LMFDB - Embedded newform 900.6.j.b.557.9

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:	$20$
Relative dimension:	$10$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 222025 x^{16} + 11247583920 x^{12} + 151104106237945 x^{8} + \cdots + 67\!\cdots\!76 $ x^20 + 222025x^16 + 11247583920x^12 + 151104106237945x^8 + 21346152874807825x^4 + 672057978863947776
Coefficient ring:	$\Z[a_1, \ldots, a_{49}]$
Coefficient ring index:	$ 2^{46}\cdot 3^{24}\cdot 5^{4} $
Twist minimal:	no (minimal twist has level 180)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			557.9
Root			$-1.85337 + 1.85337i$ of defining polynomial
Character	$\chi$	$=$	900.557
Dual form			900.6.j.b.593.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(176.224 + 176.224i) q^{7} +O(q^{10})$	$q+(176.224 + 176.224i) q^{7} -170.036i q^{11} +(80.0637 - 80.0637i) q^{13} +(-709.979 + 709.979i) q^{17} +2231.89i q^{19} +(-2424.04 - 2424.04i) q^{23} -4296.03 q^{29} -6062.44 q^{31} +(147.841 + 147.841i) q^{37} -11053.4i q^{41} +(9926.25 - 9926.25i) q^{43} +(-17559.2 + 17559.2i) q^{47} +45302.9i q^{49} +(-24150.5 - 24150.5i) q^{53} +9558.17 q^{59} -19318.5 q^{61} +(-13130.7 - 13130.7i) q^{67} +56336.8i q^{71} +(6831.56 - 6831.56i) q^{73} +(29964.4 - 29964.4i) q^{77} -4005.41i q^{79} +(17559.5 + 17559.5i) q^{83} +112023. q^{89} +28218.3 q^{91} +(-72995.8 - 72995.8i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 152 q^{7}+O(q^{10}) $ 20 * q + 152 * q^7	$ 20 q + 152 q^{7} - 252 q^{13} + 9440 q^{31} - 29988 q^{37} - 6720 q^{43} + 67200 q^{61} - 139552 q^{67} - 134828 q^{73} - 220560 q^{91} - 207132 q^{97}+O(q^{100}) $ 20 * q + 152 * q^7 - 252 * q^13 + 9440 * q^31 - 29988 * q^37 - 6720 * q^43 + 67200 * q^61 - 139552 * q^67 - 134828 * q^73 - 220560 * q^91 - 207132 * q^97

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
$7$	176.224	+	176.224i	1.35932	+	1.35932i	0.874761	+	0.484554i	$0.161018\pi$
$7$	176.224	+	176.224i	1.35932	+	1.35932i	0.484554	+	0.874761i	$0.338982\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	170.036i		−	0.423700i	−0.977302	−	0.211850i	$-0.932051\pi$
$11$		−	170.036i		−	0.423700i	0.977302	−	0.211850i	$-0.0679488\pi$
$12$		0			0
$13$	80.0637	−	80.0637i	0.131395	−	0.131395i	−0.638351	−	0.769745i	$-0.720383\pi$
$13$	80.0637	−	80.0637i	0.131395	−	0.131395i	0.769745	+	0.638351i	$0.220383\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−709.979	+	709.979i	−0.595831	+	0.595831i	−0.939200	−	0.343369i	$-0.888432\pi$
$17$	−709.979	+	709.979i	−0.595831	+	0.595831i	0.343369	+	0.939200i	$0.388432\pi$
$18$		0			0
$19$			2231.89i			1.41837i	0.705023	+	0.709185i	$0.250937\pi$
$19$			2231.89i			1.41837i	−0.705023	+	0.709185i	$0.749063\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−2424.04	−	2424.04i	−0.955478	−	0.955478i	0.0435720	−	0.999050i	$-0.486126\pi$
$23$	−2424.04	−	2424.04i	−0.955478	−	0.955478i	−0.999050	+	0.0435720i	$0.986126\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−4296.03			−0.948575			−0.474288	−	0.880370i	$-0.657294\pi$
$29$	−4296.03			−0.948575			−0.474288	+	0.880370i	$0.657294\pi$
$30$		0			0
$31$	−6062.44			−1.13304			−0.566518	−	0.824050i	$-0.691710\pi$
$31$	−6062.44			−1.13304			−0.566518	+	0.824050i	$0.691710\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	147.841	+	147.841i	0.0177538	+	0.0177538i	0.715928	−	0.698174i	$-0.246004\pi$
$37$	147.841	+	147.841i	0.0177538	+	0.0177538i	−0.698174	+	0.715928i	$0.746004\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	11053.4i		−	1.02692i	−0.858113	−	0.513461i	$-0.828363\pi$
$41$		−	11053.4i		−	1.02692i	0.858113	−	0.513461i	$-0.171637\pi$
$42$		0			0
$43$	9926.25	−	9926.25i	0.818680	−	0.818680i	−0.167237	−	0.985917i	$-0.553485\pi$
$43$	9926.25	−	9926.25i	0.818680	−	0.818680i	0.985917	+	0.167237i	$0.0534845\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−17559.2	+	17559.2i	−1.15947	+	1.15947i	−0.174886	+	0.984589i	$0.555956\pi$
$47$	−17559.2	+	17559.2i	−1.15947	+	1.15947i	−0.984589	+	0.174886i	$0.944044\pi$
$48$		0			0
$49$			45302.9i			2.69548i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−24150.5	−	24150.5i	−1.18096	−	1.18096i	−0.979495	−	0.201469i	$-0.935428\pi$
$53$	−24150.5	−	24150.5i	−1.18096	−	1.18096i	−0.201469	−	0.979495i	$-0.564572\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	9558.17			0.357474			0.178737	−	0.983897i	$-0.442799\pi$
$59$	9558.17			0.357474			0.178737	+	0.983897i	$0.442799\pi$
$60$		0			0
$61$	−19318.5			−0.664736			−0.332368	−	0.943150i	$-0.607848\pi$
$61$	−19318.5			−0.664736			−0.332368	+	0.943150i	$0.607848\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−13130.7	−	13130.7i	−0.357355	−	0.357355i	0.505482	−	0.862837i	$-0.331315\pi$
$67$	−13130.7	−	13130.7i	−0.357355	−	0.357355i	−0.862837	+	0.505482i	$0.831315\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			56336.8i			1.32631i	0.748481	+	0.663157i	$0.230784\pi$
$71$			56336.8i			1.32631i	−0.748481	+	0.663157i	$0.769216\pi$
$72$		0			0
$73$	6831.56	−	6831.56i	0.150042	−	0.150042i	−0.628095	−	0.778137i	$-0.716165\pi$
$73$	6831.56	−	6831.56i	0.150042	−	0.150042i	0.778137	+	0.628095i	$0.216165\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	29964.4	−	29964.4i	0.575942	−	0.575942i
$78$		0			0
$79$		−	4005.41i		−	0.0722070i	−0.999348	−	0.0361035i	$-0.988505\pi$
$79$		−	4005.41i		−	0.0722070i	0.999348	−	0.0361035i	$-0.0114946\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	17559.5	+	17559.5i	0.279780	+	0.279780i	0.833021	−	0.553241i	$-0.186609\pi$
$83$	17559.5	+	17559.5i	0.279780	+	0.279780i	−0.553241	+	0.833021i	$0.686609\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	112023.			1.49911			0.749555	−	0.661942i	$-0.230267\pi$
$89$	112023.			1.49911			0.749555	+	0.661942i	$0.230267\pi$
$90$		0			0
$91$	28218.3			0.357213
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−72995.8	−	72995.8i	−0.787714	−	0.787714i	0.193405	−	0.981119i	$-0.438047\pi$
$97$	−72995.8	−	72995.8i	−0.787714	−	0.787714i	−0.981119	+	0.193405i	$0.938047\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	107160.i		−	1.04527i	−0.852557	−	0.522635i	$-0.824949\pi$
$101$		−	107160.i		−	1.04527i	0.852557	−	0.522635i	$-0.175051\pi$
$102$		0			0
$103$	32531.0	−	32531.0i	0.302138	−	0.302138i	−0.539712	−	0.841850i	$-0.681467\pi$
$103$	32531.0	−	32531.0i	0.302138	−	0.302138i	0.841850	+	0.539712i	$0.181467\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	116018.	−	116018.i	0.979636	−	0.979636i	−0.0201608	−	0.999797i	$-0.506418\pi$
$107$	116018.	−	116018.i	0.979636	−	0.979636i	0.999797	+	0.0201608i	$0.00641781\pi$
$108$		0			0
$109$		−	221964.i		−	1.78943i	−0.446633	−	0.894717i	$-0.647377\pi$
$109$		−	221964.i		−	1.78943i	0.446633	−	0.894717i	$-0.352623\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−12580.5	−	12580.5i	−0.0926836	−	0.0926836i	0.659245	−	0.751928i	$-0.270876\pi$
$113$	−12580.5	−	12580.5i	−0.0926836	−	0.0926836i	−0.751928	+	0.659245i	$0.770876\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−250231.			−1.61984
$120$		0			0
$121$	132139.			0.820479
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	143588.	+	143588.i	0.789968	+	0.789968i	0.981489	−	0.191521i	$-0.0613419\pi$
$127$	143588.	+	143588.i	0.789968	+	0.789968i	−0.191521	+	0.981489i	$0.561342\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			198395.i			1.01007i	0.863098	+	0.505037i	$0.168521\pi$
$131$			198395.i			1.01007i	−0.863098	+	0.505037i	$0.831479\pi$
$132$		0			0
$133$	−393313.	+	393313.i	−1.92801	+	1.92801i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−16088.3	+	16088.3i	−0.0732332	+	0.0732332i	−0.742775	−	0.669541i	$-0.766491\pi$
$137$	−16088.3	+	16088.3i	−0.0732332	+	0.0732332i	0.669541	+	0.742775i	$0.266491\pi$
$138$		0			0
$139$			114813.i			0.504029i	0.967723	+	0.252015i	$0.0810931\pi$
$139$			114813.i			0.504029i	−0.967723	+	0.252015i	$0.918907\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−13613.7	−	13613.7i	−0.0556718	−	0.0556718i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−364604.			−1.34541			−0.672706	−	0.739909i	$-0.734868\pi$
$149$	−364604.			−1.34541			−0.672706	+	0.739909i	$0.734868\pi$
$150$		0			0
$151$	−263064.			−0.938900			−0.469450	−	0.882959i	$-0.655548\pi$
$151$	−263064.			−0.938900			−0.469450	+	0.882959i	$0.655548\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−77726.0	−	77726.0i	−0.251662	−	0.251662i	0.569990	−	0.821652i	$-0.306947\pi$
$157$	−77726.0	−	77726.0i	−0.251662	−	0.251662i	−0.821652	+	0.569990i	$0.806947\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	854350.i		−	2.59759i
$162$		0			0
$163$	−179100.	+	179100.i	−0.527990	+	0.527990i	−0.919973	−	0.391983i	$-0.871789\pi$
$163$	−179100.	+	179100.i	−0.527990	+	0.527990i	0.391983	+	0.919973i	$0.371789\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	40609.9	−	40609.9i	0.112679	−	0.112679i	−0.648520	−	0.761198i	$-0.724612\pi$
$167$	40609.9	−	40609.9i	0.112679	−	0.112679i	0.761198	+	0.648520i	$0.224612\pi$
$168$		0			0
$169$			358473.i			0.965471i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−313909.	−	313909.i	−0.797423	−	0.797423i	0.185266	−	0.982688i	$-0.440685\pi$
$173$	−313909.	−	313909.i	−0.797423	−	0.797423i	−0.982688	+	0.185266i	$0.940685\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−796710.			−1.85852			−0.929261	−	0.369425i	$-0.879555\pi$
$179$	−796710.			−1.85852			−0.929261	+	0.369425i	$0.879555\pi$
$180$		0			0
$181$	−396805.			−0.900287			−0.450144	−	0.892956i	$-0.648627\pi$
$181$	−396805.			−0.900287			−0.450144	+	0.892956i	$0.648627\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	120722.	+	120722.i	0.252453	+	0.252453i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	26227.3i		−	0.0520199i	−0.999662	−	0.0260100i	$-0.991720\pi$
$191$		−	26227.3i		−	0.0520199i	0.999662	−	0.0260100i	$-0.00828016\pi$
$192$		0			0
$193$	−132124.	+	132124.i	−0.255323	+	0.255323i	−0.823149	−	0.567826i	$-0.807785\pi$
$193$	−132124.	+	132124.i	−0.255323	+	0.255323i	0.567826	+	0.823149i	$0.307785\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	106552.	−	106552.i	0.195613	−	0.195613i	−0.602503	−	0.798116i	$-0.705830\pi$
$197$	106552.	−	106552.i	0.195613	−	0.195613i	0.798116	+	0.602503i	$0.205830\pi$
$198$		0			0
$199$			390342.i			0.698735i	0.936986	+	0.349368i	$0.113604\pi$
$199$			390342.i			0.698735i	−0.936986	+	0.349368i	$0.886396\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−757063.	−	757063.i	−1.28941	−	1.28941i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	379501.			0.600963
$210$		0			0
$211$	169966.			0.262818			0.131409	−	0.991328i	$-0.458050\pi$
$211$	169966.			0.262818			0.131409	+	0.991328i	$0.458050\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−1.06835e6	−	1.06835e6i	−1.54015	−	1.54015i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			113687.i			0.156578i
$222$		0			0
$223$	355461.	−	355461.i	0.478663	−	0.478663i	−0.426041	−	0.904704i	$-0.640092\pi$
$223$	355461.	−	355461.i	0.478663	−	0.478663i	0.904704	+	0.426041i	$0.140092\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	387258.	−	387258.i	0.498810	−	0.498810i	−0.412257	−	0.911067i	$-0.635260\pi$
$227$	387258.	−	387258.i	0.498810	−	0.498810i	0.911067	+	0.412257i	$0.135260\pi$
$228$		0			0
$229$			1.12442e6i			1.41690i	0.705759	+	0.708452i	$0.250606\pi$
$229$			1.12442e6i			1.41690i	−0.705759	+	0.708452i	$0.749394\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−949930.	−	949930.i	−1.14631	−	1.14631i	−0.987273	−	0.159037i	$-0.949161\pi$
$233$	−949930.	−	949930.i	−1.14631	−	1.14631i	−0.159037	−	0.987273i	$-0.550839\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	247064.			0.279778			0.139889	−	0.990167i	$-0.455325\pi$
$239$	247064.			0.279778			0.139889	+	0.990167i	$0.455325\pi$
$240$		0			0
$241$	−151301.			−0.167803			−0.0839013	−	0.996474i	$-0.526738\pi$
$241$	−151301.			−0.167803			−0.0839013	+	0.996474i	$0.526738\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	178694.	+	178694.i	0.186366	+	0.186366i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			1.06254e6i			1.06454i	0.846575	+	0.532269i	$0.178661\pi$
$251$			1.06254e6i			1.06454i	−0.846575	+	0.532269i	$0.821339\pi$
$252$		0			0
$253$	−412174.	+	412174.i	−0.404836	+	0.404836i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	685738.	−	685738.i	0.647628	−	0.647628i	−0.304792	−	0.952419i	$-0.598587\pi$
$257$	685738.	−	685738.i	0.647628	−	0.647628i	0.952419	+	0.304792i	$0.0985868\pi$
$258$		0			0
$259$			52106.4i			0.0482660i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−67005.2	−	67005.2i	−0.0597337	−	0.0597337i	0.676609	−	0.736343i	$-0.263449\pi$
$263$	−67005.2	−	67005.2i	−0.0597337	−	0.0597337i	−0.736343	+	0.676609i	$0.763449\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	614652.			0.517903			0.258952	−	0.965890i	$-0.416623\pi$
$269$	614652.			0.517903			0.258952	+	0.965890i	$0.416623\pi$
$270$		0			0
$271$	−2.08682e6			−1.72609			−0.863043	−	0.505130i	$-0.831444\pi$
$271$	−2.08682e6			−1.72609			−0.863043	+	0.505130i	$0.831444\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−1.41387e6	−	1.41387e6i	−1.10716	−	1.10716i	−0.993522	−	0.113640i	$-0.963749\pi$
$277$	−1.41387e6	−	1.41387e6i	−1.10716	−	1.10716i	−0.113640	−	0.993522i	$-0.536251\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	1.99827e6i		−	1.50969i	−0.655904	−	0.754844i	$-0.727712\pi$
$281$		−	1.99827e6i		−	1.50969i	0.655904	−	0.754844i	$-0.272288\pi$
$282$		0			0
$283$	541964.	−	541964.i	0.402258	−	0.402258i	−0.476770	−	0.879028i	$-0.658193\pi$
$283$	541964.	−	541964.i	0.402258	−	0.402258i	0.879028	+	0.476770i	$0.158193\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	1.94788e6	−	1.94788e6i	1.39591	−	1.39591i
$288$		0			0
$289$			411717.i			0.289971i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−933218.	−	933218.i	−0.635059	−	0.635059i	0.314274	−	0.949332i	$-0.398239\pi$
$293$	−933218.	−	933218.i	−0.635059	−	0.635059i	−0.949332	+	0.314274i	$0.898239\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−388156.			−0.251089
$300$		0			0
$301$	3.49849e6			2.22569
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−857534.	−	857534.i	−0.519285	−	0.519285i	0.398070	−	0.917355i	$-0.369680\pi$
$307$	−857534.	−	857534.i	−0.519285	−	0.519285i	−0.917355	+	0.398070i	$0.869680\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	701232.i		−	0.411112i	−0.978645	−	0.205556i	$-0.934100\pi$
$311$		−	701232.i		−	0.411112i	0.978645	−	0.205556i	$-0.0659003\pi$
$312$		0			0
$313$	−2.09715e6	+	2.09715e6i	−1.20995	+	1.20995i	−0.238913	+	0.971041i	$0.576791\pi$
$313$	−2.09715e6	+	2.09715e6i	−1.20995	+	1.20995i	−0.971041	+	0.238913i	$0.923209\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−1.61379e6	+	1.61379e6i	−0.901983	+	0.901983i	−0.995608	−	0.0936246i	$-0.970155\pi$
$317$	−1.61379e6	+	1.61379e6i	−0.901983	+	0.901983i	0.0936246	+	0.995608i	$0.470155\pi$
$318$		0			0
$319$			730477.i			0.401911i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−1.58460e6	−	1.58460e6i	−0.845108	−	0.845108i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−6.18873e6			−3.15218
$330$		0			0
$331$	−3.20310e6			−1.60694			−0.803471	−	0.595344i	$-0.797016\pi$
$331$	−3.20310e6			−1.60694			−0.803471	+	0.595344i	$0.797016\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	2.38594e6	+	2.38594e6i	1.14442	+	1.14442i	0.987633	+	0.156786i	$0.0501133\pi$
$337$	2.38594e6	+	2.38594e6i	1.14442	+	1.14442i	0.156786	+	0.987633i	$0.449887\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			1.03083e6i			0.480067i
$342$		0			0
$343$	−5.02166e6	+	5.02166e6i	−2.30469	+	2.30469i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	1.15353e6	−	1.15353e6i	0.514285	−	0.514285i	−0.401551	−	0.915836i	$-0.631529\pi$
$347$	1.15353e6	−	1.15353e6i	0.514285	−	0.514285i	0.915836	+	0.401551i	$0.131529\pi$
$348$		0			0
$349$		−	604337.i		−	0.265592i	−0.991143	−	0.132796i	$-0.957604\pi$
$349$		−	604337.i		−	0.265592i	0.991143	−	0.132796i	$-0.0423956\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	618215.	+	618215.i	0.264060	+	0.264060i	0.826701	−	0.562641i	$-0.190215\pi$
$353$	618215.	+	618215.i	0.264060	+	0.264060i	−0.562641	+	0.826701i	$0.690215\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−4.16504e6			−1.70562			−0.852812	−	0.522218i	$-0.825105\pi$
$359$	−4.16504e6			−1.70562			−0.852812	+	0.522218i	$0.825105\pi$
$360$		0			0
$361$	−2.50525e6			−1.01177
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	454401.	+	454401.i	0.176106	+	0.176106i	0.789656	−	0.613550i	$-0.210259\pi$
$367$	454401.	+	454401.i	0.176106	+	0.176106i	−0.613550	+	0.789656i	$0.710259\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	8.51181e6i		−	3.21061i
$372$		0			0
$373$	1.62195e6	−	1.62195e6i	0.603621	−	0.603621i	−0.337651	−	0.941271i	$-0.609632\pi$
$373$	1.62195e6	−	1.62195e6i	0.603621	−	0.603621i	0.941271	+	0.337651i	$0.109632\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−343956.	+	343956.i	−0.124638	+	0.124638i
$378$		0			0
$379$		−	995454.i		−	0.355978i	−0.984032	−	0.177989i	$-0.943041\pi$
$379$		−	995454.i		−	0.355978i	0.984032	−	0.177989i	$-0.0569592\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−2.30324e6	−	2.30324e6i	−0.802310	−	0.802310i	0.181146	−	0.983456i	$-0.442019\pi$
$383$	−2.30324e6	−	2.30324e6i	−0.802310	−	0.802310i	−0.983456	+	0.181146i	$0.942019\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−801902.			−0.268687			−0.134344	−	0.990935i	$-0.542893\pi$
$389$	−801902.			−0.268687			−0.134344	+	0.990935i	$0.542893\pi$
$390$		0			0
$391$	3.44204e6			1.13861
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	956519.	+	956519.i	0.304591	+	0.304591i	0.842807	−	0.538216i	$-0.180901\pi$
$397$	956519.	+	956519.i	0.304591	+	0.304591i	−0.538216	+	0.842807i	$0.680901\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			4.13097e6i			1.28290i	0.767166	+	0.641448i	$0.221666\pi$
$401$			4.13097e6i			1.28290i	−0.767166	+	0.641448i	$0.778334\pi$
$402$		0			0
$403$	−485382.	+	485382.i	−0.148875	+	0.148875i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	25138.3	−	25138.3i	0.00752228	−	0.00752228i
$408$		0			0
$409$			5.32990e6i			1.57547i	0.616013	+	0.787736i	$0.288747\pi$
$409$			5.32990e6i			1.57547i	−0.616013	+	0.787736i	$0.711253\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	1.68438e6	+	1.68438e6i	0.485920	+	0.485920i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	3.73508e6			1.03936			0.519679	−	0.854362i	$-0.326052\pi$
$419$	3.73508e6			1.03936			0.519679	+	0.854362i	$0.326052\pi$
$420$		0			0
$421$	543447.			0.149435			0.0747174	−	0.997205i	$-0.476195\pi$
$421$	543447.			0.149435			0.0747174	+	0.997205i	$0.476195\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−3.40439e6	−	3.40439e6i	−0.903586	−	0.903586i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			177629.i			0.0460595i	0.999735	+	0.0230298i	$0.00733125\pi$
$431$			177629.i			0.0460595i	−0.999735	+	0.0230298i	$0.992669\pi$
$432$		0			0
$433$	−1.02209e6	+	1.02209e6i	−0.261981	+	0.261981i	−0.825859	−	0.563877i	$-0.809309\pi$
$433$	−1.02209e6	+	1.02209e6i	−0.261981	+	0.261981i	0.563877	+	0.825859i	$0.309309\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	5.41021e6	−	5.41021e6i	1.35522	−	1.35522i
$438$		0			0
$439$		−	4.99207e6i		−	1.23629i	−0.786065	−	0.618144i	$-0.787885\pi$
$439$		−	4.99207e6i		−	1.23629i	0.786065	−	0.618144i	$-0.212115\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−455320.	−	455320.i	−0.110232	−	0.110232i	0.649839	−	0.760071i	$-0.274836\pi$
$443$	−455320.	−	455320.i	−0.110232	−	0.110232i	−0.760071	+	0.649839i	$0.774836\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−879068.			−0.205782			−0.102891	−	0.994693i	$-0.532809\pi$
$449$	−879068.			−0.205782			−0.102891	+	0.994693i	$0.532809\pi$
$450$		0			0
$451$	−1.87948e6			−0.435107
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−383548.	−	383548.i	−0.0859070	−	0.0859070i	0.662847	−	0.748754i	$-0.269348\pi$
$457$	−383548.	−	383548.i	−0.0859070	−	0.0859070i	−0.748754	+	0.662847i	$0.769348\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			3.53736e6i			0.775223i	0.921823	+	0.387611i	$0.126700\pi$
$461$			3.53736e6i			0.775223i	−0.921823	+	0.387611i	$0.873300\pi$
$462$		0			0
$463$	4.04433e6	−	4.04433e6i	0.876787	−	0.876787i	−0.116414	−	0.993201i	$-0.537140\pi$
$463$	4.04433e6	−	4.04433e6i	0.876787	−	0.876787i	0.993201	+	0.116414i	$0.0371400\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−1.42695e6	+	1.42695e6i	−0.302772	+	0.302772i	−0.842098	−	0.539325i	$-0.818679\pi$
$467$	−1.42695e6	+	1.42695e6i	−0.302772	+	0.302772i	0.539325	+	0.842098i	$0.318679\pi$
$468$		0			0
$469$		−	4.62788e6i		−	0.971516i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−1.68782e6	−	1.68782e6i	−0.346874	−	0.346874i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	8.25388e6			1.64369			0.821844	−	0.569713i	$-0.192946\pi$
$479$	8.25388e6			1.64369			0.821844	+	0.569713i	$0.192946\pi$
$480$		0			0
$481$	23673.4			0.00466551
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−4.22599e6	−	4.22599e6i	−0.807433	−	0.807433i	0.176812	−	0.984245i	$-0.443422\pi$
$487$	−4.22599e6	−	4.22599e6i	−0.807433	−	0.807433i	−0.984245	+	0.176812i	$0.943422\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			5.25054e6i			0.982879i	0.870912	+	0.491440i	$0.163529\pi$
$491$			5.25054e6i			0.982879i	−0.870912	+	0.491440i	$0.836471\pi$
$492$		0			0
$493$	3.05009e6	−	3.05009e6i	0.565191	−	0.565191i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−9.92790e6	+	9.92790e6i	−1.80288	+	1.80288i
$498$		0			0
$499$			8.54107e6i			1.53554i	0.640726	+	0.767770i	$0.278633\pi$
$499$			8.54107e6i			1.53554i	−0.640726	+	0.767770i	$0.721367\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	3.22171e6	+	3.22171e6i	0.567762	+	0.567762i	0.931501	−	0.363739i	$-0.118500\pi$
$503$	3.22171e6	+	3.22171e6i	0.567762	+	0.567762i	−0.363739	+	0.931501i	$0.618500\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	1.03259e7			1.76658			0.883290	−	0.468827i	$-0.155323\pi$
$509$	1.03259e7			1.76658			0.883290	+	0.468827i	$0.155323\pi$
$510$		0			0
$511$	2.40777e6			0.407909
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	2.98570e6	+	2.98570e6i	0.491269	+	0.491269i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			4.36975e6i			0.705282i	0.935759	+	0.352641i	$0.114716\pi$
$521$			4.36975e6i			0.705282i	−0.935759	+	0.352641i	$0.885284\pi$
$522$		0			0
$523$	6.98390e6	−	6.98390e6i	1.11646	−	1.11646i	0.124206	−	0.992256i	$-0.460362\pi$
$523$	6.98390e6	−	6.98390e6i	1.11646	−	1.11646i	0.992256	−	0.124206i	$-0.0396383\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	4.30421e6	−	4.30421e6i	0.675097	−	0.675097i
$528$		0			0
$529$			5.31563e6i			0.825877i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−884979.	−	884979.i	−0.134932	−	0.134932i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	7.70310e6			1.14207
$540$		0			0
$541$	−345047.			−0.0506857			−0.0253428	−	0.999679i	$-0.508068\pi$
$541$	−345047.			−0.0506857			−0.0253428	+	0.999679i	$0.508068\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−2.33286e6	−	2.33286e6i	−0.333366	−	0.333366i	0.520497	−	0.853863i	$-0.325747\pi$
$547$	−2.33286e6	−	2.33286e6i	−0.333366	−	0.333366i	−0.853863	+	0.520497i	$0.825747\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	9.58827e6i		−	1.34543i
$552$		0			0
$553$	705850.	−	705850.i	0.0981521	−	0.0981521i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	8.18880e6	−	8.18880e6i	1.11836	−	1.11836i	0.126380	−	0.991982i	$-0.459664\pi$
$557$	8.18880e6	−	8.18880e6i	1.11836	−	1.11836i	0.991982	−	0.126380i	$-0.0403357\pi$
$558$		0			0
$559$		−	1.58946e6i		−	0.215140i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	7.16576e6	+	7.16576e6i	0.952777	+	0.952777i	0.998934	−	0.0461569i	$-0.0146974\pi$
$563$	7.16576e6	+	7.16576e6i	0.952777	+	0.952777i	−0.0461569	+	0.998934i	$0.514697\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−7.00369e6			−0.906873			−0.453436	−	0.891289i	$-0.649802\pi$
$569$	−7.00369e6			−0.906873			−0.453436	+	0.891289i	$0.649802\pi$
$570$		0			0
$571$	1.26045e6			0.161784			0.0808919	−	0.996723i	$-0.474223\pi$
$571$	1.26045e6			0.161784			0.0808919	+	0.996723i	$0.474223\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	9.09434e6	+	9.09434e6i	1.13719	+	1.13719i	0.988952	+	0.148234i	$0.0473589\pi$
$577$	9.09434e6	+	9.09434e6i	1.13719	+	1.13719i	0.148234	+	0.988952i	$0.452641\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			6.18881e6i			0.760619i
$582$		0			0
$583$	−4.10645e6	+	4.10645e6i	−0.500374	+	0.500374i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−1.16175e7	+	1.16175e7i	−1.39161	+	1.39161i	−0.569878	+	0.821729i	$0.693010\pi$
$587$	−1.16175e7	+	1.16175e7i	−1.39161	+	1.39161i	−0.821729	+	0.569878i	$0.806990\pi$
$588$		0			0
$589$		−	1.35307e7i		−	1.60706i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−9.62275e6	−	9.62275e6i	−1.12373	−	1.12373i	−0.991175	−	0.132556i	$-0.957682\pi$
$593$	−9.62275e6	−	9.62275e6i	−1.12373	−	1.12373i	−0.132556	−	0.991175i	$-0.542318\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−1.51231e7			−1.72216			−0.861079	−	0.508472i	$-0.830211\pi$
$599$	−1.51231e7			−1.72216			−0.861079	+	0.508472i	$0.830211\pi$
$600$		0			0
$601$	−8.88152e6			−1.00300			−0.501500	−	0.865157i	$-0.667218\pi$
$601$	−8.88152e6			−1.00300			−0.501500	+	0.865157i	$0.667218\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	2.05434e6	+	2.05434e6i	0.226308	+	0.226308i	0.811148	−	0.584840i	$-0.198843\pi$
$607$	2.05434e6	+	2.05434e6i	0.226308	+	0.226308i	−0.584840	+	0.811148i	$0.698843\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			2.81172e6i			0.304697i
$612$		0			0
$613$	6.07408e6	−	6.07408e6i	0.652874	−	0.652874i	−0.300810	−	0.953684i	$-0.597257\pi$
$613$	6.07408e6	−	6.07408e6i	0.652874	−	0.652874i	0.953684	+	0.300810i	$0.0972569\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1.28178e7	+	1.28178e7i	−1.35550	+	1.35550i	−0.476122	+	0.879379i	$0.657958\pi$
$617$	−1.28178e7	+	1.28178e7i	−1.35550	+	1.35550i	−0.879379	+	0.476122i	$0.842042\pi$
$618$		0			0
$619$			3.58499e6i			0.376063i	0.982163	+	0.188032i	$0.0602108\pi$
$619$			3.58499e6i			0.376063i	−0.982163	+	0.188032i	$0.939789\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	1.97412e7	+	1.97412e7i	2.03776	+	2.03776i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−209928.			−0.0211565
$630$		0			0
$631$	1.33735e7			1.33713			0.668563	−	0.743655i	$-0.266910\pi$
$631$	1.33735e7			1.33713			0.668563	+	0.743655i	$0.266910\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	3.62712e6	+	3.62712e6i	0.354171	+	0.354171i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	1.30991e7i		−	1.25920i	−0.776918	−	0.629602i	$-0.783218\pi$
$641$		−	1.30991e7i		−	1.25920i	0.776918	−	0.629602i	$-0.216782\pi$
$642$		0			0
$643$	−3.40612e6	+	3.40612e6i	−0.324887	+	0.324887i	−0.850638	−	0.525751i	$-0.823784\pi$
$643$	−3.40612e6	+	3.40612e6i	−0.324887	+	0.324887i	0.525751	+	0.850638i	$0.323784\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−8.40971e6	+	8.40971e6i	−0.789806	+	0.789806i	−0.981462	−	0.191656i	$-0.938614\pi$
$647$	−8.40971e6	+	8.40971e6i	−0.789806	+	0.789806i	0.191656	+	0.981462i	$0.438614\pi$
$648$		0			0
$649$		−	1.62523e6i		−	0.151462i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	6.31448e6	+	6.31448e6i	0.579502	+	0.579502i	0.934766	−	0.355264i	$-0.115609\pi$
$653$	6.31448e6	+	6.31448e6i	0.579502	+	0.579502i	−0.355264	+	0.934766i	$0.615609\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	7.22171e6			0.647778			0.323889	−	0.946095i	$-0.395010\pi$
$659$	7.22171e6			0.647778			0.323889	+	0.946095i	$0.395010\pi$
$660$		0			0
$661$	−33624.7			−0.00299333			−0.00149666	−	0.999999i	$-0.500476\pi$
$661$	−33624.7			−0.00299333			−0.00149666	+	0.999999i	$0.500476\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	1.04138e7	+	1.04138e7i	0.906343	+	0.906343i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			3.28484e6i			0.281649i
$672$		0			0
$673$	−2.78019e6	+	2.78019e6i	−0.236612	+	0.236612i	−0.815446	−	0.578833i	$-0.803508\pi$
$673$	−2.78019e6	+	2.78019e6i	−0.236612	+	0.236612i	0.578833	+	0.815446i	$0.303508\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	1.80501e6	−	1.80501e6i	0.151359	−	0.151359i	−0.627366	−	0.778725i	$-0.715867\pi$
$677$	1.80501e6	−	1.80501e6i	0.151359	−	0.151359i	0.778725	+	0.627366i	$0.215867\pi$
$678$		0			0
$679$		−	2.57273e7i		−	2.14151i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	889355.	+	889355.i	0.0729497	+	0.0729497i	0.742640	−	0.669691i	$-0.233573\pi$
$683$	889355.	+	889355.i	0.0729497	+	0.0729497i	−0.669691	+	0.742640i	$0.733573\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−3.86716e6			−0.310344
$690$		0			0
$691$	1.15778e7			0.922426			0.461213	−	0.887289i	$-0.347414\pi$
$691$	1.15778e7			0.922426			0.461213	+	0.887289i	$0.347414\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	7.84771e6	+	7.84771e6i	0.611872	+	0.611872i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			1.17918e7i			0.906324i	0.891428	+	0.453162i	$0.149704\pi$
$701$			1.17918e7i			0.906324i	−0.891428	+	0.453162i	$0.850296\pi$
$702$		0			0
$703$	−329966.	+	329966.i	−0.0251815	+	0.0251815i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1.88841e7	−	1.88841e7i	1.42085	−	1.42085i
$708$		0			0
$709$		−	1.24482e7i		−	0.930014i	−0.885307	−	0.465007i	$-0.846052\pi$
$709$		−	1.24482e7i		−	0.930014i	0.885307	−	0.465007i	$-0.153948\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	1.46956e7	+	1.46956e7i	1.08259	+	1.08259i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	7.37065e6			0.531721			0.265860	−	0.964012i	$-0.414344\pi$
$719$	7.37065e6			0.531721			0.265860	+	0.964012i	$0.414344\pi$
$720$		0			0
$721$	1.14655e7			0.821401
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−5.03706e6	−	5.03706e6i	−0.353460	−	0.353460i	0.507935	−	0.861395i	$-0.330409\pi$
$727$	−5.03706e6	−	5.03706e6i	−0.353460	−	0.353460i	−0.861395	+	0.507935i	$0.830409\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			1.40949e7i			0.975589i
$732$		0			0
$733$	−8.88954e6	+	8.88954e6i	−0.611110	+	0.611110i	−0.943235	−	0.332125i	$-0.892234\pi$
$733$	−8.88954e6	+	8.88954e6i	−0.611110	+	0.611110i	0.332125	+	0.943235i	$0.392234\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−2.23268e6	+	2.23268e6i	−0.151411	+	0.151411i
$738$		0			0
$739$			9.89686e6i			0.666633i	0.942815	+	0.333316i	$0.108168\pi$
$739$			9.89686e6i			0.666633i	−0.942815	+	0.333316i	$0.891832\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	8.12361e6	+	8.12361e6i	0.539855	+	0.539855i	0.923486	−	0.383632i	$-0.125327\pi$
$743$	8.12361e6	+	8.12361e6i	0.539855	+	0.539855i	−0.383632	+	0.923486i	$0.625327\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	4.08902e7			2.66327
$750$		0			0
$751$	−1.63486e7			−1.05774			−0.528871	−	0.848702i	$-0.677385\pi$
$751$	−1.63486e7			−1.05774			−0.528871	+	0.848702i	$0.677385\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−1.28025e7	−	1.28025e7i	−0.811995	−	0.811995i	0.172937	−	0.984933i	$-0.444674\pi$
$757$	−1.28025e7	−	1.28025e7i	−0.811995	−	0.811995i	−0.984933	+	0.172937i	$0.944674\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			6.50168e6i			0.406971i	0.979078	+	0.203486i	$0.0652270\pi$
$761$			6.50168e6i			0.406971i	−0.979078	+	0.203486i	$0.934773\pi$
$762$		0			0
$763$	3.91154e7	−	3.91154e7i	2.43241	−	2.43241i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	765263.	−	765263.i	0.0469702	−	0.0469702i
$768$		0			0
$769$		−	1.08169e7i		−	0.659607i	−0.944050	−	0.329803i	$-0.893018\pi$
$769$		−	1.08169e7i		−	0.659607i	0.944050	−	0.329803i	$-0.106982\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	1.85198e7	+	1.85198e7i	1.11477	+	1.11477i	0.992496	+	0.122279i	$0.0390201\pi$
$773$	1.85198e7	+	1.85198e7i	1.11477	+	1.11477i	0.122279	+	0.992496i	$0.460980\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	2.46701e7			1.45656
$780$		0			0
$781$	9.57925e6			0.561958
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−1.75968e7	−	1.75968e7i	−1.01274	−	1.01274i	−0.999918	−	0.0128194i	$-0.995919\pi$
$787$	−1.75968e7	−	1.75968e7i	−1.01274	−	1.01274i	−0.0128194	−	0.999918i	$-0.504081\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	4.43399e6i		−	0.251973i
$792$		0			0
$793$	−1.54671e6	+	1.54671e6i	−0.0873427	+	0.0873427i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−2.37966e7	+	2.37966e7i	−1.32700	+	1.32700i	−0.419020	+	0.907977i	$0.637626\pi$
$797$	−2.37966e7	+	2.37966e7i	−1.32700	+	1.32700i	−0.907977	+	0.419020i	$0.862374\pi$
$798$		0			0
$799$		−	2.49334e7i		−	1.38170i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1.16161e6	−	1.16161e6i	−0.0635727	−	0.0635727i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−7.25878e6			−0.389935			−0.194967	−	0.980810i	$-0.562460\pi$
$809$	−7.25878e6			−0.389935			−0.194967	+	0.980810i	$0.562460\pi$
$810$		0			0
$811$	5.25301e6			0.280450			0.140225	−	0.990120i	$-0.455217\pi$
$811$	5.25301e6			0.280450			0.140225	+	0.990120i	$0.455217\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	2.21543e7	+	2.21543e7i	1.16119	+	1.16119i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			2.31946e7i			1.20096i	0.799640	+	0.600480i	$0.205024\pi$
$821$			2.31946e7i			1.20096i	−0.799640	+	0.600480i	$0.794976\pi$
$822$		0			0
$823$	6.04657e6	−	6.04657e6i	0.311178	−	0.311178i	−0.534188	−	0.845366i	$-0.679382\pi$
$823$	6.04657e6	−	6.04657e6i	0.311178	−	0.311178i	0.845366	+	0.534188i	$0.179382\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	4.11793e6	−	4.11793e6i	0.209370	−	0.209370i	−0.594630	−	0.804000i	$-0.702701\pi$
$827$	4.11793e6	−	4.11793e6i	0.209370	−	0.209370i	0.804000	+	0.594630i	$0.202701\pi$
$828$		0			0
$829$			1.80855e7i			0.913998i	0.889467	+	0.456999i	$0.151076\pi$
$829$			1.80855e7i			0.913998i	−0.889467	+	0.456999i	$0.848924\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−3.21641e7	−	3.21641e7i	−1.60605	−	1.60605i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−1.22177e7			−0.599219			−0.299610	−	0.954062i	$-0.596856\pi$
$839$	−1.22177e7			−0.599219			−0.299610	+	0.954062i	$0.596856\pi$
$840$		0			0
$841$	−2.05532e6			−0.100205
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	2.32861e7	+	2.32861e7i	1.11529	+	1.11529i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	716747.i		−	0.0339268i
$852$		0			0
$853$	9.27358e6	−	9.27358e6i	0.436390	−	0.436390i	−0.454405	−	0.890795i	$-0.650148\pi$
$853$	9.27358e6	−	9.27358e6i	0.436390	−	0.436390i	0.890795	+	0.454405i	$0.150148\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−3.24386e6	+	3.24386e6i	−0.150872	+	0.150872i	−0.778508	−	0.627635i	$-0.784023\pi$
$857$	−3.24386e6	+	3.24386e6i	−0.150872	+	0.150872i	0.627635	+	0.778508i	$0.284023\pi$
$858$		0			0
$859$			2.14498e7i			0.991836i	0.868369	+	0.495918i	$0.165168\pi$
$859$			2.14498e7i			0.991836i	−0.868369	+	0.495918i	$0.834832\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	1.39674e7	+	1.39674e7i	0.638396	+	0.638396i	0.950160	−	0.311764i	$-0.100920\pi$
$863$	1.39674e7	+	1.39674e7i	0.638396	+	0.638396i	−0.311764	+	0.950160i	$0.600920\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−681062.			−0.0305941
$870$		0			0
$871$	−2.10258e6			−0.0939090
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−2.04528e7	−	2.04528e7i	−0.897955	−	0.897955i	0.0973002	−	0.995255i	$-0.468979\pi$
$877$	−2.04528e7	−	2.04528e7i	−0.897955	−	0.897955i	−0.995255	+	0.0973002i	$0.968979\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	9.48325e6i		−	0.411640i	−0.978590	−	0.205820i	$-0.934014\pi$
$881$		−	9.48325e6i		−	0.411640i	0.978590	−	0.205820i	$-0.0659861\pi$
$882$		0			0
$883$	1.14074e7	−	1.14074e7i	0.492364	−	0.492364i	−0.416686	−	0.909050i	$-0.636809\pi$
$883$	1.14074e7	−	1.14074e7i	0.492364	−	0.492364i	0.909050	+	0.416686i	$0.136809\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1.37533e7	−	1.37533e7i	0.586947	−	0.586947i	−0.349857	−	0.936803i	$-0.613770\pi$
$887$	1.37533e7	−	1.37533e7i	0.586947	−	0.586947i	0.936803	+	0.349857i	$0.113770\pi$
$888$		0			0
$889$			5.06074e7i			2.14763i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−3.91904e7	−	3.91904e7i	−1.64456	−	1.64456i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	2.60444e7			1.07477
$900$		0			0
$901$	3.42927e7			1.40731
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−2.44189e7	−	2.44189e7i	−0.985616	−	0.985616i	0.0142823	−	0.999898i	$-0.495454\pi$
$907$	−2.44189e7	−	2.44189e7i	−0.985616	−	0.985616i	−0.999898	+	0.0142823i	$0.995454\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	1.90466e7i		−	0.760365i	−0.924911	−	0.380182i	$-0.875861\pi$
$911$		−	1.90466e7i		−	0.760365i	0.924911	−	0.380182i	$-0.124139\pi$
$912$		0			0
$913$	2.98574e6	−	2.98574e6i	0.118543	−	0.118543i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−3.49620e7	+	3.49620e7i	−1.37301	+	1.37301i
$918$		0			0
$919$			1.18840e7i			0.464165i	0.972696	+	0.232083i	$0.0745540\pi$
$919$			1.18840e7i			0.464165i	−0.972696	+	0.232083i	$0.925446\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	4.51053e6	+	4.51053e6i	0.174270	+	0.174270i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−3.70471e7			−1.40836			−0.704182	−	0.710020i	$-0.748686\pi$
$929$	−3.70471e7			−1.40836			−0.704182	+	0.710020i	$0.748686\pi$
$930$		0			0
$931$	−1.01111e8			−3.82318
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	4.99379e6	+	4.99379e6i	0.185815	+	0.185815i	0.793884	−	0.608069i	$-0.208056\pi$
$937$	4.99379e6	+	4.99379e6i	0.185815	+	0.185815i	−0.608069	+	0.793884i	$0.708056\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	7.41829e6i		−	0.273105i	−0.990633	−	0.136553i	$-0.956398\pi$
$941$		−	7.41829e6i		−	0.273105i	0.990633	−	0.136553i	$-0.0436022\pi$
$942$		0			0
$943$	−2.67940e7	+	2.67940e7i	−0.981202	+	0.981202i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	4.37246e6	−	4.37246e6i	0.158435	−	0.158435i	−0.623438	−	0.781873i	$-0.714265\pi$
$947$	4.37246e6	−	4.37246e6i	0.158435	−	0.158435i	0.781873	+	0.623438i	$0.214265\pi$
$948$		0			0
$949$		−	1.09392e6i		−	0.0394294i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1.47754e7	+	1.47754e7i	0.526994	+	0.526994i	0.919675	−	0.392681i	$-0.128452\pi$
$953$	1.47754e7	+	1.47754e7i	0.526994	+	0.526994i	−0.392681	+	0.919675i	$0.628452\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−5.67028e6			−0.199094
$960$		0			0
$961$	8.12406e6			0.283769
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	1.57336e7	+	1.57336e7i	0.541081	+	0.541081i	0.923846	−	0.382765i	$-0.125028\pi$
$967$	1.57336e7	+	1.57336e7i	0.541081	+	0.541081i	−0.382765	+	0.923846i	$0.625028\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	1.57310e7i		−	0.535438i	−0.963497	−	0.267719i	$-0.913730\pi$
$971$		−	1.57310e7i		−	0.535438i	0.963497	−	0.267719i	$-0.0862699\pi$
$972$		0			0
$973$	−2.02329e7	+	2.02329e7i	−0.685135	+	0.685135i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1.44882e7	+	1.44882e7i	−0.485599	+	0.485599i	−0.906914	−	0.421315i	$-0.861569\pi$
$977$	−1.44882e7	+	1.44882e7i	−0.485599	+	0.485599i	0.421315	+	0.906914i	$0.361569\pi$
$978$		0			0
$979$		−	1.90480e7i		−	0.635173i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1.54292e7	−	1.54292e7i	−0.509283	−	0.509283i	0.405024	−	0.914306i	$-0.367263\pi$
$983$	−1.54292e7	−	1.54292e7i	−0.509283	−	0.509283i	−0.914306	+	0.405024i	$0.867263\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−4.81233e7			−1.56446
$990$		0			0
$991$	−2.07735e7			−0.671934			−0.335967	−	0.941874i	$-0.609063\pi$
$991$	−2.07735e7			−0.671934			−0.335967	+	0.941874i	$0.609063\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	2.84027e7	+	2.84027e7i	0.904943	+	0.904943i	0.995859	−	0.0909159i	$-0.0289794\pi$
$997$	2.84027e7	+	2.84027e7i	0.904943	+	0.904943i	−0.0909159	+	0.995859i	$0.528979\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	900.6.j.b.557.9		20
3.2	odd	2	inner	900.6.j.b.557.10		20
5.2	odd	4		180.6.j.a.53.9	yes	20
5.3	odd	4	inner	900.6.j.b.593.9		20
5.4	even	2		180.6.j.a.17.2	✓	20
15.2	even	4		180.6.j.a.53.2	yes	20
15.8	even	4	inner	900.6.j.b.593.10		20
15.14	odd	2		180.6.j.a.17.9	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.6.j.a.17.2	✓	20	5.4	even	2
180.6.j.a.17.9	yes	20	15.14	odd	2
180.6.j.a.53.2	yes	20	15.2	even	4
180.6.j.a.53.9	yes	20	5.2	odd	4
900.6.j.b.557.9		20	1.1	even	1	trivial
900.6.j.b.557.10		20	3.2	odd	2	inner
900.6.j.b.593.9		20	5.3	odd	4	inner
900.6.j.b.593.10		20	15.8	even	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)

\(f(q)\)	\(=\)	\(q+(176.224 + 176.224i) q^{7} +O(q^{10})\)	\(q+(176.224 + 176.224i) q^{7} -170.036i q^{11} +(80.0637 - 80.0637i) q^{13} +(-709.979 + 709.979i) q^{17} +2231.89i q^{19} +(-2424.04 - 2424.04i) q^{23} -4296.03 q^{29} -6062.44 q^{31} +(147.841 + 147.841i) q^{37} -11053.4i q^{41} +(9926.25 - 9926.25i) q^{43} +(-17559.2 + 17559.2i) q^{47} +45302.9i q^{49} +(-24150.5 - 24150.5i) q^{53} +9558.17 q^{59} -19318.5 q^{61} +(-13130.7 - 13130.7i) q^{67} +56336.8i q^{71} +(6831.56 - 6831.56i) q^{73} +(29964.4 - 29964.4i) q^{77} -4005.41i q^{79} +(17559.5 + 17559.5i) q^{83} +112023. q^{89} +28218.3 q^{91} +(-72995.8 - 72995.8i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 152 q^{7}+O(q^{10}) \) 20 * q + 152 * q^7	\( 20 q + 152 q^{7} - 252 q^{13} + 9440 q^{31} - 29988 q^{37} - 6720 q^{43} + 67200 q^{61} - 139552 q^{67} - 134828 q^{73} - 220560 q^{91} - 207132 q^{97}+O(q^{100}) \) 20 * q + 152 * q^7 - 252 * q^13 + 9440 * q^31 - 29988 * q^37 - 6720 * q^43 + 67200 * q^61 - 139552 * q^67 - 134828 * q^73 - 220560 * q^91 - 207132 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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