LMFDB - Embedded newform 900.6.j.a.557.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [900,6,Mod(557,900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(900, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 2, 1]))

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

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

chi := DirichletCharacter("900.557");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	900.j (of order $4$, degree $2$, 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:	$144.345437832$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
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} + 16112194x^{12} + 72373894590801x^{8} + 60780662400876311824x^{4} + 14178241191207403341807616 $ x^16 + 16112194x^12 + 72373894590801x^8 + 60780662400876311824*x^4 + 14178241191207403341807616
Coefficient ring:	$\Z[a_1, \ldots, a_{49}]$
Coefficient ring index:	$ 2^{26}\cdot 3^{24} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			557.4
Root			$19.1476 + 19.1476i$ of defining polynomial
Character	$\chi$	$=$	900.557
Dual form			900.6.j.a.593.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-53.1577 - 53.1577i) q^{7} +O(q^{10})$	$q+(-53.1577 - 53.1577i) q^{7} +540.614i q^{11} +(-744.598 + 744.598i) q^{13} +(1654.43 - 1654.43i) q^{17} +2265.63i q^{19} +(1279.10 + 1279.10i) q^{23} -4554.02 q^{29} +4435.26 q^{31} +(7124.30 + 7124.30i) q^{37} -1360.60i q^{41} +(10634.7 - 10634.7i) q^{43} +(-4365.75 + 4365.75i) q^{47} -11155.5i q^{49} +(-23315.1 - 23315.1i) q^{53} +25023.2 q^{59} -40116.1 q^{61} +(-34777.6 - 34777.6i) q^{67} +10245.3i q^{71} +(-27434.8 + 27434.8i) q^{73} +(28737.8 - 28737.8i) q^{77} -45219.6i q^{79} +(70849.2 + 70849.2i) q^{83} -107473. q^{89} +79162.2 q^{91} +(85951.2 + 85951.2i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 5632 q^{31} - 119200 q^{61} - 138048 q^{91}+O(q^{100}) $ 16 * q + 5632 * q^31 - 119200 * q^61 - 138048 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/900\mathbb{Z}\right)^\times$.

$n$	$101$	$451$	$577$
$\chi(n)$	$-1$	$1$	$e\left(\frac{1}{4}\right)$

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			0
$3$		0			0
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−53.1577	−	53.1577i	−0.410035	−	0.410035i	0.471716	−	0.881751i	$-0.343635\pi$
$7$	−53.1577	−	53.1577i	−0.410035	−	0.410035i	−0.881751	+	0.471716i	$0.843635\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			540.614i			1.34712i	0.739133	+	0.673559i	$0.235235\pi$
$11$			540.614i			1.34712i	−0.739133	+	0.673559i	$0.764765\pi$
$12$		0			0
$13$	−744.598	+	744.598i	−1.22198	+	1.22198i	−0.255050	+	0.966928i	$0.582092\pi$
$13$	−744.598	+	744.598i	−1.22198	+	1.22198i	−0.966928	+	0.255050i	$0.917908\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1654.43	−	1654.43i	1.38844	−	1.38844i	0.559830	−	0.828608i	$-0.310867\pi$
$17$	1654.43	−	1654.43i	1.38844	−	1.38844i	0.828608	−	0.559830i	$-0.189133\pi$
$18$		0			0
$19$			2265.63i			1.43981i	0.694072	+	0.719905i	$0.255815\pi$
$19$			2265.63i			1.43981i	−0.694072	+	0.719905i	$0.744185\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1279.10	+	1279.10i	0.504180	+	0.504180i	0.912734	−	0.408554i	$-0.133967\pi$
$23$	1279.10	+	1279.10i	0.504180	+	0.504180i	−0.408554	+	0.912734i	$0.633967\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−4554.02			−1.00554			−0.502770	−	0.864420i	$-0.667686\pi$
$29$	−4554.02			−1.00554			−0.502770	+	0.864420i	$0.667686\pi$
$30$		0			0
$31$	4435.26			0.828925			0.414462	−	0.910066i	$-0.363970\pi$
$31$	4435.26			0.828925			0.414462	+	0.910066i	$0.363970\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	7124.30	+	7124.30i	0.855535	+	0.855535i	0.990808	−	0.135273i	$-0.0431912\pi$
$37$	7124.30	+	7124.30i	0.855535	+	0.855535i	−0.135273	+	0.990808i	$0.543191\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	1360.60i		−	0.126407i	−0.998001	−	0.0632033i	$-0.979868\pi$
$41$		−	1360.60i		−	0.126407i	0.998001	−	0.0632033i	$-0.0201317\pi$
$42$		0			0
$43$	10634.7	−	10634.7i	0.877107	−	0.877107i	−0.116128	−	0.993234i	$-0.537048\pi$
$43$	10634.7	−	10634.7i	0.877107	−	0.877107i	0.993234	+	0.116128i	$0.0370482\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−4365.75	+	4365.75i	−0.288280	+	0.288280i	−0.836400	−	0.548120i	$-0.815344\pi$
$47$	−4365.75	+	4365.75i	−0.288280	+	0.288280i	0.548120	+	0.836400i	$0.315344\pi$
$48$		0			0
$49$		−	11155.5i		−	0.663743i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−23315.1	−	23315.1i	−1.14011	−	1.14011i	−0.988429	−	0.151686i	$-0.951530\pi$
$53$	−23315.1	−	23315.1i	−1.14011	−	1.14011i	−0.151686	−	0.988429i	$-0.548470\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	25023.2			0.935863			0.467931	−	0.883765i	$-0.344999\pi$
$59$	25023.2			0.935863			0.467931	+	0.883765i	$0.344999\pi$
$60$		0			0
$61$	−40116.1			−1.38037			−0.690183	−	0.723635i	$-0.742470\pi$
$61$	−40116.1			−1.38037			−0.690183	+	0.723635i	$0.742470\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−34777.6	−	34777.6i	−0.946483	−	0.946483i	0.0521561	−	0.998639i	$-0.483391\pi$
$67$	−34777.6	−	34777.6i	−0.946483	−	0.946483i	−0.998639	+	0.0521561i	$0.983391\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			10245.3i			0.241202i	0.992701	+	0.120601i	$0.0384821\pi$
$71$			10245.3i			0.241202i	−0.992701	+	0.120601i	$0.961518\pi$
$72$		0			0
$73$	−27434.8	+	27434.8i	−0.602553	+	0.602553i	−0.940989	−	0.338437i	$-0.890102\pi$
$73$	−27434.8	+	27434.8i	−0.602553	+	0.602553i	0.338437	+	0.940989i	$0.390102\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	28737.8	−	28737.8i	0.552365	−	0.552365i
$78$		0			0
$79$		−	45219.6i		−	0.815190i	−0.913163	−	0.407595i	$-0.866368\pi$
$79$		−	45219.6i		−	0.815190i	0.913163	−	0.407595i	$-0.133632\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	70849.2	+	70849.2i	1.12886	+	1.12886i	0.990363	+	0.138496i	$0.0442269\pi$
$83$	70849.2	+	70849.2i	1.12886	+	1.12886i	0.138496	+	0.990363i	$0.455773\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−107473.			−1.43821			−0.719107	−	0.694900i	$-0.755449\pi$
$89$	−107473.			−1.43821			−0.719107	+	0.694900i	$0.755449\pi$
$90$		0			0
$91$	79162.2			1.00211
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	85951.2	+	85951.2i	0.927519	+	0.927519i	0.997545	−	0.0700263i	$-0.0223083\pi$
$97$	85951.2	+	85951.2i	0.927519	+	0.927519i	−0.0700263	+	0.997545i	$0.522308\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			15317.0i			0.149407i	0.997206	+	0.0747035i	$0.0238010\pi$
$101$			15317.0i			0.149407i	−0.997206	+	0.0747035i	$0.976199\pi$
$102$		0			0
$103$	−18040.8	+	18040.8i	−0.167557	+	0.167557i	−0.785905	−	0.618348i	$-0.787802\pi$
$103$	−18040.8	+	18040.8i	−0.167557	+	0.167557i	0.618348	+	0.785905i	$0.287802\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−126626.	+	126626.i	−1.06921	+	1.06921i	−0.0717877	+	0.997420i	$0.522870\pi$
$107$	−126626.	+	126626.i	−1.06921	+	1.06921i	−0.997420	+	0.0717877i	$0.977130\pi$
$108$		0			0
$109$			116529.i			0.939434i	0.882817	+	0.469717i	$0.155644\pi$
$109$			116529.i			0.939434i	−0.882817	+	0.469717i	$0.844356\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	36520.8	+	36520.8i	0.269057	+	0.269057i	0.828720	−	0.559663i	$-0.189070\pi$
$113$	36520.8	+	36520.8i	0.269057	+	0.269057i	−0.559663	+	0.828720i	$0.689070\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−175891.			−1.13862
$120$		0			0
$121$	−131213.			−0.814728
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−122586.	−	122586.i	−0.674423	−	0.674423i	0.284310	−	0.958733i	$-0.408236\pi$
$127$	−122586.	−	122586.i	−0.674423	−	0.674423i	−0.958733	+	0.284310i	$0.908236\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			339658.i			1.72927i	0.502399	+	0.864636i	$0.332451\pi$
$131$			339658.i			1.72927i	−0.502399	+	0.864636i	$0.667549\pi$
$132$		0			0
$133$	120436.	−	120436.i	0.590372	−	0.590372i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	105088.	−	105088.i	0.478357	−	0.478357i	−0.426249	−	0.904606i	$-0.640165\pi$
$137$	105088.	−	105088.i	0.478357	−	0.478357i	0.904606	+	0.426249i	$0.140165\pi$
$138$		0			0
$139$		−	356120.i		−	1.56336i	−0.623679	−	0.781681i	$-0.714363\pi$
$139$		−	356120.i		−	1.56336i	0.623679	−	0.781681i	$-0.285637\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−402540.	−	402540.i	−1.64615	−	1.64615i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−59416.0			−0.219249			−0.109625	−	0.993973i	$-0.534965\pi$
$149$	−59416.0			−0.219249			−0.109625	+	0.993973i	$0.534965\pi$
$150$		0			0
$151$	−247958.			−0.884986			−0.442493	−	0.896772i	$-0.645906\pi$
$151$	−247958.			−0.884986			−0.442493	+	0.896772i	$0.645906\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−187130.	−	187130.i	−0.605891	−	0.605891i	0.335978	−	0.941870i	$-0.390933\pi$
$157$	−187130.	−	187130.i	−0.605891	−	0.605891i	−0.941870	+	0.335978i	$0.890933\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	135988.i		−	0.413463i
$162$		0			0
$163$	−56297.1	+	56297.1i	−0.165965	+	0.165965i	−0.785203	−	0.619238i	$-0.787442\pi$
$163$	−56297.1	+	56297.1i	−0.165965	+	0.165965i	0.619238	+	0.785203i	$0.287442\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	189578.	−	189578.i	0.526013	−	0.526013i	−0.393368	−	0.919381i	$-0.628690\pi$
$167$	189578.	−	189578.i	0.526013	−	0.526013i	0.919381	+	0.393368i	$0.128690\pi$
$168$		0			0
$169$		−	737559.i		−	1.98646i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	408674.	+	408674.i	1.03815	+	1.03815i	0.999243	+	0.0389119i	$0.0123892\pi$
$173$	408674.	+	408674.i	1.03815	+	1.03815i	0.0389119	+	0.999243i	$0.487611\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−753137.			−1.75688			−0.878439	−	0.477855i	$-0.841415\pi$
$179$	−753137.			−1.75688			−0.878439	+	0.477855i	$0.841415\pi$
$180$		0			0
$181$	−220257.			−0.499727			−0.249863	−	0.968281i	$-0.580386\pi$
$181$	−220257.			−0.499727			−0.249863	+	0.968281i	$0.580386\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	894409.	+	894409.i	1.87039	+	1.87039i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			159911.i			0.317173i	0.987345	+	0.158586i	$0.0506936\pi$
$191$			159911.i			0.317173i	−0.987345	+	0.158586i	$0.949306\pi$
$192$		0			0
$193$	−188370.	+	188370.i	−0.364013	+	0.364013i	−0.865288	−	0.501275i	$-0.832865\pi$
$193$	−188370.	+	188370.i	−0.364013	+	0.364013i	0.501275	+	0.865288i	$0.332865\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−125767.	+	125767.i	−0.230887	+	0.230887i	−0.813063	−	0.582176i	$-0.802202\pi$
$197$	−125767.	+	125767.i	−0.230887	+	0.230887i	0.582176	+	0.813063i	$0.302202\pi$
$198$		0			0
$199$		−	204329.i		−	0.365760i	−0.983135	−	0.182880i	$-0.941458\pi$
$199$		−	204329.i		−	0.365760i	0.983135	−	0.182880i	$-0.0585420\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	242081.	+	242081.i	0.412307	+	0.412307i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−1.22483e6			−1.93959
$210$		0			0
$211$	−472424.			−0.730510			−0.365255	−	0.930908i	$-0.619018\pi$
$211$	−472424.			−0.730510			−0.365255	+	0.930908i	$0.619018\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−235768.	−	235768.i	−0.339888	−	0.339888i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			2.46377e6i			3.39328i
$222$		0			0
$223$	436887.	−	436887.i	0.588311	−	0.588311i	−0.348863	−	0.937174i	$-0.613432\pi$
$223$	436887.	−	436887.i	0.588311	−	0.588311i	0.937174	+	0.348863i	$0.113432\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	122133.	−	122133.i	0.157314	−	0.157314i	−0.624061	−	0.781375i	$-0.714519\pi$
$227$	122133.	−	122133.i	0.157314	−	0.157314i	0.781375	+	0.624061i	$0.214519\pi$
$228$		0			0
$229$		−	274403.i		−	0.345781i	−0.984941	−	0.172890i	$-0.944689\pi$
$229$		−	274403.i		−	0.345781i	0.984941	−	0.172890i	$-0.0553106\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−208701.	−	208701.i	−0.251846	−	0.251846i	0.569881	−	0.821727i	$-0.306989\pi$
$233$	−208701.	−	208701.i	−0.251846	−	0.251846i	−0.821727	+	0.569881i	$0.806989\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−1.37069e6			−1.55219			−0.776096	−	0.630615i	$-0.782803\pi$
$239$	−1.37069e6			−1.55219			−0.776096	+	0.630615i	$0.782803\pi$
$240$		0			0
$241$	−26377.0			−0.0292539			−0.0146269	−	0.999893i	$-0.504656\pi$
$241$	−26377.0			−0.0292539			−0.0146269	+	0.999893i	$0.504656\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−1.68698e6	−	1.68698e6i	−1.75942	−	1.75942i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	239275.i		−	0.239724i	−0.992791	−	0.119862i	$-0.961755\pi$
$251$		−	239275.i		−	0.239724i	0.992791	−	0.119862i	$-0.0382453\pi$
$252$		0			0
$253$	−691501.	+	691501.i	−0.679190	+	0.679190i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−785914.	+	785914.i	−0.742237	+	0.742237i	−0.973008	−	0.230771i	$-0.925875\pi$
$257$	−785914.	+	785914.i	−0.742237	+	0.742237i	0.230771	+	0.973008i	$0.425875\pi$
$258$		0			0
$259$		−	757422.i		−	0.701598i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−910319.	−	910319.i	−0.811530	−	0.811530i	0.173334	−	0.984863i	$-0.444546\pi$
$263$	−910319.	−	910319.i	−0.811530	−	0.811530i	−0.984863	+	0.173334i	$0.944546\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−2.23439e6			−1.88269			−0.941345	−	0.337446i	$-0.890437\pi$
$269$	−2.23439e6			−1.88269			−0.941345	+	0.337446i	$0.890437\pi$
$270$		0			0
$271$	−1.42257e6			−1.17666			−0.588331	−	0.808620i	$-0.700215\pi$
$271$	−1.42257e6			−1.17666			−0.588331	+	0.808620i	$0.700215\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	444169.	+	444169.i	0.347816	+	0.347816i	0.859295	−	0.511480i	$-0.170902\pi$
$277$	444169.	+	444169.i	0.347816	+	0.347816i	−0.511480	+	0.859295i	$0.670902\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			372453.i			0.281388i	0.990053	+	0.140694i	$0.0449334\pi$
$281$			372453.i			0.281388i	−0.990053	+	0.140694i	$0.955067\pi$
$282$		0			0
$283$	634765.	−	634765.i	0.471137	−	0.471137i	−0.431146	−	0.902282i	$-0.641891\pi$
$283$	634765.	−	634765.i	0.471137	−	0.471137i	0.902282	+	0.431146i	$0.141891\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−72326.2	+	72326.2i	−0.0518311	+	0.0518311i
$288$		0			0
$289$		−	4.05443e6i		−	2.85552i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−95284.8	−	95284.8i	−0.0648418	−	0.0648418i	0.673942	−	0.738784i	$-0.264600\pi$
$293$	−95284.8	−	95284.8i	−0.0648418	−	0.0648418i	−0.738784	+	0.673942i	$0.764600\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1.90483e6			−1.23219
$300$		0			0
$301$	−1.13063e6			−0.719288
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	1.54273e6	+	1.54273e6i	0.934210	+	0.934210i	0.997966	−	0.0637554i	$-0.0203078\pi$
$307$	1.54273e6	+	1.54273e6i	0.934210	+	0.934210i	−0.0637554	+	0.997966i	$0.520308\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			1.64625e6i			0.965152i	0.875854	+	0.482576i	$0.160299\pi$
$311$			1.64625e6i			0.965152i	−0.875854	+	0.482576i	$0.839701\pi$
$312$		0			0
$313$	93309.7	−	93309.7i	0.0538351	−	0.0538351i	−0.679677	−	0.733512i	$-0.737880\pi$
$313$	93309.7	−	93309.7i	0.0538351	−	0.0538351i	0.733512	+	0.679677i	$0.237880\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	317919.	−	317919.i	0.177692	−	0.177692i	−0.612657	−	0.790349i	$-0.709899\pi$
$317$	317919.	−	317919.i	0.177692	−	0.177692i	0.790349	+	0.612657i	$0.209899\pi$
$318$		0			0
$319$		−	2.46197e6i		−	1.35458i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	3.74833e6	+	3.74833e6i	1.99909	+	1.99909i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	464146.			0.236410
$330$		0			0
$331$	−1.34869e6			−0.676614			−0.338307	−	0.941036i	$-0.609854\pi$
$331$	−1.34869e6			−0.676614			−0.338307	+	0.941036i	$0.609854\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	1.46174e6	+	1.46174e6i	0.701124	+	0.701124i	0.964652	−	0.263528i	$-0.0848861\pi$
$337$	1.46174e6	+	1.46174e6i	0.701124	+	0.701124i	−0.263528	+	0.964652i	$0.584886\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			2.39777e6i			1.11666i
$342$		0			0
$343$	−1.48642e6	+	1.48642e6i	−0.682193	+	0.682193i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	852331.	−	852331.i	0.380001	−	0.380001i	−0.491101	−	0.871102i	$-0.663406\pi$
$347$	852331.	−	852331.i	0.380001	−	0.380001i	0.871102	+	0.491101i	$0.163406\pi$
$348$		0			0
$349$			1.98238e6i			0.871211i	0.900138	+	0.435605i	$0.143466\pi$
$349$			1.98238e6i			0.871211i	−0.900138	+	0.435605i	$0.856534\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−1.88845e6	−	1.88845e6i	−0.806621	−	0.806621i	0.177500	−	0.984121i	$-0.443199\pi$
$353$	−1.88845e6	−	1.88845e6i	−0.806621	−	0.806621i	−0.984121	+	0.177500i	$0.943199\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−4.13908e6			−1.69499			−0.847497	−	0.530800i	$-0.821891\pi$
$359$	−4.13908e6			−1.69499			−0.847497	+	0.530800i	$0.821891\pi$
$360$		0			0
$361$	−2.65699e6			−1.07305
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−2.29291e6	−	2.29291e6i	−0.888633	−	0.888633i	0.105759	−	0.994392i	$-0.466273\pi$
$367$	−2.29291e6	−	2.29291e6i	−0.888633	−	0.888633i	−0.994392	+	0.105759i	$0.966273\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			2.47876e6i			0.934973i
$372$		0			0
$373$	2.52699e6	−	2.52699e6i	0.940442	−	0.940442i	−0.0578815	−	0.998323i	$-0.518435\pi$
$373$	2.52699e6	−	2.52699e6i	0.940442	−	0.940442i	0.998323	+	0.0578815i	$0.0184345\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	3.39091e6	−	3.39091e6i	1.22875	−	1.22875i
$378$		0			0
$379$		−	1.34056e6i		−	0.479389i	−0.970848	−	0.239694i	$-0.922953\pi$
$379$		−	1.34056e6i		−	0.479389i	0.970848	−	0.239694i	$-0.0770473\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−3.29180e6	−	3.29180e6i	−1.14666	−	1.14666i	−0.987204	−	0.159459i	$-0.949025\pi$
$383$	−3.29180e6	−	3.29180e6i	−1.14666	−	1.14666i	−0.159459	−	0.987204i	$-0.550975\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−4.37280e6			−1.46516			−0.732581	−	0.680680i	$-0.761685\pi$
$389$	−4.37280e6			−1.46516			−0.732581	+	0.680680i	$0.761685\pi$
$390$		0			0
$391$	4.23237e6			1.40004
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−1.75223e6	−	1.75223e6i	−0.557974	−	0.557974i	0.370756	−	0.928730i	$-0.379098\pi$
$397$	−1.75223e6	−	1.75223e6i	−0.557974	−	0.557974i	−0.928730	+	0.370756i	$0.879098\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	1.43495e6i		−	0.445632i	−0.974860	−	0.222816i	$-0.928475\pi$
$401$		−	1.43495e6i		−	0.445632i	0.974860	−	0.222816i	$-0.0715250\pi$
$402$		0			0
$403$	−3.30249e6	+	3.30249e6i	−1.01293	+	1.01293i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−3.85150e6	+	3.85150e6i	−1.15251	+	1.15251i
$408$		0			0
$409$			1.23589e6i			0.365318i	0.983176	+	0.182659i	$0.0584704\pi$
$409$			1.23589e6i			0.365318i	−0.983176	+	0.182659i	$0.941530\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−1.33017e6	−	1.33017e6i	−0.383736	−	0.383736i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−3.18784e6			−0.887077			−0.443539	−	0.896255i	$-0.646277\pi$
$419$	−3.18784e6			−0.887077			−0.443539	+	0.896255i	$0.646277\pi$
$420$		0			0
$421$	234906.			0.0645936			0.0322968	−	0.999478i	$-0.489718\pi$
$421$	234906.			0.0645936			0.0322968	+	0.999478i	$0.489718\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	2.13248e6	+	2.13248e6i	0.565998	+	0.565998i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			199673.i			0.0517758i	0.999665	+	0.0258879i	$0.00824129\pi$
$431$			199673.i			0.0517758i	−0.999665	+	0.0258879i	$0.991759\pi$
$432$		0			0
$433$	3.83325e6	−	3.83325e6i	0.982533	−	0.982533i	−0.0173172	−	0.999850i	$-0.505513\pi$
$433$	3.83325e6	−	3.83325e6i	0.982533	−	0.982533i	0.999850	+	0.0173172i	$0.00551250\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−2.89797e6	+	2.89797e6i	−0.725923	+	0.725923i
$438$		0			0
$439$		−	4.69942e6i		−	1.16381i	−0.813256	−	0.581906i	$-0.802307\pi$
$439$		−	4.69942e6i		−	1.16381i	0.813256	−	0.581906i	$-0.197693\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	1.79107e6	+	1.79107e6i	0.433615	+	0.433615i	0.889856	−	0.456241i	$-0.150805\pi$
$443$	1.79107e6	+	1.79107e6i	0.433615	+	0.433615i	−0.456241	+	0.889856i	$0.650805\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	5.25095e6			1.22920			0.614599	−	0.788839i	$-0.289318\pi$
$449$	5.25095e6			1.22920			0.614599	+	0.788839i	$0.289318\pi$
$450$		0			0
$451$	735558.			0.170285
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−1.58768e6	−	1.58768e6i	−0.355609	−	0.355609i	0.506582	−	0.862192i	$-0.330909\pi$
$457$	−1.58768e6	−	1.58768e6i	−0.355609	−	0.355609i	−0.862192	+	0.506582i	$0.830909\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			1.78398e6i			0.390965i	0.980707	+	0.195483i	$0.0626273\pi$
$461$			1.78398e6i			0.390965i	−0.980707	+	0.195483i	$0.937373\pi$
$462$		0			0
$463$	3.23125e6	−	3.23125e6i	0.700517	−	0.700517i	−0.264004	−	0.964521i	$-0.585043\pi$
$463$	3.23125e6	−	3.23125e6i	0.700517	−	0.700517i	0.964521	+	0.264004i	$0.0850433\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	79429.9	−	79429.9i	0.0168536	−	0.0168536i	−0.698630	−	0.715483i	$-0.746207\pi$
$467$	79429.9	−	79429.9i	0.0168536	−	0.0168536i	0.715483	+	0.698630i	$0.246207\pi$
$468$		0			0
$469$			3.69739e6i			0.776182i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	5.74925e6	+	5.74925e6i	1.18157	+	1.18157i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	8.52977e6			1.69863			0.849314	−	0.527888i	$-0.177016\pi$
$479$	8.52977e6			1.69863			0.849314	+	0.527888i	$0.177016\pi$
$480$		0			0
$481$	−1.06095e7			−2.09089
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	5.16662e6	+	5.16662e6i	0.987152	+	0.987152i	0.999919	−	0.0127666i	$-0.00406384\pi$
$487$	5.16662e6	+	5.16662e6i	0.987152	+	0.987152i	−0.0127666	+	0.999919i	$0.504064\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	7.30691e6i		−	1.36782i	−0.729565	−	0.683912i	$-0.760277\pi$
$491$		−	7.30691e6i		−	1.36782i	0.729565	−	0.683912i	$-0.239723\pi$
$492$		0			0
$493$	−7.53430e6	+	7.53430e6i	−1.39613	+	1.39613i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	544618.	−	544618.i	0.0989011	−	0.0989011i
$498$		0			0
$499$		−	1.40509e6i		−	0.252611i	−0.991991	−	0.126306i	$-0.959688\pi$
$499$		−	1.40509e6i		−	0.252611i	0.991991	−	0.126306i	$-0.0403120\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	906766.	+	906766.i	0.159799	+	0.159799i	0.782478	−	0.622678i	$-0.213955\pi$
$503$	906766.	+	906766.i	0.159799	+	0.159799i	−0.622678	+	0.782478i	$0.713955\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−1.11833e6			−0.191327			−0.0956636	−	0.995414i	$-0.530497\pi$
$509$	−1.11833e6			−0.191327			−0.0956636	+	0.995414i	$0.530497\pi$
$510$		0			0
$511$	2.91674e6			0.494135
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−2.36019e6	−	2.36019e6i	−0.388347	−	0.388347i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			4.88692e6i			0.788753i	0.918949	+	0.394376i	$0.129039\pi$
$521$			4.88692e6i			0.788753i	−0.918949	+	0.394376i	$0.870961\pi$
$522$		0			0
$523$	−3.85618e6	+	3.85618e6i	−0.616458	+	0.616458i	−0.944621	−	0.328163i	$-0.893570\pi$
$523$	−3.85618e6	+	3.85618e6i	−0.616458	+	0.616458i	0.328163	+	0.944621i	$0.393570\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	7.33784e6	−	7.33784e6i	1.15091	−	1.15091i
$528$		0			0
$529$		−	3.16414e6i		−	0.491605i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	1.01310e6	+	1.01310e6i	0.154466	+	0.154466i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	6.03084e6			0.894140
$540$		0			0
$541$	9.22094e6			1.35451			0.677255	−	0.735749i	$-0.263169\pi$
$541$	9.22094e6			1.35451			0.677255	+	0.735749i	$0.263169\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−8.90650e6	−	8.90650e6i	−1.27274	−	1.27274i	−0.944646	−	0.328090i	$-0.893595\pi$
$547$	−8.90650e6	−	8.90650e6i	−1.27274	−	1.27274i	−0.328090	−	0.944646i	$-0.606405\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	1.03177e7i		−	1.44779i
$552$		0			0
$553$	−2.40377e6	+	2.40377e6i	−0.334256	+	0.334256i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−7.10687e6	+	7.10687e6i	−0.970600	+	0.970600i	−0.999580	−	0.0289800i	$-0.990774\pi$
$557$	−7.10687e6	+	7.10687e6i	−0.970600	+	0.970600i	0.0289800	+	0.999580i	$0.490774\pi$
$558$		0			0
$559$			1.58371e7i			2.14361i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−6.50929e6	−	6.50929e6i	−0.865491	−	0.865491i	0.126479	−	0.991969i	$-0.459633\pi$
$563$	−6.50929e6	−	6.50929e6i	−0.865491	−	0.865491i	−0.991969	+	0.126479i	$0.959633\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	8.05713e6			1.04328			0.521639	−	0.853167i	$-0.325321\pi$
$569$	8.05713e6			1.04328			0.521639	+	0.853167i	$0.325321\pi$
$570$		0			0
$571$	1.28050e7			1.64358			0.821789	−	0.569793i	$-0.192976\pi$
$571$	1.28050e7			1.64358			0.821789	+	0.569793i	$0.192976\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	3.23483e6	+	3.23483e6i	0.404494	+	0.404494i	0.879813	−	0.475320i	$-0.157668\pi$
$577$	3.23483e6	+	3.23483e6i	0.404494	+	0.404494i	−0.475320	+	0.879813i	$0.657668\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	7.53236e6i		−	0.925743i
$582$		0			0
$583$	1.26045e7	−	1.26045e7i	1.53587	−	1.53587i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	5.88015e6	−	5.88015e6i	0.704358	−	0.704358i	−0.260985	−	0.965343i	$-0.584047\pi$
$587$	5.88015e6	−	5.88015e6i	0.704358	−	0.704358i	0.965343	+	0.260985i	$0.0840473\pi$
$588$		0			0
$589$			1.00487e7i			1.19349i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−1.07735e7	−	1.07735e7i	−1.25811	−	1.25811i	−0.951993	−	0.306119i	$-0.900969\pi$
$593$	−1.07735e7	−	1.07735e7i	−1.25811	−	1.25811i	−0.306119	−	0.951993i	$-0.599031\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−6.78362e6			−0.772493			−0.386246	−	0.922396i	$-0.626229\pi$
$599$	−6.78362e6			−0.772493			−0.386246	+	0.922396i	$0.626229\pi$
$600$		0			0
$601$	3.04350e6			0.343706			0.171853	−	0.985123i	$-0.445025\pi$
$601$	3.04350e6			0.343706			0.171853	+	0.985123i	$0.445025\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	5.66065e6	+	5.66065e6i	0.623583	+	0.623583i	0.946446	−	0.322863i	$-0.104645\pi$
$607$	5.66065e6	+	5.66065e6i	0.623583	+	0.623583i	−0.322863	+	0.946446i	$0.604645\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	6.50146e6i		−	0.704543i
$612$		0			0
$613$	4.52027e6	−	4.52027e6i	0.485862	−	0.485862i	−0.421135	−	0.906998i	$-0.638368\pi$
$613$	4.52027e6	−	4.52027e6i	0.485862	−	0.485862i	0.906998	+	0.421135i	$0.138368\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−4.01198e6	+	4.01198e6i	−0.424274	+	0.424274i	−0.886672	−	0.462398i	$-0.846989\pi$
$617$	−4.01198e6	+	4.01198e6i	−0.424274	+	0.424274i	0.462398	+	0.886672i	$0.346989\pi$
$618$		0			0
$619$			1.13209e7i			1.18755i	0.804630	+	0.593776i	$0.202364\pi$
$619$			1.13209e7i			1.18755i	−0.804630	+	0.593776i	$0.797636\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	5.71300e6	+	5.71300e6i	0.589718	+	0.589718i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$	2.35733e7			2.37571
$630$		0			0
$631$	−5.30746e6			−0.530657			−0.265328	−	0.964158i	$-0.585480\pi$
$631$	−5.30746e6			−0.530657			−0.265328	+	0.964158i	$0.585480\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	8.30638e6	+	8.30638e6i	0.811079	+	0.811079i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			1.46710e7i			1.41031i	0.709052	+	0.705156i	$0.249123\pi$
$641$			1.46710e7i			1.41031i	−0.709052	+	0.705156i	$0.750877\pi$
$642$		0			0
$643$	4.54761e6	−	4.54761e6i	0.433766	−	0.433766i	−0.456141	−	0.889907i	$-0.650769\pi$
$643$	4.54761e6	−	4.54761e6i	0.433766	−	0.433766i	0.889907	+	0.456141i	$0.150769\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−9.16861e6	+	9.16861e6i	−0.861078	+	0.861078i	−0.991463	−	0.130385i	$-0.958379\pi$
$647$	−9.16861e6	+	9.16861e6i	−0.861078	+	0.861078i	0.130385	+	0.991463i	$0.458379\pi$
$648$		0			0
$649$			1.35279e7i			1.26072i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	9.15310e6	+	9.15310e6i	0.840011	+	0.840011i	0.988860	−	0.148849i	$-0.0475568\pi$
$653$	9.15310e6	+	9.15310e6i	0.840011	+	0.840011i	−0.148849	+	0.988860i	$0.547557\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	9.19398e6			0.824689			0.412344	−	0.911028i	$-0.364710\pi$
$659$	9.19398e6			0.824689			0.412344	+	0.911028i	$0.364710\pi$
$660$		0			0
$661$	3.89018e6			0.346311			0.173155	−	0.984895i	$-0.444604\pi$
$661$	3.89018e6			0.346311			0.173155	+	0.984895i	$0.444604\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−5.82505e6	−	5.82505e6i	−0.506973	−	0.506973i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	2.16873e7i		−	1.85952i
$672$		0			0
$673$	−6.60598e6	+	6.60598e6i	−0.562211	+	0.562211i	−0.929935	−	0.367724i	$-0.880137\pi$
$673$	−6.60598e6	+	6.60598e6i	−0.562211	+	0.562211i	0.367724	+	0.929935i	$0.380137\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	1.23121e7	−	1.23121e7i	1.03243	−	1.03243i	0.0329739	−	0.999456i	$-0.489502\pi$
$677$	1.23121e7	−	1.23121e7i	1.03243	−	1.03243i	0.999456	−	0.0329739i	$-0.0104978\pi$
$678$		0			0
$679$		−	9.13793e6i		−	0.760630i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−7.79590e6	−	7.79590e6i	−0.639462	−	0.639462i	0.310961	−	0.950423i	$-0.399349\pi$
$683$	−7.79590e6	−	7.79590e6i	−0.639462	−	0.639462i	−0.950423	+	0.310961i	$0.899349\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	3.47208e7			2.78639
$690$		0			0
$691$	7.97218e6			0.635158			0.317579	−	0.948232i	$-0.397130\pi$
$691$	7.97218e6			0.635158			0.317579	+	0.948232i	$0.397130\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−2.25101e6	−	2.25101e6i	−0.175508	−	0.175508i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	5.97246e6i		−	0.459048i	−0.973303	−	0.229524i	$-0.926283\pi$
$701$		−	5.97246e6i		−	0.459048i	0.973303	−	0.229524i	$-0.0737170\pi$
$702$		0			0
$703$	−1.61410e7	+	1.61410e7i	−1.23181	+	1.23181i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	814217.	−	814217.i	0.0612621	−	0.0612621i
$708$		0			0
$709$			5.80745e6i			0.433881i	0.976185	+	0.216940i	$0.0696077\pi$
$709$			5.80745e6i			0.433881i	−0.976185	+	0.216940i	$0.930392\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	5.67315e6	+	5.67315e6i	0.417927	+	0.417927i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	1.90921e6			0.137731			0.0688654	−	0.997626i	$-0.478062\pi$
$719$	1.90921e6			0.137731			0.0688654	+	0.997626i	$0.478062\pi$
$720$		0			0
$721$	1.91801e6			0.137408
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	4.90786e6	+	4.90786e6i	0.344394	+	0.344394i	0.858016	−	0.513622i	$-0.171697\pi$
$727$	4.90786e6	+	4.90786e6i	0.344394	+	0.344394i	−0.513622	+	0.858016i	$0.671697\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	3.51886e7i		−	2.43561i
$732$		0			0
$733$	−1.12824e7	+	1.12824e7i	−0.775604	+	0.775604i	−0.979080	−	0.203476i	$-0.934776\pi$
$733$	−1.12824e7	+	1.12824e7i	−0.775604	+	0.775604i	0.203476	+	0.979080i	$0.434776\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	1.88013e7	−	1.88013e7i	1.27502	−	1.27502i
$738$		0			0
$739$		−	9.09279e6i		−	0.612472i	−0.951956	−	0.306236i	$-0.900930\pi$
$739$		−	9.09279e6i		−	0.612472i	0.951956	−	0.306236i	$-0.0990697\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	9.09450e6	+	9.09450e6i	0.604375	+	0.604375i	0.941471	−	0.337095i	$-0.109444\pi$
$743$	9.09450e6	+	9.09450e6i	0.604375	+	0.604375i	−0.337095	+	0.941471i	$0.609444\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	1.34622e7			0.876825
$750$		0			0
$751$	1.22164e7			0.790394			0.395197	−	0.918596i	$-0.370676\pi$
$751$	1.22164e7			0.790394			0.395197	+	0.918596i	$0.370676\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−1.78724e7	−	1.78724e7i	−1.13356	−	1.13356i	−0.989581	−	0.143978i	$-0.954010\pi$
$757$	−1.78724e7	−	1.78724e7i	−1.13356	−	1.13356i	−0.143978	−	0.989581i	$-0.545990\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	32938.4i		−	0.00206177i	−0.999999	−	0.00103089i	$-0.999672\pi$
$761$		−	32938.4i		−	0.00206177i	0.999999	−	0.00103089i	$-0.000328141\pi$
$762$		0			0
$763$	6.19438e6	−	6.19438e6i	0.385200	−	0.385200i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1.86322e7	+	1.86322e7i	−1.14360	+	1.14360i
$768$		0			0
$769$			2.64362e7i			1.61207i	0.591870	+	0.806033i	$0.298390\pi$
$769$			2.64362e7i			1.61207i	−0.591870	+	0.806033i	$0.701610\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	5.30090e6	+	5.30090e6i	0.319081	+	0.319081i	0.848414	−	0.529333i	$-0.177558\pi$
$773$	5.30090e6	+	5.30090e6i	0.319081	+	0.319081i	−0.529333	+	0.848414i	$0.677558\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	3.08261e6			0.182002
$780$		0			0
$781$	−5.53877e6			−0.324927
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−1.11065e7	−	1.11065e7i	−0.639208	−	0.639208i	0.311152	−	0.950360i	$-0.399285\pi$
$787$	−1.11065e7	−	1.11065e7i	−0.639208	−	0.639208i	−0.950360	+	0.311152i	$0.899285\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	3.88272e6i		−	0.220645i
$792$		0			0
$793$	2.98704e7	−	2.98704e7i	1.68678	−	1.68678i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1.51828e7	+	1.51828e7i	−0.846656	+	0.846656i	−0.989714	−	0.143058i	$-0.954306\pi$
$797$	−1.51828e7	+	1.51828e7i	−0.846656	+	0.846656i	0.143058	+	0.989714i	$0.454306\pi$
$798$		0			0
$799$			1.44457e7i			0.800517i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1.48317e7	−	1.48317e7i	−0.811710	−	0.811710i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−5.98692e6			−0.321612			−0.160806	−	0.986986i	$-0.551409\pi$
$809$	−5.98692e6			−0.321612			−0.160806	+	0.986986i	$0.551409\pi$
$810$		0			0
$811$	9.69659e6			0.517686			0.258843	−	0.965919i	$-0.416659\pi$
$811$	9.69659e6			0.517686			0.258843	+	0.965919i	$0.416659\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	2.40942e7	+	2.40942e7i	1.26287	+	1.26287i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			2.39957e7i			1.24244i	0.783636	+	0.621220i	$0.213363\pi$
$821$			2.39957e7i			1.24244i	−0.783636	+	0.621220i	$0.786637\pi$
$822$		0			0
$823$	−1.70610e7	+	1.70610e7i	−0.878022	+	0.878022i	−0.993330	−	0.115307i	$-0.963215\pi$
$823$	−1.70610e7	+	1.70610e7i	−0.878022	+	0.878022i	0.115307	+	0.993330i	$0.463215\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−2.07936e7	+	2.07936e7i	−1.05722	+	1.05722i	−0.0589625	+	0.998260i	$0.518779\pi$
$827$	−2.07936e7	+	2.07936e7i	−1.05722	+	1.05722i	−0.998260	+	0.0589625i	$0.981221\pi$
$828$		0			0
$829$			1.04142e7i			0.526305i	0.964754	+	0.263153i	$0.0847623\pi$
$829$			1.04142e7i			0.526305i	−0.964754	+	0.263153i	$0.915238\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−1.84560e7	−	1.84560e7i	−0.921565	−	0.921565i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	7.26427e6			0.356277			0.178138	−	0.984005i	$-0.442993\pi$
$839$	7.26427e6			0.356277			0.178138	+	0.984005i	$0.442993\pi$
$840$		0			0
$841$	227913.			0.0111117
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	6.97496e6	+	6.97496e6i	0.334067	+	0.334067i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			1.82254e7i			0.862687i
$852$		0			0
$853$	2.52834e7	−	2.52834e7i	1.18977	−	1.18977i	0.212636	−	0.977131i	$-0.431795\pi$
$853$	2.52834e7	−	2.52834e7i	1.18977	−	1.18977i	0.977131	−	0.212636i	$-0.0682049\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	875598.	−	875598.i	0.0407242	−	0.0407242i	−0.686451	−	0.727176i	$-0.740833\pi$
$857$	875598.	−	875598.i	0.0407242	−	0.0407242i	0.727176	+	0.686451i	$0.240833\pi$
$858$		0			0
$859$			1.76413e7i			0.815731i	0.913042	+	0.407865i	$0.133727\pi$
$859$			1.76413e7i			0.815731i	−0.913042	+	0.407865i	$0.866273\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	1.90490e7	+	1.90490e7i	0.870652	+	0.870652i	0.992543	−	0.121891i	$-0.0388959\pi$
$863$	1.90490e7	+	1.90490e7i	0.870652	+	0.870652i	−0.121891	+	0.992543i	$0.538896\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	2.44463e7			1.09816
$870$		0			0
$871$	5.17907e7			2.31316
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−2.20695e6	−	2.20695e6i	−0.0968932	−	0.0968932i	0.656999	−	0.753892i	$-0.271826\pi$
$877$	−2.20695e6	−	2.20695e6i	−0.0968932	−	0.0968932i	−0.753892	+	0.656999i	$0.771826\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	7.42358e6i		−	0.322236i	−0.986935	−	0.161118i	$-0.948490\pi$
$881$		−	7.42358e6i		−	0.322236i	0.986935	−	0.161118i	$-0.0515099\pi$
$882$		0			0
$883$	−1.14031e6	+	1.14031e6i	−0.0492179	+	0.0492179i	−0.731287	−	0.682069i	$-0.761080\pi$
$883$	−1.14031e6	+	1.14031e6i	−0.0492179	+	0.0492179i	0.682069	+	0.731287i	$0.261080\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	3.00968e7	−	3.00968e7i	1.28443	−	1.28443i	0.346316	−	0.938118i	$-0.387432\pi$
$887$	3.00968e7	−	3.00968e7i	1.28443	−	1.28443i	0.938118	−	0.346316i	$-0.112568\pi$
$888$		0			0
$889$			1.30328e7i			0.553074i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−9.89118e6	−	9.89118e6i	−0.415068	−	0.415068i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−2.01983e7			−0.833518
$900$		0			0
$901$	−7.71466e7			−3.16595
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	943761.	+	943761.i	0.0380929	+	0.0380929i	0.725897	−	0.687804i	$-0.241425\pi$
$907$	943761.	+	943761.i	0.0380929	+	0.0380929i	−0.687804	+	0.725897i	$0.741425\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	4.29726e7i		−	1.71552i	−0.514050	−	0.857760i	$-0.671855\pi$
$911$		−	4.29726e7i		−	1.71552i	0.514050	−	0.857760i	$-0.328145\pi$
$912$		0			0
$913$	−3.83021e7	+	3.83021e7i	−1.52071	+	1.52071i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1.80554e7	−	1.80554e7i	0.709062	−	0.709062i
$918$		0			0
$919$		−	3.52785e7i		−	1.37791i	−0.724804	−	0.688955i	$-0.758070\pi$
$919$		−	3.52785e7i		−	1.37791i	0.724804	−	0.688955i	$-0.241930\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−7.62865e6	−	7.62865e6i	−0.294743	−	0.294743i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−3.75506e7			−1.42751			−0.713753	−	0.700398i	$-0.753006\pi$
$929$	−3.75506e7			−1.42751			−0.713753	+	0.700398i	$0.753006\pi$
$930$		0			0
$931$	2.52743e7			0.955664
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−1.80918e7	−	1.80918e7i	−0.673182	−	0.673182i	0.285266	−	0.958448i	$-0.407918\pi$
$937$	−1.80918e7	−	1.80918e7i	−0.673182	−	0.673182i	−0.958448	+	0.285266i	$0.907918\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			3.68014e7i			1.35485i	0.735594	+	0.677423i	$0.236903\pi$
$941$			3.68014e7i			1.35485i	−0.735594	+	0.677423i	$0.763097\pi$
$942$		0			0
$943$	1.74034e6	−	1.74034e6i	0.0637317	−	0.0637317i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1.46623e7	−	1.46623e7i	0.531284	−	0.531284i	−0.389671	−	0.920954i	$-0.627411\pi$
$947$	1.46623e7	−	1.46623e7i	0.531284	−	0.531284i	0.920954	+	0.389671i	$0.127411\pi$
$948$		0			0
$949$		−	4.08558e7i		−	1.47261i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	2.80395e7	+	2.80395e7i	1.00009	+	1.00009i	1.00000		8.73085e-5i	$2.77911e-5\pi$
$953$	2.80395e7	+	2.80395e7i	1.00009	+	1.00009i	8.73085e−5		1.00000i	$0.499972\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−1.11725e7			−0.392286
$960$		0			0
$961$	−8.95759e6			−0.312884
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−2.44518e6	−	2.44518e6i	−0.0840901	−	0.0840901i	0.663811	−	0.747901i	$-0.268938\pi$
$967$	−2.44518e6	−	2.44518e6i	−0.0840901	−	0.0840901i	−0.747901	+	0.663811i	$0.768938\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	4.90773e7i		−	1.67045i	−0.549910	−	0.835224i	$-0.685338\pi$
$971$		−	4.90773e7i		−	1.67045i	0.549910	−	0.835224i	$-0.314662\pi$
$972$		0			0
$973$	−1.89305e7	+	1.89305e7i	−0.641033	+	0.641033i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1.26540e7	+	1.26540e7i	−0.424121	+	0.424121i	−0.886620	−	0.462499i	$-0.846953\pi$
$977$	−1.26540e7	+	1.26540e7i	−0.424121	+	0.424121i	0.462499	+	0.886620i	$0.346953\pi$
$978$		0			0
$979$		−	5.81013e7i		−	1.93744i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	3.28130e6	+	3.28130e6i	0.108308	+	0.108308i	0.759184	−	0.650876i	$-0.225598\pi$
$983$	3.28130e6	+	3.28130e6i	0.108308	+	0.108308i	−0.650876	+	0.759184i	$0.725598\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	2.72056e7			0.884439
$990$		0			0
$991$	−8.02332e6			−0.259520			−0.129760	−	0.991545i	$-0.541421\pi$
$991$	−8.02332e6			−0.259520			−0.129760	+	0.991545i	$0.541421\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	1.28213e7	+	1.28213e7i	0.408501	+	0.408501i	0.881216	−	0.472714i	$-0.156726\pi$
$997$	1.28213e7	+	1.28213e7i	0.408501	+	0.408501i	−0.472714	+	0.881216i	$0.656726\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	900.6.j.a.557.4	yes	16
3.2	odd	2	inner	900.6.j.a.557.3	✓	16
5.2	odd	4	inner	900.6.j.a.593.6	yes	16
5.3	odd	4	inner	900.6.j.a.593.4	yes	16
5.4	even	2	inner	900.6.j.a.557.6	yes	16
15.2	even	4	inner	900.6.j.a.593.5	yes	16
15.8	even	4	inner	900.6.j.a.593.3	yes	16
15.14	odd	2	inner	900.6.j.a.557.5	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
900.6.j.a.557.3	✓	16	3.2	odd	2	inner
900.6.j.a.557.4	yes	16	1.1	even	1	trivial
900.6.j.a.557.5	yes	16	15.14	odd	2	inner
900.6.j.a.557.6	yes	16	5.4	even	2	inner
900.6.j.a.593.3	yes	16	15.8	even	4	inner
900.6.j.a.593.4	yes	16	5.3	odd	4	inner
900.6.j.a.593.5	yes	16	15.2	even	4	inner
900.6.j.a.593.6	yes	16	5.2	odd	4	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:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	900.j (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-53.1577 - 53.1577i) q^{7} +O(q^{10})\)	\(q+(-53.1577 - 53.1577i) q^{7} +540.614i q^{11} +(-744.598 + 744.598i) q^{13} +(1654.43 - 1654.43i) q^{17} +2265.63i q^{19} +(1279.10 + 1279.10i) q^{23} -4554.02 q^{29} +4435.26 q^{31} +(7124.30 + 7124.30i) q^{37} -1360.60i q^{41} +(10634.7 - 10634.7i) q^{43} +(-4365.75 + 4365.75i) q^{47} -11155.5i q^{49} +(-23315.1 - 23315.1i) q^{53} +25023.2 q^{59} -40116.1 q^{61} +(-34777.6 - 34777.6i) q^{67} +10245.3i q^{71} +(-27434.8 + 27434.8i) q^{73} +(28737.8 - 28737.8i) q^{77} -45219.6i q^{79} +(70849.2 + 70849.2i) q^{83} -107473. q^{89} +79162.2 q^{91} +(85951.2 + 85951.2i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 5632 q^{31} - 119200 q^{61} - 138048 q^{91}+O(q^{100}) \) 16 * q + 5632 * q^31 - 119200 * q^61 - 138048 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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