LMFDB - Embedded newform 1098.2.d.a.487.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1098,2,Mod(487,1098)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1098, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1098.487");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1098 = 2 \cdot 3^{2} \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1098.d (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:	$8.76757414194$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 122)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			487.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1098.487
Dual form			1098.2.d.a.487.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} +3.00000 q^{5} +3.00000i q^{7} -1.00000i q^{8} +O(q^{10})$	$q+1.00000i q^{2} -1.00000 q^{4} +3.00000 q^{5} +3.00000i q^{7} -1.00000i q^{8} +3.00000i q^{10} +3.00000i q^{11} -5.00000 q^{13} -3.00000 q^{14} +1.00000 q^{16} +6.00000i q^{17} -2.00000 q^{19} -3.00000 q^{20} -3.00000 q^{22} -9.00000i q^{23} +4.00000 q^{25} -5.00000i q^{26} -3.00000i q^{28} +6.00000i q^{29} +1.00000i q^{32} -6.00000 q^{34} +9.00000i q^{35} +6.00000i q^{37} -2.00000i q^{38} -3.00000i q^{40} +9.00000 q^{41} -3.00000i q^{44} +9.00000 q^{46} -2.00000 q^{49} +4.00000i q^{50} +5.00000 q^{52} +9.00000i q^{55} +3.00000 q^{56} -6.00000 q^{58} -3.00000i q^{59} +(-5.00000 + 6.00000i) q^{61} -1.00000 q^{64} -15.0000 q^{65} +3.00000i q^{67} -6.00000i q^{68} -9.00000 q^{70} -7.00000 q^{73} -6.00000 q^{74} +2.00000 q^{76} -9.00000 q^{77} -15.0000i q^{79} +3.00000 q^{80} +9.00000i q^{82} +12.0000 q^{83} +18.0000i q^{85} +3.00000 q^{88} -15.0000i q^{91} +9.00000i q^{92} -6.00000 q^{95} +10.0000 q^{97} -2.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 6 q^{5}+O(q^{10}) $ 2 * q - 2 * q^4 + 6 * q^5	$ 2 q - 2 q^{4} + 6 q^{5} - 10 q^{13} - 6 q^{14} + 2 q^{16} - 4 q^{19} - 6 q^{20} - 6 q^{22} + 8 q^{25} - 12 q^{34} + 18 q^{41} + 18 q^{46} - 4 q^{49} + 10 q^{52} + 6 q^{56} - 12 q^{58} - 10 q^{61} - 2 q^{64} - 30 q^{65} - 18 q^{70} - 14 q^{73} - 12 q^{74} + 4 q^{76} - 18 q^{77} + 6 q^{80} + 24 q^{83} + 6 q^{88} - 12 q^{95} + 20 q^{97}+O(q^{100}) $ 2 * q - 2 * q^4 + 6 * q^5 - 10 * q^13 - 6 * q^14 + 2 * q^16 - 4 * q^19 - 6 * q^20 - 6 * q^22 + 8 * q^25 - 12 * q^34 + 18 * q^41 + 18 * q^46 - 4 * q^49 + 10 * q^52 + 6 * q^56 - 12 * q^58 - 10 * q^61 - 2 * q^64 - 30 * q^65 - 18 * q^70 - 14 * q^73 - 12 * q^74 + 4 * q^76 - 18 * q^77 + 6 * q^80 + 24 * q^83 + 6 * q^88 - 12 * q^95 + 20 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1098\mathbb{Z}\right)^\times$.

$n$	$245$	$307$
$\chi(n)$	$1$	$-1$

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 $

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$	3.00000			1.34164			0.670820	−	0.741620i	$-0.265942\pi$
$5$	3.00000			1.34164			0.670820	+	0.741620i	$0.265942\pi$
$6$		0			0
$7$			3.00000i			1.13389i	0.823754	+	0.566947i	$0.191875\pi$
$7$			3.00000i			1.13389i	−0.823754	+	0.566947i	$0.808125\pi$
$8$		−	1.00000i		−	0.353553i
$9$		0			0
$10$			3.00000i			0.948683i
$11$			3.00000i			0.904534i	0.891883	+	0.452267i	$0.149385\pi$
$11$			3.00000i			0.904534i	−0.891883	+	0.452267i	$0.850615\pi$
$12$		0			0
$13$	−5.00000			−1.38675			−0.693375	−	0.720577i	$-0.743877\pi$
$13$	−5.00000			−1.38675			−0.693375	+	0.720577i	$0.743877\pi$
$14$	−3.00000			−0.801784
$15$		0			0
$16$	1.00000			0.250000
$17$			6.00000i			1.45521i	0.685994	+	0.727607i	$0.259367\pi$
$17$			6.00000i			1.45521i	−0.685994	+	0.727607i	$0.740633\pi$
$18$		0			0
$19$	−2.00000			−0.458831			−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000			−0.458831			−0.229416	+	0.973329i	$0.573682\pi$
$20$	−3.00000			−0.670820
$21$		0			0
$22$	−3.00000			−0.639602
$23$		−	9.00000i		−	1.87663i	−0.345782	−	0.938315i	$-0.612386\pi$
$23$		−	9.00000i		−	1.87663i	0.345782	−	0.938315i	$-0.387614\pi$
$24$		0			0
$25$	4.00000			0.800000
$26$		−	5.00000i		−	0.980581i
$27$		0			0
$28$		−	3.00000i		−	0.566947i
$29$			6.00000i			1.11417i	0.830455	+	0.557086i	$0.188081\pi$
$29$			6.00000i			1.11417i	−0.830455	+	0.557086i	$0.811919\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$			1.00000i			0.176777i
$33$		0			0
$34$	−6.00000			−1.02899
$35$			9.00000i			1.52128i
$36$		0			0
$37$			6.00000i			0.986394i	0.869918	+	0.493197i	$0.164172\pi$
$37$			6.00000i			0.986394i	−0.869918	+	0.493197i	$0.835828\pi$
$38$		−	2.00000i		−	0.324443i
$39$		0			0
$40$		−	3.00000i		−	0.474342i
$41$	9.00000			1.40556			0.702782	−	0.711405i	$-0.251941\pi$
$41$	9.00000			1.40556			0.702782	+	0.711405i	$0.251941\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		−	3.00000i		−	0.452267i
$45$		0			0
$46$	9.00000			1.32698
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		0			0
$49$	−2.00000			−0.285714
$50$			4.00000i			0.565685i
$51$		0			0
$52$	5.00000			0.693375
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$			9.00000i			1.21356i
$56$	3.00000			0.400892
$57$		0			0
$58$	−6.00000			−0.787839
$59$		−	3.00000i		−	0.390567i	−0.980747	−	0.195283i	$-0.937437\pi$
$59$		−	3.00000i		−	0.390567i	0.980747	−	0.195283i	$-0.0625627\pi$
$60$		0			0
$61$	−5.00000	+	6.00000i	−0.640184	+	0.768221i
$61$	−5.00000	+	6.00000i	−0.640184	+	0.768221i
$62$		0			0
$63$		0			0
$64$	−1.00000			−0.125000
$65$	−15.0000			−1.86052
$66$		0			0
$67$			3.00000i			0.366508i	0.983066	+	0.183254i	$0.0586631\pi$
$67$			3.00000i			0.366508i	−0.983066	+	0.183254i	$0.941337\pi$
$68$		−	6.00000i		−	0.727607i
$69$		0			0
$70$	−9.00000			−1.07571
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	−7.00000			−0.819288			−0.409644	−	0.912245i	$-0.634347\pi$
$73$	−7.00000			−0.819288			−0.409644	+	0.912245i	$0.634347\pi$
$74$	−6.00000			−0.697486
$75$		0			0
$76$	2.00000			0.229416
$77$	−9.00000			−1.02565
$78$		0			0
$79$		−	15.0000i		−	1.68763i	−0.536633	−	0.843816i	$-0.680304\pi$
$79$		−	15.0000i		−	1.68763i	0.536633	−	0.843816i	$-0.319696\pi$
$80$	3.00000			0.335410
$81$		0			0
$82$			9.00000i			0.993884i
$83$	12.0000			1.31717			0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000			1.31717			0.658586	+	0.752506i	$0.271155\pi$
$84$		0			0
$85$			18.0000i			1.95237i
$86$		0			0
$87$		0			0
$88$	3.00000			0.319801
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$		−	15.0000i		−	1.57243i
$92$			9.00000i			0.938315i
$93$		0			0
$94$		0			0
$95$	−6.00000			−0.615587
$96$		0			0
$97$	10.0000			1.01535			0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000			1.01535			0.507673	+	0.861550i	$0.330506\pi$
$98$		−	2.00000i		−	0.202031i
$99$		0			0
$100$	−4.00000			−0.400000
$101$			18.0000i			1.79107i	0.444994	+	0.895533i	$0.353206\pi$
$101$			18.0000i			1.79107i	−0.444994	+	0.895533i	$0.646794\pi$
$102$		0			0
$103$	−14.0000			−1.37946			−0.689730	−	0.724066i	$-0.742271\pi$
$103$	−14.0000			−1.37946			−0.689730	+	0.724066i	$0.742271\pi$
$104$			5.00000i			0.490290i
$105$		0			0
$106$		0			0
$107$	12.0000			1.16008			0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000			1.16008			0.580042	+	0.814587i	$0.303036\pi$
$108$		0			0
$109$	11.0000			1.05361			0.526804	−	0.849987i	$-0.323390\pi$
$109$	11.0000			1.05361			0.526804	+	0.849987i	$0.323390\pi$
$110$	−9.00000			−0.858116
$111$		0			0
$112$			3.00000i			0.283473i
$113$	−3.00000			−0.282216			−0.141108	−	0.989994i	$-0.545067\pi$
$113$	−3.00000			−0.282216			−0.141108	+	0.989994i	$0.545067\pi$
$114$		0			0
$115$		−	27.0000i		−	2.51776i
$116$		−	6.00000i		−	0.557086i
$117$		0			0
$118$	3.00000			0.276172
$119$	−18.0000			−1.65006
$120$		0			0
$121$	2.00000			0.181818
$122$	−6.00000	−	5.00000i	−0.543214	−	0.452679i
$123$		0			0
$124$		0			0
$125$	−3.00000			−0.268328
$126$		0			0
$127$	−2.00000			−0.177471			−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000			−0.177471			−0.0887357	+	0.996055i	$0.528283\pi$
$128$		−	1.00000i		−	0.0883883i
$129$		0			0
$130$		−	15.0000i		−	1.31559i
$131$	−18.0000			−1.57267			−0.786334	−	0.617802i	$-0.788023\pi$
$131$	−18.0000			−1.57267			−0.786334	+	0.617802i	$0.788023\pi$
$132$		0			0
$133$		−	6.00000i		−	0.520266i
$134$	−3.00000			−0.259161
$135$		0			0
$136$	6.00000			0.514496
$137$	3.00000			0.256307			0.128154	−	0.991754i	$-0.459095\pi$
$137$	3.00000			0.256307			0.128154	+	0.991754i	$0.459095\pi$
$138$		0			0
$139$			15.0000i			1.27228i	0.771572	+	0.636142i	$0.219471\pi$
$139$			15.0000i			1.27228i	−0.771572	+	0.636142i	$0.780529\pi$
$140$		−	9.00000i		−	0.760639i
$141$		0			0
$142$		0			0
$143$		−	15.0000i		−	1.25436i
$144$		0			0
$145$			18.0000i			1.49482i
$146$		−	7.00000i		−	0.579324i
$147$		0			0
$148$		−	6.00000i		−	0.493197i
$149$	21.0000			1.72039			0.860194	−	0.509968i	$-0.170343\pi$
$149$	21.0000			1.72039			0.860194	+	0.509968i	$0.170343\pi$
$150$		0			0
$151$			3.00000i			0.244137i	0.992522	+	0.122068i	$0.0389527\pi$
$151$			3.00000i			0.244137i	−0.992522	+	0.122068i	$0.961047\pi$
$152$			2.00000i			0.162221i
$153$		0			0
$154$		−	9.00000i		−	0.725241i
$155$		0			0
$156$		0			0
$157$		−	18.0000i		−	1.43656i	−0.695756	−	0.718278i	$-0.744931\pi$
$157$		−	18.0000i		−	1.43656i	0.695756	−	0.718278i	$-0.255069\pi$
$158$	15.0000			1.19334
$159$		0			0
$160$			3.00000i			0.237171i
$161$	27.0000			2.12790
$162$		0			0
$163$	−2.00000			−0.156652			−0.0783260	−	0.996928i	$-0.524958\pi$
$163$	−2.00000			−0.156652			−0.0783260	+	0.996928i	$0.524958\pi$
$164$	−9.00000			−0.702782
$165$		0			0
$166$			12.0000i			0.931381i
$167$	6.00000			0.464294			0.232147	−	0.972681i	$-0.425425\pi$
$167$	6.00000			0.464294			0.232147	+	0.972681i	$0.425425\pi$
$168$		0			0
$169$	12.0000			0.923077
$170$	−18.0000			−1.38054
$171$		0			0
$172$		0			0
$173$		−	24.0000i		−	1.82469i	−0.409426	−	0.912343i	$-0.634271\pi$
$173$		−	24.0000i		−	1.82469i	0.409426	−	0.912343i	$-0.365729\pi$
$174$		0			0
$175$			12.0000i			0.907115i
$176$			3.00000i			0.226134i
$177$		0			0
$178$		0			0
$179$	24.0000			1.79384			0.896922	−	0.442189i	$-0.145798\pi$
$179$	24.0000			1.79384			0.896922	+	0.442189i	$0.145798\pi$
$180$		0			0
$181$			12.0000i			0.891953i	0.895045	+	0.445976i	$0.147144\pi$
$181$			12.0000i			0.891953i	−0.895045	+	0.445976i	$0.852856\pi$
$182$	15.0000			1.11187
$183$		0			0
$184$	−9.00000			−0.663489
$185$			18.0000i			1.32339i
$186$		0			0
$187$	−18.0000			−1.31629
$188$		0			0
$189$		0			0
$190$		−	6.00000i		−	0.435286i
$191$		−	3.00000i		−	0.217072i	−0.994092	−	0.108536i	$-0.965384\pi$
$191$		−	3.00000i		−	0.217072i	0.994092	−	0.108536i	$-0.0346163\pi$
$192$		0			0
$193$		−	12.0000i		−	0.863779i	−0.901927	−	0.431889i	$-0.857847\pi$
$193$		−	12.0000i		−	0.863779i	0.901927	−	0.431889i	$-0.142153\pi$
$194$			10.0000i			0.717958i
$195$		0			0
$196$	2.00000			0.142857
$197$	3.00000			0.213741			0.106871	−	0.994273i	$-0.465917\pi$
$197$	3.00000			0.213741			0.106871	+	0.994273i	$0.465917\pi$
$198$		0			0
$199$	20.0000			1.41776			0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000			1.41776			0.708881	+	0.705328i	$0.249200\pi$
$200$		−	4.00000i		−	0.282843i
$201$		0			0
$202$	−18.0000			−1.26648
$203$	−18.0000			−1.26335
$204$		0			0
$205$	27.0000			1.88576
$206$		−	14.0000i		−	0.975426i
$207$		0			0
$208$	−5.00000			−0.346688
$209$		−	6.00000i		−	0.415029i
$210$		0			0
$211$		−	12.0000i		−	0.826114i	−0.910705	−	0.413057i	$-0.864461\pi$
$211$		−	12.0000i		−	0.826114i	0.910705	−	0.413057i	$-0.135539\pi$
$212$		0			0
$213$		0			0
$214$			12.0000i			0.820303i
$215$		0			0
$216$		0			0
$217$		0			0
$218$			11.0000i			0.745014i
$219$		0			0
$220$		−	9.00000i		−	0.606780i
$221$		−	30.0000i		−	2.01802i
$222$		0			0
$223$			9.00000i			0.602685i	0.953516	+	0.301342i	$0.0974347\pi$
$223$			9.00000i			0.602685i	−0.953516	+	0.301342i	$0.902565\pi$
$224$	−3.00000			−0.200446
$225$		0			0
$226$		−	3.00000i		−	0.199557i
$227$			3.00000i			0.199117i	0.995032	+	0.0995585i	$0.0317430\pi$
$227$			3.00000i			0.199117i	−0.995032	+	0.0995585i	$0.968257\pi$
$228$		0			0
$229$	13.0000			0.859064			0.429532	−	0.903052i	$-0.358679\pi$
$229$	13.0000			0.859064			0.429532	+	0.903052i	$0.358679\pi$
$230$	27.0000			1.78033
$231$		0			0
$232$	6.00000			0.393919
$233$			6.00000i			0.393073i	0.980497	+	0.196537i	$0.0629694\pi$
$233$			6.00000i			0.393073i	−0.980497	+	0.196537i	$0.937031\pi$
$234$		0			0
$235$		0			0
$236$			3.00000i			0.195283i
$237$		0			0
$238$		−	18.0000i		−	1.16677i
$239$	6.00000			0.388108			0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000			0.388108			0.194054	+	0.980991i	$0.437836\pi$
$240$		0			0
$241$	1.00000			0.0644157			0.0322078	−	0.999481i	$-0.489746\pi$
$241$	1.00000			0.0644157			0.0322078	+	0.999481i	$0.489746\pi$
$242$			2.00000i			0.128565i
$243$		0			0
$244$	5.00000	−	6.00000i	0.320092	−	0.384111i
$245$	−6.00000			−0.383326
$246$		0			0
$247$	10.0000			0.636285
$248$		0			0
$249$		0			0
$250$		−	3.00000i		−	0.189737i
$251$		−	24.0000i		−	1.51487i	−0.652913	−	0.757433i	$-0.726453\pi$
$251$		−	24.0000i		−	1.51487i	0.652913	−	0.757433i	$-0.273547\pi$
$252$		0			0
$253$	27.0000			1.69748
$254$		−	2.00000i		−	0.125491i
$255$		0			0
$256$	1.00000			0.0625000
$257$	6.00000			0.374270			0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000			0.374270			0.187135	+	0.982334i	$0.440080\pi$
$258$		0			0
$259$	−18.0000			−1.11847
$260$	15.0000			0.930261
$261$		0			0
$262$		−	18.0000i		−	1.11204i
$263$	−18.0000			−1.10993			−0.554964	−	0.831875i	$-0.687268\pi$
$263$	−18.0000			−1.10993			−0.554964	+	0.831875i	$0.687268\pi$
$264$		0			0
$265$		0			0
$266$	6.00000			0.367884
$267$		0			0
$268$		−	3.00000i		−	0.183254i
$269$	6.00000			0.365826			0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000			0.365826			0.182913	+	0.983129i	$0.441447\pi$
$270$		0			0
$271$	2.00000			0.121491			0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000			0.121491			0.0607457	+	0.998153i	$0.480652\pi$
$272$			6.00000i			0.363803i
$273$		0			0
$274$			3.00000i			0.181237i
$275$			12.0000i			0.723627i
$276$		0			0
$277$			18.0000i			1.08152i	0.841178	+	0.540758i	$0.181862\pi$
$277$			18.0000i			1.08152i	−0.841178	+	0.540758i	$0.818138\pi$
$278$	−15.0000			−0.899640
$279$		0			0
$280$	9.00000			0.537853
$281$			12.0000i			0.715860i	0.933748	+	0.357930i	$0.116517\pi$
$281$			12.0000i			0.715860i	−0.933748	+	0.357930i	$0.883483\pi$
$282$		0			0
$283$	14.0000			0.832214			0.416107	−	0.909316i	$-0.363394\pi$
$283$	14.0000			0.832214			0.416107	+	0.909316i	$0.363394\pi$
$284$		0			0
$285$		0			0
$286$	15.0000			0.886969
$287$			27.0000i			1.59376i
$288$		0			0
$289$	−19.0000			−1.11765
$290$	−18.0000			−1.05700
$291$		0			0
$292$	7.00000			0.409644
$293$	30.0000			1.75262			0.876309	−	0.481749i	$-0.159998\pi$
$293$	30.0000			1.75262			0.876309	+	0.481749i	$0.159998\pi$
$294$		0			0
$295$		−	9.00000i		−	0.524000i
$296$	6.00000			0.348743
$297$		0			0
$298$			21.0000i			1.21650i
$299$			45.0000i			2.60242i
$300$		0			0
$301$		0			0
$302$	−3.00000			−0.172631
$303$		0			0
$304$	−2.00000			−0.114708
$305$	−15.0000	+	18.0000i	−0.858898	+	1.03068i
$306$		0			0
$307$		−	27.0000i		−	1.54097i	−0.637457	−	0.770486i	$-0.720014\pi$
$307$		−	27.0000i		−	1.54097i	0.637457	−	0.770486i	$-0.279986\pi$
$308$	9.00000			0.512823
$309$		0			0
$310$		0			0
$311$			3.00000i			0.170114i	0.996376	+	0.0850572i	$0.0271073\pi$
$311$			3.00000i			0.170114i	−0.996376	+	0.0850572i	$0.972893\pi$
$312$		0			0
$313$		−	12.0000i		−	0.678280i	−0.940736	−	0.339140i	$-0.889864\pi$
$313$		−	12.0000i		−	0.678280i	0.940736	−	0.339140i	$-0.110136\pi$
$314$	18.0000			1.01580
$315$		0			0
$316$			15.0000i			0.843816i
$317$	−18.0000			−1.01098			−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000			−1.01098			−0.505490	+	0.862832i	$0.668688\pi$
$318$		0			0
$319$	−18.0000			−1.00781
$320$	−3.00000			−0.167705
$321$		0			0
$322$			27.0000i			1.50465i
$323$		−	12.0000i		−	0.667698i
$324$		0			0
$325$	−20.0000			−1.10940
$326$		−	2.00000i		−	0.110770i
$327$		0			0
$328$		−	9.00000i		−	0.496942i
$329$		0			0
$330$		0			0
$331$			3.00000i			0.164895i	0.996595	+	0.0824475i	$0.0262737\pi$
$331$			3.00000i			0.164895i	−0.996595	+	0.0824475i	$0.973726\pi$
$332$	−12.0000			−0.658586
$333$		0			0
$334$			6.00000i			0.328305i
$335$			9.00000i			0.491723i
$336$		0			0
$337$		−	12.0000i		−	0.653682i	−0.945079	−	0.326841i	$-0.894016\pi$
$337$		−	12.0000i		−	0.653682i	0.945079	−	0.326841i	$-0.105984\pi$
$338$			12.0000i			0.652714i
$339$		0			0
$340$		−	18.0000i		−	0.976187i
$341$		0			0
$342$		0			0
$343$			15.0000i			0.809924i
$344$		0			0
$345$		0			0
$346$	24.0000			1.29025
$347$	6.00000			0.322097			0.161048	−	0.986947i	$-0.448512\pi$
$347$	6.00000			0.322097			0.161048	+	0.986947i	$0.448512\pi$
$348$		0			0
$349$			6.00000i			0.321173i	0.987022	+	0.160586i	$0.0513385\pi$
$349$			6.00000i			0.321173i	−0.987022	+	0.160586i	$0.948662\pi$
$350$	−12.0000			−0.641427
$351$		0			0
$352$	−3.00000			−0.159901
$353$	3.00000			0.159674			0.0798369	−	0.996808i	$-0.474560\pi$
$353$	3.00000			0.159674			0.0798369	+	0.996808i	$0.474560\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$			24.0000i			1.26844i
$359$		−	24.0000i		−	1.26667i	−0.773877	−	0.633336i	$-0.781685\pi$
$359$		−	24.0000i		−	1.26667i	0.773877	−	0.633336i	$-0.218315\pi$
$360$		0			0
$361$	−15.0000			−0.789474
$362$	−12.0000			−0.630706
$363$		0			0
$364$			15.0000i			0.786214i
$365$	−21.0000			−1.09919
$366$		0			0
$367$	−8.00000			−0.417597			−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000			−0.417597			−0.208798	+	0.977959i	$0.566955\pi$
$368$		−	9.00000i		−	0.469157i
$369$		0			0
$370$	−18.0000			−0.935775
$371$		0			0
$372$		0			0
$373$		−	30.0000i		−	1.55334i	−0.629907	−	0.776671i	$-0.716907\pi$
$373$		−	30.0000i		−	1.55334i	0.629907	−	0.776671i	$-0.283093\pi$
$374$		−	18.0000i		−	0.930758i
$375$		0			0
$376$		0			0
$377$		−	30.0000i		−	1.54508i
$378$		0			0
$379$	2.00000			0.102733			0.0513665	−	0.998680i	$-0.483642\pi$
$379$	2.00000			0.102733			0.0513665	+	0.998680i	$0.483642\pi$
$380$	6.00000			0.307794
$381$		0			0
$382$	3.00000			0.153493
$383$		−	9.00000i		−	0.459879i	−0.973205	−	0.229939i	$-0.926147\pi$
$383$		−	9.00000i		−	0.459879i	0.973205	−	0.229939i	$-0.0738528\pi$
$384$		0			0
$385$	−27.0000			−1.37605
$386$	12.0000			0.610784
$387$		0			0
$388$	−10.0000			−0.507673
$389$		−	12.0000i		−	0.608424i	−0.952604	−	0.304212i	$-0.901607\pi$
$389$		−	12.0000i		−	0.608424i	0.952604	−	0.304212i	$-0.0983931\pi$
$390$		0			0
$391$	54.0000			2.73090
$392$			2.00000i			0.101015i
$393$		0			0
$394$			3.00000i			0.151138i
$395$		−	45.0000i		−	2.26420i
$396$		0			0
$397$		−	18.0000i		−	0.903394i	−0.892171	−	0.451697i	$-0.850819\pi$
$397$		−	18.0000i		−	0.903394i	0.892171	−	0.451697i	$-0.149181\pi$
$398$			20.0000i			1.00251i
$399$		0			0
$400$	4.00000			0.200000
$401$			24.0000i			1.19850i	0.800561	+	0.599251i	$0.204535\pi$
$401$			24.0000i			1.19850i	−0.800561	+	0.599251i	$0.795465\pi$
$402$		0			0
$403$		0			0
$404$		−	18.0000i		−	0.895533i
$405$		0			0
$406$		−	18.0000i		−	0.893325i
$407$	−18.0000			−0.892227
$408$		0			0
$409$			12.0000i			0.593362i	0.954977	+	0.296681i	$0.0958798\pi$
$409$			12.0000i			0.593362i	−0.954977	+	0.296681i	$0.904120\pi$
$410$			27.0000i			1.33343i
$411$		0			0
$412$	14.0000			0.689730
$413$	9.00000			0.442861
$414$		0			0
$415$	36.0000			1.76717
$416$		−	5.00000i		−	0.245145i
$417$		0			0
$418$	6.00000			0.293470
$419$		−	12.0000i		−	0.586238i	−0.956076	−	0.293119i	$-0.905307\pi$
$419$		−	12.0000i		−	0.586238i	0.956076	−	0.293119i	$-0.0946933\pi$
$420$		0			0
$421$		−	6.00000i		−	0.292422i	−0.989253	−	0.146211i	$-0.953292\pi$
$421$		−	6.00000i		−	0.292422i	0.989253	−	0.146211i	$-0.0467079\pi$
$422$	12.0000			0.584151
$423$		0			0
$424$		0			0
$425$			24.0000i			1.16417i
$426$		0			0
$427$	−18.0000	−	15.0000i	−0.871081	−	0.725901i
$428$	−12.0000			−0.580042
$429$		0			0
$430$		0			0
$431$	12.0000			0.578020			0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000			0.578020			0.289010	+	0.957326i	$0.406674\pi$
$432$		0			0
$433$		−	36.0000i		−	1.73005i	−0.501729	−	0.865025i	$-0.667303\pi$
$433$		−	36.0000i		−	1.73005i	0.501729	−	0.865025i	$-0.332697\pi$
$434$		0			0
$435$		0			0
$436$	−11.0000			−0.526804
$437$			18.0000i			0.861057i
$438$		0			0
$439$	−10.0000			−0.477274			−0.238637	−	0.971109i	$-0.576701\pi$
$439$	−10.0000			−0.477274			−0.238637	+	0.971109i	$0.576701\pi$
$440$	9.00000			0.429058
$441$		0			0
$442$	30.0000			1.42695
$443$		0			0			−	1.00000i	$-0.5\pi$
$443$		0			0				1.00000i	$0.5\pi$
$444$		0			0
$445$		0			0
$446$	−9.00000			−0.426162
$447$		0			0
$448$		−	3.00000i		−	0.141737i
$449$	−33.0000			−1.55737			−0.778683	−	0.627417i	$-0.784112\pi$
$449$	−33.0000			−1.55737			−0.778683	+	0.627417i	$0.784112\pi$
$450$		0			0
$451$			27.0000i			1.27138i
$452$	3.00000			0.141108
$453$		0			0
$454$	−3.00000			−0.140797
$455$		−	45.0000i		−	2.10963i
$456$		0			0
$457$			18.0000i			0.842004i	0.907060	+	0.421002i	$0.138322\pi$
$457$			18.0000i			0.842004i	−0.907060	+	0.421002i	$0.861678\pi$
$458$			13.0000i			0.607450i
$459$		0			0
$460$			27.0000i			1.25888i
$461$	18.0000			0.838344			0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000			0.838344			0.419172	+	0.907907i	$0.362320\pi$
$462$		0			0
$463$	−32.0000			−1.48717			−0.743583	−	0.668644i	$-0.766875\pi$
$463$	−32.0000			−1.48717			−0.743583	+	0.668644i	$0.766875\pi$
$464$			6.00000i			0.278543i
$465$		0			0
$466$	−6.00000			−0.277945
$467$			9.00000i			0.416470i	0.978079	+	0.208235i	$0.0667719\pi$
$467$			9.00000i			0.416470i	−0.978079	+	0.208235i	$0.933228\pi$
$468$		0			0
$469$	−9.00000			−0.415581
$470$		0			0
$471$		0			0
$472$	−3.00000			−0.138086
$473$		0			0
$474$		0			0
$475$	−8.00000			−0.367065
$476$	18.0000			0.825029
$477$		0			0
$478$			6.00000i			0.274434i
$479$	−12.0000			−0.548294			−0.274147	−	0.961688i	$-0.588395\pi$
$479$	−12.0000			−0.548294			−0.274147	+	0.961688i	$0.588395\pi$
$480$		0			0
$481$		−	30.0000i		−	1.36788i
$482$			1.00000i			0.0455488i
$483$		0			0
$484$	−2.00000			−0.0909091
$485$	30.0000			1.36223
$486$		0			0
$487$	2.00000			0.0906287			0.0453143	−	0.998973i	$-0.485571\pi$
$487$	2.00000			0.0906287			0.0453143	+	0.998973i	$0.485571\pi$
$488$	6.00000	+	5.00000i	0.271607	+	0.226339i
$489$		0			0
$490$		−	6.00000i		−	0.271052i
$491$	−6.00000			−0.270776			−0.135388	−	0.990793i	$-0.543228\pi$
$491$	−6.00000			−0.270776			−0.135388	+	0.990793i	$0.543228\pi$
$492$		0			0
$493$	−36.0000			−1.62136
$494$			10.0000i			0.449921i
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$			27.0000i			1.20869i	0.796724	+	0.604343i	$0.206564\pi$
$499$			27.0000i			1.20869i	−0.796724	+	0.604343i	$0.793436\pi$
$500$	3.00000			0.134164
$501$		0			0
$502$	24.0000			1.07117
$503$	−6.00000			−0.267527			−0.133763	−	0.991013i	$-0.542706\pi$
$503$	−6.00000			−0.267527			−0.133763	+	0.991013i	$0.542706\pi$
$504$		0			0
$505$			54.0000i			2.40297i
$506$			27.0000i			1.20030i
$507$		0			0
$508$	2.00000			0.0887357
$509$		−	18.0000i		−	0.797836i	−0.916987	−	0.398918i	$-0.869386\pi$
$509$		−	18.0000i		−	0.797836i	0.916987	−	0.398918i	$-0.130614\pi$
$510$		0			0
$511$		−	21.0000i		−	0.928985i
$512$			1.00000i			0.0441942i
$513$		0			0
$514$			6.00000i			0.264649i
$515$	−42.0000			−1.85074
$516$		0			0
$517$		0			0
$518$		−	18.0000i		−	0.790875i
$519$		0			0
$520$			15.0000i			0.657794i
$521$			18.0000i			0.788594i	0.918983	+	0.394297i	$0.129012\pi$
$521$			18.0000i			0.788594i	−0.918983	+	0.394297i	$0.870988\pi$
$522$		0			0
$523$			33.0000i			1.44299i	0.692420	+	0.721495i	$0.256545\pi$
$523$			33.0000i			1.44299i	−0.692420	+	0.721495i	$0.743455\pi$
$524$	18.0000			0.786334
$525$		0			0
$526$		−	18.0000i		−	0.784837i
$527$		0			0
$528$		0			0
$529$	−58.0000			−2.52174
$530$		0			0
$531$		0			0
$532$			6.00000i			0.260133i
$533$	−45.0000			−1.94917
$534$		0			0
$535$	36.0000			1.55642
$536$	3.00000			0.129580
$537$		0			0
$538$			6.00000i			0.258678i
$539$		−	6.00000i		−	0.258438i
$540$		0			0
$541$			12.0000i			0.515920i	0.966156	+	0.257960i	$0.0830503\pi$
$541$			12.0000i			0.515920i	−0.966156	+	0.257960i	$0.916950\pi$
$542$			2.00000i			0.0859074i
$543$		0			0
$544$	−6.00000			−0.257248
$545$	33.0000			1.41356
$546$		0			0
$547$		−	15.0000i		−	0.641354i	−0.947189	−	0.320677i	$-0.896090\pi$
$547$		−	15.0000i		−	0.641354i	0.947189	−	0.320677i	$-0.103910\pi$
$548$	−3.00000			−0.128154
$549$		0			0
$550$	−12.0000			−0.511682
$551$		−	12.0000i		−	0.511217i
$552$		0			0
$553$	45.0000			1.91359
$554$	−18.0000			−0.764747
$555$		0			0
$556$		−	15.0000i		−	0.636142i
$557$		−	24.0000i		−	1.01691i	−0.861088	−	0.508456i	$-0.830216\pi$
$557$		−	24.0000i		−	1.01691i	0.861088	−	0.508456i	$-0.169784\pi$
$558$		0			0
$559$		0			0
$560$			9.00000i			0.380319i
$561$		0			0
$562$	−12.0000			−0.506189
$563$	−18.0000			−0.758610			−0.379305	−	0.925272i	$-0.623837\pi$
$563$	−18.0000			−0.758610			−0.379305	+	0.925272i	$0.623837\pi$
$564$		0			0
$565$	−9.00000			−0.378633
$566$			14.0000i			0.588464i
$567$		0			0
$568$		0			0
$569$	27.0000			1.13190			0.565949	−	0.824440i	$-0.308510\pi$
$569$	27.0000			1.13190			0.565949	+	0.824440i	$0.308510\pi$
$570$		0			0
$571$	4.00000			0.167395			0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000			0.167395			0.0836974	+	0.996491i	$0.473327\pi$
$572$			15.0000i			0.627182i
$573$		0			0
$574$	−27.0000			−1.12696
$575$		−	36.0000i		−	1.50130i
$576$		0			0
$577$			6.00000i			0.249783i	0.992170	+	0.124892i	$0.0398583\pi$
$577$			6.00000i			0.249783i	−0.992170	+	0.124892i	$0.960142\pi$
$578$		−	19.0000i		−	0.790296i
$579$		0			0
$580$		−	18.0000i		−	0.747409i
$581$			36.0000i			1.49353i
$582$		0			0
$583$		0			0
$584$			7.00000i			0.289662i
$585$		0			0
$586$			30.0000i			1.23929i
$587$		−	12.0000i		−	0.495293i	−0.968850	−	0.247647i	$-0.920343\pi$
$587$		−	12.0000i		−	0.495293i	0.968850	−	0.247647i	$-0.0796572\pi$
$588$		0			0
$589$		0			0
$590$	9.00000			0.370524
$591$		0			0
$592$			6.00000i			0.246598i
$593$			6.00000i			0.246390i	0.992382	+	0.123195i	$0.0393141\pi$
$593$			6.00000i			0.246390i	−0.992382	+	0.123195i	$0.960686\pi$
$594$		0			0
$595$	−54.0000			−2.21378
$596$	−21.0000			−0.860194
$597$		0			0
$598$	−45.0000			−1.84019
$599$		−	15.0000i		−	0.612883i	−0.951889	−	0.306442i	$-0.900862\pi$
$599$		−	15.0000i		−	0.612883i	0.951889	−	0.306442i	$-0.0991384\pi$
$600$		0			0
$601$	19.0000			0.775026			0.387513	−	0.921864i	$-0.373334\pi$
$601$	19.0000			0.775026			0.387513	+	0.921864i	$0.373334\pi$
$602$		0			0
$603$		0			0
$604$		−	3.00000i		−	0.122068i
$605$	6.00000			0.243935
$606$		0			0
$607$	−40.0000			−1.62355			−0.811775	−	0.583970i	$-0.801498\pi$
$607$	−40.0000			−1.62355			−0.811775	+	0.583970i	$0.801498\pi$
$608$		−	2.00000i		−	0.0811107i
$609$		0			0
$610$	−18.0000	−	15.0000i	−0.728799	−	0.607332i
$611$		0			0
$612$		0			0
$613$	−2.00000			−0.0807792			−0.0403896	−	0.999184i	$-0.512860\pi$
$613$	−2.00000			−0.0807792			−0.0403896	+	0.999184i	$0.512860\pi$
$614$	27.0000			1.08963
$615$		0			0
$616$			9.00000i			0.362620i
$617$		−	36.0000i		−	1.44931i	−0.689114	−	0.724653i	$-0.742000\pi$
$617$		−	36.0000i		−	1.44931i	0.689114	−	0.724653i	$-0.258000\pi$
$618$		0			0
$619$	−8.00000			−0.321547			−0.160774	−	0.986991i	$-0.551399\pi$
$619$	−8.00000			−0.321547			−0.160774	+	0.986991i	$0.551399\pi$
$620$		0			0
$621$		0			0
$622$	−3.00000			−0.120289
$623$		0			0
$624$		0			0
$625$	−29.0000			−1.16000
$626$	12.0000			0.479616
$627$		0			0
$628$			18.0000i			0.718278i
$629$	−36.0000			−1.43541
$630$		0			0
$631$		−	39.0000i		−	1.55257i	−0.630385	−	0.776283i	$-0.717103\pi$
$631$		−	39.0000i		−	1.55257i	0.630385	−	0.776283i	$-0.282897\pi$
$632$	−15.0000			−0.596668
$633$		0			0
$634$		−	18.0000i		−	0.714871i
$635$	−6.00000			−0.238103
$636$		0			0
$637$	10.0000			0.396214
$638$		−	18.0000i		−	0.712627i
$639$		0			0
$640$		−	3.00000i		−	0.118585i
$641$			12.0000i			0.473972i	0.971513	+	0.236986i	$0.0761595\pi$
$641$			12.0000i			0.473972i	−0.971513	+	0.236986i	$0.923841\pi$
$642$		0			0
$643$			36.0000i			1.41970i	0.704352	+	0.709851i	$0.251238\pi$
$643$			36.0000i			1.41970i	−0.704352	+	0.709851i	$0.748762\pi$
$644$	−27.0000			−1.06395
$645$		0			0
$646$	12.0000			0.472134
$647$			21.0000i			0.825595i	0.910823	+	0.412798i	$0.135448\pi$
$647$			21.0000i			0.825595i	−0.910823	+	0.412798i	$0.864552\pi$
$648$		0			0
$649$	9.00000			0.353281
$650$		−	20.0000i		−	0.784465i
$651$		0			0
$652$	2.00000			0.0783260
$653$			36.0000i			1.40879i	0.709809	+	0.704394i	$0.248781\pi$
$653$			36.0000i			1.40879i	−0.709809	+	0.704394i	$0.751219\pi$
$654$		0			0
$655$	−54.0000			−2.10995
$656$	9.00000			0.351391
$657$		0			0
$658$		0			0
$659$	−18.0000			−0.701180			−0.350590	−	0.936529i	$-0.614019\pi$
$659$	−18.0000			−0.701180			−0.350590	+	0.936529i	$0.614019\pi$
$660$		0			0
$661$			30.0000i			1.16686i	0.812162	+	0.583432i	$0.198291\pi$
$661$			30.0000i			1.16686i	−0.812162	+	0.583432i	$0.801709\pi$
$662$	−3.00000			−0.116598
$663$		0			0
$664$		−	12.0000i		−	0.465690i
$665$		−	18.0000i		−	0.698010i
$666$		0			0
$667$	54.0000			2.09089
$668$	−6.00000			−0.232147
$669$		0			0
$670$	−9.00000			−0.347700
$671$	−18.0000	−	15.0000i	−0.694882	−	0.579069i
$672$		0			0
$673$			6.00000i			0.231283i	0.993291	+	0.115642i	$0.0368924\pi$
$673$			6.00000i			0.231283i	−0.993291	+	0.115642i	$0.963108\pi$
$674$	12.0000			0.462223
$675$		0			0
$676$	−12.0000			−0.461538
$677$			36.0000i			1.38359i	0.722093	+	0.691796i	$0.243180\pi$
$677$			36.0000i			1.38359i	−0.722093	+	0.691796i	$0.756820\pi$
$678$		0			0
$679$			30.0000i			1.15129i
$680$	18.0000			0.690268
$681$		0			0
$682$		0			0
$683$	30.0000			1.14792			0.573959	−	0.818884i	$-0.305407\pi$
$683$	30.0000			1.14792			0.573959	+	0.818884i	$0.305407\pi$
$684$		0			0
$685$	9.00000			0.343872
$686$	−15.0000			−0.572703
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	28.0000			1.06517			0.532585	−	0.846376i	$-0.321221\pi$
$691$	28.0000			1.06517			0.532585	+	0.846376i	$0.321221\pi$
$692$			24.0000i			0.912343i
$693$		0			0
$694$			6.00000i			0.227757i
$695$			45.0000i			1.70695i
$696$		0			0
$697$			54.0000i			2.04540i
$698$	−6.00000			−0.227103
$699$		0			0
$700$		−	12.0000i		−	0.453557i
$701$			24.0000i			0.906467i	0.891392	+	0.453234i	$0.149730\pi$
$701$			24.0000i			0.906467i	−0.891392	+	0.453234i	$0.850270\pi$
$702$		0			0
$703$		−	12.0000i		−	0.452589i
$704$		−	3.00000i		−	0.113067i
$705$		0			0
$706$			3.00000i			0.112906i
$707$	−54.0000			−2.03088
$708$		0			0
$709$			6.00000i			0.225335i	0.993633	+	0.112667i	$0.0359394\pi$
$709$			6.00000i			0.225335i	−0.993633	+	0.112667i	$0.964061\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		−	45.0000i		−	1.68290i
$716$	−24.0000			−0.896922
$717$		0			0
$718$	24.0000			0.895672
$719$	18.0000			0.671287			0.335643	−	0.941989i	$-0.391046\pi$
$719$	18.0000			0.671287			0.335643	+	0.941989i	$0.391046\pi$
$720$		0			0
$721$		−	42.0000i		−	1.56416i
$722$		−	15.0000i		−	0.558242i
$723$		0			0
$724$		−	12.0000i		−	0.445976i
$725$			24.0000i			0.891338i
$726$		0			0
$727$	−8.00000			−0.296704			−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000			−0.296704			−0.148352	+	0.988935i	$0.547397\pi$
$728$	−15.0000			−0.555937
$729$		0			0
$730$		−	21.0000i		−	0.777245i
$731$		0			0
$732$		0			0
$733$	−13.0000			−0.480166			−0.240083	−	0.970752i	$-0.577175\pi$
$733$	−13.0000			−0.480166			−0.240083	+	0.970752i	$0.577175\pi$
$734$		−	8.00000i		−	0.295285i
$735$		0			0
$736$	9.00000			0.331744
$737$	−9.00000			−0.331519
$738$		0			0
$739$			9.00000i			0.331070i	0.986204	+	0.165535i	$0.0529351\pi$
$739$			9.00000i			0.331070i	−0.986204	+	0.165535i	$0.947065\pi$
$740$		−	18.0000i		−	0.661693i
$741$		0			0
$742$		0			0
$743$		−	3.00000i		−	0.110059i	−0.998485	−	0.0550297i	$-0.982475\pi$
$743$		−	3.00000i		−	0.110059i	0.998485	−	0.0550297i	$-0.0175253\pi$
$744$		0			0
$745$	63.0000			2.30814
$746$	30.0000			1.09838
$747$		0			0
$748$	18.0000			0.658145
$749$			36.0000i			1.31541i
$750$		0			0
$751$	−4.00000			−0.145962			−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000			−0.145962			−0.0729810	+	0.997333i	$0.523251\pi$
$752$		0			0
$753$		0			0
$754$	30.0000			1.09254
$755$			9.00000i			0.327544i
$756$		0			0
$757$	−43.0000			−1.56286			−0.781431	−	0.623992i	$-0.785510\pi$
$757$	−43.0000			−1.56286			−0.781431	+	0.623992i	$0.785510\pi$
$758$			2.00000i			0.0726433i
$759$		0			0
$760$			6.00000i			0.217643i
$761$			30.0000i			1.08750i	0.839248	+	0.543750i	$0.182996\pi$
$761$			30.0000i			1.08750i	−0.839248	+	0.543750i	$0.817004\pi$
$762$		0			0
$763$			33.0000i			1.19468i
$764$			3.00000i			0.108536i
$765$		0			0
$766$	9.00000			0.325183
$767$			15.0000i			0.541619i
$768$		0			0
$769$			42.0000i			1.51456i	0.653091	+	0.757279i	$0.273472\pi$
$769$			42.0000i			1.51456i	−0.653091	+	0.757279i	$0.726528\pi$
$770$		−	27.0000i		−	0.973012i
$771$		0			0
$772$			12.0000i			0.431889i
$773$	6.00000			0.215805			0.107903	−	0.994161i	$-0.465587\pi$
$773$	6.00000			0.215805			0.107903	+	0.994161i	$0.465587\pi$
$774$		0			0
$775$		0			0
$776$		−	10.0000i		−	0.358979i
$777$		0			0
$778$	12.0000			0.430221
$779$	−18.0000			−0.644917
$780$		0			0
$781$		0			0
$782$			54.0000i			1.93104i
$783$		0			0
$784$	−2.00000			−0.0714286
$785$		−	54.0000i		−	1.92734i
$786$		0			0
$787$			24.0000i			0.855508i	0.903895	+	0.427754i	$0.140695\pi$
$787$			24.0000i			0.855508i	−0.903895	+	0.427754i	$0.859305\pi$
$788$	−3.00000			−0.106871
$789$		0			0
$790$	45.0000			1.60103
$791$		−	9.00000i		−	0.320003i
$792$		0			0
$793$	25.0000	−	30.0000i	0.887776	−	1.06533i
$794$	18.0000			0.638796
$795$		0			0
$796$	−20.0000			−0.708881
$797$	−9.00000			−0.318796			−0.159398	−	0.987214i	$-0.550955\pi$
$797$	−9.00000			−0.318796			−0.159398	+	0.987214i	$0.550955\pi$
$798$		0			0
$799$		0			0
$800$			4.00000i			0.141421i
$801$		0			0
$802$	−24.0000			−0.847469
$803$		−	21.0000i		−	0.741074i
$804$		0			0
$805$	81.0000			2.85487
$806$		0			0
$807$		0			0
$808$	18.0000			0.633238
$809$	27.0000			0.949269			0.474635	−	0.880183i	$-0.342580\pi$
$809$	27.0000			0.949269			0.474635	+	0.880183i	$0.342580\pi$
$810$		0			0
$811$		−	45.0000i		−	1.58016i	−0.613001	−	0.790082i	$-0.710038\pi$
$811$		−	45.0000i		−	1.58016i	0.613001	−	0.790082i	$-0.289962\pi$
$812$	18.0000			0.631676
$813$		0			0
$814$		−	18.0000i		−	0.630900i
$815$	−6.00000			−0.210171
$816$		0			0
$817$		0			0
$818$	−12.0000			−0.419570
$819$		0			0
$820$	−27.0000			−0.942881
$821$		−	36.0000i		−	1.25641i	−0.778048	−	0.628204i	$-0.783790\pi$
$821$		−	36.0000i		−	1.25641i	0.778048	−	0.628204i	$-0.216210\pi$
$822$		0			0
$823$			12.0000i			0.418294i	0.977884	+	0.209147i	$0.0670687\pi$
$823$			12.0000i			0.418294i	−0.977884	+	0.209147i	$0.932931\pi$
$824$			14.0000i			0.487713i
$825$		0			0
$826$			9.00000i			0.313150i
$827$	42.0000			1.46048			0.730242	−	0.683189i	$-0.239408\pi$
$827$	42.0000			1.46048			0.730242	+	0.683189i	$0.239408\pi$
$828$		0			0
$829$	−25.0000			−0.868286			−0.434143	−	0.900844i	$-0.642949\pi$
$829$	−25.0000			−0.868286			−0.434143	+	0.900844i	$0.642949\pi$
$830$			36.0000i			1.24958i
$831$		0			0
$832$	5.00000			0.173344
$833$		−	12.0000i		−	0.415775i
$834$		0			0
$835$	18.0000			0.622916
$836$			6.00000i			0.207514i
$837$		0			0
$838$	12.0000			0.414533
$839$	48.0000			1.65714			0.828572	−	0.559883i	$-0.189154\pi$
$839$	48.0000			1.65714			0.828572	+	0.559883i	$0.189154\pi$
$840$		0			0
$841$	−7.00000			−0.241379
$842$	6.00000			0.206774
$843$		0			0
$844$			12.0000i			0.413057i
$845$	36.0000			1.23844
$846$		0			0
$847$			6.00000i			0.206162i
$848$		0			0
$849$		0			0
$850$	−24.0000			−0.823193
$851$	54.0000			1.85110
$852$		0			0
$853$	17.0000			0.582069			0.291034	−	0.956713i	$-0.406001\pi$
$853$	17.0000			0.582069			0.291034	+	0.956713i	$0.406001\pi$
$854$	15.0000	−	18.0000i	0.513289	−	0.615947i
$855$		0			0
$856$		−	12.0000i		−	0.410152i
$857$	−39.0000			−1.33221			−0.666107	−	0.745856i	$-0.732041\pi$
$857$	−39.0000			−1.33221			−0.666107	+	0.745856i	$0.732041\pi$
$858$		0			0
$859$	14.0000			0.477674			0.238837	−	0.971060i	$-0.423234\pi$
$859$	14.0000			0.477674			0.238837	+	0.971060i	$0.423234\pi$
$860$		0			0
$861$		0			0
$862$			12.0000i			0.408722i
$863$	−24.0000			−0.816970			−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000			−0.816970			−0.408485	+	0.912765i	$0.633943\pi$
$864$		0			0
$865$		−	72.0000i		−	2.44807i
$866$	36.0000			1.22333
$867$		0			0
$868$		0			0
$869$	45.0000			1.52652
$870$		0			0
$871$		−	15.0000i		−	0.508256i
$872$		−	11.0000i		−	0.372507i
$873$		0			0
$874$	−18.0000			−0.608859
$875$		−	9.00000i		−	0.304256i
$876$		0			0
$877$		−	12.0000i		−	0.405211i	−0.979260	−	0.202606i	$-0.935059\pi$
$877$		−	12.0000i		−	0.405211i	0.979260	−	0.202606i	$-0.0649409\pi$
$878$		−	10.0000i		−	0.337484i
$879$		0			0
$880$			9.00000i			0.303390i
$881$	15.0000			0.505363			0.252681	−	0.967550i	$-0.418688\pi$
$881$	15.0000			0.505363			0.252681	+	0.967550i	$0.418688\pi$
$882$		0			0
$883$		−	21.0000i		−	0.706706i	−0.935490	−	0.353353i	$-0.885041\pi$
$883$		−	21.0000i		−	0.706706i	0.935490	−	0.353353i	$-0.114959\pi$
$884$			30.0000i			1.00901i
$885$		0			0
$886$		0			0
$887$		−	12.0000i		−	0.402921i	−0.979497	−	0.201460i	$-0.935431\pi$
$887$		−	12.0000i		−	0.402921i	0.979497	−	0.201460i	$-0.0645687\pi$
$888$		0			0
$889$		−	6.00000i		−	0.201234i
$890$		0			0
$891$		0			0
$892$		−	9.00000i		−	0.301342i
$893$		0			0
$894$		0			0
$895$	72.0000			2.40669
$896$	3.00000			0.100223
$897$		0			0
$898$		−	33.0000i		−	1.10122i
$899$		0			0
$900$		0			0
$901$		0			0
$902$	−27.0000			−0.899002
$903$		0			0
$904$			3.00000i			0.0997785i
$905$			36.0000i			1.19668i
$906$		0			0
$907$		−	12.0000i		−	0.398453i	−0.979953	−	0.199227i	$-0.936157\pi$
$907$		−	12.0000i		−	0.398453i	0.979953	−	0.199227i	$-0.0638430\pi$
$908$		−	3.00000i		−	0.0995585i
$909$		0			0
$910$	45.0000			1.49174
$911$	42.0000			1.39152			0.695761	−	0.718273i	$-0.255067\pi$
$911$	42.0000			1.39152			0.695761	+	0.718273i	$0.255067\pi$
$912$		0			0
$913$			36.0000i			1.19143i
$914$	−18.0000			−0.595387
$915$		0			0
$916$	−13.0000			−0.429532
$917$		−	54.0000i		−	1.78324i
$918$		0			0
$919$	−52.0000			−1.71532			−0.857661	−	0.514216i	$-0.828083\pi$
$919$	−52.0000			−1.71532			−0.857661	+	0.514216i	$0.828083\pi$
$920$	−27.0000			−0.890164
$921$		0			0
$922$			18.0000i			0.592798i
$923$		0			0
$924$		0			0
$925$			24.0000i			0.789115i
$926$		−	32.0000i		−	1.05159i
$927$		0			0
$928$	−6.00000			−0.196960
$929$	−15.0000			−0.492134			−0.246067	−	0.969253i	$-0.579138\pi$
$929$	−15.0000			−0.492134			−0.246067	+	0.969253i	$0.579138\pi$
$930$		0			0
$931$	4.00000			0.131095
$932$		−	6.00000i		−	0.196537i
$933$		0			0
$934$	−9.00000			−0.294489
$935$	−54.0000			−1.76599
$936$		0			0
$937$	−29.0000			−0.947389			−0.473694	−	0.880689i	$-0.657080\pi$
$937$	−29.0000			−0.947389			−0.473694	+	0.880689i	$0.657080\pi$
$938$		−	9.00000i		−	0.293860i
$939$		0			0
$940$		0			0
$941$			6.00000i			0.195594i	0.995206	+	0.0977972i	$0.0311797\pi$
$941$			6.00000i			0.195594i	−0.995206	+	0.0977972i	$0.968820\pi$
$942$		0			0
$943$		−	81.0000i		−	2.63772i
$944$		−	3.00000i		−	0.0976417i
$945$		0			0
$946$		0			0
$947$		−	15.0000i		−	0.487435i	−0.969846	−	0.243717i	$-0.921633\pi$
$947$		−	15.0000i		−	0.487435i	0.969846	−	0.243717i	$-0.0783669\pi$
$948$		0			0
$949$	35.0000			1.13615
$950$		−	8.00000i		−	0.259554i
$951$		0			0
$952$			18.0000i			0.583383i
$953$			30.0000i			0.971795i	0.874016	+	0.485898i	$0.161507\pi$
$953$			30.0000i			0.971795i	−0.874016	+	0.485898i	$0.838493\pi$
$954$		0			0
$955$		−	9.00000i		−	0.291233i
$956$	−6.00000			−0.194054
$957$		0			0
$958$		−	12.0000i		−	0.387702i
$959$			9.00000i			0.290625i
$960$		0			0
$961$	31.0000			1.00000
$962$	30.0000			0.967239
$963$		0			0
$964$	−1.00000			−0.0322078
$965$		−	36.0000i		−	1.15888i
$966$		0			0
$967$	−58.0000			−1.86515			−0.932577	−	0.360971i	$-0.882445\pi$
$967$	−58.0000			−1.86515			−0.932577	+	0.360971i	$0.882445\pi$
$968$		−	2.00000i		−	0.0642824i
$969$		0			0
$970$			30.0000i			0.963242i
$971$	−12.0000			−0.385098			−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000			−0.385098			−0.192549	+	0.981287i	$0.561675\pi$
$972$		0			0
$973$	−45.0000			−1.44263
$974$			2.00000i			0.0640841i
$975$		0			0
$976$	−5.00000	+	6.00000i	−0.160046	+	0.192055i
$977$	−30.0000			−0.959785			−0.479893	−	0.877327i	$-0.659324\pi$
$977$	−30.0000			−0.959785			−0.479893	+	0.877327i	$0.659324\pi$
$978$		0			0
$979$		0			0
$980$	6.00000			0.191663
$981$		0			0
$982$		−	6.00000i		−	0.191468i
$983$			48.0000i			1.53096i	0.643458	+	0.765481i	$0.277499\pi$
$983$			48.0000i			1.53096i	−0.643458	+	0.765481i	$0.722501\pi$
$984$		0			0
$985$	9.00000			0.286764
$986$		−	36.0000i		−	1.14647i
$987$		0			0
$988$	−10.0000			−0.318142
$989$		0			0
$990$		0			0
$991$	16.0000			0.508257			0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000			0.508257			0.254128	+	0.967170i	$0.418211\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	60.0000			1.90213
$996$		0			0
$997$		−	18.0000i		−	0.570066i	−0.958518	−	0.285033i	$-0.907995\pi$
$997$		−	18.0000i		−	0.570066i	0.958518	−	0.285033i	$-0.0920045\pi$
$998$	−27.0000			−0.854670
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1098.2.d.a.487.2		2
3.2	odd	2		122.2.b.a.121.1	✓	2
12.11	even	2		976.2.h.a.609.1		2
15.2	even	4		3050.2.c.c.3049.2		2
15.8	even	4		3050.2.c.b.3049.1		2
15.14	odd	2		3050.2.d.c.1951.2		2
61.60	even	2	inner	1098.2.d.a.487.1		2
183.11	even	4		7442.2.a.c.1.1		1
183.50	even	4		7442.2.a.a.1.1		1
183.182	odd	2		122.2.b.a.121.2	yes	2
732.731	even	2		976.2.h.a.609.2		2
915.182	even	4		3050.2.c.b.3049.2		2
915.548	even	4		3050.2.c.c.3049.1		2
915.914	odd	2		3050.2.d.c.1951.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
122.2.b.a.121.1	✓	2	3.2	odd	2
122.2.b.a.121.2	yes	2	183.182	odd	2
976.2.h.a.609.1		2	12.11	even	2
976.2.h.a.609.2		2	732.731	even	2
1098.2.d.a.487.1		2	61.60	even	2	inner
1098.2.d.a.487.2		2	1.1	even	1	trivial
3050.2.c.b.3049.1		2	15.8	even	4
3050.2.c.b.3049.2		2	915.182	even	4
3050.2.c.c.3049.1		2	915.548	even	4
3050.2.c.c.3049.2		2	15.2	even	4
3050.2.d.c.1951.1		2	915.914	odd	2
3050.2.d.c.1951.2		2	15.14	odd	2
7442.2.a.a.1.1		1	183.50	even	4
7442.2.a.c.1.1		1	183.11	even	4

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:	\( N \)	\(=\)	\( 1098 = 2 \cdot 3^{2} \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1098.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +3.00000 q^{5} +3.00000i q^{7} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} +3.00000 q^{5} +3.00000i q^{7} -1.00000i q^{8} +3.00000i q^{10} +3.00000i q^{11} -5.00000 q^{13} -3.00000 q^{14} +1.00000 q^{16} +6.00000i q^{17} -2.00000 q^{19} -3.00000 q^{20} -3.00000 q^{22} -9.00000i q^{23} +4.00000 q^{25} -5.00000i q^{26} -3.00000i q^{28} +6.00000i q^{29} +1.00000i q^{32} -6.00000 q^{34} +9.00000i q^{35} +6.00000i q^{37} -2.00000i q^{38} -3.00000i q^{40} +9.00000 q^{41} -3.00000i q^{44} +9.00000 q^{46} -2.00000 q^{49} +4.00000i q^{50} +5.00000 q^{52} +9.00000i q^{55} +3.00000 q^{56} -6.00000 q^{58} -3.00000i q^{59} +(-5.00000 + 6.00000i) q^{61} -1.00000 q^{64} -15.0000 q^{65} +3.00000i q^{67} -6.00000i q^{68} -9.00000 q^{70} -7.00000 q^{73} -6.00000 q^{74} +2.00000 q^{76} -9.00000 q^{77} -15.0000i q^{79} +3.00000 q^{80} +9.00000i q^{82} +12.0000 q^{83} +18.0000i q^{85} +3.00000 q^{88} -15.0000i q^{91} +9.00000i q^{92} -6.00000 q^{95} +10.0000 q^{97} -2.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 6 q^{5}+O(q^{10}) \) 2 * q - 2 * q^4 + 6 * q^5	\( 2 q - 2 q^{4} + 6 q^{5} - 10 q^{13} - 6 q^{14} + 2 q^{16} - 4 q^{19} - 6 q^{20} - 6 q^{22} + 8 q^{25} - 12 q^{34} + 18 q^{41} + 18 q^{46} - 4 q^{49} + 10 q^{52} + 6 q^{56} - 12 q^{58} - 10 q^{61} - 2 q^{64} - 30 q^{65} - 18 q^{70} - 14 q^{73} - 12 q^{74} + 4 q^{76} - 18 q^{77} + 6 q^{80} + 24 q^{83} + 6 q^{88} - 12 q^{95} + 20 q^{97}+O(q^{100}) \) 2 * q - 2 * q^4 + 6 * q^5 - 10 * q^13 - 6 * q^14 + 2 * q^16 - 4 * q^19 - 6 * q^20 - 6 * q^22 + 8 * q^25 - 12 * q^34 + 18 * q^41 + 18 * q^46 - 4 * q^49 + 10 * q^52 + 6 * q^56 - 12 * q^58 - 10 * q^61 - 2 * q^64 - 30 * q^65 - 18 * q^70 - 14 * q^73 - 12 * q^74 + 4 * q^76 - 18 * q^77 + 6 * q^80 + 24 * q^83 + 6 * q^88 - 12 * q^95 + 20 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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