LMFDB - Embedded newform 90.6.c.d.19.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [90,6,Mod(19,90)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(90, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("90.19");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 90 = 2 \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	90.c (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:	$14.4345437832$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{19})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 9x^{2} + 25 $ x^4 - 9*x^2 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			19.4
Root			$2.17945 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	90.19
Dual form			90.6.c.d.19.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.00000i q^{2} -16.0000 q^{4} +(43.5890 - 35.0000i) q^{5} +17.4356i q^{7} -64.0000i q^{8} +O(q^{10})$	\(q+4.00000i q^{2} -16.0000 q^{4} +(43.5890 - 35.0000i) q^{5} +17.4356i q^{7} -64.0000i q^{8} +(140.000 + 174.356i) q^{10} -645.117 q^{11} -1098.44i q^{13} -69.7424 q^{14} +256.000 q^{16} -1166.00i q^{17} +2244.00 q^{19} +(-697.424 + 560.000i) q^{20} -2580.47i q^{22} +500.000i q^{23} +(675.000 - 3051.23i) q^{25} +4393.77 q^{26} -278.970i q^{28} -470.761 q^{29} +3856.00 q^{31} +1024.00i q^{32} +4664.00 q^{34} +(610.246 + 760.000i) q^{35} -6991.67i q^{37} +8976.00i q^{38} +(-2240.00 - 2789.70i) q^{40} -9589.58 q^{41} -5300.42i q^{43} +10321.9 q^{44} -2000.00 q^{46} -19900.0i q^{47} +16503.0 q^{49} +(12204.9 + 2700.00i) q^{50} +17575.1i q^{52} +1146.00i q^{53} +(-28120.0 + 22579.1i) q^{55} +1115.88 q^{56} -1883.04i q^{58} -37364.5 q^{59} -38158.0 q^{61} +15424.0i q^{62} -4096.00 q^{64} +(-38445.5 - 47880.0i) q^{65} +36231.2i q^{67} +18656.0i q^{68} +(-3040.00 + 2440.98i) q^{70} +16633.6 q^{71} +69393.7i q^{73} +27966.7 q^{74} -35904.0 q^{76} -11248.0i q^{77} +20664.0 q^{79} +(11158.8 - 8960.00i) q^{80} -38358.3i q^{82} +96968.0i q^{83} +(-40810.0 - 50824.8i) q^{85} +21201.7 q^{86} +41287.5i q^{88} +59699.5 q^{89} +19152.0 q^{91} -8000.00i q^{92} +79600.0 q^{94} +(97813.7 - 78540.0i) q^{95} -5823.49i q^{97} +66012.0i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 64 q^{4}+O(q^{10}) $ 4 * q - 64 * q^4	$ 4 q - 64 q^{4} + 560 q^{10} + 1024 q^{16} + 8976 q^{19} + 2700 q^{25} + 15424 q^{31} + 18656 q^{34} - 8960 q^{40} - 8000 q^{46} + 66012 q^{49} - 112480 q^{55} - 152632 q^{61} - 16384 q^{64} - 12160 q^{70} - 143616 q^{76} + 82656 q^{79} - 163240 q^{85} + 76608 q^{91} + 318400 q^{94}+O(q^{100}) $ 4 * q - 64 * q^4 + 560 * q^10 + 1024 * q^16 + 8976 * q^19 + 2700 * q^25 + 15424 * q^31 + 18656 * q^34 - 8960 * q^40 - 8000 * q^46 + 66012 * q^49 - 112480 * q^55 - 152632 * q^61 - 16384 * q^64 - 12160 * q^70 - 143616 * q^76 + 82656 * q^79 - 163240 * q^85 + 76608 * q^91 + 318400 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/90\mathbb{Z}\right)^\times$.

$n$	$11$	$37$
$\chi(n)$	$1$	$-1$

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 $

$2$			4.00000i			0.707107i
$2$			4.00000i			0.707107i
$3$		0			0
$3$		0			0
$4$	−16.0000			−0.500000
$5$	43.5890	−	35.0000i	0.779744	−	0.626099i
$5$	43.5890	−	35.0000i	0.779744	−	0.626099i
$6$		0			0
$7$			17.4356i			0.134491i	0.997736	+	0.0672453i	$0.0214210\pi$
$7$			17.4356i			0.134491i	−0.997736	+	0.0672453i	$0.978579\pi$
$8$		−	64.0000i		−	0.353553i
$9$		0			0
$10$	140.000	+	174.356i	0.442719	+	0.551362i
$11$	−645.117			−1.60752			−0.803761	−	0.594953i	$-0.797171\pi$
$11$	−645.117			−1.60752			−0.803761	+	0.594953i	$0.797171\pi$
$12$		0			0
$13$		−	1098.44i		−	1.80268i	−0.433111	−	0.901341i	$-0.642584\pi$
$13$		−	1098.44i		−	1.80268i	0.433111	−	0.901341i	$-0.357416\pi$
$14$	−69.7424			−0.0950992
$15$		0			0
$16$	256.000			0.250000
$17$		−	1166.00i		−	0.978535i	−0.872134	−	0.489267i	$-0.837264\pi$
$17$		−	1166.00i		−	0.978535i	0.872134	−	0.489267i	$-0.162736\pi$
$18$		0			0
$19$	2244.00			1.42606			0.713032	−	0.701132i	$-0.247322\pi$
$19$	2244.00			1.42606			0.713032	+	0.701132i	$0.247322\pi$
$20$	−697.424	+	560.000i	−0.389872	+	0.313050i
$21$		0			0
$22$		−	2580.47i		−	1.13669i
$23$			500.000i			0.197084i	0.995133	+	0.0985418i	$0.0314178\pi$
$23$			500.000i			0.197084i	−0.995133	+	0.0985418i	$0.968582\pi$
$24$		0			0
$25$	675.000	−	3051.23i	0.216000	−	0.976393i
$26$	4393.77			1.27469
$27$		0			0
$28$		−	278.970i		−	0.0672453i
$29$	−470.761			−0.103945			−0.0519727	−	0.998649i	$-0.516551\pi$
$29$	−470.761			−0.103945			−0.0519727	+	0.998649i	$0.516551\pi$
$30$		0			0
$31$	3856.00			0.720664			0.360332	−	0.932824i	$-0.382663\pi$
$31$	3856.00			0.720664			0.360332	+	0.932824i	$0.382663\pi$
$32$			1024.00i			0.176777i
$33$		0			0
$34$	4664.00			0.691929
$35$	610.246	+	760.000i	0.0842044	+	0.104868i
$36$		0			0
$37$		−	6991.67i		−	0.839609i	−0.907615	−	0.419804i	$-0.862099\pi$
$37$		−	6991.67i		−	0.839609i	0.907615	−	0.419804i	$-0.137901\pi$
$38$			8976.00i			1.00838i
$39$		0			0
$40$	−2240.00	−	2789.70i	−0.221359	−	0.275681i
$41$	−9589.58			−0.890922			−0.445461	−	0.895301i	$-0.646960\pi$
$41$	−9589.58			−0.890922			−0.445461	+	0.895301i	$0.646960\pi$
$42$		0			0
$43$		−	5300.42i		−	0.437159i	−0.975819	−	0.218579i	$-0.929858\pi$
$43$		−	5300.42i		−	0.437159i	0.975819	−	0.218579i	$-0.0701423\pi$
$44$	10321.9			0.803761
$45$		0			0
$46$	−2000.00			−0.139359
$47$		−	19900.0i		−	1.31404i	−0.753873	−	0.657020i	$-0.771817\pi$
$47$		−	19900.0i		−	1.31404i	0.753873	−	0.657020i	$-0.228183\pi$
$48$		0			0
$49$	16503.0			0.981912
$50$	12204.9	+	2700.00i	0.690414	+	0.152735i
$51$		0			0
$52$			17575.1i			0.901341i
$53$			1146.00i			0.0560396i	0.999607	+	0.0280198i	$0.00892015\pi$
$53$			1146.00i			0.0560396i	−0.999607	+	0.0280198i	$0.991080\pi$
$54$		0			0
$55$	−28120.0	+	22579.1i	−1.25345	+	1.00647i
$56$	1115.88			0.0475496
$57$		0			0
$58$		−	1883.04i		−	0.0735005i
$59$	−37364.5			−1.39743			−0.698713	−	0.715402i	$-0.746244\pi$
$59$	−37364.5			−1.39743			−0.698713	+	0.715402i	$0.746244\pi$
$60$		0			0
$61$	−38158.0			−1.31299			−0.656494	−	0.754331i	$-0.727961\pi$
$61$	−38158.0			−1.31299			−0.656494	+	0.754331i	$0.727961\pi$
$62$			15424.0i			0.509586i
$63$		0			0
$64$	−4096.00			−0.125000
$65$	−38445.5	−	47880.0i	−1.12866	−	1.40563i
$66$		0			0
$67$			36231.2i			0.986042i	0.870017	+	0.493021i	$0.164107\pi$
$67$			36231.2i			0.986042i	−0.870017	+	0.493021i	$0.835893\pi$
$68$			18656.0i			0.489267i
$69$		0			0
$70$	−3040.00	+	2440.98i	−0.0741530	+	0.0595415i
$71$	16633.6			0.391597			0.195798	−	0.980644i	$-0.437270\pi$
$71$	16633.6			0.391597			0.195798	+	0.980644i	$0.437270\pi$
$72$		0			0
$73$			69393.7i			1.52410i	0.647520	+	0.762049i	$0.275806\pi$
$73$			69393.7i			1.52410i	−0.647520	+	0.762049i	$0.724194\pi$
$74$	27966.7			0.593693
$75$		0			0
$76$	−35904.0			−0.713032
$77$		−	11248.0i		−	0.216196i
$78$		0			0
$79$	20664.0			0.372517			0.186259	−	0.982501i	$-0.440364\pi$
$79$	20664.0			0.372517			0.186259	+	0.982501i	$0.440364\pi$
$80$	11158.8	−	8960.00i	0.194936	−	0.156525i
$81$		0			0
$82$		−	38358.3i		−	0.629977i
$83$			96968.0i			1.54502i	0.635005	+	0.772508i	$0.280998\pi$
$83$			96968.0i			1.54502i	−0.635005	+	0.772508i	$0.719002\pi$
$84$		0			0
$85$	−40810.0	−	50824.8i	−0.612660	−	0.763006i
$86$	21201.7			0.309118
$87$		0			0
$88$			41287.5i			0.568345i
$89$	59699.5			0.798906			0.399453	−	0.916754i	$-0.369200\pi$
$89$	59699.5			0.798906			0.399453	+	0.916754i	$0.369200\pi$
$90$		0			0
$91$	19152.0			0.242444
$92$		−	8000.00i		−	0.0985418i
$93$		0			0
$94$	79600.0			0.929166
$95$	97813.7	−	78540.0i	1.11196	−	0.892857i
$96$		0			0
$97$		−	5823.49i		−	0.0628426i	−0.999506	−	0.0314213i	$-0.989997\pi$
$97$		−	5823.49i		−	0.0628426i	0.999506	−	0.0314213i	$-0.0100034\pi$
$98$			66012.0i			0.694317i
$99$		0			0
$100$	−10800.0	+	48819.7i	−0.108000	+	0.488197i
$101$	−102922.			−1.00394			−0.501968	−	0.864886i	$-0.667391\pi$
$101$	−102922.			−1.00394			−0.501968	+	0.864886i	$0.667391\pi$
$102$		0			0
$103$		−	178558.i		−	1.65839i	−0.558961	−	0.829194i	$-0.688800\pi$
$103$		−	178558.i		−	1.65839i	0.558961	−	0.829194i	$-0.311200\pi$
$104$	−70300.3			−0.637344
$105$		0			0
$106$	−4584.00			−0.0396260
$107$			7512.00i			0.0634302i	0.999497	+	0.0317151i	$0.0100969\pi$
$107$			7512.00i			0.0634302i	−0.999497	+	0.0317151i	$0.989903\pi$
$108$		0			0
$109$	58238.0			0.469505			0.234752	−	0.972055i	$-0.424572\pi$
$109$	58238.0			0.469505			0.234752	+	0.972055i	$0.424572\pi$
$110$	−90316.4	−	112480.i	−0.711680	−	0.886326i
$111$		0			0
$112$			4463.51i			0.0336226i
$113$			217314.i			1.60100i	0.599332	+	0.800500i	$0.295433\pi$
$113$			217314.i			1.60100i	−0.599332	+	0.800500i	$0.704567\pi$
$114$		0			0
$115$	17500.0	+	21794.5i	0.123394	+	0.153675i
$116$	7532.18			0.0519727
$117$		0			0
$118$		−	149458.i		−	0.988130i
$119$	20329.9			0.131604
$120$		0			0
$121$	255125.			1.58413
$122$		−	152632.i		−	0.928423i
$123$		0			0
$124$	−61696.0			−0.360332
$125$	−77370.5	−	156625.i	−0.442894	−	0.896574i
$126$		0			0
$127$		−	212348.i		−	1.16826i	−0.811660	−	0.584129i	$-0.801436\pi$
$127$		−	212348.i		−	1.16826i	0.811660	−	0.584129i	$-0.198564\pi$
$128$		−	16384.0i		−	0.0883883i
$129$		0			0
$130$	191520.	−	153782.i	0.993930	−	0.798081i
$131$	279266.			1.42180			0.710902	−	0.703291i	$-0.248287\pi$
$131$	279266.			1.42180			0.710902	+	0.703291i	$0.248287\pi$
$132$		0			0
$133$			39125.5i			0.191792i
$134$	−144925.			−0.697237
$135$		0			0
$136$	−74624.0			−0.345964
$137$		−	337914.i		−	1.53817i	−0.639146	−	0.769086i	$-0.720712\pi$
$137$		−	337914.i		−	1.53817i	0.639146	−	0.769086i	$-0.279288\pi$
$138$		0			0
$139$	−92620.0			−0.406600			−0.203300	−	0.979116i	$-0.565167\pi$
$139$	−92620.0			−0.406600			−0.203300	+	0.979116i	$0.565167\pi$
$140$	−9763.93	−	12160.0i	−0.0421022	−	0.0524341i
$141$		0			0
$142$			66534.2i			0.276901i
$143$			708624.i			2.89785i
$144$		0			0
$145$	−20520.0	+	16476.6i	−0.0810508	+	0.0650802i
$146$	−277575.			−1.07770
$147$		0			0
$148$			111867.i			0.419804i
$149$	29588.2			0.109182			0.0545912	−	0.998509i	$-0.482614\pi$
$149$	29588.2			0.109182			0.0545912	+	0.998509i	$0.482614\pi$
$150$		0			0
$151$	40384.0			0.144134			0.0720671	−	0.997400i	$-0.477040\pi$
$151$	40384.0			0.144134			0.0720671	+	0.997400i	$0.477040\pi$
$152$		−	143616.i		−	0.504190i
$153$		0			0
$154$	44992.0			0.152874
$155$	168079.	−	134960.i	0.561933	−	0.451207i
$156$		0			0
$157$			81476.5i			0.263805i	0.991263	+	0.131903i	$0.0421086\pi$
$157$			81476.5i			0.263805i	−0.991263	+	0.131903i	$0.957891\pi$
$158$			82656.0i			0.263410i
$159$		0			0
$160$	35840.0	+	44635.1i	0.110680	+	0.137840i
$161$	−8717.80			−0.0265059
$162$		0			0
$163$			458382.i			1.35132i	0.737213	+	0.675660i	$0.236141\pi$
$163$			458382.i			1.35132i	−0.737213	+	0.675660i	$0.763859\pi$
$164$	153433.			0.445461
$165$		0			0
$166$	−387872.			−1.09249
$167$			194436.i			0.539493i	0.962931	+	0.269746i	$0.0869398\pi$
$167$			194436.i			0.539493i	−0.962931	+	0.269746i	$0.913060\pi$
$168$		0			0
$169$	−835283.			−2.24966
$170$	203299.	−	163240.i	0.539527	−	0.433216i
$171$		0			0
$172$			84806.7i			0.218579i
$173$			322222.i			0.818540i	0.912413	+	0.409270i	$0.134217\pi$
$173$			322222.i			0.818540i	−0.912413	+	0.409270i	$0.865783\pi$
$174$		0			0
$175$	53200.0	+	11769.0i	0.131316	+	0.0290500i
$176$	−165150.			−0.401880
$177$		0			0
$178$			238798.i			0.564912i
$179$	430258.			1.00368			0.501842	−	0.864960i	$-0.332656\pi$
$179$	430258.			1.00368			0.501842	+	0.864960i	$0.332656\pi$
$180$		0			0
$181$	242522.			0.550243			0.275122	−	0.961409i	$-0.411282\pi$
$181$	242522.			0.550243			0.275122	+	0.961409i	$0.411282\pi$
$182$			76608.0i			0.171433i
$183$		0			0
$184$	32000.0			0.0696796
$185$	−244709.	−	304760.i	−0.525678	−	0.654679i
$186$		0			0
$187$			752206.i			1.57302i
$188$			318400.i			0.657020i
$189$		0			0
$190$	314160.	+	391255.i	0.631345	+	0.786277i
$191$	290930.			0.577040			0.288520	−	0.957474i	$-0.406837\pi$
$191$	290930.			0.577040			0.288520	+	0.957474i	$0.406837\pi$
$192$		0			0
$193$		−	723054.i		−	1.39726i	−0.715483	−	0.698631i	$-0.753793\pi$
$193$		−	723054.i		−	1.39726i	0.715483	−	0.698631i	$-0.246207\pi$
$194$	23294.0			0.0444364
$195$		0			0
$196$	−264048.			−0.490956
$197$		−	599638.i		−	1.10084i	−0.834888	−	0.550419i	$-0.814468\pi$
$197$		−	599638.i		−	1.10084i	0.834888	−	0.550419i	$-0.185532\pi$
$198$		0			0
$199$	−650184.			−1.16387			−0.581934	−	0.813236i	$-0.697704\pi$
$199$	−650184.			−1.16387			−0.581934	+	0.813236i	$0.697704\pi$
$200$	−195279.	−	43200.0i	−0.345207	−	0.0763675i
$201$		0			0
$202$		−	411689.i		−	0.709890i
$203$		−	8208.00i		−	0.0139797i
$204$		0			0
$205$	−418000.	+	335635.i	−0.694691	+	0.557806i
$206$	714232.			1.17266
$207$		0			0
$208$		−	281201.i		−	0.450670i
$209$	−1.44764e6			−2.29243
$210$		0			0
$211$	986316.			1.52514			0.762570	−	0.646905i	$-0.223937\pi$
$211$	986316.			1.52514			0.762570	+	0.646905i	$0.223937\pi$
$212$		−	18336.0i		−	0.0280198i
$213$		0			0
$214$	−30048.0			−0.0448519
$215$	−185515.	−	231040.i	−0.273705	−	0.340872i
$216$		0			0
$217$			67231.7i			0.0969225i
$218$			232952.i			0.331990i
$219$		0			0
$220$	449920.	−	361266.i	0.626727	−	0.503234i
$221$	−1.28078e6			−1.76399
$222$		0			0
$223$			694407.i			0.935087i	0.883970	+	0.467544i	$0.154861\pi$
$223$			694407.i			0.935087i	−0.883970	+	0.467544i	$0.845139\pi$
$224$	−17854.1			−0.0237748
$225$		0			0
$226$	−869256.			−1.13208
$227$			435400.i			0.560820i	0.959880	+	0.280410i	$0.0904705\pi$
$227$			435400.i			0.560820i	−0.959880	+	0.280410i	$0.909530\pi$
$228$		0			0
$229$	1.16123e6			1.46328			0.731641	−	0.681690i	$-0.238755\pi$
$229$	1.16123e6			1.46328			0.731641	+	0.681690i	$0.238755\pi$
$230$	−87178.0	+	70000.0i	−0.108664	+	0.0872526i
$231$		0			0
$232$			30128.7i			0.0367503i
$233$		−	1.21303e6i		−	1.46380i	−0.681409	−	0.731902i	$-0.738633\pi$
$233$		−	1.21303e6i		−	1.46380i	0.681409	−	0.731902i	$-0.261367\pi$
$234$		0			0
$235$	−696500.	−	867421.i	−0.822719	−	1.02461i
$236$	597832.			0.698713
$237$		0			0
$238$			81319.6i			0.0930578i
$239$	1.40053e6			1.58598			0.792991	−	0.609234i	$-0.208523\pi$
$239$	1.40053e6			1.58598			0.792991	+	0.609234i	$0.208523\pi$
$240$		0			0
$241$	471490.			0.522914			0.261457	−	0.965215i	$-0.415797\pi$
$241$	471490.			0.522914			0.261457	+	0.965215i	$0.415797\pi$
$242$			1.02050e6i			1.12015i
$243$		0			0
$244$	610528.			0.656494
$245$	719349.	−	577605.i	0.765640	−	0.614774i
$246$		0			0
$247$		−	2.46491e6i		−	2.57074i
$248$		−	246784.i		−	0.254793i
$249$		0			0
$250$	626500.	−	309482.i	0.633973	−	0.313174i
$251$	−601371.			−0.602502			−0.301251	−	0.953545i	$-0.597404\pi$
$251$	−601371.			−0.602502			−0.301251	+	0.953545i	$0.597404\pi$
$252$		0			0
$253$		−	322559.i		−	0.316816i
$254$	849392.			0.826084
$255$		0			0
$256$	65536.0			0.0625000
$257$			124206.i			0.117303i	0.998279	+	0.0586516i	$0.0186801\pi$
$257$			124206.i			0.117303i	−0.998279	+	0.0586516i	$0.981320\pi$
$258$		0			0
$259$	121904.			0.112919
$260$	615128.	+	766080.i	0.564329	+	0.702815i
$261$		0			0
$262$			1.11706e6i			1.00537i
$263$			429956.i			0.383296i	0.981464	+	0.191648i	$0.0613832\pi$
$263$			429956.i			0.383296i	−0.981464	+	0.191648i	$0.938617\pi$
$264$		0			0
$265$	40110.0	+	49953.0i	0.0350863	+	0.0436965i
$266$	−156502.			−0.135617
$267$		0			0
$268$		−	579699.i		−	0.493021i
$269$	−1.43755e6			−1.21127			−0.605636	−	0.795742i	$-0.707081\pi$
$269$	−1.43755e6			−1.21127			−0.605636	+	0.795742i	$0.707081\pi$
$270$		0			0
$271$	1.54473e6			1.27770			0.638850	−	0.769331i	$-0.279411\pi$
$271$	1.54473e6			1.27770			0.638850	+	0.769331i	$0.279411\pi$
$272$		−	298496.i		−	0.244634i
$273$		0			0
$274$	1.35166e6			1.08765
$275$	−435454.	+	1.96840e6i	−0.347225	+	1.56957i
$276$		0			0
$277$			211128.i			0.165328i	0.996577	+	0.0826639i	$0.0263428\pi$
$277$			211128.i			0.165328i	−0.996577	+	0.0826639i	$0.973657\pi$
$278$		−	370480.i		−	0.287510i
$279$		0			0
$280$	48640.0	−	39055.7i	0.0370765	−	0.0297707i
$281$	1.29051e6			0.974982			0.487491	−	0.873128i	$-0.337912\pi$
$281$	1.29051e6			0.974982			0.487491	+	0.873128i	$0.337912\pi$
$282$		0			0
$283$		−	1.69959e6i		−	1.26147i	−0.775998	−	0.630736i	$-0.782753\pi$
$283$		−	1.69959e6i		−	1.26147i	0.775998	−	0.630736i	$-0.217247\pi$
$284$	−266137.			−0.195798
$285$		0			0
$286$	−2.83450e6			−2.04909
$287$		−	167200.i		−	0.119821i
$288$		0			0
$289$	60301.0			0.0424698
$290$	−65906.6	−	82080.0i	−0.0460186	−	0.0573116i
$291$		0			0
$292$		−	1.11030e6i		−	0.762049i
$293$		−	2.49148e6i		−	1.69546i	−0.530424	−	0.847732i	$-0.677967\pi$
$293$		−	2.49148e6i		−	1.69546i	0.530424	−	0.847732i	$-0.322033\pi$
$294$		0			0
$295$	−1.62868e6	+	1.30776e6i	−1.08963	+	0.874927i
$296$	−447467.			−0.296846
$297$		0			0
$298$			118353.i			0.0772037i
$299$	549221.			0.355279
$300$		0			0
$301$	92416.0			0.0587937
$302$			161536.i			0.101918i
$303$		0			0
$304$	574464.			0.356516
$305$	−1.66327e6	+	1.33553e6i	−1.02379	+	0.822061i
$306$		0			0
$307$			3.16791e6i			1.91834i	0.282822	+	0.959172i	$0.408729\pi$
$307$			3.16791e6i			1.91834i	−0.282822	+	0.959172i	$0.591271\pi$
$308$			179968.i			0.108098i
$309$		0			0
$310$	539840.	+	672317.i	0.319052	+	0.397347i
$311$	2.23653e6			1.31122			0.655608	−	0.755101i	$-0.272412\pi$
$311$	2.23653e6			1.31122			0.655608	+	0.755101i	$0.272412\pi$
$312$		0			0
$313$			2.37351e6i			1.36940i	0.728826	+	0.684699i	$0.240066\pi$
$313$			2.37351e6i			1.36940i	−0.728826	+	0.684699i	$0.759934\pi$
$314$	−325906.			−0.186538
$315$		0			0
$316$	−330624.			−0.186259
$317$			987502.i			0.551937i	0.961167	+	0.275969i	$0.0889986\pi$
$317$			987502.i			0.551937i	−0.961167	+	0.275969i	$0.911001\pi$
$318$		0			0
$319$	303696.			0.167095
$320$	−178541.	+	143360.i	−0.0974679	+	0.0782624i
$321$		0			0
$322$		−	34871.2i		−	0.0187425i
$323$		−	2.61650e6i		−	1.39545i
$324$		0			0
$325$	−3.35160e6	−	741449.i	−1.76013	−	0.389379i
$326$	−1.83353e6			−0.955528
$327$		0			0
$328$			613733.i			0.314989i
$329$	346968.			0.176726
$330$		0			0
$331$	−918644.			−0.460869			−0.230434	−	0.973088i	$-0.574015\pi$
$331$	−918644.			−0.460869			−0.230434	+	0.973088i	$0.574015\pi$
$332$		−	1.55149e6i		−	0.772508i
$333$		0			0
$334$	−777744.			−0.381479
$335$	1.26809e6	+	1.57928e6i	0.617360	+	0.768860i
$336$		0			0
$337$			355547.i			0.170538i	0.996358	+	0.0852691i	$0.0271750\pi$
$337$			355547.i			0.170538i	−0.996358	+	0.0852691i	$0.972825\pi$
$338$		−	3.34113e6i		−	1.59075i
$339$		0			0
$340$	652960.	+	813196.i	0.306330	+	0.381503i
$341$	−2.48757e6			−1.15848
$342$		0			0
$343$			580780.i			0.266548i
$344$	−339227.			−0.154559
$345$		0			0
$346$	−1.28889e6			−0.578795
$347$			1.85760e6i			0.828187i	0.910234	+	0.414094i	$0.135901\pi$
$347$			1.85760e6i			0.828187i	−0.910234	+	0.414094i	$0.864099\pi$
$348$		0			0
$349$	−950578.			−0.417757			−0.208879	−	0.977942i	$-0.566981\pi$
$349$	−950578.			−0.417757			−0.208879	+	0.977942i	$0.566981\pi$
$350$	−47076.1	+	212800.i	−0.0205414	+	0.0928542i
$351$		0			0
$352$		−	660600.i		−	0.284172i
$353$		−	767570.i		−	0.327855i	−0.986472	−	0.163927i	$-0.947584\pi$
$353$		−	767570.i		−	0.327855i	0.986472	−	0.163927i	$-0.0524162\pi$
$354$		0			0
$355$	725040.	−	582175.i	0.305345	−	0.245178i
$356$	−955192.			−0.399453
$357$		0			0
$358$			1.72103e6i			0.709711i
$359$	160896.			0.0658883			0.0329441	−	0.999457i	$-0.489512\pi$
$359$	160896.			0.0658883			0.0329441	+	0.999457i	$0.489512\pi$
$360$		0			0
$361$	2.55944e6			1.03366
$362$			970088.i			0.389081i
$363$		0			0
$364$	−306432.			−0.121222
$365$	2.42878e6	+	3.02480e6i	0.954236	+	1.18841i
$366$		0			0
$367$			202131.i			0.0783371i	0.999233	+	0.0391685i	$0.0124709\pi$
$367$			202131.i			0.0783371i	−0.999233	+	0.0391685i	$0.987529\pi$
$368$			128000.i			0.0492709i
$369$		0			0
$370$	1.21904e6	−	978834.i	0.462928	−	0.371711i
$371$	−19981.2			−0.00753679
$372$		0			0
$373$			1.48933e6i			0.554267i	0.960831	+	0.277134i	$0.0893845\pi$
$373$			1.48933e6i			0.554267i	−0.960831	+	0.277134i	$0.910616\pi$
$374$	−3.00883e6			−1.11229
$375$		0			0
$376$	−1.27360e6			−0.464583
$377$			517104.i			0.187381i
$378$		0			0
$379$	−2.69222e6			−0.962748			−0.481374	−	0.876515i	$-0.659862\pi$
$379$	−2.69222e6			−0.962748			−0.481374	+	0.876515i	$0.659862\pi$
$380$	−1.56502e6	+	1.25664e6i	−0.555982	+	0.446428i
$381$		0			0
$382$			1.16372e6i			0.408029i
$383$		−	565100.i		−	0.196847i	−0.995145	−	0.0984234i	$-0.968620\pi$
$383$		−	565100.i		−	0.196847i	0.995145	−	0.0984234i	$-0.0313799\pi$
$384$		0			0
$385$	−393680.	−	490289.i	−0.135360	−	0.168578i
$386$	2.89222e6			0.988013
$387$		0			0
$388$			93175.8i			0.0314213i
$389$	5.46587e6			1.83141			0.915704	−	0.401853i	$-0.131634\pi$
$389$	5.46587e6			1.83141			0.915704	+	0.401853i	$0.131634\pi$
$390$		0			0
$391$	583000.			0.192853
$392$		−	1.05619e6i		−	0.347158i
$393$		0			0
$394$	2.39855e6			0.778410
$395$	900723.	−	723240.i	0.290468	−	0.233233i
$396$		0			0
$397$		−	4.00762e6i		−	1.27618i	−0.769963	−	0.638088i	$-0.779726\pi$
$397$		−	4.00762e6i		−	1.27618i	0.769963	−	0.638088i	$-0.220274\pi$
$398$		−	2.60074e6i		−	0.822979i
$399$		0			0
$400$	172800.	−	781115.i	0.0540000	−	0.244098i
$401$	3.92967e6			1.22038			0.610190	−	0.792255i	$-0.291093\pi$
$401$	3.92967e6			1.22038			0.610190	+	0.792255i	$0.291093\pi$
$402$		0			0
$403$		−	4.23559e6i		−	1.29913i
$404$	1.64676e6			0.501968
$405$		0			0
$406$	32832.0			0.00988513
$407$			4.51045e6i			1.34969i
$408$		0			0
$409$	1.15028e6			0.340013			0.170007	−	0.985443i	$-0.445621\pi$
$409$	1.15028e6			0.340013			0.170007	+	0.985443i	$0.445621\pi$
$410$	−1.34254e6	−	1.67200e6i	−0.394428	−	0.491221i
$411$		0			0
$412$			2.85693e6i			0.829194i
$413$		−	651472.i		−	0.187941i
$414$		0			0
$415$	3.39388e6	+	4.22674e6i	0.967334	+	1.20472i
$416$	1.12481e6			0.318672
$417$		0			0
$418$		−	5.79057e6i		−	1.62099i
$419$	−4.95898e6			−1.37993			−0.689965	−	0.723842i	$-0.742374\pi$
$419$	−4.95898e6			−1.37993			−0.689965	+	0.723842i	$0.742374\pi$
$420$		0			0
$421$	2.89077e6			0.794893			0.397447	−	0.917625i	$-0.369896\pi$
$421$	2.89077e6			0.794893			0.397447	+	0.917625i	$0.369896\pi$
$422$			3.94526e6i			1.07844i
$423$		0			0
$424$	73344.0			0.0198130
$425$	−3.55773e6	−	787050.i	−0.955435	−	0.211364i
$426$		0			0
$427$		−	665307.i		−	0.176585i
$428$		−	120192.i		−	0.0317151i
$429$		0			0
$430$	924160.	−	742059.i	0.241033	−	0.193538i
$431$	−1.98417e6			−0.514501			−0.257250	−	0.966345i	$-0.582816\pi$
$431$	−1.98417e6			−0.514501			−0.257250	+	0.966345i	$0.582816\pi$
$432$		0			0
$433$		−	3.75622e6i		−	0.962789i	−0.876504	−	0.481395i	$-0.840130\pi$
$433$		−	3.75622e6i		−	0.962789i	0.876504	−	0.481395i	$-0.159870\pi$
$434$	−268927.			−0.0685345
$435$		0			0
$436$	−931808.			−0.234752
$437$			1.12200e6i			0.281054i
$438$		0			0
$439$	−1.94897e6			−0.482662			−0.241331	−	0.970443i	$-0.577584\pi$
$439$	−1.94897e6			−0.482662			−0.241331	+	0.970443i	$0.577584\pi$
$440$	1.44506e6	+	1.79968e6i	0.355840	+	0.443163i
$441$		0			0
$442$		−	5.12314e6i		−	1.24733i
$443$		−	5.74198e6i		−	1.39012i	−0.718952	−	0.695060i	$-0.755378\pi$
$443$		−	5.74198e6i		−	1.39012i	0.718952	−	0.695060i	$-0.244622\pi$
$444$		0			0
$445$	2.60224e6	−	2.08948e6i	0.622942	−	0.500194i
$446$	−2.77763e6			−0.661207
$447$		0			0
$448$		−	71416.2i		−	0.0168113i
$449$	−538655.			−0.126094			−0.0630471	−	0.998011i	$-0.520082\pi$
$449$	−538655.			−0.126094			−0.0630471	+	0.998011i	$0.520082\pi$
$450$		0			0
$451$	6.18640e6			1.43218
$452$		−	3.47702e6i		−	0.800500i
$453$		0			0
$454$	−1.74160e6			−0.396560
$455$	834816.	−	670320.i	0.189044	−	0.151794i
$456$		0			0
$457$		−	1.55459e6i		−	0.348198i	−0.984728	−	0.174099i	$-0.944299\pi$
$457$		−	1.55459e6i		−	0.348198i	0.984728	−	0.174099i	$-0.0557013\pi$
$458$			4.64490e6i			1.03470i
$459$		0			0
$460$	−280000.	−	348712.i	−0.0616969	−	0.0768373i
$461$	−4.22445e6			−0.925802			−0.462901	−	0.886410i	$-0.653191\pi$
$461$	−4.22445e6			−0.925802			−0.462901	+	0.886410i	$0.653191\pi$
$462$		0			0
$463$			4.77347e6i			1.03486i	0.855726	+	0.517430i	$0.173111\pi$
$463$			4.77347e6i			1.03486i	−0.855726	+	0.517430i	$0.826889\pi$
$464$	−120515.			−0.0259864
$465$		0			0
$466$	4.85214e6			1.03507
$467$		−	9.11683e6i		−	1.93442i	−0.253970	−	0.967212i	$-0.581736\pi$
$467$		−	9.11683e6i		−	1.93442i	0.253970	−	0.967212i	$-0.418264\pi$
$468$		0			0
$469$	−631712.			−0.132613
$470$	3.46968e6	−	2.78600e6i	0.724511	−	0.581750i
$471$		0			0
$472$			2.39133e6i			0.494065i
$473$			3.41939e6i			0.702742i
$474$		0			0
$475$	1.51470e6	−	6.84696e6i	0.308030	−	1.39240i
$476$	−325278.			−0.0658018
$477$		0			0
$478$			5.60213e6i			1.12146i
$479$	−4.22182e6			−0.840739			−0.420369	−	0.907353i	$-0.638099\pi$
$479$	−4.22182e6			−0.840739			−0.420369	+	0.907353i	$0.638099\pi$
$480$		0			0
$481$	−7.67995e6			−1.51355
$482$			1.88596e6i			0.369756i
$483$		0			0
$484$	−4.08200e6			−0.792063
$485$	−203822.	−	253840.i	−0.0393457	−	0.0490011i
$486$		0			0
$487$		−	501465.i		−	0.0958117i	−0.998852	−	0.0479058i	$-0.984745\pi$
$487$		−	501465.i		−	0.0958117i	0.998852	−	0.0479058i	$-0.0152547\pi$
$488$			2.44211e6i			0.464212i
$489$		0			0
$490$	2.31042e6	+	2.87740e6i	0.434711	+	0.541389i
$491$	−2.61435e6			−0.489395			−0.244697	−	0.969600i	$-0.578689\pi$
$491$	−2.61435e6			−0.489395			−0.244697	+	0.969600i	$0.578689\pi$
$492$		0			0
$493$			548907.i			0.101714i
$494$	9.85962e6			1.81779
$495$		0			0
$496$	987136.			0.180166
$497$			290016.i			0.0526661i
$498$		0			0
$499$	−5.68240e6			−1.02160			−0.510799	−	0.859700i	$-0.670650\pi$
$499$	−5.68240e6			−1.02160			−0.510799	+	0.859700i	$0.670650\pi$
$500$	1.23793e6	+	2.50600e6i	0.221447	+	0.448287i
$501$		0			0
$502$		−	2.40548e6i		−	0.426033i
$503$			3.36030e6i			0.592186i	0.955159	+	0.296093i	$0.0956838\pi$
$503$			3.36030e6i			0.592186i	−0.955159	+	0.296093i	$0.904316\pi$
$504$		0			0
$505$	−4.48628e6	+	3.60228e6i	−0.782813	+	0.628564i
$506$	1.29023e6			0.224023
$507$		0			0
$508$			3.39757e6i			0.584129i
$509$	5.60242e6			0.958476			0.479238	−	0.877685i	$-0.340913\pi$
$509$	5.60242e6			0.958476			0.479238	+	0.877685i	$0.340913\pi$
$510$		0			0
$511$	−1.20992e6			−0.204977
$512$			262144.i			0.0441942i
$513$		0			0
$514$	−496824.			−0.0829459
$515$	−6.24953e6	−	7.78316e6i	−1.03832	−	1.29312i
$516$		0			0
$517$			1.28378e7i			2.11235i
$518$			487616.i			0.0798461i
$519$		0			0
$520$	−3.06432e6	+	2.46051e6i	−0.496965	+	0.399041i
$521$	8.49019e6			1.37032			0.685162	−	0.728391i	$-0.259731\pi$
$521$	8.49019e6			1.37032			0.685162	+	0.728391i	$0.259731\pi$
$522$		0			0
$523$			3.31350e6i			0.529703i	0.964289	+	0.264851i	$0.0853229\pi$
$523$			3.31350e6i			0.529703i	−0.964289	+	0.264851i	$0.914677\pi$
$524$	−4.46826e6			−0.710902
$525$		0			0
$526$	−1.71982e6			−0.271031
$527$		−	4.49610e6i		−	0.705195i
$528$		0			0
$529$	6.18634e6			0.961158
$530$	−199812.	+	160440.i	−0.0308981	+	0.0248098i
$531$		0			0
$532$		−	626008.i		−	0.0958960i
$533$			1.05336e7i			1.60605i
$534$		0			0
$535$	262920.	+	327440.i	0.0397136	+	0.0494593i
$536$	2.31879e6			0.348618
$537$		0			0
$538$		−	5.75019e6i		−	0.856498i
$539$	−1.06464e7			−1.57845
$540$		0			0
$541$	−4.76958e6			−0.700627			−0.350314	−	0.936632i	$-0.613925\pi$
$541$	−4.76958e6			−0.700627			−0.350314	+	0.936632i	$0.613925\pi$
$542$			6.17891e6i			0.903470i
$543$		0			0
$544$	1.19398e6			0.172982
$545$	2.53854e6	−	2.03833e6i	0.366093	−	0.293957i
$546$		0			0
$547$		−	1.10402e6i		−	0.157765i	−0.996884	−	0.0788823i	$-0.974865\pi$
$547$		−	1.10402e6i		−	0.157765i	0.996884	−	0.0788823i	$-0.0251351\pi$
$548$			5.40662e6i			0.769086i
$549$		0			0
$550$	−7.87360e6	−	1.74182e6i	−1.10986	−	0.245525i
$551$	−1.05639e6			−0.148233
$552$		0			0
$553$			360289.i			0.0501001i
$554$	−844511.			−0.116904
$555$		0			0
$556$	1.48192e6			0.203300
$557$		−	5.14466e6i		−	0.702617i	−0.936260	−	0.351308i	$-0.885737\pi$
$557$		−	5.14466e6i		−	0.702617i	0.936260	−	0.351308i	$-0.114263\pi$
$558$		0			0
$559$	−5.82221e6			−0.788058
$560$	156223.	+	194560.i	0.0210511	+	0.0262170i
$561$		0			0
$562$			5.16205e6i			0.689416i
$563$			9.51894e6i			1.26566i	0.774290	+	0.632831i	$0.218107\pi$
$563$			9.51894e6i			1.26566i	−0.774290	+	0.632831i	$0.781893\pi$
$564$		0			0
$565$	7.60599e6	+	9.47250e6i	1.00239	+	1.24837i
$566$	6.79835e6			0.891995
$567$		0			0
$568$		−	1.06455e6i		−	0.138450i
$569$	1.29027e6			0.167070			0.0835352	−	0.996505i	$-0.473379\pi$
$569$	1.29027e6			0.167070			0.0835352	+	0.996505i	$0.473379\pi$
$570$		0			0
$571$	−1.11726e7			−1.43405			−0.717024	−	0.697048i	$-0.754496\pi$
$571$	−1.11726e7			−1.43405			−0.717024	+	0.697048i	$0.754496\pi$
$572$		−	1.13380e7i		−	1.44892i
$573$		0			0
$574$	668800.			0.0847260
$575$	1.52561e6	+	337500.i	0.192431	+	0.0425701i
$576$		0			0
$577$			129791.i			0.0162294i	0.999967	+	0.00811472i	$0.00258302\pi$
$577$			129791.i			0.0162294i	−0.999967	+	0.00811472i	$0.997417\pi$
$578$			241204.i			0.0300307i
$579$		0			0
$580$	328320.	−	263626.i	0.0405254	−	0.0325401i
$581$	−1.69069e6			−0.207790
$582$		0			0
$583$		−	739304.i		−	0.0900848i
$584$	4.44119e6			0.538850
$585$		0			0
$586$	9.96593e6			1.19887
$587$		−	9.30142e6i		−	1.11418i	−0.830453	−	0.557088i	$-0.811919\pi$
$587$		−	9.30142e6i		−	1.11418i	0.830453	−	0.557088i	$-0.188081\pi$
$588$		0			0
$589$	8.65286e6			1.02771
$590$	−5.23103e6	−	6.51472e6i	−0.618667	−	0.770488i
$591$		0			0
$592$		−	1.78987e6i		−	0.209902i
$593$			6.91802e6i			0.807876i	0.914786	+	0.403938i	$0.132359\pi$
$593$			6.91802e6i			0.807876i	−0.914786	+	0.403938i	$0.867641\pi$
$594$		0			0
$595$	886160.	−	711547.i	0.102617	−	0.0823969i
$596$	−473411.			−0.0545912
$597$		0			0
$598$			2.19689e6i			0.251220i
$599$	1.57759e7			1.79650			0.898252	−	0.439481i	$-0.144838\pi$
$599$	1.57759e7			1.79650			0.898252	+	0.439481i	$0.144838\pi$
$600$		0			0
$601$	−6.95761e6			−0.785731			−0.392866	−	0.919596i	$-0.628516\pi$
$601$	−6.95761e6			−0.785731			−0.392866	+	0.919596i	$0.628516\pi$
$602$			369664.i			0.0415734i
$603$		0			0
$604$	−646144.			−0.0720671
$605$	1.11206e7	−	8.92938e6i	1.23521	−	0.991819i
$606$		0			0
$607$		−	6.13874e6i		−	0.676251i	−0.941101	−	0.338125i	$-0.890207\pi$
$607$		−	6.13874e6i		−	0.676251i	0.941101	−	0.338125i	$-0.109793\pi$
$608$			2.29786e6i			0.252095i
$609$		0			0
$610$	−5.34212e6	−	6.65307e6i	−0.581285	−	0.723932i
$611$	−2.18590e7			−2.36879
$612$		0			0
$613$		−	5.89321e6i		−	0.633433i	−0.948520	−	0.316717i	$-0.897420\pi$
$613$		−	5.89321e6i		−	0.633433i	0.948520	−	0.316717i	$-0.102580\pi$
$614$	−1.26716e7			−1.35647
$615$		0			0
$616$	−719872.			−0.0764370
$617$			9.14737e6i			0.967349i	0.875248	+	0.483675i	$0.160698\pi$
$617$			9.14737e6i			0.967349i	−0.875248	+	0.483675i	$0.839302\pi$
$618$		0			0
$619$	1.23318e7			1.29360			0.646799	−	0.762661i	$-0.276107\pi$
$619$	1.23318e7			1.29360			0.646799	+	0.762661i	$0.276107\pi$
$620$	−2.68927e6	+	2.15936e6i	−0.280967	+	0.225603i
$621$		0			0
$622$			8.94613e6i			0.927170i
$623$			1.04090e6i			0.107445i
$624$		0			0
$625$	−8.85438e6	−	4.11916e6i	−0.906688	−	0.421802i
$626$	−9.49403e6			−0.968311
$627$		0			0
$628$		−	1.30362e6i		−	0.131903i
$629$	−8.15229e6			−0.821586
$630$		0			0
$631$	6.87452e6			0.687336			0.343668	−	0.939091i	$-0.388331\pi$
$631$	6.87452e6			0.687336			0.343668	+	0.939091i	$0.388331\pi$
$632$		−	1.32250e6i		−	0.131705i
$633$		0			0
$634$	−3.95001e6			−0.390279
$635$	−7.43218e6	−	9.25604e6i	−0.731446	−	0.910942i
$636$		0			0
$637$		−	1.81276e7i		−	1.77007i
$638$			1.21478e6i			0.118154i
$639$		0			0
$640$	−573440.	−	714162.i	−0.0553399	−	0.0689202i
$641$	9.05120e6			0.870084			0.435042	−	0.900410i	$-0.356734\pi$
$641$	9.05120e6			0.870084			0.435042	+	0.900410i	$0.356734\pi$
$642$		0			0
$643$			8.76840e6i			0.836359i	0.908364	+	0.418179i	$0.137332\pi$
$643$			8.76840e6i			0.836359i	−0.908364	+	0.418179i	$0.862668\pi$
$644$	139485.			0.0132529
$645$		0			0
$646$	1.04660e7			0.986734
$647$		−	1.49567e6i		−	0.140467i	−0.997531	−	0.0702335i	$-0.977626\pi$
$647$		−	1.49567e6i		−	0.140467i	0.997531	−	0.0702335i	$-0.0223744\pi$
$648$		0			0
$649$	2.41045e7			2.24639
$650$	2.96579e6	−	1.34064e7i	0.275333	−	1.24460i
$651$		0			0
$652$		−	7.33411e6i		−	0.675660i
$653$			1.17639e7i			1.07962i	0.841788	+	0.539809i	$0.181503\pi$
$653$			1.17639e7i			1.07962i	−0.841788	+	0.539809i	$0.818497\pi$
$654$		0			0
$655$	1.21729e7	−	9.77431e6i	1.10864	−	0.890190i
$656$	−2.45493e6			−0.222731
$657$		0			0
$658$			1.38787e6i			0.124964i
$659$	3.36184e6			0.301553			0.150777	−	0.988568i	$-0.451823\pi$
$659$	3.36184e6			0.301553			0.150777	+	0.988568i	$0.451823\pi$
$660$		0			0
$661$	−1.28140e7			−1.14072			−0.570361	−	0.821394i	$-0.693197\pi$
$661$	−1.28140e7			−1.14072			−0.570361	+	0.821394i	$0.693197\pi$
$662$		−	3.67458e6i		−	0.325883i
$663$		0			0
$664$	6.20595e6			0.546246
$665$	1.36939e6	+	1.70544e6i	0.120081	+	0.149549i
$666$		0			0
$667$		−	235381.i		−	0.0204859i
$668$		−	3.11098e6i		−	0.269746i
$669$		0			0
$670$	−6.31712e6	+	5.07236e6i	−0.543666	+	0.436539i
$671$	2.46164e7			2.11066
$672$		0			0
$673$		−	1.07843e7i		−	0.917813i	−0.888485	−	0.458907i	$-0.848241\pi$
$673$		−	1.07843e7i		−	0.917813i	0.888485	−	0.458907i	$-0.151759\pi$
$674$	−1.42219e6			−0.120589
$675$		0			0
$676$	1.33645e7			1.12483
$677$			1.72031e7i			1.44257i	0.692640	+	0.721283i	$0.256447\pi$
$677$			1.72031e7i			1.44257i	−0.692640	+	0.721283i	$0.743553\pi$
$678$		0			0
$679$	101536.			0.00845173
$680$	−3.25278e6	+	2.61184e6i	−0.269763	+	0.216608i
$681$		0			0
$682$		−	9.95029e6i		−	0.819171i
$683$			5.61037e6i			0.460193i	0.973168	+	0.230096i	$0.0739041\pi$
$683$			5.61037e6i			0.460193i	−0.973168	+	0.230096i	$0.926096\pi$
$684$		0			0
$685$	−1.18270e7	−	1.47293e7i	−0.963048	−	1.19938i
$686$	−2.32312e6			−0.188478
$687$		0			0
$688$		−	1.35691e6i		−	0.109290i
$689$	1.25882e6			0.101022
$690$		0			0
$691$	−3.89064e6			−0.309974			−0.154987	−	0.987916i	$-0.549534\pi$
$691$	−3.89064e6			−0.309974			−0.154987	+	0.987916i	$0.549534\pi$
$692$		−	5.15555e6i		−	0.409270i
$693$		0			0
$694$	−7.43040e6			−0.585617
$695$	−4.03721e6	+	3.24170e6i	−0.317044	+	0.254572i
$696$		0			0
$697$			1.11814e7i			0.871798i
$698$		−	3.80231e6i		−	0.295399i
$699$		0			0
$700$	−851200.	−	188304.i	−0.0656578	−	0.0145250i
$701$	2.02753e7			1.55838			0.779190	−	0.626788i	$-0.215631\pi$
$701$	2.02753e7			1.55838			0.779190	+	0.626788i	$0.215631\pi$
$702$		0			0
$703$		−	1.56893e7i		−	1.19734i
$704$	2.64240e6			0.200940
$705$		0			0
$706$	3.07028e6			0.231828
$707$		−	1.79451e6i		−	0.135020i
$708$		0			0
$709$	1.60535e6			0.119938			0.0599688	−	0.998200i	$-0.480900\pi$
$709$	1.60535e6			0.119938			0.0599688	+	0.998200i	$0.480900\pi$
$710$	2.32870e6	+	2.90016e6i	0.173367	+	0.215912i
$711$		0			0
$712$		−	3.82077e6i		−	0.282456i
$713$			1.92800e6i			0.142031i
$714$		0			0
$715$	2.48018e7	+	3.08882e7i	1.81434	+	2.25958i
$716$	−6.88413e6			−0.501842
$717$		0			0
$718$			643583.i			0.0465901i
$719$	1.04392e7			0.753090			0.376545	−	0.926398i	$-0.377112\pi$
$719$	1.04392e7			0.753090			0.376545	+	0.926398i	$0.377112\pi$
$720$		0			0
$721$	3.11326e6			0.223037
$722$			1.02377e7i			0.730906i
$723$		0			0
$724$	−3.88035e6			−0.275122
$725$	−317764.	+	1.43640e6i	−0.0224522	+	0.101492i
$726$		0			0
$727$			2.36879e7i			1.66223i	0.556102	+	0.831114i	$0.312297\pi$
$727$			2.36879e7i			1.66223i	−0.556102	+	0.831114i	$0.687703\pi$
$728$		−	1.22573e6i		−	0.0857167i
$729$		0			0
$730$	−1.20992e7	+	9.71511e6i	−0.840329	+	0.674747i
$731$	−6.18029e6			−0.427775
$732$		0			0
$733$			1.14116e7i			0.784487i	0.919861	+	0.392244i	$0.128301\pi$
$733$			1.14116e7i			0.784487i	−0.919861	+	0.392244i	$0.871699\pi$
$734$	−808523.			−0.0553927
$735$		0			0
$736$	−512000.			−0.0348398
$737$		−	2.33733e7i		−	1.58508i
$738$		0			0
$739$	−2.57126e7			−1.73195			−0.865975	−	0.500088i	$-0.833301\pi$
$739$	−2.57126e7			−1.73195			−0.865975	+	0.500088i	$0.833301\pi$
$740$	3.91534e6	+	4.87616e6i	0.262839	+	0.327340i
$741$		0			0
$742$		−	79924.8i		−	0.00532932i
$743$			7.87830e6i			0.523553i	0.965129	+	0.261776i	$0.0843083\pi$
$743$			7.87830e6i			0.523553i	−0.965129	+	0.261776i	$0.915692\pi$
$744$		0			0
$745$	1.28972e6	−	1.03559e6i	0.0851343	−	0.0683590i
$746$	−5.95732e6			−0.391926
$747$		0			0
$748$		−	1.20353e7i		−	0.786508i
$749$	−130976.			−0.00853076
$750$		0			0
$751$	−2.69649e7			−1.74461			−0.872306	−	0.488961i	$-0.837376\pi$
$751$	−2.69649e7			−1.74461			−0.872306	+	0.488961i	$0.837376\pi$
$752$		−	5.09440e6i		−	0.328510i
$753$		0			0
$754$	−2.06842e6			−0.132498
$755$	1.76030e6	−	1.41344e6i	0.112388	−	0.0902423i
$756$		0			0
$757$			3.92135e6i			0.248712i	0.992238	+	0.124356i	$0.0396864\pi$
$757$			3.92135e6i			0.248712i	−0.992238	+	0.124356i	$0.960314\pi$
$758$		−	1.07689e7i		−	0.680765i
$759$		0			0
$760$	−5.02656e6	−	6.26008e6i	−0.315673	−	0.393139i
$761$	−2.84849e7			−1.78301			−0.891504	−	0.453013i	$-0.850349\pi$
$761$	−2.84849e7			−1.78301			−0.891504	+	0.453013i	$0.850349\pi$
$762$		0			0
$763$			1.01541e6i			0.0631440i
$764$	−4.65489e6			−0.288520
$765$		0			0
$766$	2.26040e6			0.139192
$767$			4.10427e7i			2.51911i
$768$		0			0
$769$	5.18935e6			0.316444			0.158222	−	0.987404i	$-0.449424\pi$
$769$	5.18935e6			0.316444			0.158222	+	0.987404i	$0.449424\pi$
$770$	1.96116e6	−	1.57472e6i	0.119202	−	0.0957142i
$771$		0			0
$772$			1.15689e7i			0.698631i
$773$		−	1.88767e7i		−	1.13626i	−0.822939	−	0.568130i	$-0.807667\pi$
$773$		−	1.88767e7i		−	1.13626i	0.822939	−	0.568130i	$-0.192333\pi$
$774$		0			0
$775$	2.60280e6	−	1.17655e7i	0.155663	−	0.703651i
$776$	−372703.			−0.0222182
$777$		0			0
$778$			2.18635e7i			1.29500i
$779$	−2.15190e7			−1.27051
$780$		0			0
$781$	−1.07306e7			−0.629501
$782$			2.33200e6i			0.136368i
$783$		0			0
$784$	4.22477e6			0.245478
$785$	2.85168e6	+	3.55148e6i	0.165168	+	0.205700i
$786$		0			0
$787$		−	1.73159e7i		−	0.996570i	−0.867013	−	0.498285i	$-0.833963\pi$
$787$		−	1.73159e7i		−	0.996570i	0.867013	−	0.498285i	$-0.166037\pi$
$788$			9.59421e6i			0.550419i
$789$		0			0
$790$	2.89296e6	+	3.60289e6i	0.164921	+	0.205392i
$791$	−3.78900e6			−0.215319
$792$		0			0
$793$			4.19144e7i			2.36690i
$794$	1.60305e7			0.902393
$795$		0			0
$796$	1.04029e7			0.581934
$797$		−	2.62525e7i		−	1.46394i	−0.681335	−	0.731972i	$-0.738600\pi$
$797$		−	2.62525e7i		−	1.46394i	0.681335	−	0.731972i	$-0.261400\pi$
$798$		0			0
$799$	−2.32034e7			−1.28583
$800$	3.12446e6	+	691200.i	0.172604	+	0.0381838i
$801$		0			0
$802$			1.57187e7i			0.862939i
$803$		−	4.47670e7i		−	2.45002i
$804$		0			0
$805$	−380000.	+	305123.i	−0.0206678	+	0.0165953i
$806$	1.69424e7			0.918622
$807$		0			0
$808$			6.58703e6i			0.354945i
$809$	−3.02992e6			−0.162765			−0.0813824	−	0.996683i	$-0.525933\pi$
$809$	−3.02992e6			−0.162765			−0.0813824	+	0.996683i	$0.525933\pi$
$810$		0			0
$811$	9.81951e6			0.524249			0.262124	−	0.965034i	$-0.415577\pi$
$811$	9.81951e6			0.524249			0.262124	+	0.965034i	$0.415577\pi$
$812$			131328.i			0.00698984i
$813$		0			0
$814$	−1.80418e7			−0.954374
$815$	1.60434e7	+	1.99804e7i	0.846060	+	1.05368i
$816$		0			0
$817$		−	1.18941e7i		−	0.623416i
$818$			4.60113e6i			0.240426i
$819$		0			0
$820$	6.68800e6	−	5.37016e6i	0.347345	−	0.278903i
$821$	−5.80196e6			−0.300411			−0.150206	−	0.988655i	$-0.547994\pi$
$821$	−5.80196e6			−0.300411			−0.150206	+	0.988655i	$0.547994\pi$
$822$		0			0
$823$		−	1.31473e7i		−	0.676606i	−0.941037	−	0.338303i	$-0.890147\pi$
$823$		−	1.31473e7i		−	0.676606i	0.941037	−	0.338303i	$-0.109853\pi$
$824$	−1.14277e7			−0.586329
$825$		0			0
$826$	2.60589e6			0.132894
$827$		−	2.77989e7i		−	1.41340i	−0.707515	−	0.706698i	$-0.750184\pi$
$827$		−	2.77989e7i		−	1.41340i	0.707515	−	0.706698i	$-0.249816\pi$
$828$		0			0
$829$	1.08971e7			0.550714			0.275357	−	0.961342i	$-0.411204\pi$
$829$	1.08971e7			0.550714			0.275357	+	0.961342i	$0.411204\pi$
$830$	−1.69069e7	+	1.35755e7i	−0.851863	+	0.684008i
$831$		0			0
$832$			4.49922e6i			0.225335i
$833$		−	1.92425e7i		−	0.960835i
$834$		0			0
$835$	6.80526e6	+	8.47527e6i	0.337776	+	0.420666i
$836$	2.31623e7			1.14621
$837$		0			0
$838$		−	1.98359e7i		−	0.975758i
$839$	1.13976e7			0.558995			0.279497	−	0.960146i	$-0.409832\pi$
$839$	1.13976e7			0.558995			0.279497	+	0.960146i	$0.409832\pi$
$840$		0			0
$841$	−2.02895e7			−0.989195
$842$			1.15631e7i			0.562075i
$843$		0			0
$844$	−1.57811e7			−0.762570
$845$	−3.64091e7	+	2.92349e7i	−1.75416	+	1.40851i
$846$		0			0
$847$			4.44826e6i			0.213050i
$848$			293376.i			0.0140099i
$849$		0			0
$850$	3.14820e6	−	1.42309e7i	0.149457	−	0.675594i
$851$	3.49584e6			0.165473
$852$		0			0
$853$			4.97544e6i			0.234131i	0.993124	+	0.117065i	$0.0373487\pi$
$853$			4.97544e6i			0.234131i	−0.993124	+	0.117065i	$0.962651\pi$
$854$	2.66123e6			0.124864
$855$		0			0
$856$	480768.			0.0224260
$857$			2.38632e7i			1.10988i	0.831889	+	0.554942i	$0.187259\pi$
$857$			2.38632e7i			1.10988i	−0.831889	+	0.554942i	$0.812741\pi$
$858$		0			0
$859$	−1.49286e7			−0.690298			−0.345149	−	0.938548i	$-0.612172\pi$
$859$	−1.49286e7			−0.690298			−0.345149	+	0.938548i	$0.612172\pi$
$860$	2.96824e6	+	3.69664e6i	0.136852	+	0.170436i
$861$		0			0
$862$		−	7.93668e6i		−	0.363807i
$863$			9.90376e6i			0.452661i	0.974051	+	0.226330i	$0.0726729\pi$
$863$			9.90376e6i			0.452661i	−0.974051	+	0.226330i	$0.927327\pi$
$864$		0			0
$865$	1.12778e7	+	1.40453e7i	0.512487	+	0.638251i
$866$	1.50249e7			0.680795
$867$		0			0
$868$		−	1.07571e6i		−	0.0484612i
$869$	−1.33307e7			−0.598830
$870$		0			0
$871$	3.97979e7			1.77752
$872$		−	3.72723e6i		−	0.165995i
$873$		0			0
$874$	−4.48800e6			−0.198735
$875$	2.73085e6	−	1.34900e6i	0.120581	−	0.0595651i
$876$		0			0
$877$			3.84422e6i			0.168775i	0.996433	+	0.0843877i	$0.0268934\pi$
$877$			3.84422e6i			0.168775i	−0.996433	+	0.0843877i	$0.973107\pi$
$878$		−	7.79587e6i		−	0.341294i
$879$		0			0
$880$	−7.19872e6	+	5.78025e6i	−0.313364	+	0.251617i
$881$	−2.94025e7			−1.27627			−0.638137	−	0.769922i	$-0.720295\pi$
$881$	−2.94025e7			−1.27627			−0.638137	+	0.769922i	$0.720295\pi$
$882$		0			0
$883$		−	6.24156e6i		−	0.269396i	−0.990887	−	0.134698i	$-0.956994\pi$
$883$		−	6.24156e6i		−	0.269396i	0.990887	−	0.134698i	$-0.0430065\pi$
$884$	2.04925e7			0.881993
$885$		0			0
$886$	2.29679e7			0.982963
$887$			1.67076e7i			0.713025i	0.934291	+	0.356512i	$0.116034\pi$
$887$			1.67076e7i			0.713025i	−0.934291	+	0.356512i	$0.883966\pi$
$888$		0			0
$889$	3.70242e6			0.157120
$890$	8.35793e6	+	1.04090e7i	0.353691	+	0.440486i
$891$		0			0
$892$		−	1.11105e7i		−	0.467544i
$893$		−	4.46556e7i		−	1.87390i
$894$		0			0
$895$	1.87545e7	−	1.50590e7i	0.782615	−	0.628405i
$896$	285665.			0.0118874
$897$		0			0
$898$		−	2.15462e6i		−	0.0891621i
$899$	−1.81525e6			−0.0749098
$900$		0			0
$901$	1.33624e6			0.0548367
$902$			2.47456e7i			1.01270i
$903$		0			0
$904$	1.39081e7			0.566039
$905$	1.05713e7	−	8.48827e6i	0.429049	−	0.344507i
$906$		0			0
$907$		−	7.14197e6i		−	0.288270i	−0.989558	−	0.144135i	$-0.953960\pi$
$907$		−	7.14197e6i		−	0.288270i	0.989558	−	0.144135i	$-0.0460400\pi$
$908$		−	6.96640e6i		−	0.280410i
$909$		0			0
$910$	2.68128e6	+	3.33927e6i	0.107334	+	0.133674i
$911$	3.51083e7			1.40157			0.700784	−	0.713374i	$-0.252834\pi$
$911$	3.51083e7			1.40157			0.700784	+	0.713374i	$0.252834\pi$
$912$		0			0
$913$		−	6.25557e7i		−	2.48365i
$914$	6.21837e6			0.246213
$915$		0			0
$916$	−1.85796e7			−0.731641
$917$			4.86917e6i			0.191219i
$918$		0			0
$919$	−2.66789e7			−1.04203			−0.521014	−	0.853548i	$-0.674446\pi$
$919$	−2.66789e7			−1.04203			−0.521014	+	0.853548i	$0.674446\pi$
$920$	1.39485e6	−	1.12000e6i	0.0543322	−	0.0436263i
$921$		0			0
$922$		−	1.68978e7i		−	0.654641i
$923$		−	1.82710e7i		−	0.705925i
$924$		0			0
$925$	−2.13332e7	−	4.71938e6i	−0.819788	−	0.181355i
$926$	−1.90939e7			−0.731756
$927$		0			0
$928$		−	482059.i		−	0.0183751i
$929$	6.04426e6			0.229776			0.114888	−	0.993378i	$-0.463349\pi$
$929$	6.04426e6			0.229776			0.114888	+	0.993378i	$0.463349\pi$
$930$		0			0
$931$	3.70327e7			1.40027
$932$			1.94085e7i			0.731902i
$933$		0			0
$934$	3.64673e7			1.36784
$935$	2.63272e7	+	3.27879e7i	0.984864	+	1.22655i
$936$		0			0
$937$		−	3.60422e6i		−	0.134110i	−0.997749	−	0.0670551i	$-0.978640\pi$
$937$		−	3.60422e6i		−	0.134110i	0.997749	−	0.0670551i	$-0.0213603\pi$
$938$		−	2.52685e6i		−	0.0937718i
$939$		0			0
$940$	1.11440e7	+	1.38787e7i	0.411359	+	0.512307i
$941$	−1.30244e7			−0.479495			−0.239747	−	0.970835i	$-0.577065\pi$
$941$	−1.30244e7			−0.479495			−0.239747	+	0.970835i	$0.577065\pi$
$942$		0			0
$943$		−	4.79479e6i		−	0.175586i
$944$	−9.56531e6			−0.349357
$945$		0			0
$946$	−1.36776e7			−0.496914
$947$			3.17698e7i			1.15117i	0.817742	+	0.575585i	$0.195226\pi$
$947$			3.17698e7i			1.15117i	−0.817742	+	0.575585i	$0.804774\pi$
$948$		0			0
$949$	7.62250e7			2.74746
$950$	2.73878e7	+	6.05880e6i	0.984575	+	0.217810i
$951$		0			0
$952$		−	1.30111e6i		−	0.0465289i
$953$		−	2.25254e7i		−	0.803417i	−0.915768	−	0.401708i	$-0.868417\pi$
$953$		−	2.25254e7i		−	0.803417i	0.915768	−	0.401708i	$-0.131583\pi$
$954$		0			0
$955$	1.26814e7	−	1.01826e7i	0.449943	−	0.361284i
$956$	−2.24085e7			−0.792991
$957$		0			0
$958$		−	1.68873e7i		−	0.594492i
$959$	5.89173e6			0.206869
$960$		0			0
$961$	−1.37604e7			−0.480643
$962$		−	3.07198e7i		−	1.07024i
$963$		0			0
$964$	−7.54384e6			−0.261457
$965$	−2.53069e7	−	3.15172e7i	−0.874824	−	1.08951i
$966$		0			0
$967$		−	169805.i		−	0.00583963i	−0.999996	−	0.00291981i	$-0.999071\pi$
$967$		−	169805.i		−	0.00583963i	0.999996	−	0.00291981i	$-0.000929407\pi$
$968$		−	1.63280e7i		−	0.560073i
$969$		0			0
$970$	1.01536e6	−	815288.i	0.0346490	−	0.0278216i
$971$	−164296.			−0.00559214			−0.00279607	−	0.999996i	$-0.500890\pi$
$971$	−164296.			−0.00559214			−0.00279607	+	0.999996i	$0.500890\pi$
$972$		0			0
$973$		−	1.61488e6i		−	0.0546839i
$974$	2.00586e6			0.0677491
$975$		0			0
$976$	−9.76845e6			−0.328247
$977$		−	2.00989e7i		−	0.673653i	−0.941567	−	0.336827i	$-0.890646\pi$
$977$		−	2.00989e7i		−	0.673653i	0.941567	−	0.336827i	$-0.109354\pi$
$978$		0			0
$979$	−3.85132e7			−1.28426
$980$	−1.15096e7	+	9.24168e6i	−0.382820	+	0.307387i
$981$		0			0
$982$		−	1.04574e7i		−	0.346054i
$983$			1.39716e7i			0.461172i	0.973052	+	0.230586i	$0.0740644\pi$
$983$			1.39716e7i			0.461172i	−0.973052	+	0.230586i	$0.925936\pi$
$984$		0			0
$985$	−2.09873e7	−	2.61376e7i	−0.689234	−	0.858372i
$986$	−2.19563e6			−0.0719228
$987$		0			0
$988$			3.94385e7i			1.28537i
$989$	2.65021e6			0.0861568
$990$		0			0
$991$	140392.			0.00454107			0.00227054	−	0.999997i	$-0.499277\pi$
$991$	140392.			0.00454107			0.00227054	+	0.999997i	$0.499277\pi$
$992$			3.94854e6i			0.127397i
$993$		0			0
$994$	−1.16006e6			−0.0372405
$995$	−2.83409e7	+	2.27564e7i	−0.907518	+	0.728696i
$996$		0			0
$997$			1.79809e7i			0.572894i	0.958096	+	0.286447i	$0.0924742\pi$
$997$			1.79809e7i			0.572894i	−0.958096	+	0.286447i	$0.907526\pi$
$998$		−	2.27296e7i		−	0.722379i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	90.6.c.d.19.4	yes	4
3.2	odd	2	inner	90.6.c.d.19.1	✓	4
4.3	odd	2		720.6.f.j.289.3		4
5.2	odd	4		450.6.a.z.1.1		2
5.3	odd	4		450.6.a.be.1.2		2
5.4	even	2	inner	90.6.c.d.19.2	yes	4
12.11	even	2		720.6.f.j.289.2		4
15.2	even	4		450.6.a.be.1.1		2
15.8	even	4		450.6.a.z.1.2		2
15.14	odd	2	inner	90.6.c.d.19.3	yes	4
20.19	odd	2		720.6.f.j.289.4		4
60.59	even	2		720.6.f.j.289.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
90.6.c.d.19.1	✓	4	3.2	odd	2	inner
90.6.c.d.19.2	yes	4	5.4	even	2	inner
90.6.c.d.19.3	yes	4	15.14	odd	2	inner
90.6.c.d.19.4	yes	4	1.1	even	1	trivial
450.6.a.z.1.1		2	5.2	odd	4
450.6.a.z.1.2		2	15.8	even	4
450.6.a.be.1.1		2	15.2	even	4
450.6.a.be.1.2		2	5.3	odd	4
720.6.f.j.289.1		4	60.59	even	2
720.6.f.j.289.2		4	12.11	even	2
720.6.f.j.289.3		4	4.3	odd	2
720.6.f.j.289.4		4	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	90.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+4.00000i q^{2} -16.0000 q^{4} +(43.5890 - 35.0000i) q^{5} +17.4356i q^{7} -64.0000i q^{8} +O(q^{10})\)	\(q+4.00000i q^{2} -16.0000 q^{4} +(43.5890 - 35.0000i) q^{5} +17.4356i q^{7} -64.0000i q^{8} +(140.000 + 174.356i) q^{10} -645.117 q^{11} -1098.44i q^{13} -69.7424 q^{14} +256.000 q^{16} -1166.00i q^{17} +2244.00 q^{19} +(-697.424 + 560.000i) q^{20} -2580.47i q^{22} +500.000i q^{23} +(675.000 - 3051.23i) q^{25} +4393.77 q^{26} -278.970i q^{28} -470.761 q^{29} +3856.00 q^{31} +1024.00i q^{32} +4664.00 q^{34} +(610.246 + 760.000i) q^{35} -6991.67i q^{37} +8976.00i q^{38} +(-2240.00 - 2789.70i) q^{40} -9589.58 q^{41} -5300.42i q^{43} +10321.9 q^{44} -2000.00 q^{46} -19900.0i q^{47} +16503.0 q^{49} +(12204.9 + 2700.00i) q^{50} +17575.1i q^{52} +1146.00i q^{53} +(-28120.0 + 22579.1i) q^{55} +1115.88 q^{56} -1883.04i q^{58} -37364.5 q^{59} -38158.0 q^{61} +15424.0i q^{62} -4096.00 q^{64} +(-38445.5 - 47880.0i) q^{65} +36231.2i q^{67} +18656.0i q^{68} +(-3040.00 + 2440.98i) q^{70} +16633.6 q^{71} +69393.7i q^{73} +27966.7 q^{74} -35904.0 q^{76} -11248.0i q^{77} +20664.0 q^{79} +(11158.8 - 8960.00i) q^{80} -38358.3i q^{82} +96968.0i q^{83} +(-40810.0 - 50824.8i) q^{85} +21201.7 q^{86} +41287.5i q^{88} +59699.5 q^{89} +19152.0 q^{91} -8000.00i q^{92} +79600.0 q^{94} +(97813.7 - 78540.0i) q^{95} -5823.49i q^{97} +66012.0i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 64 q^{4}+O(q^{10}) \) 4 * q - 64 * q^4	\( 4 q - 64 q^{4} + 560 q^{10} + 1024 q^{16} + 8976 q^{19} + 2700 q^{25} + 15424 q^{31} + 18656 q^{34} - 8960 q^{40} - 8000 q^{46} + 66012 q^{49} - 112480 q^{55} - 152632 q^{61} - 16384 q^{64} - 12160 q^{70} - 143616 q^{76} + 82656 q^{79} - 163240 q^{85} + 76608 q^{91} + 318400 q^{94}+O(q^{100}) \) 4 * q - 64 * q^4 + 560 * q^10 + 1024 * q^16 + 8976 * q^19 + 2700 * q^25 + 15424 * q^31 + 18656 * q^34 - 8960 * q^40 - 8000 * q^46 + 66012 * q^49 - 112480 * q^55 - 152632 * q^61 - 16384 * q^64 - 12160 * q^70 - 143616 * q^76 + 82656 * q^79 - 163240 * q^85 + 76608 * q^91 + 318400 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more