LMFDB - Embedded newform 448.6.a.s.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [448,6,Mod(1,448)] 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("448.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(448, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	448.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-14,0,-34] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$71.8519512762$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{61}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 15 $ x^2 - x - 15
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 224)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$4.40512$ of defining polynomial
Character	$\chi$	$=$	448.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-14.8102 q^{3} +37.6717 q^{5} +49.0000 q^{7} -23.6565 q^{9} -149.271 q^{11} +627.702 q^{13} -557.928 q^{15} -637.343 q^{17} -1306.13 q^{19} -725.702 q^{21} +376.964 q^{23} -1705.84 q^{25} +3949.25 q^{27} +6216.40 q^{29} -6036.18 q^{31} +2210.75 q^{33} +1845.92 q^{35} +3634.66 q^{37} -9296.43 q^{39} +12772.8 q^{41} -3347.17 q^{43} -891.182 q^{45} -14347.3 q^{47} +2401.00 q^{49} +9439.22 q^{51} +4587.95 q^{53} -5623.32 q^{55} +19344.1 q^{57} +11398.0 q^{59} -3085.32 q^{61} -1159.17 q^{63} +23646.6 q^{65} +35278.7 q^{67} -5582.93 q^{69} -48903.1 q^{71} +15156.6 q^{73} +25263.9 q^{75} -7314.30 q^{77} -23630.2 q^{79} -52740.8 q^{81} -28021.0 q^{83} -24009.8 q^{85} -92066.4 q^{87} -121536. q^{89} +30757.4 q^{91} +89397.3 q^{93} -49204.3 q^{95} -646.205 q^{97} +3531.24 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 14 q^{3} - 34 q^{5} + 98 q^{7} - 266 q^{9} + 420 q^{11} + 490 q^{13} - 616 q^{15} - 1056 q^{17} + 1246 q^{19} - 686 q^{21} + 504 q^{23} + 306 q^{25} + 3556 q^{27} + 3904 q^{29} - 2044 q^{31} + 2672 q^{33}+ \cdots - 134428 q^{99}+O(q^{100}) $ 2 * q - 14 * q^3 - 34 * q^5 + 98 * q^7 - 266 * q^9 + 420 * q^11 + 490 * q^13 - 616 * q^15 - 1056 * q^17 + 1246 * q^19 - 686 * q^21 + 504 * q^23 + 306 * q^25 + 3556 * q^27 + 3904 * q^29 - 2044 * q^31 + 2672 * q^33 - 1666 * q^35 + 7488 * q^37 - 9408 * q^39 + 7832 * q^41 + 10332 * q^43 + 16478 * q^45 - 41972 * q^47 + 4802 * q^49 + 9100 * q^51 - 32812 * q^53 - 46424 * q^55 + 21412 * q^57 + 48398 * q^59 + 718 * q^61 - 13034 * q^63 + 33516 * q^65 + 12824 * q^67 - 5480 * q^69 - 103992 * q^71 - 54100 * q^73 + 26894 * q^75 + 20580 * q^77 - 64568 * q^79 + 5830 * q^81 - 47810 * q^83 + 5996 * q^85 - 93940 * q^87 - 17388 * q^89 + 24010 * q^91 + 92632 * q^93 - 232120 * q^95 - 97296 * q^97 - 134428 * 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
$3$	−14.8102	−0.950078	−0.475039	−	0.879965i	$-0.657566\pi$
$3$	−14.8102	−0.950078	−0.475039	+	0.879965i	$0.657566\pi$
$4$	0	0
$5$	37.6717	0.673893	0.336946	−	0.941524i	$-0.390606\pi$
$5$	37.6717	0.673893	0.336946	+	0.941524i	$0.390606\pi$
$6$	0	0
$7$	49.0000	0.377964
$7$	49.0000	0.377964
$8$	0	0
$9$	−23.6565	−0.0973519
$10$	0	0
$11$	−149.271	−0.371959	−0.185980	−	0.982554i	$-0.559546\pi$
$11$	−149.271	−0.371959	−0.185980	+	0.982554i	$0.559546\pi$
$12$	0	0
$13$	627.702	1.03014	0.515069	−	0.857149i	$-0.327766\pi$
$13$	627.702	1.03014	0.515069	+	0.857149i	$0.327766\pi$
$14$	0	0
$15$	−557.928	−0.640251
$16$	0	0
$17$	−637.343	−0.534874	−0.267437	−	0.963575i	$-0.586177\pi$
$17$	−637.343	−0.534874	−0.267437	+	0.963575i	$0.586177\pi$
$18$	0	0
$19$	−1306.13	−0.830048	−0.415024	−	0.909811i	$-0.636227\pi$
$19$	−1306.13	−0.830048	−0.415024	+	0.909811i	$0.636227\pi$
$20$	0	0
$21$	−725.702	−0.359096
$22$	0	0
$23$	376.964	0.148587	0.0742934	−	0.997236i	$-0.476330\pi$
$23$	376.964	0.148587	0.0742934	+	0.997236i	$0.476330\pi$
$24$	0	0
$25$	−1705.84	−0.545869
$26$	0	0
$27$	3949.25	1.04257
$28$	0	0
$29$	6216.40	1.37260	0.686300	−	0.727319i	$-0.259234\pi$
$29$	6216.40	1.37260	0.686300	+	0.727319i	$0.259234\pi$
$30$	0	0
$31$	−6036.18	−1.12813	−0.564063	−	0.825731i	$-0.690763\pi$
$31$	−6036.18	−1.12813	−0.564063	+	0.825731i	$0.690763\pi$
$32$	0	0
$33$	2210.75	0.353390
$34$	0	0
$35$	1845.92	0.254708
$36$	0	0
$37$	3634.66	0.436475	0.218237	−	0.975896i	$-0.429969\pi$
$37$	3634.66	0.436475	0.218237	+	0.975896i	$0.429969\pi$
$38$	0	0
$39$	−9296.43	−0.978711
$40$	0	0
$41$	12772.8	1.18666	0.593331	−	0.804958i	$-0.297812\pi$
$41$	12772.8	1.18666	0.593331	+	0.804958i	$0.297812\pi$
$42$	0	0
$43$	−3347.17	−0.276062	−0.138031	−	0.990428i	$-0.544077\pi$
$43$	−3347.17	−0.276062	−0.138031	+	0.990428i	$0.544077\pi$
$44$	0	0
$45$	−891.182	−0.0656047
$46$	0	0
$47$	−14347.3	−0.947382	−0.473691	−	0.880691i	$-0.657079\pi$
$47$	−14347.3	−0.947382	−0.473691	+	0.880691i	$0.657079\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	9439.22	0.508172
$52$	0	0
$53$	4587.95	0.224352	0.112176	−	0.993688i	$-0.464218\pi$
$53$	4587.95	0.224352	0.112176	+	0.993688i	$0.464218\pi$
$54$	0	0
$55$	−5623.32	−0.250660
$56$	0	0
$57$	19344.1	0.788610
$58$	0	0
$59$	11398.0	0.426284	0.213142	−	0.977021i	$-0.431630\pi$
$59$	11398.0	0.426284	0.213142	+	0.977021i	$0.431630\pi$
$60$	0	0
$61$	−3085.32	−0.106164	−0.0530818	−	0.998590i	$-0.516904\pi$
$61$	−3085.32	−0.106164	−0.0530818	+	0.998590i	$0.516904\pi$
$62$	0	0
$63$	−1159.17	−0.0367955
$64$	0	0
$65$	23646.6	0.694202
$66$	0	0
$67$	35278.7	0.960120	0.480060	−	0.877236i	$-0.340615\pi$
$67$	35278.7	0.960120	0.480060	+	0.877236i	$0.340615\pi$
$68$	0	0
$69$	−5582.93	−0.141169
$70$	0	0
$71$	−48903.1	−1.15131	−0.575653	−	0.817694i	$-0.695252\pi$
$71$	−48903.1	−1.15131	−0.575653	+	0.817694i	$0.695252\pi$
$72$	0	0
$73$	15156.6	0.332885	0.166443	−	0.986051i	$-0.446772\pi$
$73$	15156.6	0.332885	0.166443	+	0.986051i	$0.446772\pi$
$74$	0	0
$75$	25263.9	0.518618
$76$	0	0
$77$	−7314.30	−0.140587
$78$	0	0
$79$	−23630.2	−0.425991	−0.212996	−	0.977053i	$-0.568322\pi$
$79$	−23630.2	−0.425991	−0.212996	+	0.977053i	$0.568322\pi$
$80$	0	0
$81$	−52740.8	−0.893171
$82$	0	0
$83$	−28021.0	−0.446466	−0.223233	−	0.974765i	$-0.571661\pi$
$83$	−28021.0	−0.446466	−0.223233	+	0.974765i	$0.571661\pi$
$84$	0	0
$85$	−24009.8	−0.360447
$86$	0	0
$87$	−92066.4	−1.30408
$88$	0	0
$89$	−121536.	−1.62642	−0.813208	−	0.581973i	$-0.802281\pi$
$89$	−121536.	−1.62642	−0.813208	+	0.581973i	$0.802281\pi$
$90$	0	0
$91$	30757.4	0.389355
$92$	0	0
$93$	89397.3	1.07181
$94$	0	0
$95$	−49204.3	−0.559363
$96$	0	0
$97$	−646.205	−0.00697335	−0.00348667	−	0.999994i	$-0.501110\pi$
$97$	−646.205	−0.00697335	−0.00348667	+	0.999994i	$0.501110\pi$
$98$	0	0
$99$	3531.24	0.0362109
$100$	0	0
$101$	−73325.8	−0.715243	−0.357621	−	0.933867i	$-0.616412\pi$
$101$	−73325.8	−0.715243	−0.357621	+	0.933867i	$0.616412\pi$
$102$	0	0
$103$	−179916.	−1.67101	−0.835503	−	0.549486i	$-0.814824\pi$
$103$	−179916.	−1.67101	−0.835503	+	0.549486i	$0.814824\pi$
$104$	0	0
$105$	−27338.5	−0.241992
$106$	0	0
$107$	−229269.	−1.93591	−0.967957	−	0.251115i	$-0.919203\pi$
$107$	−229269.	−1.93591	−0.967957	+	0.251115i	$0.919203\pi$
$108$	0	0
$109$	38061.4	0.306844	0.153422	−	0.988161i	$-0.450971\pi$
$109$	38061.4	0.306844	0.153422	+	0.988161i	$0.450971\pi$
$110$	0	0
$111$	−53830.2	−0.414685
$112$	0	0
$113$	50424.1	0.371485	0.185743	−	0.982598i	$-0.440531\pi$
$113$	50424.1	0.371485	0.185743	+	0.982598i	$0.440531\pi$
$114$	0	0
$115$	14200.9	0.100132
$116$	0	0
$117$	−14849.2	−0.100286
$118$	0	0
$119$	−31229.8	−0.202163
$120$	0	0
$121$	−138769.	−0.861646
$122$	0	0
$123$	−189169.	−1.12742
$124$	0	0
$125$	−181986.	−1.04175
$126$	0	0
$127$	163141.	0.897541	0.448771	−	0.893647i	$-0.351862\pi$
$127$	163141.	0.897541	0.448771	+	0.893647i	$0.351862\pi$
$128$	0	0
$129$	49572.5	0.262281
$130$	0	0
$131$	−147002.	−0.748417	−0.374208	−	0.927345i	$-0.622086\pi$
$131$	−147002.	−0.748417	−0.374208	+	0.927345i	$0.622086\pi$
$132$	0	0
$133$	−64000.5	−0.313728
$134$	0	0
$135$	148775.	0.702580
$136$	0	0
$137$	5291.80	0.0240881	0.0120440	−	0.999927i	$-0.496166\pi$
$137$	5291.80	0.0240881	0.0120440	+	0.999927i	$0.496166\pi$
$138$	0	0
$139$	−173621.	−0.762195	−0.381097	−	0.924535i	$-0.624454\pi$
$139$	−173621.	−0.762195	−0.381097	+	0.924535i	$0.624454\pi$
$140$	0	0
$141$	212487.	0.900087
$142$	0	0
$143$	−93698.0	−0.383169
$144$	0	0
$145$	234183.	0.924985
$146$	0	0
$147$	−35559.4	−0.135725
$148$	0	0
$149$	−352741.	−1.30164	−0.650820	−	0.759232i	$-0.725575\pi$
$149$	−352741.	−1.30164	−0.650820	+	0.759232i	$0.725575\pi$
$150$	0	0
$151$	−474621.	−1.69397	−0.846983	−	0.531619i	$-0.821584\pi$
$151$	−474621.	−1.69397	−0.846983	+	0.531619i	$0.821584\pi$
$152$	0	0
$153$	15077.3	0.0520710
$154$	0	0
$155$	−227393.	−0.760237
$156$	0	0
$157$	79394.0	0.257062	0.128531	−	0.991705i	$-0.458974\pi$
$157$	79394.0	0.257062	0.128531	+	0.991705i	$0.458974\pi$
$158$	0	0
$159$	−67948.7	−0.213151
$160$	0	0
$161$	18471.2	0.0561605
$162$	0	0
$163$	−566730.	−1.67073	−0.835367	−	0.549692i	$-0.814745\pi$
$163$	−566730.	−1.67073	−0.835367	+	0.549692i	$0.814745\pi$
$164$	0	0
$165$	83282.7	0.238147
$166$	0	0
$167$	537999.	1.49276	0.746381	−	0.665519i	$-0.231790\pi$
$167$	537999.	1.49276	0.746381	+	0.665519i	$0.231790\pi$
$168$	0	0
$169$	22717.1	0.0611837
$170$	0	0
$171$	30898.5	0.0808067
$172$	0	0
$173$	182499.	0.463603	0.231801	−	0.972763i	$-0.425538\pi$
$173$	182499.	0.463603	0.231801	+	0.972763i	$0.425538\pi$
$174$	0	0
$175$	−83586.1	−0.206319
$176$	0	0
$177$	−168807.	−0.405003
$178$	0	0
$179$	−270286.	−0.630510	−0.315255	−	0.949007i	$-0.602090\pi$
$179$	−270286.	−0.630510	−0.315255	+	0.949007i	$0.602090\pi$
$180$	0	0
$181$	−391356.	−0.887924	−0.443962	−	0.896046i	$-0.646427\pi$
$181$	−391356.	−0.887924	−0.443962	+	0.896046i	$0.646427\pi$
$182$	0	0
$183$	45694.4	0.100864
$184$	0	0
$185$	136924.	0.294137
$186$	0	0
$187$	95137.2	0.198951
$188$	0	0
$189$	193513.	0.394054
$190$	0	0
$191$	489469.	0.970826	0.485413	−	0.874285i	$-0.338669\pi$
$191$	489469.	0.970826	0.485413	+	0.874285i	$0.338669\pi$
$192$	0	0
$193$	620214.	1.19853	0.599265	−	0.800551i	$-0.295460\pi$
$193$	620214.	1.19853	0.599265	+	0.800551i	$0.295460\pi$
$194$	0	0
$195$	−350213.	−0.659546
$196$	0	0
$197$	614756.	1.12859	0.564296	−	0.825572i	$-0.309147\pi$
$197$	614756.	1.12859	0.564296	+	0.825572i	$0.309147\pi$
$198$	0	0
$199$	−6443.77	−0.0115347	−0.00576736	−	0.999983i	$-0.501836\pi$
$199$	−6443.77	−0.0115347	−0.00576736	+	0.999983i	$0.501836\pi$
$200$	0	0
$201$	−522486.	−0.912189
$202$	0	0
$203$	304603.	0.518794
$204$	0	0
$205$	481175.	0.799683
$206$	0	0
$207$	−8917.65	−0.0144652
$208$	0	0
$209$	194968.	0.308744
$210$	0	0
$211$	−533104.	−0.824340	−0.412170	−	0.911107i	$-0.635229\pi$
$211$	−533104.	−0.824340	−0.412170	+	0.911107i	$0.635229\pi$
$212$	0	0
$213$	724268.	1.09383
$214$	0	0
$215$	−126094.	−0.186036
$216$	0	0
$217$	−295773.	−0.426392
$218$	0	0
$219$	−224473.	−0.316267
$220$	0	0
$221$	−400062.	−0.550994
$222$	0	0
$223$	67470.5	0.0908556	0.0454278	−	0.998968i	$-0.485535\pi$
$223$	67470.5	0.0908556	0.0454278	+	0.998968i	$0.485535\pi$
$224$	0	0
$225$	40354.2	0.0531413
$226$	0	0
$227$	489721.	0.630788	0.315394	−	0.948961i	$-0.397863\pi$
$227$	489721.	0.630788	0.315394	+	0.948961i	$0.397863\pi$
$228$	0	0
$229$	−373971.	−0.471248	−0.235624	−	0.971844i	$-0.575713\pi$
$229$	−373971.	−0.471248	−0.235624	+	0.971844i	$0.575713\pi$
$230$	0	0
$231$	108327.	0.133569
$232$	0	0
$233$	−579217.	−0.698958	−0.349479	−	0.936944i	$-0.613641\pi$
$233$	−579217.	−0.698958	−0.349479	+	0.936944i	$0.613641\pi$
$234$	0	0
$235$	−540487.	−0.638434
$236$	0	0
$237$	349970.	0.404725
$238$	0	0
$239$	−165669.	−0.187605	−0.0938027	−	0.995591i	$-0.529902\pi$
$239$	−165669.	−0.187605	−0.0938027	+	0.995591i	$0.529902\pi$
$240$	0	0
$241$	574223.	0.636851	0.318426	−	0.947948i	$-0.396846\pi$
$241$	574223.	0.636851	0.318426	+	0.947948i	$0.396846\pi$
$242$	0	0
$243$	−178563.	−0.193988
$244$	0	0
$245$	90449.9	0.0962704
$246$	0	0
$247$	−819862.	−0.855063
$248$	0	0
$249$	414998.	0.424178
$250$	0	0
$251$	940094.	0.941861	0.470931	−	0.882170i	$-0.343918\pi$
$251$	940094.	0.941861	0.470931	+	0.882170i	$0.343918\pi$
$252$	0	0
$253$	−56270.0	−0.0552682
$254$	0	0
$255$	355592.	0.342453
$256$	0	0
$257$	129585.	0.122383	0.0611915	−	0.998126i	$-0.480510\pi$
$257$	129585.	0.122383	0.0611915	+	0.998126i	$0.480510\pi$
$258$	0	0
$259$	178098.	0.164972
$260$	0	0
$261$	−147058.	−0.133625
$262$	0	0
$263$	−1.42670e6	−1.27187	−0.635936	−	0.771742i	$-0.719386\pi$
$263$	−1.42670e6	−1.27187	−0.635936	+	0.771742i	$0.719386\pi$
$264$	0	0
$265$	172836.	0.151189
$266$	0	0
$267$	1.79999e6	1.54522
$268$	0	0
$269$	−1.00662e6	−0.848173	−0.424087	−	0.905622i	$-0.639405\pi$
$269$	−1.00662e6	−0.848173	−0.424087	+	0.905622i	$0.639405\pi$
$270$	0	0
$271$	−1.14104e6	−0.943793	−0.471897	−	0.881654i	$-0.656430\pi$
$271$	−1.14104e6	−0.943793	−0.471897	+	0.881654i	$0.656430\pi$
$272$	0	0
$273$	−455525.	−0.369918
$274$	0	0
$275$	254633.	0.203041
$276$	0	0
$277$	−2.20121e6	−1.72370	−0.861850	−	0.507163i	$-0.830694\pi$
$277$	−2.20121e6	−1.72370	−0.861850	+	0.507163i	$0.830694\pi$
$278$	0	0
$279$	142795.	0.109825
$280$	0	0
$281$	359913.	0.271914	0.135957	−	0.990715i	$-0.456589\pi$
$281$	359913.	0.271914	0.135957	+	0.990715i	$0.456589\pi$
$282$	0	0
$283$	2.10587e6	1.56302	0.781510	−	0.623893i	$-0.214450\pi$
$283$	2.10587e6	1.56302	0.781510	+	0.623893i	$0.214450\pi$
$284$	0	0
$285$	728727.	0.531438
$286$	0	0
$287$	625868.	0.448516
$288$	0	0
$289$	−1.01365e6	−0.713910
$290$	0	0
$291$	9570.46	0.00662522
$292$	0	0
$293$	−2.47171e6	−1.68201	−0.841003	−	0.541030i	$-0.818035\pi$
$293$	−2.47171e6	−1.68201	−0.841003	+	0.541030i	$0.818035\pi$
$294$	0	0
$295$	429383.	0.287269
$296$	0	0
$297$	−589510.	−0.387793
$298$	0	0
$299$	236621.	0.153065
$300$	0	0
$301$	−164011.	−0.104342
$302$	0	0
$303$	1.08597e6	0.679537
$304$	0	0
$305$	−116229.	−0.0715429
$306$	0	0
$307$	2.46333e6	1.49169	0.745843	−	0.666122i	$-0.232047\pi$
$307$	2.46333e6	1.49169	0.745843	+	0.666122i	$0.232047\pi$
$308$	0	0
$309$	2.66461e6	1.58759
$310$	0	0
$311$	1.87297e6	1.09807	0.549036	−	0.835799i	$-0.314995\pi$
$311$	1.87297e6	1.09807	0.549036	+	0.835799i	$0.314995\pi$
$312$	0	0
$313$	−482265.	−0.278244	−0.139122	−	0.990275i	$-0.544428\pi$
$313$	−482265.	−0.278244	−0.139122	+	0.990275i	$0.544428\pi$
$314$	0	0
$315$	−43667.9	−0.0247963
$316$	0	0
$317$	790018.	0.441559	0.220780	−	0.975324i	$-0.429140\pi$
$317$	790018.	0.441559	0.220780	+	0.975324i	$0.429140\pi$
$318$	0	0
$319$	−927931.	−0.510551
$320$	0	0
$321$	3.39553e6	1.83927
$322$	0	0
$323$	832455.	0.443971
$324$	0	0
$325$	−1.07076e6	−0.562320
$326$	0	0
$327$	−563698.	−0.291526
$328$	0	0
$329$	−703017.	−0.358077
$330$	0	0
$331$	3.54749e6	1.77972	0.889859	−	0.456236i	$-0.150803\pi$
$331$	3.54749e6	1.77972	0.889859	+	0.456236i	$0.150803\pi$
$332$	0	0
$333$	−85983.3	−0.0424916
$334$	0	0
$335$	1.32901e6	0.647018
$336$	0	0
$337$	−3.78817e6	−1.81700	−0.908500	−	0.417884i	$-0.862772\pi$
$337$	−3.78817e6	−1.81700	−0.908500	+	0.417884i	$0.862772\pi$
$338$	0	0
$339$	−746793.	−0.352940
$340$	0	0
$341$	901030.	0.419617
$342$	0	0
$343$	117649.	0.0539949
$344$	0	0
$345$	−210319.	−0.0951328
$346$	0	0
$347$	−352917.	−0.157344	−0.0786719	−	0.996901i	$-0.525068\pi$
$347$	−352917.	−0.157344	−0.0786719	+	0.996901i	$0.525068\pi$
$348$	0	0
$349$	−1.43108e6	−0.628926	−0.314463	−	0.949270i	$-0.601824\pi$
$349$	−1.43108e6	−0.628926	−0.314463	+	0.949270i	$0.601824\pi$
$350$	0	0
$351$	2.47895e6	1.07399
$352$	0	0
$353$	−2.23269e6	−0.953657	−0.476828	−	0.878996i	$-0.658214\pi$
$353$	−2.23269e6	−0.953657	−0.476828	+	0.878996i	$0.658214\pi$
$354$	0	0
$355$	−1.84227e6	−0.775857
$356$	0	0
$357$	462522.	0.192071
$358$	0	0
$359$	−3.95264e6	−1.61864	−0.809322	−	0.587365i	$-0.800165\pi$
$359$	−3.95264e6	−1.61864	−0.809322	+	0.587365i	$0.800165\pi$
$360$	0	0
$361$	−770119.	−0.311021
$362$	0	0
$363$	2.05520e6	0.818631
$364$	0	0
$365$	570975.	0.224329
$366$	0	0
$367$	1.67542e6	0.649320	0.324660	−	0.945831i	$-0.394750\pi$
$367$	1.67542e6	0.649320	0.324660	+	0.945831i	$0.394750\pi$
$368$	0	0
$369$	−302160.	−0.115524
$370$	0	0
$371$	224810.	0.0847969
$372$	0	0
$373$	3.64764e6	1.35750	0.678751	−	0.734369i	$-0.262522\pi$
$373$	3.64764e6	1.35750	0.678751	+	0.734369i	$0.262522\pi$
$374$	0	0
$375$	2.69526e6	0.989743
$376$	0	0
$377$	3.90205e6	1.41397
$378$	0	0
$379$	4.23267e6	1.51362	0.756809	−	0.653636i	$-0.226757\pi$
$379$	4.23267e6	1.51362	0.756809	+	0.653636i	$0.226757\pi$
$380$	0	0
$381$	−2.41616e6	−0.852734
$382$	0	0
$383$	4.11709e6	1.43415	0.717074	−	0.696997i	$-0.245481\pi$
$383$	4.11709e6	1.43415	0.717074	+	0.696997i	$0.245481\pi$
$384$	0	0
$385$	−275543.	−0.0947408
$386$	0	0
$387$	79182.4	0.0268752
$388$	0	0
$389$	1.41160e6	0.472974	0.236487	−	0.971635i	$-0.424004\pi$
$389$	1.41160e6	0.472974	0.236487	+	0.971635i	$0.424004\pi$
$390$	0	0
$391$	−240256.	−0.0794752
$392$	0	0
$393$	2.17713e6	0.711054
$394$	0	0
$395$	−890193.	−0.287072
$396$	0	0
$397$	3.99388e6	1.27180	0.635900	−	0.771771i	$-0.280629\pi$
$397$	3.99388e6	1.27180	0.635900	+	0.771771i	$0.280629\pi$
$398$	0	0
$399$	947863.	0.298067
$400$	0	0
$401$	−1.72629e6	−0.536110	−0.268055	−	0.963404i	$-0.586381\pi$
$401$	−1.72629e6	−0.536110	−0.268055	+	0.963404i	$0.586381\pi$
$402$	0	0
$403$	−3.78892e6	−1.16213
$404$	0	0
$405$	−1.98684e6	−0.601901
$406$	0	0
$407$	−542551.	−0.162351
$408$	0	0
$409$	4.99081e6	1.47524	0.737620	−	0.675216i	$-0.235949\pi$
$409$	4.99081e6	1.47524	0.737620	+	0.675216i	$0.235949\pi$
$410$	0	0
$411$	−78372.9	−0.0228855
$412$	0	0
$413$	558502.	0.161120
$414$	0	0
$415$	−1.05560e6	−0.300870
$416$	0	0
$417$	2.57137e6	0.724144
$418$	0	0
$419$	−1.02457e6	−0.285105	−0.142553	−	0.989787i	$-0.545531\pi$
$419$	−1.02457e6	−0.285105	−0.142553	+	0.989787i	$0.545531\pi$
$420$	0	0
$421$	−118203.	−0.0325029	−0.0162514	−	0.999868i	$-0.505173\pi$
$421$	−118203.	−0.0325029	−0.0162514	+	0.999868i	$0.505173\pi$
$422$	0	0
$423$	339407.	0.0922294
$424$	0	0
$425$	1.08721e6	0.291971
$426$	0	0
$427$	−151181.	−0.0401261
$428$	0	0
$429$	1.38769e6	0.364040
$430$	0	0
$431$	−99057.3	−0.0256858	−0.0128429	−	0.999918i	$-0.504088\pi$
$431$	−99057.3	−0.0256858	−0.0128429	+	0.999918i	$0.504088\pi$
$432$	0	0
$433$	4.05998e6	1.04065	0.520325	−	0.853968i	$-0.325811\pi$
$433$	4.05998e6	1.04065	0.520325	+	0.853968i	$0.325811\pi$
$434$	0	0
$435$	−3.46830e6	−0.878807
$436$	0	0
$437$	−492365.	−0.123334
$438$	0	0
$439$	1.77648e6	0.439946	0.219973	−	0.975506i	$-0.429403\pi$
$439$	1.77648e6	0.439946	0.219973	+	0.975506i	$0.429403\pi$
$440$	0	0
$441$	−56799.3	−0.0139074
$442$	0	0
$443$	−3.49343e6	−0.845752	−0.422876	−	0.906188i	$-0.638979\pi$
$443$	−3.49343e6	−0.845752	−0.422876	+	0.906188i	$0.638979\pi$
$444$	0	0
$445$	−4.57849e6	−1.09603
$446$	0	0
$447$	5.22419e6	1.23666
$448$	0	0
$449$	−4.18634e6	−0.979983	−0.489991	−	0.871727i	$-0.663000\pi$
$449$	−4.18634e6	−0.979983	−0.489991	+	0.871727i	$0.663000\pi$
$450$	0	0
$451$	−1.90662e6	−0.441390
$452$	0	0
$453$	7.02926e6	1.60940
$454$	0	0
$455$	1.15869e6	0.262384
$456$	0	0
$457$	−8.42202e6	−1.88636	−0.943182	−	0.332275i	$-0.892184\pi$
$457$	−8.42202e6	−1.88636	−0.943182	+	0.332275i	$0.892184\pi$
$458$	0	0
$459$	−2.51703e6	−0.557643
$460$	0	0
$461$	−8.99005e6	−1.97020	−0.985099	−	0.171988i	$-0.944981\pi$
$461$	−8.99005e6	−1.97020	−0.985099	+	0.171988i	$0.944981\pi$
$462$	0	0
$463$	990479.	0.214730	0.107365	−	0.994220i	$-0.465759\pi$
$463$	990479.	0.214730	0.107365	+	0.994220i	$0.465759\pi$
$464$	0	0
$465$	3.36775e6	0.722284
$466$	0	0
$467$	−5.43975e6	−1.15422	−0.577108	−	0.816668i	$-0.695819\pi$
$467$	−5.43975e6	−1.15422	−0.577108	+	0.816668i	$0.695819\pi$
$468$	0	0
$469$	1.72866e6	0.362891
$470$	0	0
$471$	−1.17584e6	−0.244229
$472$	0	0
$473$	499637.	0.102684
$474$	0	0
$475$	2.22805e6	0.453097
$476$	0	0
$477$	−108535.	−0.0218410
$478$	0	0
$479$	8.32397e6	1.65765	0.828823	−	0.559512i	$-0.189011\pi$
$479$	8.32397e6	1.65765	0.828823	+	0.559512i	$0.189011\pi$
$480$	0	0
$481$	2.28148e6	0.449629
$482$	0	0
$483$	−273564.	−0.0533569
$484$	0	0
$485$	−24343.7	−0.00469929
$486$	0	0
$487$	−7.77528e6	−1.48557	−0.742786	−	0.669529i	$-0.766496\pi$
$487$	−7.77528e6	−1.48557	−0.742786	+	0.669529i	$0.766496\pi$
$488$	0	0
$489$	8.39342e6	1.58733
$490$	0	0
$491$	−5.72471e6	−1.07164	−0.535821	−	0.844332i	$-0.679998\pi$
$491$	−5.72471e6	−1.07164	−0.535821	+	0.844332i	$0.679998\pi$
$492$	0	0
$493$	−3.96198e6	−0.734167
$494$	0	0
$495$	133028.	0.0244023
$496$	0	0
$497$	−2.39625e6	−0.435153
$498$	0	0
$499$	4.43278e6	0.796938	0.398469	−	0.917182i	$-0.369542\pi$
$499$	4.43278e6	0.796938	0.398469	+	0.917182i	$0.369542\pi$
$500$	0	0
$501$	−7.96790e6	−1.41824
$502$	0	0
$503$	−5.69126e6	−1.00297	−0.501485	−	0.865166i	$-0.667213\pi$
$503$	−5.69126e6	−1.00297	−0.501485	+	0.865166i	$0.667213\pi$
$504$	0	0
$505$	−2.76231e6	−0.481997
$506$	0	0
$507$	−336446.	−0.0581293
$508$	0	0
$509$	8.16835e6	1.39746	0.698731	−	0.715384i	$-0.253748\pi$
$509$	8.16835e6	1.39746	0.698731	+	0.715384i	$0.253748\pi$
$510$	0	0
$511$	742673.	0.125819
$512$	0	0
$513$	−5.15824e6	−0.865383
$514$	0	0
$515$	−6.77777e6	−1.12608
$516$	0	0
$517$	2.14164e6	0.352387
$518$	0	0
$519$	−2.70286e6	−0.440459
$520$	0	0
$521$	−5.28498e6	−0.853000	−0.426500	−	0.904488i	$-0.640254\pi$
$521$	−5.28498e6	−0.853000	−0.426500	+	0.904488i	$0.640254\pi$
$522$	0	0
$523$	6.45225e6	1.03147	0.515736	−	0.856748i	$-0.327519\pi$
$523$	6.45225e6	1.03147	0.515736	+	0.856748i	$0.327519\pi$
$524$	0	0
$525$	1.23793e6	0.196019
$526$	0	0
$527$	3.84712e6	0.603405
$528$	0	0
$529$	−6.29424e6	−0.977922
$530$	0	0
$531$	−269637.	−0.0414995
$532$	0	0
$533$	8.01753e6	1.22243
$534$	0	0
$535$	−8.63697e6	−1.30460
$536$	0	0
$537$	4.00301e6	0.599033
$538$	0	0
$539$	−358401.	−0.0531370
$540$	0	0
$541$	−850524.	−0.124938	−0.0624688	−	0.998047i	$-0.519897\pi$
$541$	−850524.	−0.124938	−0.0624688	+	0.998047i	$0.519897\pi$
$542$	0	0
$543$	5.79608e6	0.843597
$544$	0	0
$545$	1.43384e6	0.206780
$546$	0	0
$547$	5.85068e6	0.836061	0.418030	−	0.908433i	$-0.362721\pi$
$547$	5.85068e6	0.836061	0.418030	+	0.908433i	$0.362721\pi$
$548$	0	0
$549$	72987.9	0.0103352
$550$	0	0
$551$	−8.11943e6	−1.13932
$552$	0	0
$553$	−1.15788e6	−0.161009
$554$	0	0
$555$	−2.02788e6	−0.279453
$556$	0	0
$557$	−3.50904e6	−0.479238	−0.239619	−	0.970867i	$-0.577022\pi$
$557$	−3.50904e6	−0.479238	−0.239619	+	0.970867i	$0.577022\pi$
$558$	0	0
$559$	−2.10103e6	−0.284382
$560$	0	0
$561$	−1.40901e6	−0.189019
$562$	0	0
$563$	−3.04945e6	−0.405462	−0.202731	−	0.979234i	$-0.564982\pi$
$563$	−3.04945e6	−0.405462	−0.202731	+	0.979234i	$0.564982\pi$
$564$	0	0
$565$	1.89956e6	0.250341
$566$	0	0
$567$	−2.58430e6	−0.337587
$568$	0	0
$569$	−3.30346e6	−0.427748	−0.213874	−	0.976861i	$-0.568608\pi$
$569$	−3.30346e6	−0.427748	−0.213874	+	0.976861i	$0.568608\pi$
$570$	0	0
$571$	1.43527e7	1.84222	0.921112	−	0.389297i	$-0.127282\pi$
$571$	1.43527e7	1.84222	0.921112	+	0.389297i	$0.127282\pi$
$572$	0	0
$573$	−7.24915e6	−0.922361
$574$	0	0
$575$	−643040.	−0.0811089
$576$	0	0
$577$	−1.11546e7	−1.39481	−0.697404	−	0.716679i	$-0.745661\pi$
$577$	−1.11546e7	−1.39481	−0.697404	+	0.716679i	$0.745661\pi$
$578$	0	0
$579$	−9.18553e6	−1.13870
$580$	0	0
$581$	−1.37303e6	−0.168748
$582$	0	0
$583$	−684850.	−0.0834496
$584$	0	0
$585$	−559397.	−0.0675819
$586$	0	0
$587$	−8.85459e6	−1.06065	−0.530326	−	0.847794i	$-0.677931\pi$
$587$	−8.85459e6	−1.06065	−0.530326	+	0.847794i	$0.677931\pi$
$588$	0	0
$589$	7.88405e6	0.936399
$590$	0	0
$591$	−9.10469e6	−1.07225
$592$	0	0
$593$	5.01381e6	0.585505	0.292753	−	0.956188i	$-0.405429\pi$
$593$	5.01381e6	0.585505	0.292753	+	0.956188i	$0.405429\pi$
$594$	0	0
$595$	−1.17648e6	−0.136236
$596$	0	0
$597$	95433.8	0.0109589
$598$	0	0
$599$	−1.15050e6	−0.131014	−0.0655072	−	0.997852i	$-0.520867\pi$
$599$	−1.15050e6	−0.131014	−0.0655072	+	0.997852i	$0.520867\pi$
$600$	0	0
$601$	1.59933e7	1.80614	0.903072	−	0.429489i	$-0.141306\pi$
$601$	1.59933e7	1.80614	0.903072	+	0.429489i	$0.141306\pi$
$602$	0	0
$603$	−834570.	−0.0934694
$604$	0	0
$605$	−5.22767e6	−0.580657
$606$	0	0
$607$	−1.17435e7	−1.29367	−0.646836	−	0.762629i	$-0.723908\pi$
$607$	−1.17435e7	−1.29367	−0.646836	+	0.762629i	$0.723908\pi$
$608$	0	0
$609$	−4.51125e6	−0.492894
$610$	0	0
$611$	−9.00582e6	−0.975934
$612$	0	0
$613$	−1.45792e7	−1.56705	−0.783526	−	0.621359i	$-0.786581\pi$
$613$	−1.45792e7	−1.56705	−0.783526	+	0.621359i	$0.786581\pi$
$614$	0	0
$615$	−7.12632e6	−0.759761
$616$	0	0
$617$	−1.55808e7	−1.64769	−0.823845	−	0.566815i	$-0.808175\pi$
$617$	−1.55808e7	−1.64769	−0.823845	+	0.566815i	$0.808175\pi$
$618$	0	0
$619$	1.31769e7	1.38225	0.691125	−	0.722735i	$-0.257115\pi$
$619$	1.31769e7	1.38225	0.691125	+	0.722735i	$0.257115\pi$
$620$	0	0
$621$	1.48872e6	0.154912
$622$	0	0
$623$	−5.95529e6	−0.614728
$624$	0	0
$625$	−1.52499e6	−0.156159
$626$	0	0
$627$	−2.88753e6	−0.293331
$628$	0	0
$629$	−2.31652e6	−0.233459
$630$	0	0
$631$	−1.39854e7	−1.39830	−0.699150	−	0.714975i	$-0.746438\pi$
$631$	−1.39854e7	−1.39830	−0.699150	+	0.714975i	$0.746438\pi$
$632$	0	0
$633$	7.89541e6	0.783187
$634$	0	0
$635$	6.14582e6	0.604847
$636$	0	0
$637$	1.50711e6	0.147163
$638$	0	0
$639$	1.15688e6	0.112082
$640$	0	0
$641$	1.99947e7	1.92208	0.961038	−	0.276416i	$-0.0891467\pi$
$641$	1.99947e7	1.92208	0.961038	+	0.276416i	$0.0891467\pi$
$642$	0	0
$643$	−1.12846e7	−1.07637	−0.538183	−	0.842828i	$-0.680889\pi$
$643$	−1.12846e7	−1.07637	−0.538183	+	0.842828i	$0.680889\pi$
$644$	0	0
$645$	1.86748e6	0.176749
$646$	0	0
$647$	−6.66736e6	−0.626172	−0.313086	−	0.949725i	$-0.601363\pi$
$647$	−6.66736e6	−0.626172	−0.313086	+	0.949725i	$0.601363\pi$
$648$	0	0
$649$	−1.70140e6	−0.158560
$650$	0	0
$651$	4.38047e6	0.405106
$652$	0	0
$653$	4.11346e6	0.377506	0.188753	−	0.982025i	$-0.439555\pi$
$653$	4.11346e6	0.377506	0.188753	+	0.982025i	$0.439555\pi$
$654$	0	0
$655$	−5.53780e6	−0.504353
$656$	0	0
$657$	−358552.	−0.0324070
$658$	0	0
$659$	−1.01860e7	−0.913673	−0.456836	−	0.889551i	$-0.651018\pi$
$659$	−1.01860e7	−0.913673	−0.456836	+	0.889551i	$0.651018\pi$
$660$	0	0
$661$	−8.38745e6	−0.746666	−0.373333	−	0.927697i	$-0.621785\pi$
$661$	−8.38745e6	−0.746666	−0.373333	+	0.927697i	$0.621785\pi$
$662$	0	0
$663$	5.92502e6	0.523487
$664$	0	0
$665$	−2.41101e6	−0.211419
$666$	0	0
$667$	2.34336e6	0.203950
$668$	0	0
$669$	−999255.	−0.0863199
$670$	0	0
$671$	460550.	0.0394885
$672$	0	0
$673$	1.98643e7	1.69058	0.845290	−	0.534308i	$-0.179428\pi$
$673$	1.98643e7	1.69058	0.845290	+	0.534308i	$0.179428\pi$
$674$	0	0
$675$	−6.73679e6	−0.569106
$676$	0	0
$677$	5.35862e6	0.449346	0.224673	−	0.974434i	$-0.427869\pi$
$677$	5.35862e6	0.449346	0.224673	+	0.974434i	$0.427869\pi$
$678$	0	0
$679$	−31664.1	−0.00263568
$680$	0	0
$681$	−7.25288e6	−0.599298
$682$	0	0
$683$	1.18486e7	0.971889	0.485945	−	0.873990i	$-0.338476\pi$
$683$	1.18486e7	0.971889	0.485945	+	0.873990i	$0.338476\pi$
$684$	0	0
$685$	199351.	0.0162328
$686$	0	0
$687$	5.53860e6	0.447722
$688$	0	0
$689$	2.87987e6	0.231113
$690$	0	0
$691$	1.79391e7	1.42924	0.714619	−	0.699513i	$-0.246600\pi$
$691$	1.79391e7	1.42924	0.714619	+	0.699513i	$0.246600\pi$
$692$	0	0
$693$	173031.	0.0136864
$694$	0	0
$695$	−6.54062e6	−0.513638
$696$	0	0
$697$	−8.14068e6	−0.634715
$698$	0	0
$699$	8.57834e6	0.664065
$700$	0	0
$701$	1.47768e7	1.13575	0.567877	−	0.823114i	$-0.307765\pi$
$701$	1.47768e7	1.13575	0.567877	+	0.823114i	$0.307765\pi$
$702$	0	0
$703$	−4.74734e6	−0.362295
$704$	0	0
$705$	8.00475e6	0.606562
$706$	0	0
$707$	−3.59297e6	−0.270336
$708$	0	0
$709$	2.09017e6	0.156159	0.0780794	−	0.996947i	$-0.475121\pi$
$709$	2.09017e6	0.156159	0.0780794	+	0.996947i	$0.475121\pi$
$710$	0	0
$711$	559009.	0.0414710
$712$	0	0
$713$	−2.27542e6	−0.167625
$714$	0	0
$715$	−3.52977e6	−0.258215
$716$	0	0
$717$	2.45359e6	0.178240
$718$	0	0
$719$	−1.23653e6	−0.0892037	−0.0446019	−	0.999005i	$-0.514202\pi$
$719$	−1.23653e6	−0.0892037	−0.0446019	+	0.999005i	$0.514202\pi$
$720$	0	0
$721$	−8.81591e6	−0.631581
$722$	0	0
$723$	−8.50438e6	−0.605058
$724$	0	0
$725$	−1.06042e7	−0.749259
$726$	0	0
$727$	−7.50721e6	−0.526796	−0.263398	−	0.964687i	$-0.584843\pi$
$727$	−7.50721e6	−0.526796	−0.263398	+	0.964687i	$0.584843\pi$
$728$	0	0
$729$	1.54606e7	1.07747
$730$	0	0
$731$	2.13330e6	0.147658
$732$	0	0
$733$	2.22220e7	1.52765	0.763823	−	0.645425i	$-0.223320\pi$
$733$	2.22220e7	1.52765	0.763823	+	0.645425i	$0.223320\pi$
$734$	0	0
$735$	−1.33959e6	−0.0914644
$736$	0	0
$737$	−5.26610e6	−0.357125
$738$	0	0
$739$	1.52607e7	1.02793	0.513964	−	0.857812i	$-0.328176\pi$
$739$	1.52607e7	1.02793	0.513964	+	0.857812i	$0.328176\pi$
$740$	0	0
$741$	1.21424e7	0.812377
$742$	0	0
$743$	5.74605e6	0.381854	0.190927	−	0.981604i	$-0.438851\pi$
$743$	5.74605e6	0.381854	0.190927	+	0.981604i	$0.438851\pi$
$744$	0	0
$745$	−1.32884e7	−0.877165
$746$	0	0
$747$	662879.	0.0434643
$748$	0	0
$749$	−1.12342e7	−0.731707
$750$	0	0
$751$	2.64874e7	1.71372	0.856858	−	0.515553i	$-0.172413\pi$
$751$	2.64874e7	1.71372	0.856858	+	0.515553i	$0.172413\pi$
$752$	0	0
$753$	−1.39230e7	−0.894842
$754$	0	0
$755$	−1.78798e7	−1.14155
$756$	0	0
$757$	3.06970e6	0.194695	0.0973477	−	0.995250i	$-0.468964\pi$
$757$	3.06970e6	0.194695	0.0973477	+	0.995250i	$0.468964\pi$
$758$	0	0
$759$	833372.	0.0525091
$760$	0	0
$761$	2.03017e7	1.27078	0.635390	−	0.772191i	$-0.280839\pi$
$761$	2.03017e7	1.27078	0.635390	+	0.772191i	$0.280839\pi$
$762$	0	0
$763$	1.86501e6	0.115976
$764$	0	0
$765$	567989.	0.0350902
$766$	0	0
$767$	7.15455e6	0.439131
$768$	0	0
$769$	1.07950e7	0.658274	0.329137	−	0.944282i	$-0.393242\pi$
$769$	1.07950e7	0.658274	0.329137	+	0.944282i	$0.393242\pi$
$770$	0	0
$771$	−1.91918e6	−0.116273
$772$	0	0
$773$	−1.21883e6	−0.0733662	−0.0366831	−	0.999327i	$-0.511679\pi$
$773$	−1.21883e6	−0.0733662	−0.0366831	+	0.999327i	$0.511679\pi$
$774$	0	0
$775$	1.02968e7	0.615809
$776$	0	0
$777$	−2.63768e6	−0.156736
$778$	0	0
$779$	−1.66830e7	−0.984986
$780$	0	0
$781$	7.29984e6	0.428239
$782$	0	0
$783$	2.45501e7	1.43103
$784$	0	0
$785$	2.99091e6	0.173232
$786$	0	0
$787$	−8.06011e6	−0.463878	−0.231939	−	0.972730i	$-0.574507\pi$
$787$	−8.06011e6	−0.463878	−0.231939	+	0.972730i	$0.574507\pi$
$788$	0	0
$789$	2.11298e7	1.20838
$790$	0	0
$791$	2.47078e6	0.140408
$792$	0	0
$793$	−1.93666e6	−0.109363
$794$	0	0
$795$	−2.55975e6	−0.143641
$796$	0	0
$797$	−8.16016e6	−0.455043	−0.227522	−	0.973773i	$-0.573062\pi$
$797$	−8.16016e6	−0.455043	−0.227522	+	0.973773i	$0.573062\pi$
$798$	0	0
$799$	9.14415e6	0.506730
$800$	0	0
$801$	2.87513e6	0.158335
$802$	0	0
$803$	−2.26245e6	−0.123820
$804$	0	0
$805$	695844.	0.0378462
$806$	0	0
$807$	1.49083e7	0.805831
$808$	0	0
$809$	−2.78986e7	−1.49869	−0.749343	−	0.662182i	$-0.769631\pi$
$809$	−2.78986e7	−1.49869	−0.749343	+	0.662182i	$0.769631\pi$
$810$	0	0
$811$	4.61578e6	0.246429	0.123215	−	0.992380i	$-0.460680\pi$
$811$	4.61578e6	0.246429	0.123215	+	0.992380i	$0.460680\pi$
$812$	0	0
$813$	1.68991e7	0.896677
$814$	0	0
$815$	−2.13497e7	−1.12590
$816$	0	0
$817$	4.37185e6	0.229145
$818$	0	0
$819$	−727613.	−0.0379045
$820$	0	0
$821$	−2.87016e7	−1.48610	−0.743050	−	0.669236i	$-0.766622\pi$
$821$	−2.87016e7	−1.48610	−0.743050	+	0.669236i	$0.766622\pi$
$822$	0	0
$823$	2.90189e7	1.49342	0.746710	−	0.665150i	$-0.231633\pi$
$823$	2.90189e7	1.49342	0.746710	+	0.665150i	$0.231633\pi$
$824$	0	0
$825$	−3.77118e6	−0.192905
$826$	0	0
$827$	4.18186e6	0.212621	0.106310	−	0.994333i	$-0.466096\pi$
$827$	4.18186e6	0.212621	0.106310	+	0.994333i	$0.466096\pi$
$828$	0	0
$829$	−4.54599e6	−0.229743	−0.114871	−	0.993380i	$-0.536646\pi$
$829$	−4.54599e6	−0.229743	−0.114871	+	0.993380i	$0.536646\pi$
$830$	0	0
$831$	3.26004e7	1.63765
$832$	0	0
$833$	−1.53026e6	−0.0764105
$834$	0	0
$835$	2.02674e7	1.00596
$836$	0	0
$837$	−2.38384e7	−1.17615
$838$	0	0
$839$	544689.	0.0267143	0.0133571	−	0.999911i	$-0.495748\pi$
$839$	544689.	0.0267143	0.0133571	+	0.999911i	$0.495748\pi$
$840$	0	0
$841$	1.81324e7	0.884028
$842$	0	0
$843$	−5.33041e6	−0.258340
$844$	0	0
$845$	855793.	0.0412313
$846$	0	0
$847$	−6.79968e6	−0.325672
$848$	0	0
$849$	−3.11884e7	−1.48499
$850$	0	0
$851$	1.37013e6	0.0648544
$852$	0	0
$853$	−8.73893e6	−0.411231	−0.205615	−	0.978633i	$-0.565920\pi$
$853$	−8.73893e6	−0.411231	−0.205615	+	0.978633i	$0.565920\pi$
$854$	0	0
$855$	1.16400e6	0.0544550
$856$	0	0
$857$	−1.09388e6	−0.0508768	−0.0254384	−	0.999676i	$-0.508098\pi$
$857$	−1.09388e6	−0.0508768	−0.0254384	+	0.999676i	$0.508098\pi$
$858$	0	0
$859$	1.98806e7	0.919276	0.459638	−	0.888106i	$-0.347979\pi$
$859$	1.98806e7	0.919276	0.459638	+	0.888106i	$0.347979\pi$
$860$	0	0
$861$	−9.26927e6	−0.426125
$862$	0	0
$863$	1.53027e7	0.699424	0.349712	−	0.936857i	$-0.386279\pi$
$863$	1.53027e7	0.699424	0.349712	+	0.936857i	$0.386279\pi$
$864$	0	0
$865$	6.87507e6	0.312419
$866$	0	0
$867$	1.50124e7	0.678270
$868$	0	0
$869$	3.52732e6	0.158451
$870$	0	0
$871$	2.21445e7	0.989055
$872$	0	0
$873$	15287.0	0.000678868 0
$874$	0	0
$875$	−8.91732e6	−0.393744
$876$	0	0
$877$	−3.51846e7	−1.54474	−0.772368	−	0.635176i	$-0.780928\pi$
$877$	−3.51846e7	−1.54474	−0.772368	+	0.635176i	$0.780928\pi$
$878$	0	0
$879$	3.66066e7	1.59804
$880$	0	0
$881$	1.04493e7	0.453572	0.226786	−	0.973945i	$-0.427178\pi$
$881$	1.04493e7	0.453572	0.226786	+	0.973945i	$0.427178\pi$
$882$	0	0
$883$	−2.42290e6	−0.104577	−0.0522883	−	0.998632i	$-0.516651\pi$
$883$	−2.42290e6	−0.104577	−0.0522883	+	0.998632i	$0.516651\pi$
$884$	0	0
$885$	−6.35926e6	−0.272928
$886$	0	0
$887$	−8.69859e6	−0.371227	−0.185614	−	0.982623i	$-0.559427\pi$
$887$	−8.69859e6	−0.371227	−0.185614	+	0.982623i	$0.559427\pi$
$888$	0	0
$889$	7.99392e6	0.339239
$890$	0	0
$891$	7.87270e6	0.332223
$892$	0	0
$893$	1.87394e7	0.786372
$894$	0	0
$895$	−1.01822e7	−0.424896
$896$	0	0
$897$	−3.50442e6	−0.145424
$898$	0	0
$899$	−3.75233e7	−1.54847
$900$	0	0
$901$	−2.92410e6	−0.120000
$902$	0	0
$903$	2.42905e6	0.0991327
$904$	0	0
$905$	−1.47431e7	−0.598365
$906$	0	0
$907$	1.66178e7	0.670743	0.335372	−	0.942086i	$-0.391138\pi$
$907$	1.66178e7	0.670743	0.335372	+	0.942086i	$0.391138\pi$
$908$	0	0
$909$	1.73463e6	0.0696302
$910$	0	0
$911$	−7.78722e6	−0.310876	−0.155438	−	0.987846i	$-0.549679\pi$
$911$	−7.78722e6	−0.310876	−0.155438	+	0.987846i	$0.549679\pi$
$912$	0	0
$913$	4.18274e6	0.166067
$914$	0	0
$915$	1.72139e6	0.0679713
$916$	0	0
$917$	−7.20307e6	−0.282875
$918$	0	0
$919$	−3.61606e7	−1.41236	−0.706181	−	0.708031i	$-0.749584\pi$
$919$	−3.61606e7	−1.41236	−0.706181	+	0.708031i	$0.749584\pi$
$920$	0	0
$921$	−3.64826e7	−1.41722
$922$	0	0
$923$	−3.06966e7	−1.18600
$924$	0	0
$925$	−6.20014e6	−0.238258
$926$	0	0
$927$	4.25620e6	0.162676
$928$	0	0
$929$	4.64147e7	1.76448	0.882240	−	0.470801i	$-0.156035\pi$
$929$	4.64147e7	1.76448	0.882240	+	0.470801i	$0.156035\pi$
$930$	0	0
$931$	−3.13602e6	−0.118578
$932$	0	0
$933$	−2.77392e7	−1.04325
$934$	0	0
$935$	3.58398e6	0.134072
$936$	0	0
$937$	−2.63671e7	−0.981100	−0.490550	−	0.871413i	$-0.663204\pi$
$937$	−2.63671e7	−0.981100	−0.490550	+	0.871413i	$0.663204\pi$
$938$	0	0
$939$	7.14247e6	0.264353
$940$	0	0
$941$	3.40567e6	0.125380	0.0626900	−	0.998033i	$-0.480032\pi$
$941$	3.40567e6	0.125380	0.0626900	+	0.998033i	$0.480032\pi$
$942$	0	0
$943$	4.81489e6	0.176322
$944$	0	0
$945$	7.28998e6	0.265550
$946$	0	0
$947$	4.15081e7	1.50403	0.752017	−	0.659144i	$-0.229081\pi$
$947$	4.15081e7	1.50403	0.752017	+	0.659144i	$0.229081\pi$
$948$	0	0
$949$	9.51382e6	0.342917
$950$	0	0
$951$	−1.17004e7	−0.419516
$952$	0	0
$953$	3.02395e7	1.07855	0.539277	−	0.842128i	$-0.318697\pi$
$953$	3.02395e7	1.07855	0.539277	+	0.842128i	$0.318697\pi$
$954$	0	0
$955$	1.84391e7	0.654233
$956$	0	0
$957$	1.37429e7	0.485063
$958$	0	0
$959$	259298.	0.00910444
$960$	0	0
$961$	7.80632e6	0.272670
$962$	0	0
$963$	5.42371e6	0.188465
$964$	0	0
$965$	2.33646e7	0.807680
$966$	0	0
$967$	6.09163e6	0.209492	0.104746	−	0.994499i	$-0.466597\pi$
$967$	6.09163e6	0.209492	0.104746	+	0.994499i	$0.466597\pi$
$968$	0	0
$969$	−1.23289e7	−0.421807
$970$	0	0
$971$	−2.12760e7	−0.724171	−0.362086	−	0.932145i	$-0.617935\pi$
$971$	−2.12760e7	−0.724171	−0.362086	+	0.932145i	$0.617935\pi$
$972$	0	0
$973$	−8.50744e6	−0.288083
$974$	0	0
$975$	1.58582e7	0.534248
$976$	0	0
$977$	−1.66948e7	−0.559559	−0.279779	−	0.960064i	$-0.590261\pi$
$977$	−1.66948e7	−0.559559	−0.279779	+	0.960064i	$0.590261\pi$
$978$	0	0
$979$	1.81419e7	0.604960
$980$	0	0
$981$	−900399.	−0.0298719
$982$	0	0
$983$	3.55038e7	1.17190	0.585951	−	0.810347i	$-0.300721\pi$
$983$	3.55038e7	1.17190	0.585951	+	0.810347i	$0.300721\pi$
$984$	0	0
$985$	2.31589e7	0.760551
$986$	0	0
$987$	1.04119e7	0.340201
$988$	0	0
$989$	−1.26176e6	−0.0410192
$990$	0	0
$991$	−2.44851e7	−0.791988	−0.395994	−	0.918253i	$-0.629600\pi$
$991$	−2.44851e7	−0.791988	−0.395994	+	0.918253i	$0.629600\pi$
$992$	0	0
$993$	−5.25392e7	−1.69087
$994$	0	0
$995$	−242748.	−0.00777316
$996$	0	0
$997$	7.11209e6	0.226600	0.113300	−	0.993561i	$-0.463858\pi$
$997$	7.11209e6	0.226600	0.113300	+	0.993561i	$0.463858\pi$
$998$	0	0
$999$	1.43542e7	0.455055

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.6.a.s.1.1		2
4.3	odd	2		448.6.a.y.1.2		2
8.3	odd	2		224.6.a.c.1.1	✓	2
8.5	even	2		224.6.a.d.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.6.a.c.1.1	✓	2	8.3	odd	2
224.6.a.d.1.2	yes	2	8.5	even	2
448.6.a.s.1.1		2	1.1	even	1	trivial
448.6.a.y.1.2		2	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:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	448.a (trivial)

\(f(q)\)	\(=\)	\(q-14.8102 q^{3} +37.6717 q^{5} +49.0000 q^{7} -23.6565 q^{9} -149.271 q^{11} +627.702 q^{13} -557.928 q^{15} -637.343 q^{17} -1306.13 q^{19} -725.702 q^{21} +376.964 q^{23} -1705.84 q^{25} +3949.25 q^{27} +6216.40 q^{29} -6036.18 q^{31} +2210.75 q^{33} +1845.92 q^{35} +3634.66 q^{37} -9296.43 q^{39} +12772.8 q^{41} -3347.17 q^{43} -891.182 q^{45} -14347.3 q^{47} +2401.00 q^{49} +9439.22 q^{51} +4587.95 q^{53} -5623.32 q^{55} +19344.1 q^{57} +11398.0 q^{59} -3085.32 q^{61} -1159.17 q^{63} +23646.6 q^{65} +35278.7 q^{67} -5582.93 q^{69} -48903.1 q^{71} +15156.6 q^{73} +25263.9 q^{75} -7314.30 q^{77} -23630.2 q^{79} -52740.8 q^{81} -28021.0 q^{83} -24009.8 q^{85} -92066.4 q^{87} -121536. q^{89} +30757.4 q^{91} +89397.3 q^{93} -49204.3 q^{95} -646.205 q^{97} +3531.24 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 14 q^{3} - 34 q^{5} + 98 q^{7} - 266 q^{9} + 420 q^{11} + 490 q^{13} - 616 q^{15} - 1056 q^{17} + 1246 q^{19} - 686 q^{21} + 504 q^{23} + 306 q^{25} + 3556 q^{27} + 3904 q^{29} - 2044 q^{31} + 2672 q^{33}+ \cdots - 134428 q^{99}+O(q^{100}) \) 2 * q - 14 * q^3 - 34 * q^5 + 98 * q^7 - 266 * q^9 + 420 * q^11 + 490 * q^13 - 616 * q^15 - 1056 * q^17 + 1246 * q^19 - 686 * q^21 + 504 * q^23 + 306 * q^25 + 3556 * q^27 + 3904 * q^29 - 2044 * q^31 + 2672 * q^33 - 1666 * q^35 + 7488 * q^37 - 9408 * q^39 + 7832 * q^41 + 10332 * q^43 + 16478 * q^45 - 41972 * q^47 + 4802 * q^49 + 9100 * q^51 - 32812 * q^53 - 46424 * q^55 + 21412 * q^57 + 48398 * q^59 + 718 * q^61 - 13034 * q^63 + 33516 * q^65 + 12824 * q^67 - 5480 * q^69 - 103992 * q^71 - 54100 * q^73 + 26894 * q^75 + 20580 * q^77 - 64568 * q^79 + 5830 * q^81 - 47810 * q^83 + 5996 * q^85 - 93940 * q^87 - 17388 * q^89 + 24010 * q^91 + 92632 * q^93 - 232120 * q^95 - 97296 * q^97 - 134428 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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