LMFDB - Embedded newform 784.6.a.z.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [784,6,Mod(1,784)] 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("784.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(784, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	784.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,0,0,762] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	yes
Analytic conductor:	$125.740914733$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{39}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 39 $ x^2 - 39
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 49)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-6.24500$ of defining polynomial
Character	$\chi$	$=$	784.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-24.9800 q^{3} +74.9400 q^{5} +381.000 q^{9} +284.000 q^{11} +524.580 q^{13} -1872.00 q^{15} -149.880 q^{17} -2173.26 q^{19} -1496.00 q^{23} +2491.00 q^{25} -3447.24 q^{27} -4366.00 q^{29} +6444.84 q^{31} -7094.32 q^{33} -12630.0 q^{37} -13104.0 q^{39} -9442.44 q^{41} +1356.00 q^{43} +28552.1 q^{45} -10042.0 q^{47} +3744.00 q^{51} +14150.0 q^{53} +21283.0 q^{55} +54288.0 q^{57} +37395.0 q^{59} -35596.5 q^{61} +39312.0 q^{65} +3644.00 q^{67} +37370.1 q^{69} -35632.0 q^{71} +40767.3 q^{73} -62225.2 q^{75} +54616.0 q^{79} -6471.00 q^{81} +524.580 q^{83} -11232.0 q^{85} +109063. q^{87} -20383.7 q^{89} -160992. q^{93} -162864. q^{95} +183603. q^{97} +108204. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 762 q^{9} + 568 q^{11} - 3744 q^{15} - 2992 q^{23} + 4982 q^{25} - 8732 q^{29} - 25260 q^{37} - 26208 q^{39} + 2712 q^{43} + 7488 q^{51} + 28300 q^{53} + 108576 q^{57} + 78624 q^{65} + 7288 q^{67}+ \cdots + 216408 q^{99}+O(q^{100}) $ 2 * q + 762 * q^9 + 568 * q^11 - 3744 * q^15 - 2992 * q^23 + 4982 * q^25 - 8732 * q^29 - 25260 * q^37 - 26208 * q^39 + 2712 * q^43 + 7488 * q^51 + 28300 * q^53 + 108576 * q^57 + 78624 * q^65 + 7288 * q^67 - 71264 * q^71 + 109232 * q^79 - 12942 * q^81 - 22464 * q^85 - 321984 * q^93 - 325728 * q^95 + 216408 * 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
$3$	−24.9800	−1.60247	−0.801234	−	0.598352i	$-0.795823\pi$
$3$	−24.9800	−1.60247	−0.801234	+	0.598352i	$0.795823\pi$
$4$	0	0
$5$	74.9400	1.34057	0.670284	−	0.742105i	$-0.266172\pi$
$5$	74.9400	1.34057	0.670284	+	0.742105i	$0.266172\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	381.000	1.56790
$10$	0	0
$11$	284.000	0.707680	0.353840	−	0.935306i	$-0.384876\pi$
$11$	284.000	0.707680	0.353840	+	0.935306i	$0.384876\pi$
$12$	0	0
$13$	524.580	0.860901	0.430450	−	0.902614i	$-0.358355\pi$
$13$	524.580	0.860901	0.430450	+	0.902614i	$0.358355\pi$
$14$	0	0
$15$	−1872.00	−2.14821
$16$	0	0
$17$	−149.880	−0.125783	−0.0628914	−	0.998020i	$-0.520032\pi$
$17$	−149.880	−0.125783	−0.0628914	+	0.998020i	$0.520032\pi$
$18$	0	0
$19$	−2173.26	−1.38111	−0.690554	−	0.723281i	$-0.742633\pi$
$19$	−2173.26	−1.38111	−0.690554	+	0.723281i	$0.742633\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1496.00	−0.589674	−0.294837	−	0.955548i	$-0.595265\pi$
$23$	−1496.00	−0.589674	−0.294837	+	0.955548i	$0.595265\pi$
$24$	0	0
$25$	2491.00	0.797120
$26$	0	0
$27$	−3447.24	−0.910043
$28$	0	0
$29$	−4366.00	−0.964026	−0.482013	−	0.876164i	$-0.660094\pi$
$29$	−4366.00	−0.964026	−0.482013	+	0.876164i	$0.660094\pi$
$30$	0	0
$31$	6444.84	1.20450	0.602251	−	0.798307i	$-0.294271\pi$
$31$	6444.84	1.20450	0.602251	+	0.798307i	$0.294271\pi$
$32$	0	0
$33$	−7094.32	−1.13403
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−12630.0	−1.51670	−0.758349	−	0.651849i	$-0.773994\pi$
$37$	−12630.0	−1.51670	−0.758349	+	0.651849i	$0.773994\pi$
$38$	0	0
$39$	−13104.0	−1.37957
$40$	0	0
$41$	−9442.44	−0.877252	−0.438626	−	0.898670i	$-0.644535\pi$
$41$	−9442.44	−0.877252	−0.438626	+	0.898670i	$0.644535\pi$
$42$	0	0
$43$	1356.00	0.111838	0.0559189	−	0.998435i	$-0.482191\pi$
$43$	1356.00	0.111838	0.0559189	+	0.998435i	$0.482191\pi$
$44$	0	0
$45$	28552.1	2.10188
$46$	0	0
$47$	−10042.0	−0.663092	−0.331546	−	0.943439i	$-0.607570\pi$
$47$	−10042.0	−0.663092	−0.331546	+	0.943439i	$0.607570\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	3744.00	0.201563
$52$	0	0
$53$	14150.0	0.691937	0.345969	−	0.938246i	$-0.387550\pi$
$53$	14150.0	0.691937	0.345969	+	0.938246i	$0.387550\pi$
$54$	0	0
$55$	21283.0	0.948692
$56$	0	0
$57$	54288.0	2.21318
$58$	0	0
$59$	37395.0	1.39857	0.699285	−	0.714843i	$-0.253502\pi$
$59$	37395.0	1.39857	0.699285	+	0.714843i	$0.253502\pi$
$60$	0	0
$61$	−35596.5	−1.22485	−0.612425	−	0.790529i	$-0.709806\pi$
$61$	−35596.5	−1.22485	−0.612425	+	0.790529i	$0.709806\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	39312.0	1.15410
$66$	0	0
$67$	3644.00	0.0991725	0.0495863	−	0.998770i	$-0.484210\pi$
$67$	3644.00	0.0991725	0.0495863	+	0.998770i	$0.484210\pi$
$68$	0	0
$69$	37370.1	0.944933
$70$	0	0
$71$	−35632.0	−0.838869	−0.419435	−	0.907786i	$-0.637772\pi$
$71$	−35632.0	−0.838869	−0.419435	+	0.907786i	$0.637772\pi$
$72$	0	0
$73$	40767.3	0.895376	0.447688	−	0.894190i	$-0.352248\pi$
$73$	40767.3	0.895376	0.447688	+	0.894190i	$0.352248\pi$
$74$	0	0
$75$	−62225.2	−1.27736
$76$	0	0
$77$	0	0
$78$	0	0
$79$	54616.0	0.984583	0.492291	−	0.870431i	$-0.336159\pi$
$79$	54616.0	0.984583	0.492291	+	0.870431i	$0.336159\pi$
$80$	0	0
$81$	−6471.00	−0.109587
$82$	0	0
$83$	524.580	0.00835827	0.00417913	−	0.999991i	$-0.498670\pi$
$83$	524.580	0.00835827	0.00417913	+	0.999991i	$0.498670\pi$
$84$	0	0
$85$	−11232.0	−0.168620
$86$	0	0
$87$	109063.	1.54482
$88$	0	0
$89$	−20383.7	−0.272777	−0.136388	−	0.990655i	$-0.543550\pi$
$89$	−20383.7	−0.272777	−0.136388	+	0.990655i	$0.543550\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−160992.	−1.93018
$94$	0	0
$95$	−162864.	−1.85147
$96$	0	0
$97$	183603.	1.98130	0.990650	−	0.136427i	$-0.0435619\pi$
$97$	183603.	1.98130	0.990650	+	0.136427i	$0.0435619\pi$
$98$	0	0
$99$	108204.	1.10957
$100$	0	0
$101$	75914.2	0.740491	0.370245	−	0.928934i	$-0.379274\pi$
$101$	75914.2	0.740491	0.370245	+	0.928934i	$0.379274\pi$
$102$	0	0
$103$	10941.2	0.101619	0.0508093	−	0.998708i	$-0.483820\pi$
$103$	10941.2	0.101619	0.0508093	+	0.998708i	$0.483820\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−218188.	−1.84235	−0.921173	−	0.389152i	$-0.872768\pi$
$107$	−218188.	−1.84235	−0.921173	+	0.389152i	$0.872768\pi$
$108$	0	0
$109$	−96030.0	−0.774178	−0.387089	−	0.922042i	$-0.626519\pi$
$109$	−96030.0	−0.774178	−0.387089	+	0.922042i	$0.626519\pi$
$110$	0	0
$111$	315497.	2.43046
$112$	0	0
$113$	−137422.	−1.01242	−0.506209	−	0.862411i	$-0.668954\pi$
$113$	−137422.	−1.01242	−0.506209	+	0.862411i	$0.668954\pi$
$114$	0	0
$115$	−112110.	−0.790498
$116$	0	0
$117$	199865.	1.34981
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−80395.0	−0.499190
$122$	0	0
$123$	235872.	1.40577
$124$	0	0
$125$	−47511.9	−0.271974
$126$	0	0
$127$	−170368.	−0.937300	−0.468650	−	0.883384i	$-0.655259\pi$
$127$	−170368.	−0.937300	−0.468650	+	0.883384i	$0.655259\pi$
$128$	0	0
$129$	−33872.9	−0.179216
$130$	0	0
$131$	348246.	1.77300	0.886498	−	0.462732i	$-0.153131\pi$
$131$	348246.	1.77300	0.886498	+	0.462732i	$0.153131\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−258336.	−1.21997
$136$	0	0
$137$	−75562.0	−0.343955	−0.171978	−	0.985101i	$-0.555016\pi$
$137$	−75562.0	−0.343955	−0.171978	+	0.985101i	$0.555016\pi$
$138$	0	0
$139$	−97047.3	−0.426036	−0.213018	−	0.977048i	$-0.568329\pi$
$139$	−97047.3	−0.426036	−0.213018	+	0.977048i	$0.568329\pi$
$140$	0	0
$141$	250848.	1.06258
$142$	0	0
$143$	148981.	0.609242
$144$	0	0
$145$	−327188.	−1.29234
$146$	0	0
$147$	0	0
$148$	0	0
$149$	361030.	1.33223	0.666113	−	0.745851i	$-0.267957\pi$
$149$	361030.	1.33223	0.666113	+	0.745851i	$0.267957\pi$
$150$	0	0
$151$	−32280.0	−0.115210	−0.0576051	−	0.998339i	$-0.518346\pi$
$151$	−32280.0	−0.115210	−0.0576051	+	0.998339i	$0.518346\pi$
$152$	0	0
$153$	−57104.3	−0.197215
$154$	0	0
$155$	482976.	1.61472
$156$	0	0
$157$	−132869.	−0.430203	−0.215101	−	0.976592i	$-0.569008\pi$
$157$	−132869.	−0.430203	−0.215101	+	0.976592i	$0.569008\pi$
$158$	0	0
$159$	−353467.	−1.10881
$160$	0	0
$161$	0	0
$162$	0	0
$163$	61364.0	0.180903	0.0904513	−	0.995901i	$-0.471169\pi$
$163$	61364.0	0.180903	0.0904513	+	0.995901i	$0.471169\pi$
$164$	0	0
$165$	−531648.	−1.52025
$166$	0	0
$167$	−380845.	−1.05671	−0.528356	−	0.849023i	$-0.677192\pi$
$167$	−380845.	−1.05671	−0.528356	+	0.849023i	$0.677192\pi$
$168$	0	0
$169$	−96109.0	−0.258849
$170$	0	0
$171$	−828012.	−2.16544
$172$	0	0
$173$	517311.	1.31412	0.657062	−	0.753837i	$-0.271799\pi$
$173$	517311.	1.31412	0.657062	+	0.753837i	$0.271799\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−934128.	−2.24116
$178$	0	0
$179$	−610564.	−1.42429	−0.712145	−	0.702032i	$-0.752276\pi$
$179$	−610564.	−1.42429	−0.712145	+	0.702032i	$0.752276\pi$
$180$	0	0
$181$	−433828.	−0.984285	−0.492142	−	0.870515i	$-0.663786\pi$
$181$	−433828.	−0.984285	−0.492142	+	0.870515i	$0.663786\pi$
$182$	0	0
$183$	889200.	1.96278
$184$	0	0
$185$	−946492.	−2.03324
$186$	0	0
$187$	−42565.9	−0.0890139
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−341192.	−0.676730	−0.338365	−	0.941015i	$-0.609874\pi$
$191$	−341192.	−0.676730	−0.338365	+	0.941015i	$0.609874\pi$
$192$	0	0
$193$	−616158.	−1.19069	−0.595345	−	0.803470i	$-0.702985\pi$
$193$	−616158.	−1.19069	−0.595345	+	0.803470i	$0.702985\pi$
$194$	0	0
$195$	−982013.	−1.84940
$196$	0	0
$197$	231478.	0.424956	0.212478	−	0.977166i	$-0.431847\pi$
$197$	231478.	0.424956	0.212478	+	0.977166i	$0.431847\pi$
$198$	0	0
$199$	405126.	0.725199	0.362599	−	0.931945i	$-0.381889\pi$
$199$	405126.	0.725199	0.362599	+	0.931945i	$0.381889\pi$
$200$	0	0
$201$	−91027.1	−0.158921
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−707616.	−1.17602
$206$	0	0
$207$	−569976.	−0.924551
$208$	0	0
$209$	−617206.	−0.977382
$210$	0	0
$211$	−776820.	−1.20120	−0.600599	−	0.799551i	$-0.705071\pi$
$211$	−776820.	−1.20120	−0.600599	+	0.799551i	$0.705071\pi$
$212$	0	0
$213$	890087.	1.34426
$214$	0	0
$215$	101619.	0.149926
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−1.01837e6	−1.43481
$220$	0	0
$221$	−78624.0	−0.108287
$222$	0	0
$223$	−81834.5	−0.110198	−0.0550990	−	0.998481i	$-0.517547\pi$
$223$	−81834.5	−0.110198	−0.0550990	+	0.998481i	$0.517547\pi$
$224$	0	0
$225$	949071.	1.24981
$226$	0	0
$227$	−753671.	−0.970772	−0.485386	−	0.874300i	$-0.661321\pi$
$227$	−753671.	−0.970772	−0.485386	+	0.874300i	$0.661321\pi$
$228$	0	0
$229$	26303.9	0.0331461	0.0165730	−	0.999863i	$-0.494724\pi$
$229$	26303.9	0.0331461	0.0165730	+	0.999863i	$0.494724\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	47542.0	0.0573704	0.0286852	−	0.999588i	$-0.490868\pi$
$233$	47542.0	0.0573704	0.0286852	+	0.999588i	$0.490868\pi$
$234$	0	0
$235$	−752544.	−0.888919
$236$	0	0
$237$	−1.36431e6	−1.57776
$238$	0	0
$239$	−1.08899e6	−1.23319	−0.616595	−	0.787281i	$-0.711488\pi$
$239$	−1.08899e6	−1.23319	−0.616595	+	0.787281i	$0.711488\pi$
$240$	0	0
$241$	−1.34937e6	−1.49654	−0.748270	−	0.663395i	$-0.769115\pi$
$241$	−1.34937e6	−1.49654	−0.748270	+	0.663395i	$0.769115\pi$
$242$	0	0
$243$	999325.	1.08565
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1.14005e6	−1.18900
$248$	0	0
$249$	−13104.0	−0.0133939
$250$	0	0
$251$	−630020.	−0.631205	−0.315602	−	0.948892i	$-0.602207\pi$
$251$	−630020.	−0.631205	−0.315602	+	0.948892i	$0.602207\pi$
$252$	0	0
$253$	−424864.	−0.417300
$254$	0	0
$255$	280575.	0.270208
$256$	0	0
$257$	−1.93405e6	−1.82656	−0.913282	−	0.407327i	$-0.866461\pi$
$257$	−1.93405e6	−1.82656	−0.913282	+	0.407327i	$0.866461\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−1.66345e6	−1.51150
$262$	0	0
$263$	1.11712e6	0.995888	0.497944	−	0.867209i	$-0.334089\pi$
$263$	1.11712e6	0.995888	0.497944	+	0.867209i	$0.334089\pi$
$264$	0	0
$265$	1.06040e6	0.927588
$266$	0	0
$267$	509184.	0.437116
$268$	0	0
$269$	1.52061e6	1.28126	0.640629	−	0.767851i	$-0.278674\pi$
$269$	1.52061e6	1.28126	0.640629	+	0.767851i	$0.278674\pi$
$270$	0	0
$271$	−1.02038e6	−0.843995	−0.421997	−	0.906597i	$-0.638671\pi$
$271$	−1.02038e6	−0.843995	−0.421997	+	0.906597i	$0.638671\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	707444.	0.564105
$276$	0	0
$277$	1.89642e6	1.48503	0.742516	−	0.669829i	$-0.233633\pi$
$277$	1.89642e6	1.48503	0.742516	+	0.669829i	$0.233633\pi$
$278$	0	0
$279$	2.45548e6	1.88854
$280$	0	0
$281$	1.31911e6	0.996587	0.498293	−	0.867008i	$-0.333960\pi$
$281$	1.31911e6	0.996587	0.498293	+	0.867008i	$0.333960\pi$
$282$	0	0
$283$	−478792.	−0.355370	−0.177685	−	0.984087i	$-0.556861\pi$
$283$	−478792.	−0.355370	−0.177685	+	0.984087i	$0.556861\pi$
$284$	0	0
$285$	4.06834e6	2.96692
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−1.39739e6	−0.984179
$290$	0	0
$291$	−4.58640e6	−3.17497
$292$	0	0
$293$	1.50187e6	1.02203	0.511015	−	0.859572i	$-0.329270\pi$
$293$	1.50187e6	1.02203	0.511015	+	0.859572i	$0.329270\pi$
$294$	0	0
$295$	2.80238e6	1.87488
$296$	0	0
$297$	−979016.	−0.644019
$298$	0	0
$299$	−784771.	−0.507651
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−1.89634e6	−1.18661
$304$	0	0
$305$	−2.66760e6	−1.64199
$306$	0	0
$307$	1.19657e6	0.724588	0.362294	−	0.932064i	$-0.381994\pi$
$307$	1.19657e6	0.724588	0.362294	+	0.932064i	$0.381994\pi$
$308$	0	0
$309$	−273312.	−0.162841
$310$	0	0
$311$	1.31475e6	0.770799	0.385400	−	0.922750i	$-0.374064\pi$
$311$	1.31475e6	0.770799	0.385400	+	0.922750i	$0.374064\pi$
$312$	0	0
$313$	−2.65273e6	−1.53049	−0.765247	−	0.643737i	$-0.777383\pi$
$313$	−2.65273e6	−1.53049	−0.765247	+	0.643737i	$0.777383\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1.59992e6	−0.894233	−0.447116	−	0.894476i	$-0.647549\pi$
$317$	−1.59992e6	−0.894233	−0.447116	+	0.894476i	$0.647549\pi$
$318$	0	0
$319$	−1.23994e6	−0.682221
$320$	0	0
$321$	5.45033e6	2.95230
$322$	0	0
$323$	325728.	0.173720
$324$	0	0
$325$	1.30673e6	0.686241
$326$	0	0
$327$	2.39883e6	1.24059
$328$	0	0
$329$	0	0
$330$	0	0
$331$	111708.	0.0560421	0.0280210	−	0.999607i	$-0.491079\pi$
$331$	111708.	0.0560421	0.0280210	+	0.999607i	$0.491079\pi$
$332$	0	0
$333$	−4.81203e6	−2.37803
$334$	0	0
$335$	273081.	0.132947
$336$	0	0
$337$	1.59301e6	0.764087	0.382043	−	0.924144i	$-0.375220\pi$
$337$	1.59301e6	0.764087	0.382043	+	0.924144i	$0.375220\pi$
$338$	0	0
$339$	3.43280e6	1.62237
$340$	0	0
$341$	1.83033e6	0.852402
$342$	0	0
$343$	0	0
$344$	0	0
$345$	2.80051e6	1.26675
$346$	0	0
$347$	3.33676e6	1.48765	0.743827	−	0.668372i	$-0.233009\pi$
$347$	3.33676e6	1.48765	0.743827	+	0.668372i	$0.233009\pi$
$348$	0	0
$349$	−1.60259e6	−0.704303	−0.352151	−	0.935943i	$-0.614550\pi$
$349$	−1.60259e6	−0.704303	−0.352151	+	0.935943i	$0.614550\pi$
$350$	0	0
$351$	−1.80835e6	−0.783457
$352$	0	0
$353$	1.80965e6	0.772962	0.386481	−	0.922297i	$-0.373691\pi$
$353$	1.80965e6	0.772962	0.386481	+	0.922297i	$0.373691\pi$
$354$	0	0
$355$	−2.67026e6	−1.12456
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−920792.	−0.377073	−0.188536	−	0.982066i	$-0.560374\pi$
$359$	−920792.	−0.377073	−0.188536	+	0.982066i	$0.560374\pi$
$360$	0	0
$361$	2.24696e6	0.907458
$362$	0	0
$363$	2.00827e6	0.799935
$364$	0	0
$365$	3.05510e6	1.20031
$366$	0	0
$367$	−1.34802e6	−0.522434	−0.261217	−	0.965280i	$-0.584124\pi$
$367$	−1.34802e6	−0.522434	−0.261217	+	0.965280i	$0.584124\pi$
$368$	0	0
$369$	−3.59757e6	−1.37544
$370$	0	0
$371$	0	0
$372$	0	0
$373$	3.13773e6	1.16773	0.583867	−	0.811849i	$-0.301539\pi$
$373$	3.13773e6	1.16773	0.583867	+	0.811849i	$0.301539\pi$
$374$	0	0
$375$	1.18685e6	0.435830
$376$	0	0
$377$	−2.29032e6	−0.829931
$378$	0	0
$379$	1.83188e6	0.655088	0.327544	−	0.944836i	$-0.393779\pi$
$379$	1.83188e6	0.655088	0.327544	+	0.944836i	$0.393779\pi$
$380$	0	0
$381$	4.25579e6	1.50199
$382$	0	0
$383$	27727.8	0.00965869	0.00482935	−	0.999988i	$-0.498463\pi$
$383$	27727.8	0.00965869	0.00482935	+	0.999988i	$0.498463\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	516636.	0.175351
$388$	0	0
$389$	−548342.	−0.183729	−0.0918645	−	0.995772i	$-0.529283\pi$
$389$	−548342.	−0.183729	−0.0918645	+	0.995772i	$0.529283\pi$
$390$	0	0
$391$	224220.	0.0741708
$392$	0	0
$393$	−8.69918e6	−2.84117
$394$	0	0
$395$	4.09292e6	1.31990
$396$	0	0
$397$	−403852.	−0.128601	−0.0643007	−	0.997931i	$-0.520482\pi$
$397$	−403852.	−0.128601	−0.0643007	+	0.997931i	$0.520482\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−4.16427e6	−1.29324	−0.646619	−	0.762813i	$-0.723818\pi$
$401$	−4.16427e6	−1.29324	−0.646619	+	0.762813i	$0.723818\pi$
$402$	0	0
$403$	3.38083e6	1.03696
$404$	0	0
$405$	−484937.	−0.146909
$406$	0	0
$407$	−3.58692e6	−1.07334
$408$	0	0
$409$	−2.54121e6	−0.751161	−0.375581	−	0.926790i	$-0.622557\pi$
$409$	−2.54121e6	−0.751161	−0.375581	+	0.926790i	$0.622557\pi$
$410$	0	0
$411$	1.88754e6	0.551177
$412$	0	0
$413$	0	0
$414$	0	0
$415$	39312.0	0.0112048
$416$	0	0
$417$	2.42424e6	0.682709
$418$	0	0
$419$	2.30133e6	0.640389	0.320195	−	0.947352i	$-0.396252\pi$
$419$	2.30133e6	0.640389	0.320195	+	0.947352i	$0.396252\pi$
$420$	0	0
$421$	−2.79991e6	−0.769909	−0.384955	−	0.922936i	$-0.625783\pi$
$421$	−2.79991e6	−0.769909	−0.384955	+	0.922936i	$0.625783\pi$
$422$	0	0
$423$	−3.82599e6	−1.03966
$424$	0	0
$425$	−373351.	−0.100264
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−3.72154e6	−0.976290
$430$	0	0
$431$	−954320.	−0.247458	−0.123729	−	0.992316i	$-0.539485\pi$
$431$	−954320.	−0.247458	−0.123729	+	0.992316i	$0.539485\pi$
$432$	0	0
$433$	519334.	0.133115	0.0665575	−	0.997783i	$-0.478798\pi$
$433$	519334.	0.133115	0.0665575	+	0.997783i	$0.478798\pi$
$434$	0	0
$435$	8.17315e6	2.07093
$436$	0	0
$437$	3.25120e6	0.814403
$438$	0	0
$439$	−5.98081e6	−1.48115	−0.740574	−	0.671974i	$-0.765446\pi$
$439$	−5.98081e6	−1.48115	−0.740574	+	0.671974i	$0.765446\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−3.74820e6	−0.907432	−0.453716	−	0.891146i	$-0.649902\pi$
$443$	−3.74820e6	−0.907432	−0.453716	+	0.891146i	$0.649902\pi$
$444$	0	0
$445$	−1.52755e6	−0.365676
$446$	0	0
$447$	−9.01853e6	−2.13485
$448$	0	0
$449$	99458.0	0.0232822	0.0116411	−	0.999932i	$-0.496294\pi$
$449$	99458.0	0.0232822	0.0116411	+	0.999932i	$0.496294\pi$
$450$	0	0
$451$	−2.68165e6	−0.620813
$452$	0	0
$453$	806354.	0.184621
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−161814.	−0.0362431	−0.0181216	−	0.999836i	$-0.505769\pi$
$457$	−161814.	−0.0362431	−0.0181216	+	0.999836i	$0.505769\pi$
$458$	0	0
$459$	516672.	0.114468
$460$	0	0
$461$	−4.49198e6	−0.984431	−0.492215	−	0.870473i	$-0.663813\pi$
$461$	−4.49198e6	−0.984431	−0.492215	+	0.870473i	$0.663813\pi$
$462$	0	0
$463$	−3.59382e6	−0.779118	−0.389559	−	0.921001i	$-0.627373\pi$
$463$	−3.59382e6	−0.779118	−0.389559	+	0.921001i	$0.627373\pi$
$464$	0	0
$465$	−1.20647e7	−2.58753
$466$	0	0
$467$	−2.05223e6	−0.435446	−0.217723	−	0.976011i	$-0.569863\pi$
$467$	−2.05223e6	−0.435446	−0.217723	+	0.976011i	$0.569863\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	3.31906e6	0.689386
$472$	0	0
$473$	385104.	0.0791453
$474$	0	0
$475$	−5.41359e6	−1.10091
$476$	0	0
$477$	5.39115e6	1.08489
$478$	0	0
$479$	1.99985e6	0.398252	0.199126	−	0.979974i	$-0.436190\pi$
$479$	1.99985e6	0.398252	0.199126	+	0.979974i	$0.436190\pi$
$480$	0	0
$481$	−6.62544e6	−1.30573
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1.37592e7	2.65607
$486$	0	0
$487$	−2.17126e6	−0.414848	−0.207424	−	0.978251i	$-0.566508\pi$
$487$	−2.17126e6	−0.414848	−0.207424	+	0.978251i	$0.566508\pi$
$488$	0	0
$489$	−1.53287e6	−0.289890
$490$	0	0
$491$	−3.04555e6	−0.570114	−0.285057	−	0.958511i	$-0.592013\pi$
$491$	−3.04555e6	−0.570114	−0.285057	+	0.958511i	$0.592013\pi$
$492$	0	0
$493$	654376.	0.121258
$494$	0	0
$495$	8.10881e6	1.48746
$496$	0	0
$497$	0	0
$498$	0	0
$499$	7.99225e6	1.43687	0.718436	−	0.695594i	$-0.244859\pi$
$499$	7.99225e6	1.43687	0.718436	+	0.695594i	$0.244859\pi$
$500$	0	0
$501$	9.51350e6	1.69335
$502$	0	0
$503$	−9.47811e6	−1.67033	−0.835164	−	0.550001i	$-0.814627\pi$
$503$	−9.47811e6	−1.67033	−0.835164	+	0.550001i	$0.814627\pi$
$504$	0	0
$505$	5.68901e6	0.992677
$506$	0	0
$507$	2.40080e6	0.414798
$508$	0	0
$509$	1.01003e7	1.72799	0.863995	−	0.503500i	$-0.167955\pi$
$509$	1.01003e7	1.72799	0.863995	+	0.503500i	$0.167955\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	7.49174e6	1.25687
$514$	0	0
$515$	819936.	0.136227
$516$	0	0
$517$	−2.85192e6	−0.469257
$518$	0	0
$519$	−1.29224e7	−2.10584
$520$	0	0
$521$	5.11975e6	0.826332	0.413166	−	0.910656i	$-0.364423\pi$
$521$	5.11975e6	0.826332	0.413166	+	0.910656i	$0.364423\pi$
$522$	0	0
$523$	−3.51476e6	−0.561877	−0.280939	−	0.959726i	$-0.590646\pi$
$523$	−3.51476e6	−0.561877	−0.280939	+	0.959726i	$0.590646\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−965952.	−0.151506
$528$	0	0
$529$	−4.19833e6	−0.652285
$530$	0	0
$531$	1.42475e7	2.19282
$532$	0	0
$533$	−4.95331e6	−0.755227
$534$	0	0
$535$	−1.63510e7	−2.46979
$536$	0	0
$537$	1.52519e7	2.28238
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−3.47547e6	−0.510530	−0.255265	−	0.966871i	$-0.582163\pi$
$541$	−3.47547e6	−0.510530	−0.255265	+	0.966871i	$0.582163\pi$
$542$	0	0
$543$	1.08370e7	1.57728
$544$	0	0
$545$	−7.19649e6	−1.03784
$546$	0	0
$547$	7.85765e6	1.12286	0.561429	−	0.827525i	$-0.310252\pi$
$547$	7.85765e6	1.12286	0.561429	+	0.827525i	$0.310252\pi$
$548$	0	0
$549$	−1.35623e7	−1.92044
$550$	0	0
$551$	9.48845e6	1.33142
$552$	0	0
$553$	0	0
$554$	0	0
$555$	2.36434e7	3.25819
$556$	0	0
$557$	9.06537e6	1.23808	0.619039	−	0.785361i	$-0.287522\pi$
$557$	9.06537e6	1.23808	0.619039	+	0.785361i	$0.287522\pi$
$558$	0	0
$559$	711330.	0.0962813
$560$	0	0
$561$	1.06330e6	0.142642
$562$	0	0
$563$	−5.80957e6	−0.772455	−0.386227	−	0.922404i	$-0.626222\pi$
$563$	−5.80957e6	−0.772455	−0.386227	+	0.922404i	$0.626222\pi$
$564$	0	0
$565$	−1.02984e7	−1.35722
$566$	0	0
$567$	0	0
$568$	0	0
$569$	1.05559e7	1.36684	0.683418	−	0.730027i	$-0.260493\pi$
$569$	1.05559e7	1.36684	0.683418	+	0.730027i	$0.260493\pi$
$570$	0	0
$571$	−7.99584e6	−1.02630	−0.513150	−	0.858299i	$-0.671521\pi$
$571$	−7.99584e6	−1.02630	−0.513150	+	0.858299i	$0.671521\pi$
$572$	0	0
$573$	8.52297e6	1.08444
$574$	0	0
$575$	−3.72654e6	−0.470041
$576$	0	0
$577$	−2.60941e6	−0.326289	−0.163145	−	0.986602i	$-0.552164\pi$
$577$	−2.60941e6	−0.326289	−0.163145	+	0.986602i	$0.552164\pi$
$578$	0	0
$579$	1.53916e7	1.90804
$580$	0	0
$581$	0	0
$582$	0	0
$583$	4.01860e6	0.489670
$584$	0	0
$585$	1.49779e7	1.80951
$586$	0	0
$587$	−4.41749e6	−0.529151	−0.264576	−	0.964365i	$-0.585232\pi$
$587$	−4.41749e6	−0.529151	−0.264576	+	0.964365i	$0.585232\pi$
$588$	0	0
$589$	−1.40063e7	−1.66355
$590$	0	0
$591$	−5.78232e6	−0.680978
$592$	0	0
$593$	−5.54106e6	−0.647077	−0.323539	−	0.946215i	$-0.604873\pi$
$593$	−5.54106e6	−0.647077	−0.323539	+	0.946215i	$0.604873\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−1.01200e7	−1.16211
$598$	0	0
$599$	−5.08611e6	−0.579187	−0.289594	−	0.957150i	$-0.593520\pi$
$599$	−5.08611e6	−0.579187	−0.289594	+	0.957150i	$0.593520\pi$
$600$	0	0
$601$	2.41307e6	0.272511	0.136255	−	0.990674i	$-0.456493\pi$
$601$	2.41307e6	0.272511	0.136255	+	0.990674i	$0.456493\pi$
$602$	0	0
$603$	1.38836e6	0.155493
$604$	0	0
$605$	−6.02480e6	−0.669197
$606$	0	0
$607$	−7.62170e6	−0.839614	−0.419807	−	0.907613i	$-0.637902\pi$
$607$	−7.62170e6	−0.839614	−0.419807	+	0.907613i	$0.637902\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−5.26781e6	−0.570856
$612$	0	0
$613$	3.60126e6	0.387082	0.193541	−	0.981092i	$-0.438003\pi$
$613$	3.60126e6	0.387082	0.193541	+	0.981092i	$0.438003\pi$
$614$	0	0
$615$	1.76762e7	1.88453
$616$	0	0
$617$	7.22901e6	0.764480	0.382240	−	0.924063i	$-0.375153\pi$
$617$	7.22901e6	0.764480	0.382240	+	0.924063i	$0.375153\pi$
$618$	0	0
$619$	1.25832e7	1.31998	0.659988	−	0.751276i	$-0.270561\pi$
$619$	1.25832e7	1.31998	0.659988	+	0.751276i	$0.270561\pi$
$620$	0	0
$621$	5.15707e6	0.536629
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−1.13449e7	−1.16172
$626$	0	0
$627$	1.54178e7	1.56622
$628$	0	0
$629$	1.89298e6	0.190774
$630$	0	0
$631$	−5.98350e6	−0.598249	−0.299125	−	0.954214i	$-0.596695\pi$
$631$	−5.98350e6	−0.598249	−0.299125	+	0.954214i	$0.596695\pi$
$632$	0	0
$633$	1.94050e7	1.92488
$634$	0	0
$635$	−1.27674e7	−1.25651
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−1.35758e7	−1.31526
$640$	0	0
$641$	−7.74000e6	−0.744040	−0.372020	−	0.928225i	$-0.621335\pi$
$641$	−7.74000e6	−0.744040	−0.372020	+	0.928225i	$0.621335\pi$
$642$	0	0
$643$	−6.81377e6	−0.649920	−0.324960	−	0.945728i	$-0.605351\pi$
$643$	−6.81377e6	−0.649920	−0.324960	+	0.945728i	$0.605351\pi$
$644$	0	0
$645$	−2.53843e6	−0.240252
$646$	0	0
$647$	1.07121e7	1.00603	0.503017	−	0.864276i	$-0.332223\pi$
$647$	1.07121e7	1.00603	0.503017	+	0.864276i	$0.332223\pi$
$648$	0	0
$649$	1.06202e7	0.989739
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.34167e7	1.23129	0.615647	−	0.788022i	$-0.288895\pi$
$653$	1.34167e7	1.23129	0.615647	+	0.788022i	$0.288895\pi$
$654$	0	0
$655$	2.60976e7	2.37682
$656$	0	0
$657$	1.55324e7	1.40386
$658$	0	0
$659$	−1.38574e7	−1.24299	−0.621494	−	0.783419i	$-0.713474\pi$
$659$	−1.38574e7	−1.24299	−0.621494	+	0.783419i	$0.713474\pi$
$660$	0	0
$661$	−1.20734e7	−1.07479	−0.537396	−	0.843330i	$-0.680592\pi$
$661$	−1.20734e7	−1.07479	−0.537396	+	0.843330i	$0.680592\pi$
$662$	0	0
$663$	1.96403e6	0.173526
$664$	0	0
$665$	0	0
$666$	0	0
$667$	6.53154e6	0.568461
$668$	0	0
$669$	2.04422e6	0.176589
$670$	0	0
$671$	−1.01094e7	−0.866801
$672$	0	0
$673$	−5.32490e6	−0.453183	−0.226592	−	0.973990i	$-0.572758\pi$
$673$	−5.32490e6	−0.453183	−0.226592	+	0.973990i	$0.572758\pi$
$674$	0	0
$675$	−8.58707e6	−0.725414
$676$	0	0
$677$	−2.34518e7	−1.96655	−0.983274	−	0.182135i	$-0.941699\pi$
$677$	−2.34518e7	−1.96655	−0.983274	+	0.182135i	$0.941699\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	1.88267e7	1.55563
$682$	0	0
$683$	−1.82270e7	−1.49508	−0.747538	−	0.664219i	$-0.768764\pi$
$683$	−1.82270e7	−1.49508	−0.747538	+	0.664219i	$0.768764\pi$
$684$	0	0
$685$	−5.66261e6	−0.461095
$686$	0	0
$687$	−657072.	−0.0531155
$688$	0	0
$689$	7.42280e6	0.595690
$690$	0	0
$691$	−7.60858e6	−0.606190	−0.303095	−	0.952960i	$-0.598020\pi$
$691$	−7.60858e6	−0.606190	−0.303095	+	0.952960i	$0.598020\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−7.27272e6	−0.571130
$696$	0	0
$697$	1.41523e6	0.110343
$698$	0	0
$699$	−1.18760e6	−0.0919341
$700$	0	0
$701$	314162.	0.0241467	0.0120734	−	0.999927i	$-0.496157\pi$
$701$	314162.	0.0241467	0.0120734	+	0.999927i	$0.496157\pi$
$702$	0	0
$703$	2.74483e7	2.09472
$704$	0	0
$705$	1.87985e7	1.42446
$706$	0	0
$707$	0	0
$708$	0	0
$709$	2.48285e7	1.85496	0.927482	−	0.373869i	$-0.121969\pi$
$709$	2.48285e7	1.85496	0.927482	+	0.373869i	$0.121969\pi$
$710$	0	0
$711$	2.08087e7	1.54373
$712$	0	0
$713$	−9.64148e6	−0.710264
$714$	0	0
$715$	1.11646e7	0.816730
$716$	0	0
$717$	2.72030e7	1.97615
$718$	0	0
$719$	−1.71932e7	−1.24032	−0.620160	−	0.784475i	$-0.712932\pi$
$719$	−1.71932e7	−1.24032	−0.620160	+	0.784475i	$0.712932\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	3.37072e7	2.39816
$724$	0	0
$725$	−1.08757e7	−0.768444
$726$	0	0
$727$	2.12927e7	1.49415	0.747076	−	0.664739i	$-0.231457\pi$
$727$	2.12927e7	1.49415	0.747076	+	0.664739i	$0.231457\pi$
$728$	0	0
$729$	−2.33907e7	−1.63014
$730$	0	0
$731$	−203237.	−0.0140673
$732$	0	0
$733$	−1.96024e7	−1.34757	−0.673783	−	0.738930i	$-0.735332\pi$
$733$	−1.96024e7	−1.34757	−0.673783	+	0.738930i	$0.735332\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.03490e6	0.0701824
$738$	0	0
$739$	−1.44181e7	−0.971173	−0.485587	−	0.874189i	$-0.661394\pi$
$739$	−1.44181e7	−0.971173	−0.485587	+	0.874189i	$0.661394\pi$
$740$	0	0
$741$	2.84784e7	1.90533
$742$	0	0
$743$	−5.57521e6	−0.370501	−0.185250	−	0.982691i	$-0.559310\pi$
$743$	−5.57521e6	−0.370501	−0.185250	+	0.982691i	$0.559310\pi$
$744$	0	0
$745$	2.70556e7	1.78594
$746$	0	0
$747$	199865.	0.0131049
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−508800.	−0.0329190	−0.0164595	−	0.999865i	$-0.505239\pi$
$751$	−508800.	−0.0329190	−0.0164595	+	0.999865i	$0.505239\pi$
$752$	0	0
$753$	1.57379e7	1.01149
$754$	0	0
$755$	−2.41906e6	−0.154447
$756$	0	0
$757$	1.10466e7	0.700631	0.350316	−	0.936632i	$-0.386074\pi$
$757$	1.10466e7	0.700631	0.350316	+	0.936632i	$0.386074\pi$
$758$	0	0
$759$	1.06131e7	0.668710
$760$	0	0
$761$	−6.77173e6	−0.423875	−0.211937	−	0.977283i	$-0.567977\pi$
$761$	−6.77173e6	−0.423875	−0.211937	+	0.977283i	$0.567977\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−4.27939e6	−0.264380
$766$	0	0
$767$	1.96167e7	1.20403
$768$	0	0
$769$	−2.48053e7	−1.51261	−0.756307	−	0.654216i	$-0.772999\pi$
$769$	−2.48053e7	−1.51261	−0.756307	+	0.654216i	$0.772999\pi$
$770$	0	0
$771$	4.83126e7	2.92701
$772$	0	0
$773$	−1.62952e7	−0.980867	−0.490434	−	0.871479i	$-0.663162\pi$
$773$	−1.62952e7	−0.980867	−0.490434	+	0.871479i	$0.663162\pi$
$774$	0	0
$775$	1.60541e7	0.960133
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2.05209e7	1.21158
$780$	0	0
$781$	−1.01195e7	−0.593651
$782$	0	0
$783$	1.50506e7	0.877305
$784$	0	0
$785$	−9.95717e6	−0.576716
$786$	0	0
$787$	1.44214e7	0.829984	0.414992	−	0.909825i	$-0.363784\pi$
$787$	1.44214e7	0.829984	0.414992	+	0.909825i	$0.363784\pi$
$788$	0	0
$789$	−2.79056e7	−1.59588
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−1.86732e7	−1.05447
$794$	0	0
$795$	−2.64888e7	−1.48643
$796$	0	0
$797$	−1.10109e6	−0.0614014	−0.0307007	−	0.999529i	$-0.509774\pi$
$797$	−1.10109e6	−0.0614014	−0.0307007	+	0.999529i	$0.509774\pi$
$798$	0	0
$799$	1.50509e6	0.0834055
$800$	0	0
$801$	−7.76618e6	−0.427687
$802$	0	0
$803$	1.15779e7	0.633639
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−3.79848e7	−2.05317
$808$	0	0
$809$	−2.45146e7	−1.31690	−0.658450	−	0.752625i	$-0.728787\pi$
$809$	−2.45146e7	−1.31690	−0.658450	+	0.752625i	$0.728787\pi$
$810$	0	0
$811$	−3.03580e7	−1.62077	−0.810383	−	0.585900i	$-0.800741\pi$
$811$	−3.03580e7	−1.62077	−0.810383	+	0.585900i	$0.800741\pi$
$812$	0	0
$813$	2.54892e7	1.35247
$814$	0	0
$815$	4.59862e6	0.242512
$816$	0	0
$817$	−2.94694e6	−0.154460
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−2.54599e7	−1.31825	−0.659126	−	0.752032i	$-0.729074\pi$
$821$	−2.54599e7	−1.31825	−0.659126	+	0.752032i	$0.729074\pi$
$822$	0	0
$823$	−5.95232e6	−0.306328	−0.153164	−	0.988201i	$-0.548946\pi$
$823$	−5.95232e6	−0.306328	−0.153164	+	0.988201i	$0.548946\pi$
$824$	0	0
$825$	−1.76719e7	−0.903961
$826$	0	0
$827$	7.85900e6	0.399580	0.199790	−	0.979839i	$-0.435974\pi$
$827$	7.85900e6	0.399580	0.199790	+	0.979839i	$0.435974\pi$
$828$	0	0
$829$	1.12642e7	0.569262	0.284631	−	0.958637i	$-0.408129\pi$
$829$	1.12642e7	0.569262	0.284631	+	0.958637i	$0.408129\pi$
$830$	0	0
$831$	−4.73726e7	−2.37971
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−2.85405e7	−1.41659
$836$	0	0
$837$	−2.22169e7	−1.09615
$838$	0	0
$839$	7.80470e6	0.382782	0.191391	−	0.981514i	$-0.438700\pi$
$839$	7.80470e6	0.382782	0.191391	+	0.981514i	$0.438700\pi$
$840$	0	0
$841$	−1.44919e6	−0.0706539
$842$	0	0
$843$	−3.29514e7	−1.59700
$844$	0	0
$845$	−7.20241e6	−0.347005
$846$	0	0
$847$	0	0
$848$	0	0
$849$	1.19602e7	0.569468
$850$	0	0
$851$	1.88945e7	0.894357
$852$	0	0
$853$	1.50581e7	0.708593	0.354296	−	0.935133i	$-0.384720\pi$
$853$	1.50581e7	0.708593	0.354296	+	0.935133i	$0.384720\pi$
$854$	0	0
$855$	−6.20512e7	−2.90292
$856$	0	0
$857$	2.28736e7	1.06386	0.531928	−	0.846789i	$-0.321468\pi$
$857$	2.28736e7	1.06386	0.531928	+	0.846789i	$0.321468\pi$
$858$	0	0
$859$	−8.22863e6	−0.380491	−0.190246	−	0.981737i	$-0.560928\pi$
$859$	−8.22863e6	−0.380491	−0.190246	+	0.981737i	$0.560928\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−4.09902e6	−0.187350	−0.0936748	−	0.995603i	$-0.529861\pi$
$863$	−4.09902e6	−0.187350	−0.0936748	+	0.995603i	$0.529861\pi$
$864$	0	0
$865$	3.87672e7	1.76167
$866$	0	0
$867$	3.49069e7	1.57711
$868$	0	0
$869$	1.55109e7	0.696769
$870$	0	0
$871$	1.91157e6	0.0853777
$872$	0	0
$873$	6.99527e7	3.10648
$874$	0	0
$875$	0	0
$876$	0	0
$877$	787778.	0.0345864	0.0172932	−	0.999850i	$-0.494495\pi$
$877$	787778.	0.0345864	0.0172932	+	0.999850i	$0.494495\pi$
$878$	0	0
$879$	−3.75168e7	−1.63777
$880$	0	0
$881$	1.73321e7	0.752336	0.376168	−	0.926551i	$-0.377242\pi$
$881$	1.73321e7	0.752336	0.376168	+	0.926551i	$0.377242\pi$
$882$	0	0
$883$	−1.23991e7	−0.535164	−0.267582	−	0.963535i	$-0.586225\pi$
$883$	−1.23991e7	−0.535164	−0.267582	+	0.963535i	$0.586225\pi$
$884$	0	0
$885$	−7.00035e7	−3.00443
$886$	0	0
$887$	1.67555e7	0.715071	0.357535	−	0.933900i	$-0.383617\pi$
$887$	1.67555e7	0.715071	0.357535	+	0.933900i	$0.383617\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−1.83776e6	−0.0775524
$892$	0	0
$893$	2.18238e7	0.915801
$894$	0	0
$895$	−4.57557e7	−1.90936
$896$	0	0
$897$	1.96036e7	0.813494
$898$	0	0
$899$	−2.81382e7	−1.16117
$900$	0	0
$901$	−2.12080e6	−0.0870338
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−3.25110e7	−1.31950
$906$	0	0
$907$	3.96543e7	1.60056	0.800280	−	0.599627i	$-0.204684\pi$
$907$	3.96543e7	1.60056	0.800280	+	0.599627i	$0.204684\pi$
$908$	0	0
$909$	2.89233e7	1.16102
$910$	0	0
$911$	−2.99138e7	−1.19419	−0.597097	−	0.802169i	$-0.703679\pi$
$911$	−2.99138e7	−1.19419	−0.597097	+	0.802169i	$0.703679\pi$
$912$	0	0
$913$	148981.	0.00591498
$914$	0	0
$915$	6.66366e7	2.63124
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−1.98997e7	−0.777245	−0.388622	−	0.921397i	$-0.627049\pi$
$919$	−1.98997e7	−0.777245	−0.388622	+	0.921397i	$0.627049\pi$
$920$	0	0
$921$	−2.98902e7	−1.16113
$922$	0	0
$923$	−1.86918e7	−0.722183
$924$	0	0
$925$	−3.14613e7	−1.20899
$926$	0	0
$927$	4.16861e6	0.159328
$928$	0	0
$929$	1.19663e7	0.454904	0.227452	−	0.973789i	$-0.426961\pi$
$929$	1.19663e7	0.454904	0.227452	+	0.973789i	$0.426961\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−3.28424e7	−1.23518
$934$	0	0
$935$	−3.18989e6	−0.119329
$936$	0	0
$937$	−6.18165e6	−0.230015	−0.115007	−	0.993365i	$-0.536689\pi$
$937$	−6.18165e6	−0.230015	−0.115007	+	0.993365i	$0.536689\pi$
$938$	0	0
$939$	6.62651e7	2.45257
$940$	0	0
$941$	−2.99426e7	−1.10234	−0.551171	−	0.834393i	$-0.685819\pi$
$941$	−2.99426e7	−1.10234	−0.551171	+	0.834393i	$0.685819\pi$
$942$	0	0
$943$	1.41259e7	0.517293
$944$	0	0
$945$	0	0
$946$	0	0
$947$	4.84279e7	1.75477	0.877386	−	0.479785i	$-0.159286\pi$
$947$	4.84279e7	1.75477	0.877386	+	0.479785i	$0.159286\pi$
$948$	0	0
$949$	2.13857e7	0.770830
$950$	0	0
$951$	3.99660e7	1.43298
$952$	0	0
$953$	2.26780e7	0.808860	0.404430	−	0.914569i	$-0.367470\pi$
$953$	2.26780e7	0.808860	0.404430	+	0.914569i	$0.367470\pi$
$954$	0	0
$955$	−2.55689e7	−0.907202
$956$	0	0
$957$	3.09738e7	1.09324
$958$	0	0
$959$	0	0
$960$	0	0
$961$	1.29068e7	0.450827
$962$	0	0
$963$	−8.31296e7	−2.88862
$964$	0	0
$965$	−4.61749e7	−1.59620
$966$	0	0
$967$	3.60431e6	0.123953	0.0619764	−	0.998078i	$-0.480260\pi$
$967$	3.60431e6	0.123953	0.0619764	+	0.998078i	$0.480260\pi$
$968$	0	0
$969$	−8.13668e6	−0.278380
$970$	0	0
$971$	−2.90807e7	−0.989821	−0.494910	−	0.868944i	$-0.664799\pi$
$971$	−2.90807e7	−0.989821	−0.494910	+	0.868944i	$0.664799\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−3.26421e7	−1.09968
$976$	0	0
$977$	−1.81960e7	−0.609873	−0.304936	−	0.952373i	$-0.598635\pi$
$977$	−1.81960e7	−0.609873	−0.304936	+	0.952373i	$0.598635\pi$
$978$	0	0
$979$	−5.78896e6	−0.193039
$980$	0	0
$981$	−3.65874e7	−1.21383
$982$	0	0
$983$	3.32808e6	0.109853	0.0549263	−	0.998490i	$-0.482508\pi$
$983$	3.32808e6	0.109853	0.0549263	+	0.998490i	$0.482508\pi$
$984$	0	0
$985$	1.73470e7	0.569682
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−2.02858e6	−0.0659478
$990$	0	0
$991$	−5.25420e7	−1.69951	−0.849753	−	0.527181i	$-0.823249\pi$
$991$	−5.25420e7	−1.69951	−0.849753	+	0.527181i	$0.823249\pi$
$992$	0	0
$993$	−2.79046e6	−0.0898056
$994$	0	0
$995$	3.03601e7	0.972177
$996$	0	0
$997$	1.22999e6	0.0391889	0.0195945	−	0.999808i	$-0.493762\pi$
$997$	1.22999e6	0.0391889	0.0195945	+	0.999808i	$0.493762\pi$
$998$	0	0
$999$	4.35386e7	1.38026

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.6.a.z.1.1		2
4.3	odd	2		49.6.a.c.1.2	yes	2
7.6	odd	2	inner	784.6.a.z.1.2		2
12.11	even	2		441.6.a.u.1.1		2
28.3	even	6		49.6.c.g.30.2		4
28.11	odd	6		49.6.c.g.30.1		4
28.19	even	6		49.6.c.g.18.2		4
28.23	odd	6		49.6.c.g.18.1		4
28.27	even	2		49.6.a.c.1.1	✓	2
84.83	odd	2		441.6.a.u.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.6.a.c.1.1	✓	2	28.27	even	2
49.6.a.c.1.2	yes	2	4.3	odd	2
49.6.c.g.18.1		4	28.23	odd	6
49.6.c.g.18.2		4	28.19	even	6
49.6.c.g.30.1		4	28.11	odd	6
49.6.c.g.30.2		4	28.3	even	6
441.6.a.u.1.1		2	12.11	even	2
441.6.a.u.1.2		2	84.83	odd	2
784.6.a.z.1.1		2	1.1	even	1	trivial
784.6.a.z.1.2		2	7.6	odd	2	inner

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:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	784.a (trivial)

\(f(q)\)	\(=\)	\(q-24.9800 q^{3} +74.9400 q^{5} +381.000 q^{9} +284.000 q^{11} +524.580 q^{13} -1872.00 q^{15} -149.880 q^{17} -2173.26 q^{19} -1496.00 q^{23} +2491.00 q^{25} -3447.24 q^{27} -4366.00 q^{29} +6444.84 q^{31} -7094.32 q^{33} -12630.0 q^{37} -13104.0 q^{39} -9442.44 q^{41} +1356.00 q^{43} +28552.1 q^{45} -10042.0 q^{47} +3744.00 q^{51} +14150.0 q^{53} +21283.0 q^{55} +54288.0 q^{57} +37395.0 q^{59} -35596.5 q^{61} +39312.0 q^{65} +3644.00 q^{67} +37370.1 q^{69} -35632.0 q^{71} +40767.3 q^{73} -62225.2 q^{75} +54616.0 q^{79} -6471.00 q^{81} +524.580 q^{83} -11232.0 q^{85} +109063. q^{87} -20383.7 q^{89} -160992. q^{93} -162864. q^{95} +183603. q^{97} +108204. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 762 q^{9} + 568 q^{11} - 3744 q^{15} - 2992 q^{23} + 4982 q^{25} - 8732 q^{29} - 25260 q^{37} - 26208 q^{39} + 2712 q^{43} + 7488 q^{51} + 28300 q^{53} + 108576 q^{57} + 78624 q^{65} + 7288 q^{67}+ \cdots + 216408 q^{99}+O(q^{100}) \) 2 * q + 762 * q^9 + 568 * q^11 - 3744 * q^15 - 2992 * q^23 + 4982 * q^25 - 8732 * q^29 - 25260 * q^37 - 26208 * q^39 + 2712 * q^43 + 7488 * q^51 + 28300 * q^53 + 108576 * q^57 + 78624 * q^65 + 7288 * q^67 - 71264 * q^71 + 109232 * q^79 - 12942 * q^81 - 22464 * q^85 - 321984 * q^93 - 325728 * q^95 + 216408 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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