LMFDB - Embedded newform 448.6.a.bd.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:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 198x^{2} + 43x + 5999 $ x^4 - x^3 - 198x^2 + 43x + 5999
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 224)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-12.2094$ 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)

$f(q)$	$=$	$q-20.4188 q^{3} -21.4943 q^{5} -49.0000 q^{7} +173.928 q^{9} +O(q^{10})$	$q-20.4188 q^{3} -21.4943 q^{5} -49.0000 q^{7} +173.928 q^{9} +647.760 q^{11} +787.030 q^{13} +438.888 q^{15} -1937.22 q^{17} -964.776 q^{19} +1000.52 q^{21} -1493.66 q^{23} -2662.99 q^{25} +1410.38 q^{27} -7167.22 q^{29} -2225.21 q^{31} -13226.5 q^{33} +1053.22 q^{35} +2050.95 q^{37} -16070.2 q^{39} +9009.74 q^{41} -2603.57 q^{43} -3738.45 q^{45} +22200.2 q^{47} +2401.00 q^{49} +39555.7 q^{51} -27740.0 q^{53} -13923.1 q^{55} +19699.6 q^{57} +43836.3 q^{59} -31459.4 q^{61} -8522.45 q^{63} -16916.7 q^{65} -5417.61 q^{67} +30498.7 q^{69} -48731.4 q^{71} -7553.51 q^{73} +54375.2 q^{75} -31740.2 q^{77} -43361.7 q^{79} -71062.6 q^{81} +71834.6 q^{83} +41639.1 q^{85} +146346. q^{87} +1138.04 q^{89} -38564.5 q^{91} +45436.2 q^{93} +20737.2 q^{95} +135062. q^{97} +112663. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 18 q^{3} + 30 q^{5} - 196 q^{7} + 696 q^{9}+O(q^{10}) $ 4 * q + 18 * q^3 + 30 * q^5 - 196 * q^7 + 696 * q^9	$ 4 q + 18 q^{3} + 30 q^{5} - 196 q^{7} + 696 q^{9} + 484 q^{11} - 686 q^{13} + 2184 q^{15} - 1700 q^{17} + 654 q^{19} - 882 q^{21} - 136 q^{23} + 3304 q^{25} + 14388 q^{27} - 3812 q^{29} - 12748 q^{31} - 1536 q^{33} - 1470 q^{35} - 820 q^{37} - 34704 q^{39} + 22340 q^{41} + 32924 q^{43} + 59070 q^{45} + 2620 q^{47} + 9604 q^{49} + 55788 q^{51} - 22984 q^{53} - 24360 q^{55} + 15132 q^{57} + 108158 q^{59} - 4258 q^{61} - 34104 q^{63} - 95028 q^{65} + 109496 q^{67} + 82392 q^{69} - 54600 q^{71} - 12384 q^{73} + 315198 q^{75} - 23716 q^{77} - 78184 q^{79} + 114384 q^{81} + 115582 q^{83} - 101652 q^{85} + 17772 q^{87} - 31560 q^{89} + 33614 q^{91} - 218040 q^{93} + 67032 q^{95} + 41068 q^{97} + 301284 q^{99}+O(q^{100}) $ 4 * q + 18 * q^3 + 30 * q^5 - 196 * q^7 + 696 * q^9 + 484 * q^11 - 686 * q^13 + 2184 * q^15 - 1700 * q^17 + 654 * q^19 - 882 * q^21 - 136 * q^23 + 3304 * q^25 + 14388 * q^27 - 3812 * q^29 - 12748 * q^31 - 1536 * q^33 - 1470 * q^35 - 820 * q^37 - 34704 * q^39 + 22340 * q^41 + 32924 * q^43 + 59070 * q^45 + 2620 * q^47 + 9604 * q^49 + 55788 * q^51 - 22984 * q^53 - 24360 * q^55 + 15132 * q^57 + 108158 * q^59 - 4258 * q^61 - 34104 * q^63 - 95028 * q^65 + 109496 * q^67 + 82392 * q^69 - 54600 * q^71 - 12384 * q^73 + 315198 * q^75 - 23716 * q^77 - 78184 * q^79 + 114384 * q^81 + 115582 * q^83 - 101652 * q^85 + 17772 * q^87 - 31560 * q^89 + 33614 * q^91 - 218040 * q^93 + 67032 * q^95 + 41068 * q^97 + 301284 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−20.4188	−1.30987	−0.654933	−	0.755687i	$-0.727303\pi$
$3$	−20.4188	−1.30987	−0.654933	+	0.755687i	$0.727303\pi$
$4$	0	0
$5$	−21.4943	−0.384502	−0.192251	−	0.981346i	$-0.561579\pi$
$5$	−21.4943	−0.384502	−0.192251	+	0.981346i	$0.561579\pi$
$6$	0	0
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	0	0
$9$	173.928	0.715751
$10$	0	0
$11$	647.760	1.61411	0.807053	−	0.590479i	$-0.201061\pi$
$11$	647.760	1.61411	0.807053	+	0.590479i	$0.201061\pi$
$12$	0	0
$13$	787.030	1.29162	0.645808	−	0.763500i	$-0.276521\pi$
$13$	787.030	1.29162	0.645808	+	0.763500i	$0.276521\pi$
$14$	0	0
$15$	438.888	0.503646
$16$	0	0
$17$	−1937.22	−1.62576	−0.812879	−	0.582432i	$-0.802101\pi$
$17$	−1937.22	−1.62576	−0.812879	+	0.582432i	$0.802101\pi$
$18$	0	0
$19$	−964.776	−0.613116	−0.306558	−	0.951852i	$-0.599177\pi$
$19$	−964.776	−0.613116	−0.306558	+	0.951852i	$0.599177\pi$
$20$	0	0
$21$	1000.52	0.495083
$22$	0	0
$23$	−1493.66	−0.588751	−0.294375	−	0.955690i	$-0.595112\pi$
$23$	−1493.66	−0.588751	−0.294375	+	0.955690i	$0.595112\pi$
$24$	0	0
$25$	−2662.99	−0.852158
$26$	0	0
$27$	1410.38	0.372328
$28$	0	0
$29$	−7167.22	−1.58254	−0.791272	−	0.611464i	$-0.790581\pi$
$29$	−7167.22	−1.58254	−0.791272	+	0.611464i	$0.790581\pi$
$30$	0	0
$31$	−2225.21	−0.415879	−0.207940	−	0.978142i	$-0.566676\pi$
$31$	−2225.21	−0.415879	−0.207940	+	0.978142i	$0.566676\pi$
$32$	0	0
$33$	−13226.5	−2.11426
$34$	0	0
$35$	1053.22	0.145328
$36$	0	0
$37$	2050.95	0.246292	0.123146	−	0.992389i	$-0.460702\pi$
$37$	2050.95	0.246292	0.123146	+	0.992389i	$0.460702\pi$
$38$	0	0
$39$	−16070.2	−1.69184
$40$	0	0
$41$	9009.74	0.837053	0.418526	−	0.908205i	$-0.362547\pi$
$41$	9009.74	0.837053	0.418526	+	0.908205i	$0.362547\pi$
$42$	0	0
$43$	−2603.57	−0.214733	−0.107366	−	0.994220i	$-0.534242\pi$
$43$	−2603.57	−0.214733	−0.107366	+	0.994220i	$0.534242\pi$
$44$	0	0
$45$	−3738.45	−0.275208
$46$	0	0
$47$	22200.2	1.46593	0.732965	−	0.680266i	$-0.238136\pi$
$47$	22200.2	1.46593	0.732965	+	0.680266i	$0.238136\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	39555.7	2.12953
$52$	0	0
$53$	−27740.0	−1.35649	−0.678246	−	0.734835i	$-0.737260\pi$
$53$	−27740.0	−1.35649	−0.678246	+	0.734835i	$0.737260\pi$
$54$	0	0
$55$	−13923.1	−0.620627
$56$	0	0
$57$	19699.6	0.803100
$58$	0	0
$59$	43836.3	1.63947	0.819736	−	0.572742i	$-0.194120\pi$
$59$	43836.3	1.63947	0.819736	+	0.572742i	$0.194120\pi$
$60$	0	0
$61$	−31459.4	−1.08249	−0.541247	−	0.840863i	$-0.682048\pi$
$61$	−31459.4	−1.08249	−0.541247	+	0.840863i	$0.682048\pi$
$62$	0	0
$63$	−8522.45	−0.270529
$64$	0	0
$65$	−16916.7	−0.496628
$66$	0	0
$67$	−5417.61	−0.147442	−0.0737210	−	0.997279i	$-0.523487\pi$
$67$	−5417.61	−0.147442	−0.0737210	+	0.997279i	$0.523487\pi$
$68$	0	0
$69$	30498.7	0.771185
$70$	0	0
$71$	−48731.4	−1.14726	−0.573631	−	0.819114i	$-0.694466\pi$
$71$	−48731.4	−1.14726	−0.573631	+	0.819114i	$0.694466\pi$
$72$	0	0
$73$	−7553.51	−0.165898	−0.0829491	−	0.996554i	$-0.526434\pi$
$73$	−7553.51	−0.165898	−0.0829491	+	0.996554i	$0.526434\pi$
$74$	0	0
$75$	54375.2	1.11621
$76$	0	0
$77$	−31740.2	−0.610075
$78$	0	0
$79$	−43361.7	−0.781697	−0.390849	−	0.920455i	$-0.627818\pi$
$79$	−43361.7	−0.781697	−0.390849	+	0.920455i	$0.627818\pi$
$80$	0	0
$81$	−71062.6	−1.20345
$82$	0	0
$83$	71834.6	1.14456	0.572280	−	0.820059i	$-0.306059\pi$
$83$	71834.6	1.14456	0.572280	+	0.820059i	$0.306059\pi$
$84$	0	0
$85$	41639.1	0.625107
$86$	0	0
$87$	146346.	2.07292
$88$	0	0
$89$	1138.04	0.0152294	0.00761472	−	0.999971i	$-0.497576\pi$
$89$	1138.04	0.0152294	0.00761472	+	0.999971i	$0.497576\pi$
$90$	0	0
$91$	−38564.5	−0.488185
$92$	0	0
$93$	45436.2	0.544746
$94$	0	0
$95$	20737.2	0.235744
$96$	0	0
$97$	135062.	1.45749	0.728744	−	0.684786i	$-0.240104\pi$
$97$	135062.	1.45749	0.728744	+	0.684786i	$0.240104\pi$
$98$	0	0
$99$	112663.	1.15530
$100$	0	0
$101$	20108.2	0.196142	0.0980709	−	0.995179i	$-0.468733\pi$
$101$	20108.2	0.196142	0.0980709	+	0.995179i	$0.468733\pi$
$102$	0	0
$103$	−14829.7	−0.137733	−0.0688665	−	0.997626i	$-0.521938\pi$
$103$	−14829.7	−0.137733	−0.0688665	+	0.997626i	$0.521938\pi$
$104$	0	0
$105$	−21505.5	−0.190360
$106$	0	0
$107$	168416.	1.42208	0.711041	−	0.703151i	$-0.248224\pi$
$107$	168416.	1.42208	0.711041	+	0.703151i	$0.248224\pi$
$108$	0	0
$109$	175717.	1.41660	0.708299	−	0.705913i	$-0.249463\pi$
$109$	175717.	1.41660	0.708299	+	0.705913i	$0.249463\pi$
$110$	0	0
$111$	−41877.9	−0.322610
$112$	0	0
$113$	−150362.	−1.10775	−0.553875	−	0.832600i	$-0.686852\pi$
$113$	−150362.	−1.10775	−0.553875	+	0.832600i	$0.686852\pi$
$114$	0	0
$115$	32105.1	0.226376
$116$	0	0
$117$	136886.	0.924475
$118$	0	0
$119$	94923.6	0.614479
$120$	0	0
$121$	258542.	1.60534
$122$	0	0
$123$	−183968.	−1.09643
$124$	0	0
$125$	124409.	0.712158
$126$	0	0
$127$	56065.1	0.308449	0.154224	−	0.988036i	$-0.450712\pi$
$127$	56065.1	0.308449	0.154224	+	0.988036i	$0.450712\pi$
$128$	0	0
$129$	53161.8	0.281271
$130$	0	0
$131$	−99935.1	−0.508791	−0.254396	−	0.967100i	$-0.581877\pi$
$131$	−99935.1	−0.508791	−0.254396	+	0.967100i	$0.581877\pi$
$132$	0	0
$133$	47274.0	0.231736
$134$	0	0
$135$	−30315.1	−0.143161
$136$	0	0
$137$	246538.	1.12223	0.561116	−	0.827737i	$-0.310372\pi$
$137$	246538.	1.12223	0.561116	+	0.827737i	$0.310372\pi$
$138$	0	0
$139$	258431.	1.13451	0.567255	−	0.823542i	$-0.308005\pi$
$139$	258431.	1.13451	0.567255	+	0.823542i	$0.308005\pi$
$140$	0	0
$141$	−453303.	−1.92017
$142$	0	0
$143$	509807.	2.08480
$144$	0	0
$145$	154054.	0.608491
$146$	0	0
$147$	−49025.5	−0.187124
$148$	0	0
$149$	−246723.	−0.910425	−0.455213	−	0.890383i	$-0.650437\pi$
$149$	−246723.	−0.910425	−0.455213	+	0.890383i	$0.650437\pi$
$150$	0	0
$151$	248510.	0.886955	0.443478	−	0.896285i	$-0.353745\pi$
$151$	248510.	0.886955	0.443478	+	0.896285i	$0.353745\pi$
$152$	0	0
$153$	−336935.	−1.16364
$154$	0	0
$155$	47829.4	0.159906
$156$	0	0
$157$	−360954.	−1.16870	−0.584350	−	0.811502i	$-0.698650\pi$
$157$	−360954.	−1.16870	−0.584350	+	0.811502i	$0.698650\pi$
$158$	0	0
$159$	566418.	1.77682
$160$	0	0
$161$	73189.2	0.222527
$162$	0	0
$163$	486179.	1.43327	0.716634	−	0.697449i	$-0.245682\pi$
$163$	486179.	1.43327	0.716634	+	0.697449i	$0.245682\pi$
$164$	0	0
$165$	284294.	0.812939
$166$	0	0
$167$	275768.	0.765160	0.382580	−	0.923922i	$-0.375036\pi$
$167$	275768.	0.765160	0.382580	+	0.923922i	$0.375036\pi$
$168$	0	0
$169$	248124.	0.668270
$170$	0	0
$171$	−167801.	−0.438838
$172$	0	0
$173$	−144055.	−0.365942	−0.182971	−	0.983118i	$-0.558572\pi$
$173$	−144055.	−0.365942	−0.182971	+	0.983118i	$0.558572\pi$
$174$	0	0
$175$	130487.	0.322086
$176$	0	0
$177$	−895085.	−2.14749
$178$	0	0
$179$	−196486.	−0.458353	−0.229176	−	0.973385i	$-0.573603\pi$
$179$	−196486.	−0.458353	−0.229176	+	0.973385i	$0.573603\pi$
$180$	0	0
$181$	−7419.09	−0.0168327	−0.00841636	−	0.999965i	$-0.502679\pi$
$181$	−7419.09	−0.0168327	−0.00841636	+	0.999965i	$0.502679\pi$
$182$	0	0
$183$	642363.	1.41792
$184$	0	0
$185$	−44083.7	−0.0946997
$186$	0	0
$187$	−1.25485e6	−2.62415
$188$	0	0
$189$	−69108.4	−0.140727
$190$	0	0
$191$	−801615.	−1.58995	−0.794973	−	0.606644i	$-0.792515\pi$
$191$	−801615.	−1.58995	−0.794973	+	0.606644i	$0.792515\pi$
$192$	0	0
$193$	−304970.	−0.589336	−0.294668	−	0.955600i	$-0.595209\pi$
$193$	−304970.	−0.589336	−0.294668	+	0.955600i	$0.595209\pi$
$194$	0	0
$195$	345418.	0.650517
$196$	0	0
$197$	382141.	0.701549	0.350775	−	0.936460i	$-0.385918\pi$
$197$	382141.	0.701549	0.350775	+	0.936460i	$0.385918\pi$
$198$	0	0
$199$	142214.	0.254572	0.127286	−	0.991866i	$-0.459373\pi$
$199$	142214.	0.254572	0.127286	+	0.991866i	$0.459373\pi$
$200$	0	0
$201$	110621.	0.193129
$202$	0	0
$203$	351194.	0.598146
$204$	0	0
$205$	−193658.	−0.321848
$206$	0	0
$207$	−259788.	−0.421399
$208$	0	0
$209$	−624943.	−0.989634
$210$	0	0
$211$	782360.	1.20976	0.604882	−	0.796315i	$-0.293220\pi$
$211$	782360.	1.20976	0.604882	+	0.796315i	$0.293220\pi$
$212$	0	0
$213$	995036.	1.50276
$214$	0	0
$215$	55961.9	0.0825651
$216$	0	0
$217$	109035.	0.157188
$218$	0	0
$219$	154234.	0.217304
$220$	0	0
$221$	−1.52465e6	−2.09985
$222$	0	0
$223$	680812.	0.916780	0.458390	−	0.888751i	$-0.348426\pi$
$223$	680812.	0.916780	0.458390	+	0.888751i	$0.348426\pi$
$224$	0	0
$225$	−463168.	−0.609933
$226$	0	0
$227$	−1.03738e6	−1.33621	−0.668103	−	0.744069i	$-0.732894\pi$
$227$	−1.03738e6	−1.33621	−0.668103	+	0.744069i	$0.732894\pi$
$228$	0	0
$229$	323871.	0.408116	0.204058	−	0.978959i	$-0.434587\pi$
$229$	323871.	0.408116	0.204058	+	0.978959i	$0.434587\pi$
$230$	0	0
$231$	648097.	0.799117
$232$	0	0
$233$	1.30328e6	1.57271	0.786353	−	0.617777i	$-0.211966\pi$
$233$	1.30328e6	1.57271	0.786353	+	0.617777i	$0.211966\pi$
$234$	0	0
$235$	−477179.	−0.563653
$236$	0	0
$237$	885394.	1.02392
$238$	0	0
$239$	−86138.1	−0.0975440	−0.0487720	−	0.998810i	$-0.515531\pi$
$239$	−86138.1	−0.0975440	−0.0487720	+	0.998810i	$0.515531\pi$
$240$	0	0
$241$	−1.25796e6	−1.39516	−0.697581	−	0.716506i	$-0.745740\pi$
$241$	−1.25796e6	−1.39516	−0.697581	+	0.716506i	$0.745740\pi$
$242$	0	0
$243$	1.10829e6	1.20403
$244$	0	0
$245$	−51607.8	−0.0549288
$246$	0	0
$247$	−759308.	−0.791910
$248$	0	0
$249$	−1.46678e6	−1.49922
$250$	0	0
$251$	1.15904e6	1.16122	0.580609	−	0.814183i	$-0.302815\pi$
$251$	1.15904e6	1.16122	0.580609	+	0.814183i	$0.302815\pi$
$252$	0	0
$253$	−967531.	−0.950306
$254$	0	0
$255$	−850221.	−0.818807
$256$	0	0
$257$	1.53094e6	1.44586	0.722930	−	0.690921i	$-0.242795\pi$
$257$	1.53094e6	1.44586	0.722930	+	0.690921i	$0.242795\pi$
$258$	0	0
$259$	−100496.	−0.0930896
$260$	0	0
$261$	−1.24658e6	−1.13271
$262$	0	0
$263$	−1.06192e6	−0.946680	−0.473340	−	0.880880i	$-0.656952\pi$
$263$	−1.06192e6	−0.946680	−0.473340	+	0.880880i	$0.656952\pi$
$264$	0	0
$265$	596253.	0.521574
$266$	0	0
$267$	−23237.5	−0.0199485
$268$	0	0
$269$	367406.	0.309575	0.154787	−	0.987948i	$-0.450531\pi$
$269$	367406.	0.309575	0.154787	+	0.987948i	$0.450531\pi$
$270$	0	0
$271$	−2.12719e6	−1.75947	−0.879737	−	0.475461i	$-0.842281\pi$
$271$	−2.12719e6	−1.75947	−0.879737	+	0.475461i	$0.842281\pi$
$272$	0	0
$273$	787441.	0.639457
$274$	0	0
$275$	−1.72498e6	−1.37547
$276$	0	0
$277$	1.24666e6	0.976223	0.488111	−	0.872781i	$-0.337686\pi$
$277$	1.24666e6	0.976223	0.488111	+	0.872781i	$0.337686\pi$
$278$	0	0
$279$	−387026.	−0.297666
$280$	0	0
$281$	1.79523e6	1.35630	0.678149	−	0.734925i	$-0.262782\pi$
$281$	1.79523e6	1.35630	0.678149	+	0.734925i	$0.262782\pi$
$282$	0	0
$283$	2.12563e6	1.57769	0.788844	−	0.614593i	$-0.210680\pi$
$283$	2.12563e6	1.57769	0.788844	+	0.614593i	$0.210680\pi$
$284$	0	0
$285$	−423429.	−0.308793
$286$	0	0
$287$	−441477.	−0.316376
$288$	0	0
$289$	2.33295e6	1.64309
$290$	0	0
$291$	−2.75781e6	−1.90911
$292$	0	0
$293$	1.14547e6	0.779494	0.389747	−	0.920922i	$-0.372562\pi$
$293$	1.14547e6	0.779494	0.389747	+	0.920922i	$0.372562\pi$
$294$	0	0
$295$	−942231.	−0.630380
$296$	0	0
$297$	913585.	0.600977
$298$	0	0
$299$	−1.17555e6	−0.760439
$300$	0	0
$301$	127575.	0.0811613
$302$	0	0
$303$	−410586.	−0.256920
$304$	0	0
$305$	676198.	0.416221
$306$	0	0
$307$	1.60698e6	0.973114	0.486557	−	0.873649i	$-0.338253\pi$
$307$	1.60698e6	0.973114	0.486557	+	0.873649i	$0.338253\pi$
$308$	0	0
$309$	302804.	0.180412
$310$	0	0
$311$	2.38046e6	1.39560	0.697799	−	0.716294i	$-0.254163\pi$
$311$	2.38046e6	1.39560	0.697799	+	0.716294i	$0.254163\pi$
$312$	0	0
$313$	−600241.	−0.346310	−0.173155	−	0.984895i	$-0.555396\pi$
$313$	−600241.	−0.346310	−0.173155	+	0.984895i	$0.555396\pi$
$314$	0	0
$315$	183184.	0.104019
$316$	0	0
$317$	3.01742e6	1.68650	0.843252	−	0.537519i	$-0.180638\pi$
$317$	3.01742e6	1.68650	0.843252	+	0.537519i	$0.180638\pi$
$318$	0	0
$319$	−4.64264e6	−2.55440
$320$	0	0
$321$	−3.43886e6	−1.86274
$322$	0	0
$323$	1.86898e6	0.996778
$324$	0	0
$325$	−2.09586e6	−1.10066
$326$	0	0
$327$	−3.58792e6	−1.85555
$328$	0	0
$329$	−1.08781e6	−0.554069
$330$	0	0
$331$	−1.68509e6	−0.845380	−0.422690	−	0.906274i	$-0.638914\pi$
$331$	−1.68509e6	−0.845380	−0.422690	+	0.906274i	$0.638914\pi$
$332$	0	0
$333$	356716.	0.176284
$334$	0	0
$335$	116448.	0.0566917
$336$	0	0
$337$	1.82977e6	0.877652	0.438826	−	0.898572i	$-0.355395\pi$
$337$	1.82977e6	0.877652	0.438826	+	0.898572i	$0.355395\pi$
$338$	0	0
$339$	3.07021e6	1.45101
$340$	0	0
$341$	−1.44140e6	−0.671273
$342$	0	0
$343$	−117649.	−0.0539949
$344$	0	0
$345$	−655548.	−0.296522
$346$	0	0
$347$	−861788.	−0.384217	−0.192109	−	0.981374i	$-0.561533\pi$
$347$	−861788.	−0.384217	−0.192109	+	0.981374i	$0.561533\pi$
$348$	0	0
$349$	4.11641e6	1.80907	0.904534	−	0.426401i	$-0.140219\pi$
$349$	4.11641e6	1.80907	0.904534	+	0.426401i	$0.140219\pi$
$350$	0	0
$351$	1.11001e6	0.480904
$352$	0	0
$353$	−1.32890e6	−0.567619	−0.283809	−	0.958881i	$-0.591598\pi$
$353$	−1.32890e6	−0.567619	−0.283809	+	0.958881i	$0.591598\pi$
$354$	0	0
$355$	1.04745e6	0.441125
$356$	0	0
$357$	−1.93823e6	−0.804886
$358$	0	0
$359$	3.94367e6	1.61497	0.807485	−	0.589888i	$-0.200828\pi$
$359$	3.94367e6	1.61497	0.807485	+	0.589888i	$0.200828\pi$
$360$	0	0
$361$	−1.54531e6	−0.624089
$362$	0	0
$363$	−5.27911e6	−2.10278
$364$	0	0
$365$	162357.	0.0637881
$366$	0	0
$367$	−3.50971e6	−1.36021	−0.680104	−	0.733115i	$-0.738065\pi$
$367$	−3.50971e6	−1.36021	−0.680104	+	0.733115i	$0.738065\pi$
$368$	0	0
$369$	1.56704e6	0.599122
$370$	0	0
$371$	1.35926e6	0.512706
$372$	0	0
$373$	−4.31380e6	−1.60542	−0.802709	−	0.596371i	$-0.796609\pi$
$373$	−4.31380e6	−1.60542	−0.802709	+	0.596371i	$0.796609\pi$
$374$	0	0
$375$	−2.54028e6	−0.932833
$376$	0	0
$377$	−5.64082e6	−2.04404
$378$	0	0
$379$	−5.00478e6	−1.78973	−0.894864	−	0.446339i	$-0.852728\pi$
$379$	−5.00478e6	−1.78973	−0.894864	+	0.446339i	$0.852728\pi$
$380$	0	0
$381$	−1.14478e6	−0.404027
$382$	0	0
$383$	2.30249e6	0.802050	0.401025	−	0.916067i	$-0.368654\pi$
$383$	2.30249e6	0.802050	0.401025	+	0.916067i	$0.368654\pi$
$384$	0	0
$385$	682234.	0.234575
$386$	0	0
$387$	−452833.	−0.153695
$388$	0	0
$389$	4.52146e6	1.51497	0.757485	−	0.652852i	$-0.226428\pi$
$389$	4.52146e6	1.51497	0.757485	+	0.652852i	$0.226428\pi$
$390$	0	0
$391$	2.89354e6	0.957166
$392$	0	0
$393$	2.04055e6	0.666449
$394$	0	0
$395$	932030.	0.300564
$396$	0	0
$397$	4.05864e6	1.29242	0.646212	−	0.763158i	$-0.276352\pi$
$397$	4.05864e6	1.29242	0.646212	+	0.763158i	$0.276352\pi$
$398$	0	0
$399$	−965279.	−0.303543
$400$	0	0
$401$	1.20112e6	0.373015	0.186507	−	0.982454i	$-0.440283\pi$
$401$	1.20112e6	0.373015	0.186507	+	0.982454i	$0.440283\pi$
$402$	0	0
$403$	−1.75131e6	−0.537156
$404$	0	0
$405$	1.52744e6	0.462729
$406$	0	0
$407$	1.32852e6	0.397541
$408$	0	0
$409$	−2.72738e6	−0.806190	−0.403095	−	0.915158i	$-0.632066\pi$
$409$	−2.72738e6	−0.806190	−0.403095	+	0.915158i	$0.632066\pi$
$410$	0	0
$411$	−5.03402e6	−1.46998
$412$	0	0
$413$	−2.14798e6	−0.619662
$414$	0	0
$415$	−1.54403e6	−0.440085
$416$	0	0
$417$	−5.27686e6	−1.48606
$418$	0	0
$419$	−838850.	−0.233426	−0.116713	−	0.993166i	$-0.537236\pi$
$419$	−838850.	−0.233426	−0.116713	+	0.993166i	$0.537236\pi$
$420$	0	0
$421$	−4.09743e6	−1.12670	−0.563348	−	0.826220i	$-0.690487\pi$
$421$	−4.09743e6	−1.12670	−0.563348	+	0.826220i	$0.690487\pi$
$422$	0	0
$423$	3.86124e6	1.04924
$424$	0	0
$425$	5.15880e6	1.38540
$426$	0	0
$427$	1.54151e6	0.409145
$428$	0	0
$429$	−1.04096e7	−2.73082
$430$	0	0
$431$	−6.21333e6	−1.61113	−0.805566	−	0.592506i	$-0.798139\pi$
$431$	−6.21333e6	−1.61113	−0.805566	+	0.592506i	$0.798139\pi$
$432$	0	0
$433$	5.48637e6	1.40626	0.703130	−	0.711061i	$-0.251785\pi$
$433$	5.48637e6	1.40626	0.703130	+	0.711061i	$0.251785\pi$
$434$	0	0
$435$	−3.14561e6	−0.797043
$436$	0	0
$437$	1.44104e6	0.360972
$438$	0	0
$439$	−2.71784e6	−0.673073	−0.336537	−	0.941670i	$-0.609256\pi$
$439$	−2.71784e6	−0.673073	−0.336537	+	0.941670i	$0.609256\pi$
$440$	0	0
$441$	417600.	0.102250
$442$	0	0
$443$	−7.50234e6	−1.81630	−0.908150	−	0.418645i	$-0.862505\pi$
$443$	−7.50234e6	−1.81630	−0.908150	+	0.418645i	$0.862505\pi$
$444$	0	0
$445$	−24461.5	−0.00585575
$446$	0	0
$447$	5.03779e6	1.19254
$448$	0	0
$449$	1.28575e6	0.300981	0.150491	−	0.988611i	$-0.451915\pi$
$449$	1.28575e6	0.300981	0.150491	+	0.988611i	$0.451915\pi$
$450$	0	0
$451$	5.83615e6	1.35109
$452$	0	0
$453$	−5.07428e6	−1.16179
$454$	0	0
$455$	828917.	0.187708
$456$	0	0
$457$	−968058.	−0.216826	−0.108413	−	0.994106i	$-0.534577\pi$
$457$	−968058.	−0.216826	−0.108413	+	0.994106i	$0.534577\pi$
$458$	0	0
$459$	−2.73221e6	−0.605315
$460$	0	0
$461$	−112166.	−0.0245814	−0.0122907	−	0.999924i	$-0.503912\pi$
$461$	−112166.	−0.0245814	−0.0122907	+	0.999924i	$0.503912\pi$
$462$	0	0
$463$	256452.	0.0555973	0.0277986	−	0.999614i	$-0.491150\pi$
$463$	256452.	0.0555973	0.0277986	+	0.999614i	$0.491150\pi$
$464$	0	0
$465$	−976619.	−0.209456
$466$	0	0
$467$	2.22246e6	0.471565	0.235782	−	0.971806i	$-0.424235\pi$
$467$	2.22246e6	0.471565	0.235782	+	0.971806i	$0.424235\pi$
$468$	0	0
$469$	265463.	0.0557278
$470$	0	0
$471$	7.37026e6	1.53084
$472$	0	0
$473$	−1.68649e6	−0.346601
$474$	0	0
$475$	2.56919e6	0.522472
$476$	0	0
$477$	−4.82476e6	−0.970911
$478$	0	0
$479$	1.97880e6	0.394061	0.197031	−	0.980397i	$-0.436870\pi$
$479$	1.97880e6	0.394061	0.197031	+	0.980397i	$0.436870\pi$
$480$	0	0
$481$	1.61416e6	0.318114
$482$	0	0
$483$	−1.49444e6	−0.291480
$484$	0	0
$485$	−2.90307e6	−0.560407
$486$	0	0
$487$	9.11381e6	1.74132	0.870658	−	0.491889i	$-0.163693\pi$
$487$	9.11381e6	1.74132	0.870658	+	0.491889i	$0.163693\pi$
$488$	0	0
$489$	−9.92720e6	−1.87739
$490$	0	0
$491$	2.69944e6	0.505325	0.252662	−	0.967555i	$-0.418694\pi$
$491$	2.69944e6	0.505325	0.252662	+	0.967555i	$0.418694\pi$
$492$	0	0
$493$	1.38845e7	2.57283
$494$	0	0
$495$	−2.42162e6	−0.444215
$496$	0	0
$497$	2.38784e6	0.433624
$498$	0	0
$499$	435566.	0.0783074	0.0391537	−	0.999233i	$-0.487534\pi$
$499$	435566.	0.0783074	0.0391537	+	0.999233i	$0.487534\pi$
$500$	0	0
$501$	−5.63085e6	−1.00226
$502$	0	0
$503$	−92382.1	−0.0162805	−0.00814025	−	0.999967i	$-0.502591\pi$
$503$	−92382.1	−0.0162805	−0.00814025	+	0.999967i	$0.502591\pi$
$504$	0	0
$505$	−432212.	−0.0754169
$506$	0	0
$507$	−5.06639e6	−0.875344
$508$	0	0
$509$	2.86197e6	0.489634	0.244817	−	0.969569i	$-0.421272\pi$
$509$	2.86197e6	0.489634	0.244817	+	0.969569i	$0.421272\pi$
$510$	0	0
$511$	370122.	0.0627036
$512$	0	0
$513$	−1.36070e6	−0.228280
$514$	0	0
$515$	318753.	0.0529586
$516$	0	0
$517$	1.43804e7	2.36617
$518$	0	0
$519$	2.94143e6	0.479336
$520$	0	0
$521$	4.45599e6	0.719200	0.359600	−	0.933107i	$-0.382913\pi$
$521$	4.45599e6	0.719200	0.359600	+	0.933107i	$0.382913\pi$
$522$	0	0
$523$	4.35927e6	0.696882	0.348441	−	0.937331i	$-0.386711\pi$
$523$	4.35927e6	0.696882	0.348441	+	0.937331i	$0.386711\pi$
$524$	0	0
$525$	−2.66438e6	−0.421889
$526$	0	0
$527$	4.31072e6	0.676119
$528$	0	0
$529$	−4.20533e6	−0.653373
$530$	0	0
$531$	7.62434e6	1.17345
$532$	0	0
$533$	7.09094e6	1.08115
$534$	0	0
$535$	−3.61999e6	−0.546793
$536$	0	0
$537$	4.01202e6	0.600381
$538$	0	0
$539$	1.55527e6	0.230587
$540$	0	0
$541$	−1.30648e6	−0.191915	−0.0959576	−	0.995385i	$-0.530591\pi$
$541$	−1.30648e6	−0.191915	−0.0959576	+	0.995385i	$0.530591\pi$
$542$	0	0
$543$	151489.	0.0220486
$544$	0	0
$545$	−3.77691e6	−0.544685
$546$	0	0
$547$	1.33330e7	1.90528	0.952640	−	0.304102i	$-0.0983563\pi$
$547$	1.33330e7	1.90528	0.952640	+	0.304102i	$0.0983563\pi$
$548$	0	0
$549$	−5.47166e6	−0.774797
$550$	0	0
$551$	6.91476e6	0.970283
$552$	0	0
$553$	2.12472e6	0.295454
$554$	0	0
$555$	900136.	0.124044
$556$	0	0
$557$	1.06595e7	1.45579	0.727894	−	0.685690i	$-0.240499\pi$
$557$	1.06595e7	1.45579	0.727894	+	0.685690i	$0.240499\pi$
$558$	0	0
$559$	−2.04909e6	−0.277352
$560$	0	0
$561$	2.56226e7	3.43728
$562$	0	0
$563$	−9.43316e6	−1.25426	−0.627128	−	0.778916i	$-0.715770\pi$
$563$	−9.43316e6	−1.25426	−0.627128	+	0.778916i	$0.715770\pi$
$564$	0	0
$565$	3.23193e6	0.425932
$566$	0	0
$567$	3.48207e6	0.454862
$568$	0	0
$569$	−1.64441e6	−0.212927	−0.106463	−	0.994317i	$-0.533953\pi$
$569$	−1.64441e6	−0.212927	−0.106463	+	0.994317i	$0.533953\pi$
$570$	0	0
$571$	9.16956e6	1.17695	0.588475	−	0.808515i	$-0.299728\pi$
$571$	9.16956e6	1.17695	0.588475	+	0.808515i	$0.299728\pi$
$572$	0	0
$573$	1.63680e7	2.08262
$574$	0	0
$575$	3.97760e6	0.501709
$576$	0	0
$577$	546566.	0.0683444	0.0341722	−	0.999416i	$-0.489121\pi$
$577$	546566.	0.0683444	0.0341722	+	0.999416i	$0.489121\pi$
$578$	0	0
$579$	6.22711e6	0.771952
$580$	0	0
$581$	−3.51989e6	−0.432603
$582$	0	0
$583$	−1.79689e7	−2.18952
$584$	0	0
$585$	−2.94228e6	−0.355462
$586$	0	0
$587$	−2.60611e6	−0.312175	−0.156087	−	0.987743i	$-0.549888\pi$
$587$	−2.60611e6	−0.312175	−0.156087	+	0.987743i	$0.549888\pi$
$588$	0	0
$589$	2.14683e6	0.254982
$590$	0	0
$591$	−7.80286e6	−0.918936
$592$	0	0
$593$	2.08057e6	0.242966	0.121483	−	0.992594i	$-0.461235\pi$
$593$	2.08057e6	0.242966	0.121483	+	0.992594i	$0.461235\pi$
$594$	0	0
$595$	−2.04032e6	−0.236268
$596$	0	0
$597$	−2.90385e6	−0.333455
$598$	0	0
$599$	1.40857e7	1.60403	0.802015	−	0.597304i	$-0.203761\pi$
$599$	1.40857e7	1.60403	0.802015	+	0.597304i	$0.203761\pi$
$600$	0	0
$601$	−9.33576e6	−1.05430	−0.527149	−	0.849773i	$-0.676739\pi$
$601$	−9.33576e6	−1.05430	−0.527149	+	0.849773i	$0.676739\pi$
$602$	0	0
$603$	−942272.	−0.105532
$604$	0	0
$605$	−5.55717e6	−0.617256
$606$	0	0
$607$	−1.05476e7	−1.16193	−0.580967	−	0.813927i	$-0.697325\pi$
$607$	−1.05476e7	−1.16193	−0.580967	+	0.813927i	$0.697325\pi$
$608$	0	0
$609$	−7.17096e6	−0.783491
$610$	0	0
$611$	1.74723e7	1.89342
$612$	0	0
$613$	−8.90712e6	−0.957384	−0.478692	−	0.877983i	$-0.658889\pi$
$613$	−8.90712e6	−0.957384	−0.478692	+	0.877983i	$0.658889\pi$
$614$	0	0
$615$	3.95427e6	0.421578
$616$	0	0
$617$	2.75653e6	0.291507	0.145754	−	0.989321i	$-0.453439\pi$
$617$	2.75653e6	0.291507	0.145754	+	0.989321i	$0.453439\pi$
$618$	0	0
$619$	792026.	0.0830831	0.0415416	−	0.999137i	$-0.486773\pi$
$619$	792026.	0.0830831	0.0415416	+	0.999137i	$0.486773\pi$
$620$	0	0
$621$	−2.10662e6	−0.219208
$622$	0	0
$623$	−55764.1	−0.00575619
$624$	0	0
$625$	5.64777e6	0.578332
$626$	0	0
$627$	1.27606e7	1.29629
$628$	0	0
$629$	−3.97313e6	−0.400411
$630$	0	0
$631$	−3.72493e6	−0.372430	−0.186215	−	0.982509i	$-0.559622\pi$
$631$	−3.72493e6	−0.372430	−0.186215	+	0.982509i	$0.559622\pi$
$632$	0	0
$633$	−1.59749e7	−1.58463
$634$	0	0
$635$	−1.20508e6	−0.118599
$636$	0	0
$637$	1.88966e6	0.184516
$638$	0	0
$639$	−8.47573e6	−0.821155
$640$	0	0
$641$	−2.67519e6	−0.257163	−0.128582	−	0.991699i	$-0.541042\pi$
$641$	−2.67519e6	−0.257163	−0.128582	+	0.991699i	$0.541042\pi$
$642$	0	0
$643$	1.82378e7	1.73958	0.869792	−	0.493418i	$-0.164253\pi$
$643$	1.82378e7	1.73958	0.869792	+	0.493418i	$0.164253\pi$
$644$	0	0
$645$	−1.14268e6	−0.108149
$646$	0	0
$647$	8.16473e6	0.766798	0.383399	−	0.923583i	$-0.374753\pi$
$647$	8.16473e6	0.766798	0.383399	+	0.923583i	$0.374753\pi$
$648$	0	0
$649$	2.83954e7	2.64628
$650$	0	0
$651$	−2.22637e6	−0.205895
$652$	0	0
$653$	−2.04750e7	−1.87906	−0.939528	−	0.342471i	$-0.888736\pi$
$653$	−2.04750e7	−1.87906	−0.939528	+	0.342471i	$0.888736\pi$
$654$	0	0
$655$	2.14804e6	0.195631
$656$	0	0
$657$	−1.31376e6	−0.118742
$658$	0	0
$659$	1.15848e7	1.03914	0.519572	−	0.854427i	$-0.326092\pi$
$659$	1.15848e7	1.03914	0.519572	+	0.854427i	$0.326092\pi$
$660$	0	0
$661$	−1.60142e7	−1.42562	−0.712809	−	0.701359i	$-0.752577\pi$
$661$	−1.60142e7	−1.42562	−0.712809	+	0.701359i	$0.752577\pi$
$662$	0	0
$663$	3.11315e7	2.75053
$664$	0	0
$665$	−1.01612e6	−0.0891029
$666$	0	0
$667$	1.07054e7	0.931724
$668$	0	0
$669$	−1.39014e7	−1.20086
$670$	0	0
$671$	−2.03781e7	−1.74726
$672$	0	0
$673$	−1.14532e7	−0.974739	−0.487370	−	0.873196i	$-0.662044\pi$
$673$	−1.14532e7	−0.974739	−0.487370	+	0.873196i	$0.662044\pi$
$674$	0	0
$675$	−3.75583e6	−0.317282
$676$	0	0
$677$	2.11530e7	1.77378	0.886891	−	0.461979i	$-0.152860\pi$
$677$	2.11530e7	1.77378	0.886891	+	0.461979i	$0.152860\pi$
$678$	0	0
$679$	−6.61805e6	−0.550879
$680$	0	0
$681$	2.11821e7	1.75025
$682$	0	0
$683$	−1.37808e6	−0.113037	−0.0565187	−	0.998402i	$-0.518000\pi$
$683$	−1.37808e6	−0.113037	−0.0565187	+	0.998402i	$0.518000\pi$
$684$	0	0
$685$	−5.29917e6	−0.431500
$686$	0	0
$687$	−6.61306e6	−0.534577
$688$	0	0
$689$	−2.18323e7	−1.75207
$690$	0	0
$691$	−1.77460e7	−1.41386	−0.706930	−	0.707284i	$-0.749920\pi$
$691$	−1.77460e7	−1.41386	−0.706930	+	0.707284i	$0.749920\pi$
$692$	0	0
$693$	−5.52050e6	−0.436662
$694$	0	0
$695$	−5.55480e6	−0.436221
$696$	0	0
$697$	−1.74538e7	−1.36085
$698$	0	0
$699$	−2.66114e7	−2.06004
$700$	0	0
$701$	−841648.	−0.0646898	−0.0323449	−	0.999477i	$-0.510297\pi$
$701$	−841648.	−0.0646898	−0.0323449	+	0.999477i	$0.510297\pi$
$702$	0	0
$703$	−1.97870e6	−0.151005
$704$	0	0
$705$	9.74342e6	0.738310
$706$	0	0
$707$	−985303.	−0.0741347
$708$	0	0
$709$	6.61436e6	0.494165	0.247082	−	0.968994i	$-0.420528\pi$
$709$	6.61436e6	0.494165	0.247082	+	0.968994i	$0.420528\pi$
$710$	0	0
$711$	−7.54180e6	−0.559501
$712$	0	0
$713$	3.32370e6	0.244849
$714$	0	0
$715$	−1.09579e7	−0.801611
$716$	0	0
$717$	1.75884e6	0.127770
$718$	0	0
$719$	1.11026e7	0.800948	0.400474	−	0.916308i	$-0.368846\pi$
$719$	1.11026e7	0.800948	0.400474	+	0.916308i	$0.368846\pi$
$720$	0	0
$721$	726653.	0.0520582
$722$	0	0
$723$	2.56861e7	1.82748
$724$	0	0
$725$	1.90863e7	1.34858
$726$	0	0
$727$	−1.44518e7	−1.01411	−0.507057	−	0.861913i	$-0.669267\pi$
$727$	−1.44518e7	−1.01411	−0.507057	+	0.861913i	$0.669267\pi$
$728$	0	0
$729$	−5.36178e6	−0.373672
$730$	0	0
$731$	5.04368e6	0.349103
$732$	0	0
$733$	−2.29283e6	−0.157620	−0.0788102	−	0.996890i	$-0.525112\pi$
$733$	−2.29283e6	−0.157620	−0.0788102	+	0.996890i	$0.525112\pi$
$734$	0	0
$735$	1.05377e6	0.0719495
$736$	0	0
$737$	−3.50931e6	−0.237987
$738$	0	0
$739$	1.02688e7	0.691686	0.345843	−	0.938292i	$-0.387593\pi$
$739$	1.02688e7	0.691686	0.345843	+	0.938292i	$0.387593\pi$
$740$	0	0
$741$	1.55042e7	1.03730
$742$	0	0
$743$	1.42827e6	0.0949159	0.0474579	−	0.998873i	$-0.484888\pi$
$743$	1.42827e6	0.0949159	0.0474579	+	0.998873i	$0.484888\pi$
$744$	0	0
$745$	5.30314e6	0.350060
$746$	0	0
$747$	1.24940e7	0.819220
$748$	0	0
$749$	−8.25239e6	−0.537496
$750$	0	0
$751$	1.23504e7	0.799064	0.399532	−	0.916719i	$-0.369173\pi$
$751$	1.23504e7	0.799064	0.399532	+	0.916719i	$0.369173\pi$
$752$	0	0
$753$	−2.36662e7	−1.52104
$754$	0	0
$755$	−5.34155e6	−0.341036
$756$	0	0
$757$	−8.11463e6	−0.514670	−0.257335	−	0.966322i	$-0.582844\pi$
$757$	−8.11463e6	−0.514670	−0.257335	+	0.966322i	$0.582844\pi$
$758$	0	0
$759$	1.97558e7	1.24477
$760$	0	0
$761$	1.67506e6	0.104850	0.0524249	−	0.998625i	$-0.483305\pi$
$761$	1.67506e6	0.104850	0.0524249	+	0.998625i	$0.483305\pi$
$762$	0	0
$763$	−8.61011e6	−0.535424
$764$	0	0
$765$	7.24219e6	0.447421
$766$	0	0
$767$	3.45005e7	2.11757
$768$	0	0
$769$	−2.68837e7	−1.63935	−0.819677	−	0.572826i	$-0.805847\pi$
$769$	−2.68837e7	−1.63935	−0.819677	+	0.572826i	$0.805847\pi$
$770$	0	0
$771$	−3.12600e7	−1.89388
$772$	0	0
$773$	400610.	0.0241142	0.0120571	−	0.999927i	$-0.496162\pi$
$773$	400610.	0.0241142	0.0120571	+	0.999927i	$0.496162\pi$
$774$	0	0
$775$	5.92573e6	0.354395
$776$	0	0
$777$	2.05202e6	0.121935
$778$	0	0
$779$	−8.69238e6	−0.513210
$780$	0	0
$781$	−3.15662e7	−1.85180
$782$	0	0
$783$	−1.01085e7	−0.589225
$784$	0	0
$785$	7.75846e6	0.449367
$786$	0	0
$787$	−1.47190e7	−0.847113	−0.423557	−	0.905870i	$-0.639218\pi$
$787$	−1.47190e7	−0.847113	−0.423557	+	0.905870i	$0.639218\pi$
$788$	0	0
$789$	2.16832e7	1.24002
$790$	0	0
$791$	7.36774e6	0.418690
$792$	0	0
$793$	−2.47595e7	−1.39817
$794$	0	0
$795$	−1.21748e7	−0.683192
$796$	0	0
$797$	−1.00867e7	−0.562474	−0.281237	−	0.959638i	$-0.590745\pi$
$797$	−1.00867e7	−0.562474	−0.281237	+	0.959638i	$0.590745\pi$
$798$	0	0
$799$	−4.30067e7	−2.38325
$800$	0	0
$801$	197937.	0.0109005
$802$	0	0
$803$	−4.89286e6	−0.267777
$804$	0	0
$805$	−1.57315e6	−0.0855620
$806$	0	0
$807$	−7.50199e6	−0.405502
$808$	0	0
$809$	1.96759e7	1.05697	0.528487	−	0.848941i	$-0.322760\pi$
$809$	1.96759e7	1.05697	0.528487	+	0.848941i	$0.322760\pi$
$810$	0	0
$811$	2.07488e7	1.10775	0.553874	−	0.832601i	$-0.313149\pi$
$811$	2.07488e7	1.10775	0.553874	+	0.832601i	$0.313149\pi$
$812$	0	0
$813$	4.34346e7	2.30468
$814$	0	0
$815$	−1.04501e7	−0.551094
$816$	0	0
$817$	2.51186e6	0.131656
$818$	0	0
$819$	−6.70743e6	−0.349419
$820$	0	0
$821$	−946675.	−0.0490166	−0.0245083	−	0.999700i	$-0.507802\pi$
$821$	−946675.	−0.0490166	−0.0245083	+	0.999700i	$0.507802\pi$
$822$	0	0
$823$	1.80402e7	0.928412	0.464206	−	0.885727i	$-0.346340\pi$
$823$	1.80402e7	0.928412	0.464206	+	0.885727i	$0.346340\pi$
$824$	0	0
$825$	3.52220e7	1.80169
$826$	0	0
$827$	−9.98123e6	−0.507482	−0.253741	−	0.967272i	$-0.581661\pi$
$827$	−9.98123e6	−0.507482	−0.253741	+	0.967272i	$0.581661\pi$
$828$	0	0
$829$	−8.24005e6	−0.416432	−0.208216	−	0.978083i	$-0.566766\pi$
$829$	−8.24005e6	−0.416432	−0.208216	+	0.978083i	$0.566766\pi$
$830$	0	0
$831$	−2.54553e7	−1.27872
$832$	0	0
$833$	−4.65126e6	−0.232251
$834$	0	0
$835$	−5.92744e6	−0.294205
$836$	0	0
$837$	−3.13839e6	−0.154843
$838$	0	0
$839$	1.34565e7	0.659973	0.329986	−	0.943986i	$-0.392956\pi$
$839$	1.34565e7	0.659973	0.329986	+	0.943986i	$0.392956\pi$
$840$	0	0
$841$	3.08579e7	1.50445
$842$	0	0
$843$	−3.66565e7	−1.77657
$844$	0	0
$845$	−5.33325e6	−0.256951
$846$	0	0
$847$	−1.26685e7	−0.606762
$848$	0	0
$849$	−4.34028e7	−2.06656
$850$	0	0
$851$	−3.06341e6	−0.145004
$852$	0	0
$853$	5.38794e6	0.253542	0.126771	−	0.991932i	$-0.459539\pi$
$853$	5.38794e6	0.253542	0.126771	+	0.991932i	$0.459539\pi$
$854$	0	0
$855$	3.60677e6	0.168734
$856$	0	0
$857$	−3.96394e6	−0.184364	−0.0921818	−	0.995742i	$-0.529384\pi$
$857$	−3.96394e6	−0.184364	−0.0921818	+	0.995742i	$0.529384\pi$
$858$	0	0
$859$	1.95904e7	0.905859	0.452930	−	0.891546i	$-0.350379\pi$
$859$	1.95904e7	0.905859	0.452930	+	0.891546i	$0.350379\pi$
$860$	0	0
$861$	9.01444e6	0.414411
$862$	0	0
$863$	3.83066e7	1.75084	0.875420	−	0.483363i	$-0.160585\pi$
$863$	3.83066e7	1.75084	0.875420	+	0.483363i	$0.160585\pi$
$864$	0	0
$865$	3.09636e6	0.140706
$866$	0	0
$867$	−4.76361e7	−2.15223
$868$	0	0
$869$	−2.80880e7	−1.26174
$870$	0	0
$871$	−4.26383e6	−0.190438
$872$	0	0
$873$	2.34911e7	1.04320
$874$	0	0
$875$	−6.09604e6	−0.269171
$876$	0	0
$877$	−1.52390e7	−0.669048	−0.334524	−	0.942387i	$-0.608576\pi$
$877$	−1.52390e7	−0.669048	−0.334524	+	0.942387i	$0.608576\pi$
$878$	0	0
$879$	−2.33890e7	−1.02103
$880$	0	0
$881$	3.13058e7	1.35889	0.679445	−	0.733726i	$-0.262220\pi$
$881$	3.13058e7	1.35889	0.679445	+	0.733726i	$0.262220\pi$
$882$	0	0
$883$	−9.58868e6	−0.413864	−0.206932	−	0.978355i	$-0.566348\pi$
$883$	−9.58868e6	−0.413864	−0.206932	+	0.978355i	$0.566348\pi$
$884$	0	0
$885$	1.92392e7	0.825714
$886$	0	0
$887$	8.49529e6	0.362551	0.181275	−	0.983432i	$-0.441977\pi$
$887$	8.49529e6	0.362551	0.181275	+	0.983432i	$0.441977\pi$
$888$	0	0
$889$	−2.74719e6	−0.116583
$890$	0	0
$891$	−4.60315e7	−1.94250
$892$	0	0
$893$	−2.14183e7	−0.898785
$894$	0	0
$895$	4.22334e6	0.176238
$896$	0	0
$897$	2.40034e7	0.996074
$898$	0	0
$899$	1.59486e7	0.658147
$900$	0	0
$901$	5.37385e7	2.20533
$902$	0	0
$903$	−2.60493e6	−0.106311
$904$	0	0
$905$	159468.	0.00647221
$906$	0	0
$907$	3.16843e6	0.127887	0.0639435	−	0.997954i	$-0.479632\pi$
$907$	3.16843e6	0.127887	0.0639435	+	0.997954i	$0.479632\pi$
$908$	0	0
$909$	3.49737e6	0.140389
$910$	0	0
$911$	2.14906e7	0.857932	0.428966	−	0.903321i	$-0.358878\pi$
$911$	2.14906e7	0.857932	0.428966	+	0.903321i	$0.358878\pi$
$912$	0	0
$913$	4.65315e7	1.84744
$914$	0	0
$915$	−1.38072e7	−0.545194
$916$	0	0
$917$	4.89682e6	0.192305
$918$	0	0
$919$	−2.43349e7	−0.950473	−0.475237	−	0.879858i	$-0.657638\pi$
$919$	−2.43349e7	−0.950473	−0.475237	+	0.879858i	$0.657638\pi$
$920$	0	0
$921$	−3.28126e7	−1.27465
$922$	0	0
$923$	−3.83531e7	−1.48182
$924$	0	0
$925$	−5.46166e6	−0.209880
$926$	0	0
$927$	−2.57929e6	−0.0985826
$928$	0	0
$929$	3.95011e7	1.50165	0.750827	−	0.660499i	$-0.229655\pi$
$929$	3.95011e7	1.50165	0.750827	+	0.660499i	$0.229655\pi$
$930$	0	0
$931$	−2.31643e6	−0.0875880
$932$	0	0
$933$	−4.86062e7	−1.82805
$934$	0	0
$935$	2.69722e7	1.00899
$936$	0	0
$937$	3.80717e7	1.41662	0.708310	−	0.705901i	$-0.249458\pi$
$937$	3.80717e7	1.41662	0.708310	+	0.705901i	$0.249458\pi$
$938$	0	0
$939$	1.22562e7	0.453620
$940$	0	0
$941$	−2.78889e7	−1.02673	−0.513367	−	0.858169i	$-0.671602\pi$
$941$	−2.78889e7	−1.02673	−0.513367	+	0.858169i	$0.671602\pi$
$942$	0	0
$943$	−1.34575e7	−0.492815
$944$	0	0
$945$	1.48544e6	0.0541097
$946$	0	0
$947$	3.95348e7	1.43253	0.716266	−	0.697828i	$-0.245850\pi$
$947$	3.95348e7	1.43253	0.716266	+	0.697828i	$0.245850\pi$
$948$	0	0
$949$	−5.94484e6	−0.214277
$950$	0	0
$951$	−6.16120e7	−2.20909
$952$	0	0
$953$	−2.46155e7	−0.877964	−0.438982	−	0.898496i	$-0.644661\pi$
$953$	−2.46155e7	−0.877964	−0.438982	+	0.898496i	$0.644661\pi$
$954$	0	0
$955$	1.72302e7	0.611337
$956$	0	0
$957$	9.47971e7	3.34592
$958$	0	0
$959$	−1.20804e7	−0.424164
$960$	0	0
$961$	−2.36776e7	−0.827045
$962$	0	0
$963$	2.92922e7	1.01786
$964$	0	0
$965$	6.55511e6	0.226601
$966$	0	0
$967$	−2.95008e7	−1.01453	−0.507267	−	0.861789i	$-0.669344\pi$
$967$	−2.95008e7	−1.01453	−0.507267	+	0.861789i	$0.669344\pi$
$968$	0	0
$969$	−3.81623e7	−1.30565
$970$	0	0
$971$	3.46448e7	1.17921	0.589604	−	0.807692i	$-0.299284\pi$
$971$	3.46448e7	1.17921	0.589604	+	0.807692i	$0.299284\pi$
$972$	0	0
$973$	−1.26631e7	−0.428804
$974$	0	0
$975$	4.27949e7	1.44172
$976$	0	0
$977$	−3.52468e7	−1.18136	−0.590681	−	0.806905i	$-0.701141\pi$
$977$	−3.52468e7	−1.18136	−0.590681	+	0.806905i	$0.701141\pi$
$978$	0	0
$979$	737179.	0.0245819
$980$	0	0
$981$	3.05620e7	1.01393
$982$	0	0
$983$	−5.34734e7	−1.76504	−0.882519	−	0.470277i	$-0.844154\pi$
$983$	−5.34734e7	−1.76504	−0.882519	+	0.470277i	$0.844154\pi$
$984$	0	0
$985$	−8.21386e6	−0.269747
$986$	0	0
$987$	2.22118e7	0.725757
$988$	0	0
$989$	3.88884e6	0.126424
$990$	0	0
$991$	−1.45979e7	−0.472177	−0.236089	−	0.971732i	$-0.575866\pi$
$991$	−1.45979e7	−0.472177	−0.236089	+	0.971732i	$0.575866\pi$
$992$	0	0
$993$	3.44074e7	1.10734
$994$	0	0
$995$	−3.05680e6	−0.0978834
$996$	0	0
$997$	−5.69407e6	−0.181420	−0.0907100	−	0.995877i	$-0.528914\pi$
$997$	−5.69407e6	−0.181420	−0.0907100	+	0.995877i	$0.528914\pi$
$998$	0	0
$999$	2.89261e6	0.0917013

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.6.a.bd.1.1		4
4.3	odd	2		448.6.a.bc.1.4		4
8.3	odd	2		224.6.a.h.1.1	yes	4
8.5	even	2		224.6.a.g.1.4	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.6.a.g.1.4	✓	4	8.5	even	2
224.6.a.h.1.1	yes	4	8.3	odd	2
448.6.a.bc.1.4		4	4.3	odd	2
448.6.a.bd.1.1		4	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)

\(f(q)\)	\(=\)	\(q-20.4188 q^{3} -21.4943 q^{5} -49.0000 q^{7} +173.928 q^{9} +O(q^{10})\)	\(q-20.4188 q^{3} -21.4943 q^{5} -49.0000 q^{7} +173.928 q^{9} +647.760 q^{11} +787.030 q^{13} +438.888 q^{15} -1937.22 q^{17} -964.776 q^{19} +1000.52 q^{21} -1493.66 q^{23} -2662.99 q^{25} +1410.38 q^{27} -7167.22 q^{29} -2225.21 q^{31} -13226.5 q^{33} +1053.22 q^{35} +2050.95 q^{37} -16070.2 q^{39} +9009.74 q^{41} -2603.57 q^{43} -3738.45 q^{45} +22200.2 q^{47} +2401.00 q^{49} +39555.7 q^{51} -27740.0 q^{53} -13923.1 q^{55} +19699.6 q^{57} +43836.3 q^{59} -31459.4 q^{61} -8522.45 q^{63} -16916.7 q^{65} -5417.61 q^{67} +30498.7 q^{69} -48731.4 q^{71} -7553.51 q^{73} +54375.2 q^{75} -31740.2 q^{77} -43361.7 q^{79} -71062.6 q^{81} +71834.6 q^{83} +41639.1 q^{85} +146346. q^{87} +1138.04 q^{89} -38564.5 q^{91} +45436.2 q^{93} +20737.2 q^{95} +135062. q^{97} +112663. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 18 q^{3} + 30 q^{5} - 196 q^{7} + 696 q^{9}+O(q^{10}) \) 4 * q + 18 * q^3 + 30 * q^5 - 196 * q^7 + 696 * q^9	\( 4 q + 18 q^{3} + 30 q^{5} - 196 q^{7} + 696 q^{9} + 484 q^{11} - 686 q^{13} + 2184 q^{15} - 1700 q^{17} + 654 q^{19} - 882 q^{21} - 136 q^{23} + 3304 q^{25} + 14388 q^{27} - 3812 q^{29} - 12748 q^{31} - 1536 q^{33} - 1470 q^{35} - 820 q^{37} - 34704 q^{39} + 22340 q^{41} + 32924 q^{43} + 59070 q^{45} + 2620 q^{47} + 9604 q^{49} + 55788 q^{51} - 22984 q^{53} - 24360 q^{55} + 15132 q^{57} + 108158 q^{59} - 4258 q^{61} - 34104 q^{63} - 95028 q^{65} + 109496 q^{67} + 82392 q^{69} - 54600 q^{71} - 12384 q^{73} + 315198 q^{75} - 23716 q^{77} - 78184 q^{79} + 114384 q^{81} + 115582 q^{83} - 101652 q^{85} + 17772 q^{87} - 31560 q^{89} + 33614 q^{91} - 218040 q^{93} + 67032 q^{95} + 41068 q^{97} + 301284 q^{99}+O(q^{100}) \) 4 * q + 18 * q^3 + 30 * q^5 - 196 * q^7 + 696 * q^9 + 484 * q^11 - 686 * q^13 + 2184 * q^15 - 1700 * q^17 + 654 * q^19 - 882 * q^21 - 136 * q^23 + 3304 * q^25 + 14388 * q^27 - 3812 * q^29 - 12748 * q^31 - 1536 * q^33 - 1470 * q^35 - 820 * q^37 - 34704 * q^39 + 22340 * q^41 + 32924 * q^43 + 59070 * q^45 + 2620 * q^47 + 9604 * q^49 + 55788 * q^51 - 22984 * q^53 - 24360 * q^55 + 15132 * q^57 + 108158 * q^59 - 4258 * q^61 - 34104 * q^63 - 95028 * q^65 + 109496 * q^67 + 82392 * q^69 - 54600 * q^71 - 12384 * q^73 + 315198 * q^75 - 23716 * q^77 - 78184 * q^79 + 114384 * q^81 + 115582 * q^83 - 101652 * q^85 + 17772 * q^87 - 31560 * q^89 + 33614 * q^91 - 218040 * q^93 + 67032 * q^95 + 41068 * q^97 + 301284 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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