LMFDB - Embedded newform 684.2.c.a.647.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [684,2,Mod(647,684)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(684, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("684.647");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 684 = 2^{2} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	684.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$5.46176749826$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			647.2
Root			$0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	684.647
Dual form			684.2.c.a.647.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{2} -2.00000 q^{4} -1.41421i q^{5} +4.00000i q^{7} +2.82843i q^{8} +O(q^{10})$	$q-1.41421i q^{2} -2.00000 q^{4} -1.41421i q^{5} +4.00000i q^{7} +2.82843i q^{8} -2.00000 q^{10} +2.82843 q^{11} +5.65685 q^{14} +4.00000 q^{16} +1.41421i q^{17} +1.00000i q^{19} +2.82843i q^{20} -4.00000i q^{22} +8.48528 q^{23} +3.00000 q^{25} -8.00000i q^{28} -1.41421i q^{29} +4.00000i q^{31} -5.65685i q^{32} +2.00000 q^{34} +5.65685 q^{35} -6.00000 q^{37} +1.41421 q^{38} +4.00000 q^{40} +7.07107i q^{41} -8.00000i q^{43} -5.65685 q^{44} -12.0000i q^{46} +2.82843 q^{47} -9.00000 q^{49} -4.24264i q^{50} -7.07107i q^{53} -4.00000i q^{55} -11.3137 q^{56} -2.00000 q^{58} +11.3137 q^{59} -2.00000 q^{61} +5.65685 q^{62} -8.00000 q^{64} +12.0000i q^{67} -2.82843i q^{68} -8.00000i q^{70} -5.65685 q^{71} +16.0000 q^{73} +8.48528i q^{74} -2.00000i q^{76} +11.3137i q^{77} +8.00000i q^{79} -5.65685i q^{80} +10.0000 q^{82} -2.82843 q^{83} +2.00000 q^{85} -11.3137 q^{86} +8.00000i q^{88} -9.89949i q^{89} -16.9706 q^{92} -4.00000i q^{94} +1.41421 q^{95} +12.0000 q^{97} +12.7279i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{4}+O(q^{10}) $ 4 * q - 8 * q^4	$ 4 q - 8 q^{4} - 8 q^{10} + 16 q^{16} + 12 q^{25} + 8 q^{34} - 24 q^{37} + 16 q^{40} - 36 q^{49} - 8 q^{58} - 8 q^{61} - 32 q^{64} + 64 q^{73} + 40 q^{82} + 8 q^{85} + 48 q^{97}+O(q^{100}) $ 4 * q - 8 * q^4 - 8 * q^10 + 16 * q^16 + 12 * q^25 + 8 * q^34 - 24 * q^37 + 16 * q^40 - 36 * q^49 - 8 * q^58 - 8 * q^61 - 32 * q^64 + 64 * q^73 + 40 * q^82 + 8 * q^85 + 48 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/684\mathbb{Z}\right)^\times$.

$n$	$325$	$343$	$533$
$\chi(n)$	$1$	$-1$	$-1$

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 $

$2$	− 1.41421i	− 1.00000i
$2$	− 1.41421i	− 1.00000i
$3$	0	0
$3$	0	0
$4$	−2.00000	−1.00000
$5$	− 1.41421i	− 0.632456i	−0.948683	−	0.316228i	$-0.897584\pi$
$5$	− 1.41421i	− 0.632456i	0.948683	−	0.316228i	$-0.102416\pi$
$6$	0	0
$7$	4.00000i	1.51186i	0.654654	+	0.755929i	$0.272814\pi$
$7$	4.00000i	1.51186i	−0.654654	+	0.755929i	$0.727186\pi$
$8$	2.82843i	1.00000i
$9$	0	0
$10$	−2.00000	−0.632456
$11$	2.82843	0.852803	0.426401	−	0.904534i	$-0.359781\pi$
$11$	2.82843	0.852803	0.426401	+	0.904534i	$0.359781\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	5.65685	1.51186
$15$	0	0
$16$	4.00000	1.00000
$17$	1.41421i	0.342997i	0.985184	+	0.171499i	$0.0548609\pi$
$17$	1.41421i	0.342997i	−0.985184	+	0.171499i	$0.945139\pi$
$18$	0	0
$19$	1.00000i	0.229416i
$19$	1.00000i	0.229416i
$20$	2.82843i	0.632456i
$21$	0	0
$22$	− 4.00000i	− 0.852803i
$23$	8.48528	1.76930	0.884652	−	0.466252i	$-0.154396\pi$
$23$	8.48528	1.76930	0.884652	+	0.466252i	$0.154396\pi$
$24$	0	0
$25$	3.00000	0.600000
$26$	0	0
$27$	0	0
$28$	− 8.00000i	− 1.51186i
$29$	− 1.41421i	− 0.262613i	−0.991342	−	0.131306i	$-0.958083\pi$
$29$	− 1.41421i	− 0.262613i	0.991342	−	0.131306i	$-0.0419172\pi$
$30$	0	0
$31$	4.00000i	0.718421i	0.933257	+	0.359211i	$0.116954\pi$
$31$	4.00000i	0.718421i	−0.933257	+	0.359211i	$0.883046\pi$
$32$	− 5.65685i	− 1.00000i
$33$	0	0
$34$	2.00000	0.342997
$35$	5.65685	0.956183
$36$	0	0
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	1.41421	0.229416
$39$	0	0
$40$	4.00000	0.632456
$41$	7.07107i	1.10432i	0.833740	+	0.552158i	$0.186195\pi$
$41$	7.07107i	1.10432i	−0.833740	+	0.552158i	$0.813805\pi$
$42$	0	0
$43$	− 8.00000i	− 1.21999i	−0.792406	−	0.609994i	$-0.791172\pi$
$43$	− 8.00000i	− 1.21999i	0.792406	−	0.609994i	$-0.208828\pi$
$44$	−5.65685	−0.852803
$45$	0	0
$46$	− 12.0000i	− 1.76930i
$47$	2.82843	0.412568	0.206284	−	0.978492i	$-0.433863\pi$
$47$	2.82843	0.412568	0.206284	+	0.978492i	$0.433863\pi$
$48$	0	0
$49$	−9.00000	−1.28571
$50$	− 4.24264i	− 0.600000i
$51$	0	0
$52$	0	0
$53$	− 7.07107i	− 0.971286i	−0.874157	−	0.485643i	$-0.838586\pi$
$53$	− 7.07107i	− 0.971286i	0.874157	−	0.485643i	$-0.161414\pi$
$54$	0	0
$55$	− 4.00000i	− 0.539360i
$56$	−11.3137	−1.51186
$57$	0	0
$58$	−2.00000	−0.262613
$59$	11.3137	1.47292	0.736460	−	0.676481i	$-0.236496\pi$
$59$	11.3137	1.47292	0.736460	+	0.676481i	$0.236496\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	5.65685	0.718421
$63$	0	0
$64$	−8.00000	−1.00000
$65$	0	0
$66$	0	0
$67$	12.0000i	1.46603i	0.680211	+	0.733017i	$0.261888\pi$
$67$	12.0000i	1.46603i	−0.680211	+	0.733017i	$0.738112\pi$
$68$	− 2.82843i	− 0.342997i
$69$	0	0
$70$	− 8.00000i	− 0.956183i
$71$	−5.65685	−0.671345	−0.335673	−	0.941979i	$-0.608964\pi$
$71$	−5.65685	−0.671345	−0.335673	+	0.941979i	$0.608964\pi$
$72$	0	0
$73$	16.0000	1.87266	0.936329	−	0.351123i	$-0.114200\pi$
$73$	16.0000	1.87266	0.936329	+	0.351123i	$0.114200\pi$
$74$	8.48528i	0.986394i
$75$	0	0
$76$	− 2.00000i	− 0.229416i
$77$	11.3137i	1.28932i
$78$	0	0
$79$	8.00000i	0.900070i	0.893011	+	0.450035i	$0.148589\pi$
$79$	8.00000i	0.900070i	−0.893011	+	0.450035i	$0.851411\pi$
$80$	− 5.65685i	− 0.632456i
$81$	0	0
$82$	10.0000	1.10432
$83$	−2.82843	−0.310460	−0.155230	−	0.987878i	$-0.549612\pi$
$83$	−2.82843	−0.310460	−0.155230	+	0.987878i	$0.549612\pi$
$84$	0	0
$85$	2.00000	0.216930
$86$	−11.3137	−1.21999
$87$	0	0
$88$	8.00000i	0.852803i
$89$	− 9.89949i	− 1.04934i	−0.851304	−	0.524672i	$-0.824188\pi$
$89$	− 9.89949i	− 1.04934i	0.851304	−	0.524672i	$-0.175812\pi$
$90$	0	0
$91$	0	0
$92$	−16.9706	−1.76930
$93$	0	0
$94$	− 4.00000i	− 0.412568i
$95$	1.41421	0.145095
$96$	0	0
$97$	12.0000	1.21842	0.609208	−	0.793011i	$-0.291488\pi$
$97$	12.0000	1.21842	0.609208	+	0.793011i	$0.291488\pi$
$98$	12.7279i	1.28571i
$99$	0	0
$100$	−6.00000	−0.600000
$101$	− 1.41421i	− 0.140720i	−0.997522	−	0.0703598i	$-0.977585\pi$
$101$	− 1.41421i	− 0.140720i	0.997522	−	0.0703598i	$-0.0224147\pi$
$102$	0	0
$103$	16.0000i	1.57653i	0.615338	+	0.788263i	$0.289020\pi$
$103$	16.0000i	1.57653i	−0.615338	+	0.788263i	$0.710980\pi$
$104$	0	0
$105$	0	0
$106$	−10.0000	−0.971286
$107$	−16.9706	−1.64061	−0.820303	−	0.571929i	$-0.806195\pi$
$107$	−16.9706	−1.64061	−0.820303	+	0.571929i	$0.806195\pi$
$108$	0	0
$109$	−8.00000	−0.766261	−0.383131	−	0.923694i	$-0.625154\pi$
$109$	−8.00000	−0.766261	−0.383131	+	0.923694i	$0.625154\pi$
$110$	−5.65685	−0.539360
$111$	0	0
$112$	16.0000i	1.51186i
$113$	7.07107i	0.665190i	0.943070	+	0.332595i	$0.107924\pi$
$113$	7.07107i	0.665190i	−0.943070	+	0.332595i	$0.892076\pi$
$114$	0	0
$115$	− 12.0000i	− 1.11901i
$116$	2.82843i	0.262613i
$117$	0	0
$118$	− 16.0000i	− 1.47292i
$119$	−5.65685	−0.518563
$120$	0	0
$121$	−3.00000	−0.272727
$122$	2.82843i	0.256074i
$123$	0	0
$124$	− 8.00000i	− 0.718421i
$125$	− 11.3137i	− 1.01193i
$126$	0	0
$127$	− 4.00000i	− 0.354943i	−0.984126	−	0.177471i	$-0.943208\pi$
$127$	− 4.00000i	− 0.354943i	0.984126	−	0.177471i	$-0.0567917\pi$
$128$	11.3137i	1.00000i
$129$	0	0
$130$	0	0
$131$	−2.82843	−0.247121	−0.123560	−	0.992337i	$-0.539431\pi$
$131$	−2.82843	−0.247121	−0.123560	+	0.992337i	$0.539431\pi$
$132$	0	0
$133$	−4.00000	−0.346844
$134$	16.9706	1.46603
$135$	0	0
$136$	−4.00000	−0.342997
$137$	− 15.5563i	− 1.32907i	−0.747258	−	0.664534i	$-0.768630\pi$
$137$	− 15.5563i	− 1.32907i	0.747258	−	0.664534i	$-0.231370\pi$
$138$	0	0
$139$	12.0000i	1.01783i	0.860818	+	0.508913i	$0.169953\pi$
$139$	12.0000i	1.01783i	−0.860818	+	0.508913i	$0.830047\pi$
$140$	−11.3137	−0.956183
$141$	0	0
$142$	8.00000i	0.671345i
$143$	0	0
$144$	0	0
$145$	−2.00000	−0.166091
$146$	− 22.6274i	− 1.87266i
$147$	0	0
$148$	12.0000	0.986394
$149$	− 15.5563i	− 1.27443i	−0.770688	−	0.637213i	$-0.780087\pi$
$149$	− 15.5563i	− 1.27443i	0.770688	−	0.637213i	$-0.219913\pi$
$150$	0	0
$151$	16.0000i	1.30206i	0.759051	+	0.651031i	$0.225663\pi$
$151$	16.0000i	1.30206i	−0.759051	+	0.651031i	$0.774337\pi$
$152$	−2.82843	−0.229416
$153$	0	0
$154$	16.0000	1.28932
$155$	5.65685	0.454369
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	11.3137	0.900070
$159$	0	0
$160$	−8.00000	−0.632456
$161$	33.9411i	2.67494i
$162$	0	0
$163$	− 12.0000i	− 0.939913i	−0.882690	−	0.469956i	$-0.844270\pi$
$163$	− 12.0000i	− 0.939913i	0.882690	−	0.469956i	$-0.155730\pi$
$164$	− 14.1421i	− 1.10432i
$165$	0	0
$166$	4.00000i	0.310460i
$167$	−11.3137	−0.875481	−0.437741	−	0.899101i	$-0.644221\pi$
$167$	−11.3137	−0.875481	−0.437741	+	0.899101i	$0.644221\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	− 2.82843i	− 0.216930i
$171$	0	0
$172$	16.0000i	1.21999i
$173$	12.7279i	0.967686i	0.875155	+	0.483843i	$0.160759\pi$
$173$	12.7279i	0.967686i	−0.875155	+	0.483843i	$0.839241\pi$
$174$	0	0
$175$	12.0000i	0.907115i
$176$	11.3137	0.852803
$177$	0	0
$178$	−14.0000	−1.04934
$179$	16.9706	1.26844	0.634220	−	0.773153i	$-0.281321\pi$
$179$	16.9706	1.26844	0.634220	+	0.773153i	$0.281321\pi$
$180$	0	0
$181$	−8.00000	−0.594635	−0.297318	−	0.954779i	$-0.596092\pi$
$181$	−8.00000	−0.594635	−0.297318	+	0.954779i	$0.596092\pi$
$182$	0	0
$183$	0	0
$184$	24.0000i	1.76930i
$185$	8.48528i	0.623850i
$186$	0	0
$187$	4.00000i	0.292509i
$188$	−5.65685	−0.412568
$189$	0	0
$190$	− 2.00000i	− 0.145095i
$191$	−19.7990	−1.43260	−0.716302	−	0.697790i	$-0.754167\pi$
$191$	−19.7990	−1.43260	−0.716302	+	0.697790i	$0.754167\pi$
$192$	0	0
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	− 16.9706i	− 1.21842i
$195$	0	0
$196$	18.0000	1.28571
$197$	4.24264i	0.302276i	0.988513	+	0.151138i	$0.0482937\pi$
$197$	4.24264i	0.302276i	−0.988513	+	0.151138i	$0.951706\pi$
$198$	0	0
$199$	8.00000i	0.567105i	0.958957	+	0.283552i	$0.0915130\pi$
$199$	8.00000i	0.567105i	−0.958957	+	0.283552i	$0.908487\pi$
$200$	8.48528i	0.600000i
$201$	0	0
$202$	−2.00000	−0.140720
$203$	5.65685	0.397033
$204$	0	0
$205$	10.0000	0.698430
$206$	22.6274	1.57653
$207$	0	0
$208$	0	0
$209$	2.82843i	0.195646i
$210$	0	0
$211$	− 4.00000i	− 0.275371i	−0.990476	−	0.137686i	$-0.956034\pi$
$211$	− 4.00000i	− 0.275371i	0.990476	−	0.137686i	$-0.0439664\pi$
$212$	14.1421i	0.971286i
$213$	0	0
$214$	24.0000i	1.64061i
$215$	−11.3137	−0.771589
$216$	0	0
$217$	−16.0000	−1.08615
$218$	11.3137i	0.766261i
$219$	0	0
$220$	8.00000i	0.539360i
$221$	0	0
$222$	0	0
$223$	− 24.0000i	− 1.60716i	−0.595198	−	0.803579i	$-0.702926\pi$
$223$	− 24.0000i	− 1.60716i	0.595198	−	0.803579i	$-0.297074\pi$
$224$	22.6274	1.51186
$225$	0	0
$226$	10.0000	0.665190
$227$	11.3137	0.750917	0.375459	−	0.926839i	$-0.377485\pi$
$227$	11.3137	0.750917	0.375459	+	0.926839i	$0.377485\pi$
$228$	0	0
$229$	20.0000	1.32164	0.660819	−	0.750546i	$-0.270209\pi$
$229$	20.0000	1.32164	0.660819	+	0.750546i	$0.270209\pi$
$230$	−16.9706	−1.11901
$231$	0	0
$232$	4.00000	0.262613
$233$	− 26.8701i	− 1.76032i	−0.474681	−	0.880158i	$-0.657437\pi$
$233$	− 26.8701i	− 1.76032i	0.474681	−	0.880158i	$-0.342563\pi$
$234$	0	0
$235$	− 4.00000i	− 0.260931i
$236$	−22.6274	−1.47292
$237$	0	0
$238$	8.00000i	0.518563i
$239$	−2.82843	−0.182956	−0.0914779	−	0.995807i	$-0.529159\pi$
$239$	−2.82843	−0.182956	−0.0914779	+	0.995807i	$0.529159\pi$
$240$	0	0
$241$	−28.0000	−1.80364	−0.901819	−	0.432113i	$-0.857768\pi$
$241$	−28.0000	−1.80364	−0.901819	+	0.432113i	$0.857768\pi$
$242$	4.24264i	0.272727i
$243$	0	0
$244$	4.00000	0.256074
$245$	12.7279i	0.813157i
$246$	0	0
$247$	0	0
$248$	−11.3137	−0.718421
$249$	0	0
$250$	−16.0000	−1.01193
$251$	−14.1421	−0.892644	−0.446322	−	0.894873i	$-0.647266\pi$
$251$	−14.1421	−0.892644	−0.446322	+	0.894873i	$0.647266\pi$
$252$	0	0
$253$	24.0000	1.50887
$254$	−5.65685	−0.354943
$255$	0	0
$256$	16.0000	1.00000
$257$	− 12.7279i	− 0.793946i	−0.917830	−	0.396973i	$-0.870061\pi$
$257$	− 12.7279i	− 0.793946i	0.917830	−	0.396973i	$-0.129939\pi$
$258$	0	0
$259$	− 24.0000i	− 1.49129i
$260$	0	0
$261$	0	0
$262$	4.00000i	0.247121i
$263$	−8.48528	−0.523225	−0.261612	−	0.965173i	$-0.584254\pi$
$263$	−8.48528	−0.523225	−0.261612	+	0.965173i	$0.584254\pi$
$264$	0	0
$265$	−10.0000	−0.614295
$266$	5.65685i	0.346844i
$267$	0	0
$268$	− 24.0000i	− 1.46603i
$269$	7.07107i	0.431131i	0.976489	+	0.215565i	$0.0691594\pi$
$269$	7.07107i	0.431131i	−0.976489	+	0.215565i	$0.930841\pi$
$270$	0	0
$271$	− 8.00000i	− 0.485965i	−0.970031	−	0.242983i	$-0.921874\pi$
$271$	− 8.00000i	− 0.485965i	0.970031	−	0.242983i	$-0.0781258\pi$
$272$	5.65685i	0.342997i
$273$	0	0
$274$	−22.0000	−1.32907
$275$	8.48528	0.511682
$276$	0	0
$277$	−20.0000	−1.20168	−0.600842	−	0.799368i	$-0.705168\pi$
$277$	−20.0000	−1.20168	−0.600842	+	0.799368i	$0.705168\pi$
$278$	16.9706	1.01783
$279$	0	0
$280$	16.0000i	0.956183i
$281$	− 15.5563i	− 0.928014i	−0.885832	−	0.464007i	$-0.846411\pi$
$281$	− 15.5563i	− 0.928014i	0.885832	−	0.464007i	$-0.153589\pi$
$282$	0	0
$283$	− 32.0000i	− 1.90220i	−0.308879	−	0.951101i	$-0.599954\pi$
$283$	− 32.0000i	− 1.90220i	0.308879	−	0.951101i	$-0.400046\pi$
$284$	11.3137	0.671345
$285$	0	0
$286$	0	0
$287$	−28.2843	−1.66957
$288$	0	0
$289$	15.0000	0.882353
$290$	2.82843i	0.166091i
$291$	0	0
$292$	−32.0000	−1.87266
$293$	− 24.0416i	− 1.40453i	−0.711917	−	0.702264i	$-0.752173\pi$
$293$	− 24.0416i	− 1.40453i	0.711917	−	0.702264i	$-0.247827\pi$
$294$	0	0
$295$	− 16.0000i	− 0.931556i
$296$	− 16.9706i	− 0.986394i
$297$	0	0
$298$	−22.0000	−1.27443
$299$	0	0
$300$	0	0
$301$	32.0000	1.84445
$302$	22.6274	1.30206
$303$	0	0
$304$	4.00000i	0.229416i
$305$	2.82843i	0.161955i
$306$	0	0
$307$	− 16.0000i	− 0.913168i	−0.889680	−	0.456584i	$-0.849073\pi$
$307$	− 16.0000i	− 0.913168i	0.889680	−	0.456584i	$-0.150927\pi$
$308$	− 22.6274i	− 1.28932i
$309$	0	0
$310$	− 8.00000i	− 0.454369i
$311$	19.7990	1.12270	0.561349	−	0.827579i	$-0.310283\pi$
$311$	19.7990	1.12270	0.561349	+	0.827579i	$0.310283\pi$
$312$	0	0
$313$	2.00000	0.113047	0.0565233	−	0.998401i	$-0.481998\pi$
$313$	2.00000	0.113047	0.0565233	+	0.998401i	$0.481998\pi$
$314$	− 2.82843i	− 0.159617i
$315$	0	0
$316$	− 16.0000i	− 0.900070i
$317$	7.07107i	0.397151i	0.980086	+	0.198575i	$0.0636315\pi$
$317$	7.07107i	0.397151i	−0.980086	+	0.198575i	$0.936369\pi$
$318$	0	0
$319$	− 4.00000i	− 0.223957i
$320$	11.3137i	0.632456i
$321$	0	0
$322$	48.0000	2.67494
$323$	−1.41421	−0.0786889
$324$	0	0
$325$	0	0
$326$	−16.9706	−0.939913
$327$	0	0
$328$	−20.0000	−1.10432
$329$	11.3137i	0.623745i
$330$	0	0
$331$	− 8.00000i	− 0.439720i	−0.975531	−	0.219860i	$-0.929440\pi$
$331$	− 8.00000i	− 0.439720i	0.975531	−	0.219860i	$-0.0705600\pi$
$332$	5.65685	0.310460
$333$	0	0
$334$	16.0000i	0.875481i
$335$	16.9706	0.927201
$336$	0	0
$337$	−20.0000	−1.08947	−0.544735	−	0.838608i	$-0.683370\pi$
$337$	−20.0000	−1.08947	−0.544735	+	0.838608i	$0.683370\pi$
$338$	18.3848i	1.00000i
$339$	0	0
$340$	−4.00000	−0.216930
$341$	11.3137i	0.612672i
$342$	0	0
$343$	− 8.00000i	− 0.431959i
$344$	22.6274	1.21999
$345$	0	0
$346$	18.0000	0.967686
$347$	−19.7990	−1.06287	−0.531433	−	0.847100i	$-0.678346\pi$
$347$	−19.7990	−1.06287	−0.531433	+	0.847100i	$0.678346\pi$
$348$	0	0
$349$	22.0000	1.17763	0.588817	−	0.808267i	$-0.299594\pi$
$349$	22.0000	1.17763	0.588817	+	0.808267i	$0.299594\pi$
$350$	16.9706	0.907115
$351$	0	0
$352$	− 16.0000i	− 0.852803i
$353$	− 7.07107i	− 0.376355i	−0.982135	−	0.188177i	$-0.939742\pi$
$353$	− 7.07107i	− 0.376355i	0.982135	−	0.188177i	$-0.0602580\pi$
$354$	0	0
$355$	8.00000i	0.424596i
$356$	19.7990i	1.04934i
$357$	0	0
$358$	− 24.0000i	− 1.26844i
$359$	−14.1421	−0.746393	−0.373197	−	0.927752i	$-0.621738\pi$
$359$	−14.1421	−0.746393	−0.373197	+	0.927752i	$0.621738\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	11.3137i	0.594635i
$363$	0	0
$364$	0	0
$365$	− 22.6274i	− 1.18437i
$366$	0	0
$367$	− 32.0000i	− 1.67039i	−0.549957	−	0.835193i	$-0.685356\pi$
$367$	− 32.0000i	− 1.67039i	0.549957	−	0.835193i	$-0.314644\pi$
$368$	33.9411	1.76930
$369$	0	0
$370$	12.0000	0.623850
$371$	28.2843	1.46845
$372$	0	0
$373$	−26.0000	−1.34623	−0.673114	−	0.739538i	$-0.735044\pi$
$373$	−26.0000	−1.34623	−0.673114	+	0.739538i	$0.735044\pi$
$374$	5.65685	0.292509
$375$	0	0
$376$	8.00000i	0.412568i
$377$	0	0
$378$	0	0
$379$	− 24.0000i	− 1.23280i	−0.787434	−	0.616399i	$-0.788591\pi$
$379$	− 24.0000i	− 1.23280i	0.787434	−	0.616399i	$-0.211409\pi$
$380$	−2.82843	−0.145095
$381$	0	0
$382$	28.0000i	1.43260i
$383$	−5.65685	−0.289052	−0.144526	−	0.989501i	$-0.546166\pi$
$383$	−5.65685	−0.289052	−0.144526	+	0.989501i	$0.546166\pi$
$384$	0	0
$385$	16.0000	0.815436
$386$	− 19.7990i	− 1.00774i
$387$	0	0
$388$	−24.0000	−1.21842
$389$	35.3553i	1.79259i	0.443461	+	0.896293i	$0.353750\pi$
$389$	35.3553i	1.79259i	−0.443461	+	0.896293i	$0.646250\pi$
$390$	0	0
$391$	12.0000i	0.606866i
$392$	− 25.4558i	− 1.28571i
$393$	0	0
$394$	6.00000	0.302276
$395$	11.3137	0.569254
$396$	0	0
$397$	−10.0000	−0.501886	−0.250943	−	0.968002i	$-0.580741\pi$
$397$	−10.0000	−0.501886	−0.250943	+	0.968002i	$0.580741\pi$
$398$	11.3137	0.567105
$399$	0	0
$400$	12.0000	0.600000
$401$	− 1.41421i	− 0.0706225i	−0.999376	−	0.0353112i	$-0.988758\pi$
$401$	− 1.41421i	− 0.0706225i	0.999376	−	0.0353112i	$-0.0112422\pi$
$402$	0	0
$403$	0	0
$404$	2.82843i	0.140720i
$405$	0	0
$406$	− 8.00000i	− 0.397033i
$407$	−16.9706	−0.841200
$408$	0	0
$409$	−4.00000	−0.197787	−0.0988936	−	0.995098i	$-0.531530\pi$
$409$	−4.00000	−0.197787	−0.0988936	+	0.995098i	$0.531530\pi$
$410$	− 14.1421i	− 0.698430i
$411$	0	0
$412$	− 32.0000i	− 1.57653i
$413$	45.2548i	2.22684i
$414$	0	0
$415$	4.00000i	0.196352i
$416$	0	0
$417$	0	0
$418$	4.00000	0.195646
$419$	31.1127	1.51995	0.759977	−	0.649950i	$-0.225210\pi$
$419$	31.1127	1.51995	0.759977	+	0.649950i	$0.225210\pi$
$420$	0	0
$421$	0	0		−	1.00000i	$-0.5\pi$
$421$	0	0			1.00000i	$0.5\pi$
$422$	−5.65685	−0.275371
$423$	0	0
$424$	20.0000	0.971286
$425$	4.24264i	0.205798i
$426$	0	0
$427$	− 8.00000i	− 0.387147i
$428$	33.9411	1.64061
$429$	0	0
$430$	16.0000i	0.771589i
$431$	−22.6274	−1.08992	−0.544962	−	0.838461i	$-0.683456\pi$
$431$	−22.6274	−1.08992	−0.544962	+	0.838461i	$0.683456\pi$
$432$	0	0
$433$	−2.00000	−0.0961139	−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000	−0.0961139	−0.0480569	+	0.998845i	$0.515303\pi$
$434$	22.6274i	1.08615i
$435$	0	0
$436$	16.0000	0.766261
$437$	8.48528i	0.405906i
$438$	0	0
$439$	24.0000i	1.14546i	0.819745	+	0.572729i	$0.194115\pi$
$439$	24.0000i	1.14546i	−0.819745	+	0.572729i	$0.805885\pi$
$440$	11.3137	0.539360
$441$	0	0
$442$	0	0
$443$	31.1127	1.47821	0.739104	−	0.673591i	$-0.235249\pi$
$443$	31.1127	1.47821	0.739104	+	0.673591i	$0.235249\pi$
$444$	0	0
$445$	−14.0000	−0.663664
$446$	−33.9411	−1.60716
$447$	0	0
$448$	− 32.0000i	− 1.51186i
$449$	− 4.24264i	− 0.200223i	−0.994976	−	0.100111i	$-0.968080\pi$
$449$	− 4.24264i	− 0.200223i	0.994976	−	0.100111i	$-0.0319199\pi$
$450$	0	0
$451$	20.0000i	0.941763i
$452$	− 14.1421i	− 0.665190i
$453$	0	0
$454$	− 16.0000i	− 0.750917i
$455$	0	0
$456$	0	0
$457$	16.0000	0.748448	0.374224	−	0.927338i	$-0.377909\pi$
$457$	16.0000	0.748448	0.374224	+	0.927338i	$0.377909\pi$
$458$	− 28.2843i	− 1.32164i
$459$	0	0
$460$	24.0000i	1.11901i
$461$	− 21.2132i	− 0.987997i	−0.869463	−	0.493999i	$-0.835535\pi$
$461$	− 21.2132i	− 0.987997i	0.869463	−	0.493999i	$-0.164465\pi$
$462$	0	0
$463$	− 16.0000i	− 0.743583i	−0.928316	−	0.371792i	$-0.878744\pi$
$463$	− 16.0000i	− 0.743583i	0.928316	−	0.371792i	$-0.121256\pi$
$464$	− 5.65685i	− 0.262613i
$465$	0	0
$466$	−38.0000	−1.76032
$467$	−19.7990	−0.916188	−0.458094	−	0.888904i	$-0.651468\pi$
$467$	−19.7990	−0.916188	−0.458094	+	0.888904i	$0.651468\pi$
$468$	0	0
$469$	−48.0000	−2.21643
$470$	−5.65685	−0.260931
$471$	0	0
$472$	32.0000i	1.47292i
$473$	− 22.6274i	− 1.04041i
$474$	0	0
$475$	3.00000i	0.137649i
$476$	11.3137	0.518563
$477$	0	0
$478$	4.00000i	0.182956i
$479$	2.82843	0.129234	0.0646171	−	0.997910i	$-0.479417\pi$
$479$	2.82843	0.129234	0.0646171	+	0.997910i	$0.479417\pi$
$480$	0	0
$481$	0	0
$482$	39.5980i	1.80364i
$483$	0	0
$484$	6.00000	0.272727
$485$	− 16.9706i	− 0.770594i
$486$	0	0
$487$	12.0000i	0.543772i	0.962329	+	0.271886i	$0.0876473\pi$
$487$	12.0000i	0.543772i	−0.962329	+	0.271886i	$0.912353\pi$
$488$	− 5.65685i	− 0.256074i
$489$	0	0
$490$	18.0000	0.813157
$491$	2.82843	0.127645	0.0638226	−	0.997961i	$-0.479671\pi$
$491$	2.82843	0.127645	0.0638226	+	0.997961i	$0.479671\pi$
$492$	0	0
$493$	2.00000	0.0900755
$494$	0	0
$495$	0	0
$496$	16.0000i	0.718421i
$497$	− 22.6274i	− 1.01498i
$498$	0	0
$499$	4.00000i	0.179065i	0.995984	+	0.0895323i	$0.0285372\pi$
$499$	4.00000i	0.179065i	−0.995984	+	0.0895323i	$0.971463\pi$
$500$	22.6274i	1.01193i
$501$	0	0
$502$	20.0000i	0.892644i
$503$	−36.7696	−1.63947	−0.819737	−	0.572741i	$-0.805880\pi$
$503$	−36.7696	−1.63947	−0.819737	+	0.572741i	$0.805880\pi$
$504$	0	0
$505$	−2.00000	−0.0889988
$506$	− 33.9411i	− 1.50887i
$507$	0	0
$508$	8.00000i	0.354943i
$509$	4.24264i	0.188052i	0.995570	+	0.0940259i	$0.0299736\pi$
$509$	4.24264i	0.188052i	−0.995570	+	0.0940259i	$0.970026\pi$
$510$	0	0
$511$	64.0000i	2.83119i
$512$	− 22.6274i	− 1.00000i
$513$	0	0
$514$	−18.0000	−0.793946
$515$	22.6274	0.997083
$516$	0	0
$517$	8.00000	0.351840
$518$	−33.9411	−1.49129
$519$	0	0
$520$	0	0
$521$	− 29.6985i	− 1.30111i	−0.759457	−	0.650557i	$-0.774535\pi$
$521$	− 29.6985i	− 1.30111i	0.759457	−	0.650557i	$-0.225465\pi$
$522$	0	0
$523$	4.00000i	0.174908i	0.996169	+	0.0874539i	$0.0278730\pi$
$523$	4.00000i	0.174908i	−0.996169	+	0.0874539i	$0.972127\pi$
$524$	5.65685	0.247121
$525$	0	0
$526$	12.0000i	0.523225i
$527$	−5.65685	−0.246416
$528$	0	0
$529$	49.0000	2.13043
$530$	14.1421i	0.614295i
$531$	0	0
$532$	8.00000	0.346844
$533$	0	0
$534$	0	0
$535$	24.0000i	1.03761i
$536$	−33.9411	−1.46603
$537$	0	0
$538$	10.0000	0.431131
$539$	−25.4558	−1.09646
$540$	0	0
$541$	−4.00000	−0.171973	−0.0859867	−	0.996296i	$-0.527404\pi$
$541$	−4.00000	−0.171973	−0.0859867	+	0.996296i	$0.527404\pi$
$542$	−11.3137	−0.485965
$543$	0	0
$544$	8.00000	0.342997
$545$	11.3137i	0.484626i
$546$	0	0
$547$	− 8.00000i	− 0.342055i	−0.985266	−	0.171028i	$-0.945291\pi$
$547$	− 8.00000i	− 0.342055i	0.985266	−	0.171028i	$-0.0547087\pi$
$548$	31.1127i	1.32907i
$549$	0	0
$550$	− 12.0000i	− 0.511682i
$551$	1.41421	0.0602475
$552$	0	0
$553$	−32.0000	−1.36078
$554$	28.2843i	1.20168i
$555$	0	0
$556$	− 24.0000i	− 1.01783i
$557$	24.0416i	1.01868i	0.860566	+	0.509338i	$0.170110\pi$
$557$	24.0416i	1.01868i	−0.860566	+	0.509338i	$0.829890\pi$
$558$	0	0
$559$	0	0
$560$	22.6274	0.956183
$561$	0	0
$562$	−22.0000	−0.928014
$563$	45.2548	1.90726	0.953632	−	0.300975i	$-0.0973122\pi$
$563$	45.2548	1.90726	0.953632	+	0.300975i	$0.0973122\pi$
$564$	0	0
$565$	10.0000	0.420703
$566$	−45.2548	−1.90220
$567$	0	0
$568$	− 16.0000i	− 0.671345i
$569$	32.5269i	1.36360i	0.731539	+	0.681800i	$0.238802\pi$
$569$	32.5269i	1.36360i	−0.731539	+	0.681800i	$0.761198\pi$
$570$	0	0
$571$	40.0000i	1.67395i	0.547243	+	0.836974i	$0.315677\pi$
$571$	40.0000i	1.67395i	−0.547243	+	0.836974i	$0.684323\pi$
$572$	0	0
$573$	0	0
$574$	40.0000i	1.66957i
$575$	25.4558	1.06158
$576$	0	0
$577$	42.0000	1.74848	0.874241	−	0.485491i	$-0.161359\pi$
$577$	42.0000	1.74848	0.874241	+	0.485491i	$0.161359\pi$
$578$	− 21.2132i	− 0.882353i
$579$	0	0
$580$	4.00000	0.166091
$581$	− 11.3137i	− 0.469372i
$582$	0	0
$583$	− 20.0000i	− 0.828315i
$584$	45.2548i	1.87266i
$585$	0	0
$586$	−34.0000	−1.40453
$587$	25.4558	1.05068	0.525338	−	0.850894i	$-0.323939\pi$
$587$	25.4558	1.05068	0.525338	+	0.850894i	$0.323939\pi$
$588$	0	0
$589$	−4.00000	−0.164817
$590$	−22.6274	−0.931556
$591$	0	0
$592$	−24.0000	−0.986394
$593$	− 9.89949i	− 0.406524i	−0.979124	−	0.203262i	$-0.934846\pi$
$593$	− 9.89949i	− 0.406524i	0.979124	−	0.203262i	$-0.0651542\pi$
$594$	0	0
$595$	8.00000i	0.327968i
$596$	31.1127i	1.27443i
$597$	0	0
$598$	0	0
$599$	−39.5980	−1.61793	−0.808965	−	0.587857i	$-0.799972\pi$
$599$	−39.5980	−1.61793	−0.808965	+	0.587857i	$0.799972\pi$
$600$	0	0
$601$	−26.0000	−1.06056	−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000	−1.06056	−0.530281	+	0.847822i	$0.677914\pi$
$602$	− 45.2548i	− 1.84445i
$603$	0	0
$604$	− 32.0000i	− 1.30206i
$605$	4.24264i	0.172488i
$606$	0	0
$607$	− 4.00000i	− 0.162355i	−0.996700	−	0.0811775i	$-0.974132\pi$
$607$	− 4.00000i	− 0.162355i	0.996700	−	0.0811775i	$-0.0258681\pi$
$608$	5.65685	0.229416
$609$	0	0
$610$	4.00000	0.161955
$611$	0	0
$612$	0	0
$613$	38.0000	1.53481	0.767403	−	0.641165i	$-0.221549\pi$
$613$	38.0000	1.53481	0.767403	+	0.641165i	$0.221549\pi$
$614$	−22.6274	−0.913168
$615$	0	0
$616$	−32.0000	−1.28932
$617$	21.2132i	0.854011i	0.904249	+	0.427006i	$0.140432\pi$
$617$	21.2132i	0.854011i	−0.904249	+	0.427006i	$0.859568\pi$
$618$	0	0
$619$	28.0000i	1.12542i	0.826656	+	0.562708i	$0.190240\pi$
$619$	28.0000i	1.12542i	−0.826656	+	0.562708i	$0.809760\pi$
$620$	−11.3137	−0.454369
$621$	0	0
$622$	− 28.0000i	− 1.12270i
$623$	39.5980	1.58646
$624$	0	0
$625$	−1.00000	−0.0400000
$626$	− 2.82843i	− 0.113047i
$627$	0	0
$628$	−4.00000	−0.159617
$629$	− 8.48528i	− 0.338330i
$630$	0	0
$631$	− 20.0000i	− 0.796187i	−0.917345	−	0.398094i	$-0.869672\pi$
$631$	− 20.0000i	− 0.796187i	0.917345	−	0.398094i	$-0.130328\pi$
$632$	−22.6274	−0.900070
$633$	0	0
$634$	10.0000	0.397151
$635$	−5.65685	−0.224485
$636$	0	0
$637$	0	0
$638$	−5.65685	−0.223957
$639$	0	0
$640$	16.0000	0.632456
$641$	9.89949i	0.391007i	0.980703	+	0.195503i	$0.0626340\pi$
$641$	9.89949i	0.391007i	−0.980703	+	0.195503i	$0.937366\pi$
$642$	0	0
$643$	− 16.0000i	− 0.630978i	−0.948929	−	0.315489i	$-0.897831\pi$
$643$	− 16.0000i	− 0.630978i	0.948929	−	0.315489i	$-0.102169\pi$
$644$	− 67.8823i	− 2.67494i
$645$	0	0
$646$	2.00000i	0.0786889i
$647$	−25.4558	−1.00077	−0.500386	−	0.865802i	$-0.666809\pi$
$647$	−25.4558	−1.00077	−0.500386	+	0.865802i	$0.666809\pi$
$648$	0	0
$649$	32.0000	1.25611
$650$	0	0
$651$	0	0
$652$	24.0000i	0.939913i
$653$	− 1.41421i	− 0.0553425i	−0.999617	−	0.0276712i	$-0.991191\pi$
$653$	− 1.41421i	− 0.0553425i	0.999617	−	0.0276712i	$-0.00880915\pi$
$654$	0	0
$655$	4.00000i	0.156293i
$656$	28.2843i	1.10432i
$657$	0	0
$658$	16.0000	0.623745
$659$	5.65685	0.220360	0.110180	−	0.993912i	$-0.464857\pi$
$659$	5.65685	0.220360	0.110180	+	0.993912i	$0.464857\pi$
$660$	0	0
$661$	42.0000	1.63361	0.816805	−	0.576913i	$-0.195743\pi$
$661$	42.0000	1.63361	0.816805	+	0.576913i	$0.195743\pi$
$662$	−11.3137	−0.439720
$663$	0	0
$664$	− 8.00000i	− 0.310460i
$665$	5.65685i	0.219363i
$666$	0	0
$667$	− 12.0000i	− 0.464642i
$668$	22.6274	0.875481
$669$	0	0
$670$	− 24.0000i	− 0.927201i
$671$	−5.65685	−0.218380
$672$	0	0
$673$	2.00000	0.0770943	0.0385472	−	0.999257i	$-0.487727\pi$
$673$	2.00000	0.0770943	0.0385472	+	0.999257i	$0.487727\pi$
$674$	28.2843i	1.08947i
$675$	0	0
$676$	26.0000	1.00000
$677$	− 43.8406i	− 1.68493i	−0.538750	−	0.842466i	$-0.681103\pi$
$677$	− 43.8406i	− 1.68493i	0.538750	−	0.842466i	$-0.318897\pi$
$678$	0	0
$679$	48.0000i	1.84207i
$680$	5.65685i	0.216930i
$681$	0	0
$682$	16.0000	0.612672
$683$	−33.9411	−1.29872	−0.649361	−	0.760481i	$-0.724963\pi$
$683$	−33.9411	−1.29872	−0.649361	+	0.760481i	$0.724963\pi$
$684$	0	0
$685$	−22.0000	−0.840577
$686$	−11.3137	−0.431959
$687$	0	0
$688$	− 32.0000i	− 1.21999i
$689$	0	0
$690$	0	0
$691$	40.0000i	1.52167i	0.648944	+	0.760836i	$0.275211\pi$
$691$	40.0000i	1.52167i	−0.648944	+	0.760836i	$0.724789\pi$
$692$	− 25.4558i	− 0.967686i
$693$	0	0
$694$	28.0000i	1.06287i
$695$	16.9706	0.643730
$696$	0	0
$697$	−10.0000	−0.378777
$698$	− 31.1127i	− 1.17763i
$699$	0	0
$700$	− 24.0000i	− 0.907115i
$701$	− 15.5563i	− 0.587555i	−0.955874	−	0.293778i	$-0.905087\pi$
$701$	− 15.5563i	− 0.587555i	0.955874	−	0.293778i	$-0.0949125\pi$
$702$	0	0
$703$	− 6.00000i	− 0.226294i
$704$	−22.6274	−0.852803
$705$	0	0
$706$	−10.0000	−0.376355
$707$	5.65685	0.212748
$708$	0	0
$709$	−52.0000	−1.95290	−0.976450	−	0.215742i	$-0.930783\pi$
$709$	−52.0000	−1.95290	−0.976450	+	0.215742i	$0.930783\pi$
$710$	11.3137	0.424596
$711$	0	0
$712$	28.0000	1.04934
$713$	33.9411i	1.27111i
$714$	0	0
$715$	0	0
$716$	−33.9411	−1.26844
$717$	0	0
$718$	20.0000i	0.746393i
$719$	2.82843	0.105483	0.0527413	−	0.998608i	$-0.483204\pi$
$719$	2.82843	0.105483	0.0527413	+	0.998608i	$0.483204\pi$
$720$	0	0
$721$	−64.0000	−2.38348
$722$	1.41421i	0.0526316i
$723$	0	0
$724$	16.0000	0.594635
$725$	− 4.24264i	− 0.157568i
$726$	0	0
$727$	48.0000i	1.78022i	0.455744	+	0.890111i	$0.349373\pi$
$727$	48.0000i	1.78022i	−0.455744	+	0.890111i	$0.650627\pi$
$728$	0	0
$729$	0	0
$730$	−32.0000	−1.18437
$731$	11.3137	0.418453
$732$	0	0
$733$	−44.0000	−1.62518	−0.812589	−	0.582838i	$-0.801942\pi$
$733$	−44.0000	−1.62518	−0.812589	+	0.582838i	$0.801942\pi$
$734$	−45.2548	−1.67039
$735$	0	0
$736$	− 48.0000i	− 1.76930i
$737$	33.9411i	1.25024i
$738$	0	0
$739$	4.00000i	0.147142i	0.997290	+	0.0735712i	$0.0234396\pi$
$739$	4.00000i	0.147142i	−0.997290	+	0.0735712i	$0.976560\pi$
$740$	− 16.9706i	− 0.623850i
$741$	0	0
$742$	− 40.0000i	− 1.46845i
$743$	−22.6274	−0.830119	−0.415060	−	0.909794i	$-0.636239\pi$
$743$	−22.6274	−0.830119	−0.415060	+	0.909794i	$0.636239\pi$
$744$	0	0
$745$	−22.0000	−0.806018
$746$	36.7696i	1.34623i
$747$	0	0
$748$	− 8.00000i	− 0.292509i
$749$	− 67.8823i	− 2.48036i
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	11.3137	0.412568
$753$	0	0
$754$	0	0
$755$	22.6274	0.823496
$756$	0	0
$757$	44.0000	1.59921	0.799604	−	0.600528i	$-0.205043\pi$
$757$	44.0000	1.59921	0.799604	+	0.600528i	$0.205043\pi$
$758$	−33.9411	−1.23280
$759$	0	0
$760$	4.00000i	0.145095i
$761$	15.5563i	0.563917i	0.959427	+	0.281959i	$0.0909841\pi$
$761$	15.5563i	0.563917i	−0.959427	+	0.281959i	$0.909016\pi$
$762$	0	0
$763$	− 32.0000i	− 1.15848i
$764$	39.5980	1.43260
$765$	0	0
$766$	8.00000i	0.289052i
$767$	0	0
$768$	0	0
$769$	−6.00000	−0.216366	−0.108183	−	0.994131i	$-0.534503\pi$
$769$	−6.00000	−0.216366	−0.108183	+	0.994131i	$0.534503\pi$
$770$	− 22.6274i	− 0.815436i
$771$	0	0
$772$	−28.0000	−1.00774
$773$	− 24.0416i	− 0.864717i	−0.901702	−	0.432359i	$-0.857681\pi$
$773$	− 24.0416i	− 0.864717i	0.901702	−	0.432359i	$-0.142319\pi$
$774$	0	0
$775$	12.0000i	0.431053i
$776$	33.9411i	1.21842i
$777$	0	0
$778$	50.0000	1.79259
$779$	−7.07107	−0.253347
$780$	0	0
$781$	−16.0000	−0.572525
$782$	16.9706	0.606866
$783$	0	0
$784$	−36.0000	−1.28571
$785$	− 2.82843i	− 0.100951i
$786$	0	0
$787$	− 52.0000i	− 1.85360i	−0.375555	−	0.926800i	$-0.622548\pi$
$787$	− 52.0000i	− 1.85360i	0.375555	−	0.926800i	$-0.377452\pi$
$788$	− 8.48528i	− 0.302276i
$789$	0	0
$790$	− 16.0000i	− 0.569254i
$791$	−28.2843	−1.00567
$792$	0	0
$793$	0	0
$794$	14.1421i	0.501886i
$795$	0	0
$796$	− 16.0000i	− 0.567105i
$797$	35.3553i	1.25235i	0.779682	+	0.626175i	$0.215381\pi$
$797$	35.3553i	1.25235i	−0.779682	+	0.626175i	$0.784619\pi$
$798$	0	0
$799$	4.00000i	0.141510i
$800$	− 16.9706i	− 0.600000i
$801$	0	0
$802$	−2.00000	−0.0706225
$803$	45.2548	1.59701
$804$	0	0
$805$	48.0000	1.69178
$806$	0	0
$807$	0	0
$808$	4.00000	0.140720
$809$	21.2132i	0.745817i	0.927868	+	0.372908i	$0.121639\pi$
$809$	21.2132i	0.745817i	−0.927868	+	0.372908i	$0.878361\pi$
$810$	0	0
$811$	56.0000i	1.96643i	0.182462	+	0.983213i	$0.441593\pi$
$811$	56.0000i	1.96643i	−0.182462	+	0.983213i	$0.558407\pi$
$812$	−11.3137	−0.397033
$813$	0	0
$814$	24.0000i	0.841200i
$815$	−16.9706	−0.594453
$816$	0	0
$817$	8.00000	0.279885
$818$	5.65685i	0.197787i
$819$	0	0
$820$	−20.0000	−0.698430
$821$	32.5269i	1.13520i	0.823305	+	0.567599i	$0.192127\pi$
$821$	32.5269i	1.13520i	−0.823305	+	0.567599i	$0.807873\pi$
$822$	0	0
$823$	36.0000i	1.25488i	0.778664	+	0.627441i	$0.215897\pi$
$823$	36.0000i	1.25488i	−0.778664	+	0.627441i	$0.784103\pi$
$824$	−45.2548	−1.57653
$825$	0	0
$826$	64.0000	2.22684
$827$	−11.3137	−0.393416	−0.196708	−	0.980462i	$-0.563025\pi$
$827$	−11.3137	−0.393416	−0.196708	+	0.980462i	$0.563025\pi$
$828$	0	0
$829$	−24.0000	−0.833554	−0.416777	−	0.909009i	$-0.636840\pi$
$829$	−24.0000	−0.833554	−0.416777	+	0.909009i	$0.636840\pi$
$830$	5.65685	0.196352
$831$	0	0
$832$	0	0
$833$	− 12.7279i	− 0.440996i
$834$	0	0
$835$	16.0000i	0.553703i
$836$	− 5.65685i	− 0.195646i
$837$	0	0
$838$	− 44.0000i	− 1.51995i
$839$	−16.9706	−0.585889	−0.292944	−	0.956129i	$-0.594635\pi$
$839$	−16.9706	−0.585889	−0.292944	+	0.956129i	$0.594635\pi$
$840$	0	0
$841$	27.0000	0.931034
$842$	0	0
$843$	0	0
$844$	8.00000i	0.275371i
$845$	18.3848i	0.632456i
$846$	0	0
$847$	− 12.0000i	− 0.412325i
$848$	− 28.2843i	− 0.971286i
$849$	0	0
$850$	6.00000	0.205798
$851$	−50.9117	−1.74523
$852$	0	0
$853$	−30.0000	−1.02718	−0.513590	−	0.858036i	$-0.671685\pi$
$853$	−30.0000	−1.02718	−0.513590	+	0.858036i	$0.671685\pi$
$854$	−11.3137	−0.387147
$855$	0	0
$856$	− 48.0000i	− 1.64061i
$857$	− 4.24264i	− 0.144926i	−0.997371	−	0.0724629i	$-0.976914\pi$
$857$	− 4.24264i	− 0.144926i	0.997371	−	0.0724629i	$-0.0230859\pi$
$858$	0	0
$859$	− 12.0000i	− 0.409435i	−0.978821	−	0.204717i	$-0.934372\pi$
$859$	− 12.0000i	− 0.409435i	0.978821	−	0.204717i	$-0.0656275\pi$
$860$	22.6274	0.771589
$861$	0	0
$862$	32.0000i	1.08992i
$863$	−33.9411	−1.15537	−0.577685	−	0.816260i	$-0.696044\pi$
$863$	−33.9411	−1.15537	−0.577685	+	0.816260i	$0.696044\pi$
$864$	0	0
$865$	18.0000	0.612018
$866$	2.82843i	0.0961139i
$867$	0	0
$868$	32.0000	1.08615
$869$	22.6274i	0.767583i
$870$	0	0
$871$	0	0
$872$	− 22.6274i	− 0.766261i
$873$	0	0
$874$	12.0000	0.405906
$875$	45.2548	1.52989
$876$	0	0
$877$	46.0000	1.55331	0.776655	−	0.629926i	$-0.216915\pi$
$877$	46.0000	1.55331	0.776655	+	0.629926i	$0.216915\pi$
$878$	33.9411	1.14546
$879$	0	0
$880$	− 16.0000i	− 0.539360i
$881$	− 38.1838i	− 1.28644i	−0.765680	−	0.643222i	$-0.777597\pi$
$881$	− 38.1838i	− 1.28644i	0.765680	−	0.643222i	$-0.222403\pi$
$882$	0	0
$883$	− 20.0000i	− 0.673054i	−0.941674	−	0.336527i	$-0.890748\pi$
$883$	− 20.0000i	− 0.673054i	0.941674	−	0.336527i	$-0.109252\pi$
$884$	0	0
$885$	0	0
$886$	− 44.0000i	− 1.47821i
$887$	50.9117	1.70945	0.854724	−	0.519083i	$-0.173727\pi$
$887$	50.9117	1.70945	0.854724	+	0.519083i	$0.173727\pi$
$888$	0	0
$889$	16.0000	0.536623
$890$	19.7990i	0.663664i
$891$	0	0
$892$	48.0000i	1.60716i
$893$	2.82843i	0.0946497i
$894$	0	0
$895$	− 24.0000i	− 0.802232i
$896$	−45.2548	−1.51186
$897$	0	0
$898$	−6.00000	−0.200223
$899$	5.65685	0.188667
$900$	0	0
$901$	10.0000	0.333148
$902$	28.2843	0.941763
$903$	0	0
$904$	−20.0000	−0.665190
$905$	11.3137i	0.376080i
$906$	0	0
$907$	− 28.0000i	− 0.929725i	−0.885383	−	0.464862i	$-0.846104\pi$
$907$	− 28.0000i	− 0.929725i	0.885383	−	0.464862i	$-0.153896\pi$
$908$	−22.6274	−0.750917
$909$	0	0
$910$	0	0
$911$	5.65685	0.187420	0.0937100	−	0.995600i	$-0.470127\pi$
$911$	5.65685	0.187420	0.0937100	+	0.995600i	$0.470127\pi$
$912$	0	0
$913$	−8.00000	−0.264761
$914$	− 22.6274i	− 0.748448i
$915$	0	0
$916$	−40.0000	−1.32164
$917$	− 11.3137i	− 0.373612i
$918$	0	0
$919$	− 20.0000i	− 0.659739i	−0.944027	−	0.329870i	$-0.892995\pi$
$919$	− 20.0000i	− 0.659739i	0.944027	−	0.329870i	$-0.107005\pi$
$920$	33.9411	1.11901
$921$	0	0
$922$	−30.0000	−0.987997
$923$	0	0
$924$	0	0
$925$	−18.0000	−0.591836
$926$	−22.6274	−0.743583
$927$	0	0
$928$	−8.00000	−0.262613
$929$	41.0122i	1.34557i	0.739840	+	0.672783i	$0.234901\pi$
$929$	41.0122i	1.34557i	−0.739840	+	0.672783i	$0.765099\pi$
$930$	0	0
$931$	− 9.00000i	− 0.294963i
$932$	53.7401i	1.76032i
$933$	0	0
$934$	28.0000i	0.916188i
$935$	5.65685	0.184999
$936$	0	0
$937$	38.0000	1.24141	0.620703	−	0.784046i	$-0.286847\pi$
$937$	38.0000	1.24141	0.620703	+	0.784046i	$0.286847\pi$
$938$	67.8823i	2.21643i
$939$	0	0
$940$	8.00000i	0.260931i
$941$	− 12.7279i	− 0.414918i	−0.978244	−	0.207459i	$-0.933481\pi$
$941$	− 12.7279i	− 0.414918i	0.978244	−	0.207459i	$-0.0665194\pi$
$942$	0	0
$943$	60.0000i	1.95387i
$944$	45.2548	1.47292
$945$	0	0
$946$	−32.0000	−1.04041
$947$	−25.4558	−0.827204	−0.413602	−	0.910458i	$-0.635729\pi$
$947$	−25.4558	−0.827204	−0.413602	+	0.910458i	$0.635729\pi$
$948$	0	0
$949$	0	0
$950$	4.24264	0.137649
$951$	0	0
$952$	− 16.0000i	− 0.518563i
$953$	− 35.3553i	− 1.14527i	−0.819810	−	0.572636i	$-0.805921\pi$
$953$	− 35.3553i	− 1.14527i	0.819810	−	0.572636i	$-0.194079\pi$
$954$	0	0
$955$	28.0000i	0.906059i
$956$	5.65685	0.182956
$957$	0	0
$958$	− 4.00000i	− 0.129234i
$959$	62.2254	2.00936
$960$	0	0
$961$	15.0000	0.483871
$962$	0	0
$963$	0	0
$964$	56.0000	1.80364
$965$	− 19.7990i	− 0.637352i
$966$	0	0
$967$	− 4.00000i	− 0.128631i	−0.997930	−	0.0643157i	$-0.979514\pi$
$967$	− 4.00000i	− 0.128631i	0.997930	−	0.0643157i	$-0.0204865\pi$
$968$	− 8.48528i	− 0.272727i
$969$	0	0
$970$	−24.0000	−0.770594
$971$	28.2843	0.907685	0.453843	−	0.891082i	$-0.350053\pi$
$971$	28.2843	0.907685	0.453843	+	0.891082i	$0.350053\pi$
$972$	0	0
$973$	−48.0000	−1.53881
$974$	16.9706	0.543772
$975$	0	0
$976$	−8.00000	−0.256074
$977$	− 32.5269i	− 1.04063i	−0.853975	−	0.520314i	$-0.825815\pi$
$977$	− 32.5269i	− 1.04063i	0.853975	−	0.520314i	$-0.174185\pi$
$978$	0	0
$979$	− 28.0000i	− 0.894884i
$980$	− 25.4558i	− 0.813157i
$981$	0	0
$982$	− 4.00000i	− 0.127645i
$983$	50.9117	1.62383	0.811915	−	0.583775i	$-0.198425\pi$
$983$	50.9117	1.62383	0.811915	+	0.583775i	$0.198425\pi$
$984$	0	0
$985$	6.00000	0.191176
$986$	− 2.82843i	− 0.0900755i
$987$	0	0
$988$	0	0
$989$	− 67.8823i	− 2.15853i
$990$	0	0
$991$	20.0000i	0.635321i	0.948205	+	0.317660i	$0.102897\pi$
$991$	20.0000i	0.635321i	−0.948205	+	0.317660i	$0.897103\pi$
$992$	22.6274	0.718421
$993$	0	0
$994$	−32.0000	−1.01498
$995$	11.3137	0.358669
$996$	0	0
$997$	−26.0000	−0.823428	−0.411714	−	0.911313i	$-0.635070\pi$
$997$	−26.0000	−0.823428	−0.411714	+	0.911313i	$0.635070\pi$
$998$	5.65685	0.179065
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	684.2.c.a.647.2	yes	4
3.2	odd	2	inner	684.2.c.a.647.4	yes	4
4.3	odd	2	inner	684.2.c.a.647.1	✓	4
12.11	even	2	inner	684.2.c.a.647.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
684.2.c.a.647.1	✓	4	4.3	odd	2	inner
684.2.c.a.647.2	yes	4	1.1	even	1	trivial
684.2.c.a.647.3	yes	4	12.11	even	2	inner
684.2.c.a.647.4	yes	4	3.2	odd	2	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:	\( N \)	\(=\)	\( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	684.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} -2.00000 q^{4} -1.41421i q^{5} +4.00000i q^{7} +2.82843i q^{8} +O(q^{10})\)	\(q-1.41421i q^{2} -2.00000 q^{4} -1.41421i q^{5} +4.00000i q^{7} +2.82843i q^{8} -2.00000 q^{10} +2.82843 q^{11} +5.65685 q^{14} +4.00000 q^{16} +1.41421i q^{17} +1.00000i q^{19} +2.82843i q^{20} -4.00000i q^{22} +8.48528 q^{23} +3.00000 q^{25} -8.00000i q^{28} -1.41421i q^{29} +4.00000i q^{31} -5.65685i q^{32} +2.00000 q^{34} +5.65685 q^{35} -6.00000 q^{37} +1.41421 q^{38} +4.00000 q^{40} +7.07107i q^{41} -8.00000i q^{43} -5.65685 q^{44} -12.0000i q^{46} +2.82843 q^{47} -9.00000 q^{49} -4.24264i q^{50} -7.07107i q^{53} -4.00000i q^{55} -11.3137 q^{56} -2.00000 q^{58} +11.3137 q^{59} -2.00000 q^{61} +5.65685 q^{62} -8.00000 q^{64} +12.0000i q^{67} -2.82843i q^{68} -8.00000i q^{70} -5.65685 q^{71} +16.0000 q^{73} +8.48528i q^{74} -2.00000i q^{76} +11.3137i q^{77} +8.00000i q^{79} -5.65685i q^{80} +10.0000 q^{82} -2.82843 q^{83} +2.00000 q^{85} -11.3137 q^{86} +8.00000i q^{88} -9.89949i q^{89} -16.9706 q^{92} -4.00000i q^{94} +1.41421 q^{95} +12.0000 q^{97} +12.7279i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4}+O(q^{10}) \) 4 * q - 8 * q^4	\( 4 q - 8 q^{4} - 8 q^{10} + 16 q^{16} + 12 q^{25} + 8 q^{34} - 24 q^{37} + 16 q^{40} - 36 q^{49} - 8 q^{58} - 8 q^{61} - 32 q^{64} + 64 q^{73} + 40 q^{82} + 8 q^{85} + 48 q^{97}+O(q^{100}) \) 4 * q - 8 * q^4 - 8 * q^10 + 16 * q^16 + 12 * q^25 + 8 * q^34 - 24 * q^37 + 16 * q^40 - 36 * q^49 - 8 * q^58 - 8 * q^61 - 32 * q^64 + 64 * q^73 + 40 * q^82 + 8 * q^85 + 48 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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