.846729i	$-0.321431\pi$
$907$	2.63621e7	1.06405	0.532025	+	0.846729i	$0.321431\pi$
$908$	0	0
$909$	1.49460e6	0.0599950
$910$	0	0
$911$	1.95110e7	0.778903	0.389452	−	0.921047i	$-0.372665\pi$
$911$	1.95110e7	0.778903	0.389452	+	0.921047i	$0.372665\pi$
$912$	0	0
$913$	−9.93585e6	−0.394483
$914$	0	0
$915$	−4.71202e7	−1.86061
$916$	0	0
$917$	9.08531e6	0.356793
$918$	0	0
$919$	1.08939e7	0.425493	0.212747	−	0.977107i	$-0.431759\pi$
$919$	1.08939e7	0.425493	0.212747	+	0.977107i	$0.431759\pi$
$920$	0	0
$921$	−3.17429e7	−1.23310
$922$	0	0
$923$	−3.66880e6	−0.141749
$924$	0	0
$925$	−2.38085e7	−0.914907
$926$	0	0
$927$	7.54904e6	0.288531
$928$	0	0
$929$	4.10689e7	1.56125	0.780627	−	0.624998i	$-0.214900\pi$
$929$	4.10689e7	1.56125	0.780627	+	0.624998i	$0.214900\pi$
$930$	0	0
$931$	−2.28288e6	−0.0863195
$932$	0	0
$933$	4.74740e7	1.78546
$934$	0	0
$935$	2.91440e6	0.109024
$936$	0	0
$937$	3.05076e7	1.13517	0.567583	−	0.823316i	$-0.307878\pi$
$937$	3.05076e7	1.13517	0.567583	+	0.823316i	$0.307878\pi$
$938$	0	0
$939$	4.60340e7	1.70378
$940$	0	0
$941$	9.50717e6	0.350007	0.175004	−	0.984568i	$-0.444006\pi$
$941$	9.50717e6	0.350007	0.175004	+	0.984568i	$0.444006\pi$
$942$	0	0
$943$	4.95958e7	1.81621
$944$	0	0
$945$	−2.84398e7	−1.03597
$946$	0	0
$947$	−3.24762e7	−1.17677	−0.588384	−	0.808582i	$-0.700236\pi$
$947$	−3.24762e7	−1.17677	−0.588384	+	0.808582i	$0.700236\pi$
$948$	0	0
$949$	−3.07175e7	−1.10719
$950$	0	0
$951$	4.70809e7	1.68808
$952$	0	0
$953$	−6.92438e6	−0.246972	−0.123486	−	0.992346i	$-0.539407\pi$
$953$	−6.92438e6	−0.246972	−0.123486	+	0.992346i	$0.539407\pi$
$954$	0	0
$955$	3.62616e7	1.28659
$956$	0	0
$957$	−8.22957e6	−0.290467
$958$	0	0
$959$	−3.40260e7	−1.19471
$960$	0	0
$961$	−2.16048e6	−0.0754642
$962$	0	0
$963$	−1.53921e7	−0.534851
$964$	0	0
$965$	1.82612e7	0.631265
$966$	0	0
$967$	3.40976e7	1.17262	0.586311	−	0.810086i	$-0.300580\pi$
$967$	3.40976e7	1.17262	0.586311	+	0.810086i	$0.300580\pi$
$968$	0	0
$969$	1.07182e6	0.0366702
$970$	0	0
$971$	−4.50092e7	−1.53198	−0.765990	−	0.642852i	$-0.777751\pi$
$971$	−4.50092e7	−1.53198	−0.765990	+	0.642852i	$0.777751\pi$
$972$	0	0
$973$	2.34898e7	0.795421
$974$	0	0
$975$	−2.05448e7	−0.692135
$976$	0	0
$977$	−1.75662e7	−0.588764	−0.294382	−	0.955688i	$-0.595114\pi$
$977$	−1.75662e7	−0.588764	−0.294382	+	0.955688i	$0.595114\pi$
$978$	0	0
$979$	2.69666e6	0.0899226
$980$	0	0
$981$	1.68374e7	0.558603
$982$	0	0
$983$	−3.94567e6	−0.130238	−0.0651188	−	0.997878i	$-0.520743\pi$
$983$	−3.94567e6	−0.130238	−0.0651188	+	0.997878i	$0.520743\pi$
$984$	0	0
$985$	−6.36076e7	−2.08890
$986$	0	0
$987$	−5.00083e7	−1.63399
$988$	0	0
$989$	6.31809e6	0.205397
$990$	0	0
$991$	−7.30525e6	−0.236293	−0.118146	−	0.992996i	$-0.537695\pi$
$991$	−7.30525e6	−0.236293	−0.118146	+	0.992996i	$0.537695\pi$
$992$	0	0
$993$	−2.15632e7	−0.693971
$994$	0	0
$995$	−3.89616e6	−0.124761
$996$	0	0
$997$	2.37141e7	0.755560	0.377780	−	0.925895i	$-0.376688\pi$
$997$	2.37141e7	0.755560	0.377780	+	0.925895i	$0.376688\pi$
$998$	0	0
$999$	−3.57726e7	−1.13406

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	152.6.a.c.1.2	✓	6
4.3	odd	2		304.6.a.n.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.6.a.c.1.2	✓	6	1.1	even	1	trivial
304.6.a.n.1.5		6	4.3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 152 = 2^{3} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	152.a (trivial)

\(f(q)\)	\(=\)	\(q-18.4790 q^{3} +70.0177 q^{5} -152.088 q^{7} +98.4744 q^{9} +259.063 q^{11} +625.486 q^{13} -1293.86 q^{15} +160.671 q^{17} -361.000 q^{19} +2810.44 q^{21} -2528.59 q^{23} +1777.48 q^{25} +2670.69 q^{27} +1719.07 q^{29} -5144.77 q^{31} -4787.23 q^{33} -10648.9 q^{35} -13394.5 q^{37} -11558.4 q^{39} -19614.0 q^{41} -2498.66 q^{43} +6894.95 q^{45} -17793.8 q^{47} +6323.77 q^{49} -2969.04 q^{51} -13506.5 q^{53} +18139.0 q^{55} +6670.93 q^{57} -27961.8 q^{59} +36418.3 q^{61} -14976.8 q^{63} +43795.1 q^{65} +33655.4 q^{67} +46725.9 q^{69} -5865.51 q^{71} -49109.8 q^{73} -32846.1 q^{75} -39400.3 q^{77} +65185.2 q^{79} -73281.1 q^{81} -38353.1 q^{83} +11249.8 q^{85} -31766.7 q^{87} +10409.3 q^{89} -95129.0 q^{91} +95070.4 q^{93} -25276.4 q^{95} +16402.2 q^{97} +25511.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{3} - 25 q^{5} - 161 q^{7} + 572 q^{9} - 205 q^{11} - 382 q^{13} - 950 q^{15} - 65 q^{17} - 2166 q^{19} - 6006 q^{21} - 5584 q^{23} - 3111 q^{25} - 6874 q^{27} - 9972 q^{29} - 7140 q^{31} - 3774 q^{33}+ \cdots + 49107 q^{99}+O(q^{100}) \) 6 * q - 4 * q^3 - 25 * q^5 - 161 * q^7 + 572 * q^9 - 205 * q^11 - 382 * q^13 - 950 * q^15 - 65 * q^17 - 2166 * q^19 - 6006 * q^21 - 5584 * q^23 - 3111 * q^25 - 6874 * q^27 - 9972 * q^29 - 7140 * q^31 - 3774 * q^33 - 14979 * q^35 - 7872 * q^37 - 46448 * q^39 - 12476 * q^41 - 45829 * q^43 - 41933 * q^45 - 61259 * q^47 + 5315 * q^49 - 72572 * q^51 + 13382 * q^53 - 55199 * q^55 + 1444 * q^57 - 56812 * q^59 + 2559 * q^61 - 97953 * q^63 + 37452 * q^65 - 36678 * q^67 - 12888 * q^69 + 14934 * q^71 + 43567 * q^73 - 57842 * q^75 + 36359 * q^77 - 155476 * q^79 + 199946 * q^81 - 149856 * q^83 + 112601 * q^85 - 95180 * q^87 + 176022 * q^89 + 76214 * q^91 + 434620 * q^93 + 9025 * q^95 + 117190 * q^97 + 49107 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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