LMFDB - Embedded newform 400.6.a.z.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,6,Mod(1,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, 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("400.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$N$	$=$	$400 = 2^{4} \cdot 5^{2}$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	400.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:	$64.1535279252$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.1595208.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - x^{3} - 20x^{2} + 33x - 3$ x^4 - x^3 - 20x^2 + 33x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$2^{8}\cdot 5^{2}$
Twist minimal:	no (minimal twist has level 40)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-4.73066$ of defining polynomial
Character	$\chi$	$=$	400.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.49000 q^{3} +188.968 q^{7} -212.860 q^{9} +501.871 q^{11} +1061.48 q^{13} -29.5861 q^{17} -1578.33 q^{19} +1037.43 q^{21} +1295.86 q^{23} -2502.67 q^{27} -3586.63 q^{29} +3526.32 q^{31} +2755.27 q^{33} +8413.79 q^{37} +5827.54 q^{39} +7015.12 q^{41} -22694.2 q^{43} +3501.99 q^{47} +18901.8 q^{49} -162.428 q^{51} +27309.1 q^{53} -8665.04 q^{57} -7925.39 q^{59} -7020.54 q^{61} -40223.6 q^{63} -17631.2 q^{67} +7114.27 q^{69} -13432.9 q^{71} +39946.8 q^{73} +94837.3 q^{77} +93321.2 q^{79} +37985.3 q^{81} +58448.5 q^{83} -19690.6 q^{87} -13989.4 q^{89} +200586. q^{91} +19359.5 q^{93} -110640. q^{97} -106828. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 4 q^{3} - 148 q^{7} + 500 q^{9} + 368 q^{11} + 440 q^{13} - 672 q^{17} + 688 q^{19} + 992 q^{21} - 4492 q^{23} - 8152 q^{27} - 2936 q^{29} - 2112 q^{31} + 26864 q^{33} + 8792 q^{37} - 1504 q^{39}+ \cdots + 133424 q^{99}+O(q^{100})$ 4 * q - 4 * q^3 - 148 * q^7 + 500 * q^9 + 368 * q^11 + 440 * q^13 - 672 * q^17 + 688 * q^19 + 992 * q^21 - 4492 * q^23 - 8152 * q^27 - 2936 * q^29 - 2112 * q^31 + 26864 * q^33 + 8792 * q^37 - 1504 * q^39 + 11800 * q^41 - 48276 * q^43 - 14724 * q^47 + 22500 * q^49 + 62400 * q^51 + 84296 * q^53 + 71024 * q^57 + 45840 * q^59 + 61928 * q^61 - 186292 * q^63 - 72700 * q^67 + 38368 * q^69 + 62816 * q^71 + 133072 * q^73 + 11440 * q^77 + 21632 * q^79 + 204836 * q^81 - 74660 * q^83 + 12472 * q^87 + 20952 * q^89 + 243808 * q^91 + 105600 * q^93 + 59456 * q^97 + 133424 * 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
$3$	5.49000	0.352184	0.176092	−	0.984374i	$-0.443654\pi$
$3$	5.49000	0.352184	0.176092	+	0.984374i	$0.443654\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	188.968	1.45761	0.728807	−	0.684720i	$-0.240075\pi$
$7$	188.968	1.45761	0.728807	+	0.684720i	$0.240075\pi$
$8$	0	0
$9$	−212.860	−0.875967
$10$	0	0
$11$	501.871	1.25058	0.625288	−	0.780394i	$-0.284982\pi$
$11$	501.871	1.25058	0.625288	+	0.780394i	$0.284982\pi$
$12$	0	0
$13$	1061.48	1.74203	0.871013	−	0.491259i	$-0.163463\pi$
$13$	1061.48	1.74203	0.871013	+	0.491259i	$0.163463\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−29.5861	−0.0248294	−0.0124147	−	0.999923i	$-0.503952\pi$
$17$	−29.5861	−0.0248294	−0.0124147	+	0.999923i	$0.503952\pi$
$18$	0	0
$19$	−1578.33	−1.00303	−0.501515	−	0.865149i	$-0.667224\pi$
$19$	−1578.33	−1.00303	−0.501515	+	0.865149i	$0.667224\pi$
$20$	0	0
$21$	1037.43	0.513347
$22$	0	0
$23$	1295.86	0.510786	0.255393	−	0.966837i	$-0.417795\pi$
$23$	1295.86	0.510786	0.255393	+	0.966837i	$0.417795\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−2502.67	−0.660685
$28$	0	0
$29$	−3586.63	−0.791938	−0.395969	−	0.918264i	$-0.629591\pi$
$29$	−3586.63	−0.791938	−0.395969	+	0.918264i	$0.629591\pi$
$30$	0	0
$31$	3526.32	0.659048	0.329524	−	0.944147i	$-0.393112\pi$
$31$	3526.32	0.659048	0.329524	+	0.944147i	$0.393112\pi$
$32$	0	0
$33$	2755.27	0.440432
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8413.79	1.01039	0.505193	−	0.863006i	$-0.331421\pi$
$37$	8413.79	1.01039	0.505193	+	0.863006i	$0.331421\pi$
$38$	0	0
$39$	5827.54	0.613513
$40$	0	0
$41$	7015.12	0.651742	0.325871	−	0.945414i	$-0.394343\pi$
$41$	7015.12	0.651742	0.325871	+	0.945414i	$0.394343\pi$
$42$	0	0
$43$	−22694.2	−1.87173	−0.935865	−	0.352358i	$-0.885380\pi$
$43$	−22694.2	−1.87173	−0.935865	+	0.352358i	$0.885380\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3501.99	0.231244	0.115622	−	0.993293i	$-0.463114\pi$
$47$	3501.99	0.231244	0.115622	+	0.993293i	$0.463114\pi$
$48$	0	0
$49$	18901.8	1.12464
$50$	0	0
$51$	−162.428	−0.00874449
$52$	0	0
$53$	27309.1	1.33542	0.667710	−	0.744422i	$-0.267275\pi$
$53$	27309.1	1.33542	0.667710	+	0.744422i	$0.267275\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−8665.04	−0.353251
$58$	0	0
$59$	−7925.39	−0.296409	−0.148204	−	0.988957i	$-0.547349\pi$
$59$	−7925.39	−0.296409	−0.148204	+	0.988957i	$0.547349\pi$
$60$	0	0
$61$	−7020.54	−0.241572	−0.120786	−	0.992679i	$-0.538541\pi$
$61$	−7020.54	−0.241572	−0.120786	+	0.992679i	$0.538541\pi$
$62$	0	0
$63$	−40223.6	−1.27682
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−17631.2	−0.479837	−0.239919	−	0.970793i	$-0.577121\pi$
$67$	−17631.2	−0.479837	−0.239919	+	0.970793i	$0.577121\pi$
$68$	0	0
$69$	7114.27	0.179890
$70$	0	0
$71$	−13432.9	−0.316246	−0.158123	−	0.987419i	$-0.550544\pi$
$71$	−13432.9	−0.316246	−0.158123	+	0.987419i	$0.550544\pi$
$72$	0	0
$73$	39946.8	0.877353	0.438677	−	0.898645i	$-0.355447\pi$
$73$	39946.8	0.877353	0.438677	+	0.898645i	$0.355447\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	94837.3	1.82286
$78$	0	0
$79$	93321.2	1.68234	0.841168	−	0.540774i	$-0.181869\pi$
$79$	93321.2	1.68234	0.841168	+	0.540774i	$0.181869\pi$
$80$	0	0
$81$	37985.3	0.643284
$82$	0	0
$83$	58448.5	0.931275	0.465638	−	0.884975i	$-0.345825\pi$
$83$	58448.5	0.931275	0.465638	+	0.884975i	$0.345825\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−19690.6	−0.278908
$88$	0	0
$89$	−13989.4	−0.187208	−0.0936038	−	0.995610i	$-0.529839\pi$
$89$	−13989.4	−0.187208	−0.0936038	+	0.995610i	$0.529839\pi$
$90$	0	0
$91$	200586.	2.53920
$92$	0	0
$93$	19359.5	0.232106
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−110640.	−1.19394	−0.596969	−	0.802264i	$-0.703629\pi$
$97$	−110640.	−1.19394	−0.596969	+	0.802264i	$0.703629\pi$
$98$	0	0
$99$	−106828.	−1.09546
$100$	0	0
$101$	198468.	1.93592	0.967960	−	0.251103i	$-0.0807934\pi$
$101$	198468.	1.93592	0.967960	+	0.251103i	$0.0807934\pi$
$102$	0	0
$103$	−134780.	−1.25179	−0.625897	−	0.779906i	$-0.715267\pi$
$103$	−134780.	−1.25179	−0.625897	+	0.779906i	$0.715267\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−39785.5	−0.335943	−0.167971	−	0.985792i	$-0.553722\pi$
$107$	−39785.5	−0.335943	−0.167971	+	0.985792i	$0.553722\pi$
$108$	0	0
$109$	92692.6	0.747272	0.373636	−	0.927575i	$-0.378111\pi$
$109$	92692.6	0.747272	0.373636	+	0.927575i	$0.378111\pi$
$110$	0	0
$111$	46191.7	0.355841
$112$	0	0
$113$	−66923.1	−0.493037	−0.246519	−	0.969138i	$-0.579287\pi$
$113$	−66923.1	−0.493037	−0.246519	+	0.969138i	$0.579287\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−225947.	−1.52596
$118$	0	0
$119$	−5590.81	−0.0361916
$120$	0	0
$121$	90823.1	0.563940
$122$	0	0
$123$	38513.0	0.229533
$124$	0	0
$125$	0	0
$126$	0	0
$127$	58998.9	0.324589	0.162295	−	0.986742i	$-0.448110\pi$
$127$	58998.9	0.324589	0.162295	+	0.986742i	$0.448110\pi$
$128$	0	0
$129$	−124591.	−0.659193
$130$	0	0
$131$	209380.	1.06600	0.533001	−	0.846115i	$-0.321064\pi$
$131$	209380.	1.06600	0.533001	+	0.846115i	$0.321064\pi$
$132$	0	0
$133$	−298254.	−1.46203
$134$	0	0
$135$	0	0
$136$	0	0
$137$	326432.	1.48591	0.742953	−	0.669344i	$-0.233425\pi$
$137$	326432.	1.48591	0.742953	+	0.669344i	$0.233425\pi$
$138$	0	0
$139$	233648.	1.02571	0.512856	−	0.858475i	$-0.328588\pi$
$139$	233648.	1.02571	0.512856	+	0.858475i	$0.328588\pi$
$140$	0	0
$141$	19225.9	0.0814403
$142$	0	0
$143$	532727.	2.17854
$144$	0	0
$145$	0	0
$146$	0	0
$147$	103771.	0.396078
$148$	0	0
$149$	145336.	0.536299	0.268149	−	0.963377i	$-0.413588\pi$
$149$	145336.	0.536299	0.268149	+	0.963377i	$0.413588\pi$
$150$	0	0
$151$	−287343.	−1.02555	−0.512777	−	0.858522i	$-0.671383\pi$
$151$	−287343.	−1.02555	−0.512777	+	0.858522i	$0.671383\pi$
$152$	0	0
$153$	6297.69	0.0217497
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−42626.2	−0.138015	−0.0690077	−	0.997616i	$-0.521983\pi$
$157$	−42626.2	−0.138015	−0.0690077	+	0.997616i	$0.521983\pi$
$158$	0	0
$159$	149927.	0.470313
$160$	0	0
$161$	244876.	0.744528
$162$	0	0
$163$	221599.	0.653279	0.326639	−	0.945149i	$-0.394084\pi$
$163$	221599.	0.653279	0.326639	+	0.945149i	$0.394084\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	334835.	0.929051	0.464525	−	0.885560i	$-0.346225\pi$
$167$	334835.	0.929051	0.464525	+	0.885560i	$0.346225\pi$
$168$	0	0
$169$	755454.	2.03466
$170$	0	0
$171$	335964.	0.878622
$172$	0	0
$173$	−268475.	−0.682006	−0.341003	−	0.940062i	$-0.610767\pi$
$173$	−268475.	−0.682006	−0.341003	+	0.940062i	$0.610767\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−43510.4	−0.104390
$178$	0	0
$179$	268052.	0.625297	0.312649	−	0.949869i	$-0.398784\pi$
$179$	268052.	0.625297	0.312649	+	0.949869i	$0.398784\pi$
$180$	0	0
$181$	−90565.5	−0.205479	−0.102739	−	0.994708i	$-0.532761\pi$
$181$	−90565.5	−0.205479	−0.102739	+	0.994708i	$0.532761\pi$
$182$	0	0
$183$	−38542.8	−0.0850776
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−14848.4	−0.0310510
$188$	0	0
$189$	−472924.	−0.963023
$190$	0	0
$191$	706725.	1.40174	0.700870	−	0.713290i	$-0.252796\pi$
$191$	706725.	1.40174	0.700870	+	0.713290i	$0.252796\pi$
$192$	0	0
$193$	236999.	0.457988	0.228994	−	0.973428i	$-0.426456\pi$
$193$	236999.	0.457988	0.228994	+	0.973428i	$0.426456\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−607901.	−1.11601	−0.558004	−	0.829838i	$-0.688433\pi$
$197$	−607901.	−1.11601	−0.558004	+	0.829838i	$0.688433\pi$
$198$	0	0
$199$	−621060.	−1.11173	−0.555867	−	0.831272i	$-0.687613\pi$
$199$	−621060.	−1.11173	−0.555867	+	0.831272i	$0.687613\pi$
$200$	0	0
$201$	−96795.1	−0.168991
$202$	0	0
$203$	−677756.	−1.15434
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−275837.	−0.447431
$208$	0	0
$209$	−792118.	−1.25437
$210$	0	0
$211$	−1.06177e6	−1.64181	−0.820906	−	0.571063i	$-0.806531\pi$
$211$	−1.06177e6	−1.64181	−0.820906	+	0.571063i	$0.806531\pi$
$212$	0	0
$213$	−73746.8	−0.111377
$214$	0	0
$215$	0	0
$216$	0	0
$217$	666360.	0.960638
$218$	0	0
$219$	219308.	0.308989
$220$	0	0
$221$	−31405.2	−0.0432534
$222$	0	0
$223$	−240721.	−0.324154	−0.162077	−	0.986778i	$-0.551819\pi$
$223$	−240721.	−0.324154	−0.162077	+	0.986778i	$0.551819\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	202494.	0.260824	0.130412	−	0.991460i	$-0.458370\pi$
$227$	202494.	0.260824	0.130412	+	0.991460i	$0.458370\pi$
$228$	0	0
$229$	−284279.	−0.358225	−0.179112	−	0.983829i	$-0.557323\pi$
$229$	−284279.	−0.358225	−0.179112	+	0.983829i	$0.557323\pi$
$230$	0	0
$231$	520657.	0.641980
$232$	0	0
$233$	345405.	0.416811	0.208406	−	0.978042i	$-0.433173\pi$
$233$	345405.	0.416811	0.208406	+	0.978042i	$0.433173\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	512333.	0.592491
$238$	0	0
$239$	−1.59920e6	−1.81095	−0.905476	−	0.424398i	$-0.860486\pi$
$239$	−1.59920e6	−1.81095	−0.905476	+	0.424398i	$0.860486\pi$
$240$	0	0
$241$	−42246.4	−0.0468541	−0.0234270	−	0.999726i	$-0.507458\pi$
$241$	−42246.4	−0.0468541	−0.0234270	+	0.999726i	$0.507458\pi$
$242$	0	0
$243$	816688.	0.887239
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1.67537e6	−1.74731
$248$	0	0
$249$	320882.	0.327980
$250$	0	0
$251$	−802841.	−0.804351	−0.402175	−	0.915563i	$-0.631746\pi$
$251$	−802841.	−0.804351	−0.402175	+	0.915563i	$0.631746\pi$
$252$	0	0
$253$	650354.	0.638776
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−941153.	−0.888848	−0.444424	−	0.895817i	$-0.646592\pi$
$257$	−941153.	−0.888848	−0.444424	+	0.895817i	$0.646592\pi$
$258$	0	0
$259$	1.58993e6	1.47275
$260$	0	0
$261$	763449.	0.693711
$262$	0	0
$263$	−1.53130e6	−1.36512	−0.682560	−	0.730830i	$-0.739133\pi$
$263$	−1.53130e6	−1.36512	−0.682560	+	0.730830i	$0.739133\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−76801.7	−0.0659315
$268$	0	0
$269$	1.25712e6	1.05924	0.529621	−	0.848234i	$-0.322334\pi$
$269$	1.25712e6	1.05924	0.529621	+	0.848234i	$0.322334\pi$
$270$	0	0
$271$	−843976.	−0.698083	−0.349041	−	0.937107i	$-0.613493\pi$
$271$	−843976.	−0.698083	−0.349041	+	0.937107i	$0.613493\pi$
$272$	0	0
$273$	1.10122e6	0.894265
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1.52070e6	1.19082	0.595408	−	0.803423i	$-0.296990\pi$
$277$	1.52070e6	1.19082	0.595408	+	0.803423i	$0.296990\pi$
$278$	0	0
$279$	−750612.	−0.577305
$280$	0	0
$281$	−1.97975e6	−1.49570	−0.747848	−	0.663870i	$-0.768913\pi$
$281$	−1.97975e6	−1.49570	−0.747848	+	0.663870i	$0.768913\pi$
$282$	0	0
$283$	−2.02901e6	−1.50597	−0.752986	−	0.658036i	$-0.771387\pi$
$283$	−2.02901e6	−1.50597	−0.752986	+	0.658036i	$0.771387\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.32563e6	0.949987
$288$	0	0
$289$	−1.41898e6	−0.999384
$290$	0	0
$291$	−607412.	−0.420485
$292$	0	0
$293$	2.23140e6	1.51848	0.759238	−	0.650813i	$-0.225572\pi$
$293$	2.23140e6	1.51848	0.759238	+	0.650813i	$0.225572\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−1.25602e6	−0.826236
$298$	0	0
$299$	1.37553e6	0.889802
$300$	0	0
$301$	−4.28847e6	−2.72826
$302$	0	0
$303$	1.08959e6	0.681799
$304$	0	0
$305$	0	0
$306$	0	0
$307$	334318.	0.202448	0.101224	−	0.994864i	$-0.467724\pi$
$307$	334318.	0.202448	0.101224	+	0.994864i	$0.467724\pi$
$308$	0	0
$309$	−739943.	−0.440861
$310$	0	0
$311$	−2.39090e6	−1.40172	−0.700858	−	0.713300i	$-0.747200\pi$
$311$	−2.39090e6	−1.40172	−0.700858	+	0.713300i	$0.747200\pi$
$312$	0	0
$313$	−1.88563e6	−1.08791	−0.543957	−	0.839113i	$-0.683075\pi$
$313$	−1.88563e6	−1.08791	−0.543957	+	0.839113i	$0.683075\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	179636.	0.100403	0.0502014	−	0.998739i	$-0.484014\pi$
$317$	179636.	0.100403	0.0502014	+	0.998739i	$0.484014\pi$
$318$	0	0
$319$	−1.80002e6	−0.990378
$320$	0	0
$321$	−218422.	−0.118313
$322$	0	0
$323$	46696.7	0.0249046
$324$	0	0
$325$	0	0
$326$	0	0
$327$	508882.	0.263177
$328$	0	0
$329$	661763.	0.337064
$330$	0	0
$331$	−168620.	−0.0845937	−0.0422968	−	0.999105i	$-0.513468\pi$
$331$	−168620.	−0.0845937	−0.0422968	+	0.999105i	$0.513468\pi$
$332$	0	0
$333$	−1.79096e6	−0.885064
$334$	0	0
$335$	0	0
$336$	0	0
$337$	1.74240e6	0.835744	0.417872	−	0.908506i	$-0.362776\pi$
$337$	1.74240e6	0.835744	0.417872	+	0.908506i	$0.362776\pi$
$338$	0	0
$339$	−367407.	−0.173640
$340$	0	0
$341$	1.76976e6	0.824190
$342$	0	0
$343$	395842.	0.181672
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−2.89447e6	−1.29046	−0.645232	−	0.763987i	$-0.723239\pi$
$347$	−2.89447e6	−1.29046	−0.645232	+	0.763987i	$0.723239\pi$
$348$	0	0
$349$	63803.8	0.0280403	0.0140202	−	0.999902i	$-0.495537\pi$
$349$	63803.8	0.0280403	0.0140202	+	0.999902i	$0.495537\pi$
$350$	0	0
$351$	−2.65654e6	−1.15093
$352$	0	0
$353$	−2.79206e6	−1.19258	−0.596290	−	0.802769i	$-0.703359\pi$
$353$	−2.79206e6	−1.19258	−0.596290	+	0.802769i	$0.703359\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−30693.6	−0.0127461
$358$	0	0
$359$	−1.48391e6	−0.607674	−0.303837	−	0.952724i	$-0.598268\pi$
$359$	−1.48391e6	−0.607674	−0.303837	+	0.952724i	$0.598268\pi$
$360$	0	0
$361$	15032.0	0.00607083
$362$	0	0
$363$	498619.	0.198610
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−1.08188e6	−0.419289	−0.209644	−	0.977778i	$-0.567231\pi$
$367$	−1.08188e6	−0.419289	−0.209644	+	0.977778i	$0.567231\pi$
$368$	0	0
$369$	−1.49324e6	−0.570904
$370$	0	0
$371$	5.16053e6	1.94652
$372$	0	0
$373$	−127750.	−0.0475434	−0.0237717	−	0.999717i	$-0.507567\pi$
$373$	−127750.	−0.0475434	−0.0237717	+	0.999717i	$0.507567\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−3.80714e6	−1.37958
$378$	0	0
$379$	1.61295e6	0.576798	0.288399	−	0.957510i	$-0.406877\pi$
$379$	1.61295e6	0.576798	0.288399	+	0.957510i	$0.406877\pi$
$380$	0	0
$381$	323904.	0.114315
$382$	0	0
$383$	3.96656e6	1.38171	0.690855	−	0.722993i	$-0.257234\pi$
$383$	3.96656e6	1.38171	0.690855	+	0.722993i	$0.257234\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4.83068e6	1.63957
$388$	0	0
$389$	−3.54852e6	−1.18898	−0.594488	−	0.804104i	$-0.702645\pi$
$389$	−3.54852e6	−1.18898	−0.594488	+	0.804104i	$0.702645\pi$
$390$	0	0
$391$	−38339.5	−0.0126825
$392$	0	0
$393$	1.14950e6	0.375428
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−580854.	−0.184965	−0.0924827	−	0.995714i	$-0.529480\pi$
$397$	−580854.	−0.184965	−0.0924827	+	0.995714i	$0.529480\pi$
$398$	0	0
$399$	−1.63741e6	−0.514903
$400$	0	0
$401$	2.10413e6	0.653449	0.326725	−	0.945120i	$-0.394055\pi$
$401$	2.10413e6	0.653449	0.326725	+	0.945120i	$0.394055\pi$
$402$	0	0
$403$	3.74313e6	1.14808
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4.22263e6	1.26356
$408$	0	0
$409$	3.23687e6	0.956790	0.478395	−	0.878145i	$-0.341219\pi$
$409$	3.23687e6	0.956790	0.478395	+	0.878145i	$0.341219\pi$
$410$	0	0
$411$	1.79211e6	0.523312
$412$	0	0
$413$	−1.49764e6	−0.432049
$414$	0	0
$415$	0	0
$416$	0	0
$417$	1.28273e6	0.361239
$418$	0	0
$419$	2.03059e6	0.565049	0.282525	−	0.959260i	$-0.408828\pi$
$419$	2.03059e6	0.565049	0.282525	+	0.959260i	$0.408828\pi$
$420$	0	0
$421$	2.15883e6	0.593625	0.296813	−	0.954936i	$-0.404076\pi$
$421$	2.15883e6	0.593625	0.296813	+	0.954936i	$0.404076\pi$
$422$	0	0
$423$	−745434.	−0.202562
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−1.32665e6	−0.352118
$428$	0	0
$429$	2.92467e6	0.767245
$430$	0	0
$431$	−5.49812e6	−1.42568	−0.712838	−	0.701328i	$-0.752591\pi$
$431$	−5.49812e6	−1.42568	−0.712838	+	0.701328i	$0.752591\pi$
$432$	0	0
$433$	3.87201e6	0.992470	0.496235	−	0.868188i	$-0.334716\pi$
$433$	3.87201e6	0.992470	0.496235	+	0.868188i	$0.334716\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2.04530e6	−0.512334
$438$	0	0
$439$	−3.15465e6	−0.781250	−0.390625	−	0.920550i	$-0.627741\pi$
$439$	−3.15465e6	−0.781250	−0.390625	+	0.920550i	$0.627741\pi$
$440$	0	0
$441$	−4.02343e6	−0.985144
$442$	0	0
$443$	4.07021e6	0.985389	0.492694	−	0.870202i	$-0.336012\pi$
$443$	4.07021e6	0.985389	0.492694	+	0.870202i	$0.336012\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	797893.	0.188876
$448$	0	0
$449$	957718.	0.224193	0.112096	−	0.993697i	$-0.464243\pi$
$449$	957718.	0.224193	0.112096	+	0.993697i	$0.464243\pi$
$450$	0	0
$451$	3.52068e6	0.815052
$452$	0	0
$453$	−1.57751e6	−0.361183
$454$	0	0
$455$	0	0
$456$	0	0
$457$	7.38167e6	1.65335	0.826673	−	0.562682i	$-0.190231\pi$
$457$	7.38167e6	1.65335	0.826673	+	0.562682i	$0.190231\pi$
$458$	0	0
$459$	74044.2	0.0164044
$460$	0	0
$461$	−2.96862e6	−0.650582	−0.325291	−	0.945614i	$-0.605462\pi$
$461$	−2.96862e6	−0.650582	−0.325291	+	0.945614i	$0.605462\pi$
$462$	0	0
$463$	−6.84593e6	−1.48416	−0.742079	−	0.670313i	$-0.766160\pi$
$463$	−6.84593e6	−1.48416	−0.742079	+	0.670313i	$0.766160\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−4.20816e6	−0.892895	−0.446447	−	0.894810i	$-0.647311\pi$
$467$	−4.20816e6	−0.892895	−0.446447	+	0.894810i	$0.647311\pi$
$468$	0	0
$469$	−3.33172e6	−0.699417
$470$	0	0
$471$	−234018.	−0.0486068
$472$	0	0
$473$	−1.13895e7	−2.34074
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−5.81301e6	−1.16978
$478$	0	0
$479$	5.73077e6	1.14123	0.570617	−	0.821217i	$-0.306704\pi$
$479$	5.73077e6	1.14123	0.570617	+	0.821217i	$0.306704\pi$
$480$	0	0
$481$	8.93110e6	1.76012
$482$	0	0
$483$	1.34437e6	0.262211
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−6.63249e6	−1.26723	−0.633613	−	0.773650i	$-0.718429\pi$
$487$	−6.63249e6	−1.26723	−0.633613	+	0.773650i	$0.718429\pi$
$488$	0	0
$489$	1.21658e6	0.230074
$490$	0	0
$491$	3.26463e6	0.611125	0.305563	−	0.952172i	$-0.401156\pi$
$491$	3.26463e6	0.611125	0.305563	+	0.952172i	$0.401156\pi$
$492$	0	0
$493$	106114.	0.0196633
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−2.53839e6	−0.460965
$498$	0	0
$499$	8.51527e6	1.53090	0.765450	−	0.643495i	$-0.222516\pi$
$499$	8.51527e6	1.53090	0.765450	+	0.643495i	$0.222516\pi$
$500$	0	0
$501$	1.83824e6	0.327196
$502$	0	0
$503$	−3.24174e6	−0.571292	−0.285646	−	0.958335i	$-0.592208\pi$
$503$	−3.24174e6	−0.571292	−0.285646	+	0.958335i	$0.592208\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	4.14744e6	0.716573
$508$	0	0
$509$	1.04569e7	1.78900	0.894499	−	0.447070i	$-0.147533\pi$
$509$	1.04569e7	1.78900	0.894499	+	0.447070i	$0.147533\pi$
$510$	0	0
$511$	7.54865e6	1.27884
$512$	0	0
$513$	3.95004e6	0.662687
$514$	0	0
$515$	0	0
$516$	0	0
$517$	1.75755e6	0.289188
$518$	0	0
$519$	−1.47393e6	−0.240191
$520$	0	0
$521$	−328142.	−0.0529625	−0.0264812	−	0.999649i	$-0.508430\pi$
$521$	−328142.	−0.0529625	−0.0264812	+	0.999649i	$0.508430\pi$
$522$	0	0
$523$	−9.42102e6	−1.50606	−0.753032	−	0.657984i	$-0.771410\pi$
$523$	−9.42102e6	−1.50606	−0.753032	+	0.657984i	$0.771410\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−104330.	−0.0163637
$528$	0	0
$529$	−4.75709e6	−0.739098
$530$	0	0
$531$	1.68700e6	0.259644
$532$	0	0
$533$	7.44643e6	1.13535
$534$	0	0
$535$	0	0
$536$	0	0
$537$	1.47160e6	0.220219
$538$	0	0
$539$	9.48624e6	1.40644
$540$	0	0
$541$	6.08041e6	0.893182	0.446591	−	0.894738i	$-0.352638\pi$
$541$	6.08041e6	0.893182	0.446591	+	0.894738i	$0.352638\pi$
$542$	0	0
$543$	−497205.	−0.0723662
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−4.32182e6	−0.617587	−0.308794	−	0.951129i	$-0.599925\pi$
$547$	−4.32182e6	−0.617587	−0.308794	+	0.951129i	$0.599925\pi$
$548$	0	0
$549$	1.49439e6	0.211609
$550$	0	0
$551$	5.66089e6	0.794338
$552$	0	0
$553$	1.76347e7	2.45220
$554$	0	0
$555$	0	0
$556$	0	0
$557$	7.41019e6	1.01203	0.506013	−	0.862526i	$-0.331119\pi$
$557$	7.41019e6	1.01203	0.506013	+	0.862526i	$0.331119\pi$
$558$	0	0
$559$	−2.40895e7	−3.26061
$560$	0	0
$561$	−81517.7	−0.0109356
$562$	0	0
$563$	−1.08039e7	−1.43652	−0.718260	−	0.695775i	$-0.755061\pi$
$563$	−1.08039e7	−1.43652	−0.718260	+	0.695775i	$0.755061\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	7.17799e6	0.937660
$568$	0	0
$569$	1.77404e6	0.229712	0.114856	−	0.993382i	$-0.463359\pi$
$569$	1.77404e6	0.229712	0.114856	+	0.993382i	$0.463359\pi$
$570$	0	0
$571$	70981.0	0.00911070	0.00455535	−	0.999990i	$-0.498550\pi$
$571$	70981.0	0.00911070	0.00455535	+	0.999990i	$0.498550\pi$
$572$	0	0
$573$	3.87992e6	0.493669
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−4.04410e6	−0.505688	−0.252844	−	0.967507i	$-0.581366\pi$
$577$	−4.04410e6	−0.505688	−0.252844	+	0.967507i	$0.581366\pi$
$578$	0	0
$579$	1.30113e6	0.161296
$580$	0	0
$581$	1.10449e7	1.35744
$582$	0	0
$583$	1.37056e7	1.67004
$584$	0	0
$585$	0	0
$586$	0	0
$587$	2.86862e6	0.343620	0.171810	−	0.985130i	$-0.445038\pi$
$587$	2.86862e6	0.343620	0.171810	+	0.985130i	$0.445038\pi$
$588$	0	0
$589$	−5.56570e6	−0.661046
$590$	0	0
$591$	−3.33737e6	−0.393040
$592$	0	0
$593$	−3.23131e6	−0.377347	−0.188674	−	0.982040i	$-0.560419\pi$
$593$	−3.23131e6	−0.377347	−0.188674	+	0.982040i	$0.560419\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−3.40962e6	−0.391534
$598$	0	0
$599$	−3.80078e6	−0.432819	−0.216409	−	0.976303i	$-0.569435\pi$
$599$	−3.80078e6	−0.432819	−0.216409	+	0.976303i	$0.569435\pi$
$600$	0	0
$601$	5.45289e6	0.615801	0.307900	−	0.951419i	$-0.400374\pi$
$601$	5.45289e6	0.615801	0.307900	+	0.951419i	$0.400374\pi$
$602$	0	0
$603$	3.75297e6	0.420321
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−1.50483e7	−1.65774	−0.828870	−	0.559441i	$-0.811016\pi$
$607$	−1.50483e7	−1.65774	−0.828870	+	0.559441i	$0.811016\pi$
$608$	0	0
$609$	−3.72088e6	−0.406539
$610$	0	0
$611$	3.71731e6	0.402833
$612$	0	0
$613$	−2.30782e6	−0.248056	−0.124028	−	0.992279i	$-0.539581\pi$
$613$	−2.30782e6	−0.248056	−0.124028	+	0.992279i	$0.539581\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	381761.	0.0403718	0.0201859	−	0.999796i	$-0.493574\pi$
$617$	381761.	0.0403718	0.0201859	+	0.999796i	$0.493574\pi$
$618$	0	0
$619$	1.07208e7	1.12460	0.562302	−	0.826932i	$-0.309916\pi$
$619$	1.07208e7	1.12460	0.562302	+	0.826932i	$0.309916\pi$
$620$	0	0
$621$	−3.24311e6	−0.337468
$622$	0	0
$623$	−2.64354e6	−0.272876
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−4.34873e6	−0.441767
$628$	0	0
$629$	−248931.	−0.0250872
$630$	0	0
$631$	4.56189e6	0.456112	0.228056	−	0.973648i	$-0.426763\pi$
$631$	4.56189e6	0.456112	0.228056	+	0.973648i	$0.426763\pi$
$632$	0	0
$633$	−5.82910e6	−0.578219
$634$	0	0
$635$	0	0
$636$	0	0
$637$	2.00639e7	1.95915
$638$	0	0
$639$	2.85933e6	0.277021
$640$	0	0
$641$	−2.70983e6	−0.260494	−0.130247	−	0.991482i	$-0.541577\pi$
$641$	−2.70983e6	−0.260494	−0.130247	+	0.991482i	$0.541577\pi$
$642$	0	0
$643$	−6.10481e6	−0.582297	−0.291149	−	0.956678i	$-0.594037\pi$
$643$	−6.10481e6	−0.582297	−0.291149	+	0.956678i	$0.594037\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1.22081e7	1.14654	0.573269	−	0.819367i	$-0.305675\pi$
$647$	1.22081e7	1.14654	0.573269	+	0.819367i	$0.305675\pi$
$648$	0	0
$649$	−3.97752e6	−0.370681
$650$	0	0
$651$	3.65832e6	0.338321
$652$	0	0
$653$	6.26514e6	0.574973	0.287487	−	0.957785i	$-0.407180\pi$
$653$	6.26514e6	0.574973	0.287487	+	0.957785i	$0.407180\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−8.50307e6	−0.768532
$658$	0	0
$659$	1.85107e7	1.66039	0.830194	−	0.557474i	$-0.188229\pi$
$659$	1.85107e7	1.66039	0.830194	+	0.557474i	$0.188229\pi$
$660$	0	0
$661$	−8.46041e6	−0.753161	−0.376581	−	0.926384i	$-0.622900\pi$
$661$	−8.46041e6	−0.753161	−0.376581	+	0.926384i	$0.622900\pi$
$662$	0	0
$663$	−172414.	−0.0152331
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−4.64777e6	−0.404511
$668$	0	0
$669$	−1.32156e6	−0.114162
$670$	0	0
$671$	−3.52340e6	−0.302104
$672$	0	0
$673$	−7.25182e6	−0.617177	−0.308588	−	0.951196i	$-0.599857\pi$
$673$	−7.25182e6	−0.617177	−0.308588	+	0.951196i	$0.599857\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−4.74893e6	−0.398221	−0.199110	−	0.979977i	$-0.563805\pi$
$677$	−4.74893e6	−0.398221	−0.199110	+	0.979977i	$0.563805\pi$
$678$	0	0
$679$	−2.09073e7	−1.74030
$680$	0	0
$681$	1.11169e6	0.0918578
$682$	0	0
$683$	1.37712e7	1.12959	0.564793	−	0.825233i	$-0.308956\pi$
$683$	1.37712e7	1.12959	0.564793	+	0.825233i	$0.308956\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−1.56069e6	−0.126161
$688$	0	0
$689$	2.89882e7	2.32634
$690$	0	0
$691$	−1.64305e7	−1.30905	−0.654526	−	0.756040i	$-0.727132\pi$
$691$	−1.64305e7	−1.30905	−0.654526	+	0.756040i	$0.727132\pi$
$692$	0	0
$693$	−2.01871e7	−1.59676
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−207550.	−0.0161823
$698$	0	0
$699$	1.89628e6	0.146794
$700$	0	0
$701$	−1.05170e7	−0.808342	−0.404171	−	0.914683i	$-0.632440\pi$
$701$	−1.05170e7	−0.808342	−0.404171	+	0.914683i	$0.632440\pi$
$702$	0	0
$703$	−1.32797e7	−1.01345
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3.75041e7	2.82182
$708$	0	0
$709$	−1.32691e6	−0.0991349	−0.0495674	−	0.998771i	$-0.515784\pi$
$709$	−1.32691e6	−0.0991349	−0.0495674	+	0.998771i	$0.515784\pi$
$710$	0	0
$711$	−1.98643e7	−1.47367
$712$	0	0
$713$	4.56962e6	0.336633
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−8.77958e6	−0.637787
$718$	0	0
$719$	−4.65753e6	−0.335995	−0.167998	−	0.985787i	$-0.553730\pi$
$719$	−4.65753e6	−0.335995	−0.167998	+	0.985787i	$0.553730\pi$
$720$	0	0
$721$	−2.54691e7	−1.82463
$722$	0	0
$723$	−231933.	−0.0165012
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−1.47884e6	−0.103773	−0.0518867	−	0.998653i	$-0.516523\pi$
$727$	−1.47884e6	−0.103773	−0.0518867	+	0.998653i	$0.516523\pi$
$728$	0	0
$729$	−4.74681e6	−0.330814
$730$	0	0
$731$	671432.	0.0464739
$732$	0	0
$733$	−1.41118e7	−0.970115	−0.485057	−	0.874482i	$-0.661201\pi$
$733$	−1.41118e7	−0.970115	−0.485057	+	0.874482i	$0.661201\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−8.84856e6	−0.600073
$738$	0	0
$739$	5.94045e6	0.400137	0.200068	−	0.979782i	$-0.435884\pi$
$739$	5.94045e6	0.400137	0.200068	+	0.979782i	$0.435884\pi$
$740$	0	0
$741$	−9.19780e6	−0.615373
$742$	0	0
$743$	1.46589e7	0.974156	0.487078	−	0.873359i	$-0.338063\pi$
$743$	1.46589e7	0.974156	0.487078	+	0.873359i	$0.338063\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−1.24413e7	−0.815766
$748$	0	0
$749$	−7.51817e6	−0.489674
$750$	0	0
$751$	6.08263e6	0.393543	0.196771	−	0.980449i	$-0.436954\pi$
$751$	6.08263e6	0.393543	0.196771	+	0.980449i	$0.436954\pi$
$752$	0	0
$753$	−4.40760e6	−0.283279
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−8.11757e6	−0.514856	−0.257428	−	0.966297i	$-0.582875\pi$
$757$	−8.11757e6	−0.514856	−0.257428	+	0.966297i	$0.582875\pi$
$758$	0	0
$759$	3.57044e6	0.224966
$760$	0	0
$761$	1.63567e7	1.02384	0.511921	−	0.859033i	$-0.328934\pi$
$761$	1.63567e7	1.02384	0.511921	+	0.859033i	$0.328934\pi$
$762$	0	0
$763$	1.75159e7	1.08923
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−8.41267e6	−0.516352
$768$	0	0
$769$	−2.09586e7	−1.27805	−0.639024	−	0.769187i	$-0.720661\pi$
$769$	−2.09586e7	−1.27805	−0.639024	+	0.769187i	$0.720661\pi$
$770$	0	0
$771$	−5.16693e6	−0.313038
$772$	0	0
$773$	8.95651e6	0.539126	0.269563	−	0.962983i	$-0.413121\pi$
$773$	8.95651e6	0.539126	0.269563	+	0.962983i	$0.413121\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	8.72873e6	0.518679
$778$	0	0
$779$	−1.10722e7	−0.653717
$780$	0	0
$781$	−6.74160e6	−0.395490
$782$	0	0
$783$	8.97614e6	0.523221
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−2.42009e7	−1.39282	−0.696411	−	0.717643i	$-0.745221\pi$
$787$	−2.42009e7	−1.39282	−0.696411	+	0.717643i	$0.745221\pi$
$788$	0	0
$789$	−8.40683e6	−0.480773
$790$	0	0
$791$	−1.26463e7	−0.718657
$792$	0	0
$793$	−7.45219e6	−0.420824
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−9.91659e6	−0.552989	−0.276494	−	0.961016i	$-0.589173\pi$
$797$	−9.91659e6	−0.552989	−0.276494	+	0.961016i	$0.589173\pi$
$798$	0	0
$799$	−103610.	−0.00574164
$800$	0	0
$801$	2.97778e6	0.163988
$802$	0	0
$803$	2.00481e7	1.09720
$804$	0	0
$805$	0	0
$806$	0	0
$807$	6.90158e6	0.373048
$808$	0	0
$809$	−9.15657e6	−0.491883	−0.245941	−	0.969285i	$-0.579097\pi$
$809$	−9.15657e6	−0.491883	−0.245941	+	0.969285i	$0.579097\pi$
$810$	0	0
$811$	−1.49145e7	−0.796264	−0.398132	−	0.917328i	$-0.630341\pi$
$811$	−1.49145e7	−0.796264	−0.398132	+	0.917328i	$0.630341\pi$
$812$	0	0
$813$	−4.63343e6	−0.245853
$814$	0	0
$815$	0	0
$816$	0	0
$817$	3.58189e7	1.87740
$818$	0	0
$819$	−4.26967e7	−2.22426
$820$	0	0
$821$	9.39549e6	0.486476	0.243238	−	0.969967i	$-0.421790\pi$
$821$	9.39549e6	0.486476	0.243238	+	0.969967i	$0.421790\pi$
$822$	0	0
$823$	1.59508e7	0.820884	0.410442	−	0.911887i	$-0.365374\pi$
$823$	1.59508e7	0.820884	0.410442	+	0.911887i	$0.365374\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.68518e7	0.856805	0.428403	−	0.903588i	$-0.359077\pi$
$827$	1.68518e7	0.856805	0.428403	+	0.903588i	$0.359077\pi$
$828$	0	0
$829$	−3.92620e6	−0.198420	−0.0992101	−	0.995067i	$-0.531632\pi$
$829$	−3.92620e6	−0.198420	−0.0992101	+	0.995067i	$0.531632\pi$
$830$	0	0
$831$	8.34865e6	0.419386
$832$	0	0
$833$	−559229.	−0.0279240
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−8.82521e6	−0.435423
$838$	0	0
$839$	1.23626e7	0.606322	0.303161	−	0.952939i	$-0.401958\pi$
$839$	1.23626e7	0.606322	0.303161	+	0.952939i	$0.401958\pi$
$840$	0	0
$841$	−7.64726e6	−0.372834
$842$	0	0
$843$	−1.08688e7	−0.526760
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1.71626e7	0.822006
$848$	0	0
$849$	−1.11392e7	−0.530379
$850$	0	0
$851$	1.09031e7	0.516091
$852$	0	0
$853$	3.17612e6	0.149460	0.0747299	−	0.997204i	$-0.476191\pi$
$853$	3.17612e6	0.149460	0.0747299	+	0.997204i	$0.476191\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1.97233e7	−0.917334	−0.458667	−	0.888608i	$-0.651673\pi$
$857$	−1.97233e7	−0.917334	−0.458667	+	0.888608i	$0.651673\pi$
$858$	0	0
$859$	−4.18456e6	−0.193494	−0.0967470	−	0.995309i	$-0.530844\pi$
$859$	−4.18456e6	−0.193494	−0.0967470	+	0.995309i	$0.530844\pi$
$860$	0	0
$861$	7.27771e6	0.334570
$862$	0	0
$863$	−2.89684e7	−1.32403	−0.662015	−	0.749490i	$-0.730299\pi$
$863$	−2.89684e7	−1.32403	−0.662015	+	0.749490i	$0.730299\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−7.79021e6	−0.351966
$868$	0	0
$869$	4.68352e7	2.10389
$870$	0	0
$871$	−1.87152e7	−0.835889
$872$	0	0
$873$	2.35508e7	1.04585
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−2.78801e7	−1.22404	−0.612019	−	0.790843i	$-0.709642\pi$
$877$	−2.78801e7	−1.22404	−0.612019	+	0.790843i	$0.709642\pi$
$878$	0	0
$879$	1.22504e7	0.534783
$880$	0	0
$881$	−4.21471e7	−1.82948	−0.914741	−	0.404040i	$-0.867606\pi$
$881$	−4.21471e7	−1.82948	−0.914741	+	0.404040i	$0.867606\pi$
$882$	0	0
$883$	−4.38615e7	−1.89314	−0.946568	−	0.322505i	$-0.895475\pi$
$883$	−4.38615e7	−1.89314	−0.946568	+	0.322505i	$0.895475\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−2.32386e7	−0.991747	−0.495874	−	0.868395i	$-0.665152\pi$
$887$	−2.32386e7	−0.991747	−0.495874	+	0.868395i	$0.665152\pi$
$888$	0	0
$889$	1.11489e7	0.473126
$890$	0	0
$891$	1.90637e7	0.804476
$892$	0	0
$893$	−5.52731e6	−0.231945
$894$	0	0
$895$	0	0
$896$	0	0
$897$	7.55168e6	0.313374
$898$	0	0
$899$	−1.26476e7	−0.521925
$900$	0	0
$901$	−807970.	−0.0331576
$902$	0	0
$903$	−2.35437e7	−0.960848
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−7.74916e6	−0.312778	−0.156389	−	0.987696i	$-0.549985\pi$
$907$	−7.74916e6	−0.312778	−0.156389	+	0.987696i	$0.549985\pi$
$908$	0	0
$909$	−4.22459e7	−1.69580
$910$	0	0
$911$	−1.54221e7	−0.615668	−0.307834	−	0.951440i	$-0.599604\pi$
$911$	−1.54221e7	−0.615668	−0.307834	+	0.951440i	$0.599604\pi$
$912$	0	0
$913$	2.93336e7	1.16463
$914$	0	0
$915$	0	0
$916$	0	0
$917$	3.95661e7	1.55382
$918$	0	0
$919$	2.40744e7	0.940302	0.470151	−	0.882586i	$-0.344199\pi$
$919$	2.40744e7	0.940302	0.470151	+	0.882586i	$0.344199\pi$
$920$	0	0
$921$	1.83540e6	0.0712989
$922$	0	0
$923$	−1.42588e7	−0.550909
$924$	0	0
$925$	0	0
$926$	0	0
$927$	2.86893e7	1.09653
$928$	0	0
$929$	−1.51619e7	−0.576387	−0.288193	−	0.957572i	$-0.593055\pi$
$929$	−1.51619e7	−0.576387	−0.288193	+	0.957572i	$0.593055\pi$
$930$	0	0
$931$	−2.98333e7	−1.12804
$932$	0	0
$933$	−1.31260e7	−0.493662
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−4.90390e6	−0.182470	−0.0912352	−	0.995829i	$-0.529082\pi$
$937$	−4.90390e6	−0.182470	−0.0912352	+	0.995829i	$0.529082\pi$
$938$	0	0
$939$	−1.03521e7	−0.383146
$940$	0	0
$941$	−1.62296e7	−0.597495	−0.298748	−	0.954332i	$-0.596569\pi$
$941$	−1.62296e7	−0.597495	−0.298748	+	0.954332i	$0.596569\pi$
$942$	0	0
$943$	9.09062e6	0.332900
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−4.16817e7	−1.51033	−0.755163	−	0.655537i	$-0.772442\pi$
$947$	−4.16817e7	−1.51033	−0.755163	+	0.655537i	$0.772442\pi$
$948$	0	0
$949$	4.24028e7	1.52837
$950$	0	0
$951$	986202.	0.0353602
$952$	0	0
$953$	−1.61135e7	−0.574723	−0.287362	−	0.957822i	$-0.592778\pi$
$953$	−1.61135e7	−0.574723	−0.287362	+	0.957822i	$0.592778\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−9.88212e6	−0.348795
$958$	0	0
$959$	6.16851e7	2.16588
$960$	0	0
$961$	−1.61942e7	−0.565655
$962$	0	0
$963$	8.46873e6	0.294274
$964$	0	0
$965$	0	0
$966$	0	0
$967$	1.51707e7	0.521722	0.260861	−	0.965376i	$-0.415994\pi$
$967$	1.51707e7	0.521722	0.260861	+	0.965376i	$0.415994\pi$
$968$	0	0
$969$	256365.	0.00877099
$970$	0	0
$971$	2.82884e7	0.962854	0.481427	−	0.876486i	$-0.340119\pi$
$971$	2.82884e7	0.962854	0.481427	+	0.876486i	$0.340119\pi$
$972$	0	0
$973$	4.41519e7	1.49509
$974$	0	0
$975$	0	0
$976$	0	0
$977$	2.88218e7	0.966017	0.483009	−	0.875616i	$-0.339544\pi$
$977$	2.88218e7	0.966017	0.483009	+	0.875616i	$0.339544\pi$
$978$	0	0
$979$	−7.02086e6	−0.234117
$980$	0	0
$981$	−1.97305e7	−0.654586
$982$	0	0
$983$	2.26448e6	0.0747453	0.0373727	−	0.999301i	$-0.488101\pi$
$983$	2.26448e6	0.0747453	0.0373727	+	0.999301i	$0.488101\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	3.63308e6	0.118709
$988$	0	0
$989$	−2.94085e7	−0.956053
$990$	0	0
$991$	−995592.	−0.0322031	−0.0161015	−	0.999870i	$-0.505125\pi$
$991$	−995592.	−0.0322031	−0.0161015	+	0.999870i	$0.505125\pi$
$992$	0	0
$993$	−925721.	−0.0297925
$994$	0	0
$995$	0	0
$996$	0	0
$997$	5.57133e7	1.77509	0.887547	−	0.460718i	$-0.152408\pi$
$997$	5.57133e7	1.77509	0.887547	+	0.460718i	$0.152408\pi$
$998$	0	0
$999$	−2.10569e7	−0.667546

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.6.a.z.1.3		4
4.3	odd	2		200.6.a.k.1.2		4
5.2	odd	4		80.6.c.d.49.3		8
5.3	odd	4		80.6.c.d.49.6		8
5.4	even	2		400.6.a.ba.1.2		4
15.2	even	4		720.6.f.n.289.8		8
15.8	even	4		720.6.f.n.289.7		8
20.3	even	4		40.6.c.a.9.3	✓	8
20.7	even	4		40.6.c.a.9.6	yes	8
20.19	odd	2		200.6.a.j.1.3		4
40.3	even	4		320.6.c.j.129.6		8
40.13	odd	4		320.6.c.i.129.3		8
40.27	even	4		320.6.c.j.129.3		8
40.37	odd	4		320.6.c.i.129.6		8
60.23	odd	4		360.6.f.b.289.7		8
60.47	odd	4		360.6.f.b.289.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.6.c.a.9.3	✓	8	20.3	even	4
40.6.c.a.9.6	yes	8	20.7	even	4
80.6.c.d.49.3		8	5.2	odd	4
80.6.c.d.49.6		8	5.3	odd	4
200.6.a.j.1.3		4	20.19	odd	2
200.6.a.k.1.2		4	4.3	odd	2
320.6.c.i.129.3		8	40.13	odd	4
320.6.c.i.129.6		8	40.37	odd	4
320.6.c.j.129.3		8	40.27	even	4
320.6.c.j.129.6		8	40.3	even	4
360.6.f.b.289.7		8	60.23	odd	4
360.6.f.b.289.8		8	60.47	odd	4
400.6.a.z.1.3		4	1.1	even	1	trivial
400.6.a.ba.1.2		4	5.4	even	2
720.6.f.n.289.7		8	15.8	even	4
720.6.f.n.289.8		8	15.2	even	4

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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	400.6.a.z.1.3

Level	$400$
Weight	$6$
Character	400.1
Self dual	yes
Analytic conductor	$64.154$
Analytic rank	$0$
Dimension	$4$
CM	no
Inner twists	$1$