LMFDB - Embedded newform 448.6.f.d.447.7

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			447.7
Root			$-4.07684 + 2.69664i$ of defining polynomial
Character	$\chi$	$=$	448.447
Dual form			448.6.f.d.447.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-6.44707 q^{3} -76.3023i q^{5} +(-120.438 + 47.9752i) q^{7} -201.435 q^{9} +O(q^{10})$	$q-6.44707 q^{3} -76.3023i q^{5} +(-120.438 + 47.9752i) q^{7} -201.435 q^{9} +334.774i q^{11} -912.099i q^{13} +491.926i q^{15} +478.534i q^{17} -2651.10 q^{19} +(776.473 - 309.299i) q^{21} +1420.85i q^{23} -2697.03 q^{25} +2865.30 q^{27} -6813.64 q^{29} +513.105 q^{31} -2158.31i q^{33} +(3660.62 + 9189.71i) q^{35} +5370.81 q^{37} +5880.36i q^{39} +697.095i q^{41} +527.614i q^{43} +15370.0i q^{45} -11640.9 q^{47} +(12203.8 - 11556.1i) q^{49} -3085.14i q^{51} -4929.75 q^{53} +25544.0 q^{55} +17091.8 q^{57} +10780.3 q^{59} -17440.4i q^{61} +(24260.5 - 9663.91i) q^{63} -69595.2 q^{65} -49231.3i q^{67} -9160.33i q^{69} -40989.1i q^{71} +59368.9i q^{73} +17388.0 q^{75} +(-16060.8 - 40319.6i) q^{77} -12641.7i q^{79} +30476.0 q^{81} +49569.1 q^{83} +36513.2 q^{85} +43928.0 q^{87} -13219.8i q^{89} +(43758.1 + 109852. i) q^{91} -3308.02 q^{93} +202285. i q^{95} -57155.4i q^{97} -67435.2i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−6.44707			−0.413579			−0.206790	−	0.978385i	$-0.566302\pi$
$3$	−6.44707			−0.413579			−0.206790	+	0.978385i	$0.566302\pi$
$4$		0			0
$5$		−	76.3023i		−	1.36494i	−0.730915	−	0.682468i	$-0.760907\pi$
$5$		−	76.3023i		−	1.36494i	0.730915	−	0.682468i	$-0.239093\pi$
$6$		0			0
$7$	−120.438	+	47.9752i	−0.929008	+	0.370060i
$7$	−120.438	+	47.9752i	−0.929008	+	0.370060i
$8$		0			0
$9$	−201.435			−0.828952
$10$		0			0
$11$			334.774i			0.834199i	0.908861	+	0.417099i	$0.136953\pi$
$11$			334.774i			0.834199i	−0.908861	+	0.417099i	$0.863047\pi$
$12$		0			0
$13$		−	912.099i		−	1.49687i	−0.663209	−	0.748434i	$-0.730806\pi$
$13$		−	912.099i		−	1.49687i	0.663209	−	0.748434i	$-0.269194\pi$
$14$		0			0
$15$			491.926i			0.564510i
$16$		0			0
$17$			478.534i			0.401597i	0.979633	+	0.200798i	$0.0643536\pi$
$17$			478.534i			0.401597i	−0.979633	+	0.200798i	$0.935646\pi$
$18$		0			0
$19$	−2651.10			−1.68478			−0.842388	−	0.538872i	$-0.818851\pi$
$19$	−2651.10			−1.68478			−0.842388	+	0.538872i	$0.818851\pi$
$20$		0			0
$21$	776.473	−	309.299i	0.384219	−	0.153049i
$22$		0			0
$23$			1420.85i			0.560054i	0.959992	+	0.280027i	$0.0903434\pi$
$23$			1420.85i			0.560054i	−0.959992	+	0.280027i	$0.909657\pi$
$24$		0			0
$25$	−2697.03			−0.863051
$26$		0			0
$27$	2865.30			0.756417
$28$		0			0
$29$	−6813.64			−1.50447			−0.752236	−	0.658894i	$-0.771025\pi$
$29$	−6813.64			−1.50447			−0.752236	+	0.658894i	$0.771025\pi$
$30$		0			0
$31$	513.105			0.0958963			0.0479482	−	0.998850i	$-0.484732\pi$
$31$	513.105			0.0958963			0.0479482	+	0.998850i	$0.484732\pi$
$32$		0			0
$33$		−	2158.31i		−	0.345007i
$34$		0			0
$35$	3660.62	+	9189.71i	0.505108	+	1.26804i
$36$		0			0
$37$	5370.81			0.644964			0.322482	−	0.946576i	$-0.395483\pi$
$37$	5370.81			0.644964			0.322482	+	0.946576i	$0.395483\pi$
$38$		0			0
$39$			5880.36i			0.619074i
$40$		0			0
$41$			697.095i			0.0647638i	0.999476	+	0.0323819i	$0.0103093\pi$
$41$			697.095i			0.0647638i	−0.999476	+	0.0323819i	$0.989691\pi$
$42$		0			0
$43$			527.614i			0.0435156i	0.999763	+	0.0217578i	$0.00692628\pi$
$43$			527.614i			0.0435156i	−0.999763	+	0.0217578i	$0.993074\pi$
$44$		0			0
$45$			15370.0i			1.13147i
$46$		0			0
$47$	−11640.9			−0.768670			−0.384335	−	0.923194i	$-0.625569\pi$
$47$	−11640.9			−0.768670			−0.384335	+	0.923194i	$0.625569\pi$
$48$		0			0
$49$	12203.8	−	11556.1i	0.726111	−	0.687577i
$50$		0			0
$51$		−	3085.14i		−	0.166092i
$52$		0			0
$53$	−4929.75			−0.241066			−0.120533	−	0.992709i	$-0.538460\pi$
$53$	−4929.75			−0.241066			−0.120533	+	0.992709i	$0.538460\pi$
$54$		0			0
$55$	25544.0			1.13863
$56$		0			0
$57$	17091.8			0.696788
$58$		0			0
$59$	10780.3			0.403181			0.201591	−	0.979470i	$-0.435389\pi$
$59$	10780.3			0.403181			0.201591	+	0.979470i	$0.435389\pi$
$60$		0			0
$61$		−	17440.4i		−	0.600112i	−0.953922	−	0.300056i	$-0.902995\pi$
$61$		−	17440.4i		−	0.600112i	0.953922	−	0.300056i	$-0.0970053\pi$
$62$		0			0
$63$	24260.5	−	9663.91i	0.770103	−	0.306762i
$64$		0			0
$65$	−69595.2			−2.04313
$66$		0			0
$67$		−	49231.3i		−	1.33984i	−0.742432	−	0.669922i	$-0.766328\pi$
$67$		−	49231.3i		−	1.33984i	0.742432	−	0.669922i	$-0.233672\pi$
$68$		0			0
$69$		−	9160.33i		−	0.231627i
$70$		0			0
$71$		−	40989.1i		−	0.964990i	−0.875899	−	0.482495i	$-0.839731\pi$
$71$		−	40989.1i		−	0.964990i	0.875899	−	0.482495i	$-0.160269\pi$
$72$		0			0
$73$			59368.9i			1.30392i	0.758252	+	0.651962i	$0.226054\pi$
$73$			59368.9i			1.30392i	−0.758252	+	0.651962i	$0.773946\pi$
$74$		0			0
$75$	17388.0			0.356940
$76$		0			0
$77$	−16060.8	−	40319.6i	−0.308703	−	0.774977i
$78$		0			0
$79$		−	12641.7i		−	0.227896i	−0.993487	−	0.113948i	$-0.963650\pi$
$79$		−	12641.7i		−	0.227896i	0.993487	−	0.113948i	$-0.0363498\pi$
$80$		0			0
$81$	30476.0			0.516113
$82$		0			0
$83$	49569.1			0.789798			0.394899	−	0.918725i	$-0.370780\pi$
$83$	49569.1			0.789798			0.394899	+	0.918725i	$0.370780\pi$
$84$		0			0
$85$	36513.2			0.548154
$86$		0			0
$87$	43928.0			0.622219
$88$		0			0
$89$		−	13219.8i		−	0.176909i	−0.996080	−	0.0884547i	$-0.971807\pi$
$89$		−	13219.8i		−	0.176909i	0.996080	−	0.0884547i	$-0.0281928\pi$
$90$		0			0
$91$	43758.1	+	109852.i	0.553931	+	1.39060i
$92$		0			0
$93$	−3308.02			−0.0396607
$94$		0			0
$95$			202285.i			2.29961i
$96$		0			0
$97$		−	57155.4i		−	0.616777i	−0.951261	−	0.308388i	$-0.900210\pi$
$97$		−	57155.4i		−	0.616777i	0.951261	−	0.308388i	$-0.0997896\pi$
$98$		0			0
$99$		−	67435.2i		−	0.691511i
$100$		0			0
$101$			77288.8i			0.753899i	0.926234	+	0.376949i	$0.123027\pi$
$101$			77288.8i			0.753899i	−0.926234	+	0.376949i	$0.876973\pi$
$102$		0			0
$103$	180104.			1.67275			0.836374	−	0.548160i	$-0.184671\pi$
$103$	180104.			1.67275			0.836374	+	0.548160i	$0.184671\pi$
$104$		0			0
$105$	−23600.2	−	59246.7i	−0.208902	−	0.524434i
$106$		0			0
$107$			214915.i			1.81471i	0.420366	+	0.907354i	$0.361902\pi$
$107$			214915.i			1.81471i	−0.420366	+	0.907354i	$0.638098\pi$
$108$		0			0
$109$	−62569.2			−0.504422			−0.252211	−	0.967672i	$-0.581158\pi$
$109$	−62569.2			−0.504422			−0.252211	+	0.967672i	$0.581158\pi$
$110$		0			0
$111$	−34626.0			−0.266744
$112$		0			0
$113$	126905.			0.934939			0.467469	−	0.884009i	$-0.345166\pi$
$113$	126905.			0.934939			0.467469	+	0.884009i	$0.345166\pi$
$114$		0			0
$115$	108414.			0.764438
$116$		0			0
$117$			183729.i			1.24083i
$118$		0			0
$119$	−22957.8	−	57633.8i	−0.148615	−	0.373087i
$120$		0			0
$121$	48977.6			0.304113
$122$		0			0
$123$		−	4494.22i		−	0.0267850i
$124$		0			0
$125$		−	32654.7i		−	0.186927i
$126$		0			0
$127$			252336.i			1.38825i	0.719852	+	0.694127i	$0.244210\pi$
$127$			252336.i			1.38825i	−0.719852	+	0.694127i	$0.755790\pi$
$128$		0			0
$129$		−	3401.56i		−	0.0179972i
$130$		0			0
$131$	309045.			1.57342			0.786708	−	0.617326i	$-0.211784\pi$
$131$	309045.			1.57342			0.786708	+	0.617326i	$0.211784\pi$
$132$		0			0
$133$	319294.	−	127187.i	1.56517	−	0.623468i
$134$		0			0
$135$		−	218629.i		−	1.03246i
$136$		0			0
$137$	18629.7			0.0848018			0.0424009	−	0.999101i	$-0.486499\pi$
$137$	18629.7			0.0848018			0.0424009	+	0.999101i	$0.486499\pi$
$138$		0			0
$139$	−34562.3			−0.151728			−0.0758639	−	0.997118i	$-0.524171\pi$
$139$	−34562.3			−0.151728			−0.0758639	+	0.997118i	$0.524171\pi$
$140$		0			0
$141$	75049.3			0.317906
$142$		0			0
$143$	305347.			1.24869
$144$		0			0
$145$			519896.i			2.05351i
$146$		0			0
$147$	−78678.4	+	74503.0i	−0.300305	+	0.284368i
$148$		0			0
$149$	−226557.			−0.836010			−0.418005	−	0.908445i	$-0.637271\pi$
$149$	−226557.			−0.836010			−0.418005	+	0.908445i	$0.637271\pi$
$150$		0			0
$151$			292164.i			1.04276i	0.853325	+	0.521380i	$0.174582\pi$
$151$			292164.i			1.04276i	−0.853325	+	0.521380i	$0.825418\pi$
$152$		0			0
$153$		−	96393.6i		−	0.332905i
$154$		0			0
$155$		−	39151.1i		−	0.130892i
$156$		0			0
$157$		−	244229.i		−	0.790767i	−0.918516	−	0.395384i	$-0.870612\pi$
$157$		−	244229.i		−	0.790767i	0.918516	−	0.395384i	$-0.129388\pi$
$158$		0			0
$159$	31782.4			0.0996999
$160$		0			0
$161$	−68165.8	−	171125.i	−0.207253	−	0.520294i
$162$		0			0
$163$			218907.i			0.645342i	0.946511	+	0.322671i	$0.104581\pi$
$163$			218907.i			0.645342i	−0.946511	+	0.322671i	$0.895419\pi$
$164$		0			0
$165$	−164684.			−0.470913
$166$		0			0
$167$	441969.			1.22631			0.613155	−	0.789962i	$-0.289900\pi$
$167$	441969.			1.22631			0.613155	+	0.789962i	$0.289900\pi$
$168$		0			0
$169$	−460631.			−1.24061
$170$		0			0
$171$	534025.			1.39660
$172$		0			0
$173$			447193.i			1.13600i	0.823028	+	0.568001i	$0.192283\pi$
$173$			447193.i			1.13600i	−0.823028	+	0.568001i	$0.807717\pi$
$174$		0			0
$175$	324826.	−	129391.i	0.801781	−	0.319381i
$176$		0			0
$177$	−69501.2			−0.166748
$178$		0			0
$179$		−	435778.i		−	1.01656i	−0.861192	−	0.508280i	$-0.830282\pi$
$179$		−	435778.i		−	1.01656i	0.861192	−	0.508280i	$-0.169718\pi$
$180$		0			0
$181$		−	715186.i		−	1.62264i	−0.584601	−	0.811321i	$-0.698749\pi$
$181$		−	715186.i		−	1.62264i	0.584601	−	0.811321i	$-0.301251\pi$
$182$		0			0
$183$			112439.i			0.248194i
$184$		0			0
$185$		−	409805.i		−	0.880335i
$186$		0			0
$187$	−160201.			−0.335012
$188$		0			0
$189$	−345092.	+	137464.i	−0.702717	+	0.279920i
$190$		0			0
$191$		−	350615.i		−	0.695420i	−0.937602	−	0.347710i	$-0.886959\pi$
$191$		−	350615.i		−	0.695420i	0.937602	−	0.347710i	$-0.113041\pi$
$192$		0			0
$193$	337927.			0.653024			0.326512	−	0.945193i	$-0.394127\pi$
$193$	337927.			0.653024			0.326512	+	0.945193i	$0.394127\pi$
$194$		0			0
$195$	448685.			0.844996
$196$		0			0
$197$	−31816.8			−0.0584105			−0.0292052	−	0.999573i	$-0.509298\pi$
$197$	−31816.8			−0.0584105			−0.0292052	+	0.999573i	$0.509298\pi$
$198$		0			0
$199$	31190.2			0.0558323			0.0279162	−	0.999610i	$-0.491113\pi$
$199$	31190.2			0.0558323			0.0279162	+	0.999610i	$0.491113\pi$
$200$		0			0
$201$			317397.i			0.554132i
$202$		0			0
$203$	820623.	−	326886.i	1.39767	−	0.556745i
$204$		0			0
$205$	53189.9			0.0883984
$206$		0			0
$207$		−	286210.i		−	0.464258i
$208$		0			0
$209$		−	887518.i		−	1.40544i
$210$		0			0
$211$			723348.i			1.11851i	0.828994	+	0.559257i	$0.188913\pi$
$211$			723348.i			1.11851i	−0.828994	+	0.559257i	$0.811087\pi$
$212$		0			0
$213$			264260.i			0.399100i
$214$		0			0
$215$	40258.2			0.0593961
$216$		0			0
$217$	−61797.5	+	24616.3i	−0.0890884	+	0.0354874i
$218$		0			0
$219$		−	382756.i		−	0.539276i
$220$		0			0
$221$	436470.			0.601137
$222$		0			0
$223$	−728622.			−0.981161			−0.490580	−	0.871396i	$-0.663215\pi$
$223$	−728622.			−0.981161			−0.490580	+	0.871396i	$0.663215\pi$
$224$		0			0
$225$	543278.			0.715428
$226$		0			0
$227$	−874014.			−1.12578			−0.562890	−	0.826532i	$-0.690311\pi$
$227$	−874014.			−1.12578			−0.562890	+	0.826532i	$0.690311\pi$
$228$		0			0
$229$			1.15837e6i			1.45968i	0.683618	+	0.729840i	$0.260405\pi$
$229$			1.15837e6i			1.45968i	−0.683618	+	0.729840i	$0.739595\pi$
$230$		0			0
$231$	103545.	+	259943.i	0.127673	+	0.320515i
$232$		0			0
$233$	−277974.			−0.335440			−0.167720	−	0.985835i	$-0.553640\pi$
$233$	−277974.			−0.335440			−0.167720	+	0.985835i	$0.553640\pi$
$234$		0			0
$235$			888223.i			1.04919i
$236$		0			0
$237$			81501.7i			0.0942532i
$238$		0			0
$239$		−	88758.8i		−	0.100512i	−0.998736	−	0.0502558i	$-0.983996\pi$
$239$		−	88758.8i		−	0.100512i	0.998736	−	0.0502558i	$-0.0160037\pi$
$240$		0			0
$241$			748467.i			0.830099i	0.909799	+	0.415050i	$0.136236\pi$
$241$			748467.i			0.830099i	−0.909799	+	0.415050i	$0.863764\pi$
$242$		0			0
$243$	−892750.			−0.969871
$244$		0			0
$245$	−881757.	−	931174.i	−0.938499	−	0.991096i
$246$		0			0
$247$			2.41806e6i			2.52189i
$248$		0			0
$249$	−319575.			−0.326644
$250$		0			0
$251$	−768117.			−0.769561			−0.384781	−	0.923008i	$-0.625723\pi$
$251$	−768117.			−0.769561			−0.384781	+	0.923008i	$0.625723\pi$
$252$		0			0
$253$	−475664.			−0.467196
$254$		0			0
$255$	−235403.			−0.226705
$256$		0			0
$257$			1.22830e6i			1.16003i	0.814604	+	0.580017i	$0.196954\pi$
$257$			1.22830e6i			1.16003i	−0.814604	+	0.580017i	$0.803046\pi$
$258$		0			0
$259$	−646851.	+	257666.i	−0.599177	+	0.238675i
$260$		0			0
$261$	1.37251e6			1.24714
$262$		0			0
$263$		−	1.41313e6i		−	1.25978i	−0.776686	−	0.629888i	$-0.783101\pi$
$263$		−	1.41313e6i		−	1.25978i	0.776686	−	0.629888i	$-0.216899\pi$
$264$		0			0
$265$			376151.i			0.329039i
$266$		0			0
$267$			85229.1i			0.0731661i
$268$		0			0
$269$		−	103713.i		−	0.0873878i	−0.999045	−	0.0436939i	$-0.986087\pi$
$269$		−	103713.i		−	0.0873878i	0.999045	−	0.0436939i	$-0.0139126\pi$
$270$		0			0
$271$	−1.31141e6			−1.08471			−0.542355	−	0.840149i	$-0.682467\pi$
$271$	−1.31141e6			−1.08471			−0.542355	+	0.840149i	$0.682467\pi$
$272$		0			0
$273$	−282112.	−	708220.i	−0.229094	−	0.575124i
$274$		0			0
$275$		−	902896.i		−	0.719956i
$276$		0			0
$277$	1.50328e6			1.17717			0.588587	−	0.808434i	$-0.299685\pi$
$277$	1.50328e6			1.17717			0.588587	+	0.808434i	$0.299685\pi$
$278$		0			0
$279$	−103357.			−0.0794934
$280$		0			0
$281$	335270.			0.253297			0.126648	−	0.991948i	$-0.459578\pi$
$281$	335270.			0.253297			0.126648	+	0.991948i	$0.459578\pi$
$282$		0			0
$283$	933526.			0.692884			0.346442	−	0.938071i	$-0.387390\pi$
$283$	933526.			0.692884			0.346442	+	0.938071i	$0.387390\pi$
$284$		0			0
$285$		−	1.30414e6i		−	0.951072i
$286$		0			0
$287$	−33443.3	−	83956.9i	−0.0239665	−	0.0601661i
$288$		0			0
$289$	1.19086e6			0.838720
$290$		0			0
$291$			368485.i			0.255086i
$292$		0			0
$293$		−	563086.i		−	0.383183i	−0.981475	−	0.191591i	$-0.938635\pi$
$293$		−	563086.i		−	0.383183i	0.981475	−	0.191591i	$-0.0613648\pi$
$294$		0			0
$295$		−	822560.i		−	0.550317i
$296$		0			0
$297$			959228.i			0.631002i
$298$		0			0
$299$	1.29596e6			0.838326
$300$		0			0
$301$	−25312.4	−	63544.9i	−0.0161034	−	0.0404264i
$302$		0			0
$303$		−	498286.i		−	0.311797i
$304$		0			0
$305$	−1.33074e6			−0.819114
$306$		0			0
$307$	−2.49384e6			−1.51016			−0.755079	−	0.655633i	$-0.772402\pi$
$307$	−2.49384e6			−1.51016			−0.755079	+	0.655633i	$0.772402\pi$
$308$		0			0
$309$	−1.16114e6			−0.691814
$310$		0			0
$311$	198558.			0.116409			0.0582046	−	0.998305i	$-0.481462\pi$
$311$	198558.			0.116409			0.0582046	+	0.998305i	$0.481462\pi$
$312$		0			0
$313$		−	2.40632e6i		−	1.38833i	−0.719817	−	0.694164i	$-0.755774\pi$
$313$		−	2.40632e6i		−	1.38833i	0.719817	−	0.694164i	$-0.244226\pi$
$314$		0			0
$315$	−737378.	−	1.85113e6i	−0.418710	−	1.05114i
$316$		0			0
$317$	−841749.			−0.470473			−0.235236	−	0.971938i	$-0.575586\pi$
$317$	−841749.			−0.470473			−0.235236	+	0.971938i	$0.575586\pi$
$318$		0			0
$319$		−	2.28103e6i		−	1.25503i
$320$		0			0
$321$		−	1.38557e6i		−	0.750526i
$322$		0			0
$323$		−	1.26864e6i		−	0.676601i
$324$		0			0
$325$			2.45996e6i			1.29187i
$326$		0			0
$327$	403388.			0.208619
$328$		0			0
$329$	1.40200e6	−	558473.i	0.714101	−	0.284454i
$330$		0			0
$331$		−	1.95105e6i		−	0.978811i	−0.872056	−	0.489405i	$-0.837214\pi$
$331$		−	1.95105e6i		−	0.978811i	0.872056	−	0.489405i	$-0.162786\pi$
$332$		0			0
$333$	−1.08187e6			−0.534644
$334$		0			0
$335$	−3.75646e6			−1.82880
$336$		0			0
$337$	−1.26826e6			−0.608322			−0.304161	−	0.952621i	$-0.598376\pi$
$337$	−1.26826e6			−0.608322			−0.304161	+	0.952621i	$0.598376\pi$
$338$		0			0
$339$	−818166.			−0.386671
$340$		0			0
$341$			171774.i			0.0799966i
$342$		0			0
$343$	−915392.	+	1.97728e6i	−0.420119	+	0.907469i
$344$		0			0
$345$	−698954.			−0.316156
$346$		0			0
$347$			1.44829e6i			0.645703i	0.946450	+	0.322852i	$0.104641\pi$
$347$			1.44829e6i			0.645703i	−0.946450	+	0.322852i	$0.895359\pi$
$348$		0			0
$349$			2.38877e6i			1.04981i	0.851161	+	0.524905i	$0.175899\pi$
$349$			2.38877e6i			1.04981i	−0.851161	+	0.524905i	$0.824101\pi$
$350$		0			0
$351$		−	2.61344e6i		−	1.13226i
$352$		0			0
$353$			126904.i			0.0542051i	0.999633	+	0.0271025i	$0.00862806\pi$
$353$			126904.i			0.0542051i	−0.999633	+	0.0271025i	$0.991372\pi$
$354$		0			0
$355$	−3.12756e6			−1.31715
$356$		0			0
$357$	148010.	+	371569.i	0.0614641	+	0.154301i
$358$		0			0
$359$		−	1.17972e6i		−	0.483105i	−0.970388	−	0.241552i	$-0.922343\pi$
$359$		−	1.17972e6i		−	0.483105i	0.970388	−	0.241552i	$-0.0776565\pi$
$360$		0			0
$361$	4.55223e6			1.83847
$362$		0			0
$363$	−315762.			−0.125775
$364$		0			0
$365$	4.52998e6			1.77977
$366$		0			0
$367$	−1.38080e6			−0.535136			−0.267568	−	0.963539i	$-0.586220\pi$
$367$	−1.38080e6			−0.535136			−0.267568	+	0.963539i	$0.586220\pi$
$368$		0			0
$369$		−	140419.i		−	0.0536861i
$370$		0			0
$371$	593731.	−	236506.i	0.223952	−	0.0892088i
$372$		0			0
$373$	−3.90118e6			−1.45186			−0.725929	−	0.687770i	$-0.758590\pi$
$373$	−3.90118e6			−1.45186			−0.725929	+	0.687770i	$0.758590\pi$
$374$		0			0
$375$			210527.i			0.0773090i
$376$		0			0
$377$			6.21471e6i			2.25200i
$378$		0			0
$379$		−	249613.i		−	0.0892627i	−0.999004	−	0.0446313i	$-0.985789\pi$
$379$		−	249613.i		−	0.0892627i	0.999004	−	0.0446313i	$-0.0142113\pi$
$380$		0			0
$381$		−	1.62682e6i		−	0.574154i
$382$		0			0
$383$	−49441.0			−0.0172223			−0.00861114	−	0.999963i	$-0.502741\pi$
$383$	−49441.0			−0.0172223			−0.00861114	+	0.999963i	$0.502741\pi$
$384$		0			0
$385$	−3.07647e6	+	1.22548e6i	−1.05779	+	0.421361i
$386$		0			0
$387$		−	106280.i		−	0.0360724i
$388$		0			0
$389$	3.79873e6			1.27281			0.636406	−	0.771355i	$-0.280420\pi$
$389$	3.79873e6			1.27281			0.636406	+	0.771355i	$0.280420\pi$
$390$		0			0
$391$	−679926.			−0.224916
$392$		0			0
$393$	−1.99243e6			−0.650732
$394$		0			0
$395$	−964589.			−0.311064
$396$		0			0
$397$		−	3.11906e6i		−	0.993225i	−0.867972	−	0.496613i	$-0.834577\pi$
$397$		−	3.11906e6i		−	0.993225i	0.867972	−	0.496613i	$-0.165423\pi$
$398$		0			0
$399$	−2.05851e6	+	819984.i	−0.647322	+	0.257853i
$400$		0			0
$401$	−108330.			−0.0336424			−0.0168212	−	0.999859i	$-0.505355\pi$
$401$	−108330.			−0.0336424			−0.0168212	+	0.999859i	$0.505355\pi$
$402$		0			0
$403$		−	468002.i		−	0.143544i
$404$		0			0
$405$		−	2.32539e6i		−	0.704462i
$406$		0			0
$407$			1.79801e6i			0.538028i
$408$		0			0
$409$		−	3.40094e6i		−	1.00529i	−0.864494	−	0.502644i	$-0.832361\pi$
$409$		−	3.40094e6i		−	1.00529i	0.864494	−	0.502644i	$-0.167639\pi$
$410$		0			0
$411$	−120107.			−0.0350723
$412$		0			0
$413$	−1.29836e6	+	517187.i	−0.374559	+	0.149201i
$414$		0			0
$415$		−	3.78224e6i		−	1.07802i
$416$		0			0
$417$	222825.			0.0627515
$418$		0			0
$419$	−3.64283e6			−1.01369			−0.506843	−	0.862038i	$-0.669188\pi$
$419$	−3.64283e6			−1.01369			−0.506843	+	0.862038i	$0.669188\pi$
$420$		0			0
$421$	−1.38888e6			−0.381908			−0.190954	−	0.981599i	$-0.561158\pi$
$421$	−1.38888e6			−0.381908			−0.190954	+	0.981599i	$0.561158\pi$
$422$		0			0
$423$	2.34488e6			0.637191
$424$		0			0
$425$		−	1.29062e6i		−	0.346599i
$426$		0			0
$427$	836707.	+	2.10049e6i	0.222077	+	0.557508i
$428$		0			0
$429$	−1.96859e6			−0.516431
$430$		0			0
$431$			7.38713e6i			1.91550i	0.287600	+	0.957751i	$0.407143\pi$
$431$			7.38713e6i			1.91550i	−0.287600	+	0.957751i	$0.592857\pi$
$432$		0			0
$433$		−	2.84317e6i		−	0.728759i	−0.931251	−	0.364379i	$-0.881281\pi$
$433$		−	2.84317e6i		−	0.728759i	0.931251	−	0.364379i	$-0.118719\pi$
$434$		0			0
$435$		−	3.35180e6i		−	0.849289i
$436$		0			0
$437$		−	3.76682e6i		−	0.943565i
$438$		0			0
$439$	−3.25165e6			−0.805272			−0.402636	−	0.915360i	$-0.631906\pi$
$439$	−3.25165e6			−0.805272			−0.402636	+	0.915360i	$0.631906\pi$
$440$		0			0
$441$	−2.45827e6	+	2.32781e6i	−0.601912	+	0.569968i
$442$		0			0
$443$			5.79357e6i			1.40261i	0.712861	+	0.701305i	$0.247399\pi$
$443$			5.79357e6i			1.40261i	−0.712861	+	0.701305i	$0.752601\pi$
$444$		0			0
$445$	−1.00870e6			−0.241470
$446$		0			0
$447$	1.46063e6			0.345757
$448$		0			0
$449$	5.24967e6			1.22890			0.614450	−	0.788956i	$-0.289378\pi$
$449$	5.24967e6			1.22890			0.614450	+	0.788956i	$0.289378\pi$
$450$		0			0
$451$	−233369.			−0.0540259
$452$		0			0
$453$		−	1.88360e6i		−	0.431264i
$454$		0			0
$455$	8.38192e6	−	3.33885e6i	1.89808	−	0.756080i
$456$		0			0
$457$	−4.80494e6			−1.07621			−0.538105	−	0.842878i	$-0.680860\pi$
$457$	−4.80494e6			−1.07621			−0.538105	+	0.842878i	$0.680860\pi$
$458$		0			0
$459$			1.37114e6i			0.303775i
$460$		0			0
$461$			7.81325e6i			1.71230i	0.516729	+	0.856149i	$0.327149\pi$
$461$			7.81325e6i			1.71230i	−0.516729	+	0.856149i	$0.672851\pi$
$462$		0			0
$463$			4.81625e6i			1.04413i	0.852904	+	0.522067i	$0.174839\pi$
$463$			4.81625e6i			1.04413i	−0.852904	+	0.522067i	$0.825161\pi$
$464$		0			0
$465$			252409.i			0.0541344i
$466$		0			0
$467$	−1.24144e6			−0.263411			−0.131705	−	0.991289i	$-0.542045\pi$
$467$	−1.24144e6			−0.263411			−0.131705	+	0.991289i	$0.542045\pi$
$468$		0			0
$469$	2.36188e6	+	5.92933e6i	0.495822	+	1.24472i
$470$		0			0
$471$			1.57456e6i			0.327045i
$472$		0			0
$473$	−176631.			−0.0363007
$474$		0			0
$475$	7.15011e6			1.45405
$476$		0			0
$477$	993027.			0.199832
$478$		0			0
$479$	−7.73156e6			−1.53967			−0.769836	−	0.638242i	$-0.779662\pi$
$479$	−7.73156e6			−1.53967			−0.769836	+	0.638242i	$0.779662\pi$
$480$		0			0
$481$		−	4.89871e6i		−	0.965426i
$482$		0			0
$483$	439469.	+	1.10325e6i	0.0857157	+	0.215183i
$484$		0			0
$485$	−4.36109e6			−0.841861
$486$		0			0
$487$		−	2.72935e6i		−	0.521479i	−0.965409	−	0.260739i	$-0.916034\pi$
$487$		−	2.72935e6i		−	0.521479i	0.965409	−	0.260739i	$-0.0839663\pi$
$488$		0			0
$489$		−	1.41130e6i		−	0.266900i
$490$		0			0
$491$			4.78104e6i			0.894991i	0.894286	+	0.447496i	$0.147684\pi$
$491$			4.78104e6i			0.894991i	−0.894286	+	0.447496i	$0.852316\pi$
$492$		0			0
$493$		−	3.26056e6i		−	0.604191i
$494$		0			0
$495$	−5.14546e6			−0.943868
$496$		0			0
$497$	1.96646e6	+	4.93666e6i	0.357104	+	0.896484i
$498$		0			0
$499$		−	1.43655e6i		−	0.258267i	−0.991627	−	0.129134i	$-0.958780\pi$
$499$		−	1.43655e6i		−	0.258267i	0.991627	−	0.129134i	$-0.0412196\pi$
$500$		0			0
$501$	−2.84940e6			−0.507177
$502$		0			0
$503$	4.30022e6			0.757828			0.378914	−	0.925432i	$-0.376298\pi$
$503$	4.30022e6			0.757828			0.378914	+	0.925432i	$0.376298\pi$
$504$		0			0
$505$	5.89731e6			1.02902
$506$		0			0
$507$	2.96972e6			0.513092
$508$		0			0
$509$			2.38009e6i			0.407192i	0.979055	+	0.203596i	$0.0652629\pi$
$509$			2.38009e6i			0.407192i	−0.979055	+	0.203596i	$0.934737\pi$
$510$		0			0
$511$	−2.84824e6	−	7.15029e6i	−0.482530	−	1.21136i
$512$		0			0
$513$	−7.59621e6			−1.27439
$514$		0			0
$515$		−	1.37423e7i		−	2.28319i
$516$		0			0
$517$		−	3.89705e6i		−	0.641224i
$518$		0			0
$519$		−	2.88308e6i		−	0.469827i
$520$		0			0
$521$		−	6.45172e6i		−	1.04131i	−0.853766	−	0.520657i	$-0.825687\pi$
$521$		−	6.45172e6i		−	1.04131i	0.853766	−	0.520657i	$-0.174313\pi$
$522$		0			0
$523$	5.18760e6			0.829302			0.414651	−	0.909981i	$-0.363904\pi$
$523$	5.18760e6			0.829302			0.414651	+	0.909981i	$0.363904\pi$
$524$		0			0
$525$	−2.09418e6	+	834191.i	−0.331600	+	0.132089i
$526$		0			0
$527$			245538.i			0.0385117i
$528$		0			0
$529$	4.41752e6			0.686340
$530$		0			0
$531$	−2.17153e6			−0.334218
$532$		0			0
$533$	635819.			0.0969428
$534$		0			0
$535$	1.63985e7			2.47696
$536$		0			0
$537$			2.80949e6i			0.420428i
$538$		0			0
$539$	3.86868e6	+	4.08549e6i	0.573576	+	0.605721i
$540$		0			0
$541$	1.02776e7			1.50973			0.754863	−	0.655882i	$-0.227703\pi$
$541$	1.02776e7			1.50973			0.754863	+	0.655882i	$0.227703\pi$
$542$		0			0
$543$			4.61085e6i			0.671091i
$544$		0			0
$545$			4.77417e6i			0.688504i
$546$		0			0
$547$		−	3.13504e6i		−	0.447997i	−0.974590	−	0.223998i	$-0.928089\pi$
$547$		−	3.13504e6i		−	0.447997i	0.974590	−	0.223998i	$-0.0719110\pi$
$548$		0			0
$549$			3.51311e6i			0.497464i
$550$		0			0
$551$	1.80636e7			2.53470
$552$		0			0
$553$	606488.	+	1.52254e6i	0.0843352	+	0.211717i
$554$		0			0
$555$			2.64204e6i			0.364088i
$556$		0			0
$557$	1.08742e7			1.48511			0.742554	−	0.669787i	$-0.233614\pi$
$557$	1.08742e7			1.48511			0.742554	+	0.669787i	$0.233614\pi$
$558$		0			0
$559$	481236.			0.0651372
$560$		0			0
$561$	1.03282e6			0.138554
$562$		0			0
$563$	4.81281e6			0.639923			0.319961	−	0.947431i	$-0.396330\pi$
$563$	4.81281e6			0.639923			0.319961	+	0.947431i	$0.396330\pi$
$564$		0			0
$565$		−	9.68315e6i		−	1.27613i
$566$		0			0
$567$	−3.67047e6	+	1.46209e6i	−0.479473	+	0.190993i
$568$		0			0
$569$	−830234.			−0.107503			−0.0537514	−	0.998554i	$-0.517118\pi$
$569$	−830234.			−0.107503			−0.0537514	+	0.998554i	$0.517118\pi$
$570$		0			0
$571$		−	4.35081e6i		−	0.558444i	−0.960227	−	0.279222i	$-0.909923\pi$
$571$		−	4.35081e6i		−	0.558444i	0.960227	−	0.279222i	$-0.0900766\pi$
$572$		0			0
$573$			2.26044e6i			0.287612i
$574$		0			0
$575$		−	3.83209e6i		−	0.483355i
$576$		0			0
$577$			9.01602e6i			1.12739i	0.825982	+	0.563697i	$0.190621\pi$
$577$			9.01602e6i			1.12739i	−0.825982	+	0.563697i	$0.809379\pi$
$578$		0			0
$579$	−2.17864e6			−0.270078
$580$		0			0
$581$	−5.97002e6	+	2.37809e6i	−0.733728	+	0.292272i
$582$		0			0
$583$		−	1.65035e6i		−	0.201097i
$584$		0			0
$585$	1.40189e7			1.69366
$586$		0			0
$587$	1.16822e7			1.39936			0.699679	−	0.714457i	$-0.253326\pi$
$587$	1.16822e7			1.39936			0.699679	+	0.714457i	$0.253326\pi$
$588$		0			0
$589$	−1.36029e6			−0.161564
$590$		0			0
$591$	205125.			0.0241574
$592$		0			0
$593$		−	7.34176e6i		−	0.857361i	−0.903456	−	0.428680i	$-0.858979\pi$
$593$		−	7.34176e6i		−	0.857361i	0.903456	−	0.428680i	$-0.141021\pi$
$594$		0			0
$595$	−4.39759e6	+	1.75173e6i	−0.509240	+	0.202850i
$596$		0			0
$597$	−201085.			−0.0230911
$598$		0			0
$599$			1.39805e7i			1.59204i	0.605268	+	0.796022i	$0.293066\pi$
$599$			1.39805e7i			1.59204i	−0.605268	+	0.796022i	$0.706934\pi$
$600$		0			0
$601$			9.66115e6i			1.09105i	0.838096	+	0.545523i	$0.183669\pi$
$601$			9.66115e6i			1.09105i	−0.838096	+	0.545523i	$0.816331\pi$
$602$		0			0
$603$			9.91692e6i			1.11067i
$604$		0			0
$605$		−	3.73710e6i		−	0.415094i
$606$		0			0
$607$	3.62733e6			0.399591			0.199795	−	0.979838i	$-0.435972\pi$
$607$	3.62733e6			0.399591			0.199795	+	0.979838i	$0.435972\pi$
$608$		0			0
$609$	−5.29061e6	+	2.10746e6i	−0.578046	+	0.230258i
$610$		0			0
$611$			1.06176e7i			1.15060i
$612$		0			0
$613$	9.03872e6			0.971529			0.485764	−	0.874090i	$-0.338541\pi$
$613$	9.03872e6			0.971529			0.485764	+	0.874090i	$0.338541\pi$
$614$		0			0
$615$	−342919.			−0.0365598
$616$		0			0
$617$	−1.23771e7			−1.30890			−0.654448	−	0.756107i	$-0.727099\pi$
$617$	−1.23771e7			−1.30890			−0.654448	+	0.756107i	$0.727099\pi$
$618$		0			0
$619$	5.58961e6			0.586348			0.293174	−	0.956059i	$-0.405289\pi$
$619$	5.58961e6			0.586348			0.293174	+	0.956059i	$0.405289\pi$
$620$		0			0
$621$			4.07118e6i			0.423634i
$622$		0			0
$623$	634224.	+	1.59217e6i	0.0654670	+	0.164350i
$624$		0			0
$625$	−1.09199e7			−1.11819
$626$		0			0
$627$			5.72189e6i			0.581260i
$628$		0			0
$629$			2.57011e6i			0.259016i
$630$		0			0
$631$			1.92095e7i			1.92062i	0.278931	+	0.960311i	$0.410020\pi$
$631$			1.92095e7i			1.92062i	−0.278931	+	0.960311i	$0.589980\pi$
$632$		0			0
$633$		−	4.66347e6i		−	0.462594i
$634$		0			0
$635$	1.92538e7			1.89488
$636$		0			0
$637$	−1.05403e7	−	1.11310e7i	−1.02921	−	1.08689i
$638$		0			0
$639$			8.25666e6i			0.799931i
$640$		0			0
$641$	1.00426e7			0.965388			0.482694	−	0.875789i	$-0.339658\pi$
$641$	1.00426e7			0.965388			0.482694	+	0.875789i	$0.339658\pi$
$642$		0			0
$643$	−1.06242e7			−1.01337			−0.506686	−	0.862131i	$-0.669130\pi$
$643$	−1.06242e7			−1.01337			−0.506686	+	0.862131i	$0.669130\pi$
$644$		0			0
$645$	−259547.			−0.0245650
$646$		0			0
$647$	−1.78505e6			−0.167644			−0.0838222	−	0.996481i	$-0.526713\pi$
$647$	−1.78505e6			−0.167644			−0.0838222	+	0.996481i	$0.526713\pi$
$648$		0			0
$649$			3.60896e6i			0.336333i
$650$		0			0
$651$	398412.	−	158703.i	0.0368451	−	0.0146769i
$652$		0			0
$653$	−2.00432e7			−1.83943			−0.919715	−	0.392586i	$-0.871581\pi$
$653$	−2.00432e7			−1.83943			−0.919715	+	0.392586i	$0.871581\pi$
$654$		0			0
$655$		−	2.35808e7i		−	2.14761i
$656$		0			0
$657$		−	1.19590e7i		−	1.08089i
$658$		0			0
$659$			1.04062e7i			0.933419i	0.884411	+	0.466710i	$0.154561\pi$
$659$			1.04062e7i			0.933419i	−0.884411	+	0.466710i	$0.845439\pi$
$660$		0			0
$661$			1.60791e7i			1.43139i	0.698411	+	0.715697i	$0.253891\pi$
$661$			1.60791e7i			1.43139i	−0.698411	+	0.715697i	$0.746109\pi$
$662$		0			0
$663$	−2.81395e6			−0.248618
$664$		0			0
$665$	−9.70466e6	−	2.43628e7i	−0.850994	−	2.13636i
$666$		0			0
$667$		−	9.68118e6i		−	0.842585i
$668$		0			0
$669$	4.69747e6			0.405788
$670$		0			0
$671$	5.83859e6			0.500612
$672$		0			0
$673$	1.22652e7			1.04384			0.521922	−	0.852993i	$-0.325215\pi$
$673$	1.22652e7			1.04384			0.521922	+	0.852993i	$0.325215\pi$
$674$		0			0
$675$	−7.72782e6			−0.652826
$676$		0			0
$677$		−	3.04232e6i		−	0.255113i	−0.991831	−	0.127557i	$-0.959287\pi$
$677$		−	3.04232e6i		−	0.255113i	0.991831	−	0.127557i	$-0.0407134\pi$
$678$		0			0
$679$	2.74204e6	+	6.88370e6i	0.228244	+	0.572990i
$680$		0			0
$681$	5.63482e6			0.465600
$682$		0			0
$683$			7.90467e6i			0.648384i	0.945991	+	0.324192i	$0.105092\pi$
$683$			7.90467e6i			0.648384i	−0.945991	+	0.324192i	$0.894908\pi$
$684$		0			0
$685$		−	1.42149e6i		−	0.115749i
$686$		0			0
$687$		−	7.46807e6i		−	0.603694i
$688$		0			0
$689$			4.49642e6i			0.360844i
$690$		0			0
$691$	−1.21588e6			−0.0968717			−0.0484359	−	0.998826i	$-0.515424\pi$
$691$	−1.21588e6			−0.0968717			−0.0484359	+	0.998826i	$0.515424\pi$
$692$		0			0
$693$	3.23522e6	+	8.12178e6i	0.255900	+	0.642419i
$694$		0			0
$695$			2.63718e6i			0.207099i
$696$		0			0
$697$	−333583.			−0.0260089
$698$		0			0
$699$	1.79212e6			0.138731
$700$		0			0
$701$	15658.1			0.00120350			0.000601748	−	1.00000i	$-0.499808\pi$
$701$	15658.1			0.00120350			0.000601748		1.00000i	$0.499808\pi$
$702$		0			0
$703$	−1.42386e7			−1.08662
$704$		0			0
$705$		−	5.72643e6i		−	0.433922i
$706$		0			0
$707$	−3.70795e6	−	9.30853e6i	−0.278988	−	0.700378i
$708$		0			0
$709$	−8.85478e6			−0.661550			−0.330775	−	0.943710i	$-0.607310\pi$
$709$	−8.85478e6			−0.661550			−0.330775	+	0.943710i	$0.607310\pi$
$710$		0			0
$711$			2.54648e6i			0.188915i
$712$		0			0
$713$			729047.i			0.0537071i
$714$		0			0
$715$		−	2.32986e7i		−	1.70438i
$716$		0			0
$717$			572234.i			0.0415696i
$718$		0			0
$719$	−3.64275e6			−0.262789			−0.131394	−	0.991330i	$-0.541945\pi$
$719$	−3.64275e6			−0.262789			−0.131394	+	0.991330i	$0.541945\pi$
$720$		0			0
$721$	−2.16914e7	+	8.64053e6i	−1.55400	+	0.619017i
$722$		0			0
$723$		−	4.82541e6i		−	0.343312i
$724$		0			0
$725$	1.83766e7			1.29844
$726$		0			0
$727$	1.57332e7			1.10403			0.552015	−	0.833834i	$-0.313859\pi$
$727$	1.57332e7			1.10403			0.552015	+	0.833834i	$0.313859\pi$
$728$		0			0
$729$	−1.65005e6			−0.114995
$730$		0			0
$731$	−252481.			−0.0174757
$732$		0			0
$733$			8.40149e6i			0.577559i	0.957396	+	0.288780i	$0.0932495\pi$
$733$			8.40149e6i			0.577559i	−0.957396	+	0.288780i	$0.906751\pi$
$734$		0			0
$735$	5.68475e6	+	6.00334e6i	0.388144	+	0.409897i
$736$		0			0
$737$	1.64813e7			1.11770
$738$		0			0
$739$			2.67625e7i			1.80267i	0.433125	+	0.901334i	$0.357411\pi$
$739$			2.67625e7i			1.80267i	−0.433125	+	0.901334i	$0.642589\pi$
$740$		0			0
$741$		−	1.55894e7i		−	1.04300i
$742$		0			0
$743$		−	4.36420e6i		−	0.290023i	−0.989430	−	0.145011i	$-0.953678\pi$
$743$		−	4.36420e6i		−	0.290023i	0.989430	−	0.145011i	$-0.0463219\pi$
$744$		0			0
$745$			1.72868e7i			1.14110i
$746$		0			0
$747$	−9.98497e6			−0.654705
$748$		0			0
$749$	−1.03106e7	−	2.58840e7i	−0.671551	−	1.68588i
$750$		0			0
$751$		−	1.22583e7i		−	0.793106i	−0.918012	−	0.396553i	$-0.870206\pi$
$751$		−	1.22583e7i		−	0.793106i	0.918012	−	0.396553i	$-0.129794\pi$
$752$		0			0
$753$	4.95210e6			0.318275
$754$		0			0
$755$	2.22928e7			1.42330
$756$		0			0
$757$	1.29285e7			0.819991			0.409995	−	0.912088i	$-0.365530\pi$
$757$	1.29285e7			0.819991			0.409995	+	0.912088i	$0.365530\pi$
$758$		0			0
$759$	3.06664e6			0.193223
$760$		0			0
$761$			2.91943e7i			1.82741i	0.406375	+	0.913707i	$0.366793\pi$
$761$			2.91943e7i			1.82741i	−0.406375	+	0.913707i	$0.633207\pi$
$762$		0			0
$763$	7.53572e6	−	3.00177e6i	0.468612	−	0.186666i
$764$		0			0
$765$	−7.35505e6			−0.454394
$766$		0			0
$767$		−	9.83269e6i		−	0.603509i
$768$		0			0
$769$		−	1.48516e7i		−	0.905645i	−0.891601	−	0.452823i	$-0.850417\pi$
$769$		−	1.48516e7i		−	0.905645i	0.891601	−	0.452823i	$-0.149583\pi$
$770$		0			0
$771$		−	7.91892e6i		−	0.479766i
$772$		0			0
$773$		−	6.86539e6i		−	0.413254i	−0.978420	−	0.206627i	$-0.933751\pi$
$773$		−	6.86539e6i		−	0.413254i	0.978420	−	0.206627i	$-0.0662486\pi$
$774$		0			0
$775$	−1.38386e6			−0.0827634
$776$		0			0
$777$	4.17029e6	−	1.66119e6i	0.247807	−	0.0987112i
$778$		0			0
$779$		−	1.84807e6i		−	0.109112i
$780$		0			0
$781$	1.37221e7			0.804994
$782$		0			0
$783$	−1.95231e7			−1.13801
$784$		0			0
$785$	−1.86352e7			−1.07935
$786$		0			0
$787$	−1.71195e7			−0.985268			−0.492634	−	0.870237i	$-0.663966\pi$
$787$	−1.71195e7			−0.985268			−0.492634	+	0.870237i	$0.663966\pi$
$788$		0			0
$789$			9.11055e6i			0.521017i
$790$		0			0
$791$	−1.52842e7	+	6.08830e6i	−0.868565	+	0.345983i
$792$		0			0
$793$	−1.59074e7			−0.898288
$794$		0			0
$795$		−	2.42507e6i		−	0.136084i
$796$		0			0
$797$			9.93315e6i			0.553913i	0.960883	+	0.276956i	$0.0893257\pi$
$797$			9.93315e6i			0.553913i	−0.960883	+	0.276956i	$0.910674\pi$
$798$		0			0
$799$		−	5.57054e6i		−	0.308696i
$800$		0			0
$801$			2.66294e6i			0.146649i
$802$		0			0
$803$	−1.98752e7			−1.08773
$804$		0			0
$805$	−1.30572e7	+	5.20120e6i	−0.710169	+	0.282888i
$806$		0			0
$807$			668642.i			0.0361418i
$808$		0			0
$809$	7.89984e6			0.424372			0.212186	−	0.977229i	$-0.431942\pi$
$809$	7.89984e6			0.424372			0.212186	+	0.977229i	$0.431942\pi$
$810$		0			0
$811$	−422897.			−0.0225778			−0.0112889	−	0.999936i	$-0.503593\pi$
$811$	−422897.			−0.0225778			−0.0112889	+	0.999936i	$0.503593\pi$
$812$		0			0
$813$	8.45472e6			0.448614
$814$		0			0
$815$	1.67031e7			0.880850
$816$		0			0
$817$		−	1.39876e6i		−	0.0733141i
$818$		0			0
$819$	−8.81444e6	−	2.21280e7i	−0.459182	−	1.15274i
$820$		0			0
$821$	−4.21022e6			−0.217995			−0.108998	−	0.994042i	$-0.534764\pi$
$821$	−4.21022e6			−0.217995			−0.108998	+	0.994042i	$0.534764\pi$
$822$		0			0
$823$		−	1.39124e7i		−	0.715984i	−0.933725	−	0.357992i	$-0.883461\pi$
$823$		−	1.39124e7i		−	0.715984i	0.933725	−	0.357992i	$-0.116539\pi$
$824$		0			0
$825$			5.82103e6i			0.297759i
$826$		0			0
$827$			7.78047e6i			0.395587i	0.980244	+	0.197794i	$0.0633776\pi$
$827$			7.78047e6i			0.395587i	−0.980244	+	0.197794i	$0.936622\pi$
$828$		0			0
$829$		−	2.12812e6i		−	0.107550i	−0.998553	−	0.0537748i	$-0.982875\pi$
$829$		−	2.12812e6i		−	0.107550i	0.998553	−	0.0537748i	$-0.0171253\pi$
$830$		0			0
$831$	−9.69174e6			−0.486855
$832$		0			0
$833$	5.52999e6	+	5.83991e6i	0.276129	+	0.291604i
$834$		0			0
$835$		−	3.37232e7i		−	1.67384i
$836$		0			0
$837$	1.47020e6			0.0725376
$838$		0			0
$839$	1.95328e7			0.957988			0.478994	−	0.877818i	$-0.341002\pi$
$839$	1.95328e7			0.957988			0.478994	+	0.877818i	$0.341002\pi$
$840$		0			0
$841$	2.59145e7			1.26344
$842$		0			0
$843$	−2.16151e6			−0.104758
$844$		0			0
$845$			3.51472e7i			1.69336i
$846$		0			0
$847$	−5.89878e6	+	2.34971e6i	−0.282523	+	0.112540i
$848$		0			0
$849$	−6.01851e6			−0.286563
$850$		0			0
$851$			7.63113e6i			0.361214i
$852$		0			0
$853$		−	1.65240e7i		−	0.777576i	−0.921327	−	0.388788i	$-0.872894\pi$
$853$		−	1.65240e7i		−	0.777576i	0.921327	−	0.388788i	$-0.127106\pi$
$854$		0			0
$855$		−	4.07473e7i		−	1.90627i
$856$		0			0
$857$		−	4.14446e7i		−	1.92760i	−0.266634	−	0.963798i	$-0.585912\pi$
$857$		−	4.14446e7i		−	1.92760i	0.266634	−	0.963798i	$-0.414088\pi$
$858$		0			0
$859$	−1.64726e7			−0.761691			−0.380845	−	0.924639i	$-0.624367\pi$
$859$	−1.64726e7			−0.761691			−0.380845	+	0.924639i	$0.624367\pi$
$860$		0			0
$861$	215611.	+	541276.i	0.00991204	+	0.0248834i
$862$		0			0
$863$			2.56771e7i			1.17360i	0.809733	+	0.586798i	$0.199612\pi$
$863$			2.56771e7i			1.17360i	−0.809733	+	0.586798i	$0.800388\pi$
$864$		0			0
$865$	3.41218e7			1.55057
$866$		0			0
$867$	−7.67757e6			−0.346877
$868$		0			0
$869$	4.23210e6			0.190111
$870$		0			0
$871$	−4.49038e7			−2.00557
$872$		0			0
$873$			1.15131e7i			0.511278i
$874$		0			0
$875$	1.56662e6	+	3.93288e6i	0.0691740	+	0.173656i
$876$		0			0
$877$	−2.56342e7			−1.12543			−0.562717	−	0.826649i	$-0.690244\pi$
$877$	−2.56342e7			−1.12543			−0.562717	+	0.826649i	$0.690244\pi$
$878$		0			0
$879$			3.63025e6i			0.158476i
$880$		0			0
$881$		−	2.41946e7i		−	1.05022i	−0.851036	−	0.525108i	$-0.824025\pi$
$881$		−	2.41946e7i		−	1.05022i	0.851036	−	0.525108i	$-0.175975\pi$
$882$		0			0
$883$			1.42568e6i			0.0615348i	0.999527	+	0.0307674i	$0.00979512\pi$
$883$			1.42568e6i			0.0615348i	−0.999527	+	0.0307674i	$0.990205\pi$
$884$		0			0
$885$			5.30310e6i			0.227600i
$886$		0			0
$887$	−2.26183e7			−0.965274			−0.482637	−	0.875821i	$-0.660321\pi$
$887$	−2.26183e7			−0.965274			−0.482637	+	0.875821i	$0.660321\pi$
$888$		0			0
$889$	−1.21059e7	−	3.03909e7i	−0.513737	−	1.28970i
$890$		0			0
$891$			1.02026e7i			0.430541i
$892$		0			0
$893$	3.08611e7			1.29504
$894$		0			0
$895$	−3.32509e7			−1.38754
$896$		0			0
$897$	−8.35513e6			−0.346715
$898$		0			0
$899$	−3.49611e6			−0.144273
$900$		0			0
$901$		−	2.35905e6i		−	0.0968113i
$902$		0			0
$903$	163191.	+	409678.i	0.00666003	+	0.0167195i
$904$		0			0
$905$	−5.45703e7			−2.21480
$906$		0			0
$907$			1.26676e7i			0.511301i	0.966769	+	0.255650i	$0.0822896\pi$
$907$			1.26676e7i			0.511301i	−0.966769	+	0.255650i	$0.917710\pi$
$908$		0			0
$909$		−	1.55687e7i		−	0.624946i
$910$		0			0
$911$			347737.i			0.0138821i	0.999976	+	0.00694105i	$0.00220942\pi$
$911$			347737.i			0.0138821i	−0.999976	+	0.00694105i	$0.997791\pi$
$912$		0			0
$913$			1.65944e7i			0.658848i
$914$		0			0
$915$	8.57938e6			0.338769
$916$		0			0
$917$	−3.72208e7	+	1.48265e7i	−1.46172	+	0.582258i
$918$		0			0
$919$			3.37846e6i			0.131956i	0.997821	+	0.0659781i	$0.0210167\pi$
$919$			3.37846e6i			0.131956i	−0.997821	+	0.0659781i	$0.978983\pi$
$920$		0			0
$921$	1.60779e7			0.624571
$922$		0			0
$923$	−3.73861e7			−1.44446
$924$		0			0
$925$	−1.44853e7			−0.556637
$926$		0			0
$927$	−3.62793e7			−1.38663
$928$		0			0
$929$			3.30634e7i			1.25692i	0.777841	+	0.628461i	$0.216315\pi$
$929$			3.30634e7i			1.25692i	−0.777841	+	0.628461i	$0.783685\pi$
$930$		0			0
$931$	−3.23534e7	+	3.06364e7i	−1.22333	+	1.15841i
$932$		0			0
$933$	−1.28012e6			−0.0481444
$934$		0			0
$935$			1.22237e7i			0.457270i
$936$		0			0
$937$		−	1.72186e7i		−	0.640692i	−0.947301	−	0.320346i	$-0.896201\pi$
$937$		−	1.72186e7i		−	0.640692i	0.947301	−	0.320346i	$-0.103799\pi$
$938$		0			0
$939$			1.55137e7i			0.574184i
$940$		0			0
$941$		−	2.81758e7i		−	1.03729i	−0.854989	−	0.518647i	$-0.826436\pi$
$941$		−	2.81758e7i		−	1.03729i	0.854989	−	0.518647i	$-0.173564\pi$
$942$		0			0
$943$	−990469.			−0.0362712
$944$		0			0
$945$	1.04888e7	+	2.63313e7i	0.382072	+	0.959164i
$946$		0			0
$947$			2.79057e6i			0.101115i	0.998721	+	0.0505577i	$0.0160999\pi$
$947$			2.79057e6i			0.101115i	−0.998721	+	0.0505577i	$0.983900\pi$
$948$		0			0
$949$	5.41503e7			1.95180
$950$		0			0
$951$	5.42681e6			0.194578
$952$		0			0
$953$	4.15030e7			1.48029			0.740146	−	0.672446i	$-0.234756\pi$
$953$	4.15030e7			1.48029			0.740146	+	0.672446i	$0.234756\pi$
$954$		0			0
$955$	−2.67527e7			−0.949204
$956$		0			0
$957$			1.47059e7i			0.519054i
$958$		0			0
$959$	−2.24373e6	+	893766.i	−0.0787816	+	0.0313818i
$960$		0			0
$961$	−2.83659e7			−0.990804
$962$		0			0
$963$		−	4.32914e7i		−	1.50431i
$964$		0			0
$965$		−	2.57846e7i		−	0.891337i
$966$		0			0
$967$			5.78401e7i			1.98913i	0.104121	+	0.994565i	$0.466797\pi$
$967$			5.78401e7i			1.98913i	−0.104121	+	0.994565i	$0.533203\pi$
$968$		0			0
$969$			8.17901e6i			0.279828i
$970$		0			0
$971$	3.86875e7			1.31681			0.658405	−	0.752664i	$-0.271232\pi$
$971$	3.86875e7			1.31681			0.658405	+	0.752664i	$0.271232\pi$
$972$		0			0
$973$	4.16262e6	−	1.65813e6i	0.140956	−	0.0561484i
$974$		0			0
$975$		−	1.58595e7i		−	0.534292i
$976$		0			0
$977$	3.97537e7			1.33242			0.666211	−	0.745764i	$-0.267915\pi$
$977$	3.97537e7			1.33242			0.666211	+	0.745764i	$0.267915\pi$
$978$		0			0
$979$	4.42565e6			0.147578
$980$		0			0
$981$	1.26036e7			0.418142
$982$		0			0
$983$	5.72858e7			1.89088			0.945439	−	0.325799i	$-0.105633\pi$
$983$	5.72858e7			1.89088			0.945439	+	0.325799i	$0.105633\pi$
$984$		0			0
$985$			2.42769e6i			0.0797266i
$986$		0			0
$987$	−9.03881e6	+	3.60051e6i	−0.295337	+	0.117644i
$988$		0			0
$989$	−749662.			−0.0243711
$990$		0			0
$991$		−	4.14746e6i		−	0.134152i	−0.997748	−	0.0670762i	$-0.978633\pi$
$991$		−	4.14746e6i		−	0.134152i	0.997748	−	0.0670762i	$-0.0213671\pi$
$992$		0			0
$993$			1.25786e7i			0.404816i
$994$		0			0
$995$		−	2.37988e6i		−	0.0762076i
$996$		0			0
$997$			4.07956e7i			1.29980i	0.760021	+	0.649898i	$0.225189\pi$
$997$			4.07956e7i			1.29980i	−0.760021	+	0.649898i	$0.774811\pi$
$998$		0			0
$999$	1.53890e7			0.487862

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.6.f.d.447.7		16
4.3	odd	2	inner	448.6.f.d.447.9		16
7.6	odd	2	inner	448.6.f.d.447.10		16
8.3	odd	2		28.6.d.b.27.5	✓	16
8.5	even	2		28.6.d.b.27.8	yes	16
24.5	odd	2		252.6.b.d.55.9		16
24.11	even	2		252.6.b.d.55.11		16
28.27	even	2	inner	448.6.f.d.447.8		16
56.13	odd	2		28.6.d.b.27.7	yes	16
56.27	even	2		28.6.d.b.27.6	yes	16
168.83	odd	2		252.6.b.d.55.12		16
168.125	even	2		252.6.b.d.55.10		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.6.d.b.27.5	✓	16	8.3	odd	2
28.6.d.b.27.6	yes	16	56.27	even	2
28.6.d.b.27.7	yes	16	56.13	odd	2
28.6.d.b.27.8	yes	16	8.5	even	2
252.6.b.d.55.9		16	24.5	odd	2
252.6.b.d.55.10		16	168.125	even	2
252.6.b.d.55.11		16	24.11	even	2
252.6.b.d.55.12		16	168.83	odd	2
448.6.f.d.447.7		16	1.1	even	1	trivial
448.6.f.d.447.8		16	28.27	even	2	inner
448.6.f.d.447.9		16	4.3	odd	2	inner
448.6.f.d.447.10		16	7.6	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-6.44707 q^{3} -76.3023i q^{5} +(-120.438 + 47.9752i) q^{7} -201.435 q^{9} +O(q^{10})\)	\(q-6.44707 q^{3} -76.3023i q^{5} +(-120.438 + 47.9752i) q^{7} -201.435 q^{9} +334.774i q^{11} -912.099i q^{13} +491.926i q^{15} +478.534i q^{17} -2651.10 q^{19} +(776.473 - 309.299i) q^{21} +1420.85i q^{23} -2697.03 q^{25} +2865.30 q^{27} -6813.64 q^{29} +513.105 q^{31} -2158.31i q^{33} +(3660.62 + 9189.71i) q^{35} +5370.81 q^{37} +5880.36i q^{39} +697.095i q^{41} +527.614i q^{43} +15370.0i q^{45} -11640.9 q^{47} +(12203.8 - 11556.1i) q^{49} -3085.14i q^{51} -4929.75 q^{53} +25544.0 q^{55} +17091.8 q^{57} +10780.3 q^{59} -17440.4i q^{61} +(24260.5 - 9663.91i) q^{63} -69595.2 q^{65} -49231.3i q^{67} -9160.33i q^{69} -40989.1i q^{71} +59368.9i q^{73} +17388.0 q^{75} +(-16060.8 - 40319.6i) q^{77} -12641.7i q^{79} +30476.0 q^{81} +49569.1 q^{83} +36513.2 q^{85} +43928.0 q^{87} -13219.8i q^{89} +(43758.1 + 109852. i) q^{91} -3308.02 q^{93} +202285. i q^{95} -57155.4i q^{97} -67435.2i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 1616 q^{9}+O(q^{10}) \) 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

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