LMFDB - Embedded newform 448.6.f.d.447.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [448,6,Mod(447,448)]

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([1, 0, 1]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("448.447");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	448.f (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$71.8519512762$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} - 674 x^{14} + 3404 x^{13} + 173721 x^{12} - 919512 x^{11} - 21981508 x^{10} + \cdots + 224266997486896 $ x^16 - 4x^15 - 674x^14 + 3404x^13 + 173721x^12 - 919512x^11 - 21981508x^10 + 94817024x^9 + 1652635052x^8 - 4373672096x^7 - 73339451096x^6 + 61523648112x^5 + 1688776741892x^4 + 257949039296x^3 - 14815785541168x^2 + 39188323305792*x + 224266997486896
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{70} $
Twist minimal:	no (minimal twist has level 28)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			447.11
Root			$10.5645 + 0.902498i$ of defining polynomial
Character	$\chi$	$=$	448.447
Dual form			448.6.f.d.447.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+15.7679 q^{3} -70.8301i q^{5} +(78.9551 + 102.826i) q^{7} +5.62572 q^{9} +O(q^{10})$	$q+15.7679 q^{3} -70.8301i q^{5} +(78.9551 + 102.826i) q^{7} +5.62572 q^{9} +279.266i q^{11} +763.841i q^{13} -1116.84i q^{15} +1075.24i q^{17} -2188.55 q^{19} +(1244.95 + 1621.34i) q^{21} -2265.35i q^{23} -1891.90 q^{25} -3742.89 q^{27} -463.760 q^{29} -6231.26 q^{31} +4403.42i q^{33} +(7283.14 - 5592.40i) q^{35} -13683.3 q^{37} +12044.1i q^{39} -5834.56i q^{41} -6789.84i q^{43} -398.470i q^{45} -9963.66 q^{47} +(-4339.19 + 16237.2i) q^{49} +16954.2i q^{51} +7098.85 q^{53} +19780.4 q^{55} -34508.8 q^{57} -6345.28 q^{59} +39093.8i q^{61} +(444.179 + 578.468i) q^{63} +54102.9 q^{65} +58864.5i q^{67} -35719.7i q^{69} +49018.6i q^{71} -16686.3i q^{73} -29831.2 q^{75} +(-28715.6 + 22049.4i) q^{77} -48977.8i q^{79} -60384.4 q^{81} +44054.8 q^{83} +76159.3 q^{85} -7312.50 q^{87} +87121.3i q^{89} +(-78542.3 + 60309.1i) q^{91} -98253.7 q^{93} +155016. i q^{95} -74128.8i q^{97} +1571.07i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 1616 q^{9}+O(q^{10}) $ 16 * q + 1616 * q^9	$ 16 q + 1616 q^{9} + 1792 q^{21} - 9776 q^{25} - 26592 q^{29} + 26272 q^{37} + 8848 q^{49} + 41888 q^{53} - 60288 q^{57} + 66688 q^{65} - 320992 q^{77} + 56336 q^{81} - 78080 q^{85} + 335616 q^{93}+O(q^{100}) $ 16 * q + 1616 * q^9 + 1792 * q^21 - 9776 * q^25 - 26592 * q^29 + 26272 * q^37 + 8848 * q^49 + 41888 * q^53 - 60288 * q^57 + 66688 * q^65 - 320992 * q^77 + 56336 * q^81 - 78080 * q^85 + 335616 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/448\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$197$
$\chi(n)$	$-1$	$-1$	$1$

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
$3$	15.7679			1.01151			0.505755	−	0.862677i	$-0.331214\pi$
$3$	15.7679			1.01151			0.505755	+	0.862677i	$0.331214\pi$
$4$		0			0
$5$		−	70.8301i		−	1.26705i	−0.773723	−	0.633524i	$-0.781608\pi$
$5$		−	70.8301i		−	1.26705i	0.773723	−	0.633524i	$-0.218392\pi$
$6$		0			0
$7$	78.9551	+	102.826i	0.609025	+	0.793151i
$7$	78.9551	+	102.826i	0.609025	+	0.793151i
$8$		0			0
$9$	5.62572			0.0231511
$10$		0			0
$11$			279.266i			0.695882i	0.937516	+	0.347941i	$0.113119\pi$
$11$			279.266i			0.695882i	−0.937516	+	0.347941i	$0.886881\pi$
$12$		0			0
$13$			763.841i			1.25356i	0.779197	+	0.626779i	$0.215627\pi$
$13$			763.841i			1.25356i	−0.779197	+	0.626779i	$0.784373\pi$
$14$		0			0
$15$		−	1116.84i		−	1.28163i
$16$		0			0
$17$			1075.24i			0.902366i	0.892431	+	0.451183i	$0.148998\pi$
$17$			1075.24i			0.902366i	−0.892431	+	0.451183i	$0.851002\pi$
$18$		0			0
$19$	−2188.55			−1.39083			−0.695414	−	0.718609i	$-0.744779\pi$
$19$	−2188.55			−1.39083			−0.695414	+	0.718609i	$0.744779\pi$
$20$		0			0
$21$	1244.95	+	1621.34i	0.616034	+	0.802280i
$22$		0			0
$23$		−	2265.35i		−	0.892925i	−0.894802	−	0.446463i	$-0.852684\pi$
$23$		−	2265.35i		−	0.892925i	0.894802	−	0.446463i	$-0.147316\pi$
$24$		0			0
$25$	−1891.90			−0.605408
$26$		0			0
$27$	−3742.89			−0.988092
$28$		0			0
$29$	−463.760			−0.102400			−0.0511998	−	0.998688i	$-0.516305\pi$
$29$	−463.760			−0.102400			−0.0511998	+	0.998688i	$0.516305\pi$
$30$		0			0
$31$	−6231.26			−1.16459			−0.582293	−	0.812979i	$-0.697844\pi$
$31$	−6231.26			−1.16459			−0.582293	+	0.812979i	$0.697844\pi$
$32$		0			0
$33$			4403.42i			0.703891i
$34$		0			0
$35$	7283.14	−	5592.40i	1.00496	−	0.771663i
$36$		0			0
$37$	−13683.3			−1.64318			−0.821592	−	0.570075i	$-0.806914\pi$
$37$	−13683.3			−1.64318			−0.821592	+	0.570075i	$0.806914\pi$
$38$		0			0
$39$			12044.1i			1.26799i
$40$		0			0
$41$		−	5834.56i		−	0.542061i	−0.962571	−	0.271031i	$-0.912635\pi$
$41$		−	5834.56i		−	0.542061i	0.962571	−	0.271031i	$-0.0873645\pi$
$42$		0			0
$43$		−	6789.84i		−	0.560000i	−0.960000	−	0.280000i	$-0.909665\pi$
$43$		−	6789.84i		−	0.560000i	0.960000	−	0.280000i	$-0.0903345\pi$
$44$		0			0
$45$		−	398.470i		−	0.0293336i
$46$		0			0
$47$	−9963.66			−0.657922			−0.328961	−	0.944344i	$-0.606698\pi$
$47$	−9963.66			−0.657922			−0.328961	+	0.944344i	$0.606698\pi$
$48$		0			0
$49$	−4339.19	+	16237.2i	−0.258177	+	0.966098i
$50$		0			0
$51$			16954.2i			0.912752i
$52$		0			0
$53$	7098.85			0.347135			0.173568	−	0.984822i	$-0.444471\pi$
$53$	7098.85			0.347135			0.173568	+	0.984822i	$0.444471\pi$
$54$		0			0
$55$	19780.4			0.881715
$56$		0			0
$57$	−34508.8			−1.40684
$58$		0			0
$59$	−6345.28			−0.237313			−0.118656	−	0.992935i	$-0.537859\pi$
$59$	−6345.28			−0.237313			−0.118656	+	0.992935i	$0.537859\pi$
$60$		0			0
$61$			39093.8i			1.34519i	0.740011	+	0.672594i	$0.234820\pi$
$61$			39093.8i			1.34519i	−0.740011	+	0.672594i	$0.765180\pi$
$62$		0			0
$63$	444.179	+	578.468i	0.0140996	+	0.0183623i
$64$		0			0
$65$	54102.9			1.58832
$66$		0			0
$67$			58864.5i			1.60202i	0.598654	+	0.801008i	$0.295702\pi$
$67$			58864.5i			1.60202i	−0.598654	+	0.801008i	$0.704298\pi$
$68$		0			0
$69$		−	35719.7i		−	0.903202i
$70$		0			0
$71$			49018.6i			1.15402i	0.816736	+	0.577012i	$0.195781\pi$
$71$			49018.6i			1.15402i	−0.816736	+	0.577012i	$0.804219\pi$
$72$		0			0
$73$		−	16686.3i		−	0.366482i	−0.983068	−	0.183241i	$-0.941341\pi$
$73$		−	16686.3i		−	0.366482i	0.983068	−	0.183241i	$-0.0586588\pi$
$74$		0			0
$75$	−29831.2			−0.612376
$76$		0			0
$77$	−28715.6	+	22049.4i	−0.551940	+	0.423809i
$78$		0			0
$79$		−	48977.8i		−	0.882940i	−0.897276	−	0.441470i	$-0.854457\pi$
$79$		−	48977.8i		−	0.882940i	0.897276	−	0.441470i	$-0.145543\pi$
$80$		0			0
$81$	−60384.4			−1.02262
$82$		0			0
$83$	44054.8			0.701936			0.350968	−	0.936387i	$-0.385853\pi$
$83$	44054.8			0.701936			0.350968	+	0.936387i	$0.385853\pi$
$84$		0			0
$85$	76159.3			1.14334
$86$		0			0
$87$	−7312.50			−0.103578
$88$		0			0
$89$			87121.3i			1.16587i	0.812520	+	0.582934i	$0.198095\pi$
$89$			87121.3i			1.16587i	−0.812520	+	0.582934i	$0.801905\pi$
$90$		0			0
$91$	−78542.3	+	60309.1i	−0.994261	+	0.763448i
$92$		0			0
$93$	−98253.7			−1.17799
$94$		0			0
$95$			155016.i			1.76224i
$96$		0			0
$97$		−	74128.8i		−	0.799941i	−0.916528	−	0.399970i	$-0.869020\pi$
$97$		−	74128.8i		−	0.799941i	0.916528	−	0.399970i	$-0.130980\pi$
$98$		0			0
$99$			1571.07i			0.0161104i
$100$		0			0
$101$		−	62346.0i		−	0.608142i	−0.952649	−	0.304071i	$-0.901654\pi$
$101$		−	62346.0i		−	0.608142i	0.952649	−	0.304071i	$-0.0983460\pi$
$102$		0			0
$103$	−42971.8			−0.399108			−0.199554	−	0.979887i	$-0.563949\pi$
$103$	−42971.8			−0.399108			−0.199554	+	0.979887i	$0.563949\pi$
$104$		0			0
$105$	114840.	−	88180.2i	1.01653	−	0.780544i
$106$		0			0
$107$			30399.9i			0.256692i	0.991729	+	0.128346i	$0.0409668\pi$
$107$			30399.9i			0.256692i	−0.991729	+	0.128346i	$0.959033\pi$
$108$		0			0
$109$	229520.			1.85035			0.925175	−	0.379540i	$-0.123918\pi$
$109$	229520.			1.85035			0.925175	+	0.379540i	$0.123918\pi$
$110$		0			0
$111$	−215756.			−1.66210
$112$		0			0
$113$	94921.6			0.699309			0.349654	−	0.936879i	$-0.386299\pi$
$113$	94921.6			0.699309			0.349654	+	0.936879i	$0.386299\pi$
$114$		0			0
$115$	−160455.			−1.13138
$116$		0			0
$117$			4297.15i			0.0290213i
$118$		0			0
$119$	−110562.	+	84895.6i	−0.715713	+	0.549564i
$120$		0			0
$121$	83061.8			0.515748
$122$		0			0
$123$		−	91998.6i		−	0.548300i
$124$		0			0
$125$		−	87340.5i		−	0.499966i
$126$		0			0
$127$		−	89882.7i		−	0.494501i	−0.968952	−	0.247250i	$-0.920473\pi$
$127$		−	89882.7i		−	0.494501i	0.968952	−	0.247250i	$-0.0795270\pi$
$128$		0			0
$129$		−	107061.i		−	0.566446i
$130$		0			0
$131$	61000.5			0.310567			0.155283	−	0.987870i	$-0.450371\pi$
$131$	61000.5			0.310567			0.155283	+	0.987870i	$0.450371\pi$
$132$		0			0
$133$	−172798.	−	225039.i	−0.847049	−	1.10314i
$134$		0			0
$135$			265109.i			1.25196i
$136$		0			0
$137$	−331632.			−1.50957			−0.754787	−	0.655970i	$-0.772260\pi$
$137$	−331632.			−1.50957			−0.754787	+	0.655970i	$0.772260\pi$
$138$		0			0
$139$	−325020.			−1.42683			−0.713416	−	0.700741i	$-0.752853\pi$
$139$	−325020.			−1.42683			−0.713416	+	0.700741i	$0.752853\pi$
$140$		0			0
$141$	−157106.			−0.665494
$142$		0			0
$143$	−213314.			−0.872328
$144$		0			0
$145$			32848.1i			0.129745i
$146$		0			0
$147$	−68419.7	+	256026.i	−0.261149	+	0.977217i
$148$		0			0
$149$	106848.			0.394278			0.197139	−	0.980376i	$-0.436835\pi$
$149$	106848.			0.394278			0.197139	+	0.980376i	$0.436835\pi$
$150$		0			0
$151$			38254.2i			0.136533i	0.997667	+	0.0682664i	$0.0217468\pi$
$151$			38254.2i			0.136533i	−0.997667	+	0.0682664i	$0.978253\pi$
$152$		0			0
$153$			6049.00i			0.0208908i
$154$		0			0
$155$			441361.i			1.47559i
$156$		0			0
$157$			204295.i			0.661467i	0.943724	+	0.330734i	$0.107296\pi$
$157$			204295.i			0.661467i	−0.943724	+	0.330734i	$0.892704\pi$
$158$		0			0
$159$	111934.			0.351130
$160$		0			0
$161$	232936.	−	178861.i	0.708225	−	0.543814i
$162$		0			0
$163$			468548.i			1.38129i	0.723193	+	0.690646i	$0.242674\pi$
$163$			468548.i			1.38129i	−0.723193	+	0.690646i	$0.757326\pi$
$164$		0			0
$165$	311895.			0.891863
$166$		0			0
$167$	541924.			1.50365			0.751825	−	0.659362i	$-0.229174\pi$
$167$	541924.			1.50365			0.751825	+	0.659362i	$0.229174\pi$
$168$		0			0
$169$	−212159.			−0.571407
$170$		0			0
$171$	−12312.2			−0.0321992
$172$		0			0
$173$			765131.i			1.94366i	0.235682	+	0.971830i	$0.424268\pi$
$173$			765131.i			1.94366i	−0.235682	+	0.971830i	$0.575732\pi$
$174$		0			0
$175$	−149375.	−	194536.i	−0.368709	−	0.480180i
$176$		0			0
$177$	−100052.			−0.240044
$178$		0			0
$179$		−	66373.8i		−	0.154833i	−0.996999	−	0.0774166i	$-0.975333\pi$
$179$		−	66373.8i		−	0.154833i	0.996999	−	0.0774166i	$-0.0246672\pi$
$180$		0			0
$181$		−	6956.59i		−	0.0157834i	−0.999969	−	0.00789169i	$-0.997488\pi$
$181$		−	6956.59i		−	0.0157834i	0.999969	−	0.00789169i	$-0.00251203\pi$
$182$		0			0
$183$			616426.i			1.36067i
$184$		0			0
$185$			969189.i			2.08199i
$186$		0			0
$187$	−300277.			−0.627940
$188$		0			0
$189$	−295520.	−	384864.i	−0.601772	−	0.783706i
$190$		0			0
$191$			603098.i			1.19620i	0.801421	+	0.598101i	$0.204078\pi$
$191$			603098.i			1.19620i	−0.801421	+	0.598101i	$0.795922\pi$
$192$		0			0
$193$	236558.			0.457135			0.228568	−	0.973528i	$-0.426596\pi$
$193$	236558.			0.457135			0.228568	+	0.973528i	$0.426596\pi$
$194$		0			0
$195$	853087.			1.60660
$196$		0			0
$197$	52126.7			0.0956961			0.0478481	−	0.998855i	$-0.484764\pi$
$197$	52126.7			0.0956961			0.0478481	+	0.998855i	$0.484764\pi$
$198$		0			0
$199$	−793615.			−1.42062			−0.710309	−	0.703890i	$-0.751445\pi$
$199$	−793615.			−1.42062			−0.710309	+	0.703890i	$0.751445\pi$
$200$		0			0
$201$			928168.i			1.62045i
$202$		0			0
$203$	−36616.2	−	47686.3i	−0.0623639	−	0.0812183i
$204$		0			0
$205$	−413262.			−0.686817
$206$		0			0
$207$		−	12744.2i		−	0.0206722i
$208$		0			0
$209$		−	611188.i		−	0.967852i
$210$		0			0
$211$		−	651221.i		−	1.00698i	−0.864000	−	0.503491i	$-0.832049\pi$
$211$		−	651221.i		−	1.00698i	0.864000	−	0.503491i	$-0.167951\pi$
$212$		0			0
$213$			772919.i			1.16731i
$214$		0			0
$215$	−480925.			−0.709547
$216$		0			0
$217$	−491990.	−	640733.i	−0.709262	−	0.923693i
$218$		0			0
$219$		−	263107.i		−	0.370700i
$220$		0			0
$221$	−821311.			−1.13117
$222$		0			0
$223$	509823.			0.686526			0.343263	−	0.939239i	$-0.388468\pi$
$223$	509823.			0.686526			0.343263	+	0.939239i	$0.388468\pi$
$224$		0			0
$225$	−10643.3			−0.0140159
$226$		0			0
$227$	−918299.			−1.18282			−0.591411	−	0.806370i	$-0.701429\pi$
$227$	−918299.			−1.18282			−0.591411	+	0.806370i	$0.701429\pi$
$228$		0			0
$229$		−	698573.i		−	0.880284i	−0.897928	−	0.440142i	$-0.854928\pi$
$229$		−	698573.i		−	0.880284i	0.897928	−	0.440142i	$-0.145072\pi$
$230$		0			0
$231$	−452784.	+	347673.i	−0.558292	+	0.428687i
$232$		0			0
$233$	−328778.			−0.396746			−0.198373	−	0.980127i	$-0.563566\pi$
$233$	−328778.			−0.396746			−0.198373	+	0.980127i	$0.563566\pi$
$234$		0			0
$235$			705727.i			0.833618i
$236$		0			0
$237$		−	772275.i		−	0.893102i
$238$		0			0
$239$		−	411410.i		−	0.465887i	−0.972490	−	0.232943i	$-0.925164\pi$
$239$		−	411410.i		−	0.465887i	0.972490	−	0.232943i	$-0.0748357\pi$
$240$		0			0
$241$		−	490281.i		−	0.543754i	−0.962332	−	0.271877i	$-0.912356\pi$
$241$		−	490281.i		−	0.543754i	0.962332	−	0.271877i	$-0.0876445\pi$
$242$		0			0
$243$	−42611.9			−0.0462930
$244$		0			0
$245$	1.15008e6	+	307345.i	1.22409	+	0.327123i
$246$		0			0
$247$		−	1.67171e6i		−	1.74348i
$248$		0			0
$249$	694650.			0.710015
$250$		0			0
$251$	891902.			0.893579			0.446789	−	0.894639i	$-0.352567\pi$
$251$	891902.			0.893579			0.446789	+	0.894639i	$0.352567\pi$
$252$		0			0
$253$	632633.			0.621371
$254$		0			0
$255$	1.20087e6			1.15650
$256$		0			0
$257$		−	297270.i		−	0.280749i	−0.990098	−	0.140375i	$-0.955169\pi$
$257$		−	297270.i		−	0.280749i	0.990098	−	0.140375i	$-0.0448307\pi$
$258$		0			0
$259$	−1.08037e6	−	1.40699e6i	−1.00074	−	1.30329i
$260$		0			0
$261$	−2608.98			−0.00237066
$262$		0			0
$263$		−	733951.i		−	0.654301i	−0.944972	−	0.327151i	$-0.893912\pi$
$263$		−	733951.i		−	0.654301i	0.944972	−	0.327151i	$-0.106088\pi$
$264$		0			0
$265$		−	502812.i		−	0.439836i
$266$		0			0
$267$			1.37372e6i			1.17929i
$268$		0			0
$269$		−	708666.i		−	0.597119i	−0.954391	−	0.298560i	$-0.903494\pi$
$269$		−	708666.i		−	0.597119i	0.954391	−	0.298560i	$-0.0965062\pi$
$270$		0			0
$271$	250560.			0.207247			0.103624	−	0.994617i	$-0.466956\pi$
$271$	250560.			0.207247			0.103624	+	0.994617i	$0.466956\pi$
$272$		0			0
$273$	−1.23845e6	+	950946.i	−1.00570	+	0.772235i
$274$		0			0
$275$		−	528343.i		−	0.421293i
$276$		0			0
$277$	−542192.			−0.424575			−0.212287	−	0.977207i	$-0.568091\pi$
$277$	−542192.			−0.424575			−0.212287	+	0.977207i	$0.568091\pi$
$278$		0			0
$279$	−35055.4			−0.0269615
$280$		0			0
$281$	945562.			0.714372			0.357186	−	0.934033i	$-0.383736\pi$
$281$	945562.			0.714372			0.357186	+	0.934033i	$0.383736\pi$
$282$		0			0
$283$	−324976.			−0.241205			−0.120602	−	0.992701i	$-0.538483\pi$
$283$	−324976.			−0.241205			−0.120602	+	0.992701i	$0.538483\pi$
$284$		0			0
$285$			2.44426e6i			1.78253i
$286$		0			0
$287$	599942.	−	460668.i	0.429937	−	0.330129i
$288$		0			0
$289$	263717.			0.185735
$290$		0			0
$291$		−	1.16885e6i		−	0.809147i
$292$		0			0
$293$		−	1.39233e6i		−	0.947490i	−0.880662	−	0.473745i	$-0.842902\pi$
$293$		−	1.39233e6i		−	0.947490i	0.880662	−	0.473745i	$-0.157098\pi$
$294$		0			0
$295$			449437.i			0.300686i
$296$		0			0
$297$		−	1.04526e6i		−	0.687595i
$298$		0			0
$299$	1.73036e6			1.11933
$300$		0			0
$301$	698169.	−	536092.i	0.444165	−	0.341054i
$302$		0			0
$303$		−	983063.i		−	0.615141i
$304$		0			0
$305$	2.76902e6			1.70442
$306$		0			0
$307$	1.57578e6			0.954221			0.477111	−	0.878843i	$-0.341684\pi$
$307$	1.57578e6			0.954221			0.477111	+	0.878843i	$0.341684\pi$
$308$		0			0
$309$	−677573.			−0.403701
$310$		0			0
$311$	−2.22461e6			−1.30423			−0.652113	−	0.758122i	$-0.726117\pi$
$311$	−2.22461e6			−1.30423			−0.652113	+	0.758122i	$0.726117\pi$
$312$		0			0
$313$			653881.i			0.377257i	0.982048	+	0.188629i	$0.0604043\pi$
$313$			653881.i			0.377257i	−0.982048	+	0.188629i	$0.939596\pi$
$314$		0			0
$315$	40972.9	−	31461.3i	0.0232659	−	0.0178649i
$316$		0			0
$317$	2.07545e6			1.16002			0.580008	−	0.814611i	$-0.303049\pi$
$317$	2.07545e6			1.16002			0.580008	+	0.814611i	$0.303049\pi$
$318$		0			0
$319$		−	129512.i		−	0.0712580i
$320$		0			0
$321$			479341.i			0.259646i
$322$		0			0
$323$		−	2.35322e6i		−	1.25504i
$324$		0			0
$325$		−	1.44511e6i		−	0.758914i
$326$		0			0
$327$	3.61904e6			1.87165
$328$		0			0
$329$	−786682.	−	1.02452e6i	−0.400691	−	0.521832i
$330$		0			0
$331$		−	2.35073e6i		−	1.17932i	−0.807651	−	0.589661i	$-0.799261\pi$
$331$		−	2.35073e6i		−	1.17932i	0.807651	−	0.589661i	$-0.200739\pi$
$332$		0			0
$333$	−76978.4			−0.0380416
$334$		0			0
$335$	4.16938e6			2.02983
$336$		0			0
$337$	−509080.			−0.244181			−0.122090	−	0.992519i	$-0.538960\pi$
$337$	−509080.			−0.244181			−0.122090	+	0.992519i	$0.538960\pi$
$338$		0			0
$339$	1.49671e6			0.707358
$340$		0			0
$341$		−	1.74018e6i		−	0.810415i
$342$		0			0
$343$	−2.01220e6	+	835830.i	−0.923498	+	0.383604i
$344$		0			0
$345$	−2.53003e6			−1.14440
$346$		0			0
$347$			2.80255e6i			1.24948i	0.780833	+	0.624740i	$0.214795\pi$
$347$			2.80255e6i			1.24948i	−0.780833	+	0.624740i	$0.785205\pi$
$348$		0			0
$349$			2.33681e6i			1.02698i	0.858097	+	0.513488i	$0.171647\pi$
$349$			2.33681e6i			1.02698i	−0.858097	+	0.513488i	$0.828353\pi$
$350$		0			0
$351$		−	2.85897e6i		−	1.23863i
$352$		0			0
$353$		−	242114.i		−	0.103415i	−0.998662	−	0.0517075i	$-0.983534\pi$
$353$		−	242114.i		−	0.103415i	0.998662	−	0.0517075i	$-0.0164664\pi$
$354$		0			0
$355$	3.47199e6			1.46220
$356$		0			0
$357$	−1.74333e6	+	1.33862e6i	−0.723950	+	0.555889i
$358$		0			0
$359$			3.59430e6i			1.47190i	0.677036	+	0.735950i	$0.263264\pi$
$359$			3.59430e6i			1.47190i	−0.677036	+	0.735950i	$0.736736\pi$
$360$		0			0
$361$	2.31367e6			0.934403
$362$		0			0
$363$	1.30971e6			0.521684
$364$		0			0
$365$	−1.18189e6			−0.464349
$366$		0			0
$367$	21874.3			0.00847754			0.00423877	−	0.999991i	$-0.498651\pi$
$367$	21874.3			0.00847754			0.00423877	+	0.999991i	$0.498651\pi$
$368$		0			0
$369$		−	32823.6i		−	0.0125493i
$370$		0			0
$371$	560490.	+	729943.i	0.211414	+	0.275331i
$372$		0			0
$373$	−202795.			−0.0754719			−0.0377360	−	0.999288i	$-0.512015\pi$
$373$	−202795.			−0.0754719			−0.0377360	+	0.999288i	$0.512015\pi$
$374$		0			0
$375$		−	1.37717e6i		−	0.505721i
$376$		0			0
$377$		−	354238.i		−	0.128364i
$378$		0			0
$379$		−	2.41277e6i		−	0.862817i	−0.902157	−	0.431409i	$-0.858017\pi$
$379$		−	2.41277e6i		−	0.862817i	0.902157	−	0.431409i	$-0.141983\pi$
$380$		0			0
$381$		−	1.41726e6i		−	0.500192i
$382$		0			0
$383$	1.17565e6			0.409526			0.204763	−	0.978812i	$-0.434358\pi$
$383$	1.17565e6			0.409526			0.204763	+	0.978812i	$0.434358\pi$
$384$		0			0
$385$	1.56176e6	+	2.03393e6i	0.536986	+	0.699333i
$386$		0			0
$387$		−	38197.7i		−	0.0129646i
$388$		0			0
$389$	54498.1			0.0182603			0.00913013	−	0.999958i	$-0.497094\pi$
$389$	54498.1			0.0182603			0.00913013	+	0.999958i	$0.497094\pi$
$390$		0			0
$391$	2.43579e6			0.805746
$392$		0			0
$393$	961848.			0.314141
$394$		0			0
$395$	−3.46910e6			−1.11873
$396$		0			0
$397$		−	2.42519e6i		−	0.772271i	−0.922442	−	0.386136i	$-0.873810\pi$
$397$		−	2.42519e6i		−	0.772271i	0.922442	−	0.386136i	$-0.126190\pi$
$398$		0			0
$399$	−2.72465e6	−	3.54839e6i	−0.856798	−	1.11583i
$400$		0			0
$401$	−4.87290e6			−1.51331			−0.756653	−	0.653817i	$-0.773167\pi$
$401$	−4.87290e6			−1.51331			−0.756653	+	0.653817i	$0.773167\pi$
$402$		0			0
$403$		−	4.75969e6i		−	1.45988i
$404$		0			0
$405$			4.27703e6i			1.29570i
$406$		0			0
$407$		−	3.82127e6i		−	1.14346i
$408$		0			0
$409$		−	1.20868e6i		−	0.357276i	−0.983915	−	0.178638i	$-0.942831\pi$
$409$		−	1.20868e6i		−	0.357276i	0.983915	−	0.178638i	$-0.0571692\pi$
$410$		0			0
$411$	−5.22912e6			−1.52695
$412$		0			0
$413$	−500992.	−	652457.i	−0.144529	−	0.188225i
$414$		0			0
$415$		−	3.12040e6i		−	0.889386i
$416$		0			0
$417$	−5.12487e6			−1.44325
$418$		0			0
$419$	1.74195e6			0.484731			0.242365	−	0.970185i	$-0.422077\pi$
$419$	1.74195e6			0.484731			0.242365	+	0.970185i	$0.422077\pi$
$420$		0			0
$421$	1.28742e6			0.354009			0.177004	−	0.984210i	$-0.443359\pi$
$421$	1.28742e6			0.354009			0.177004	+	0.984210i	$0.443359\pi$
$422$		0			0
$423$	−56052.8			−0.0152316
$424$		0			0
$425$		−	2.03425e6i		−	0.546300i
$426$		0			0
$427$	−4.01984e6	+	3.08665e6i	−1.06694	+	0.819253i
$428$		0			0
$429$	−3.36351e6			−0.882368
$430$		0			0
$431$		−	2.69469e6i		−	0.698740i	−0.936985	−	0.349370i	$-0.886396\pi$
$431$		−	2.69469e6i		−	0.698740i	0.936985	−	0.349370i	$-0.113604\pi$
$432$		0			0
$433$			5.13919e6i			1.31727i	0.752463	+	0.658635i	$0.228866\pi$
$433$			5.13919e6i			1.31727i	−0.752463	+	0.658635i	$0.771134\pi$
$434$		0			0
$435$			517945.i			0.131238i
$436$		0			0
$437$			4.95784e6i			1.24191i
$438$		0			0
$439$	−6.00197e6			−1.48639			−0.743195	−	0.669075i	$-0.766690\pi$
$439$	−6.00197e6			−1.48639			−0.743195	+	0.669075i	$0.766690\pi$
$440$		0			0
$441$	−24411.1	+	91346.0i	−0.00597710	+	0.0223662i
$442$		0			0
$443$			2.17056e6i			0.525487i	0.964866	+	0.262744i	$0.0846273\pi$
$443$			2.17056e6i			0.525487i	−0.964866	+	0.262744i	$0.915373\pi$
$444$		0			0
$445$	6.17081e6			1.47721
$446$		0			0
$447$	1.68477e6			0.398815
$448$		0			0
$449$	2.72833e6			0.638677			0.319338	−	0.947641i	$-0.396539\pi$
$449$	2.72833e6			0.638677			0.319338	+	0.947641i	$0.396539\pi$
$450$		0			0
$451$	1.62939e6			0.377211
$452$		0			0
$453$			603187.i			0.138104i
$454$		0			0
$455$	4.27170e6	+	5.56316e6i	0.967324	+	1.25978i
$456$		0			0
$457$	3.69770e6			0.828211			0.414105	−	0.910229i	$-0.364094\pi$
$457$	3.69770e6			0.828211			0.414105	+	0.910229i	$0.364094\pi$
$458$		0			0
$459$		−	4.02450e6i		−	0.891621i
$460$		0			0
$461$		−	1.13783e6i		−	0.249358i	−0.992197	−	0.124679i	$-0.960210\pi$
$461$		−	1.13783e6i		−	0.249358i	0.992197	−	0.124679i	$-0.0397901\pi$
$462$		0			0
$463$			1.37274e6i			0.297601i	0.988867	+	0.148801i	$0.0475413\pi$
$463$			1.37274e6i			0.297601i	−0.988867	+	0.148801i	$0.952459\pi$
$464$		0			0
$465$			6.95932e6i			1.49257i
$466$		0			0
$467$	1.66512e6			0.353308			0.176654	−	0.984273i	$-0.443473\pi$
$467$	1.66512e6			0.353308			0.176654	+	0.984273i	$0.443473\pi$
$468$		0			0
$469$	−6.05278e6	+	4.64765e6i	−1.27064	+	0.975667i
$470$		0			0
$471$			3.22130e6i			0.669080i
$472$		0			0
$473$	1.89617e6			0.389694
$474$		0			0
$475$	4.14053e6			0.842019
$476$		0			0
$477$	39936.2			0.00803656
$478$		0			0
$479$	−2.58576e6			−0.514932			−0.257466	−	0.966287i	$-0.582888\pi$
$479$	−2.58576e6			−0.514932			−0.257466	+	0.966287i	$0.582888\pi$
$480$		0			0
$481$		−	1.04519e7i		−	2.05983i
$482$		0			0
$483$	3.67290e6	−	2.82025e6i	0.716376	−	0.550073i
$484$		0			0
$485$	−5.25055e6			−1.01356
$486$		0			0
$487$			7.62928e6i			1.45768i	0.684686	+	0.728838i	$0.259939\pi$
$487$			7.62928e6i			1.45768i	−0.684686	+	0.728838i	$0.740061\pi$
$488$		0			0
$489$			7.38801e6i			1.39719i
$490$		0			0
$491$			5.68722e6i			1.06462i	0.846548	+	0.532312i	$0.178677\pi$
$491$			5.68722e6i			1.06462i	−0.846548	+	0.532312i	$0.821323\pi$
$492$		0			0
$493$		−	498653.i		−	0.0924019i
$494$		0			0
$495$	111279.			0.0204127
$496$		0			0
$497$	−5.04036e6	+	3.87027e6i	−0.915315	+	0.702829i
$498$		0			0
$499$		−	1.53763e6i		−	0.276440i	−0.990402	−	0.138220i	$-0.955862\pi$
$499$		−	1.53763e6i		−	0.276440i	0.990402	−	0.138220i	$-0.0441382\pi$
$500$		0			0
$501$	8.54498e6			1.52096
$502$		0			0
$503$	−8.77651e6			−1.54668			−0.773342	−	0.633988i	$-0.781417\pi$
$503$	−8.77651e6			−1.54668			−0.773342	+	0.633988i	$0.781417\pi$
$504$		0			0
$505$	−4.41597e6			−0.770545
$506$		0			0
$507$	−3.34530e6			−0.577983
$508$		0			0
$509$		−	1.03089e7i		−	1.76368i	−0.471549	−	0.881840i	$-0.656305\pi$
$509$		−	1.03089e7i		−	1.76368i	0.471549	−	0.881840i	$-0.343695\pi$
$510$		0			0
$511$	1.71578e6	−	1.31747e6i	0.290675	−	0.223196i
$512$		0			0
$513$	8.19151e6			1.37427
$514$		0			0
$515$			3.04369e6i			0.505688i
$516$		0			0
$517$		−	2.78251e6i		−	0.457836i
$518$		0			0
$519$			1.20645e7i			1.96603i
$520$		0			0
$521$		−	6.94328e6i		−	1.12065i	−0.828272	−	0.560326i	$-0.810676\pi$
$521$		−	6.94328e6i		−	1.12065i	0.828272	−	0.560326i	$-0.189324\pi$
$522$		0			0
$523$	−2.13266e6			−0.340932			−0.170466	−	0.985364i	$-0.554527\pi$
$523$	−2.13266e6			−0.340932			−0.170466	+	0.985364i	$0.554527\pi$
$524$		0			0
$525$	−2.35533e6	−	3.06741e6i	−0.372952	−	0.485707i
$526$		0			0
$527$		−	6.70010e6i		−	1.05088i
$528$		0			0
$529$	1.30454e6			0.202684
$530$		0			0
$531$	−35696.8			−0.00549405
$532$		0			0
$533$	4.45667e6			0.679505
$534$		0			0
$535$	2.15322e6			0.325241
$536$		0			0
$537$		−	1.04657e6i		−	0.156615i
$538$		0			0
$539$	−4.53449e6	−	1.21179e6i	−0.672290	−	0.179661i
$540$		0			0
$541$	−5.66703e6			−0.832458			−0.416229	−	0.909260i	$-0.636649\pi$
$541$	−5.66703e6			−0.832458			−0.416229	+	0.909260i	$0.636649\pi$
$542$		0			0
$543$		−	109691.i		−	0.0159650i
$544$		0			0
$545$		−	1.62569e7i		−	2.34448i
$546$		0			0
$547$			201461.i			0.0287887i	0.999896	+	0.0143943i	$0.00458202\pi$
$547$			201461.i			0.0287887i	−0.999896	+	0.0143943i	$0.995418\pi$
$548$		0			0
$549$			219931.i			0.0311426i
$550$		0			0
$551$	1.01496e6			0.142420
$552$		0			0
$553$	5.03617e6	−	3.86704e6i	0.700305	−	0.537732i
$554$		0			0
$555$			1.52820e7i			2.10595i
$556$		0			0
$557$	7.77856e6			1.06233			0.531167	−	0.847267i	$-0.321754\pi$
$557$	7.77856e6			1.06233			0.531167	+	0.847267i	$0.321754\pi$
$558$		0			0
$559$	5.18635e6			0.701993
$560$		0			0
$561$	−4.73473e6			−0.635168
$562$		0			0
$563$	−5.96967e6			−0.793742			−0.396871	−	0.917874i	$-0.629904\pi$
$563$	−5.96967e6			−0.793742			−0.396871	+	0.917874i	$0.629904\pi$
$564$		0			0
$565$		−	6.72331e6i		−	0.886057i
$566$		0			0
$567$	−4.76766e6	−	6.20906e6i	−0.622798	−	0.811088i
$568$		0			0
$569$	−459183.			−0.0594573			−0.0297287	−	0.999558i	$-0.509464\pi$
$569$	−459183.			−0.0594573			−0.0297287	+	0.999558i	$0.509464\pi$
$570$		0			0
$571$			1.49706e7i			1.92154i	0.277341	+	0.960772i	$0.410547\pi$
$571$			1.49706e7i			1.92154i	−0.277341	+	0.960772i	$0.589453\pi$
$572$		0			0
$573$			9.50957e6i			1.20997i
$574$		0			0
$575$			4.28581e6i			0.540584i
$576$		0			0
$577$			4.40336e6i			0.550610i	0.961357	+	0.275305i	$0.0887789\pi$
$577$			4.40336e6i			0.550610i	−0.961357	+	0.275305i	$0.911221\pi$
$578$		0			0
$579$	3.73002e6			0.462397
$580$		0			0
$581$	3.47835e6	+	4.52996e6i	0.427497	+	0.556742i
$582$		0			0
$583$			1.98246e6i			0.241565i
$584$		0			0
$585$	304368.			0.0367713
$586$		0			0
$587$	−1.55156e7			−1.85855			−0.929274	−	0.369392i	$-0.879566\pi$
$587$	−1.55156e7			−1.85855			−0.929274	+	0.369392i	$0.879566\pi$
$588$		0			0
$589$	1.36375e7			1.61974
$590$		0			0
$591$	821926.			0.0967975
$592$		0			0
$593$			6.08906e6i			0.711071i	0.934663	+	0.355536i	$0.115702\pi$
$593$			6.08906e6i			0.711071i	−0.934663	+	0.355536i	$0.884298\pi$
$594$		0			0
$595$	6.01316e6	+	7.83112e6i	0.696323	+	0.906842i
$596$		0			0
$597$	−1.25136e7			−1.43697
$598$		0			0
$599$			9.33090e6i			1.06257i	0.847194	+	0.531283i	$0.178290\pi$
$599$			9.33090e6i			1.06257i	−0.847194	+	0.531283i	$0.821710\pi$
$600$		0			0
$601$			7.97882e6i			0.901057i	0.892762	+	0.450529i	$0.148764\pi$
$601$			7.97882e6i			0.901057i	−0.892762	+	0.450529i	$0.851236\pi$
$602$		0			0
$603$			331155.i			0.0370884i
$604$		0			0
$605$		−	5.88327e6i		−	0.653477i
$606$		0			0
$607$	6.71902e6			0.740175			0.370087	−	0.928997i	$-0.379328\pi$
$607$	6.71902e6			0.740175			0.370087	+	0.928997i	$0.379328\pi$
$608$		0			0
$609$	−577359.	−	751912.i	−0.0630816	−	0.0821531i
$610$		0			0
$611$		−	7.61065e6i		−	0.824743i
$612$		0			0
$613$	4.32747e6			0.465139			0.232569	−	0.972580i	$-0.425287\pi$
$613$	4.32747e6			0.465139			0.232569	+	0.972580i	$0.425287\pi$
$614$		0			0
$615$	−6.51627e6			−0.694722
$616$		0			0
$617$	8.42458e6			0.890913			0.445457	−	0.895303i	$-0.353041\pi$
$617$	8.42458e6			0.890913			0.445457	+	0.895303i	$0.353041\pi$
$618$		0			0
$619$	8.25677e6			0.866131			0.433065	−	0.901363i	$-0.357432\pi$
$619$	8.25677e6			0.866131			0.433065	+	0.901363i	$0.357432\pi$
$620$		0			0
$621$			8.47894e6i			0.882292i
$622$		0			0
$623$	−8.95829e6	+	6.87867e6i	−0.924709	+	0.710042i
$624$		0			0
$625$	−1.20985e7			−1.23889
$626$		0			0
$627$		−	9.63713e6i		−	0.978992i
$628$		0			0
$629$		−	1.47128e7i		−	1.48275i
$630$		0			0
$631$			3.21794e6i			0.321740i	0.986976	+	0.160870i	$0.0514299\pi$
$631$			3.21794e6i			0.321740i	−0.986976	+	0.160870i	$0.948570\pi$
$632$		0			0
$633$		−	1.02684e7i		−	1.01857i
$634$		0			0
$635$	−6.36640e6			−0.626555
$636$		0			0
$637$	−1.24026e7	−	3.31445e6i	−1.21106	−	0.323640i
$638$		0			0
$639$			275765.i			0.0267169i
$640$		0			0
$641$	1.98277e7			1.90601			0.953007	−	0.302947i	$-0.0979706\pi$
$641$	1.98277e7			1.90601			0.953007	+	0.302947i	$0.0979706\pi$
$642$		0			0
$643$	−9.72661e6			−0.927757			−0.463878	−	0.885899i	$-0.653543\pi$
$643$	−9.72661e6			−0.927757			−0.463878	+	0.885899i	$0.653543\pi$
$644$		0			0
$645$	−7.58316e6			−0.717713
$646$		0			0
$647$	−2.09814e7			−1.97049			−0.985246	−	0.171144i	$-0.945254\pi$
$647$	−2.09814e7			−1.97049			−0.985246	+	0.171144i	$0.945254\pi$
$648$		0			0
$649$		−	1.77202e6i		−	0.165142i
$650$		0			0
$651$	−7.75763e6	−	1.01030e7i	−0.717425	−	0.934324i
$652$		0			0
$653$	−1.41312e6			−0.129687			−0.0648433	−	0.997895i	$-0.520655\pi$
$653$	−1.41312e6			−0.129687			−0.0648433	+	0.997895i	$0.520655\pi$
$654$		0			0
$655$		−	4.32067e6i		−	0.393503i
$656$		0			0
$657$		−	93872.3i		−	0.00848446i
$658$		0			0
$659$			304480.i			0.0273115i	0.999907	+	0.0136557i	$0.00434689\pi$
$659$			304480.i			0.0273115i	−0.999907	+	0.0136557i	$0.995653\pi$
$660$		0			0
$661$		−	4.49788e6i		−	0.400410i	−0.979754	−	0.200205i	$-0.935839\pi$
$661$		−	4.49788e6i		−	0.400410i	0.979754	−	0.200205i	$-0.0641608\pi$
$662$		0			0
$663$	−1.29503e7			−1.14419
$664$		0			0
$665$	−1.59396e7	+	1.22393e7i	−1.39773	+	1.07325i
$666$		0			0
$667$			1.05058e6i			0.0914351i
$668$		0			0
$669$	8.03882e6			0.694428
$670$		0			0
$671$	−1.09175e7			−0.936093
$672$		0			0
$673$	204773.			0.0174275			0.00871375	−	0.999962i	$-0.497226\pi$
$673$	204773.			0.0174275			0.00871375	+	0.999962i	$0.497226\pi$
$674$		0			0
$675$	7.08117e6			0.598199
$676$		0			0
$677$			2.21780e7i			1.85974i	0.367893	+	0.929868i	$0.380079\pi$
$677$			2.21780e7i			1.85974i	−0.367893	+	0.929868i	$0.619921\pi$
$678$		0			0
$679$	7.62234e6	−	5.85285e6i	0.634474	−	0.487184i
$680$		0			0
$681$	−1.44796e7			−1.19644
$682$		0			0
$683$			6.68716e6i			0.548517i	0.961656	+	0.274258i	$0.0884323\pi$
$683$			6.68716e6i			0.548517i	−0.961656	+	0.274258i	$0.911568\pi$
$684$		0			0
$685$			2.34895e7i			1.91270i
$686$		0			0
$687$		−	1.10150e7i		−	0.890416i
$688$		0			0
$689$			5.42239e6i			0.435154i
$690$		0			0
$691$	−6.62433e6			−0.527773			−0.263886	−	0.964554i	$-0.585004\pi$
$691$	−6.62433e6			−0.527773			−0.263886	+	0.964554i	$0.585004\pi$
$692$		0			0
$693$	−161546.	+	124044.i	−0.0127780	+	0.00981166i
$694$		0			0
$695$			2.30212e7i			1.80786i
$696$		0			0
$697$	6.27355e6			0.489138
$698$		0			0
$699$	−5.18412e6			−0.401312
$700$		0			0
$701$	−1.86076e7			−1.43019			−0.715096	−	0.699026i	$-0.753617\pi$
$701$	−1.86076e7			−1.43019			−0.715096	+	0.699026i	$0.753617\pi$
$702$		0			0
$703$	2.99466e7			2.28539
$704$		0			0
$705$			1.11278e7i			0.843212i
$706$		0			0
$707$	6.41076e6	−	4.92253e6i	0.482349	−	0.370374i
$708$		0			0
$709$	1.36578e7			1.02039			0.510193	−	0.860060i	$-0.329574\pi$
$709$	1.36578e7			1.02039			0.510193	+	0.860060i	$0.329574\pi$
$710$		0			0
$711$		−	275535.i		−	0.0204411i
$712$		0			0
$713$			1.41160e7i			1.03989i
$714$		0			0
$715$			1.51091e7i			1.10528i
$716$		0			0
$717$		−	6.48707e6i		−	0.471249i
$718$		0			0
$719$	−1.01996e7			−0.735803			−0.367901	−	0.929865i	$-0.619924\pi$
$719$	−1.01996e7			−0.735803			−0.367901	+	0.929865i	$0.619924\pi$
$720$		0			0
$721$	−3.39284e6	−	4.41860e6i	−0.243067	−	0.316553i
$722$		0			0
$723$		−	7.73069e6i		−	0.550013i
$724$		0			0
$725$	877387.			0.0619935
$726$		0			0
$727$	−1.00774e7			−0.707150			−0.353575	−	0.935406i	$-0.615034\pi$
$727$	−1.00774e7			−0.707150			−0.353575	+	0.935406i	$0.615034\pi$
$728$		0			0
$729$	1.40015e7			0.975789
$730$		0			0
$731$	7.30070e6			0.505326
$732$		0			0
$733$			4.06838e6i			0.279680i	0.990174	+	0.139840i	$0.0446589\pi$
$733$			4.06838e6i			0.279680i	−0.990174	+	0.139840i	$0.955341\pi$
$734$		0			0
$735$	1.81343e7	+	4.84618e6i	1.23818	+	0.330888i
$736$		0			0
$737$	−1.64388e7			−1.11481
$738$		0			0
$739$		−	7.40684e6i		−	0.498909i	−0.968386	−	0.249455i	$-0.919749\pi$
$739$		−	7.40684e6i		−	0.498909i	0.968386	−	0.249455i	$-0.0802514\pi$
$740$		0			0
$741$		−	2.63593e7i		−	1.76355i
$742$		0			0
$743$		−	9.71045e6i		−	0.645308i	−0.946517	−	0.322654i	$-0.895425\pi$
$743$		−	9.71045e6i		−	0.645308i	0.946517	−	0.322654i	$-0.104575\pi$
$744$		0			0
$745$		−	7.56808e6i		−	0.499568i
$746$		0			0
$747$	247840.			0.0162506
$748$		0			0
$749$	−3.12588e6	+	2.40022e6i	−0.203595	+	0.156332i
$750$		0			0
$751$		−	4.66373e6i		−	0.301741i	−0.988554	−	0.150870i	$-0.951792\pi$
$751$		−	4.66373e6i		−	0.301741i	0.988554	−	0.150870i	$-0.0482076\pi$
$752$		0			0
$753$	1.40634e7			0.903863
$754$		0			0
$755$	2.70955e6			0.172993
$756$		0			0
$757$	8.39699e6			0.532579			0.266289	−	0.963893i	$-0.414202\pi$
$757$	8.39699e6			0.532579			0.266289	+	0.963893i	$0.414202\pi$
$758$		0			0
$759$	9.97528e6			0.628522
$760$		0			0
$761$			1.51635e7i			0.949157i	0.880213	+	0.474579i	$0.157400\pi$
$761$			1.51635e7i			0.949157i	−0.880213	+	0.474579i	$0.842600\pi$
$762$		0			0
$763$	1.81218e7	+	2.36005e7i	1.12691	+	1.46761i
$764$		0			0
$765$	428451.			0.0264696
$766$		0			0
$767$		−	4.84678e6i		−	0.297485i
$768$		0			0
$769$		−	2.51765e7i		−	1.53525i	−0.640899	−	0.767625i	$-0.721438\pi$
$769$		−	2.51765e7i		−	1.53525i	0.640899	−	0.767625i	$-0.278562\pi$
$770$		0			0
$771$		−	4.68732e6i		−	0.283981i
$772$		0			0
$773$			2.15859e7i			1.29934i	0.760218	+	0.649668i	$0.225092\pi$
$773$			2.15859e7i			1.29934i	−0.760218	+	0.649668i	$0.774908\pi$
$774$		0			0
$775$	1.17889e7			0.705050
$776$		0			0
$777$	−1.70351e7	−	2.21853e7i	−1.01226	−	1.31829i
$778$		0			0
$779$			1.27693e7i			0.753914i
$780$		0			0
$781$	−1.36892e7			−0.803064
$782$		0			0
$783$	1.73580e6			0.101180
$784$		0			0
$785$	1.44702e7			0.838110
$786$		0			0
$787$	−6.93945e6			−0.399382			−0.199691	−	0.979859i	$-0.563994\pi$
$787$	−6.93945e6			−0.399382			−0.199691	+	0.979859i	$0.563994\pi$
$788$		0			0
$789$		−	1.15728e7i		−	0.661832i
$790$		0			0
$791$	7.49455e6	+	9.76037e6i	0.425897	+	0.554658i
$792$		0			0
$793$	−2.98614e7			−1.68627
$794$		0			0
$795$		−	7.92828e6i		−	0.444899i
$796$		0			0
$797$			3.48363e6i			0.194261i	0.995272	+	0.0971307i	$0.0309665\pi$
$797$			3.48363e6i			0.194261i	−0.995272	+	0.0971307i	$0.969034\pi$
$798$		0			0
$799$		−	1.07133e7i		−	0.593687i
$800$		0			0
$801$			490120.i			0.0269911i
$802$		0			0
$803$	4.65990e6			0.255028
$804$		0			0
$805$	−1.26687e7	−	1.64988e7i	−0.689038	−	0.897354i
$806$		0			0
$807$		−	1.11742e7i		−	0.603992i
$808$		0			0
$809$	2.76730e7			1.48657			0.743283	−	0.668977i	$-0.233267\pi$
$809$	2.76730e7			1.48657			0.743283	+	0.668977i	$0.233267\pi$
$810$		0			0
$811$	1.00384e7			0.535937			0.267968	−	0.963428i	$-0.413648\pi$
$811$	1.00384e7			0.535937			0.267968	+	0.963428i	$0.413648\pi$
$812$		0			0
$813$	3.95080e6			0.209633
$814$		0			0
$815$	3.31873e7			1.75016
$816$		0			0
$817$			1.48599e7i			0.778864i
$818$		0			0
$819$	−441857.	+	339282.i	−0.0230182	+	0.0176747i
$820$		0			0
$821$	−1.52884e6			−0.0791598			−0.0395799	−	0.999216i	$-0.512602\pi$
$821$	−1.52884e6			−0.0791598			−0.0395799	+	0.999216i	$0.512602\pi$
$822$		0			0
$823$		−	1.99273e7i		−	1.02553i	−0.858528	−	0.512766i	$-0.828621\pi$
$823$		−	1.99273e7i		−	1.02553i	0.858528	−	0.512766i	$-0.171379\pi$
$824$		0			0
$825$		−	8.33084e6i		−	0.426141i
$826$		0			0
$827$		−	2.04452e7i		−	1.03951i	−0.854317	−	0.519753i	$-0.826024\pi$
$827$		−	2.04452e7i		−	1.03951i	0.854317	−	0.519753i	$-0.173976\pi$
$828$		0			0
$829$			5.23625e6i			0.264627i	0.991208	+	0.132314i	$0.0422406\pi$
$829$			5.23625e6i			0.264627i	−0.991208	+	0.132314i	$0.957759\pi$
$830$		0			0
$831$	−8.54922e6			−0.429461
$832$		0			0
$833$	−1.74589e7	−	4.66567e6i	−0.871774	−	0.232971i
$834$		0			0
$835$		−	3.83845e7i		−	1.90520i
$836$		0			0
$837$	2.33229e7			1.15072
$838$		0			0
$839$	−1.00961e7			−0.495162			−0.247581	−	0.968867i	$-0.579636\pi$
$839$	−1.00961e7			−0.495162			−0.247581	+	0.968867i	$0.579636\pi$
$840$		0			0
$841$	−2.02961e7			−0.989514
$842$		0			0
$843$	1.49095e7			0.722594
$844$		0			0
$845$			1.50273e7i			0.723999i
$846$		0			0
$847$	6.55815e6	+	8.54087e6i	0.314104	+	0.409066i
$848$		0			0
$849$	−5.12418e6			−0.243981
$850$		0			0
$851$			3.09974e7i			1.46724i
$852$		0			0
$853$		−	2.42330e7i		−	1.14034i	−0.821526	−	0.570171i	$-0.806877\pi$
$853$		−	2.42330e7i		−	1.14034i	0.821526	−	0.570171i	$-0.193123\pi$
$854$		0			0
$855$			872074.i			0.0407979i
$856$		0			0
$857$			3.51407e7i			1.63440i	0.576353	+	0.817201i	$0.304475\pi$
$857$			3.51407e7i			1.63440i	−0.576353	+	0.817201i	$0.695525\pi$
$858$		0			0
$859$	−1.09551e7			−0.506564			−0.253282	−	0.967393i	$-0.581510\pi$
$859$	−1.09551e7			−0.506564			−0.253282	+	0.967393i	$0.581510\pi$
$860$		0			0
$861$	9.45981e6	−	7.26376e6i	0.434885	−	0.333928i
$862$		0			0
$863$			5.90417e6i			0.269856i	0.990855	+	0.134928i	$0.0430803\pi$
$863$			5.90417e6i			0.269856i	−0.990855	+	0.134928i	$0.956920\pi$
$864$		0			0
$865$	5.41943e7			2.46271
$866$		0			0
$867$	4.15826e6			0.187873
$868$		0			0
$869$	1.36778e7			0.614422
$870$		0			0
$871$	−4.49631e7			−2.00822
$872$		0			0
$873$		−	417028.i		−	0.0185195i
$874$		0			0
$875$	8.98084e6	−	6.89598e6i	0.396549	−	0.304492i
$876$		0			0
$877$	−2.60821e7			−1.14510			−0.572551	−	0.819869i	$-0.694046\pi$
$877$	−2.60821e7			−1.14510			−0.572551	+	0.819869i	$0.694046\pi$
$878$		0			0
$879$		−	2.19541e7i		−	0.958395i
$880$		0			0
$881$		−	3.01828e7i		−	1.31015i	−0.755566	−	0.655073i	$-0.772638\pi$
$881$		−	3.01828e7i		−	1.31015i	0.755566	−	0.655073i	$-0.227362\pi$
$882$		0			0
$883$		−	4.13507e7i		−	1.78476i	−0.451281	−	0.892382i	$-0.649033\pi$
$883$		−	4.13507e7i		−	1.78476i	0.451281	−	0.892382i	$-0.350967\pi$
$884$		0			0
$885$			7.08666e6i			0.304147i
$886$		0			0
$887$	1.78499e7			0.761775			0.380888	−	0.924621i	$-0.375618\pi$
$887$	1.78499e7			0.761775			0.380888	+	0.924621i	$0.375618\pi$
$888$		0			0
$889$	9.24224e6	−	7.09670e6i	0.392214	−	0.301163i
$890$		0			0
$891$		−	1.68633e7i		−	0.711619i
$892$		0			0
$893$	2.18060e7			0.915056
$894$		0			0
$895$	−4.70126e6			−0.196181
$896$		0			0
$897$	2.72842e7			1.13222
$898$		0			0
$899$	2.88981e6			0.119253
$900$		0			0
$901$			7.63296e6i			0.313243i
$902$		0			0
$903$	1.10086e7	−	8.45303e6i	0.449277	−	0.344979i
$904$		0			0
$905$	−492736.			−0.0199983
$906$		0			0
$907$		−	4.28246e7i		−	1.72852i	−0.503043	−	0.864261i	$-0.667786\pi$
$907$		−	4.28246e7i		−	1.72852i	0.503043	−	0.864261i	$-0.332214\pi$
$908$		0			0
$909$		−	350741.i		−	0.0140792i
$910$		0			0
$911$			7.01269e6i			0.279955i	0.990155	+	0.139978i	$0.0447031\pi$
$911$			7.01269e6i			0.279955i	−0.990155	+	0.139978i	$0.955297\pi$
$912$		0			0
$913$			1.23030e7i			0.488465i
$914$		0			0
$915$	4.36615e7			1.72403
$916$		0			0
$917$	4.81630e6	+	6.27241e6i	0.189143	+	0.246326i
$918$		0			0
$919$		−	3.29160e7i		−	1.28564i	−0.766019	−	0.642818i	$-0.777765\pi$
$919$		−	3.29160e7i		−	1.28564i	0.766019	−	0.642818i	$-0.222235\pi$
$920$		0			0
$921$	2.48467e7			0.965204
$922$		0			0
$923$	−3.74424e7			−1.44664
$924$		0			0
$925$	2.58874e7			0.994797
$926$		0			0
$927$	−241747.			−0.00923979
$928$		0			0
$929$		−	4.00951e7i		−	1.52423i	−0.647439	−	0.762117i	$-0.724160\pi$
$929$		−	4.00951e7i		−	1.52423i	0.647439	−	0.762117i	$-0.275840\pi$
$930$		0			0
$931$	9.49655e6	−	3.55360e7i	0.359080	−	1.34368i
$932$		0			0
$933$	−3.50774e7			−1.31924
$934$		0			0
$935$			2.12687e7i			0.795630i
$936$		0			0
$937$			4.32978e7i			1.61108i	0.592541	+	0.805540i	$0.298125\pi$
$937$			4.32978e7i			1.61108i	−0.592541	+	0.805540i	$0.701875\pi$
$938$		0			0
$939$			1.03103e7i			0.381599i
$940$		0			0
$941$		−	2.02234e7i		−	0.744526i	−0.928127	−	0.372263i	$-0.878582\pi$
$941$		−	2.02234e7i		−	0.744526i	0.928127	−	0.372263i	$-0.121418\pi$
$942$		0			0
$943$	−1.32173e7			−0.484020
$944$		0			0
$945$	−2.72600e7	+	2.09317e7i	−0.992992	+	0.762474i
$946$		0			0
$947$			5.27533e7i			1.91150i	0.294180	+	0.955750i	$0.404953\pi$
$947$			5.27533e7i			1.91150i	−0.294180	+	0.955750i	$0.595047\pi$
$948$		0			0
$949$	1.27457e7			0.459406
$950$		0			0
$951$	3.27254e7			1.17337
$952$		0			0
$953$	−6.87433e6			−0.245187			−0.122594	−	0.992457i	$-0.539121\pi$
$953$	−6.87433e6			−0.245187			−0.122594	+	0.992457i	$0.539121\pi$
$954$		0			0
$955$	4.27175e7			1.51564
$956$		0			0
$957$		−	2.04213e6i		−	0.0720781i
$958$		0			0
$959$	−2.61840e7	−	3.41002e7i	−0.919368	−	1.19732i
$960$		0			0
$961$	1.01995e7			0.356262
$962$		0			0
$963$			171021.i			0.00594271i
$964$		0			0
$965$		−	1.67554e7i		−	0.579212i
$966$		0			0
$967$		−	1.49963e6i		−	0.0515724i	−0.999667	−	0.0257862i	$-0.991791\pi$
$967$		−	1.49963e6i		−	0.0515724i	0.999667	−	0.0257862i	$-0.00820890\pi$
$968$		0			0
$969$		−	3.71053e7i		−	1.26948i
$970$		0			0
$971$	−5.04598e7			−1.71750			−0.858752	−	0.512392i	$-0.828760\pi$
$971$	−5.04598e7			−1.71750			−0.858752	+	0.512392i	$0.828760\pi$
$972$		0			0
$973$	−2.56620e7	−	3.34203e7i	−0.868976	−	1.13169i
$974$		0			0
$975$		−	2.27863e7i		−	0.767649i
$976$		0			0
$977$	−5.00768e7			−1.67842			−0.839210	−	0.543808i	$-0.816982\pi$
$977$	−5.00768e7			−1.67842			−0.839210	+	0.543808i	$0.816982\pi$
$978$		0			0
$979$	−2.43300e7			−0.811306
$980$		0			0
$981$	1.29121e6			0.0428377
$982$		0			0
$983$	4.66284e7			1.53910			0.769550	−	0.638586i	$-0.220481\pi$
$983$	4.66284e7			1.53910			0.769550	+	0.638586i	$0.220481\pi$
$984$		0			0
$985$		−	3.69214e6i		−	0.121251i
$986$		0			0
$987$	−1.24043e7	−	1.61545e7i	−0.405303	−	0.527837i
$988$		0			0
$989$	−1.53813e7			−0.500039
$990$		0			0
$991$		−	8.98041e6i		−	0.290477i	−0.989397	−	0.145239i	$-0.953605\pi$
$991$		−	8.98041e6i		−	0.290477i	0.989397	−	0.145239i	$-0.0463950\pi$
$992$		0			0
$993$		−	3.70660e7i		−	1.19290i
$994$		0			0
$995$			5.62118e7i			1.79999i
$996$		0			0
$997$			5.26562e7i			1.67769i	0.544371	+	0.838844i	$0.316768\pi$
$997$			5.26562e7i			1.67769i	−0.544371	+	0.838844i	$0.683232\pi$
$998$		0			0
$999$	5.12150e7			1.62362

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.6.f.d.447.11		16
4.3	odd	2	inner	448.6.f.d.447.5		16
7.6	odd	2	inner	448.6.f.d.447.6		16
8.3	odd	2		28.6.d.b.27.14	yes	16
8.5	even	2		28.6.d.b.27.15	yes	16
24.5	odd	2		252.6.b.d.55.1		16
24.11	even	2		252.6.b.d.55.3		16
28.27	even	2	inner	448.6.f.d.447.12		16
56.13	odd	2		28.6.d.b.27.16	yes	16
56.27	even	2		28.6.d.b.27.13	✓	16
168.83	odd	2		252.6.b.d.55.4		16
168.125	even	2		252.6.b.d.55.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.6.d.b.27.13	✓	16	56.27	even	2
28.6.d.b.27.14	yes	16	8.3	odd	2
28.6.d.b.27.15	yes	16	8.5	even	2
28.6.d.b.27.16	yes	16	56.13	odd	2
252.6.b.d.55.1		16	24.5	odd	2
252.6.b.d.55.2		16	168.125	even	2
252.6.b.d.55.3		16	24.11	even	2
252.6.b.d.55.4		16	168.83	odd	2
448.6.f.d.447.5		16	4.3	odd	2	inner
448.6.f.d.447.6		16	7.6	odd	2	inner
448.6.f.d.447.11		16	1.1	even	1	trivial
448.6.f.d.447.12		16	28.27	even	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	448.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+15.7679 q^{3} -70.8301i q^{5} +(78.9551 + 102.826i) q^{7} +5.62572 q^{9} +O(q^{10})\)	\(q+15.7679 q^{3} -70.8301i q^{5} +(78.9551 + 102.826i) q^{7} +5.62572 q^{9} +279.266i q^{11} +763.841i q^{13} -1116.84i q^{15} +1075.24i q^{17} -2188.55 q^{19} +(1244.95 + 1621.34i) q^{21} -2265.35i q^{23} -1891.90 q^{25} -3742.89 q^{27} -463.760 q^{29} -6231.26 q^{31} +4403.42i q^{33} +(7283.14 - 5592.40i) q^{35} -13683.3 q^{37} +12044.1i q^{39} -5834.56i q^{41} -6789.84i q^{43} -398.470i q^{45} -9963.66 q^{47} +(-4339.19 + 16237.2i) q^{49} +16954.2i q^{51} +7098.85 q^{53} +19780.4 q^{55} -34508.8 q^{57} -6345.28 q^{59} +39093.8i q^{61} +(444.179 + 578.468i) q^{63} +54102.9 q^{65} +58864.5i q^{67} -35719.7i q^{69} +49018.6i q^{71} -16686.3i q^{73} -29831.2 q^{75} +(-28715.6 + 22049.4i) q^{77} -48977.8i q^{79} -60384.4 q^{81} +44054.8 q^{83} +76159.3 q^{85} -7312.50 q^{87} +87121.3i q^{89} +(-78542.3 + 60309.1i) q^{91} -98253.7 q^{93} +155016. i q^{95} -74128.8i q^{97} +1571.07i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 1616 q^{9}+O(q^{10}) \) 16 * q + 1616 * q^9	\( 16 q + 1616 q^{9} + 1792 q^{21} - 9776 q^{25} - 26592 q^{29} + 26272 q^{37} + 8848 q^{49} + 41888 q^{53} - 60288 q^{57} + 66688 q^{65} - 320992 q^{77} + 56336 q^{81} - 78080 q^{85} + 335616 q^{93}+O(q^{100}) \) 16 * q + 1616 * q^9 + 1792 * q^21 - 9776 * q^25 - 26592 * q^29 + 26272 * q^37 + 8848 * q^49 + 41888 * q^53 - 60288 * q^57 + 66688 * q^65 - 320992 * q^77 + 56336 * q^81 - 78080 * q^85 + 335616 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more