LMFDB - Embedded newform 448.6.a.bb.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [448,6,Mod(1,448)]

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

lf = mfeigenbasis(mf)

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")

//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);

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	448.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:	$71.8519512762$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.367637.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 107x + 282 $ x^3 - x^2 - 107*x + 282
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 224)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-11.0251$ 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)

$f(q)$	$=$	$q-20.0502 q^{3} +33.3202 q^{5} +49.0000 q^{7} +159.011 q^{9} +660.274 q^{11} +145.779 q^{13} -668.076 q^{15} +435.001 q^{17} +1527.81 q^{19} -982.460 q^{21} +206.539 q^{23} -2014.77 q^{25} +1684.00 q^{27} +2039.43 q^{29} -1151.33 q^{31} -13238.6 q^{33} +1632.69 q^{35} -6757.90 q^{37} -2922.90 q^{39} -15327.5 q^{41} +6258.55 q^{43} +5298.27 q^{45} +12602.1 q^{47} +2401.00 q^{49} -8721.86 q^{51} -10856.1 q^{53} +22000.4 q^{55} -30632.8 q^{57} -52563.4 q^{59} +46785.0 q^{61} +7791.53 q^{63} +4857.39 q^{65} +43838.4 q^{67} -4141.16 q^{69} +34624.8 q^{71} -60821.1 q^{73} +40396.5 q^{75} +32353.4 q^{77} -22685.6 q^{79} -72404.2 q^{81} +38812.4 q^{83} +14494.3 q^{85} -40891.0 q^{87} +5983.43 q^{89} +7143.18 q^{91} +23084.5 q^{93} +50906.7 q^{95} +67354.0 q^{97} +104991. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 8 q^{3} + 14 q^{5} + 147 q^{7} + 151 q^{9} + 600 q^{11} + 974 q^{13} - 120 q^{15} + 718 q^{17} - 1056 q^{19} + 392 q^{21} + 3760 q^{23} - 147 q^{25} - 2872 q^{27} + 9134 q^{29} - 1448 q^{31} - 14872 q^{33}+ \cdots + 83128 q^{99}+O(q^{100}) $ 3 * q + 8 * q^3 + 14 * q^5 + 147 * q^7 + 151 * q^9 + 600 * q^11 + 974 * q^13 - 120 * q^15 + 718 * q^17 - 1056 * q^19 + 392 * q^21 + 3760 * q^23 - 147 * q^25 - 2872 * q^27 + 9134 * q^29 - 1448 * q^31 - 14872 * q^33 + 686 * q^35 + 4998 * q^37 - 824 * q^39 - 23186 * q^41 + 29880 * q^43 + 28350 * q^45 + 10840 * q^47 + 7203 * q^49 + 3776 * q^51 + 28006 * q^53 + 14952 * q^55 - 53504 * q^57 - 17456 * q^59 + 92294 * q^61 + 7399 * q^63 - 95300 * q^65 + 56024 * q^67 + 63480 * q^69 + 77064 * q^71 - 46346 * q^73 + 50768 * q^75 + 29400 * q^77 - 4376 * q^79 - 121973 * q^81 + 107128 * q^83 + 94348 * q^85 + 82720 * q^87 + 29814 * q^89 + 47726 * q^91 + 108432 * q^93 + 205304 * q^95 - 156482 * q^97 + 83128 * 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
$3$	−20.0502	−1.28622	−0.643111	−	0.765773i	$-0.722356\pi$
$3$	−20.0502	−1.28622	−0.643111	+	0.765773i	$0.722356\pi$
$4$	0	0
$5$	33.3202	0.596049	0.298025	−	0.954558i	$-0.403672\pi$
$5$	33.3202	0.596049	0.298025	+	0.954558i	$0.403672\pi$
$6$	0	0
$7$	49.0000	0.377964
$7$	49.0000	0.377964
$8$	0	0
$9$	159.011	0.654365
$10$	0	0
$11$	660.274	1.64529	0.822645	−	0.568555i	$-0.192497\pi$
$11$	660.274	1.64529	0.822645	+	0.568555i	$0.192497\pi$
$12$	0	0
$13$	145.779	0.239242	0.119621	−	0.992820i	$-0.461832\pi$
$13$	145.779	0.239242	0.119621	+	0.992820i	$0.461832\pi$
$14$	0	0
$15$	−668.076	−0.766652
$16$	0	0
$17$	435.001	0.365063	0.182532	−	0.983200i	$-0.441571\pi$
$17$	435.001	0.365063	0.182532	+	0.983200i	$0.441571\pi$
$18$	0	0
$19$	1527.81	0.970921	0.485461	−	0.874259i	$-0.338652\pi$
$19$	1527.81	0.970921	0.485461	+	0.874259i	$0.338652\pi$
$20$	0	0
$21$	−982.460	−0.486146
$22$	0	0
$23$	206.539	0.0814110	0.0407055	−	0.999171i	$-0.487039\pi$
$23$	206.539	0.0814110	0.0407055	+	0.999171i	$0.487039\pi$
$24$	0	0
$25$	−2014.77	−0.644725
$26$	0	0
$27$	1684.00	0.444563
$28$	0	0
$29$	2039.43	0.450312	0.225156	−	0.974323i	$-0.427711\pi$
$29$	2039.43	0.450312	0.225156	+	0.974323i	$0.427711\pi$
$30$	0	0
$31$	−1151.33	−0.215178	−0.107589	−	0.994195i	$-0.534313\pi$
$31$	−1151.33	−0.215178	−0.107589	+	0.994195i	$0.534313\pi$
$32$	0	0
$33$	−13238.6	−2.11621
$34$	0	0
$35$	1632.69	0.225286
$36$	0	0
$37$	−6757.90	−0.811536	−0.405768	−	0.913976i	$-0.632996\pi$
$37$	−6757.90	−0.811536	−0.405768	+	0.913976i	$0.632996\pi$
$38$	0	0
$39$	−2922.90	−0.307718
$40$	0	0
$41$	−15327.5	−1.42400	−0.712002	−	0.702177i	$-0.752212\pi$
$41$	−15327.5	−1.42400	−0.712002	+	0.702177i	$0.752212\pi$
$42$	0	0
$43$	6258.55	0.516182	0.258091	−	0.966121i	$-0.416907\pi$
$43$	6258.55	0.516182	0.258091	+	0.966121i	$0.416907\pi$
$44$	0	0
$45$	5298.27	0.390034
$46$	0	0
$47$	12602.1	0.832142	0.416071	−	0.909332i	$-0.363407\pi$
$47$	12602.1	0.832142	0.416071	+	0.909332i	$0.363407\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	−8721.86	−0.469552
$52$	0	0
$53$	−10856.1	−0.530867	−0.265433	−	0.964129i	$-0.585515\pi$
$53$	−10856.1	−0.530867	−0.265433	+	0.964129i	$0.585515\pi$
$54$	0	0
$55$	22000.4	0.980674
$56$	0	0
$57$	−30632.8	−1.24882
$58$	0	0
$59$	−52563.4	−1.96586	−0.982931	−	0.183973i	$-0.941104\pi$
$59$	−52563.4	−1.96586	−0.982931	+	0.183973i	$0.941104\pi$
$60$	0	0
$61$	46785.0	1.60984	0.804919	−	0.593384i	$-0.202209\pi$
$61$	46785.0	1.60984	0.804919	+	0.593384i	$0.202209\pi$
$62$	0	0
$63$	7791.53	0.247327
$64$	0	0
$65$	4857.39	0.142600
$66$	0	0
$67$	43838.4	1.19307	0.596537	−	0.802585i	$-0.296543\pi$
$67$	43838.4	1.19307	0.596537	+	0.802585i	$0.296543\pi$
$68$	0	0
$69$	−4141.16	−0.104713
$70$	0	0
$71$	34624.8	0.815157	0.407578	−	0.913170i	$-0.366373\pi$
$71$	34624.8	0.815157	0.407578	+	0.913170i	$0.366373\pi$
$72$	0	0
$73$	−60821.1	−1.33582	−0.667909	−	0.744243i	$-0.732810\pi$
$73$	−60821.1	−1.33582	−0.667909	+	0.744243i	$0.732810\pi$
$74$	0	0
$75$	40396.5	0.829259
$76$	0	0
$77$	32353.4	0.621861
$78$	0	0
$79$	−22685.6	−0.408961	−0.204480	−	0.978871i	$-0.565550\pi$
$79$	−22685.6	−0.408961	−0.204480	+	0.978871i	$0.565550\pi$
$80$	0	0
$81$	−72404.2	−1.22617
$82$	0	0
$83$	38812.4	0.618408	0.309204	−	0.950996i	$-0.399937\pi$
$83$	38812.4	0.618408	0.309204	+	0.950996i	$0.399937\pi$
$84$	0	0
$85$	14494.3	0.217596
$86$	0	0
$87$	−40891.0	−0.579201
$88$	0	0
$89$	5983.43	0.0800710	0.0400355	−	0.999198i	$-0.487253\pi$
$89$	5983.43	0.0800710	0.0400355	+	0.999198i	$0.487253\pi$
$90$	0	0
$91$	7143.18	0.0904250
$92$	0	0
$93$	23084.5	0.276766
$94$	0	0
$95$	50906.7	0.578717
$96$	0	0
$97$	67354.0	0.726832	0.363416	−	0.931627i	$-0.381610\pi$
$97$	67354.0	0.726832	0.363416	+	0.931627i	$0.381610\pi$
$98$	0	0
$99$	104991.	1.07662
$100$	0	0
$101$	99602.9	0.971558	0.485779	−	0.874082i	$-0.338536\pi$
$101$	99602.9	0.971558	0.485779	+	0.874082i	$0.338536\pi$
$102$	0	0
$103$	−38290.5	−0.355629	−0.177815	−	0.984064i	$-0.556903\pi$
$103$	−38290.5	−0.355629	−0.177815	+	0.984064i	$0.556903\pi$
$104$	0	0
$105$	−32735.7	−0.289767
$106$	0	0
$107$	83934.5	0.708730	0.354365	−	0.935107i	$-0.384697\pi$
$107$	83934.5	0.708730	0.354365	+	0.935107i	$0.384697\pi$
$108$	0	0
$109$	207724.	1.67464	0.837320	−	0.546713i	$-0.184121\pi$
$109$	207724.	1.67464	0.837320	+	0.546713i	$0.184121\pi$
$110$	0	0
$111$	135497.	1.04381
$112$	0	0
$113$	193685.	1.42692	0.713462	−	0.700694i	$-0.247126\pi$
$113$	193685.	1.42692	0.713462	+	0.700694i	$0.247126\pi$
$114$	0	0
$115$	6881.93	0.0485250
$116$	0	0
$117$	23180.5	0.156552
$118$	0	0
$119$	21315.0	0.137981
$120$	0	0
$121$	274911.	1.70698
$122$	0	0
$123$	307319.	1.83159
$124$	0	0
$125$	−171258.	−0.980337
$126$	0	0
$127$	269758.	1.48411	0.742054	−	0.670340i	$-0.233852\pi$
$127$	269758.	1.48411	0.742054	+	0.670340i	$0.233852\pi$
$128$	0	0
$129$	−125485.	−0.663924
$130$	0	0
$131$	97453.8	0.496159	0.248079	−	0.968740i	$-0.420201\pi$
$131$	97453.8	0.496159	0.248079	+	0.968740i	$0.420201\pi$
$132$	0	0
$133$	74862.5	0.366974
$134$	0	0
$135$	56111.2	0.264981
$136$	0	0
$137$	−23080.6	−0.105062	−0.0525310	−	0.998619i	$-0.516729\pi$
$137$	−23080.6	−0.105062	−0.0525310	+	0.998619i	$0.516729\pi$
$138$	0	0
$139$	−244133.	−1.07174	−0.535871	−	0.844300i	$-0.680016\pi$
$139$	−244133.	−1.07174	−0.535871	+	0.844300i	$0.680016\pi$
$140$	0	0
$141$	−252674.	−1.07032
$142$	0	0
$143$	96254.3	0.393623
$144$	0	0
$145$	67954.1	0.268408
$146$	0	0
$147$	−48140.5	−0.183746
$148$	0	0
$149$	−497458.	−1.83565	−0.917826	−	0.396982i	$-0.870058\pi$
$149$	−497458.	−1.83565	−0.917826	+	0.396982i	$0.870058\pi$
$150$	0	0
$151$	536243.	1.91390	0.956950	−	0.290252i	$-0.0937390\pi$
$151$	536243.	1.91390	0.956950	+	0.290252i	$0.0937390\pi$
$152$	0	0
$153$	69169.8	0.238885
$154$	0	0
$155$	−38362.6	−0.128256
$156$	0	0
$157$	87512.6	0.283349	0.141674	−	0.989913i	$-0.454751\pi$
$157$	87512.6	0.283349	0.141674	+	0.989913i	$0.454751\pi$
$158$	0	0
$159$	217668.	0.682812
$160$	0	0
$161$	10120.4	0.0307705
$162$	0	0
$163$	223968.	0.660262	0.330131	−	0.943935i	$-0.392907\pi$
$163$	223968.	0.660262	0.330131	+	0.943935i	$0.392907\pi$
$164$	0	0
$165$	−441114.	−1.26136
$166$	0	0
$167$	−41952.2	−0.116403	−0.0582015	−	0.998305i	$-0.518537\pi$
$167$	−41952.2	−0.116403	−0.0582015	+	0.998305i	$0.518537\pi$
$168$	0	0
$169$	−350041.	−0.942763
$170$	0	0
$171$	242937.	0.635337
$172$	0	0
$173$	512961.	1.30307	0.651537	−	0.758617i	$-0.274124\pi$
$173$	512961.	1.30307	0.651537	+	0.758617i	$0.274124\pi$
$174$	0	0
$175$	−98723.5	−0.243683
$176$	0	0
$177$	1.05391e6	2.52853
$178$	0	0
$179$	−112735.	−0.262982	−0.131491	−	0.991317i	$-0.541976\pi$
$179$	−112735.	−0.262982	−0.131491	+	0.991317i	$0.541976\pi$
$180$	0	0
$181$	−282627.	−0.641236	−0.320618	−	0.947209i	$-0.603890\pi$
$181$	−282627.	−0.641236	−0.320618	+	0.947209i	$0.603890\pi$
$182$	0	0
$183$	−938050.	−2.07061
$184$	0	0
$185$	−225175.	−0.483716
$186$	0	0
$187$	287220.	0.600635
$188$	0	0
$189$	82516.1	0.168029
$190$	0	0
$191$	330464.	0.655452	0.327726	−	0.944773i	$-0.393718\pi$
$191$	330464.	0.655452	0.327726	+	0.944773i	$0.393718\pi$
$192$	0	0
$193$	31480.1	0.0608336	0.0304168	−	0.999537i	$-0.490317\pi$
$193$	31480.1	0.0608336	0.0304168	+	0.999537i	$0.490317\pi$
$194$	0	0
$195$	−97391.7	−0.183415
$196$	0	0
$197$	−652849.	−1.19853	−0.599263	−	0.800552i	$-0.704539\pi$
$197$	−652849.	−1.19853	−0.599263	+	0.800552i	$0.704539\pi$
$198$	0	0
$199$	307902.	0.551163	0.275582	−	0.961278i	$-0.411130\pi$
$199$	307902.	0.551163	0.275582	+	0.961278i	$0.411130\pi$
$200$	0	0
$201$	−878969.	−1.53456
$202$	0	0
$203$	99932.0	0.170202
$204$	0	0
$205$	−510715.	−0.848777
$206$	0	0
$207$	32842.0	0.0532725
$208$	0	0
$209$	1.00877e6	1.59745
$210$	0	0
$211$	−171540.	−0.265252	−0.132626	−	0.991166i	$-0.542341\pi$
$211$	−171540.	−0.265252	−0.132626	+	0.991166i	$0.542341\pi$
$212$	0	0
$213$	−694234.	−1.04847
$214$	0	0
$215$	208536.	0.307670
$216$	0	0
$217$	−56415.4	−0.0813295
$218$	0	0
$219$	1.21948e6	1.71816
$220$	0	0
$221$	63414.1	0.0873384
$222$	0	0
$223$	−225202.	−0.303256	−0.151628	−	0.988438i	$-0.548452\pi$
$223$	−225202.	−0.303256	−0.151628	+	0.988438i	$0.548452\pi$
$224$	0	0
$225$	−320369.	−0.421886
$226$	0	0
$227$	−317578.	−0.409059	−0.204530	−	0.978860i	$-0.565566\pi$
$227$	−317578.	−0.409059	−0.204530	+	0.978860i	$0.565566\pi$
$228$	0	0
$229$	−82887.7	−0.104448	−0.0522242	−	0.998635i	$-0.516631\pi$
$229$	−82887.7	−0.104448	−0.0522242	+	0.998635i	$0.516631\pi$
$230$	0	0
$231$	−648693.	−0.799851
$232$	0	0
$233$	−562095.	−0.678296	−0.339148	−	0.940733i	$-0.610139\pi$
$233$	−562095.	−0.678296	−0.339148	+	0.940733i	$0.610139\pi$
$234$	0	0
$235$	419903.	0.495998
$236$	0	0
$237$	454850.	0.526014
$238$	0	0
$239$	−623000.	−0.705494	−0.352747	−	0.935719i	$-0.614752\pi$
$239$	−623000.	−0.705494	−0.352747	+	0.935719i	$0.614752\pi$
$240$	0	0
$241$	−578114.	−0.641167	−0.320583	−	0.947220i	$-0.603879\pi$
$241$	−578114.	−0.641167	−0.320583	+	0.947220i	$0.603879\pi$
$242$	0	0
$243$	1.04251e6	1.13256
$244$	0	0
$245$	80001.7	0.0851499
$246$	0	0
$247$	222722.	0.232285
$248$	0	0
$249$	−778196.	−0.795409
$250$	0	0
$251$	−1.59750e6	−1.60050	−0.800251	−	0.599665i	$-0.795300\pi$
$251$	−1.59750e6	−1.60050	−0.800251	+	0.599665i	$0.795300\pi$
$252$	0	0
$253$	136373.	0.133945
$254$	0	0
$255$	−290614.	−0.279876
$256$	0	0
$257$	394458.	0.372536	0.186268	−	0.982499i	$-0.440361\pi$
$257$	394458.	0.372536	0.186268	+	0.982499i	$0.440361\pi$
$258$	0	0
$259$	−331137.	−0.306732
$260$	0	0
$261$	324291.	0.294669
$262$	0	0
$263$	2.20714e6	1.96762	0.983810	−	0.179216i	$-0.0573560\pi$
$263$	2.20714e6	1.96762	0.983810	+	0.179216i	$0.0573560\pi$
$264$	0	0
$265$	−361728.	−0.316423
$266$	0	0
$267$	−119969.	−0.102989
$268$	0	0
$269$	2.33693e6	1.96909	0.984545	−	0.175133i	$-0.0560355\pi$
$269$	2.33693e6	1.96909	0.984545	+	0.175133i	$0.0560355\pi$
$270$	0	0
$271$	1.62553e6	1.34453	0.672266	−	0.740310i	$-0.265321\pi$
$271$	1.62553e6	1.34453	0.672266	+	0.740310i	$0.265321\pi$
$272$	0	0
$273$	−143222.	−0.116307
$274$	0	0
$275$	−1.33030e6	−1.06076
$276$	0	0
$277$	2.41218e6	1.88891	0.944454	−	0.328643i	$-0.106591\pi$
$277$	2.41218e6	1.88891	0.944454	+	0.328643i	$0.106591\pi$
$278$	0	0
$279$	−183074.	−0.140805
$280$	0	0
$281$	569517.	0.430270	0.215135	−	0.976584i	$-0.430981\pi$
$281$	569517.	0.430270	0.215135	+	0.976584i	$0.430981\pi$
$282$	0	0
$283$	−1.13388e6	−0.841592	−0.420796	−	0.907155i	$-0.638249\pi$
$283$	−1.13388e6	−0.841592	−0.420796	+	0.907155i	$0.638249\pi$
$284$	0	0
$285$	−1.02069e6	−0.744358
$286$	0	0
$287$	−751047.	−0.538223
$288$	0	0
$289$	−1.23063e6	−0.866729
$290$	0	0
$291$	−1.35046e6	−0.934866
$292$	0	0
$293$	1.34094e6	0.912514	0.456257	−	0.889848i	$-0.349190\pi$
$293$	1.34094e6	0.912514	0.456257	+	0.889848i	$0.349190\pi$
$294$	0	0
$295$	−1.75142e6	−1.17175
$296$	0	0
$297$	1.11190e6	0.731435
$298$	0	0
$299$	30109.2	0.0194769
$300$	0	0
$301$	306669.	0.195098
$302$	0	0
$303$	−1.99706e6	−1.24964
$304$	0	0
$305$	1.55889e6	0.959543
$306$	0	0
$307$	−2.80842e6	−1.70066	−0.850328	−	0.526254i	$-0.823596\pi$
$307$	−2.80842e6	−1.70066	−0.850328	+	0.526254i	$0.823596\pi$
$308$	0	0
$309$	767731.	0.457418
$310$	0	0
$311$	−1.88370e6	−1.10436	−0.552181	−	0.833724i	$-0.686204\pi$
$311$	−1.88370e6	−1.10436	−0.552181	+	0.833724i	$0.686204\pi$
$312$	0	0
$313$	1.26722e6	0.731126	0.365563	−	0.930787i	$-0.380876\pi$
$313$	1.26722e6	0.731126	0.365563	+	0.930787i	$0.380876\pi$
$314$	0	0
$315$	259615.	0.147419
$316$	0	0
$317$	2.88579e6	1.61293	0.806466	−	0.591280i	$-0.201377\pi$
$317$	2.88579e6	1.61293	0.806466	+	0.591280i	$0.201377\pi$
$318$	0	0
$319$	1.34658e6	0.740894
$320$	0	0
$321$	−1.68290e6	−0.911584
$322$	0	0
$323$	664597.	0.354447
$324$	0	0
$325$	−293711.	−0.154245
$326$	0	0
$327$	−4.16492e6	−2.15396
$328$	0	0
$329$	617501.	0.314520
$330$	0	0
$331$	2.48205e6	1.24520	0.622601	−	0.782539i	$-0.286076\pi$
$331$	2.48205e6	1.24520	0.622601	+	0.782539i	$0.286076\pi$
$332$	0	0
$333$	−1.07458e6	−0.531041
$334$	0	0
$335$	1.46070e6	0.711131
$336$	0	0
$337$	−2.52524e6	−1.21124	−0.605618	−	0.795756i	$-0.707074\pi$
$337$	−2.52524e6	−1.21124	−0.605618	+	0.795756i	$0.707074\pi$
$338$	0	0
$339$	−3.88343e6	−1.83534
$340$	0	0
$341$	−760196.	−0.354030
$342$	0	0
$343$	117649.	0.0539949
$344$	0	0
$345$	−137984.	−0.0624139
$346$	0	0
$347$	4.10373e6	1.82960	0.914799	−	0.403910i	$-0.132349\pi$
$347$	4.10373e6	1.82960	0.914799	+	0.403910i	$0.132349\pi$
$348$	0	0
$349$	818617.	0.359764	0.179882	−	0.983688i	$-0.442428\pi$
$349$	818617.	0.359764	0.179882	+	0.983688i	$0.442428\pi$
$350$	0	0
$351$	245493.	0.106358
$352$	0	0
$353$	141420.	0.0604054	0.0302027	−	0.999544i	$-0.490385\pi$
$353$	141420.	0.0604054	0.0302027	+	0.999544i	$0.490385\pi$
$354$	0	0
$355$	1.15370e6	0.485874
$356$	0	0
$357$	−427371.	−0.177474
$358$	0	0
$359$	−4.38324e6	−1.79498	−0.897488	−	0.441038i	$-0.854610\pi$
$359$	−4.38324e6	−1.79498	−0.897488	+	0.441038i	$0.854610\pi$
$360$	0	0
$361$	−141910.	−0.0573120
$362$	0	0
$363$	−5.51202e6	−2.19555
$364$	0	0
$365$	−2.02657e6	−0.796213
$366$	0	0
$367$	−3.86104e6	−1.49637	−0.748186	−	0.663489i	$-0.769075\pi$
$367$	−3.86104e6	−1.49637	−0.748186	+	0.663489i	$0.769075\pi$
$368$	0	0
$369$	−2.43724e6	−0.931819
$370$	0	0
$371$	−531951.	−0.200649
$372$	0	0
$373$	2.73188e6	1.01669	0.508346	−	0.861153i	$-0.330257\pi$
$373$	2.73188e6	1.01669	0.508346	+	0.861153i	$0.330257\pi$
$374$	0	0
$375$	3.43376e6	1.26093
$376$	0	0
$377$	297307.	0.107734
$378$	0	0
$379$	1.25353e6	0.448266	0.224133	−	0.974559i	$-0.428045\pi$
$379$	1.25353e6	0.448266	0.224133	+	0.974559i	$0.428045\pi$
$380$	0	0
$381$	−5.40871e6	−1.90889
$382$	0	0
$383$	1.26669e6	0.441240	0.220620	−	0.975360i	$-0.429192\pi$
$383$	1.26669e6	0.441240	0.220620	+	0.975360i	$0.429192\pi$
$384$	0	0
$385$	1.07802e6	0.370660
$386$	0	0
$387$	995176.	0.337771
$388$	0	0
$389$	−3.08766e6	−1.03456	−0.517279	−	0.855817i	$-0.673055\pi$
$389$	−3.08766e6	−1.03456	−0.517279	+	0.855817i	$0.673055\pi$
$390$	0	0
$391$	89844.8	0.0297202
$392$	0	0
$393$	−1.95397e6	−0.638170
$394$	0	0
$395$	−755887.	−0.243761
$396$	0	0
$397$	306775.	0.0976885	0.0488442	−	0.998806i	$-0.484446\pi$
$397$	306775.	0.0976885	0.0488442	+	0.998806i	$0.484446\pi$
$398$	0	0
$399$	−1.50101e6	−0.472009
$400$	0	0
$401$	−3.07193e6	−0.954005	−0.477002	−	0.878902i	$-0.658277\pi$
$401$	−3.07193e6	−0.954005	−0.477002	+	0.878902i	$0.658277\pi$
$402$	0	0
$403$	−167841.	−0.0514795
$404$	0	0
$405$	−2.41252e6	−0.730859
$406$	0	0
$407$	−4.46207e6	−1.33521
$408$	0	0
$409$	6.03016e6	1.78246	0.891231	−	0.453549i	$-0.149842\pi$
$409$	6.03016e6	1.78246	0.891231	+	0.453549i	$0.149842\pi$
$410$	0	0
$411$	462771.	0.135133
$412$	0	0
$413$	−2.57560e6	−0.743026
$414$	0	0
$415$	1.29324e6	0.368602
$416$	0	0
$417$	4.89492e6	1.37850
$418$	0	0
$419$	−4.43568e6	−1.23431	−0.617156	−	0.786841i	$-0.711715\pi$
$419$	−4.43568e6	−1.23431	−0.617156	+	0.786841i	$0.711715\pi$
$420$	0	0
$421$	5.03797e6	1.38532	0.692660	−	0.721264i	$-0.256438\pi$
$421$	5.03797e6	1.38532	0.692660	+	0.721264i	$0.256438\pi$
$422$	0	0
$423$	2.00386e6	0.544525
$424$	0	0
$425$	−876425.	−0.235365
$426$	0	0
$427$	2.29247e6	0.608462
$428$	0	0
$429$	−1.92992e6	−0.506286
$430$	0	0
$431$	4.63743e6	1.20250	0.601248	−	0.799062i	$-0.294670\pi$
$431$	4.63743e6	1.20250	0.601248	+	0.799062i	$0.294670\pi$
$432$	0	0
$433$	−5.03341e6	−1.29016	−0.645079	−	0.764116i	$-0.723176\pi$
$433$	−5.03341e6	−1.29016	−0.645079	+	0.764116i	$0.723176\pi$
$434$	0	0
$435$	−1.36249e6	−0.345232
$436$	0	0
$437$	315552.	0.0790437
$438$	0	0
$439$	−4.85818e6	−1.20313	−0.601565	−	0.798824i	$-0.705456\pi$
$439$	−4.85818e6	−1.20313	−0.601565	+	0.798824i	$0.705456\pi$
$440$	0	0
$441$	381785.	0.0934807
$442$	0	0
$443$	2.47332e6	0.598784	0.299392	−	0.954130i	$-0.403216\pi$
$443$	2.47332e6	0.598784	0.299392	+	0.954130i	$0.403216\pi$
$444$	0	0
$445$	199369.	0.0477263
$446$	0	0
$447$	9.97413e6	2.36106
$448$	0	0
$449$	6.11819e6	1.43221	0.716106	−	0.697992i	$-0.245923\pi$
$449$	6.11819e6	1.43221	0.716106	+	0.697992i	$0.245923\pi$
$450$	0	0
$451$	−1.01203e7	−2.34290
$452$	0	0
$453$	−1.07518e7	−2.46170
$454$	0	0
$455$	238012.	0.0538978
$456$	0	0
$457$	650177.	0.145627	0.0728133	−	0.997346i	$-0.476802\pi$
$457$	650177.	0.145627	0.0728133	+	0.997346i	$0.476802\pi$
$458$	0	0
$459$	732542.	0.162293
$460$	0	0
$461$	−6.43756e6	−1.41081	−0.705405	−	0.708804i	$-0.749235\pi$
$461$	−6.43756e6	−1.41081	−0.705405	+	0.708804i	$0.749235\pi$
$462$	0	0
$463$	−196947.	−0.0426969	−0.0213485	−	0.999772i	$-0.506796\pi$
$463$	−196947.	−0.0426969	−0.0213485	+	0.999772i	$0.506796\pi$
$464$	0	0
$465$	769179.	0.164966
$466$	0	0
$467$	4.89949e6	1.03958	0.519790	−	0.854294i	$-0.326010\pi$
$467$	4.89949e6	1.03958	0.519790	+	0.854294i	$0.326010\pi$
$468$	0	0
$469$	2.14808e6	0.450940
$470$	0	0
$471$	−1.75465e6	−0.364449
$472$	0	0
$473$	4.13236e6	0.849269
$474$	0	0
$475$	−3.07817e6	−0.625977
$476$	0	0
$477$	−1.72624e6	−0.347381
$478$	0	0
$479$	4.52409e6	0.900934	0.450467	−	0.892793i	$-0.351258\pi$
$479$	4.52409e6	0.900934	0.450467	+	0.892793i	$0.351258\pi$
$480$	0	0
$481$	−985162.	−0.194153
$482$	0	0
$483$	−202917.	−0.0395776
$484$	0	0
$485$	2.24425e6	0.433228
$486$	0	0
$487$	−2.09327e6	−0.399948	−0.199974	−	0.979801i	$-0.564086\pi$
$487$	−2.09327e6	−0.399948	−0.199974	+	0.979801i	$0.564086\pi$
$488$	0	0
$489$	−4.49060e6	−0.849243
$490$	0	0
$491$	2.85600e6	0.534631	0.267315	−	0.963609i	$-0.413863\pi$
$491$	2.85600e6	0.534631	0.267315	+	0.963609i	$0.413863\pi$
$492$	0	0
$493$	887154.	0.164392
$494$	0	0
$495$	3.49831e6	0.641719
$496$	0	0
$497$	1.69661e6	0.308100
$498$	0	0
$499$	7.92264e6	1.42436	0.712178	−	0.701999i	$-0.247709\pi$
$499$	7.92264e6	1.42436	0.712178	+	0.701999i	$0.247709\pi$
$500$	0	0
$501$	841151.	0.149720
$502$	0	0
$503$	2.28311e6	0.402352	0.201176	−	0.979555i	$-0.435524\pi$
$503$	2.28311e6	0.402352	0.201176	+	0.979555i	$0.435524\pi$
$504$	0	0
$505$	3.31879e6	0.579096
$506$	0	0
$507$	7.01840e6	1.21260
$508$	0	0
$509$	−8.32454e6	−1.42418	−0.712092	−	0.702087i	$-0.752252\pi$
$509$	−8.32454e6	−1.42418	−0.712092	+	0.702087i	$0.752252\pi$
$510$	0	0
$511$	−2.98023e6	−0.504891
$512$	0	0
$513$	2.57283e6	0.431636
$514$	0	0
$515$	−1.27584e6	−0.211973
$516$	0	0
$517$	8.32082e6	1.36911
$518$	0	0
$519$	−1.02850e7	−1.67604
$520$	0	0
$521$	562984.	0.0908660	0.0454330	−	0.998967i	$-0.485533\pi$
$521$	562984.	0.0908660	0.0454330	+	0.998967i	$0.485533\pi$
$522$	0	0
$523$	5.35275e6	0.855703	0.427852	−	0.903849i	$-0.359271\pi$
$523$	5.35275e6	0.855703	0.427852	+	0.903849i	$0.359271\pi$
$524$	0	0
$525$	1.97943e6	0.313430
$526$	0	0
$527$	−500831.	−0.0785534
$528$	0	0
$529$	−6.39368e6	−0.993372
$530$	0	0
$531$	−8.35814e6	−1.28639
$532$	0	0
$533$	−2.23443e6	−0.340682
$534$	0	0
$535$	2.79671e6	0.422438
$536$	0	0
$537$	2.26036e6	0.338253
$538$	0	0
$539$	1.58532e6	0.235041
$540$	0	0
$541$	1.24141e7	1.82357	0.911783	−	0.410673i	$-0.134706\pi$
$541$	1.24141e7	1.82357	0.911783	+	0.410673i	$0.134706\pi$
$542$	0	0
$543$	5.66673e6	0.824771
$544$	0	0
$545$	6.92142e6	0.998168
$546$	0	0
$547$	7.19845e6	1.02866	0.514328	−	0.857593i	$-0.328041\pi$
$547$	7.19845e6	1.02866	0.514328	+	0.857593i	$0.328041\pi$
$548$	0	0
$549$	7.43932e6	1.05342
$550$	0	0
$551$	3.11585e6	0.437218
$552$	0	0
$553$	−1.11159e6	−0.154573
$554$	0	0
$555$	4.51480e6	0.622165
$556$	0	0
$557$	9.45495e6	1.29128	0.645641	−	0.763641i	$-0.276590\pi$
$557$	9.45495e6	1.29128	0.645641	+	0.763641i	$0.276590\pi$
$558$	0	0
$559$	912367.	0.123492
$560$	0	0
$561$	−5.75882e6	−0.772549
$562$	0	0
$563$	8.80283e6	1.17045	0.585223	−	0.810873i	$-0.301007\pi$
$563$	8.80283e6	1.17045	0.585223	+	0.810873i	$0.301007\pi$
$564$	0	0
$565$	6.45363e6	0.850517
$566$	0	0
$567$	−3.54781e6	−0.463449
$568$	0	0
$569$	8.86214e6	1.14751	0.573757	−	0.819025i	$-0.305485\pi$
$569$	8.86214e6	1.14751	0.573757	+	0.819025i	$0.305485\pi$
$570$	0	0
$571$	−8.40188e6	−1.07842	−0.539208	−	0.842173i	$-0.681276\pi$
$571$	−8.40188e6	−1.07842	−0.539208	+	0.842173i	$0.681276\pi$
$572$	0	0
$573$	−6.62587e6	−0.843056
$574$	0	0
$575$	−416128.	−0.0524877
$576$	0	0
$577$	−3.94823e6	−0.493700	−0.246850	−	0.969054i	$-0.579396\pi$
$577$	−3.94823e6	−0.493700	−0.246850	+	0.969054i	$0.579396\pi$
$578$	0	0
$579$	−631183.	−0.0782454
$580$	0	0
$581$	1.90181e6	0.233736
$582$	0	0
$583$	−7.16803e6	−0.873430
$584$	0	0
$585$	772377.	0.0933125
$586$	0	0
$587$	5.22309e6	0.625652	0.312826	−	0.949811i	$-0.398724\pi$
$587$	5.22309e6	0.625652	0.312826	+	0.949811i	$0.398724\pi$
$588$	0	0
$589$	−1.75901e6	−0.208920
$590$	0	0
$591$	1.30898e7	1.54157
$592$	0	0
$593$	−9.33848e6	−1.09053	−0.545267	−	0.838262i	$-0.683572\pi$
$593$	−9.33848e6	−1.09053	−0.545267	+	0.838262i	$0.683572\pi$
$594$	0	0
$595$	710221.	0.0822434
$596$	0	0
$597$	−6.17350e6	−0.708918
$598$	0	0
$599$	65978.1	0.00751333	0.00375667	−	0.999993i	$-0.498804\pi$
$599$	65978.1	0.00751333	0.00375667	+	0.999993i	$0.498804\pi$
$600$	0	0
$601$	4.69907e6	0.530671	0.265336	−	0.964156i	$-0.414517\pi$
$601$	4.69907e6	0.530671	0.265336	+	0.964156i	$0.414517\pi$
$602$	0	0
$603$	6.97077e6	0.780706
$604$	0	0
$605$	9.16008e6	1.01744
$606$	0	0
$607$	−2.04851e6	−0.225666	−0.112833	−	0.993614i	$-0.535992\pi$
$607$	−2.04851e6	−0.225666	−0.112833	+	0.993614i	$0.535992\pi$
$608$	0	0
$609$	−2.00366e6	−0.218917
$610$	0	0
$611$	1.83712e6	0.199083
$612$	0	0
$613$	1.14241e7	1.22792	0.613962	−	0.789336i	$-0.289575\pi$
$613$	1.14241e7	1.22792	0.613962	+	0.789336i	$0.289575\pi$
$614$	0	0
$615$	1.02399e7	1.09172
$616$	0	0
$617$	−5.25878e6	−0.556125	−0.278062	−	0.960563i	$-0.589692\pi$
$617$	−5.25878e6	−0.556125	−0.278062	+	0.960563i	$0.589692\pi$
$618$	0	0
$619$	1.14617e7	1.20233	0.601163	−	0.799126i	$-0.294704\pi$
$619$	1.14617e7	1.20233	0.601163	+	0.799126i	$0.294704\pi$
$620$	0	0
$621$	347813.	0.0361923
$622$	0	0
$623$	293188.	0.0302640
$624$	0	0
$625$	589799.	0.0603954
$626$	0	0
$627$	−2.02260e7	−2.05467
$628$	0	0
$629$	−2.93969e6	−0.296262
$630$	0	0
$631$	1.24372e7	1.24351	0.621757	−	0.783210i	$-0.286419\pi$
$631$	1.24372e7	1.24351	0.621757	+	0.783210i	$0.286419\pi$
$632$	0	0
$633$	3.43941e6	0.341173
$634$	0	0
$635$	8.98840e6	0.884602
$636$	0	0
$637$	350016.	0.0341774
$638$	0	0
$639$	5.50571e6	0.533410
$640$	0	0
$641$	1.02997e7	0.990102	0.495051	−	0.868864i	$-0.335149\pi$
$641$	1.02997e7	0.990102	0.495051	+	0.868864i	$0.335149\pi$
$642$	0	0
$643$	−1.23234e7	−1.17545	−0.587726	−	0.809060i	$-0.699977\pi$
$643$	−1.23234e7	−1.17545	−0.587726	+	0.809060i	$0.699977\pi$
$644$	0	0
$645$	−4.18119e6	−0.395731
$646$	0	0
$647$	4.45009e6	0.417935	0.208967	−	0.977923i	$-0.432990\pi$
$647$	4.45009e6	0.417935	0.208967	+	0.977923i	$0.432990\pi$
$648$	0	0
$649$	−3.47062e7	−3.23441
$650$	0	0
$651$	1.13114e6	0.104608
$652$	0	0
$653$	−9.78800e6	−0.898279	−0.449139	−	0.893462i	$-0.648269\pi$
$653$	−9.78800e6	−0.898279	−0.449139	+	0.893462i	$0.648269\pi$
$654$	0	0
$655$	3.24718e6	0.295735
$656$	0	0
$657$	−9.67121e6	−0.874112
$658$	0	0
$659$	7.70009e6	0.690688	0.345344	−	0.938476i	$-0.387762\pi$
$659$	7.70009e6	0.690688	0.345344	+	0.938476i	$0.387762\pi$
$660$	0	0
$661$	6.47088e6	0.576049	0.288025	−	0.957623i	$-0.407001\pi$
$661$	6.47088e6	0.576049	0.288025	+	0.957623i	$0.407001\pi$
$662$	0	0
$663$	−1.27147e6	−0.112337
$664$	0	0
$665$	2.49443e6	0.218734
$666$	0	0
$667$	421222.	0.0366604
$668$	0	0
$669$	4.51534e6	0.390054
$670$	0	0
$671$	3.08909e7	2.64865
$672$	0	0
$673$	−1.26002e7	−1.07236	−0.536180	−	0.844104i	$-0.680133\pi$
$673$	−1.26002e7	−1.07236	−0.536180	+	0.844104i	$0.680133\pi$
$674$	0	0
$675$	−3.39287e6	−0.286621
$676$	0	0
$677$	1.92476e6	0.161401	0.0807004	−	0.996738i	$-0.474284\pi$
$677$	1.92476e6	0.161401	0.0807004	+	0.996738i	$0.474284\pi$
$678$	0	0
$679$	3.30034e6	0.274717
$680$	0	0
$681$	6.36751e6	0.526141
$682$	0	0
$683$	2.96929e6	0.243558	0.121779	−	0.992557i	$-0.461140\pi$
$683$	2.96929e6	0.243558	0.121779	+	0.992557i	$0.461140\pi$
$684$	0	0
$685$	−769050.	−0.0626222
$686$	0	0
$687$	1.66192e6	0.134344
$688$	0	0
$689$	−1.58260e6	−0.127006
$690$	0	0
$691$	−1.44542e7	−1.15159	−0.575795	−	0.817594i	$-0.695308\pi$
$691$	−1.44542e7	−1.15159	−0.575795	+	0.817594i	$0.695308\pi$
$692$	0	0
$693$	5.14454e6	0.406924
$694$	0	0
$695$	−8.13456e6	−0.638811
$696$	0	0
$697$	−6.66747e6	−0.519852
$698$	0	0
$699$	1.12701e7	0.872439
$700$	0	0
$701$	715005.	0.0549558	0.0274779	−	0.999622i	$-0.491252\pi$
$701$	715005.	0.0549558	0.0274779	+	0.999622i	$0.491252\pi$
$702$	0	0
$703$	−1.03248e7	−0.787937
$704$	0	0
$705$	−8.41915e6	−0.637963
$706$	0	0
$707$	4.88054e6	0.367214
$708$	0	0
$709$	−1.75184e7	−1.30882	−0.654408	−	0.756141i	$-0.727082\pi$
$709$	−1.75184e7	−1.30882	−0.654408	+	0.756141i	$0.727082\pi$
$710$	0	0
$711$	−3.60725e6	−0.267610
$712$	0	0
$713$	−237796.	−0.0175178
$714$	0	0
$715$	3.20721e6	0.234618
$716$	0	0
$717$	1.24913e7	0.907421
$718$	0	0
$719$	−1.52406e7	−1.09946	−0.549731	−	0.835342i	$-0.685270\pi$
$719$	−1.52406e7	−1.09946	−0.549731	+	0.835342i	$0.685270\pi$
$720$	0	0
$721$	−1.87623e6	−0.134415
$722$	0	0
$723$	1.15913e7	0.824682
$724$	0	0
$725$	−4.10897e6	−0.290327
$726$	0	0
$727$	−2.01811e7	−1.41615	−0.708073	−	0.706140i	$-0.750435\pi$
$727$	−2.01811e7	−1.41615	−0.708073	+	0.706140i	$0.750435\pi$
$728$	0	0
$729$	−3.30825e6	−0.230558
$730$	0	0
$731$	2.72247e6	0.188439
$732$	0	0
$733$	−1.02074e6	−0.0701706	−0.0350853	−	0.999384i	$-0.511170\pi$
$733$	−1.02074e6	−0.0701706	−0.0350853	+	0.999384i	$0.511170\pi$
$734$	0	0
$735$	−1.60405e6	−0.109522
$736$	0	0
$737$	2.89454e7	1.96295
$738$	0	0
$739$	−1.36743e7	−0.921070	−0.460535	−	0.887642i	$-0.652342\pi$
$739$	−1.36743e7	−0.921070	−0.460535	+	0.887642i	$0.652342\pi$
$740$	0	0
$741$	−4.46563e6	−0.298770
$742$	0	0
$743$	−1.13203e7	−0.752289	−0.376144	−	0.926561i	$-0.622750\pi$
$743$	−1.13203e7	−0.752289	−0.376144	+	0.926561i	$0.622750\pi$
$744$	0	0
$745$	−1.65754e7	−1.09414
$746$	0	0
$747$	6.17158e6	0.404664
$748$	0	0
$749$	4.11279e6	0.267875
$750$	0	0
$751$	5.87893e6	0.380363	0.190181	−	0.981749i	$-0.439092\pi$
$751$	5.87893e6	0.380363	0.190181	+	0.981749i	$0.439092\pi$
$752$	0	0
$753$	3.20302e7	2.05860
$754$	0	0
$755$	1.78677e7	1.14078
$756$	0	0
$757$	−2.35139e7	−1.49137	−0.745686	−	0.666298i	$-0.767878\pi$
$757$	−2.35139e7	−1.49137	−0.745686	+	0.666298i	$0.767878\pi$
$758$	0	0
$759$	−2.73430e6	−0.172283
$760$	0	0
$761$	−1.30171e7	−0.814803	−0.407402	−	0.913249i	$-0.633565\pi$
$761$	−1.30171e7	−0.814803	−0.407402	+	0.913249i	$0.633565\pi$
$762$	0	0
$763$	1.01785e7	0.632954
$764$	0	0
$765$	2.30475e6	0.142387
$766$	0	0
$767$	−7.66265e6	−0.470317
$768$	0	0
$769$	2.15837e6	0.131617	0.0658083	−	0.997832i	$-0.479037\pi$
$769$	2.15837e6	0.131617	0.0658083	+	0.997832i	$0.479037\pi$
$770$	0	0
$771$	−7.90896e6	−0.479163
$772$	0	0
$773$	−2.08330e7	−1.25401	−0.627007	−	0.779013i	$-0.715720\pi$
$773$	−2.08330e7	−1.25401	−0.627007	+	0.779013i	$0.715720\pi$
$774$	0	0
$775$	2.31967e6	0.138730
$776$	0	0
$777$	6.63937e6	0.394525
$778$	0	0
$779$	−2.34174e7	−1.38260
$780$	0	0
$781$	2.28618e7	1.34117
$782$	0	0
$783$	3.43440e6	0.200192
$784$	0	0
$785$	2.91594e6	0.168890
$786$	0	0
$787$	−1.77822e7	−1.02341	−0.511703	−	0.859162i	$-0.670985\pi$
$787$	−1.77822e7	−1.02341	−0.511703	+	0.859162i	$0.670985\pi$
$788$	0	0
$789$	−4.42537e7	−2.53079
$790$	0	0
$791$	9.49058e6	0.539326
$792$	0	0
$793$	6.82029e6	0.385141
$794$	0	0
$795$	7.25273e6	0.406990
$796$	0	0
$797$	−1.03372e6	−0.0576447	−0.0288223	−	0.999585i	$-0.509176\pi$
$797$	−1.03372e6	−0.0576447	−0.0288223	+	0.999585i	$0.509176\pi$
$798$	0	0
$799$	5.48191e6	0.303784
$800$	0	0
$801$	951430.	0.0523957
$802$	0	0
$803$	−4.01586e7	−2.19781
$804$	0	0
$805$	337214.	0.0183407
$806$	0	0
$807$	−4.68560e7	−2.53269
$808$	0	0
$809$	−3.22789e7	−1.73399	−0.866996	−	0.498315i	$-0.833952\pi$
$809$	−3.22789e7	−1.73399	−0.866996	+	0.498315i	$0.833952\pi$
$810$	0	0
$811$	−1.57854e7	−0.842757	−0.421378	−	0.906885i	$-0.638454\pi$
$811$	−1.57854e7	−0.842757	−0.421378	+	0.906885i	$0.638454\pi$
$812$	0	0
$813$	−3.25922e7	−1.72937
$814$	0	0
$815$	7.46264e6	0.393549
$816$	0	0
$817$	9.56184e6	0.501172
$818$	0	0
$819$	1.13584e6	0.0591710
$820$	0	0
$821$	2.23707e7	1.15830	0.579150	−	0.815221i	$-0.303384\pi$
$821$	2.23707e7	1.15830	0.579150	+	0.815221i	$0.303384\pi$
$822$	0	0
$823$	−3.49932e7	−1.80087	−0.900437	−	0.434986i	$-0.856753\pi$
$823$	−3.49932e7	−1.80087	−0.900437	+	0.434986i	$0.856753\pi$
$824$	0	0
$825$	2.66727e7	1.36437
$826$	0	0
$827$	1.97157e7	1.00242	0.501208	−	0.865327i	$-0.332889\pi$
$827$	1.97157e7	1.00242	0.501208	+	0.865327i	$0.332889\pi$
$828$	0	0
$829$	8.34998e6	0.421987	0.210994	−	0.977487i	$-0.432330\pi$
$829$	8.34998e6	0.421987	0.210994	+	0.977487i	$0.432330\pi$
$830$	0	0
$831$	−4.83648e7	−2.42955
$832$	0	0
$833$	1.04444e6	0.0521519
$834$	0	0
$835$	−1.39786e6	−0.0693819
$836$	0	0
$837$	−1.93885e6	−0.0956600
$838$	0	0
$839$	3.00623e7	1.47441	0.737203	−	0.675671i	$-0.236146\pi$
$839$	3.00623e7	1.47441	0.737203	+	0.675671i	$0.236146\pi$
$840$	0	0
$841$	−1.63519e7	−0.797219
$842$	0	0
$843$	−1.14189e7	−0.553422
$844$	0	0
$845$	−1.16634e7	−0.561934
$846$	0	0
$847$	1.34706e7	0.645178
$848$	0	0
$849$	2.27346e7	1.08247
$850$	0	0
$851$	−1.39577e6	−0.0660680
$852$	0	0
$853$	−1.83935e7	−0.865550	−0.432775	−	0.901502i	$-0.642466\pi$
$853$	−1.83935e7	−0.865550	−0.432775	+	0.901502i	$0.642466\pi$
$854$	0	0
$855$	8.09472e6	0.378692
$856$	0	0
$857$	3.39540e7	1.57921	0.789603	−	0.613618i	$-0.210287\pi$
$857$	3.39540e7	1.57921	0.789603	+	0.613618i	$0.210287\pi$
$858$	0	0
$859$	2.03765e7	0.942206	0.471103	−	0.882078i	$-0.343856\pi$
$859$	2.03765e7	0.942206	0.471103	+	0.882078i	$0.343856\pi$
$860$	0	0
$861$	1.50586e7	0.692274
$862$	0	0
$863$	3.05608e7	1.39681	0.698405	−	0.715702i	$-0.253893\pi$
$863$	3.05608e7	1.39681	0.698405	+	0.715702i	$0.253893\pi$
$864$	0	0
$865$	1.70920e7	0.776697
$866$	0	0
$867$	2.46744e7	1.11481
$868$	0	0
$869$	−1.49787e7	−0.672859
$870$	0	0
$871$	6.39073e6	0.285434
$872$	0	0
$873$	1.07100e7	0.475613
$874$	0	0
$875$	−8.39164e6	−0.370533
$876$	0	0
$877$	2.99046e7	1.31292	0.656461	−	0.754360i	$-0.272052\pi$
$877$	2.99046e7	1.31292	0.656461	+	0.754360i	$0.272052\pi$
$878$	0	0
$879$	−2.68861e7	−1.17369
$880$	0	0
$881$	1.36000e7	0.590336	0.295168	−	0.955445i	$-0.404624\pi$
$881$	1.36000e7	0.590336	0.295168	+	0.955445i	$0.404624\pi$
$882$	0	0
$883$	−1.56685e7	−0.676279	−0.338139	−	0.941096i	$-0.609797\pi$
$883$	−1.56685e7	−0.676279	−0.338139	+	0.941096i	$0.609797\pi$
$884$	0	0
$885$	3.51163e7	1.50713
$886$	0	0
$887$	3.68184e7	1.57129	0.785644	−	0.618679i	$-0.212332\pi$
$887$	3.68184e7	1.57129	0.785644	+	0.618679i	$0.212332\pi$
$888$	0	0
$889$	1.32182e7	0.560940
$890$	0	0
$891$	−4.78066e7	−2.01741
$892$	0	0
$893$	1.92535e7	0.807944
$894$	0	0
$895$	−3.75635e6	−0.156750
$896$	0	0
$897$	−603695.	−0.0250516
$898$	0	0
$899$	−2.34806e6	−0.0968971
$900$	0	0
$901$	−4.72243e6	−0.193800
$902$	0	0
$903$	−6.14877e6	−0.250940
$904$	0	0
$905$	−9.41719e6	−0.382208
$906$	0	0
$907$	−2.91515e7	−1.17664	−0.588318	−	0.808630i	$-0.700210\pi$
$907$	−2.91515e7	−1.17664	−0.588318	+	0.808630i	$0.700210\pi$
$908$	0	0
$909$	1.58379e7	0.635754
$910$	0	0
$911$	8.88578e6	0.354731	0.177366	−	0.984145i	$-0.443242\pi$
$911$	8.88578e6	0.354731	0.177366	+	0.984145i	$0.443242\pi$
$912$	0	0
$913$	2.56268e7	1.01746
$914$	0	0
$915$	−3.12560e7	−1.23419
$916$	0	0
$917$	4.77524e6	0.187530
$918$	0	0
$919$	2.37925e6	0.0929292	0.0464646	−	0.998920i	$-0.485205\pi$
$919$	2.37925e6	0.0929292	0.0464646	+	0.998920i	$0.485205\pi$
$920$	0	0
$921$	5.63094e7	2.18742
$922$	0	0
$923$	5.04757e6	0.195020
$924$	0	0
$925$	1.36156e7	0.523218
$926$	0	0
$927$	−6.08859e6	−0.232711
$928$	0	0
$929$	−1.12549e7	−0.427859	−0.213930	−	0.976849i	$-0.568626\pi$
$929$	−1.12549e7	−0.427859	−0.213930	+	0.976849i	$0.568626\pi$
$930$	0	0
$931$	3.66826e6	0.138703
$932$	0	0
$933$	3.77686e7	1.42045
$934$	0	0
$935$	9.57022e6	0.358008
$936$	0	0
$937$	−3.54517e7	−1.31913	−0.659566	−	0.751647i	$-0.729260\pi$
$937$	−3.54517e7	−1.31913	−0.659566	+	0.751647i	$0.729260\pi$
$938$	0	0
$939$	−2.54081e7	−0.940390
$940$	0	0
$941$	−4.94573e7	−1.82077	−0.910387	−	0.413758i	$-0.864216\pi$
$941$	−4.94573e7	−1.82077	−0.910387	+	0.413758i	$0.864216\pi$
$942$	0	0
$943$	−3.16573e6	−0.115930
$944$	0	0
$945$	2.74945e6	0.100154
$946$	0	0
$947$	2.43591e7	0.882645	0.441323	−	0.897349i	$-0.354509\pi$
$947$	2.43591e7	0.882645	0.441323	+	0.897349i	$0.354509\pi$
$948$	0	0
$949$	−8.86645e6	−0.319584
$950$	0	0
$951$	−5.78606e7	−2.07459
$952$	0	0
$953$	358667.	0.0127926	0.00639631	−	0.999980i	$-0.497964\pi$
$953$	358667.	0.0127926	0.00639631	+	0.999980i	$0.497964\pi$
$954$	0	0
$955$	1.10111e7	0.390682
$956$	0	0
$957$	−2.69992e7	−0.952954
$958$	0	0
$959$	−1.13095e6	−0.0397097
$960$	0	0
$961$	−2.73036e7	−0.953699
$962$	0	0
$963$	1.33465e7	0.463768
$964$	0	0
$965$	1.04892e6	0.0362598
$966$	0	0
$967$	1.36902e7	0.470809	0.235404	−	0.971898i	$-0.424359\pi$
$967$	1.36902e7	0.470809	0.235404	+	0.971898i	$0.424359\pi$
$968$	0	0
$969$	−1.33253e7	−0.455898
$970$	0	0
$971$	3.29263e6	0.112071	0.0560357	−	0.998429i	$-0.482154\pi$
$971$	3.29263e6	0.112071	0.0560357	+	0.998429i	$0.482154\pi$
$972$	0	0
$973$	−1.19625e7	−0.405080
$974$	0	0
$975$	5.88897e6	0.198394
$976$	0	0
$977$	−2.04796e7	−0.686411	−0.343205	−	0.939260i	$-0.611513\pi$
$977$	−2.04796e7	−0.686411	−0.343205	+	0.939260i	$0.611513\pi$
$978$	0	0
$979$	3.95071e6	0.131740
$980$	0	0
$981$	3.30304e7	1.09583
$982$	0	0
$983$	−4.59364e6	−0.151626	−0.0758128	−	0.997122i	$-0.524155\pi$
$983$	−4.59364e6	−0.151626	−0.0758128	+	0.997122i	$0.524155\pi$
$984$	0	0
$985$	−2.17530e7	−0.714380
$986$	0	0
$987$	−1.23810e7	−0.404542
$988$	0	0
$989$	1.29264e6	0.0420229
$990$	0	0
$991$	−5.37835e7	−1.73966	−0.869831	−	0.493350i	$-0.835772\pi$
$991$	−5.37835e7	−1.73966	−0.869831	+	0.493350i	$0.835772\pi$
$992$	0	0
$993$	−4.97656e7	−1.60161
$994$	0	0
$995$	1.02594e7	0.328521
$996$	0	0
$997$	2.76551e7	0.881126	0.440563	−	0.897722i	$-0.354779\pi$
$997$	2.76551e7	0.881126	0.440563	+	0.897722i	$0.354779\pi$
$998$	0	0
$999$	−1.13803e7	−0.360779

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.6.a.bb.1.1		3
4.3	odd	2		448.6.a.ba.1.3		3
8.3	odd	2		224.6.a.f.1.1	yes	3
8.5	even	2		224.6.a.e.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.6.a.e.1.3	✓	3	8.5	even	2
224.6.a.f.1.1	yes	3	8.3	odd	2
448.6.a.ba.1.3		3	4.3	odd	2
448.6.a.bb.1.1		3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-20.0502 q^{3} +33.3202 q^{5} +49.0000 q^{7} +159.011 q^{9} +660.274 q^{11} +145.779 q^{13} -668.076 q^{15} +435.001 q^{17} +1527.81 q^{19} -982.460 q^{21} +206.539 q^{23} -2014.77 q^{25} +1684.00 q^{27} +2039.43 q^{29} -1151.33 q^{31} -13238.6 q^{33} +1632.69 q^{35} -6757.90 q^{37} -2922.90 q^{39} -15327.5 q^{41} +6258.55 q^{43} +5298.27 q^{45} +12602.1 q^{47} +2401.00 q^{49} -8721.86 q^{51} -10856.1 q^{53} +22000.4 q^{55} -30632.8 q^{57} -52563.4 q^{59} +46785.0 q^{61} +7791.53 q^{63} +4857.39 q^{65} +43838.4 q^{67} -4141.16 q^{69} +34624.8 q^{71} -60821.1 q^{73} +40396.5 q^{75} +32353.4 q^{77} -22685.6 q^{79} -72404.2 q^{81} +38812.4 q^{83} +14494.3 q^{85} -40891.0 q^{87} +5983.43 q^{89} +7143.18 q^{91} +23084.5 q^{93} +50906.7 q^{95} +67354.0 q^{97} +104991. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 8 q^{3} + 14 q^{5} + 147 q^{7} + 151 q^{9} + 600 q^{11} + 974 q^{13} - 120 q^{15} + 718 q^{17} - 1056 q^{19} + 392 q^{21} + 3760 q^{23} - 147 q^{25} - 2872 q^{27} + 9134 q^{29} - 1448 q^{31} - 14872 q^{33}+ \cdots + 83128 q^{99}+O(q^{100}) \) 3 * q + 8 * q^3 + 14 * q^5 + 147 * q^7 + 151 * q^9 + 600 * q^11 + 974 * q^13 - 120 * q^15 + 718 * q^17 - 1056 * q^19 + 392 * q^21 + 3760 * q^23 - 147 * q^25 - 2872 * q^27 + 9134 * q^29 - 1448 * q^31 - 14872 * q^33 + 686 * q^35 + 4998 * q^37 - 824 * q^39 - 23186 * q^41 + 29880 * q^43 + 28350 * q^45 + 10840 * q^47 + 7203 * q^49 + 3776 * q^51 + 28006 * q^53 + 14952 * q^55 - 53504 * q^57 - 17456 * q^59 + 92294 * q^61 + 7399 * q^63 - 95300 * q^65 + 56024 * q^67 + 63480 * q^69 + 77064 * q^71 - 46346 * q^73 + 50768 * q^75 + 29400 * q^77 - 4376 * q^79 - 121973 * q^81 + 107128 * q^83 + 94348 * q^85 + 82720 * q^87 + 29814 * q^89 + 47726 * q^91 + 108432 * q^93 + 205304 * q^95 - 156482 * q^97 + 83128 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more