LMFDB - Embedded newform 1456.2.j.a.1119.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1456,2,Mod(1119,1456)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1456, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1456.1119");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1456 = 2^{4} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1456.j (of order $2$, degree $1$, 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:	$11.6262185343$
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} - 20 x^{13} - 9 x^{12} + 42 x^{11} + 200 x^{10} - 26 x^{9} - 305 x^{8} - 922 x^{7} + 3202 x^{6} + \cdots + 1444 $ x^16 - 20x^13 - 9x^12 + 42x^11 + 200x^10 - 26x^9 - 305x^8 - 922x^7 + 3202x^6 + 1994x^5 + 5245x^4 - 12388x^3 + 11552x^2 - 5776*x + 1444
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1119.8
Root			$-0.615409 - 2.42121i$ of defining polynomial
Character	$\chi$	$=$	1456.1119
Dual form			1456.2.j.a.1119.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.623212 q^{3} +2.24100i q^{5} +(0.559261 - 2.58597i) q^{7} -2.61161 q^{9} +O(q^{10})$	$q-0.623212 q^{3} +2.24100i q^{5} +(0.559261 - 2.58597i) q^{7} -2.61161 q^{9} +0.278096i q^{11} +1.00000i q^{13} -1.39662i q^{15} -3.26307i q^{17} +3.42332 q^{19} +(-0.348538 + 1.61161i) q^{21} -4.39165i q^{23} -0.0220707 q^{25} +3.49722 q^{27} +1.89246 q^{29} +5.44315 q^{31} -0.173313i q^{33} +(5.79515 + 1.25330i) q^{35} -4.13346 q^{37} -0.623212i q^{39} +2.00000i q^{41} +4.17444i q^{43} -5.85260i q^{45} +8.72316 q^{47} +(-6.37446 - 2.89246i) q^{49} +2.03358i q^{51} +7.98606 q^{53} -0.623212 q^{55} -2.13346 q^{57} +8.45194 q^{59} -0.697076i q^{61} +(-1.46057 + 6.75353i) q^{63} -2.24100 q^{65} -5.15818i q^{67} +2.73693i q^{69} +1.52452i q^{71} -6.06768i q^{73} +0.0137548 q^{75} +(0.719147 + 0.155528i) q^{77} -4.86467i q^{79} +5.65531 q^{81} +12.4376 q^{83} +7.31253 q^{85} -1.17940 q^{87} +12.3787i q^{89} +(2.58597 + 0.559261i) q^{91} -3.39224 q^{93} +7.67166i q^{95} +2.14740i q^{97} -0.726277i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{9}+O(q^{10}) $ 16 * q + 16 * q^9	$ 16 q + 16 q^{9} + 8 q^{21} + 4 q^{25} + 4 q^{29} + 4 q^{49} - 36 q^{53} + 32 q^{57} + 4 q^{65} - 20 q^{77} + 16 q^{81} - 56 q^{85} - 24 q^{93}+O(q^{100}) $ 16 * q + 16 * q^9 + 8 * q^21 + 4 * q^25 + 4 * q^29 + 4 * q^49 - 36 * q^53 + 32 * q^57 + 4 * q^65 - 20 * q^77 + 16 * q^81 - 56 * q^85 - 24 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1456\mathbb{Z}\right)^\times$.

$n$	$561$	$911$	$1093$	$1249$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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
$3$	−0.623212			−0.359812			−0.179906	−	0.983684i	$-0.557579\pi$
$3$	−0.623212			−0.359812			−0.179906	+	0.983684i	$0.557579\pi$
$4$		0			0
$5$			2.24100i			1.00220i	0.865388	+	0.501102i	$0.167072\pi$
$5$			2.24100i			1.00220i	−0.865388	+	0.501102i	$0.832928\pi$
$6$		0			0
$7$	0.559261	−	2.58597i	0.211381	−	0.977404i
$7$	0.559261	−	2.58597i	0.211381	−	0.977404i
$8$		0			0
$9$	−2.61161			−0.870536
$10$		0			0
$11$			0.278096i			0.0838491i	0.999121	+	0.0419245i	$0.0133489\pi$
$11$			0.278096i			0.0838491i	−0.999121	+	0.0419245i	$0.986651\pi$
$12$		0			0
$13$			1.00000i			0.277350i
$13$			1.00000i			0.277350i
$14$		0			0
$15$		−	1.39662i		−	0.360605i
$16$		0			0
$17$		−	3.26307i		−	0.791410i	−0.918378	−	0.395705i	$-0.870500\pi$
$17$		−	3.26307i		−	0.791410i	0.918378	−	0.395705i	$-0.129500\pi$
$18$		0			0
$19$	3.42332			0.785364			0.392682	−	0.919674i	$-0.371547\pi$
$19$	3.42332			0.785364			0.392682	+	0.919674i	$0.371547\pi$
$20$		0			0
$21$	−0.348538	+	1.61161i	−0.0760572	+	0.351681i
$22$		0			0
$23$		−	4.39165i		−	0.915723i	−0.889024	−	0.457861i	$-0.848616\pi$
$23$		−	4.39165i		−	0.915723i	0.889024	−	0.457861i	$-0.151384\pi$
$24$		0			0
$25$	−0.0220707			−0.00441415
$26$		0			0
$27$	3.49722			0.673041
$28$		0			0
$29$	1.89246			0.351421			0.175710	−	0.984442i	$-0.443778\pi$
$29$	1.89246			0.351421			0.175710	+	0.984442i	$0.443778\pi$
$30$		0			0
$31$	5.44315			0.977619			0.488810	−	0.872390i	$-0.337431\pi$
$31$	5.44315			0.977619			0.488810	+	0.872390i	$0.337431\pi$
$32$		0			0
$33$		−	0.173313i		−	0.0301699i
$34$		0			0
$35$	5.79515	+	1.25330i	0.979559	+	0.211847i
$36$		0			0
$37$	−4.13346			−0.679536			−0.339768	−	0.940509i	$-0.610349\pi$
$37$	−4.13346			−0.679536			−0.339768	+	0.940509i	$0.610349\pi$
$38$		0			0
$39$		−	0.623212i		−	0.0997938i
$40$		0			0
$41$			2.00000i			0.312348i	0.987730	+	0.156174i	$0.0499160\pi$
$41$			2.00000i			0.312348i	−0.987730	+	0.156174i	$0.950084\pi$
$42$		0			0
$43$			4.17444i			0.636596i	0.947991	+	0.318298i	$0.103111\pi$
$43$			4.17444i			0.636596i	−0.947991	+	0.318298i	$0.896889\pi$
$44$		0			0
$45$		−	5.85260i		−	0.872455i
$46$		0			0
$47$	8.72316			1.27240			0.636202	−	0.771523i	$-0.280505\pi$
$47$	8.72316			1.27240			0.636202	+	0.771523i	$0.280505\pi$
$48$		0			0
$49$	−6.37446	−	2.89246i	−0.910636	−	0.413209i
$50$		0			0
$51$			2.03358i			0.284759i
$52$		0			0
$53$	7.98606			1.09697			0.548485	−	0.836160i	$-0.315205\pi$
$53$	7.98606			1.09697			0.548485	+	0.836160i	$0.315205\pi$
$54$		0			0
$55$	−0.623212			−0.0840339
$56$		0			0
$57$	−2.13346			−0.282583
$58$		0			0
$59$	8.45194			1.10035			0.550175	−	0.835050i	$-0.314561\pi$
$59$	8.45194			1.10035			0.550175	+	0.835050i	$0.314561\pi$
$60$		0			0
$61$		−	0.697076i		−	0.0892515i	−0.999004	−	0.0446257i	$-0.985790\pi$
$61$		−	0.697076i		−	0.0892515i	0.999004	−	0.0446257i	$-0.0142095\pi$
$62$		0			0
$63$	−1.46057	+	6.75353i	−0.184014	+	0.850865i
$64$		0			0
$65$	−2.24100			−0.277962
$66$		0			0
$67$		−	5.15818i		−	0.630172i	−0.949063	−	0.315086i	$-0.897967\pi$
$67$		−	5.15818i		−	0.630172i	0.949063	−	0.315086i	$-0.102033\pi$
$68$		0			0
$69$			2.73693i			0.329488i
$70$		0			0
$71$			1.52452i			0.180927i	0.995900	+	0.0904636i	$0.0288349\pi$
$71$			1.52452i			0.180927i	−0.995900	+	0.0904636i	$0.971165\pi$
$72$		0			0
$73$		−	6.06768i		−	0.710169i	−0.934834	−	0.355084i	$-0.884452\pi$
$73$		−	6.06768i		−	0.710169i	0.934834	−	0.355084i	$-0.115548\pi$
$74$		0			0
$75$	0.0137548			0.00158826
$76$		0			0
$77$	0.719147	+	0.155528i	0.0819544	+	0.0177241i
$78$		0			0
$79$		−	4.86467i		−	0.547318i	−0.961827	−	0.273659i	$-0.911766\pi$
$79$		−	4.86467i		−	0.547318i	0.961827	−	0.273659i	$-0.0882340\pi$
$80$		0			0
$81$	5.65531			0.628368
$82$		0			0
$83$	12.4376			1.36520			0.682602	−	0.730791i	$-0.260848\pi$
$83$	12.4376			1.36520			0.682602	+	0.730791i	$0.260848\pi$
$84$		0			0
$85$	7.31253			0.793155
$86$		0			0
$87$	−1.17940			−0.126445
$88$		0			0
$89$			12.3787i			1.31214i	0.754698	+	0.656072i	$0.227783\pi$
$89$			12.3787i			1.31214i	−0.754698	+	0.656072i	$0.772217\pi$
$90$		0			0
$91$	2.58597	+	0.559261i	0.271083	+	0.0586264i
$92$		0			0
$93$	−3.39224			−0.351759
$94$		0			0
$95$			7.67166i			0.787096i
$96$		0			0
$97$			2.14740i			0.218035i	0.994040	+	0.109018i	$0.0347705\pi$
$97$			2.14740i			0.218035i	−0.994040	+	0.109018i	$0.965230\pi$
$98$		0			0
$99$		−	0.726277i		−	0.0729936i
$100$		0			0
$101$			2.30677i			0.229532i	0.993393	+	0.114766i	$0.0366119\pi$
$101$			2.30677i			0.229532i	−0.993393	+	0.114766i	$0.963388\pi$
$102$		0			0
$103$	−10.5997			−1.04442			−0.522209	−	0.852818i	$-0.674892\pi$
$103$	−10.5997			−1.04442			−0.522209	+	0.852818i	$0.674892\pi$
$104$		0			0
$105$	−3.61161	−	0.781073i	−0.352457	−	0.0762249i
$106$		0			0
$107$		−	6.74972i		−	0.652520i	−0.945280	−	0.326260i	$-0.894211\pi$
$107$		−	6.74972i		−	0.652520i	0.945280	−	0.326260i	$-0.105789\pi$
$108$		0			0
$109$	3.57175			0.342112			0.171056	−	0.985261i	$-0.445282\pi$
$109$	3.57175			0.342112			0.171056	+	0.985261i	$0.445282\pi$
$110$		0			0
$111$	2.57602			0.244505
$112$		0			0
$113$	0.806990			0.0759152			0.0379576	−	0.999279i	$-0.487915\pi$
$113$	0.806990			0.0759152			0.0379576	+	0.999279i	$0.487915\pi$
$114$		0			0
$115$	9.84168			0.917742
$116$		0			0
$117$		−	2.61161i		−	0.241443i
$118$		0			0
$119$	−8.43819	−	1.82491i	−0.773527	−	0.167289i
$120$		0			0
$121$	10.9227			0.992969
$122$		0			0
$123$		−	1.24642i		−	0.112386i
$124$		0			0
$125$			11.1555i			0.997781i
$126$		0			0
$127$		−	12.9199i		−	1.14645i	−0.819396	−	0.573227i	$-0.805691\pi$
$127$		−	12.9199i		−	1.14645i	0.819396	−	0.573227i	$-0.194309\pi$
$128$		0			0
$129$		−	2.60156i		−	0.229055i
$130$		0			0
$131$	17.1768			1.50074			0.750370	−	0.661018i	$-0.229875\pi$
$131$	17.1768			1.50074			0.750370	+	0.661018i	$0.229875\pi$
$132$		0			0
$133$	1.91453	−	8.85260i	0.166011	−	0.767618i
$134$		0			0
$135$			7.83727i			0.674524i
$136$		0			0
$137$	12.0537			1.02982			0.514911	−	0.857244i	$-0.327825\pi$
$137$	12.0537			1.02982			0.514911	+	0.857244i	$0.327825\pi$
$138$		0			0
$139$	−0.0807752			−0.00685126			−0.00342563	−	0.999994i	$-0.501090\pi$
$139$	−0.0807752			−0.00685126			−0.00342563	+	0.999994i	$0.501090\pi$
$140$		0			0
$141$	−5.43638			−0.457826
$142$		0			0
$143$	−0.278096			−0.0232555
$144$		0			0
$145$			4.24100i			0.352196i
$146$		0			0
$147$	3.97264	+	1.80262i	0.327658	+	0.148677i
$148$		0			0
$149$	−5.52761			−0.452839			−0.226420	−	0.974030i	$-0.572702\pi$
$149$	−5.52761			−0.452839			−0.226420	+	0.974030i	$0.572702\pi$
$150$		0			0
$151$		−	13.1286i		−	1.06839i	−0.845362	−	0.534194i	$-0.820615\pi$
$151$		−	13.1286i		−	1.06839i	0.845362	−	0.534194i	$-0.179385\pi$
$152$		0			0
$153$			8.52185i			0.688951i
$154$		0			0
$155$			12.1981i			0.979775i
$156$		0			0
$157$		−	13.4858i		−	1.07629i	−0.842853	−	0.538144i	$-0.819126\pi$
$157$		−	13.4858i		−	1.07629i	0.842853	−	0.538144i	$-0.180874\pi$
$158$		0			0
$159$	−4.97701			−0.394703
$160$		0			0
$161$	−11.3567	−	2.45608i	−0.895031	−	0.193566i
$162$		0			0
$163$		−	16.4618i		−	1.28939i	−0.764439	−	0.644696i	$-0.776984\pi$
$163$		−	16.4618i		−	1.28939i	0.764439	−	0.644696i	$-0.223016\pi$
$164$		0			0
$165$	0.388393			0.0302364
$166$		0			0
$167$	−0.0738978			−0.00571839			−0.00285919	−	0.999996i	$-0.500910\pi$
$167$	−0.0738978			−0.00571839			−0.00285919	+	0.999996i	$0.500910\pi$
$168$		0			0
$169$	−1.00000			−0.0769231
$170$		0			0
$171$	−8.94038			−0.683688
$172$		0			0
$173$		−	9.66535i		−	0.734843i	−0.930055	−	0.367422i	$-0.880241\pi$
$173$		−	9.66535i		−	0.734843i	0.930055	−	0.367422i	$-0.119759\pi$
$174$		0			0
$175$	−0.0123433	+	0.0570742i	−0.000933065	+	0.00431440i
$176$		0			0
$177$	−5.26735			−0.395919
$178$		0			0
$179$		−	21.4990i		−	1.60691i	−0.595365	−	0.803455i	$-0.702993\pi$
$179$		−	21.4990i		−	1.60691i	0.595365	−	0.803455i	$-0.297007\pi$
$180$		0			0
$181$			21.7095i			1.61365i	0.590788	+	0.806827i	$0.298817\pi$
$181$			21.7095i			1.61365i	−0.590788	+	0.806827i	$0.701183\pi$
$182$		0			0
$183$			0.434426i			0.0321137i
$184$		0			0
$185$		−	9.26307i		−	0.681034i
$186$		0			0
$187$	0.907446			0.0663590
$188$		0			0
$189$	1.95586	−	9.04370i	0.142268	−	0.657832i
$190$		0			0
$191$			9.87085i			0.714230i	0.934060	+	0.357115i	$0.116240\pi$
$191$			9.87085i			0.714230i	−0.934060	+	0.357115i	$0.883760\pi$
$192$		0			0
$193$	−5.56984			−0.400926			−0.200463	−	0.979701i	$-0.564245\pi$
$193$	−5.56984			−0.400926			−0.200463	+	0.979701i	$0.564245\pi$
$194$		0			0
$195$	1.39662			0.100014
$196$		0			0
$197$	−8.31106			−0.592138			−0.296069	−	0.955167i	$-0.595676\pi$
$197$	−8.31106			−0.592138			−0.296069	+	0.955167i	$0.595676\pi$
$198$		0			0
$199$	−26.6108			−1.88639			−0.943194	−	0.332241i	$-0.892195\pi$
$199$	−26.6108			−1.88639			−0.943194	+	0.332241i	$0.892195\pi$
$200$		0			0
$201$			3.21464i			0.226743i
$202$		0			0
$203$	1.05838	−	4.89384i	0.0742836	−	0.343480i
$204$		0			0
$205$	−4.48200			−0.313036
$206$		0			0
$207$			11.4693i			0.797169i
$208$		0			0
$209$			0.952012i			0.0658521i
$210$		0			0
$211$		−	0.504583i		−	0.0347370i	−0.999849	−	0.0173685i	$-0.994471\pi$
$211$		−	0.504583i		−	0.0347370i	0.999849	−	0.0173685i	$-0.00552884\pi$
$212$		0			0
$213$		−	0.950100i		−	0.0650998i
$214$		0			0
$215$	−9.35491			−0.637999
$216$		0			0
$217$	3.04414	−	14.0758i	0.206650	−	0.955529i
$218$		0			0
$219$			3.78146i			0.255527i
$220$		0			0
$221$	3.26307			0.219498
$222$		0			0
$223$	−25.8168			−1.72882			−0.864409	−	0.502790i	$-0.832307\pi$
$223$	−25.8168			−1.72882			−0.864409	+	0.502790i	$0.832307\pi$
$224$		0			0
$225$	0.0576401			0.00384267
$226$		0			0
$227$	10.2160			0.678058			0.339029	−	0.940776i	$-0.389902\pi$
$227$	10.2160			0.678058			0.339029	+	0.940776i	$0.389902\pi$
$228$		0			0
$229$		−	11.2256i		−	0.741808i	−0.928671	−	0.370904i	$-0.879048\pi$
$229$		−	11.2256i		−	0.741808i	0.928671	−	0.370904i	$-0.120952\pi$
$230$		0			0
$231$	−0.448181	−	0.0969270i	−0.0294882	−	0.00637733i
$232$		0			0
$233$	9.20690			0.603164			0.301582	−	0.953440i	$-0.402485\pi$
$233$	9.20690			0.603164			0.301582	+	0.953440i	$0.402485\pi$
$234$		0			0
$235$			19.5486i			1.27521i
$236$		0			0
$237$			3.03172i			0.196932i
$238$		0			0
$239$			18.8515i			1.21940i	0.792632	+	0.609700i	$0.208710\pi$
$239$			18.8515i			1.21940i	−0.792632	+	0.609700i	$0.791290\pi$
$240$		0			0
$241$			6.21312i			0.400222i	0.979773	+	0.200111i	$0.0641303\pi$
$241$			6.21312i			0.400222i	−0.979773	+	0.200111i	$0.935870\pi$
$242$		0			0
$243$	−14.0161			−0.899135
$244$		0			0
$245$	6.48200	−	14.2851i	0.414119	−	0.912644i
$246$		0			0
$247$			3.42332i			0.217821i
$248$		0			0
$249$	−7.75126			−0.491216
$250$		0			0
$251$	2.31782			0.146299			0.0731497	−	0.997321i	$-0.476695\pi$
$251$	2.31782			0.146299			0.0731497	+	0.997321i	$0.476695\pi$
$252$		0			0
$253$	1.22130			0.0767825
$254$		0			0
$255$	−4.55726			−0.285387
$256$		0			0
$257$			3.73737i			0.233131i	0.993183	+	0.116565i	$0.0371885\pi$
$257$			3.73737i			0.233131i	−0.993183	+	0.116565i	$0.962812\pi$
$258$		0			0
$259$	−2.31168	+	10.6890i	−0.143641	+	0.664181i
$260$		0			0
$261$	−4.94236			−0.305924
$262$		0			0
$263$			17.9880i			1.10919i	0.832121	+	0.554594i	$0.187127\pi$
$263$			17.9880i			1.10919i	−0.832121	+	0.554594i	$0.812873\pi$
$264$		0			0
$265$			17.8967i			1.09939i
$266$		0			0
$267$		−	7.71458i		−	0.472125i
$268$		0			0
$269$			7.66107i			0.467103i	0.972344	+	0.233552i	$0.0750348\pi$
$269$			7.66107i			0.467103i	−0.972344	+	0.233552i	$0.924965\pi$
$270$		0			0
$271$	−6.73250			−0.408970			−0.204485	−	0.978870i	$-0.565552\pi$
$271$	−6.73250			−0.408970			−0.204485	+	0.978870i	$0.565552\pi$
$272$		0			0
$273$	−1.61161	−	0.348538i	−0.0975389	−	0.0210945i
$274$		0			0
$275$		−	0.00613778i		−	0.000370122i
$276$		0			0
$277$	−24.2150			−1.45494			−0.727470	−	0.686139i	$-0.759304\pi$
$277$	−24.2150			−1.45494			−0.727470	+	0.686139i	$0.759304\pi$
$278$		0			0
$279$	−14.2154			−0.851052
$280$		0			0
$281$	−3.34810			−0.199731			−0.0998654	−	0.995001i	$-0.531841\pi$
$281$	−3.34810			−0.199731			−0.0998654	+	0.995001i	$0.531841\pi$
$282$		0			0
$283$	−15.5082			−0.921866			−0.460933	−	0.887435i	$-0.652485\pi$
$283$	−15.5082			−0.921866			−0.460933	+	0.887435i	$0.652485\pi$
$284$		0			0
$285$		−	4.78107i		−	0.283206i
$286$		0			0
$287$	5.17193	+	1.11852i	0.305290	+	0.0660242i
$288$		0			0
$289$	6.35238			0.373670
$290$		0			0
$291$		−	1.33828i		−	0.0784516i
$292$		0			0
$293$			24.2990i			1.41956i	0.704421	+	0.709782i	$0.251207\pi$
$293$			24.2990i			1.41956i	−0.704421	+	0.709782i	$0.748793\pi$
$294$		0			0
$295$			18.9408i			1.10278i
$296$		0			0
$297$			0.972563i			0.0564338i
$298$		0			0
$299$	4.39165			0.253976
$300$		0			0
$301$	10.7950	+	2.33460i	0.622211	+	0.134564i
$302$		0			0
$303$		−	1.43761i		−	0.0825884i
$304$		0			0
$305$	1.56215			0.0894482
$306$		0			0
$307$	−11.1637			−0.637144			−0.318572	−	0.947899i	$-0.603203\pi$
$307$	−11.1637			−0.637144			−0.318572	+	0.947899i	$0.603203\pi$
$308$		0			0
$309$	6.60585			0.375794
$310$		0			0
$311$	−9.28623			−0.526574			−0.263287	−	0.964718i	$-0.584807\pi$
$311$	−9.28623			−0.526574			−0.263287	+	0.964718i	$0.584807\pi$
$312$		0			0
$313$			30.3230i			1.71396i	0.515350	+	0.856980i	$0.327662\pi$
$313$			30.3230i			1.71396i	−0.515350	+	0.856980i	$0.672338\pi$
$314$		0			0
$315$	−15.1346	−	3.27313i	−0.852741	−	0.184420i
$316$		0			0
$317$	−7.74935			−0.435247			−0.217623	−	0.976033i	$-0.569830\pi$
$317$	−7.74935			−0.435247			−0.217623	+	0.976033i	$0.569830\pi$
$318$		0			0
$319$			0.526285i			0.0294663i
$320$		0			0
$321$			4.20651i			0.234784i
$322$		0			0
$323$		−	11.1705i		−	0.621545i
$324$		0			0
$325$		−	0.0220707i		−	0.00122426i
$326$		0			0
$327$	−2.22596			−0.123096
$328$		0			0
$329$	4.87852	−	22.5578i	0.268962	−	1.24365i
$330$		0			0
$331$		−	5.98393i		−	0.328907i	−0.986385	−	0.164453i	$-0.947414\pi$
$331$		−	5.98393i		−	0.328907i	0.986385	−	0.164453i	$-0.0525860\pi$
$332$		0			0
$333$	10.7950			0.591560
$334$		0			0
$335$	11.5595			0.631561
$336$		0			0
$337$	4.15700			0.226446			0.113223	−	0.993570i	$-0.463883\pi$
$337$	4.15700			0.226446			0.113223	+	0.993570i	$0.463883\pi$
$338$		0			0
$339$	−0.502926			−0.0273152
$340$		0			0
$341$			1.51372i			0.0819725i
$342$		0			0
$343$	−11.0448	+	14.8665i	−0.596363	+	0.802715i
$344$		0			0
$345$	−6.13346			−0.330214
$346$		0			0
$347$			10.0572i			0.539901i	0.962874	+	0.269950i	$0.0870073\pi$
$347$			10.0572i			0.539901i	−0.962874	+	0.269950i	$0.912993\pi$
$348$		0			0
$349$			16.0235i			0.857721i	0.903371	+	0.428860i	$0.141085\pi$
$349$			16.0235i			0.857721i	−0.903371	+	0.428860i	$0.858915\pi$
$350$		0			0
$351$			3.49722i			0.186668i
$352$		0			0
$353$		−	31.7547i		−	1.69013i	−0.534663	−	0.845065i	$-0.679562\pi$
$353$		−	31.7547i		−	1.69013i	0.534663	−	0.845065i	$-0.320438\pi$
$354$		0			0
$355$	−3.41645			−0.181326
$356$		0			0
$357$	5.25878	+	1.13730i	0.278324	+	0.0601925i
$358$		0			0
$359$		−	33.9243i		−	1.79046i	−0.445607	−	0.895229i	$-0.647012\pi$
$359$		−	33.9243i		−	1.79046i	0.445607	−	0.895229i	$-0.352988\pi$
$360$		0			0
$361$	−7.28085			−0.383203
$362$		0			0
$363$	−6.80714			−0.357282
$364$		0			0
$365$	13.5977			0.711734
$366$		0			0
$367$	16.1054			0.840693			0.420347	−	0.907364i	$-0.361908\pi$
$367$	16.1054			0.840693			0.420347	+	0.907364i	$0.361908\pi$
$368$		0			0
$369$		−	5.22321i		−	0.271910i
$370$		0			0
$371$	4.46629	−	20.6517i	0.231878	−	1.07218i
$372$		0			0
$373$	4.79068			0.248052			0.124026	−	0.992279i	$-0.460419\pi$
$373$	4.79068			0.248052			0.124026	+	0.992279i	$0.460419\pi$
$374$		0			0
$375$		−	6.95226i		−	0.359013i
$376$		0			0
$377$			1.89246i			0.0974666i
$378$		0			0
$379$			7.23128i			0.371446i	0.982602	+	0.185723i	$0.0594627\pi$
$379$			7.23128i			0.371446i	−0.982602	+	0.185723i	$0.940537\pi$
$380$		0			0
$381$			8.05183i			0.412508i
$382$		0			0
$383$	−16.1477			−0.825109			−0.412555	−	0.910933i	$-0.635363\pi$
$383$	−16.1477			−0.825109			−0.412555	+	0.910933i	$0.635363\pi$
$384$		0			0
$385$	−0.348538	+	1.61161i	−0.0177631	+	0.0821351i
$386$		0			0
$387$		−	10.9020i		−	0.554179i
$388$		0			0
$389$	−28.6360			−1.45190			−0.725952	−	0.687746i	$-0.758600\pi$
$389$	−28.6360			−1.45190			−0.725952	+	0.687746i	$0.758600\pi$
$390$		0			0
$391$	−14.3303			−0.724713
$392$		0			0
$393$	−10.7048			−0.539984
$394$		0			0
$395$	10.9017			0.548525
$396$		0			0
$397$		−	9.28233i		−	0.465867i	−0.972493	−	0.232933i	$-0.925168\pi$
$397$		−	9.28233i		−	0.465867i	0.972493	−	0.232933i	$-0.0748324\pi$
$398$		0			0
$399$	−1.19316	+	5.51705i	−0.0597326	+	0.276198i
$400$		0			0
$401$	15.9280			0.795405			0.397703	−	0.917514i	$-0.369808\pi$
$401$	15.9280			0.795405			0.397703	+	0.917514i	$0.369808\pi$
$402$		0			0
$403$			5.44315i			0.271143i
$404$		0			0
$405$			12.6735i			0.629753i
$406$		0			0
$407$		−	1.14950i		−	0.0569785i
$408$		0			0
$409$		−	39.0038i		−	1.92861i	−0.264787	−	0.964307i	$-0.585302\pi$
$409$		−	39.0038i		−	1.92861i	0.264787	−	0.964307i	$-0.414698\pi$
$410$		0			0
$411$	−7.51204			−0.370542
$412$		0			0
$413$	4.72684	−	21.8565i	0.232593	−	1.07549i
$414$		0			0
$415$			27.8726i			1.36821i
$416$		0			0
$417$	0.0503401			0.00246517
$418$		0			0
$419$	−1.56925			−0.0766629			−0.0383315	−	0.999265i	$-0.512204\pi$
$419$	−1.56925			−0.0766629			−0.0383315	+	0.999265i	$0.512204\pi$
$420$		0			0
$421$	33.0177			1.60919			0.804593	−	0.593827i	$-0.202383\pi$
$421$	33.0177			1.60919			0.804593	+	0.593827i	$0.202383\pi$
$422$		0			0
$423$	−22.7815			−1.10767
$424$		0			0
$425$			0.0720183i			0.00349340i
$426$		0			0
$427$	−1.80262	−	0.389847i	−0.0872347	−	0.0188660i
$428$		0			0
$429$	0.173313			0.00836762
$430$		0			0
$431$		−	15.2600i		−	0.735048i	−0.930014	−	0.367524i	$-0.880206\pi$
$431$		−	15.2600i		−	0.735048i	0.930014	−	0.367524i	$-0.119794\pi$
$432$		0			0
$433$			11.8646i			0.570178i	0.958501	+	0.285089i	$0.0920231\pi$
$433$			11.8646i			0.570178i	−0.958501	+	0.285089i	$0.907977\pi$
$434$		0			0
$435$		−	2.64304i		−	0.126724i
$436$		0			0
$437$		−	15.0340i		−	0.719176i
$438$		0			0
$439$	−29.8471			−1.42453			−0.712263	−	0.701913i	$-0.752330\pi$
$439$	−29.8471			−1.42453			−0.712263	+	0.701913i	$0.752330\pi$
$440$		0			0
$441$	16.6476	+	7.55397i	0.792741	+	0.359713i
$442$		0			0
$443$			29.2408i			1.38927i	0.719361	+	0.694637i	$0.244435\pi$
$443$			29.2408i			1.38927i	−0.719361	+	0.694637i	$0.755565\pi$
$444$		0			0
$445$	−27.7407			−1.31504
$446$		0			0
$447$	3.44487			0.162937
$448$		0			0
$449$	−29.5420			−1.39417			−0.697086	−	0.716988i	$-0.745520\pi$
$449$	−29.5420			−1.39417			−0.697086	+	0.716988i	$0.745520\pi$
$450$		0			0
$451$	−0.556192			−0.0261900
$452$		0			0
$453$			8.18188i			0.384418i
$454$		0			0
$455$	−1.25330	+	5.79515i	−0.0587557	+	0.271681i
$456$		0			0
$457$	37.0781			1.73444			0.867221	−	0.497923i	$-0.165904\pi$
$457$	37.0781			1.73444			0.867221	+	0.497923i	$0.165904\pi$
$458$		0			0
$459$		−	11.4117i		−	0.532651i
$460$		0			0
$461$			11.9088i			0.554647i	0.960777	+	0.277323i	$0.0894473\pi$
$461$			11.9088i			0.554647i	−0.960777	+	0.277323i	$0.910553\pi$
$462$		0			0
$463$		−	38.7883i		−	1.80264i	−0.433151	−	0.901321i	$-0.642598\pi$
$463$		−	38.7883i		−	1.80264i	0.433151	−	0.901321i	$-0.357402\pi$
$464$		0			0
$465$		−	7.60200i		−	0.352534i
$466$		0			0
$467$	27.2116			1.25920			0.629600	−	0.776919i	$-0.283219\pi$
$467$	27.2116			1.25920			0.629600	+	0.776919i	$0.283219\pi$
$468$		0			0
$469$	−13.3389	−	2.88477i	−0.615933	−	0.133206i
$470$		0			0
$471$			8.40454i			0.387261i
$472$		0			0
$473$	−1.16089			−0.0533780
$474$		0			0
$475$	−0.0755553			−0.00346671
$476$		0			0
$477$	−20.8565			−0.954951
$478$		0			0
$479$	−11.2854			−0.515644			−0.257822	−	0.966192i	$-0.583005\pi$
$479$	−11.2854			−0.515644			−0.257822	+	0.966192i	$0.583005\pi$
$480$		0			0
$481$		−	4.13346i		−	0.188469i
$482$		0			0
$483$	7.07762	+	1.53066i	0.322043	+	0.0696474i
$484$		0			0
$485$	−4.81231			−0.218516
$486$		0			0
$487$		−	7.28869i		−	0.330282i	−0.986270	−	0.165141i	$-0.947192\pi$
$487$		−	7.28869i		−	0.330282i	0.986270	−	0.165141i	$-0.0528079\pi$
$488$		0			0
$489$			10.2592i			0.463938i
$490$		0			0
$491$		−	33.9080i		−	1.53025i	−0.643883	−	0.765124i	$-0.722678\pi$
$491$		−	33.9080i		−	1.53025i	0.643883	−	0.765124i	$-0.277322\pi$
$492$		0			0
$493$		−	6.17523i		−	0.278118i
$494$		0			0
$495$	1.62759			0.0731545
$496$		0			0
$497$	3.94236	+	0.852604i	0.176839	+	0.0382445i
$498$		0			0
$499$			30.1527i			1.34982i	0.737899	+	0.674911i	$0.235818\pi$
$499$			30.1527i			1.34982i	−0.737899	+	0.674911i	$0.764182\pi$
$500$		0			0
$501$	0.0460540			0.00205754
$502$		0			0
$503$	7.65717			0.341416			0.170708	−	0.985322i	$-0.445394\pi$
$503$	7.65717			0.341416			0.170708	+	0.985322i	$0.445394\pi$
$504$		0			0
$505$	−5.16947			−0.230038
$506$		0			0
$507$	0.623212			0.0276778
$508$		0			0
$509$			23.2189i			1.02916i	0.857443	+	0.514579i	$0.172052\pi$
$509$			23.2189i			1.02916i	−0.857443	+	0.514579i	$0.827948\pi$
$510$		0			0
$511$	−15.6908	−	3.39342i	−0.694122	−	0.150116i
$512$		0			0
$513$	11.9721			0.528582
$514$		0			0
$515$		−	23.7538i		−	1.04672i
$516$		0			0
$517$			2.42588i			0.106690i
$518$		0			0
$519$			6.02357i			0.264405i
$520$		0			0
$521$		−	37.1071i		−	1.62569i	−0.582480	−	0.812845i	$-0.697918\pi$
$521$		−	37.1071i		−	1.62569i	0.582480	−	0.812845i	$-0.302082\pi$
$522$		0			0
$523$	−6.90498			−0.301934			−0.150967	−	0.988539i	$-0.548239\pi$
$523$	−6.90498			−0.301934			−0.150967	+	0.988539i	$0.548239\pi$
$524$		0			0
$525$	0.00769249	−	0.0355693i	0.000335728	−	0.00155237i
$526$		0			0
$527$		−	17.7614i		−	0.773698i
$528$		0			0
$529$	3.71339			0.161452
$530$		0			0
$531$	−22.0732			−0.957893
$532$		0			0
$533$	−2.00000			−0.0866296
$534$		0			0
$535$	15.1261			0.653959
$536$		0			0
$537$			13.3984i			0.578185i
$538$		0			0
$539$	0.804381	−	1.77271i	0.0346471	−	0.0763560i
$540$		0			0
$541$	8.61545			0.370407			0.185204	−	0.982700i	$-0.440706\pi$
$541$	8.61545			0.370407			0.185204	+	0.982700i	$0.440706\pi$
$542$		0			0
$543$		−	13.5296i		−	0.580612i
$544$		0			0
$545$			8.00429i			0.342866i
$546$		0			0
$547$			34.6884i			1.48317i	0.670858	+	0.741586i	$0.265926\pi$
$547$			34.6884i			1.48317i	−0.670858	+	0.741586i	$0.734074\pi$
$548$		0			0
$549$			1.82049i			0.0776966i
$550$		0			0
$551$	6.47850			0.275993
$552$		0			0
$553$	−12.5799	−	2.72062i	−0.534951	−	0.115693i
$554$		0			0
$555$			5.77286i			0.245044i
$556$		0			0
$557$	32.8449			1.39168			0.695841	−	0.718196i	$-0.255032\pi$
$557$	32.8449			1.39168			0.695841	+	0.718196i	$0.255032\pi$
$558$		0			0
$559$	−4.17444			−0.176560
$560$		0			0
$561$	−0.565531			−0.0238768
$562$		0			0
$563$	40.0707			1.68878			0.844390	−	0.535729i	$-0.179963\pi$
$563$	40.0707			1.68878			0.844390	+	0.535729i	$0.179963\pi$
$564$		0			0
$565$			1.80846i			0.0760826i
$566$		0			0
$567$	3.16279	−	14.6244i	0.132825	−	0.614169i
$568$		0			0
$569$	14.3303			0.600758			0.300379	−	0.953820i	$-0.402887\pi$
$569$	14.3303			0.600758			0.300379	+	0.953820i	$0.402887\pi$
$570$		0			0
$571$			40.5615i			1.69745i	0.528836	+	0.848724i	$0.322629\pi$
$571$			40.5615i			1.69745i	−0.528836	+	0.848724i	$0.677371\pi$
$572$		0			0
$573$		−	6.15163i		−	0.256988i
$574$		0			0
$575$			0.0969270i			0.00404214i
$576$		0			0
$577$		−	22.0086i		−	0.916229i	−0.888893	−	0.458114i	$-0.848525\pi$
$577$		−	22.0086i		−	0.916229i	0.888893	−	0.458114i	$-0.151475\pi$
$578$		0			0
$579$	3.47119			0.144258
$580$		0			0
$581$	6.95586	−	32.1632i	0.288578	−	1.33436i
$582$		0			0
$583$			2.22089i			0.0919799i
$584$		0			0
$585$	5.85260			0.241975
$586$		0			0
$587$	36.3407			1.49994			0.749971	−	0.661471i	$-0.230067\pi$
$587$	36.3407			1.49994			0.749971	+	0.661471i	$0.230067\pi$
$588$		0			0
$589$	18.6337			0.767787
$590$		0			0
$591$	5.17955			0.213058
$592$		0			0
$593$			28.0336i			1.15120i	0.817731	+	0.575601i	$0.195232\pi$
$593$			28.0336i			1.15120i	−0.817731	+	0.575601i	$0.804768\pi$
$594$		0			0
$595$	4.08961	−	18.9100i	0.167658	−	0.775233i
$596$		0			0
$597$	16.5842			0.678745
$598$		0			0
$599$			5.43462i			0.222052i	0.993817	+	0.111026i	$0.0354138\pi$
$599$			5.43462i			0.222052i	−0.993817	+	0.111026i	$0.964586\pi$
$600$		0			0
$601$			32.9803i			1.34529i	0.739964	+	0.672646i	$0.234842\pi$
$601$			32.9803i			1.34529i	−0.739964	+	0.672646i	$0.765158\pi$
$602$		0			0
$603$			13.4711i			0.548587i
$604$		0			0
$605$			24.4777i			0.995158i
$606$		0			0
$607$	39.4783			1.60238			0.801188	−	0.598413i	$-0.204202\pi$
$607$	39.4783			1.60238			0.801188	+	0.598413i	$0.204202\pi$
$608$		0			0
$609$	−0.659594	+	3.04990i	−0.0267281	+	0.123588i
$610$		0			0
$611$			8.72316i			0.352901i
$612$		0			0
$613$	−35.2116			−1.42218			−0.711092	−	0.703099i	$-0.751799\pi$
$613$	−35.2116			−1.42218			−0.711092	+	0.703099i	$0.751799\pi$
$614$		0			0
$615$	2.79323			0.112634
$616$		0			0
$617$	−36.0163			−1.44996			−0.724980	−	0.688770i	$-0.758151\pi$
$617$	−36.0163			−1.44996			−0.724980	+	0.688770i	$0.758151\pi$
$618$		0			0
$619$	−26.5362			−1.06658			−0.533289	−	0.845933i	$-0.679044\pi$
$619$	−26.5362			−1.06658			−0.533289	+	0.845933i	$0.679044\pi$
$620$		0			0
$621$		−	15.3586i		−	0.616319i
$622$		0			0
$623$	32.0110	+	6.92294i	1.28249	+	0.277362i
$624$		0			0
$625$	−25.1099			−1.00439
$626$		0			0
$627$		−	0.593306i		−	0.0236943i
$628$		0			0
$629$			13.4878i			0.537792i
$630$		0			0
$631$		−	11.7044i		−	0.465947i	−0.972483	−	0.232973i	$-0.925155\pi$
$631$		−	11.7044i		−	0.465947i	0.972483	−	0.232973i	$-0.0748455\pi$
$632$		0			0
$633$			0.314463i			0.0124988i
$634$		0			0
$635$	28.9534			1.14898
$636$		0			0
$637$	2.89246	−	6.37446i	0.114603	−	0.252565i
$638$		0			0
$639$		−	3.98145i		−	0.157504i
$640$		0			0
$641$	−19.2952			−0.762114			−0.381057	−	0.924552i	$-0.624440\pi$
$641$	−19.2952			−0.762114			−0.381057	+	0.924552i	$0.624440\pi$
$642$		0			0
$643$	40.5747			1.60011			0.800055	−	0.599926i	$-0.204803\pi$
$643$	40.5747			1.60011			0.800055	+	0.599926i	$0.204803\pi$
$644$		0			0
$645$	5.83009			0.229560
$646$		0			0
$647$	6.25095			0.245750			0.122875	−	0.992422i	$-0.460789\pi$
$647$	6.25095			0.245750			0.122875	+	0.992422i	$0.460789\pi$
$648$		0			0
$649$			2.35045i			0.0922633i
$650$		0			0
$651$	−1.89715	+	8.77222i	−0.0743550	+	0.343810i
$652$		0			0
$653$	−4.74034			−0.185504			−0.0927519	−	0.995689i	$-0.529566\pi$
$653$	−4.74034			−0.185504			−0.0927519	+	0.995689i	$0.529566\pi$
$654$		0			0
$655$			38.4931i			1.50405i
$656$		0			0
$657$			15.8464i			0.618227i
$658$		0			0
$659$			27.6940i			1.07881i	0.842048	+	0.539403i	$0.181350\pi$
$659$			27.6940i			1.07881i	−0.842048	+	0.539403i	$0.818650\pi$
$660$		0			0
$661$		−	24.1776i		−	0.940398i	−0.882560	−	0.470199i	$-0.844182\pi$
$661$		−	24.1776i		−	0.940398i	0.882560	−	0.470199i	$-0.155818\pi$
$662$		0			0
$663$	−2.03358			−0.0789779
$664$		0			0
$665$	19.8387	+	4.29046i	0.769310	+	0.166377i
$666$		0			0
$667$		−	8.31103i		−	0.321804i
$668$		0			0
$669$	16.0893			0.622049
$670$		0			0
$671$	0.193854			0.00748365
$672$		0			0
$673$	33.0413			1.27365			0.636824	−	0.771009i	$-0.280248\pi$
$673$	33.0413			1.27365			0.636824	+	0.771009i	$0.280248\pi$
$674$		0			0
$675$	−0.0771863			−0.00297090
$676$		0			0
$677$			39.7287i			1.52690i	0.645868	+	0.763449i	$0.276496\pi$
$677$			39.7287i			1.52690i	−0.645868	+	0.763449i	$0.723504\pi$
$678$		0			0
$679$	5.55310	+	1.20095i	0.213108	+	0.0460884i
$680$		0			0
$681$	−6.36672			−0.243973
$682$		0			0
$683$		−	43.4846i		−	1.66389i	−0.554856	−	0.831946i	$-0.687227\pi$
$683$		−	43.4846i		−	1.66389i	0.554856	−	0.831946i	$-0.312773\pi$
$684$		0			0
$685$			27.0124i			1.03209i
$686$		0			0
$687$			6.99592i			0.266911i
$688$		0			0
$689$			7.98606i			0.304245i
$690$		0			0
$691$	26.1372			0.994305			0.497153	−	0.867663i	$-0.334379\pi$
$691$	26.1372			0.994305			0.497153	+	0.867663i	$0.334379\pi$
$692$		0			0
$693$	−1.87813	−	0.406178i	−0.0713442	−	0.0154294i
$694$		0			0
$695$		−	0.181017i		−	0.00686637i
$696$		0			0
$697$	6.52614			0.247195
$698$		0			0
$699$	−5.73785			−0.217026
$700$		0			0
$701$	−27.7407			−1.04775			−0.523876	−	0.851794i	$-0.675515\pi$
$701$	−27.7407			−1.04775			−0.523876	+	0.851794i	$0.675515\pi$
$702$		0			0
$703$	−14.1502			−0.533684
$704$		0			0
$705$		−	12.1829i		−	0.458835i
$706$		0			0
$707$	5.96523	+	1.29009i	0.224346	+	0.0485187i
$708$		0			0
$709$	−19.8472			−0.745379			−0.372689	−	0.927956i	$-0.621564\pi$
$709$	−19.8472			−0.745379			−0.372689	+	0.927956i	$0.621564\pi$
$710$		0			0
$711$			12.7046i			0.476460i
$712$		0			0
$713$		−	23.9044i		−	0.895228i
$714$		0			0
$715$		−	0.623212i		−	0.0233068i
$716$		0			0
$717$		−	11.7485i		−	0.438755i
$718$		0			0
$719$	21.0276			0.784198			0.392099	−	0.919923i	$-0.371749\pi$
$719$	21.0276			0.784198			0.392099	+	0.919923i	$0.371749\pi$
$720$		0			0
$721$	−5.92798	+	27.4104i	−0.220770	+	1.02082i
$722$		0			0
$723$		−	3.87209i		−	0.144005i
$724$		0			0
$725$	−0.0417680			−0.00155122
$726$		0			0
$727$	36.4380			1.35141			0.675705	−	0.737172i	$-0.263839\pi$
$727$	36.4380			1.35141			0.675705	+	0.737172i	$0.263839\pi$
$728$		0			0
$729$	−8.23091			−0.304848
$730$		0			0
$731$	13.6215			0.503809
$732$		0			0
$733$		−	21.5958i		−	0.797657i	−0.917025	−	0.398829i	$-0.869417\pi$
$733$		−	21.5958i		−	0.797657i	0.917025	−	0.398829i	$-0.130583\pi$
$734$		0			0
$735$	−4.03966	+	8.90267i	−0.149005	+	0.328380i
$736$		0			0
$737$	1.43447			0.0528393
$738$		0			0
$739$			0.0222896i			0.000819935i	1.00000		0.000409967i	$0.000130497\pi$
$739$			0.0222896i			0.000819935i	−1.00000		0.000409967i	$0.999870\pi$
$740$		0			0
$741$		−	2.13346i		−	0.0783745i
$742$		0			0
$743$		−	37.6474i		−	1.38115i	−0.723260	−	0.690575i	$-0.757357\pi$
$743$		−	37.6474i		−	1.38115i	0.723260	−	0.690575i	$-0.242643\pi$
$744$		0			0
$745$		−	12.3874i		−	0.453838i
$746$		0			0
$747$	−32.4821			−1.18846
$748$		0			0
$749$	−17.4546	−	3.77485i	−0.637776	−	0.137930i
$750$		0			0
$751$		−	37.3076i		−	1.36137i	−0.732575	−	0.680686i	$-0.761682\pi$
$751$		−	37.3076i		−	1.36137i	0.732575	−	0.680686i	$-0.238318\pi$
$752$		0			0
$753$	−1.44449			−0.0526402
$754$		0			0
$755$	29.4211			1.07074
$756$		0			0
$757$	−0.331193			−0.0120374			−0.00601871	−	0.999982i	$-0.501916\pi$
$757$	−0.331193			−0.0120374			−0.00601871	+	0.999982i	$0.501916\pi$
$758$		0			0
$759$	−0.761129			−0.0276272
$760$		0			0
$761$		−	5.91833i		−	0.214539i	−0.994230	−	0.107270i	$-0.965789\pi$
$761$		−	5.91833i		−	0.214539i	0.994230	−	0.107270i	$-0.0342108\pi$
$762$		0			0
$763$	1.99754	−	9.23643i	0.0723158	−	0.334381i
$764$		0			0
$765$	−19.0974			−0.690470
$766$		0			0
$767$			8.45194i			0.305182i
$768$		0			0
$769$		−	4.81897i		−	0.173776i	−0.996218	−	0.0868882i	$-0.972308\pi$
$769$		−	4.81897i		−	0.173776i	0.996218	−	0.0868882i	$-0.0276923\pi$
$770$		0			0
$771$		−	2.32918i		−	0.0838832i
$772$		0			0
$773$		−	28.4618i		−	1.02370i	−0.859075	−	0.511850i	$-0.828960\pi$
$773$		−	28.4618i		−	1.02370i	0.859075	−	0.511850i	$-0.171040\pi$
$774$		0			0
$775$	−0.120134			−0.00431535
$776$		0			0
$777$	1.44067	−	6.66151i	0.0516837	−	0.238980i
$778$		0			0
$779$			6.84665i			0.245307i
$780$		0			0
$781$	−0.423963			−0.0151706
$782$		0			0
$783$	6.61835			0.236521
$784$		0			0
$785$	30.2217			1.07866
$786$		0			0
$787$	3.93761			0.140361			0.0701803	−	0.997534i	$-0.477643\pi$
$787$	3.93761			0.140361			0.0701803	+	0.997534i	$0.477643\pi$
$788$		0			0
$789$		−	11.2104i		−	0.399099i
$790$		0			0
$791$	0.451318	−	2.08685i	0.0160470	−	0.0741998i
$792$		0			0
$793$	0.697076			0.0247539
$794$		0			0
$795$		−	11.1535i		−	0.395573i
$796$		0			0
$797$			6.60879i			0.234095i	0.993126	+	0.117048i	$0.0373431\pi$
$797$			6.60879i			0.234095i	−0.993126	+	0.117048i	$0.962657\pi$
$798$		0			0
$799$		−	28.4643i		−	1.00699i
$800$		0			0
$801$		−	32.3284i		−	1.14227i
$802$		0			0
$803$	1.68740			0.0595470
$804$		0			0
$805$	5.50407	−	25.4503i	0.193993	−	0.897004i
$806$		0			0
$807$		−	4.77447i		−	0.168069i
$808$		0			0
$809$	−8.29180			−0.291524			−0.145762	−	0.989320i	$-0.546563\pi$
$809$	−8.29180			−0.291524			−0.145762	+	0.989320i	$0.546563\pi$
$810$		0			0
$811$	−34.3701			−1.20690			−0.603449	−	0.797402i	$-0.706207\pi$
$811$	−34.3701			−1.20690			−0.603449	+	0.797402i	$0.706207\pi$
$812$		0			0
$813$	4.19578			0.147152
$814$		0			0
$815$	36.8910			1.29223
$816$		0			0
$817$			14.2905i			0.499960i
$818$		0			0
$819$	−6.75353	−	1.46057i	−0.235987	−	0.0510364i
$820$		0			0
$821$	0.573224			0.0200057			0.0100028	−	0.999950i	$-0.496816\pi$
$821$	0.573224			0.0200057			0.0100028	+	0.999950i	$0.496816\pi$
$822$		0			0
$823$			16.3588i			0.570231i	0.958493	+	0.285116i	$0.0920320\pi$
$823$			16.3588i			0.570231i	−0.958493	+	0.285116i	$0.907968\pi$
$824$		0			0
$825$			0.00382514i			0.000133174i
$826$		0			0
$827$		−	20.4243i		−	0.710223i	−0.934824	−	0.355112i	$-0.884443\pi$
$827$		−	20.4243i		−	0.710223i	0.934824	−	0.355112i	$-0.115557\pi$
$828$		0			0
$829$			41.5059i			1.44156i	0.693164	+	0.720780i	$0.256216\pi$
$829$			41.5059i			1.44156i	−0.693164	+	0.720780i	$0.743784\pi$
$830$		0			0
$831$	15.0911			0.523505
$832$		0			0
$833$	−9.43829	+	20.8003i	−0.327017	+	0.720687i
$834$		0			0
$835$		−	0.165605i		−	0.00573099i
$836$		0			0
$837$	19.0359			0.657977
$838$		0			0
$839$	−36.4181			−1.25729			−0.628646	−	0.777692i	$-0.716391\pi$
$839$	−36.4181			−1.25729			−0.628646	+	0.777692i	$0.716391\pi$
$840$		0			0
$841$	−25.4186			−0.876503
$842$		0			0
$843$	2.08658			0.0718655
$844$		0			0
$845$		−	2.24100i		−	0.0770927i
$846$		0			0
$847$	6.10862	−	28.2456i	0.209895	−	0.970532i
$848$		0			0
$849$	9.66489			0.331698
$850$		0			0
$851$			18.1527i			0.622267i
$852$		0			0
$853$		−	1.46421i		−	0.0501336i	−0.999686	−	0.0250668i	$-0.992020\pi$
$853$		−	1.46421i		−	0.0501336i	0.999686	−	0.0250668i	$-0.00797985\pi$
$854$		0			0
$855$		−	20.0354i		−	0.685195i
$856$		0			0
$857$			0.613540i			0.0209581i	0.999945	+	0.0104791i	$0.00333565\pi$
$857$			0.613540i			0.0209581i	−0.999945	+	0.0104791i	$0.996664\pi$
$858$		0			0
$859$	−38.9461			−1.32882			−0.664411	−	0.747367i	$-0.731318\pi$
$859$	−38.9461			−1.32882			−0.664411	+	0.747367i	$0.731318\pi$
$860$		0			0
$861$	−3.22321	−	0.697076i	−0.109847	−	0.0237563i
$862$		0			0
$863$			11.3421i			0.386090i	0.981190	+	0.193045i	$0.0618363\pi$
$863$			11.3421i			0.386090i	−0.981190	+	0.193045i	$0.938164\pi$
$864$		0			0
$865$	21.6600			0.736463
$866$		0			0
$867$	−3.95888			−0.134451
$868$		0			0
$869$	1.35285			0.0458921
$870$		0			0
$871$	5.15818			0.174778
$872$		0			0
$873$		−	5.60815i		−	0.189807i
$874$		0			0
$875$	28.8478	+	6.23885i	0.975235	+	0.210912i
$876$		0			0
$877$	−24.4089			−0.824231			−0.412116	−	0.911132i	$-0.635210\pi$
$877$	−24.4089			−0.824231			−0.412116	+	0.911132i	$0.635210\pi$
$878$		0			0
$879$		−	15.1435i		−	0.510776i
$880$		0			0
$881$			21.1598i			0.712891i	0.934316	+	0.356445i	$0.116011\pi$
$881$			21.1598i			0.712891i	−0.934316	+	0.356445i	$0.883989\pi$
$882$		0			0
$883$			24.5548i			0.826334i	0.910655	+	0.413167i	$0.135577\pi$
$883$			24.5548i			0.826334i	−0.910655	+	0.413167i	$0.864423\pi$
$884$		0			0
$885$		−	11.8041i		−	0.396791i
$886$		0			0
$887$	49.5167			1.66261			0.831305	−	0.555817i	$-0.187594\pi$
$887$	49.5167			1.66261			0.831305	+	0.555817i	$0.187594\pi$
$888$		0			0
$889$	−33.4104	−	7.22559i	−1.12055	−	0.242338i
$890$		0			0
$891$			1.57272i			0.0526880i
$892$		0			0
$893$	29.8622			0.999301
$894$		0			0
$895$	48.1792			1.61045
$896$		0			0
$897$	−2.73693			−0.0913835
$898$		0			0
$899$	10.3009			0.343556
$900$		0			0
$901$		−	26.0591i		−	0.868153i
$902$		0			0
$903$	−6.72755	−	1.45495i	−0.223879	−	0.0484177i
$904$		0			0
$905$	−48.6509			−1.61721
$906$		0			0
$907$		−	45.7719i		−	1.51983i	−0.650021	−	0.759916i	$-0.725240\pi$
$907$		−	45.7719i		−	1.51983i	0.650021	−	0.759916i	$-0.274760\pi$
$908$		0			0
$909$		−	6.02438i		−	0.199816i
$910$		0			0
$911$		−	3.75708i		−	0.124478i	−0.998061	−	0.0622388i	$-0.980176\pi$
$911$		−	3.75708i		−	0.124478i	0.998061	−	0.0622388i	$-0.0198241\pi$
$912$		0			0
$913$			3.45884i			0.114471i
$914$		0			0
$915$	−0.973549			−0.0321845
$916$		0			0
$917$	9.60629	−	44.4185i	0.317228	−	1.46683i
$918$		0			0
$919$			44.3833i			1.46407i	0.681268	+	0.732034i	$0.261429\pi$
$919$			44.3833i			1.46407i	−0.681268	+	0.732034i	$0.738571\pi$
$920$		0			0
$921$	6.95733			0.229252
$922$		0			0
$923$	−1.52452			−0.0501802
$924$		0			0
$925$	0.0912284			0.00299957
$926$		0			0
$927$	27.6822			0.909202
$928$		0			0
$929$			44.0057i			1.44378i	0.692007	+	0.721891i	$0.256727\pi$
$929$			44.0057i			1.44378i	−0.692007	+	0.721891i	$0.743273\pi$
$930$		0			0
$931$	−21.8218	−	9.90183i	−0.715181	−	0.324519i
$932$		0			0
$933$	5.78729			0.189467
$934$		0			0
$935$			2.03358i			0.0665053i
$936$		0			0
$937$			9.05656i			0.295865i	0.988997	+	0.147932i	$0.0472618\pi$
$937$			9.05656i			0.295865i	−0.988997	+	0.147932i	$0.952738\pi$
$938$		0			0
$939$		−	18.8977i		−	0.616703i
$940$		0			0
$941$			18.2457i			0.594794i	0.954754	+	0.297397i	$0.0961185\pi$
$941$			18.2457i			0.594794i	−0.954754	+	0.297397i	$0.903881\pi$
$942$		0			0
$943$	8.78331			0.286024
$944$		0			0
$945$	20.2669	+	4.38307i	0.659283	+	0.142581i
$946$		0			0
$947$		−	11.7490i		−	0.381792i	−0.981610	−	0.190896i	$-0.938861\pi$
$947$		−	11.7490i		−	0.381792i	0.981610	−	0.190896i	$-0.0611393\pi$
$948$		0			0
$949$	6.06768			0.196965
$950$		0			0
$951$	4.82949			0.156607
$952$		0			0
$953$	27.8925			0.903525			0.451763	−	0.892138i	$-0.350795\pi$
$953$	27.8925			0.903525			0.451763	+	0.892138i	$0.350795\pi$
$954$		0			0
$955$	−22.1206			−0.715804
$956$		0			0
$957$		−	0.327987i		−	0.0106023i
$958$		0			0
$959$	6.74119	−	31.1706i	0.217684	−	1.00655i
$960$		0			0
$961$	−1.37208			−0.0442607
$962$		0			0
$963$			17.6276i			0.568042i
$964$		0			0
$965$		−	12.4820i		−	0.401810i
$966$		0			0
$967$			33.3956i			1.07393i	0.843604	+	0.536966i	$0.180430\pi$
$967$			33.3956i			1.07393i	−0.843604	+	0.536966i	$0.819570\pi$
$968$		0			0
$969$			6.96162i			0.223639i
$970$		0			0
$971$	−35.2825			−1.13227			−0.566135	−	0.824313i	$-0.691562\pi$
$971$	−35.2825			−1.13227			−0.566135	+	0.824313i	$0.691562\pi$
$972$		0			0
$973$	−0.0451744	+	0.208882i	−0.00144822	+	0.00669645i
$974$		0			0
$975$			0.0137548i			0.000440505i
$976$		0			0
$977$	−9.61589			−0.307640			−0.153820	−	0.988099i	$-0.549158\pi$
$977$	−9.61589			−0.307640			−0.153820	+	0.988099i	$0.549158\pi$
$978$		0			0
$979$	−3.44248			−0.110022
$980$		0			0
$981$	−9.32801			−0.297820
$982$		0			0
$983$	16.0952			0.513356			0.256678	−	0.966497i	$-0.417372\pi$
$983$	16.0952			0.513356			0.256678	+	0.966497i	$0.417372\pi$
$984$		0			0
$985$		−	18.6251i		−	0.593444i
$986$		0			0
$987$	−3.04035	+	14.0583i	−0.0967755	+	0.447481i
$988$		0			0
$989$	18.3327			0.582946
$990$		0			0
$991$		−	24.5102i		−	0.778592i	−0.921113	−	0.389296i	$-0.872718\pi$
$991$		−	24.5102i		−	0.778592i	0.921113	−	0.389296i	$-0.127282\pi$
$992$		0			0
$993$			3.72926i			0.118344i
$994$		0			0
$995$		−	59.6347i		−	1.89055i
$996$		0			0
$997$			61.0954i			1.93491i	0.253039	+	0.967456i	$0.418570\pi$
$997$			61.0954i			1.93491i	−0.253039	+	0.967456i	$0.581430\pi$
$998$		0			0
$999$	−14.4556			−0.457355

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1456.2.j.a.1119.8	yes	16
4.3	odd	2	inner	1456.2.j.a.1119.10	yes	16
7.6	odd	2	inner	1456.2.j.a.1119.9	yes	16
28.27	even	2	inner	1456.2.j.a.1119.7	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1456.2.j.a.1119.7	✓	16	28.27	even	2	inner
1456.2.j.a.1119.8	yes	16	1.1	even	1	trivial
1456.2.j.a.1119.9	yes	16	7.6	odd	2	inner
1456.2.j.a.1119.10	yes	16	4.3	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:	\( N \)	\(=\)	\( 1456 = 2^{4} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1456.j (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.623212 q^{3} +2.24100i q^{5} +(0.559261 - 2.58597i) q^{7} -2.61161 q^{9} +O(q^{10})\)	\(q-0.623212 q^{3} +2.24100i q^{5} +(0.559261 - 2.58597i) q^{7} -2.61161 q^{9} +0.278096i q^{11} +1.00000i q^{13} -1.39662i q^{15} -3.26307i q^{17} +3.42332 q^{19} +(-0.348538 + 1.61161i) q^{21} -4.39165i q^{23} -0.0220707 q^{25} +3.49722 q^{27} +1.89246 q^{29} +5.44315 q^{31} -0.173313i q^{33} +(5.79515 + 1.25330i) q^{35} -4.13346 q^{37} -0.623212i q^{39} +2.00000i q^{41} +4.17444i q^{43} -5.85260i q^{45} +8.72316 q^{47} +(-6.37446 - 2.89246i) q^{49} +2.03358i q^{51} +7.98606 q^{53} -0.623212 q^{55} -2.13346 q^{57} +8.45194 q^{59} -0.697076i q^{61} +(-1.46057 + 6.75353i) q^{63} -2.24100 q^{65} -5.15818i q^{67} +2.73693i q^{69} +1.52452i q^{71} -6.06768i q^{73} +0.0137548 q^{75} +(0.719147 + 0.155528i) q^{77} -4.86467i q^{79} +5.65531 q^{81} +12.4376 q^{83} +7.31253 q^{85} -1.17940 q^{87} +12.3787i q^{89} +(2.58597 + 0.559261i) q^{91} -3.39224 q^{93} +7.67166i q^{95} +2.14740i q^{97} -0.726277i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{9}+O(q^{10}) \) 16 * q + 16 * q^9	\( 16 q + 16 q^{9} + 8 q^{21} + 4 q^{25} + 4 q^{29} + 4 q^{49} - 36 q^{53} + 32 q^{57} + 4 q^{65} - 20 q^{77} + 16 q^{81} - 56 q^{85} - 24 q^{93}+O(q^{100}) \) 16 * q + 16 * q^9 + 8 * q^21 + 4 * q^25 + 4 * q^29 + 4 * q^49 - 36 * q^53 + 32 * q^57 + 4 * q^65 - 20 * q^77 + 16 * q^81 - 56 * q^85 - 24 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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