LMFDB - Embedded newform 448.6.a.u.1.2

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:	$2$
Coefficient field:	$\Q(\sqrt{57}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 14 $ x^2 - x - 14
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 7)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.27492$ 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)

$f(q)$	$=$	$q+19.6495 q^{3} +46.7492 q^{5} -49.0000 q^{7} +143.103 q^{9} +O(q^{10})$	$q+19.6495 q^{3} +46.7492 q^{5} -49.0000 q^{7} +143.103 q^{9} +666.090 q^{11} +650.640 q^{13} +918.598 q^{15} +1186.89 q^{17} -1565.05 q^{19} -962.826 q^{21} +1100.15 q^{23} -939.515 q^{25} -1962.93 q^{27} -2396.72 q^{29} +2048.46 q^{31} +13088.3 q^{33} -2290.71 q^{35} -1077.54 q^{37} +12784.7 q^{39} +1098.21 q^{41} +16564.3 q^{43} +6689.95 q^{45} +8298.39 q^{47} +2401.00 q^{49} +23321.9 q^{51} -5519.18 q^{53} +31139.1 q^{55} -30752.5 q^{57} -14230.4 q^{59} +14234.7 q^{61} -7012.05 q^{63} +30416.9 q^{65} +19730.4 q^{67} +21617.5 q^{69} -64562.7 q^{71} +28567.0 q^{73} -18461.0 q^{75} -32638.4 q^{77} +30633.4 q^{79} -73344.6 q^{81} -675.946 q^{83} +55486.3 q^{85} -47094.4 q^{87} +125971. q^{89} -31881.3 q^{91} +40251.1 q^{93} -73164.9 q^{95} -22906.8 q^{97} +95319.4 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} + 18 q^{5} - 98 q^{7} + 558 q^{9}+O(q^{10}) $ 2 * q - 6 * q^3 + 18 * q^5 - 98 * q^7 + 558 * q^9	$ 2 q - 6 q^{3} + 18 q^{5} - 98 q^{7} + 558 q^{9} + 396 q^{11} + 350 q^{13} + 1656 q^{15} + 1800 q^{17} - 3266 q^{19} + 294 q^{21} - 2088 q^{23} - 3238 q^{25} - 6372 q^{27} - 6696 q^{29} + 20 q^{31} + 20016 q^{33} - 882 q^{35} - 6232 q^{37} + 20496 q^{39} - 6048 q^{41} - 3020 q^{43} - 5238 q^{45} - 11700 q^{47} + 4802 q^{49} + 7596 q^{51} - 9468 q^{53} + 38904 q^{55} + 12876 q^{57} - 43938 q^{59} + 64754 q^{61} - 27342 q^{63} + 39060 q^{65} + 24784 q^{67} + 103392 q^{69} - 97416 q^{71} + 17452 q^{73} + 40494 q^{75} - 19404 q^{77} - 51256 q^{79} - 61074 q^{81} + 117558 q^{83} + 37860 q^{85} + 63180 q^{87} + 84276 q^{89} - 17150 q^{91} + 92280 q^{93} - 24264 q^{95} + 20776 q^{97} - 16740 q^{99}+O(q^{100}) $ 2 * q - 6 * q^3 + 18 * q^5 - 98 * q^7 + 558 * q^9 + 396 * q^11 + 350 * q^13 + 1656 * q^15 + 1800 * q^17 - 3266 * q^19 + 294 * q^21 - 2088 * q^23 - 3238 * q^25 - 6372 * q^27 - 6696 * q^29 + 20 * q^31 + 20016 * q^33 - 882 * q^35 - 6232 * q^37 + 20496 * q^39 - 6048 * q^41 - 3020 * q^43 - 5238 * q^45 - 11700 * q^47 + 4802 * q^49 + 7596 * q^51 - 9468 * q^53 + 38904 * q^55 + 12876 * q^57 - 43938 * q^59 + 64754 * q^61 - 27342 * q^63 + 39060 * q^65 + 24784 * q^67 + 103392 * q^69 - 97416 * q^71 + 17452 * q^73 + 40494 * q^75 - 19404 * q^77 - 51256 * q^79 - 61074 * q^81 + 117558 * q^83 + 37860 * q^85 + 63180 * q^87 + 84276 * q^89 - 17150 * q^91 + 92280 * q^93 - 24264 * q^95 + 20776 * q^97 - 16740 * 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
$3$	19.6495	1.26052	0.630258	−	0.776386i	$-0.282949\pi$
$3$	19.6495	1.26052	0.630258	+	0.776386i	$0.282949\pi$
$4$	0	0
$5$	46.7492	0.836275	0.418137	−	0.908384i	$-0.362683\pi$
$5$	46.7492	0.836275	0.418137	+	0.908384i	$0.362683\pi$
$6$	0	0
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	0	0
$9$	143.103	0.588901
$10$	0	0
$11$	666.090	1.65978	0.829891	−	0.557926i	$-0.188403\pi$
$11$	666.090	1.65978	0.829891	+	0.557926i	$0.188403\pi$
$12$	0	0
$13$	650.640	1.06778	0.533890	−	0.845554i	$-0.320729\pi$
$13$	650.640	1.06778	0.533890	+	0.845554i	$0.320729\pi$
$14$	0	0
$15$	918.598	1.05414
$16$	0	0
$17$	1186.89	0.996069	0.498035	−	0.867157i	$-0.334055\pi$
$17$	1186.89	0.996069	0.498035	+	0.867157i	$0.334055\pi$
$18$	0	0
$19$	−1565.05	−0.994591	−0.497296	−	0.867581i	$-0.665674\pi$
$19$	−1565.05	−0.994591	−0.497296	+	0.867581i	$0.665674\pi$
$20$	0	0
$21$	−962.826	−0.476430
$22$	0	0
$23$	1100.15	0.433644	0.216822	−	0.976211i	$-0.430431\pi$
$23$	1100.15	0.433644	0.216822	+	0.976211i	$0.430431\pi$
$24$	0	0
$25$	−939.515	−0.300645
$26$	0	0
$27$	−1962.93	−0.518197
$28$	0	0
$29$	−2396.72	−0.529203	−0.264602	−	0.964358i	$-0.585240\pi$
$29$	−2396.72	−0.529203	−0.264602	+	0.964358i	$0.585240\pi$
$30$	0	0
$31$	2048.46	0.382844	0.191422	−	0.981508i	$-0.438690\pi$
$31$	2048.46	0.382844	0.191422	+	0.981508i	$0.438690\pi$
$32$	0	0
$33$	13088.3	2.09218
$34$	0	0
$35$	−2290.71	−0.316082
$36$	0	0
$37$	−1077.54	−0.129399	−0.0646995	−	0.997905i	$-0.520609\pi$
$37$	−1077.54	−0.129399	−0.0646995	+	0.997905i	$0.520609\pi$
$38$	0	0
$39$	12784.7	1.34595
$40$	0	0
$41$	1098.21	0.102029	0.0510147	−	0.998698i	$-0.483754\pi$
$41$	1098.21	0.102029	0.0510147	+	0.998698i	$0.483754\pi$
$42$	0	0
$43$	16564.3	1.36616	0.683081	−	0.730343i	$-0.260640\pi$
$43$	16564.3	1.36616	0.683081	+	0.730343i	$0.260640\pi$
$44$	0	0
$45$	6689.95	0.492483
$46$	0	0
$47$	8298.39	0.547960	0.273980	−	0.961735i	$-0.411660\pi$
$47$	8298.39	0.547960	0.273980	+	0.961735i	$0.411660\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	23321.9	1.25556
$52$	0	0
$53$	−5519.18	−0.269889	−0.134944	−	0.990853i	$-0.543086\pi$
$53$	−5519.18	−0.269889	−0.134944	+	0.990853i	$0.543086\pi$
$54$	0	0
$55$	31139.1	1.38803
$56$	0	0
$57$	−30752.5	−1.25370
$58$	0	0
$59$	−14230.4	−0.532216	−0.266108	−	0.963943i	$-0.585738\pi$
$59$	−14230.4	−0.532216	−0.266108	+	0.963943i	$0.585738\pi$
$60$	0	0
$61$	14234.7	0.489807	0.244904	−	0.969547i	$-0.421244\pi$
$61$	14234.7	0.489807	0.244904	+	0.969547i	$0.421244\pi$
$62$	0	0
$63$	−7012.05	−0.222584
$64$	0	0
$65$	30416.9	0.892958
$66$	0	0
$67$	19730.4	0.536970	0.268485	−	0.963284i	$-0.413477\pi$
$67$	19730.4	0.536970	0.268485	+	0.963284i	$0.413477\pi$
$68$	0	0
$69$	21617.5	0.546615
$70$	0	0
$71$	−64562.7	−1.51997	−0.759986	−	0.649940i	$-0.774794\pi$
$71$	−64562.7	−1.51997	−0.759986	+	0.649940i	$0.774794\pi$
$72$	0	0
$73$	28567.0	0.627418	0.313709	−	0.949519i	$-0.398428\pi$
$73$	28567.0	0.627418	0.313709	+	0.949519i	$0.398428\pi$
$74$	0	0
$75$	−18461.0	−0.378968
$76$	0	0
$77$	−32638.4	−0.627339
$78$	0	0
$79$	30633.4	0.552239	0.276119	−	0.961123i	$-0.410951\pi$
$79$	30633.4	0.552239	0.276119	+	0.961123i	$0.410951\pi$
$80$	0	0
$81$	−73344.6	−1.24210
$82$	0	0
$83$	−675.946	−0.0107700	−0.00538501	−	0.999986i	$-0.501714\pi$
$83$	−675.946	−0.0107700	−0.00538501	+	0.999986i	$0.501714\pi$
$84$	0	0
$85$	55486.3	0.832987
$86$	0	0
$87$	−47094.4	−0.667069
$88$	0	0
$89$	125971.	1.68576	0.842882	−	0.538098i	$-0.180857\pi$
$89$	125971.	1.68576	0.842882	+	0.538098i	$0.180857\pi$
$90$	0	0
$91$	−31881.3	−0.403583
$92$	0	0
$93$	40251.1	0.482582
$94$	0	0
$95$	−73164.9	−0.831751
$96$	0	0
$97$	−22906.8	−0.247192	−0.123596	−	0.992333i	$-0.539443\pi$
$97$	−22906.8	−0.247192	−0.123596	+	0.992333i	$0.539443\pi$
$98$	0	0
$99$	95319.4	0.977447
$100$	0	0
$101$	181474.	1.77015	0.885077	−	0.465444i	$-0.154105\pi$
$101$	181474.	1.77015	0.885077	+	0.465444i	$0.154105\pi$
$102$	0	0
$103$	−64772.0	−0.601581	−0.300791	−	0.953690i	$-0.597251\pi$
$103$	−64772.0	−0.601581	−0.300791	+	0.953690i	$0.597251\pi$
$104$	0	0
$105$	−45011.3	−0.398427
$106$	0	0
$107$	−148170.	−1.25112	−0.625562	−	0.780175i	$-0.715130\pi$
$107$	−148170.	−1.25112	−0.625562	+	0.780175i	$0.715130\pi$
$108$	0	0
$109$	111294.	0.897237	0.448618	−	0.893723i	$-0.351916\pi$
$109$	111294.	0.897237	0.448618	+	0.893723i	$0.351916\pi$
$110$	0	0
$111$	−21173.2	−0.163110
$112$	0	0
$113$	−43175.5	−0.318084	−0.159042	−	0.987272i	$-0.550840\pi$
$113$	−43175.5	−0.318084	−0.159042	+	0.987272i	$0.550840\pi$
$114$	0	0
$115$	51431.2	0.362646
$116$	0	0
$117$	93108.5	0.628817
$118$	0	0
$119$	−58157.8	−0.376479
$120$	0	0
$121$	282625.	1.75488
$122$	0	0
$123$	21579.3	0.128610
$124$	0	0
$125$	−190013.	−1.08770
$126$	0	0
$127$	131449.	0.723182	0.361591	−	0.932337i	$-0.382234\pi$
$127$	131449.	0.723182	0.361591	+	0.932337i	$0.382234\pi$
$128$	0	0
$129$	325480.	1.72207
$130$	0	0
$131$	−349458.	−1.77916	−0.889582	−	0.456775i	$-0.849005\pi$
$131$	−349458.	−1.77916	−0.889582	+	0.456775i	$0.849005\pi$
$132$	0	0
$133$	76687.5	0.375920
$134$	0	0
$135$	−91765.2	−0.433355
$136$	0	0
$137$	386434.	1.75903	0.879516	−	0.475869i	$-0.157866\pi$
$137$	386434.	1.75903	0.879516	+	0.475869i	$0.157866\pi$
$138$	0	0
$139$	17289.3	0.0758997	0.0379498	−	0.999280i	$-0.487917\pi$
$139$	17289.3	0.0758997	0.0379498	+	0.999280i	$0.487917\pi$
$140$	0	0
$141$	163059.	0.690713
$142$	0	0
$143$	433384.	1.77228
$144$	0	0
$145$	−112045.	−0.442559
$146$	0	0
$147$	47178.5	0.180074
$148$	0	0
$149$	112171.	0.413917	0.206959	−	0.978350i	$-0.433643\pi$
$149$	112171.	0.413917	0.206959	+	0.978350i	$0.433643\pi$
$150$	0	0
$151$	−30495.4	−0.108841	−0.0544205	−	0.998518i	$-0.517331\pi$
$151$	−30495.4	−0.108841	−0.0544205	+	0.998518i	$0.517331\pi$
$152$	0	0
$153$	169848.	0.586586
$154$	0	0
$155$	95763.6	0.320163
$156$	0	0
$157$	−523509.	−1.69502	−0.847510	−	0.530780i	$-0.821899\pi$
$157$	−523509.	−1.69502	−0.847510	+	0.530780i	$0.821899\pi$
$158$	0	0
$159$	−108449.	−0.340199
$160$	0	0
$161$	−53907.5	−0.163902
$162$	0	0
$163$	−439646.	−1.29609	−0.648043	−	0.761604i	$-0.724412\pi$
$163$	−439646.	−1.29609	−0.648043	+	0.761604i	$0.724412\pi$
$164$	0	0
$165$	611869.	1.74964
$166$	0	0
$167$	−279353.	−0.775107	−0.387554	−	0.921847i	$-0.626680\pi$
$167$	−279353.	−0.775107	−0.387554	+	0.921847i	$0.626680\pi$
$168$	0	0
$169$	52038.8	0.140156
$170$	0	0
$171$	−223964.	−0.585716
$172$	0	0
$173$	−99699.4	−0.253266	−0.126633	−	0.991950i	$-0.540417\pi$
$173$	−99699.4	−0.253266	−0.126633	+	0.991950i	$0.540417\pi$
$174$	0	0
$175$	46036.2	0.113633
$176$	0	0
$177$	−279621.	−0.670867
$178$	0	0
$179$	329980.	0.769760	0.384880	−	0.922967i	$-0.374243\pi$
$179$	329980.	0.769760	0.384880	+	0.922967i	$0.374243\pi$
$180$	0	0
$181$	505810.	1.14760	0.573800	−	0.818995i	$-0.305469\pi$
$181$	505810.	1.14760	0.573800	+	0.818995i	$0.305469\pi$
$182$	0	0
$183$	279706.	0.617410
$184$	0	0
$185$	−50374.3	−0.108213
$186$	0	0
$187$	790578.	1.65326
$188$	0	0
$189$	96183.4	0.195860
$190$	0	0
$191$	63835.6	0.126613	0.0633067	−	0.997994i	$-0.479835\pi$
$191$	63835.6	0.126613	0.0633067	+	0.997994i	$0.479835\pi$
$192$	0	0
$193$	469355.	0.907001	0.453501	−	0.891256i	$-0.350175\pi$
$193$	469355.	0.907001	0.453501	+	0.891256i	$0.350175\pi$
$194$	0	0
$195$	597676.	1.12559
$196$	0	0
$197$	−268021.	−0.492043	−0.246021	−	0.969264i	$-0.579123\pi$
$197$	−268021.	−0.492043	−0.246021	+	0.969264i	$0.579123\pi$
$198$	0	0
$199$	605167.	1.08328	0.541642	−	0.840609i	$-0.317803\pi$
$199$	605167.	1.08328	0.541642	+	0.840609i	$0.317803\pi$
$200$	0	0
$201$	387693.	0.676859
$202$	0	0
$203$	117439.	0.200020
$204$	0	0
$205$	51340.4	0.0853246
$206$	0	0
$207$	157435.	0.255374
$208$	0	0
$209$	−1.04246e6	−1.65080
$210$	0	0
$211$	335389.	0.518612	0.259306	−	0.965795i	$-0.416506\pi$
$211$	335389.	0.518612	0.259306	+	0.965795i	$0.416506\pi$
$212$	0	0
$213$	−1.26862e6	−1.91595
$214$	0	0
$215$	774367.	1.14249
$216$	0	0
$217$	−100374.	−0.144702
$218$	0	0
$219$	561327.	0.790871
$220$	0	0
$221$	772240.	1.06358
$222$	0	0
$223$	−1.02526e6	−1.38061	−0.690305	−	0.723518i	$-0.742524\pi$
$223$	−1.02526e6	−1.38061	−0.690305	+	0.723518i	$0.742524\pi$
$224$	0	0
$225$	−134447.	−0.177050
$226$	0	0
$227$	−504226.	−0.649473	−0.324736	−	0.945805i	$-0.605276\pi$
$227$	−504226.	−0.649473	−0.324736	+	0.945805i	$0.605276\pi$
$228$	0	0
$229$	1.11939e6	1.41057	0.705283	−	0.708925i	$-0.250820\pi$
$229$	1.11939e6	1.41057	0.705283	+	0.708925i	$0.250820\pi$
$230$	0	0
$231$	−641328.	−0.790770
$232$	0	0
$233$	770703.	0.930031	0.465015	−	0.885303i	$-0.346049\pi$
$233$	770703.	0.930031	0.465015	+	0.885303i	$0.346049\pi$
$234$	0	0
$235$	387943.	0.458245
$236$	0	0
$237$	601930.	0.696106
$238$	0	0
$239$	171646.	0.194374	0.0971871	−	0.995266i	$-0.469015\pi$
$239$	171646.	0.194374	0.0971871	+	0.995266i	$0.469015\pi$
$240$	0	0
$241$	−383779.	−0.425637	−0.212818	−	0.977092i	$-0.568264\pi$
$241$	−383779.	−0.425637	−0.212818	+	0.977092i	$0.568264\pi$
$242$	0	0
$243$	−964193.	−1.04749
$244$	0	0
$245$	112245.	0.119468
$246$	0	0
$247$	−1.01828e6	−1.06201
$248$	0	0
$249$	−13282.0	−0.0135758
$250$	0	0
$251$	−1.57046e6	−1.57342	−0.786708	−	0.617325i	$-0.788216\pi$
$251$	−1.57046e6	−1.57342	−0.786708	+	0.617325i	$0.788216\pi$
$252$	0	0
$253$	732801.	0.719755
$254$	0	0
$255$	1.09028e6	1.04999
$256$	0	0
$257$	790656.	0.746715	0.373357	−	0.927688i	$-0.378207\pi$
$257$	790656.	0.746715	0.373357	+	0.927688i	$0.378207\pi$
$258$	0	0
$259$	52799.7	0.0489082
$260$	0	0
$261$	−342978.	−0.311648
$262$	0	0
$263$	−464416.	−0.414017	−0.207008	−	0.978339i	$-0.566373\pi$
$263$	−464416.	−0.414017	−0.207008	+	0.978339i	$0.566373\pi$
$264$	0	0
$265$	−258017.	−0.225701
$266$	0	0
$267$	2.47527e6	2.12493
$268$	0	0
$269$	−1.99959e6	−1.68484	−0.842422	−	0.538818i	$-0.818871\pi$
$269$	−1.99959e6	−1.68484	−0.842422	+	0.538818i	$0.818871\pi$
$270$	0	0
$271$	−1.61296e6	−1.33414	−0.667070	−	0.744995i	$-0.732452\pi$
$271$	−1.61296e6	−1.33414	−0.667070	+	0.744995i	$0.732452\pi$
$272$	0	0
$273$	−626452.	−0.508723
$274$	0	0
$275$	−625801.	−0.499005
$276$	0	0
$277$	2.08119e6	1.62972	0.814860	−	0.579658i	$-0.196814\pi$
$277$	2.08119e6	1.62972	0.814860	+	0.579658i	$0.196814\pi$
$278$	0	0
$279$	293140.	0.225457
$280$	0	0
$281$	−982035.	−0.741927	−0.370964	−	0.928647i	$-0.620973\pi$
$281$	−982035.	−0.741927	−0.370964	+	0.928647i	$0.620973\pi$
$282$	0	0
$283$	−1.39622e6	−1.03630	−0.518152	−	0.855289i	$-0.673380\pi$
$283$	−1.39622e6	−1.03630	−0.518152	+	0.855289i	$0.673380\pi$
$284$	0	0
$285$	−1.43765e6	−1.04844
$286$	0	0
$287$	−53812.3	−0.0385635
$288$	0	0
$289$	−11140.3	−0.00784609
$290$	0	0
$291$	−450107.	−0.311590
$292$	0	0
$293$	−2.56205e6	−1.74348	−0.871742	−	0.489965i	$-0.837010\pi$
$293$	−2.56205e6	−1.74348	−0.871742	+	0.489965i	$0.837010\pi$
$294$	0	0
$295$	−665260.	−0.445078
$296$	0	0
$297$	−1.30749e6	−0.860094
$298$	0	0
$299$	715803.	0.463037
$300$	0	0
$301$	−811651.	−0.516361
$302$	0	0
$303$	3.56588e6	2.23131
$304$	0	0
$305$	665463.	0.409613
$306$	0	0
$307$	−884855.	−0.535829	−0.267915	−	0.963443i	$-0.586334\pi$
$307$	−884855.	−0.535829	−0.267915	+	0.963443i	$0.586334\pi$
$308$	0	0
$309$	−1.27274e6	−0.758303
$310$	0	0
$311$	−1.80120e6	−1.05599	−0.527997	−	0.849246i	$-0.677057\pi$
$311$	−1.80120e6	−1.05599	−0.527997	+	0.849246i	$0.677057\pi$
$312$	0	0
$313$	950366.	0.548315	0.274158	−	0.961685i	$-0.411601\pi$
$313$	950366.	0.548315	0.274158	+	0.961685i	$0.411601\pi$
$314$	0	0
$315$	−327807.	−0.186141
$316$	0	0
$317$	−3.04277e6	−1.70068	−0.850338	−	0.526237i	$-0.823602\pi$
$317$	−3.04277e6	−1.70068	−0.850338	+	0.526237i	$0.823602\pi$
$318$	0	0
$319$	−1.59643e6	−0.878362
$320$	0	0
$321$	−2.91146e6	−1.57706
$322$	0	0
$323$	−1.85755e6	−0.990682
$324$	0	0
$325$	−611286.	−0.321023
$326$	0	0
$327$	2.18688e6	1.13098
$328$	0	0
$329$	−406621.	−0.207110
$330$	0	0
$331$	−2.19616e6	−1.10178	−0.550889	−	0.834579i	$-0.685711\pi$
$331$	−2.19616e6	−1.10178	−0.550889	+	0.834579i	$0.685711\pi$
$332$	0	0
$333$	−154200.	−0.0762032
$334$	0	0
$335$	922382.	0.449054
$336$	0	0
$337$	−2.41491e6	−1.15832	−0.579158	−	0.815216i	$-0.696618\pi$
$337$	−2.41491e6	−1.15832	−0.579158	+	0.815216i	$0.696618\pi$
$338$	0	0
$339$	−848377.	−0.400950
$340$	0	0
$341$	1.36446e6	0.635438
$342$	0	0
$343$	−117649.	−0.0539949
$344$	0	0
$345$	1.01060e6	0.457121
$346$	0	0
$347$	−1.08833e6	−0.485219	−0.242609	−	0.970124i	$-0.578003\pi$
$347$	−1.08833e6	−0.485219	−0.242609	+	0.970124i	$0.578003\pi$
$348$	0	0
$349$	−2.79267e6	−1.22731	−0.613657	−	0.789573i	$-0.710302\pi$
$349$	−2.79267e6	−1.22731	−0.613657	+	0.789573i	$0.710302\pi$
$350$	0	0
$351$	−1.27716e6	−0.553321
$352$	0	0
$353$	2.53134e6	1.08122	0.540610	−	0.841273i	$-0.318193\pi$
$353$	2.53134e6	1.08122	0.540610	+	0.841273i	$0.318193\pi$
$354$	0	0
$355$	−3.01825e6	−1.27111
$356$	0	0
$357$	−1.14277e6	−0.474558
$358$	0	0
$359$	−1.09028e6	−0.446480	−0.223240	−	0.974763i	$-0.571663\pi$
$359$	−1.09028e6	−0.446480	−0.223240	+	0.974763i	$0.571663\pi$
$360$	0	0
$361$	−26712.8	−0.0107883
$362$	0	0
$363$	5.55343e6	2.21205
$364$	0	0
$365$	1.33548e6	0.524694
$366$	0	0
$367$	−188070.	−0.0728879	−0.0364439	−	0.999336i	$-0.511603\pi$
$367$	−188070.	−0.0728879	−0.0364439	+	0.999336i	$0.511603\pi$
$368$	0	0
$369$	157157.	0.0600853
$370$	0	0
$371$	270440.	0.102008
$372$	0	0
$373$	1.79371e6	0.667545	0.333772	−	0.942654i	$-0.391678\pi$
$373$	1.79371e6	0.667545	0.333772	+	0.942654i	$0.391678\pi$
$374$	0	0
$375$	−3.73366e6	−1.37106
$376$	0	0
$377$	−1.55940e6	−0.565073
$378$	0	0
$379$	3.58806e6	1.28310	0.641551	−	0.767080i	$-0.278291\pi$
$379$	3.58806e6	1.28310	0.641551	+	0.767080i	$0.278291\pi$
$380$	0	0
$381$	2.58291e6	0.911583
$382$	0	0
$383$	3.42457e6	1.19291	0.596457	−	0.802645i	$-0.296575\pi$
$383$	3.42457e6	1.19291	0.596457	+	0.802645i	$0.296575\pi$
$384$	0	0
$385$	−1.52582e6	−0.524627
$386$	0	0
$387$	2.37040e6	0.804534
$388$	0	0
$389$	−8625.27	−0.00289001	−0.00144500	−	0.999999i	$-0.500460\pi$
$389$	−8625.27	−0.00289001	−0.00144500	+	0.999999i	$0.500460\pi$
$390$	0	0
$391$	1.30576e6	0.431940
$392$	0	0
$393$	−6.86667e6	−2.24267
$394$	0	0
$395$	1.43208e6	0.461823
$396$	0	0
$397$	−1.25709e6	−0.400306	−0.200153	−	0.979765i	$-0.564144\pi$
$397$	−1.25709e6	−0.400306	−0.200153	+	0.979765i	$0.564144\pi$
$398$	0	0
$399$	1.50687e6	0.473853
$400$	0	0
$401$	1.42670e6	0.443070	0.221535	−	0.975152i	$-0.428893\pi$
$401$	1.42670e6	0.443070	0.221535	+	0.975152i	$0.428893\pi$
$402$	0	0
$403$	1.33281e6	0.408794
$404$	0	0
$405$	−3.42880e6	−1.03873
$406$	0	0
$407$	−717741.	−0.214774
$408$	0	0
$409$	−3.06529e6	−0.906073	−0.453036	−	0.891492i	$-0.649659\pi$
$409$	−3.06529e6	−0.906073	−0.453036	+	0.891492i	$0.649659\pi$
$410$	0	0
$411$	7.59324e6	2.21729
$412$	0	0
$413$	697291.	0.201159
$414$	0	0
$415$	−31599.9	−0.00900670
$416$	0	0
$417$	339726.	0.0956728
$418$	0	0
$419$	−248240.	−0.0690776	−0.0345388	−	0.999403i	$-0.510996\pi$
$419$	−248240.	−0.0690776	−0.0345388	+	0.999403i	$0.510996\pi$
$420$	0	0
$421$	−5.96280e6	−1.63963	−0.819814	−	0.572630i	$-0.805923\pi$
$421$	−5.96280e6	−1.63963	−0.819814	+	0.572630i	$0.805923\pi$
$422$	0	0
$423$	1.18752e6	0.322695
$424$	0	0
$425$	−1.11510e6	−0.299463
$426$	0	0
$427$	−697503.	−0.185130
$428$	0	0
$429$	8.51579e6	2.23399
$430$	0	0
$431$	4.93538e6	1.27976	0.639879	−	0.768476i	$-0.278985\pi$
$431$	4.93538e6	1.27976	0.639879	+	0.768476i	$0.278985\pi$
$432$	0	0
$433$	4.15513e6	1.06504	0.532519	−	0.846418i	$-0.321245\pi$
$433$	4.15513e6	1.06504	0.532519	+	0.846418i	$0.321245\pi$
$434$	0	0
$435$	−2.20162e6	−0.557853
$436$	0	0
$437$	−1.72180e6	−0.431299
$438$	0	0
$439$	227955.	0.0564531	0.0282265	−	0.999602i	$-0.491014\pi$
$439$	227955.	0.0564531	0.0282265	+	0.999602i	$0.491014\pi$
$440$	0	0
$441$	343590.	0.0841287
$442$	0	0
$443$	−1.98462e6	−0.480472	−0.240236	−	0.970715i	$-0.577225\pi$
$443$	−1.98462e6	−0.480472	−0.240236	+	0.970715i	$0.577225\pi$
$444$	0	0
$445$	5.88906e6	1.40976
$446$	0	0
$447$	2.20410e6	0.521749
$448$	0	0
$449$	2.61077e6	0.611157	0.305579	−	0.952167i	$-0.401150\pi$
$449$	2.61077e6	0.611157	0.305579	+	0.952167i	$0.401150\pi$
$450$	0	0
$451$	731506.	0.169347
$452$	0	0
$453$	−599220.	−0.137196
$454$	0	0
$455$	−1.49043e6	−0.337506
$456$	0	0
$457$	−4.09917e6	−0.918132	−0.459066	−	0.888402i	$-0.651816\pi$
$457$	−4.09917e6	−0.918132	−0.459066	+	0.888402i	$0.651816\pi$
$458$	0	0
$459$	−2.32979e6	−0.516160
$460$	0	0
$461$	2.62378e6	0.575009	0.287505	−	0.957779i	$-0.407174\pi$
$461$	2.62378e6	0.575009	0.287505	+	0.957779i	$0.407174\pi$
$462$	0	0
$463$	4.28563e6	0.929100	0.464550	−	0.885547i	$-0.346216\pi$
$463$	4.28563e6	0.929100	0.464550	+	0.885547i	$0.346216\pi$
$464$	0	0
$465$	1.88171e6	0.403571
$466$	0	0
$467$	990118.	0.210085	0.105042	−	0.994468i	$-0.466502\pi$
$467$	990118.	0.210085	0.105042	+	0.994468i	$0.466502\pi$
$468$	0	0
$469$	−966792.	−0.202955
$470$	0	0
$471$	−1.02867e7	−2.13660
$472$	0	0
$473$	1.10333e7	2.26753
$474$	0	0
$475$	1.47039e6	0.299019
$476$	0	0
$477$	−789812.	−0.158938
$478$	0	0
$479$	−2.57976e6	−0.513737	−0.256868	−	0.966446i	$-0.582691\pi$
$479$	−2.57976e6	−0.513737	−0.256868	+	0.966446i	$0.582691\pi$
$480$	0	0
$481$	−701093.	−0.138170
$482$	0	0
$483$	−1.05926e6	−0.206601
$484$	0	0
$485$	−1.07087e6	−0.206720
$486$	0	0
$487$	−3.21474e6	−0.614219	−0.307109	−	0.951674i	$-0.599362\pi$
$487$	−3.21474e6	−0.614219	−0.307109	+	0.951674i	$0.599362\pi$
$488$	0	0
$489$	−8.63882e6	−1.63374
$490$	0	0
$491$	−7.86108e6	−1.47156	−0.735781	−	0.677220i	$-0.763185\pi$
$491$	−7.86108e6	−1.47156	−0.735781	+	0.677220i	$0.763185\pi$
$492$	0	0
$493$	−2.84465e6	−0.527123
$494$	0	0
$495$	4.45610e6	0.817415
$496$	0	0
$497$	3.16357e6	0.574495
$498$	0	0
$499$	−1.35382e6	−0.243395	−0.121697	−	0.992567i	$-0.538834\pi$
$499$	−1.35382e6	−0.243395	−0.121697	+	0.992567i	$0.538834\pi$
$500$	0	0
$501$	−5.48914e6	−0.977035
$502$	0	0
$503$	−3.85775e6	−0.679851	−0.339926	−	0.940452i	$-0.610402\pi$
$503$	−3.85775e6	−0.679851	−0.339926	+	0.940452i	$0.610402\pi$
$504$	0	0
$505$	8.48376e6	1.48034
$506$	0	0
$507$	1.02254e6	0.176669
$508$	0	0
$509$	1.06060e7	1.81451	0.907253	−	0.420585i	$-0.138175\pi$
$509$	1.06060e7	1.81451	0.907253	+	0.420585i	$0.138175\pi$
$510$	0	0
$511$	−1.39978e6	−0.237142
$512$	0	0
$513$	3.07208e6	0.515394
$514$	0	0
$515$	−3.02804e6	−0.503087
$516$	0	0
$517$	5.52747e6	0.909495
$518$	0	0
$519$	−1.95904e6	−0.319246
$520$	0	0
$521$	−9.17989e6	−1.48164	−0.740821	−	0.671703i	$-0.765563\pi$
$521$	−9.17989e6	−1.48164	−0.740821	+	0.671703i	$0.765563\pi$
$522$	0	0
$523$	9.05585e6	1.44769	0.723844	−	0.689964i	$-0.242373\pi$
$523$	9.05585e6	1.44769	0.723844	+	0.689964i	$0.242373\pi$
$524$	0	0
$525$	904589.	0.143236
$526$	0	0
$527$	2.43130e6	0.381339
$528$	0	0
$529$	−5.22601e6	−0.811953
$530$	0	0
$531$	−2.03642e6	−0.313422
$532$	0	0
$533$	714539.	0.108945
$534$	0	0
$535$	−6.92681e6	−1.04628
$536$	0	0
$537$	6.48394e6	0.970295
$538$	0	0
$539$	1.59928e6	0.237112
$540$	0	0
$541$	−1.21783e7	−1.78894	−0.894468	−	0.447132i	$-0.852445\pi$
$541$	−1.21783e7	−1.78894	−0.894468	+	0.447132i	$0.852445\pi$
$542$	0	0
$543$	9.93891e6	1.44657
$544$	0	0
$545$	5.20292e6	0.750336
$546$	0	0
$547$	−9.00451e6	−1.28674	−0.643372	−	0.765554i	$-0.722465\pi$
$547$	−9.00451e6	−1.28674	−0.643372	+	0.765554i	$0.722465\pi$
$548$	0	0
$549$	2.03703e6	0.288448
$550$	0	0
$551$	3.75099e6	0.526341
$552$	0	0
$553$	−1.50103e6	−0.208727
$554$	0	0
$555$	−989830.	−0.136404
$556$	0	0
$557$	−1.54461e6	−0.210950	−0.105475	−	0.994422i	$-0.533636\pi$
$557$	−1.54461e6	−0.210950	−0.105475	+	0.994422i	$0.533636\pi$
$558$	0	0
$559$	1.07774e7	1.45876
$560$	0	0
$561$	1.55345e7	2.08396
$562$	0	0
$563$	−1.18748e7	−1.57890	−0.789449	−	0.613816i	$-0.789634\pi$
$563$	−1.18748e7	−1.57890	−0.789449	+	0.613816i	$0.789634\pi$
$564$	0	0
$565$	−2.01842e6	−0.266005
$566$	0	0
$567$	3.59388e6	0.469468
$568$	0	0
$569$	−1.54308e6	−0.199806	−0.0999031	−	0.994997i	$-0.531853\pi$
$569$	−1.54308e6	−0.199806	−0.0999031	+	0.994997i	$0.531853\pi$
$570$	0	0
$571$	−7.01812e6	−0.900804	−0.450402	−	0.892826i	$-0.648719\pi$
$571$	−7.01812e6	−0.900804	−0.450402	+	0.892826i	$0.648719\pi$
$572$	0	0
$573$	1.25434e6	0.159598
$574$	0	0
$575$	−1.03361e6	−0.130373
$576$	0	0
$577$	−5.37543e6	−0.672162	−0.336081	−	0.941833i	$-0.609102\pi$
$577$	−5.37543e6	−0.672162	−0.336081	+	0.941833i	$0.609102\pi$
$578$	0	0
$579$	9.22259e6	1.14329
$580$	0	0
$581$	33121.4	0.00407069
$582$	0	0
$583$	−3.67627e6	−0.447957
$584$	0	0
$585$	4.35274e6	0.525864
$586$	0	0
$587$	−2.06682e6	−0.247575	−0.123788	−	0.992309i	$-0.539504\pi$
$587$	−2.06682e6	−0.247575	−0.123788	+	0.992309i	$0.539504\pi$
$588$	0	0
$589$	−3.20594e6	−0.380774
$590$	0	0
$591$	−5.26648e6	−0.620228
$592$	0	0
$593$	−5.46947e6	−0.638717	−0.319358	−	0.947634i	$-0.603467\pi$
$593$	−5.46947e6	−0.638717	−0.319358	+	0.947634i	$0.603467\pi$
$594$	0	0
$595$	−2.71883e6	−0.314840
$596$	0	0
$597$	1.18912e7	1.36550
$598$	0	0
$599$	−1.09943e7	−1.25199	−0.625996	−	0.779826i	$-0.715307\pi$
$599$	−1.09943e7	−1.25199	−0.625996	+	0.779826i	$0.715307\pi$
$600$	0	0
$601$	1.58788e7	1.79322	0.896608	−	0.442826i	$-0.146024\pi$
$601$	1.58788e7	1.79322	0.896608	+	0.442826i	$0.146024\pi$
$602$	0	0
$603$	2.82348e6	0.316222
$604$	0	0
$605$	1.32125e7	1.46756
$606$	0	0
$607$	−5.33262e6	−0.587447	−0.293724	−	0.955890i	$-0.594895\pi$
$607$	−5.33262e6	−0.587447	−0.293724	+	0.955890i	$0.594895\pi$
$608$	0	0
$609$	2.30762e6	0.252128
$610$	0	0
$611$	5.39926e6	0.585102
$612$	0	0
$613$	8.91838e6	0.958594	0.479297	−	0.877653i	$-0.340892\pi$
$613$	8.91838e6	0.958594	0.479297	+	0.877653i	$0.340892\pi$
$614$	0	0
$615$	1.00881e6	0.107553
$616$	0	0
$617$	9.63586e6	1.01901	0.509504	−	0.860468i	$-0.329829\pi$
$617$	9.63586e6	1.01901	0.509504	+	0.860468i	$0.329829\pi$
$618$	0	0
$619$	1.21747e7	1.27712	0.638560	−	0.769572i	$-0.279530\pi$
$619$	1.21747e7	1.27712	0.638560	+	0.769572i	$0.279530\pi$
$620$	0	0
$621$	−2.15952e6	−0.224713
$622$	0	0
$623$	−6.17260e6	−0.637159
$624$	0	0
$625$	−5.94695e6	−0.608968
$626$	0	0
$627$	−2.04839e7	−2.08087
$628$	0	0
$629$	−1.27893e6	−0.128890
$630$	0	0
$631$	1.30854e7	1.30832	0.654161	−	0.756356i	$-0.273022\pi$
$631$	1.30854e7	1.30832	0.654161	+	0.756356i	$0.273022\pi$
$632$	0	0
$633$	6.59023e6	0.653719
$634$	0	0
$635$	6.14513e6	0.604779
$636$	0	0
$637$	1.56219e6	0.152540
$638$	0	0
$639$	−9.23911e6	−0.895113
$640$	0	0
$641$	−441107.	−0.0424032	−0.0212016	−	0.999775i	$-0.506749\pi$
$641$	−441107.	−0.0424032	−0.0212016	+	0.999775i	$0.506749\pi$
$642$	0	0
$643$	−4.18888e6	−0.399550	−0.199775	−	0.979842i	$-0.564021\pi$
$643$	−4.18888e6	−0.399550	−0.199775	+	0.979842i	$0.564021\pi$
$644$	0	0
$645$	1.52159e7	1.44012
$646$	0	0
$647$	−1.87822e7	−1.76395	−0.881973	−	0.471300i	$-0.843785\pi$
$647$	−1.87822e7	−1.76395	−0.881973	+	0.471300i	$0.843785\pi$
$648$	0	0
$649$	−9.47874e6	−0.883362
$650$	0	0
$651$	−1.97231e6	−0.182399
$652$	0	0
$653$	1.51733e6	0.139251	0.0696254	−	0.997573i	$-0.477820\pi$
$653$	1.51733e6	0.139251	0.0696254	+	0.997573i	$0.477820\pi$
$654$	0	0
$655$	−1.63368e7	−1.48787
$656$	0	0
$657$	4.08802e6	0.369487
$658$	0	0
$659$	1.84809e7	1.65772	0.828859	−	0.559458i	$-0.188991\pi$
$659$	1.84809e7	1.65772	0.828859	+	0.559458i	$0.188991\pi$
$660$	0	0
$661$	1.03952e7	0.925403	0.462702	−	0.886514i	$-0.346880\pi$
$661$	1.03952e7	0.925403	0.462702	+	0.886514i	$0.346880\pi$
$662$	0	0
$663$	1.51741e7	1.34066
$664$	0	0
$665$	3.58508e6	0.314372
$666$	0	0
$667$	−2.63676e6	−0.229486
$668$	0	0
$669$	−2.01458e7	−1.74028
$670$	0	0
$671$	9.48162e6	0.812973
$672$	0	0
$673$	−1.10398e7	−0.939556	−0.469778	−	0.882785i	$-0.655666\pi$
$673$	−1.10398e7	−0.939556	−0.469778	+	0.882785i	$0.655666\pi$
$674$	0	0
$675$	1.84420e6	0.155793
$676$	0	0
$677$	8.23485e6	0.690532	0.345266	−	0.938505i	$-0.387789\pi$
$677$	8.23485e6	0.690532	0.345266	+	0.938505i	$0.387789\pi$
$678$	0	0
$679$	1.12243e6	0.0934298
$680$	0	0
$681$	−9.90780e6	−0.818671
$682$	0	0
$683$	1.98437e7	1.62768	0.813842	−	0.581086i	$-0.197372\pi$
$683$	1.98437e7	1.62768	0.813842	+	0.581086i	$0.197372\pi$
$684$	0	0
$685$	1.80655e7	1.47103
$686$	0	0
$687$	2.19955e7	1.77804
$688$	0	0
$689$	−3.59100e6	−0.288182
$690$	0	0
$691$	−2.37019e6	−0.188837	−0.0944185	−	0.995533i	$-0.530099\pi$
$691$	−2.37019e6	−0.188837	−0.0944185	+	0.995533i	$0.530099\pi$
$692$	0	0
$693$	−4.67065e6	−0.369440
$694$	0	0
$695$	808259.	0.0634730
$696$	0	0
$697$	1.30346e6	0.101628
$698$	0	0
$699$	1.51439e7	1.17232
$700$	0	0
$701$	1.56833e7	1.20543	0.602714	−	0.797957i	$-0.294086\pi$
$701$	1.56833e7	1.20543	0.602714	+	0.797957i	$0.294086\pi$
$702$	0	0
$703$	1.68641e6	0.128699
$704$	0	0
$705$	7.62288e6	0.577626
$706$	0	0
$707$	−8.89223e6	−0.669056
$708$	0	0
$709$	−3.68544e6	−0.275343	−0.137671	−	0.990478i	$-0.543962\pi$
$709$	−3.68544e6	−0.275343	−0.137671	+	0.990478i	$0.543962\pi$
$710$	0	0
$711$	4.38373e6	0.325214
$712$	0	0
$713$	2.25361e6	0.166018
$714$	0	0
$715$	2.02604e7	1.48212
$716$	0	0
$717$	3.37276e6	0.245012
$718$	0	0
$719$	1.56427e7	1.12847	0.564233	−	0.825615i	$-0.309172\pi$
$719$	1.56427e7	1.12847	0.564233	+	0.825615i	$0.309172\pi$
$720$	0	0
$721$	3.17383e6	0.227376
$722$	0	0
$723$	−7.54107e6	−0.536522
$724$	0	0
$725$	2.25175e6	0.159102
$726$	0	0
$727$	−1.85908e7	−1.30456	−0.652279	−	0.757979i	$-0.726187\pi$
$727$	−1.85908e7	−1.30456	−0.652279	+	0.757979i	$0.726187\pi$
$728$	0	0
$729$	−1.12318e6	−0.0782767
$730$	0	0
$731$	1.96601e7	1.36079
$732$	0	0
$733$	−2.49466e7	−1.71495	−0.857476	−	0.514524i	$-0.827969\pi$
$733$	−2.49466e7	−1.71495	−0.857476	+	0.514524i	$0.827969\pi$
$734$	0	0
$735$	2.20555e6	0.150591
$736$	0	0
$737$	1.31422e7	0.891253
$738$	0	0
$739$	−2.42944e7	−1.63642	−0.818211	−	0.574918i	$-0.805034\pi$
$739$	−2.42944e7	−1.63642	−0.818211	+	0.574918i	$0.805034\pi$
$740$	0	0
$741$	−2.00088e7	−1.33868
$742$	0	0
$743$	−4.16541e6	−0.276812	−0.138406	−	0.990376i	$-0.544198\pi$
$743$	−4.16541e6	−0.276812	−0.138406	+	0.990376i	$0.544198\pi$
$744$	0	0
$745$	5.24388e6	0.346148
$746$	0	0
$747$	−96729.9	−0.00634248
$748$	0	0
$749$	7.26032e6	0.472880
$750$	0	0
$751$	2.36434e7	1.52972	0.764858	−	0.644199i	$-0.222809\pi$
$751$	2.36434e7	1.52972	0.764858	+	0.644199i	$0.222809\pi$
$752$	0	0
$753$	−3.08588e7	−1.98332
$754$	0	0
$755$	−1.42564e6	−0.0910209
$756$	0	0
$757$	−1.50108e7	−0.952062	−0.476031	−	0.879429i	$-0.657925\pi$
$757$	−1.50108e7	−0.952062	−0.476031	+	0.879429i	$0.657925\pi$
$758$	0	0
$759$	1.43992e7	0.907262
$760$	0	0
$761$	2.92191e6	0.182897	0.0914483	−	0.995810i	$-0.470850\pi$
$761$	2.92191e6	0.182897	0.0914483	+	0.995810i	$0.470850\pi$
$762$	0	0
$763$	−5.45343e6	−0.339124
$764$	0	0
$765$	7.94025e6	0.490547
$766$	0	0
$767$	−9.25887e6	−0.568290
$768$	0	0
$769$	1.42847e7	0.871073	0.435536	−	0.900171i	$-0.356559\pi$
$769$	1.42847e7	0.871073	0.435536	+	0.900171i	$0.356559\pi$
$770$	0	0
$771$	1.55360e7	0.941246
$772$	0	0
$773$	1.09012e7	0.656186	0.328093	−	0.944645i	$-0.393594\pi$
$773$	1.09012e7	0.656186	0.328093	+	0.944645i	$0.393594\pi$
$774$	0	0
$775$	−1.92455e6	−0.115100
$776$	0	0
$777$	1.03749e6	0.0616496
$778$	0	0
$779$	−1.71875e6	−0.101478
$780$	0	0
$781$	−4.30045e7	−2.52282
$782$	0	0
$783$	4.70459e6	0.274231
$784$	0	0
$785$	−2.44736e7	−1.41750
$786$	0	0
$787$	−2.56449e7	−1.47593	−0.737963	−	0.674841i	$-0.764212\pi$
$787$	−2.56449e7	−1.47593	−0.737963	+	0.674841i	$0.764212\pi$
$788$	0	0
$789$	−9.12555e6	−0.521875
$790$	0	0
$791$	2.11560e6	0.120224
$792$	0	0
$793$	9.26169e6	0.523007
$794$	0	0
$795$	−5.06991e6	−0.284500
$796$	0	0
$797$	−7.73086e6	−0.431104	−0.215552	−	0.976492i	$-0.569155\pi$
$797$	−7.73086e6	−0.431104	−0.215552	+	0.976492i	$0.569155\pi$
$798$	0	0
$799$	9.84931e6	0.545807
$800$	0	0
$801$	1.80269e7	0.992748
$802$	0	0
$803$	1.90282e7	1.04138
$804$	0	0
$805$	−2.52013e6	−0.137067
$806$	0	0
$807$	−3.92909e7	−2.12377
$808$	0	0
$809$	−1.87811e7	−1.00890	−0.504452	−	0.863440i	$-0.668305\pi$
$809$	−1.87811e7	−1.00890	−0.504452	+	0.863440i	$0.668305\pi$
$810$	0	0
$811$	−9.00729e6	−0.480886	−0.240443	−	0.970663i	$-0.577293\pi$
$811$	−9.00729e6	−0.480886	−0.240443	+	0.970663i	$0.577293\pi$
$812$	0	0
$813$	−3.16939e7	−1.68170
$814$	0	0
$815$	−2.05531e7	−1.08388
$816$	0	0
$817$	−2.59240e7	−1.35877
$818$	0	0
$819$	−4.56231e6	−0.237671
$820$	0	0
$821$	9.27965e6	0.480478	0.240239	−	0.970714i	$-0.422774\pi$
$821$	9.27965e6	0.480478	0.240239	+	0.970714i	$0.422774\pi$
$822$	0	0
$823$	1.08308e7	0.557393	0.278697	−	0.960379i	$-0.410098\pi$
$823$	1.08308e7	0.557393	0.278697	+	0.960379i	$0.410098\pi$
$824$	0	0
$825$	−1.22967e7	−0.629004
$826$	0	0
$827$	−2.05230e7	−1.04346	−0.521731	−	0.853110i	$-0.674714\pi$
$827$	−2.05230e7	−1.04346	−0.521731	+	0.853110i	$0.674714\pi$
$828$	0	0
$829$	1.42216e7	0.718724	0.359362	−	0.933198i	$-0.382994\pi$
$829$	1.42216e7	0.718724	0.359362	+	0.933198i	$0.382994\pi$
$830$	0	0
$831$	4.08944e7	2.05429
$832$	0	0
$833$	2.84973e6	0.142296
$834$	0	0
$835$	−1.30595e7	−0.648202
$836$	0	0
$837$	−4.02097e6	−0.198389
$838$	0	0
$839$	9.29934e6	0.456087	0.228043	−	0.973651i	$-0.426767\pi$
$839$	9.29934e6	0.456087	0.228043	+	0.973651i	$0.426767\pi$
$840$	0	0
$841$	−1.47669e7	−0.719944
$842$	0	0
$843$	−1.92965e7	−0.935211
$844$	0	0
$845$	2.43277e6	0.117209
$846$	0	0
$847$	−1.38486e7	−0.663281
$848$	0	0
$849$	−2.74350e7	−1.30628
$850$	0	0
$851$	−1.18546e6	−0.0561131
$852$	0	0
$853$	3.07436e6	0.144671	0.0723357	−	0.997380i	$-0.476955\pi$
$853$	3.07436e6	0.144671	0.0723357	+	0.997380i	$0.476955\pi$
$854$	0	0
$855$	−1.04701e7	−0.489819
$856$	0	0
$857$	3.45835e7	1.60848	0.804242	−	0.594302i	$-0.202572\pi$
$857$	3.45835e7	1.60848	0.804242	+	0.594302i	$0.202572\pi$
$858$	0	0
$859$	1.63022e7	0.753814	0.376907	−	0.926251i	$-0.376988\pi$
$859$	1.63022e7	0.753814	0.376907	+	0.926251i	$0.376988\pi$
$860$	0	0
$861$	−1.05738e6	−0.0486099
$862$	0	0
$863$	−2.56962e7	−1.17447	−0.587235	−	0.809416i	$-0.699784\pi$
$863$	−2.56962e7	−1.17447	−0.587235	+	0.809416i	$0.699784\pi$
$864$	0	0
$865$	−4.66087e6	−0.211800
$866$	0	0
$867$	−218902.	−0.00989012
$868$	0	0
$869$	2.04046e7	0.916596
$870$	0	0
$871$	1.28374e7	0.573366
$872$	0	0
$873$	−3.27803e6	−0.145572
$874$	0	0
$875$	9.31062e6	0.411111
$876$	0	0
$877$	−3.30060e7	−1.44908	−0.724542	−	0.689230i	$-0.757949\pi$
$877$	−3.30060e7	−1.44908	−0.724542	+	0.689230i	$0.757949\pi$
$878$	0	0
$879$	−5.03429e7	−2.19769
$880$	0	0
$881$	2.26705e6	0.0984060	0.0492030	−	0.998789i	$-0.484332\pi$
$881$	2.26705e6	0.0984060	0.0492030	+	0.998789i	$0.484332\pi$
$882$	0	0
$883$	1.97779e7	0.853649	0.426825	−	0.904334i	$-0.359632\pi$
$883$	1.97779e7	0.853649	0.426825	+	0.904334i	$0.359632\pi$
$884$	0	0
$885$	−1.30720e7	−0.561029
$886$	0	0
$887$	36468.3	0.00155635	0.000778173	−	1.00000i	$-0.499752\pi$
$887$	36468.3	0.00155635	0.000778173		1.00000i	$0.499752\pi$
$888$	0	0
$889$	−6.44100e6	−0.273337
$890$	0	0
$891$	−4.88541e7	−2.06161
$892$	0	0
$893$	−1.29874e7	−0.544997
$894$	0	0
$895$	1.54263e7	0.643730
$896$	0	0
$897$	1.40652e7	0.583665
$898$	0	0
$899$	−4.90958e6	−0.202602
$900$	0	0
$901$	−6.55068e6	−0.268828
$902$	0	0
$903$	−1.59485e7	−0.650881
$904$	0	0
$905$	2.36462e7	0.959709
$906$	0	0
$907$	1.62298e7	0.655079	0.327540	−	0.944837i	$-0.393781\pi$
$907$	1.62298e7	0.655079	0.327540	+	0.944837i	$0.393781\pi$
$908$	0	0
$909$	2.59695e7	1.04245
$910$	0	0
$911$	2.61699e7	1.04474	0.522368	−	0.852720i	$-0.325049\pi$
$911$	2.61699e7	1.04474	0.522368	+	0.852720i	$0.325049\pi$
$912$	0	0
$913$	−450241.	−0.0178759
$914$	0	0
$915$	1.30760e7	0.516324
$916$	0	0
$917$	1.71234e7	0.672461
$918$	0	0
$919$	−4.05973e6	−0.158565	−0.0792826	−	0.996852i	$-0.525263\pi$
$919$	−4.05973e6	−0.158565	−0.0792826	+	0.996852i	$0.525263\pi$
$920$	0	0
$921$	−1.73870e7	−0.675421
$922$	0	0
$923$	−4.20070e7	−1.62300
$924$	0	0
$925$	1.01237e6	0.0389031
$926$	0	0
$927$	−9.26907e6	−0.354272
$928$	0	0
$929$	3.69518e6	0.140474	0.0702370	−	0.997530i	$-0.477624\pi$
$929$	3.69518e6	0.140474	0.0702370	+	0.997530i	$0.477624\pi$
$930$	0	0
$931$	−3.75769e6	−0.142084
$932$	0	0
$933$	−3.53927e7	−1.33110
$934$	0	0
$935$	3.69589e7	1.38258
$936$	0	0
$937$	−1.91384e7	−0.712126	−0.356063	−	0.934462i	$-0.615881\pi$
$937$	−1.91384e7	−0.712126	−0.356063	+	0.934462i	$0.615881\pi$
$938$	0	0
$939$	1.86742e7	0.691160
$940$	0	0
$941$	−1.20954e7	−0.445294	−0.222647	−	0.974899i	$-0.571470\pi$
$941$	−1.20954e7	−0.445294	−0.222647	+	0.974899i	$0.571470\pi$
$942$	0	0
$943$	1.20820e6	0.0442445
$944$	0	0
$945$	4.49649e6	0.163793
$946$	0	0
$947$	−1.95969e7	−0.710087	−0.355044	−	0.934850i	$-0.615534\pi$
$947$	−1.95969e7	−0.710087	−0.355044	+	0.934850i	$0.615534\pi$
$948$	0	0
$949$	1.85868e7	0.669945
$950$	0	0
$951$	−5.97890e7	−2.14373
$952$	0	0
$953$	4.23265e7	1.50966	0.754832	−	0.655918i	$-0.227718\pi$
$953$	4.23265e7	1.50966	0.754832	+	0.655918i	$0.227718\pi$
$954$	0	0
$955$	2.98426e6	0.105884
$956$	0	0
$957$	−3.13691e7	−1.10719
$958$	0	0
$959$	−1.89353e7	−0.664852
$960$	0	0
$961$	−2.44330e7	−0.853430
$962$	0	0
$963$	−2.12035e7	−0.736788
$964$	0	0
$965$	2.19419e7	0.758502
$966$	0	0
$967$	−1.39211e6	−0.0478749	−0.0239375	−	0.999713i	$-0.507620\pi$
$967$	−1.39211e6	−0.0478749	−0.0239375	+	0.999713i	$0.507620\pi$
$968$	0	0
$969$	−3.64999e7	−1.24877
$970$	0	0
$971$	4.41877e7	1.50402	0.752009	−	0.659153i	$-0.229085\pi$
$971$	4.41877e7	1.50402	0.752009	+	0.659153i	$0.229085\pi$
$972$	0	0
$973$	−847175.	−0.0286874
$974$	0	0
$975$	−1.20115e7	−0.404654
$976$	0	0
$977$	−5.60457e7	−1.87848	−0.939239	−	0.343264i	$-0.888467\pi$
$977$	−5.60457e7	−1.87848	−0.939239	+	0.343264i	$0.888467\pi$
$978$	0	0
$979$	8.39082e7	2.79800
$980$	0	0
$981$	1.59266e7	0.528384
$982$	0	0
$983$	5.00560e7	1.65224	0.826118	−	0.563497i	$-0.190544\pi$
$983$	5.00560e7	1.65224	0.826118	+	0.563497i	$0.190544\pi$
$984$	0	0
$985$	−1.25298e7	−0.411483
$986$	0	0
$987$	−7.98990e6	−0.261065
$988$	0	0
$989$	1.82233e7	0.592428
$990$	0	0
$991$	−1.59116e7	−0.514670	−0.257335	−	0.966322i	$-0.582844\pi$
$991$	−1.59116e7	−0.514670	−0.257335	+	0.966322i	$0.582844\pi$
$992$	0	0
$993$	−4.31534e7	−1.38881
$994$	0	0
$995$	2.82911e7	0.905924
$996$	0	0
$997$	−4.25995e7	−1.35727	−0.678635	−	0.734476i	$-0.737428\pi$
$997$	−4.25995e7	−1.35727	−0.678635	+	0.734476i	$0.737428\pi$
$998$	0	0
$999$	2.11514e6	0.0670542

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.6.a.u.1.2		2
4.3	odd	2		448.6.a.w.1.1		2
8.3	odd	2		7.6.a.b.1.1	✓	2
8.5	even	2		112.6.a.h.1.1		2
24.5	odd	2		1008.6.a.bq.1.2		2
24.11	even	2		63.6.a.f.1.2		2
40.3	even	4		175.6.b.c.99.2		4
40.19	odd	2		175.6.a.c.1.2		2
40.27	even	4		175.6.b.c.99.3		4
56.3	even	6		49.6.c.d.30.2		4
56.11	odd	6		49.6.c.e.30.2		4
56.13	odd	2		784.6.a.v.1.2		2
56.19	even	6		49.6.c.d.18.2		4
56.27	even	2		49.6.a.f.1.1		2
56.51	odd	6		49.6.c.e.18.2		4
88.43	even	2		847.6.a.c.1.2		2
168.83	odd	2		441.6.a.l.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.6.a.b.1.1	✓	2	8.3	odd	2
49.6.a.f.1.1		2	56.27	even	2
49.6.c.d.18.2		4	56.19	even	6
49.6.c.d.30.2		4	56.3	even	6
49.6.c.e.18.2		4	56.51	odd	6
49.6.c.e.30.2		4	56.11	odd	6
63.6.a.f.1.2		2	24.11	even	2
112.6.a.h.1.1		2	8.5	even	2
175.6.a.c.1.2		2	40.19	odd	2
175.6.b.c.99.2		4	40.3	even	4
175.6.b.c.99.3		4	40.27	even	4
441.6.a.l.1.2		2	168.83	odd	2
448.6.a.u.1.2		2	1.1	even	1	trivial
448.6.a.w.1.1		2	4.3	odd	2
784.6.a.v.1.2		2	56.13	odd	2
847.6.a.c.1.2		2	88.43	even	2
1008.6.a.bq.1.2		2	24.5	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+19.6495 q^{3} +46.7492 q^{5} -49.0000 q^{7} +143.103 q^{9} +O(q^{10})\)	\(q+19.6495 q^{3} +46.7492 q^{5} -49.0000 q^{7} +143.103 q^{9} +666.090 q^{11} +650.640 q^{13} +918.598 q^{15} +1186.89 q^{17} -1565.05 q^{19} -962.826 q^{21} +1100.15 q^{23} -939.515 q^{25} -1962.93 q^{27} -2396.72 q^{29} +2048.46 q^{31} +13088.3 q^{33} -2290.71 q^{35} -1077.54 q^{37} +12784.7 q^{39} +1098.21 q^{41} +16564.3 q^{43} +6689.95 q^{45} +8298.39 q^{47} +2401.00 q^{49} +23321.9 q^{51} -5519.18 q^{53} +31139.1 q^{55} -30752.5 q^{57} -14230.4 q^{59} +14234.7 q^{61} -7012.05 q^{63} +30416.9 q^{65} +19730.4 q^{67} +21617.5 q^{69} -64562.7 q^{71} +28567.0 q^{73} -18461.0 q^{75} -32638.4 q^{77} +30633.4 q^{79} -73344.6 q^{81} -675.946 q^{83} +55486.3 q^{85} -47094.4 q^{87} +125971. q^{89} -31881.3 q^{91} +40251.1 q^{93} -73164.9 q^{95} -22906.8 q^{97} +95319.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} + 18 q^{5} - 98 q^{7} + 558 q^{9}+O(q^{10}) \) 2 * q - 6 * q^3 + 18 * q^5 - 98 * q^7 + 558 * q^9	\( 2 q - 6 q^{3} + 18 q^{5} - 98 q^{7} + 558 q^{9} + 396 q^{11} + 350 q^{13} + 1656 q^{15} + 1800 q^{17} - 3266 q^{19} + 294 q^{21} - 2088 q^{23} - 3238 q^{25} - 6372 q^{27} - 6696 q^{29} + 20 q^{31} + 20016 q^{33} - 882 q^{35} - 6232 q^{37} + 20496 q^{39} - 6048 q^{41} - 3020 q^{43} - 5238 q^{45} - 11700 q^{47} + 4802 q^{49} + 7596 q^{51} - 9468 q^{53} + 38904 q^{55} + 12876 q^{57} - 43938 q^{59} + 64754 q^{61} - 27342 q^{63} + 39060 q^{65} + 24784 q^{67} + 103392 q^{69} - 97416 q^{71} + 17452 q^{73} + 40494 q^{75} - 19404 q^{77} - 51256 q^{79} - 61074 q^{81} + 117558 q^{83} + 37860 q^{85} + 63180 q^{87} + 84276 q^{89} - 17150 q^{91} + 92280 q^{93} - 24264 q^{95} + 20776 q^{97} - 16740 q^{99}+O(q^{100}) \) 2 * q - 6 * q^3 + 18 * q^5 - 98 * q^7 + 558 * q^9 + 396 * q^11 + 350 * q^13 + 1656 * q^15 + 1800 * q^17 - 3266 * q^19 + 294 * q^21 - 2088 * q^23 - 3238 * q^25 - 6372 * q^27 - 6696 * q^29 + 20 * q^31 + 20016 * q^33 - 882 * q^35 - 6232 * q^37 + 20496 * q^39 - 6048 * q^41 - 3020 * q^43 - 5238 * q^45 - 11700 * q^47 + 4802 * q^49 + 7596 * q^51 - 9468 * q^53 + 38904 * q^55 + 12876 * q^57 - 43938 * q^59 + 64754 * q^61 - 27342 * q^63 + 39060 * q^65 + 24784 * q^67 + 103392 * q^69 - 97416 * q^71 + 17452 * q^73 + 40494 * q^75 - 19404 * q^77 - 51256 * q^79 - 61074 * q^81 + 117558 * q^83 + 37860 * q^85 + 63180 * q^87 + 84276 * q^89 - 17150 * q^91 + 92280 * q^93 - 24264 * q^95 + 20776 * q^97 - 16740 * q^99

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