LMFDB - Embedded newform 900.6.j.a.557.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("900.557");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	900.j (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$144.345437832$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 16112194x^{12} + 72373894590801x^{8} + 60780662400876311824x^{4} + 14178241191207403341807616 $ x^16 + 16112194x^12 + 72373894590801x^8 + 60780662400876311824*x^4 + 14178241191207403341807616
Coefficient ring:	$\Z[a_1, \ldots, a_{49}]$
Coefficient ring index:	$ 2^{26}\cdot 3^{24} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			557.2
Root			$-37.4215 - 37.4215i$ of defining polynomial
Character	$\chi$	$=$	900.557
Dual form			900.6.j.a.593.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-104.844 - 104.844i) q^{7} +O(q^{10})$	$q+(-104.844 - 104.844i) q^{7} +421.820i q^{11} +(459.817 - 459.817i) q^{13} +(-535.895 + 535.895i) q^{17} -1817.63i q^{19} +(2931.55 + 2931.55i) q^{23} -3145.46 q^{29} -3731.26 q^{31} +(8266.20 + 8266.20i) q^{37} -2489.14i q^{41} +(5547.73 - 5547.73i) q^{43} +(-18661.1 + 18661.1i) q^{47} +5177.53i q^{49} +(-5831.36 - 5831.36i) q^{53} +31760.5 q^{59} +25216.1 q^{61} +(-6883.71 - 6883.71i) q^{67} -8320.46i q^{71} +(-27112.6 + 27112.6i) q^{73} +(44225.3 - 44225.3i) q^{77} +3779.58i q^{79} +(23893.1 + 23893.1i) q^{83} -11869.1 q^{89} -96418.2 q^{91} +(-69598.6 - 69598.6i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 5632 q^{31} - 119200 q^{61} - 138048 q^{91}+O(q^{100}) $ 16 * q + 5632 * q^31 - 119200 * q^61 - 138048 * q^91

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−104.844	−	104.844i	−0.808721	−	0.808721i	0.175720	−	0.984440i	$-0.443775\pi$
$7$	−104.844	−	104.844i	−0.808721	−	0.808721i	−0.984440	+	0.175720i	$0.943775\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			421.820i			1.05110i	0.850761	+	0.525552i	$0.176141\pi$
$11$			421.820i			1.05110i	−0.850761	+	0.525552i	$0.823859\pi$
$12$		0			0
$13$	459.817	−	459.817i	0.754618	−	0.754618i	−0.220720	−	0.975337i	$-0.570841\pi$
$13$	459.817	−	459.817i	0.754618	−	0.754618i	0.975337	+	0.220720i	$0.0708406\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−535.895	+	535.895i	−0.449735	+	0.449735i	−0.895266	−	0.445531i	$-0.853015\pi$
$17$	−535.895	+	535.895i	−0.449735	+	0.449735i	0.445531	+	0.895266i	$0.353015\pi$
$18$		0			0
$19$		−	1817.63i		−	1.15511i	−0.816353	−	0.577553i	$-0.804008\pi$
$19$		−	1817.63i		−	1.15511i	0.816353	−	0.577553i	$-0.195992\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2931.55	+	2931.55i	1.15552	+	1.15552i	0.985428	+	0.170092i	$0.0544066\pi$
$23$	2931.55	+	2931.55i	1.15552	+	1.15552i	0.170092	+	0.985428i	$0.445593\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−3145.46			−0.694527			−0.347263	−	0.937768i	$-0.612889\pi$
$29$	−3145.46			−0.694527			−0.347263	+	0.937768i	$0.612889\pi$
$30$		0			0
$31$	−3731.26			−0.697351			−0.348676	−	0.937243i	$-0.613369\pi$
$31$	−3731.26			−0.697351			−0.348676	+	0.937243i	$0.613369\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	8266.20	+	8266.20i	0.992662	+	0.992662i	0.999973	−	0.00731132i	$-0.00232729\pi$
$37$	8266.20	+	8266.20i	0.992662	+	0.992662i	−0.00731132	+	0.999973i	$0.502327\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	2489.14i		−	0.231254i	−0.993293	−	0.115627i	$-0.963112\pi$
$41$		−	2489.14i		−	0.231254i	0.993293	−	0.115627i	$-0.0368878\pi$
$42$		0			0
$43$	5547.73	−	5547.73i	0.457556	−	0.457556i	−0.440296	−	0.897853i	$-0.645127\pi$
$43$	5547.73	−	5547.73i	0.457556	−	0.457556i	0.897853	+	0.440296i	$0.145127\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−18661.1	+	18661.1i	−1.23223	+	1.23223i	−0.269127	+	0.963105i	$0.586735\pi$
$47$	−18661.1	+	18661.1i	−1.23223	+	1.23223i	−0.963105	+	0.269127i	$0.913265\pi$
$48$		0			0
$49$			5177.53i			0.308058i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−5831.36	−	5831.36i	−0.285154	−	0.285154i	0.550006	−	0.835161i	$-0.314625\pi$
$53$	−5831.36	−	5831.36i	−0.285154	−	0.285154i	−0.835161	+	0.550006i	$0.814625\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	31760.5			1.18784			0.593919	−	0.804525i	$-0.297580\pi$
$59$	31760.5			1.18784			0.593919	+	0.804525i	$0.297580\pi$
$60$		0			0
$61$	25216.1			0.867668			0.433834	−	0.900993i	$-0.357160\pi$
$61$	25216.1			0.867668			0.433834	+	0.900993i	$0.357160\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−6883.71	−	6883.71i	−0.187342	−	0.187342i	0.607204	−	0.794546i	$-0.292291\pi$
$67$	−6883.71	−	6883.71i	−0.187342	−	0.187342i	−0.794546	+	0.607204i	$0.792291\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	8320.46i		−	0.195885i	−0.995192	−	0.0979426i	$-0.968774\pi$
$71$		−	8320.46i		−	0.195885i	0.995192	−	0.0979426i	$-0.0312262\pi$
$72$		0			0
$73$	−27112.6	+	27112.6i	−0.595476	+	0.595476i	−0.939105	−	0.343629i	$-0.888344\pi$
$73$	−27112.6	+	27112.6i	−0.595476	+	0.595476i	0.343629	+	0.939105i	$0.388344\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	44225.3	−	44225.3i	0.850049	−	0.850049i
$78$		0			0
$79$			3779.58i			0.0681359i	0.999420	+	0.0340679i	$0.0108463\pi$
$79$			3779.58i			0.0681359i	−0.999420	+	0.0340679i	$0.989154\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	23893.1	+	23893.1i	0.380695	+	0.380695i	0.871352	−	0.490658i	$-0.163243\pi$
$83$	23893.1	+	23893.1i	0.380695	+	0.380695i	−0.490658	+	0.871352i	$0.663243\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−11869.1			−0.158834			−0.0794169	−	0.996841i	$-0.525306\pi$
$89$	−11869.1			−0.158834			−0.0794169	+	0.996841i	$0.525306\pi$
$90$		0			0
$91$	−96418.2			−1.22055
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−69598.6	−	69598.6i	−0.751054	−	0.751054i	0.223622	−	0.974676i	$-0.428212\pi$
$97$	−69598.6	−	69598.6i	−0.751054	−	0.751054i	−0.974676	+	0.223622i	$0.928212\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	69213.4i		−	0.675128i	−0.941302	−	0.337564i	$-0.890397\pi$
$101$		−	69213.4i		−	0.675128i	0.941302	−	0.337564i	$-0.109603\pi$
$102$		0			0
$103$	64694.8	−	64694.8i	0.600864	−	0.600864i	−0.339678	−	0.940542i	$-0.610318\pi$
$103$	64694.8	−	64694.8i	0.600864	−	0.600864i	0.940542	+	0.339678i	$0.110318\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	107933.	−	107933.i	0.911373	−	0.911373i	−0.0850077	−	0.996380i	$-0.527091\pi$
$107$	107933.	−	107933.i	0.911373	−	0.911373i	0.996380	+	0.0850077i	$0.0270915\pi$
$108$		0			0
$109$		−	210133.i		−	1.69405i	−0.531550	−	0.847027i	$-0.678390\pi$
$109$		−	210133.i		−	1.69405i	0.531550	−	0.847027i	$-0.321610\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−145664.	−	145664.i	−1.07314	−	1.07314i	−0.997105	−	0.0760373i	$-0.975773\pi$
$113$	−145664.	−	145664.i	−1.07314	−	1.07314i	−0.0760373	−	0.997105i	$-0.524227\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	112371.			0.727420
$120$		0			0
$121$	−16881.3			−0.104820
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	43624.3	+	43624.3i	0.240004	+	0.240004i	0.816852	−	0.576847i	$-0.195717\pi$
$127$	43624.3	+	43624.3i	0.240004	+	0.240004i	−0.576847	+	0.816852i	$0.695717\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			57827.6i			0.294413i	0.989106	+	0.147207i	$0.0470282\pi$
$131$			57827.6i			0.294413i	−0.989106	+	0.147207i	$0.952972\pi$
$132$		0			0
$133$	−190568.	+	190568.i	−0.934158	+	0.934158i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−248175.	+	248175.i	−1.12968	+	1.12968i	−0.139456	+	0.990228i	$0.544535\pi$
$137$	−248175.	+	248175.i	−1.12968	+	1.12968i	−0.990228	+	0.139456i	$0.955465\pi$
$138$		0			0
$139$		−	45792.0i		−	0.201026i	−0.994936	−	0.100513i	$-0.967952\pi$
$139$		−	45792.0i		−	0.201026i	0.994936	−	0.100513i	$-0.0320484\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	193960.	+	193960.i	0.793182	+	0.793182i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	521385.			1.92394			0.961972	−	0.273148i	$-0.0880648\pi$
$149$	521385.			1.92394			0.961972	+	0.273148i	$0.0880648\pi$
$150$		0			0
$151$	−51961.7			−0.185456			−0.0927280	−	0.995691i	$-0.529559\pi$
$151$	−51961.7			−0.185456			−0.0927280	+	0.995691i	$0.529559\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	230653.	+	230653.i	0.746809	+	0.746809i	0.973879	−	0.227070i	$-0.0729145\pi$
$157$	230653.	+	230653.i	0.746809	+	0.746809i	−0.227070	+	0.973879i	$0.572914\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	614711.i		−	1.86899i
$162$		0			0
$163$	166654.	−	166654.i	0.491301	−	0.491301i	−0.417415	−	0.908716i	$-0.637064\pi$
$163$	166654.	−	166654.i	0.491301	−	0.491301i	0.908716	+	0.417415i	$0.137064\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−84828.2	+	84828.2i	−0.235369	+	0.235369i	−0.814929	−	0.579560i	$-0.803224\pi$
$167$	−84828.2	+	84828.2i	−0.235369	+	0.235369i	0.579560	+	0.814929i	$0.303224\pi$
$168$		0			0
$169$		−	51570.9i		−	0.138895i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	14842.6	+	14842.6i	0.0377047	+	0.0377047i	0.725708	−	0.688003i	$-0.241512\pi$
$173$	14842.6	+	14842.6i	0.0377047	+	0.0377047i	−0.688003	+	0.725708i	$0.741512\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	546214.			1.27418			0.637089	−	0.770790i	$-0.280138\pi$
$179$	546214.			1.27418			0.637089	+	0.770790i	$0.280138\pi$
$180$		0			0
$181$	−383587.			−0.870298			−0.435149	−	0.900359i	$-0.643304\pi$
$181$	−383587.			−0.870298			−0.435149	+	0.900359i	$0.643304\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−226051.	−	226051.i	−0.472719	−	0.472719i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	646903.i		−	1.28309i	−0.767087	−	0.641543i	$-0.778295\pi$
$191$		−	646903.i		−	1.28309i	0.767087	−	0.641543i	$-0.221705\pi$
$192$		0			0
$193$	573393.	−	573393.i	1.10805	−	1.10805i	0.114642	−	0.993407i	$-0.463428\pi$
$193$	573393.	−	573393.i	1.10805	−	1.10805i	0.993407	−	0.114642i	$-0.0365720\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−106481.	+	106481.i	−0.195481	+	0.195481i	−0.798060	−	0.602578i	$-0.794140\pi$
$197$	−106481.	+	106481.i	−0.195481	+	0.195481i	0.602578	+	0.798060i	$0.294140\pi$
$198$		0			0
$199$		−	963815.i		−	1.72529i	−0.505813	−	0.862643i	$-0.668807\pi$
$199$		−	963815.i		−	1.72529i	0.505813	−	0.862643i	$-0.331193\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	329783.	+	329783.i	0.561678	+	0.561678i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	766714.			1.21414
$210$		0			0
$211$	−341760.			−0.528463			−0.264232	−	0.964459i	$-0.585118\pi$
$211$	−341760.			−0.528463			−0.264232	+	0.964459i	$0.585118\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	391201.	+	391201.i	0.563962	+	0.563962i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			492827.i			0.678756i
$222$		0			0
$223$	−634760.	+	634760.i	−0.854766	+	0.854766i	−0.990716	−	0.135950i	$-0.956591\pi$
$223$	−634760.	+	634760.i	−0.854766	+	0.854766i	0.135950	+	0.990716i	$0.456591\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	729989.	−	729989.i	0.940269	−	0.940269i	−0.0580454	−	0.998314i	$-0.518487\pi$
$227$	729989.	−	729989.i	0.940269	−	0.940269i	0.998314	+	0.0580454i	$0.0184868\pi$
$228$		0			0
$229$		−	1.35238e6i		−	1.70416i	−0.523408	−	0.852082i	$-0.675340\pi$
$229$		−	1.35238e6i		−	1.70416i	0.523408	−	0.852082i	$-0.324660\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−387437.	−	387437.i	−0.467532	−	0.467532i	0.433582	−	0.901114i	$-0.357249\pi$
$233$	−387437.	−	387437.i	−0.467532	−	0.467532i	−0.901114	+	0.433582i	$0.857249\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	1.63055e6			1.84646			0.923229	−	0.384250i	$-0.125540\pi$
$239$	1.63055e6			1.84646			0.923229	+	0.384250i	$0.125540\pi$
$240$		0			0
$241$	−1.23502e6			−1.36972			−0.684861	−	0.728674i	$-0.740137\pi$
$241$	−1.23502e6			−1.36972			−0.684861	+	0.728674i	$0.740137\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−835778.	−	835778.i	−0.871663	−	0.871663i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	1.70966e6i		−	1.71287i	−0.516256	−	0.856435i	$-0.672675\pi$
$251$		−	1.70966e6i		−	1.71287i	0.516256	−	0.856435i	$-0.327325\pi$
$252$		0			0
$253$	−1.23659e6	+	1.23659e6i	−1.21457	+	1.21457i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−39867.3	+	39867.3i	−0.0376516	+	0.0376516i	−0.725682	−	0.688030i	$-0.758476\pi$
$257$	−39867.3	+	39867.3i	−0.0376516	+	0.0376516i	0.688030	+	0.725682i	$0.258476\pi$
$258$		0			0
$259$		−	1.73332e6i		−	1.60557i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−1.22081e6	−	1.22081e6i	−1.08833	−	1.08833i	−0.995701	−	0.0926271i	$-0.970474\pi$
$263$	−1.22081e6	−	1.22081e6i	−1.08833	−	1.08833i	−0.0926271	−	0.995701i	$-0.529526\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	1.34125e6			1.13013			0.565067	−	0.825045i	$-0.308850\pi$
$269$	1.34125e6			1.13013			0.565067	+	0.825045i	$0.308850\pi$
$270$		0			0
$271$	−205762.			−0.170193			−0.0850964	−	0.996373i	$-0.527120\pi$
$271$	−205762.			−0.170193			−0.0850964	+	0.996373i	$0.527120\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−618468.	−	618468.i	−0.484304	−	0.484304i	0.422199	−	0.906503i	$-0.361258\pi$
$277$	−618468.	−	618468.i	−0.484304	−	0.484304i	−0.906503	+	0.422199i	$0.861258\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	2.33582e6i		−	1.76471i	−0.470584	−	0.882355i	$-0.655957\pi$
$281$		−	2.33582e6i		−	1.76471i	0.470584	−	0.882355i	$-0.344043\pi$
$282$		0			0
$283$	1.46794e6	−	1.46794e6i	1.08954	−	1.08954i	0.0939633	−	0.995576i	$-0.470046\pi$
$283$	1.46794e6	−	1.46794e6i	1.08954	−	1.08954i	0.995576	−	0.0939633i	$-0.0299537\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−260971.	+	260971.i	−0.187020	+	0.187020i
$288$		0			0
$289$			845491.i			0.595476i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	378835.	+	378835.i	0.257799	+	0.257799i	0.824158	−	0.566360i	$-0.191649\pi$
$293$	378835.	+	378835.i	0.257799	+	0.257799i	−0.566360	+	0.824158i	$0.691649\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	2.69595e6			1.74395
$300$		0			0
$301$	−1.16329e6			−0.740070
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	1.51547e6	+	1.51547e6i	0.917700	+	0.917700i	0.996862	−	0.0791617i	$-0.0252243\pi$
$307$	1.51547e6	+	1.51547e6i	0.917700	+	0.917700i	−0.0791617	+	0.996862i	$0.525224\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	539454.i		−	0.316266i	−0.987418	−	0.158133i	$-0.949452\pi$
$311$		−	539454.i		−	0.316266i	0.987418	−	0.158133i	$-0.0505475\pi$
$312$		0			0
$313$	357593.	−	357593.i	0.206314	−	0.206314i	−0.596385	−	0.802699i	$-0.703397\pi$
$313$	357593.	−	357593.i	0.206314	−	0.206314i	0.802699	+	0.596385i	$0.203397\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	1.22199e6	−	1.22199e6i	0.682996	−	0.682996i	−0.277678	−	0.960674i	$-0.589565\pi$
$317$	1.22199e6	−	1.22199e6i	0.682996	−	0.682996i	0.960674	+	0.277678i	$0.0895650\pi$
$318$		0			0
$319$		−	1.32682e6i		−	0.730020i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	974059.	+	974059.i	0.519492	+	0.519492i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	3.91301e6			1.99306
$330$		0			0
$331$	68206.2			0.0342179			0.0171090	−	0.999854i	$-0.494554\pi$
$331$	68206.2			0.0342179			0.0171090	+	0.999854i	$0.494554\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−831089.	−	831089.i	−0.398633	−	0.398633i	0.479118	−	0.877751i	$-0.340957\pi$
$337$	−831089.	−	831089.i	−0.398633	−	0.398633i	−0.877751	+	0.479118i	$0.840957\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	1.57392e6i		−	0.732989i
$342$		0			0
$343$	−1.21928e6	+	1.21928e6i	−0.559588	+	0.559588i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−1.76834e6	+	1.76834e6i	−0.788392	+	0.788392i	−0.981230	−	0.192839i	$-0.938231\pi$
$347$	−1.76834e6	+	1.76834e6i	−0.788392	+	0.788392i	0.192839	+	0.981230i	$0.438231\pi$
$348$		0			0
$349$		−	4.02818e6i		−	1.77030i	−0.465310	−	0.885148i	$-0.654057\pi$
$349$		−	4.02818e6i		−	1.77030i	0.465310	−	0.885148i	$-0.345943\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	2.86012e6	+	2.86012e6i	1.22165	+	1.22165i	0.967044	+	0.254609i	$0.0819467\pi$
$353$	2.86012e6	+	2.86012e6i	1.22165	+	1.22165i	0.254609	+	0.967044i	$0.418053\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	1.37112e6			0.561487			0.280744	−	0.959783i	$-0.409419\pi$
$359$	1.37112e6			0.561487			0.280744	+	0.959783i	$0.409419\pi$
$360$		0			0
$361$	−827686.			−0.334270
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	2.61273e6	+	2.61273e6i	1.01258	+	1.01258i	0.999920	+	0.0126613i	$0.00403033\pi$
$367$	2.61273e6	+	2.61273e6i	1.01258	+	1.01258i	0.0126613	+	0.999920i	$0.495970\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			1.22277e6i			0.461220i
$372$		0			0
$373$	2.96115e6	−	2.96115e6i	1.10202	−	1.10202i	0.107849	−	0.994167i	$-0.465604\pi$
$373$	2.96115e6	−	2.96115e6i	1.10202	−	1.10202i	0.994167	−	0.107849i	$-0.0343963\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−1.44634e6	+	1.44634e6i	−0.524102	+	0.524102i
$378$		0			0
$379$		−	1.55289e6i		−	0.555319i	−0.960680	−	0.277659i	$-0.910441\pi$
$379$		−	1.55289e6i		−	0.555319i	0.960680	−	0.277659i	$-0.0895587\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−1.56359e6	−	1.56359e6i	−0.544661	−	0.544661i	0.380231	−	0.924892i	$-0.375845\pi$
$383$	−1.56359e6	−	1.56359e6i	−0.544661	−	0.544661i	−0.924892	+	0.380231i	$0.875845\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−1.56349e6			−0.523869			−0.261934	−	0.965086i	$-0.584360\pi$
$389$	−1.56349e6			−0.523869			−0.261934	+	0.965086i	$0.584360\pi$
$390$		0			0
$391$	−3.14200e6			−1.03936
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−1.14380e6	−	1.14380e6i	−0.364227	−	0.364227i	0.501139	−	0.865367i	$-0.332914\pi$
$397$	−1.14380e6	−	1.14380e6i	−0.364227	−	0.364227i	−0.865367	+	0.501139i	$0.832914\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	4.58219e6i		−	1.42302i	−0.702675	−	0.711511i	$-0.748011\pi$
$401$		−	4.58219e6i		−	1.42302i	0.702675	−	0.711511i	$-0.251989\pi$
$402$		0			0
$403$	−1.71570e6	+	1.71570e6i	−0.526234	+	0.526234i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−3.48685e6	+	3.48685e6i	−1.04339	+	1.04339i
$408$		0			0
$409$		−	1.57340e6i		−	0.465082i	−0.972587	−	0.232541i	$-0.925296\pi$
$409$		−	1.57340e6i		−	0.465082i	0.972587	−	0.232541i	$-0.0747041\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−3.32989e6	−	3.32989e6i	−0.960628	−	0.960628i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	141734.			0.0394403			0.0197201	−	0.999806i	$-0.493722\pi$
$419$	141734.			0.0394403			0.0197201	+	0.999806i	$0.493722\pi$
$420$		0			0
$421$	−1.88839e6			−0.519262			−0.259631	−	0.965708i	$-0.583601\pi$
$421$	−1.88839e6			−0.519262			−0.259631	+	0.965708i	$0.583601\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−2.64376e6	−	2.64376e6i	−0.701701	−	0.701701i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			2.31421e6i			0.600079i	0.953927	+	0.300040i	$0.0969999\pi$
$431$			2.31421e6i			0.600079i	−0.953927	+	0.300040i	$0.903000\pi$
$432$		0			0
$433$	−1.59952e6	+	1.59952e6i	−0.409987	+	0.409987i	−0.881734	−	0.471747i	$-0.843624\pi$
$433$	−1.59952e6	+	1.59952e6i	−0.409987	+	0.409987i	0.471747	+	0.881734i	$0.343624\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	5.32848e6	−	5.32848e6i	1.33475	−	1.33475i
$438$		0			0
$439$			1.95630e6i			0.484478i	0.970217	+	0.242239i	$0.0778818\pi$
$439$			1.95630e6i			0.484478i	−0.970217	+	0.242239i	$0.922118\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−205417.	−	205417.i	−0.0497311	−	0.0497311i	0.681804	−	0.731535i	$-0.261196\pi$
$443$	−205417.	−	205417.i	−0.0497311	−	0.0497311i	−0.731535	+	0.681804i	$0.761196\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	1.40910e6			0.329856			0.164928	−	0.986306i	$-0.447261\pi$
$449$	1.40910e6			0.329856			0.164928	+	0.986306i	$0.447261\pi$
$450$		0			0
$451$	1.04997e6			0.243072
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	4.05767e6	+	4.05767e6i	0.908838	+	0.908838i	0.996179	−	0.0873406i	$-0.0278369\pi$
$457$	4.05767e6	+	4.05767e6i	0.908838	+	0.908838i	−0.0873406	+	0.996179i	$0.527837\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	829247.i		−	0.181732i	−0.995863	−	0.0908660i	$-0.971036\pi$
$461$		−	829247.i		−	0.181732i	0.995863	−	0.0908660i	$-0.0289635\pi$
$462$		0			0
$463$	−1.30197e6	+	1.30197e6i	−0.282260	+	0.282260i	−0.834010	−	0.551750i	$-0.813960\pi$
$463$	−1.30197e6	+	1.30197e6i	−0.282260	+	0.282260i	0.551750	+	0.834010i	$0.313960\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	5.25405e6	−	5.25405e6i	1.11481	−	1.11481i	0.122323	−	0.992490i	$-0.460965\pi$
$467$	5.25405e6	−	5.25405e6i	1.11481	−	1.11481i	0.992490	−	0.122323i	$-0.0390345\pi$
$468$		0			0
$469$			1.44343e6i			0.303015i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	2.34015e6	+	2.34015e6i	0.480939	+	0.480939i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	5.72389e6			1.13986			0.569931	−	0.821693i	$-0.306970\pi$
$479$	5.72389e6			1.13986			0.569931	+	0.821693i	$0.306970\pi$
$480$		0			0
$481$	7.60188e6			1.49816
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−2.20555e6	−	2.20555e6i	−0.421399	−	0.421399i	0.464286	−	0.885685i	$-0.346311\pi$
$487$	−2.20555e6	−	2.20555e6i	−0.421399	−	0.421399i	−0.885685	+	0.464286i	$0.846311\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			2.00486e6i			0.375301i	0.982236	+	0.187650i	$0.0600872\pi$
$491$			2.00486e6i			0.375301i	−0.982236	+	0.187650i	$0.939913\pi$
$492$		0			0
$493$	1.68563e6	−	1.68563e6i	0.312353	−	0.312353i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−872351.	+	872351.i	−0.158416	+	0.158416i
$498$		0			0
$499$			2.86192e6i			0.514525i	0.966342	+	0.257262i	$0.0828205\pi$
$499$			2.86192e6i			0.514525i	−0.966342	+	0.257262i	$0.917180\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−353941.	−	353941.i	−0.0623750	−	0.0623750i	0.675231	−	0.737606i	$-0.264044\pi$
$503$	−353941.	−	353941.i	−0.0623750	−	0.0623750i	−0.737606	+	0.675231i	$0.764044\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−1.09390e7			−1.87148			−0.935739	−	0.352692i	$-0.885266\pi$
$509$	−1.09390e7			−1.87148			−0.935739	+	0.352692i	$0.885266\pi$
$510$		0			0
$511$	5.68519e6			0.963148
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−7.87162e6	−	7.87162e6i	−1.29520	−	1.29520i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			3.40542e6i			0.549637i	0.961496	+	0.274818i	$0.0886177\pi$
$521$			3.40542e6i			0.549637i	−0.961496	+	0.274818i	$0.911382\pi$
$522$		0			0
$523$	879348.	−	879348.i	0.140575	−	0.140575i	−0.633318	−	0.773892i	$-0.718307\pi$
$523$	879348.	−	879348.i	0.140575	−	0.140575i	0.773892	+	0.633318i	$0.218307\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	1.99956e6	−	1.99956e6i	0.313624	−	0.313624i
$528$		0			0
$529$			1.07516e7i			1.67046i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−1.14455e6	−	1.14455e6i	−0.174509	−	0.174509i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−2.18399e6			−0.323801
$540$		0			0
$541$	−9.62740e6			−1.41422			−0.707108	−	0.707105i	$-0.750001\pi$
$541$	−9.62740e6			−1.41422			−0.707108	+	0.707105i	$0.750001\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−596519.	−	596519.i	−0.0852425	−	0.0852425i	0.663200	−	0.748442i	$-0.269198\pi$
$547$	−596519.	−	596519.i	−0.0852425	−	0.0852425i	−0.748442	+	0.663200i	$0.769198\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			5.71729e6i			0.802252i
$552$		0			0
$553$	396266.	−	396266.i	0.0551029	−	0.0551029i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−6.03589e6	+	6.03589e6i	−0.824334	+	0.824334i	−0.986726	−	0.162392i	$-0.948079\pi$
$557$	−6.03589e6	+	6.03589e6i	−0.824334	+	0.824334i	0.162392	+	0.986726i	$0.448079\pi$
$558$		0			0
$559$		−	5.10189e6i		−	0.690560i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−2.36832e6	−	2.36832e6i	−0.314897	−	0.314897i	0.531906	−	0.846803i	$-0.321476\pi$
$563$	−2.36832e6	−	2.36832e6i	−0.314897	−	0.314897i	−0.846803	+	0.531906i	$0.821476\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−1.27269e7			−1.64794			−0.823969	−	0.566635i	$-0.808245\pi$
$569$	−1.27269e7			−1.64794			−0.823969	+	0.566635i	$0.808245\pi$
$570$		0			0
$571$	3.18077e6			0.408265			0.204132	−	0.978943i	$-0.434563\pi$
$571$	3.18077e6			0.408265			0.204132	+	0.978943i	$0.434563\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	9.14160e6	+	9.14160e6i	1.14310	+	1.14310i	0.987881	+	0.155215i	$0.0496069\pi$
$577$	9.14160e6	+	9.14160e6i	1.14310	+	1.14310i	0.155215	+	0.987881i	$0.450393\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	5.01009e6i		−	0.615751i
$582$		0			0
$583$	2.45978e6	−	2.45978e6i	0.299727	−	0.299727i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−1.87790e6	+	1.87790e6i	−0.224946	+	0.224946i	−0.810577	−	0.585632i	$-0.800847\pi$
$587$	−1.87790e6	+	1.87790e6i	−0.224946	+	0.224946i	0.585632	+	0.810577i	$0.300847\pi$
$588$		0			0
$589$			6.78206e6i			0.805515i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	6.54112e6	+	6.54112e6i	0.763863	+	0.763863i	0.977018	−	0.213155i	$-0.0683739\pi$
$593$	6.54112e6	+	6.54112e6i	0.763863	+	0.763863i	−0.213155	+	0.977018i	$0.568374\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−1.58028e7			−1.79956			−0.899781	−	0.436342i	$-0.856274\pi$
$599$	−1.58028e7			−1.79956			−0.899781	+	0.436342i	$0.856274\pi$
$600$		0			0
$601$	1.05086e6			0.118675			0.0593376	−	0.998238i	$-0.481101\pi$
$601$	1.05086e6			0.118675			0.0593376	+	0.998238i	$0.481101\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	9.35963e6	+	9.35963e6i	1.03107	+	1.03107i	0.999502	+	0.0315653i	$0.0100492\pi$
$607$	9.35963e6	+	9.35963e6i	1.03107	+	1.03107i	0.0315653	+	0.999502i	$0.489951\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			1.71614e7i			1.85973i
$612$		0			0
$613$	−9.23664e6	+	9.23664e6i	−0.992802	+	0.992802i	−0.999974	−	0.00717221i	$-0.997717\pi$
$613$	−9.23664e6	+	9.23664e6i	−0.992802	+	0.992802i	0.00717221	+	0.999974i	$0.497717\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1.20036e7	+	1.20036e7i	−1.26940	+	1.26940i	−0.323008	+	0.946396i	$0.604694\pi$
$617$	−1.20036e7	+	1.20036e7i	−1.26940	+	1.26940i	−0.946396	+	0.323008i	$0.895306\pi$
$618$		0			0
$619$		−	945266.i		−	0.0991579i	−0.998770	−	0.0495789i	$-0.984212\pi$
$619$		−	945266.i		−	0.0991579i	0.998770	−	0.0495789i	$-0.0157879\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	1.24440e6	+	1.24440e6i	0.128452	+	0.128452i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−8.85962e6			−0.892870
$630$		0			0
$631$	1.55090e7			1.55064			0.775320	−	0.631569i	$-0.217589\pi$
$631$	1.55090e7			1.55064			0.775320	+	0.631569i	$0.217589\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	2.38072e6	+	2.38072e6i	0.232466	+	0.232466i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			1.28007e7i			1.23052i	0.788324	+	0.615261i	$0.210949\pi$
$641$			1.28007e7i			1.23052i	−0.788324	+	0.615261i	$0.789051\pi$
$642$		0			0
$643$	3.87836e6	−	3.87836e6i	0.369931	−	0.369931i	−0.497521	−	0.867452i	$-0.665756\pi$
$643$	3.87836e6	−	3.87836e6i	0.369931	−	0.369931i	0.867452	+	0.497521i	$0.165756\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	4.94391e6	−	4.94391e6i	0.464312	−	0.464312i	−0.435754	−	0.900066i	$-0.643518\pi$
$647$	4.94391e6	−	4.94391e6i	0.464312	−	0.464312i	0.900066	+	0.435754i	$0.143518\pi$
$648$		0			0
$649$			1.33972e7i			1.24854i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−6.51144e6	−	6.51144e6i	−0.597578	−	0.597578i	0.342090	−	0.939667i	$-0.388865\pi$
$653$	−6.51144e6	−	6.51144e6i	−0.597578	−	0.597578i	−0.939667	+	0.342090i	$0.888865\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−7.82829e6			−0.702188			−0.351094	−	0.936340i	$-0.614190\pi$
$659$	−7.82829e6			−0.702188			−0.351094	+	0.936340i	$0.614190\pi$
$660$		0			0
$661$	−421749.			−0.0375449			−0.0187724	−	0.999824i	$-0.505976\pi$
$661$	−421749.			−0.0375449			−0.0187724	+	0.999824i	$0.505976\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−9.22107e6	−	9.22107e6i	−0.802540	−	0.802540i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			1.06367e7i			0.912009i
$672$		0			0
$673$	6.75246e6	−	6.75246e6i	0.574677	−	0.574677i	−0.358755	−	0.933432i	$-0.616798\pi$
$673$	6.75246e6	−	6.75246e6i	0.574677	−	0.574677i	0.933432	+	0.358755i	$0.116798\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	1.00244e7	−	1.00244e7i	0.840597	−	0.840597i	−0.148339	−	0.988937i	$-0.547393\pi$
$677$	1.00244e7	−	1.00244e7i	0.840597	−	0.840597i	0.988937	+	0.148339i	$0.0473928\pi$
$678$		0			0
$679$			1.45940e7i			1.21479i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	1.03144e7	+	1.03144e7i	0.846042	+	0.846042i	0.989637	−	0.143595i	$-0.0458661\pi$
$683$	1.03144e7	+	1.03144e7i	0.846042	+	0.846042i	−0.143595	+	0.989637i	$0.545866\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−5.36272e6			−0.430365
$690$		0			0
$691$	2.09202e7			1.66675			0.833376	−	0.552707i	$-0.186405\pi$
$691$	2.09202e7			1.66675			0.833376	+	0.552707i	$0.186405\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	1.33392e6	+	1.33392e6i	0.104003	+	0.104003i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	7.43232e6i		−	0.571254i	−0.958341	−	0.285627i	$-0.907798\pi$
$701$		−	7.43232e6i		−	0.571254i	0.958341	−	0.285627i	$-0.0922019\pi$
$702$		0			0
$703$	1.50249e7	−	1.50249e7i	1.14663	−	1.14663i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−7.25660e6	+	7.25660e6i	−0.545990	+	0.545990i
$708$		0			0
$709$			1.57053e7i			1.17336i	0.809820	+	0.586679i	$0.199565\pi$
$709$			1.57053e7i			1.17336i	−0.809820	+	0.586679i	$0.800435\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−1.09384e7	−	1.09384e7i	−0.805804	−	0.805804i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−9.10822e6			−0.657069			−0.328535	−	0.944492i	$-0.606555\pi$
$719$	−9.10822e6			−0.657069			−0.328535	+	0.944492i	$0.606555\pi$
$720$		0			0
$721$	−1.35657e7			−0.971863
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−1.74643e7	−	1.74643e7i	−1.22551	−	1.22551i	−0.965646	−	0.259862i	$-0.916323\pi$
$727$	−1.74643e7	−	1.74643e7i	−1.22551	−	1.22551i	−0.259862	−	0.965646i	$-0.583677\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			5.94600e6i			0.411558i
$732$		0			0
$733$	−1.55860e7	+	1.55860e7i	−1.07145	+	1.07145i	−0.0742113	+	0.997243i	$0.523644\pi$
$733$	−1.55860e7	+	1.55860e7i	−1.07145	+	1.07145i	−0.997243	+	0.0742113i	$0.976356\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	2.90369e6	−	2.90369e6i	0.196916	−	0.196916i
$738$		0			0
$739$			2.16991e7i			1.46161i	0.682588	+	0.730803i	$0.260854\pi$
$739$			2.16991e7i			1.46161i	−0.682588	+	0.730803i	$0.739146\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	1.68015e7	+	1.68015e7i	1.11654	+	1.11654i	0.992245	+	0.124298i	$0.0396679\pi$
$743$	1.68015e7	+	1.68015e7i	1.11654	+	1.11654i	0.124298	+	0.992245i	$0.460332\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−2.26323e7			−1.47409
$750$		0			0
$751$	1.63487e7			1.05775			0.528874	−	0.848700i	$-0.322614\pi$
$751$	1.63487e7			1.05775			0.528874	+	0.848700i	$0.322614\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−5.40044e6	−	5.40044e6i	−0.342523	−	0.342523i	0.514792	−	0.857315i	$-0.327869\pi$
$757$	−5.40044e6	−	5.40044e6i	−0.342523	−	0.342523i	−0.857315	+	0.514792i	$0.827869\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	8.82146e6i		−	0.552178i	−0.961132	−	0.276089i	$-0.910962\pi$
$761$		−	8.82146e6i		−	0.552178i	0.961132	−	0.276089i	$-0.0890384\pi$
$762$		0			0
$763$	−2.20311e7	+	2.20311e7i	−1.37002	+	1.37002i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1.46040e7	−	1.46040e7i	0.896363	−	0.896363i
$768$		0			0
$769$			678955.i			0.0414024i	0.999786	+	0.0207012i	$0.00658986\pi$
$769$			678955.i			0.0414024i	−0.999786	+	0.0207012i	$0.993410\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−1.60240e7	−	1.60240e7i	−0.964543	−	0.964543i	0.0348495	−	0.999393i	$-0.488905\pi$
$773$	−1.60240e7	−	1.60240e7i	−0.964543	−	0.964543i	−0.999393	+	0.0348495i	$0.988905\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−4.52434e6			−0.267123
$780$		0			0
$781$	3.50974e6			0.205896
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	2.16145e7	+	2.16145e7i	1.24397	+	1.24397i	0.958341	+	0.285627i	$0.0922020\pi$
$787$	2.16145e7	+	2.16145e7i	1.24397	+	1.24397i	0.285627	+	0.958341i	$0.407798\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			3.05441e7i			1.73574i
$792$		0			0
$793$	1.15948e7	−	1.15948e7i	0.654757	−	0.654757i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	2.21157e7	−	2.21157e7i	1.23326	−	1.23326i	0.270558	−	0.962704i	$-0.412792\pi$
$797$	2.21157e7	−	2.21157e7i	1.23326	−	1.23326i	0.962704	−	0.270558i	$-0.0872081\pi$
$798$		0			0
$799$		−	2.00007e7i		−	1.10836i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1.14367e7	−	1.14367e7i	−0.625908	−	0.625908i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−3.36808e7			−1.80930			−0.904651	−	0.426154i	$-0.859868\pi$
$809$	−3.36808e7			−1.80930			−0.904651	+	0.426154i	$0.859868\pi$
$810$		0			0
$811$	3.49003e6			0.186327			0.0931637	−	0.995651i	$-0.470302\pi$
$811$	3.49003e6			0.186327			0.0931637	+	0.995651i	$0.470302\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−1.00837e7	−	1.00837e7i	−0.528526	−	0.528526i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	1.58252e6i		−	0.0819393i	−0.999160	−	0.0409696i	$-0.986955\pi$
$821$		−	1.58252e6i		−	0.0819393i	0.999160	−	0.0409696i	$-0.0130447\pi$
$822$		0			0
$823$	−1.04473e7	+	1.04473e7i	−0.537655	+	0.537655i	−0.922839	−	0.385185i	$-0.874138\pi$
$823$	−1.04473e7	+	1.04473e7i	−0.537655	+	0.537655i	0.385185	+	0.922839i	$0.374138\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1.47317e7	+	1.47317e7i	−0.749014	+	0.749014i	−0.974294	−	0.225280i	$-0.927670\pi$
$827$	−1.47317e7	+	1.47317e7i	−0.749014	+	0.749014i	0.225280	+	0.974294i	$0.427670\pi$
$828$		0			0
$829$			2.14226e7i			1.08265i	0.840815	+	0.541323i	$0.182076\pi$
$829$			2.14226e7i			1.08265i	−0.840815	+	0.541323i	$0.817924\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−2.77461e6	−	2.77461e6i	−0.138544	−	0.138544i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	2.07965e7			1.01996			0.509982	−	0.860185i	$-0.329652\pi$
$839$	2.07965e7			1.01996			0.509982	+	0.860185i	$0.329652\pi$
$840$		0			0
$841$	−1.06172e7			−0.517632
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	1.76990e6	+	1.76990e6i	0.0847698	+	0.0847698i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			4.84655e7i			2.29408i
$852$		0			0
$853$	7.98005e6	−	7.98005e6i	0.375520	−	0.375520i	−0.493963	−	0.869483i	$-0.664452\pi$
$853$	7.98005e6	−	7.98005e6i	0.375520	−	0.375520i	0.869483	+	0.493963i	$0.164452\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−2.94596e7	+	2.94596e7i	−1.37017	+	1.37017i	−0.509992	+	0.860179i	$0.670352\pi$
$857$	−2.94596e7	+	2.94596e7i	−1.37017	+	1.37017i	−0.860179	+	0.509992i	$0.829648\pi$
$858$		0			0
$859$		−	2.77973e7i		−	1.28534i	−0.766141	−	0.642672i	$-0.777826\pi$
$859$		−	2.77973e7i		−	1.28534i	0.766141	−	0.642672i	$-0.222174\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−1.38825e7	−	1.38825e7i	−0.634512	−	0.634512i	0.314684	−	0.949196i	$-0.398101\pi$
$863$	−1.38825e7	−	1.38825e7i	−0.634512	−	0.634512i	−0.949196	+	0.314684i	$0.898101\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−1.59430e6			−0.0716179
$870$		0			0
$871$	−6.33050e6			−0.282743
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	7.98704e6	+	7.98704e6i	0.350661	+	0.350661i	0.860355	−	0.509695i	$-0.170242\pi$
$877$	7.98704e6	+	7.98704e6i	0.350661	+	0.350661i	−0.509695	+	0.860355i	$0.670242\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	4.21649e7i		−	1.83025i	−0.403166	−	0.915127i	$-0.632090\pi$
$881$		−	4.21649e7i		−	1.83025i	0.403166	−	0.915127i	$-0.367910\pi$
$882$		0			0
$883$	2.43938e7	−	2.43938e7i	1.05288	−	1.05288i	0.0543537	−	0.998522i	$-0.482690\pi$
$883$	2.43938e7	−	2.43938e7i	1.05288	−	1.05288i	0.998522	−	0.0543537i	$-0.0173099\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−6.21892e6	+	6.21892e6i	−0.265403	+	0.265403i	−0.827245	−	0.561842i	$-0.810093\pi$
$887$	−6.21892e6	+	6.21892e6i	−0.265403	+	0.265403i	0.561842	+	0.827245i	$0.310093\pi$
$888$		0			0
$889$		−	9.14749e6i		−	0.388193i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	3.39190e7	+	3.39190e7i	1.42336	+	1.42336i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	1.17365e7			0.484329
$900$		0			0
$901$	6.24999e6			0.256488
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	1.33687e6	+	1.33687e6i	0.0539601	+	0.0539601i	0.733572	−	0.679612i	$-0.237852\pi$
$907$	1.33687e6	+	1.33687e6i	0.0539601	+	0.0539601i	−0.679612	+	0.733572i	$0.737852\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	7.68607e6i		−	0.306838i	−0.988161	−	0.153419i	$-0.950972\pi$
$911$		−	7.68607e6i		−	0.306838i	0.988161	−	0.153419i	$-0.0490284\pi$
$912$		0			0
$913$	−1.00786e7	+	1.00786e7i	−0.400150	+	0.400150i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	6.06288e6	−	6.06288e6i	0.238098	−	0.238098i
$918$		0			0
$919$			1.44638e7i			0.564930i	0.959278	+	0.282465i	$0.0911521\pi$
$919$			1.44638e7i			0.564930i	−0.959278	+	0.282465i	$0.908848\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−3.82589e6	−	3.82589e6i	−0.147818	−	0.147818i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−3.67185e7			−1.39587			−0.697936	−	0.716160i	$-0.745898\pi$
$929$	−3.67185e7			−1.39587			−0.697936	+	0.716160i	$0.745898\pi$
$930$		0			0
$931$	9.41084e6			0.355839
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−2.54516e6	−	2.54516e6i	−0.0947033	−	0.0947033i	0.658168	−	0.752871i	$-0.271332\pi$
$937$	−2.54516e6	−	2.54516e6i	−0.0947033	−	0.0947033i	−0.752871	+	0.658168i	$0.771332\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	2.22494e7i		−	0.819112i	−0.912285	−	0.409556i	$-0.865684\pi$
$941$		−	2.22494e7i		−	0.819112i	0.912285	−	0.409556i	$-0.134316\pi$
$942$		0			0
$943$	7.29704e6	−	7.29704e6i	0.267219	−	0.267219i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	4.71279e6	−	4.71279e6i	0.170767	−	0.170767i	−0.616549	−	0.787316i	$-0.711470\pi$
$947$	4.71279e6	−	4.71279e6i	0.170767	−	0.170767i	0.787316	+	0.616549i	$0.211470\pi$
$948$		0			0
$949$			2.49337e7i			0.898714i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	3.27276e7	+	3.27276e7i	1.16730	+	1.16730i	0.982841	+	0.184457i	$0.0590527\pi$
$953$	3.27276e7	+	3.27276e7i	1.16730	+	1.16730i	0.184457	+	0.982841i	$0.440947\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	5.20394e7			1.82720
$960$		0			0
$961$	−1.47068e7			−0.513701
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	1.08360e7	+	1.08360e7i	0.372650	+	0.372650i	0.868441	−	0.495792i	$-0.165122\pi$
$967$	1.08360e7	+	1.08360e7i	0.372650	+	0.372650i	−0.495792	+	0.868441i	$0.665122\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	1.12837e7i		−	0.384063i	−0.981389	−	0.192031i	$-0.938492\pi$
$971$		−	1.12837e7i		−	0.384063i	0.981389	−	0.192031i	$-0.0615075\pi$
$972$		0			0
$973$	−4.80102e6	+	4.80102e6i	−0.162574	+	0.162574i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−5.37670e6	+	5.37670e6i	−0.180210	+	0.180210i	−0.791447	−	0.611237i	$-0.790672\pi$
$977$	−5.37670e6	+	5.37670e6i	−0.180210	+	0.180210i	0.611237	+	0.791447i	$0.290672\pi$
$978$		0			0
$979$		−	5.00663e6i		−	0.166951i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	2.24854e7	+	2.24854e7i	0.742192	+	0.742192i	0.972999	−	0.230808i	$-0.0741368\pi$
$983$	2.24854e7	+	2.24854e7i	0.742192	+	0.742192i	−0.230808	+	0.972999i	$0.574137\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	3.25269e7			1.05743
$990$		0			0
$991$	−1.46627e7			−0.474275			−0.237137	−	0.971476i	$-0.576209\pi$
$991$	−1.46627e7			−0.474275			−0.237137	+	0.971476i	$0.576209\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	266637.	+	266637.i	0.00849536	+	0.00849536i	0.711342	−	0.702846i	$-0.248088\pi$
$997$	266637.	+	266637.i	0.00849536	+	0.00849536i	−0.702846	+	0.711342i	$0.748088\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	900.6.j.a.557.2	yes	16
3.2	odd	2	inner	900.6.j.a.557.1	✓	16
5.2	odd	4	inner	900.6.j.a.593.8	yes	16
5.3	odd	4	inner	900.6.j.a.593.2	yes	16
5.4	even	2	inner	900.6.j.a.557.8	yes	16
15.2	even	4	inner	900.6.j.a.593.7	yes	16
15.8	even	4	inner	900.6.j.a.593.1	yes	16
15.14	odd	2	inner	900.6.j.a.557.7	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
900.6.j.a.557.1	✓	16	3.2	odd	2	inner
900.6.j.a.557.2	yes	16	1.1	even	1	trivial
900.6.j.a.557.7	yes	16	15.14	odd	2	inner
900.6.j.a.557.8	yes	16	5.4	even	2	inner
900.6.j.a.593.1	yes	16	15.8	even	4	inner
900.6.j.a.593.2	yes	16	5.3	odd	4	inner
900.6.j.a.593.7	yes	16	15.2	even	4	inner
900.6.j.a.593.8	yes	16	5.2	odd	4	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	900.j (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-104.844 - 104.844i) q^{7} +O(q^{10})\)	\(q+(-104.844 - 104.844i) q^{7} +421.820i q^{11} +(459.817 - 459.817i) q^{13} +(-535.895 + 535.895i) q^{17} -1817.63i q^{19} +(2931.55 + 2931.55i) q^{23} -3145.46 q^{29} -3731.26 q^{31} +(8266.20 + 8266.20i) q^{37} -2489.14i q^{41} +(5547.73 - 5547.73i) q^{43} +(-18661.1 + 18661.1i) q^{47} +5177.53i q^{49} +(-5831.36 - 5831.36i) q^{53} +31760.5 q^{59} +25216.1 q^{61} +(-6883.71 - 6883.71i) q^{67} -8320.46i q^{71} +(-27112.6 + 27112.6i) q^{73} +(44225.3 - 44225.3i) q^{77} +3779.58i q^{79} +(23893.1 + 23893.1i) q^{83} -11869.1 q^{89} -96418.2 q^{91} +(-69598.6 - 69598.6i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 5632 q^{31} - 119200 q^{61} - 138048 q^{91}+O(q^{100}) \) 16 * q + 5632 * q^31 - 119200 * q^61 - 138048 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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