LMFDB - Embedded newform 912.3.o.d.721.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [912,3,Mod(721,912)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(912, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("912.721");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 912 = 2^{4} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	912.o (of order $2$, degree $1$, not 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:	$24.8502001097$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.184143974400.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 22x^{6} + 80x^{5} + 215x^{4} - 568x^{3} - 1022x^{2} + 1320x + 2628 $ x^8 - 4x^7 - 22x^6 + 80x^5 + 215x^4 - 568x^3 - 1022x^2 + 1320*x + 2628
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{2}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 114)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			721.4
Root			$2.51885 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	912.721
Dual form			912.3.o.d.721.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{3} +9.91350 q^{5} -3.01452 q^{7} -3.00000 q^{9} +O(q^{10})$	$q-1.73205i q^{3} +9.91350 q^{5} -3.01452 q^{7} -3.00000 q^{9} +10.8190 q^{11} +15.4135i q^{13} -17.1707i q^{15} +0.115542 q^{17} +(-1.92003 + 18.9027i) q^{19} +5.22130i q^{21} +26.9636 q^{23} +73.2775 q^{25} +5.19615i q^{27} -34.0027i q^{29} +27.7631i q^{31} -18.7391i q^{33} -29.8845 q^{35} +51.6116i q^{37} +26.6969 q^{39} -15.3745i q^{41} -50.2194 q^{43} -29.7405 q^{45} +25.0210 q^{47} -39.9127 q^{49} -0.200124i q^{51} -84.8577i q^{53} +107.254 q^{55} +(32.7405 + 3.32559i) q^{57} +60.0742i q^{59} +16.6264 q^{61} +9.04356 q^{63} +152.802i q^{65} +4.59832i q^{67} -46.7023i q^{69} -73.9759i q^{71} -9.08734 q^{73} -126.920i q^{75} -32.6141 q^{77} -96.6761i q^{79} +9.00000 q^{81} +61.7513 q^{83} +1.14542 q^{85} -58.8945 q^{87} +39.9731i q^{89} -46.4643i q^{91} +48.0871 q^{93} +(-19.0342 + 187.392i) q^{95} -35.6440i q^{97} -32.4570 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{5} + 12 q^{7} - 24 q^{9}+O(q^{10}) $ 8 * q + 4 * q^5 + 12 * q^7 - 24 * q^9	$ 8 q + 4 q^{5} + 12 q^{7} - 24 q^{9} - 4 q^{11} + 4 q^{17} + 36 q^{19} + 56 q^{23} + 140 q^{25} - 236 q^{35} + 96 q^{39} - 100 q^{43} - 12 q^{45} + 188 q^{47} - 36 q^{49} - 28 q^{55} + 36 q^{57} - 180 q^{61} - 36 q^{63} - 356 q^{73} + 68 q^{77} + 72 q^{81} - 136 q^{83} + 148 q^{85} + 144 q^{87} + 168 q^{93} + 140 q^{95} + 12 q^{99}+O(q^{100}) $ 8 * q + 4 * q^5 + 12 * q^7 - 24 * q^9 - 4 * q^11 + 4 * q^17 + 36 * q^19 + 56 * q^23 + 140 * q^25 - 236 * q^35 + 96 * q^39 - 100 * q^43 - 12 * q^45 + 188 * q^47 - 36 * q^49 - 28 * q^55 + 36 * q^57 - 180 * q^61 - 36 * q^63 - 356 * q^73 + 68 * q^77 + 72 * q^81 - 136 * q^83 + 148 * q^85 + 144 * q^87 + 168 * q^93 + 140 * q^95 + 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/912\mathbb{Z}\right)^\times$.

$n$	$97$	$229$	$305$	$799$
$\chi(n)$	$-1$	$1$	$1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$		−	1.73205i		−	0.577350i
$3$		−	1.73205i		−	0.577350i
$4$		0			0
$5$	9.91350			1.98270			0.991350	−	0.131244i	$-0.0418972\pi$
$5$	9.91350			1.98270			0.991350	+	0.131244i	$0.0418972\pi$
$6$		0			0
$7$	−3.01452			−0.430646			−0.215323	−	0.976543i	$-0.569080\pi$
$7$	−3.01452			−0.430646			−0.215323	+	0.976543i	$0.569080\pi$
$8$		0			0
$9$	−3.00000			−0.333333
$10$		0			0
$11$	10.8190			0.983546			0.491773	−	0.870723i	$-0.336349\pi$
$11$	10.8190			0.983546			0.491773	+	0.870723i	$0.336349\pi$
$12$		0			0
$13$			15.4135i			1.18565i	0.805330	+	0.592826i	$0.201988\pi$
$13$			15.4135i			1.18565i	−0.805330	+	0.592826i	$0.798012\pi$
$14$		0			0
$15$		−	17.1707i		−	1.14471i
$16$		0			0
$17$	0.115542			0.00679657			0.00339829	−	0.999994i	$-0.498918\pi$
$17$	0.115542			0.00679657			0.00339829	+	0.999994i	$0.498918\pi$
$18$		0			0
$19$	−1.92003	+	18.9027i	−0.101054	+	0.994881i
$19$	−1.92003	+	18.9027i	−0.101054	+	0.994881i
$20$		0			0
$21$			5.22130i			0.248634i
$22$		0			0
$23$	26.9636			1.17233			0.586165	−	0.810192i	$-0.300637\pi$
$23$	26.9636			1.17233			0.586165	+	0.810192i	$0.300637\pi$
$24$		0			0
$25$	73.2775			2.93110
$26$		0			0
$27$			5.19615i			0.192450i
$28$		0			0
$29$		−	34.0027i		−	1.17251i	−0.810127	−	0.586254i	$-0.800602\pi$
$29$		−	34.0027i		−	1.17251i	0.810127	−	0.586254i	$-0.199398\pi$
$30$		0			0
$31$			27.7631i			0.895584i	0.894138	+	0.447792i	$0.147790\pi$
$31$			27.7631i			0.895584i	−0.894138	+	0.447792i	$0.852210\pi$
$32$		0			0
$33$		−	18.7391i		−	0.567851i
$34$		0			0
$35$	−29.8845			−0.853842
$36$		0			0
$37$			51.6116i			1.39491i	0.716630	+	0.697454i	$0.245684\pi$
$37$			51.6116i			1.39491i	−0.716630	+	0.697454i	$0.754316\pi$
$38$		0			0
$39$	26.6969			0.684537
$40$		0			0
$41$		−	15.3745i		−	0.374987i	−0.982266	−	0.187494i	$-0.939964\pi$
$41$		−	15.3745i		−	0.374987i	0.982266	−	0.187494i	$-0.0600364\pi$
$42$		0			0
$43$	−50.2194			−1.16789			−0.583947	−	0.811792i	$-0.698492\pi$
$43$	−50.2194			−1.16789			−0.583947	+	0.811792i	$0.698492\pi$
$44$		0			0
$45$	−29.7405			−0.660900
$46$		0			0
$47$	25.0210			0.532363			0.266181	−	0.963923i	$-0.414238\pi$
$47$	25.0210			0.532363			0.266181	+	0.963923i	$0.414238\pi$
$48$		0			0
$49$	−39.9127			−0.814544
$50$		0			0
$51$		−	0.200124i		−	0.00392400i
$52$		0			0
$53$		−	84.8577i		−	1.60109i	−0.599273	−	0.800545i	$-0.704544\pi$
$53$		−	84.8577i		−	1.60109i	0.599273	−	0.800545i	$-0.295456\pi$
$54$		0			0
$55$	107.254			1.95008
$56$		0			0
$57$	32.7405	+	3.32559i	0.574395	+	0.0583436i
$58$		0			0
$59$			60.0742i			1.01821i	0.860705	+	0.509104i	$0.170023\pi$
$59$			60.0742i			1.01821i	−0.860705	+	0.509104i	$0.829977\pi$
$60$		0			0
$61$	16.6264			0.272564			0.136282	−	0.990670i	$-0.456485\pi$
$61$	16.6264			0.272564			0.136282	+	0.990670i	$0.456485\pi$
$62$		0			0
$63$	9.04356			0.143549
$64$		0			0
$65$			152.802i			2.35079i
$66$		0			0
$67$			4.59832i			0.0686317i	0.999411	+	0.0343158i	$0.0109252\pi$
$67$			4.59832i			0.0686317i	−0.999411	+	0.0343158i	$0.989075\pi$
$68$		0			0
$69$		−	46.7023i		−	0.676845i
$70$		0			0
$71$		−	73.9759i		−	1.04191i	−0.853583	−	0.520957i	$-0.825575\pi$
$71$		−	73.9759i		−	1.04191i	0.853583	−	0.520957i	$-0.174425\pi$
$72$		0			0
$73$	−9.08734			−0.124484			−0.0622420	−	0.998061i	$-0.519825\pi$
$73$	−9.08734			−0.124484			−0.0622420	+	0.998061i	$0.519825\pi$
$74$		0			0
$75$		−	126.920i		−	1.69227i
$76$		0			0
$77$	−32.6141			−0.423560
$78$		0			0
$79$		−	96.6761i		−	1.22375i	−0.790955	−	0.611874i	$-0.790416\pi$
$79$		−	96.6761i		−	1.22375i	0.790955	−	0.611874i	$-0.209584\pi$
$80$		0			0
$81$	9.00000			0.111111
$82$		0			0
$83$	61.7513			0.743992			0.371996	−	0.928234i	$-0.378673\pi$
$83$	61.7513			0.743992			0.371996	+	0.928234i	$0.378673\pi$
$84$		0			0
$85$	1.14542			0.0134756
$86$		0			0
$87$	−58.8945			−0.676948
$88$		0			0
$89$			39.9731i			0.449136i	0.974458	+	0.224568i	$0.0720972\pi$
$89$			39.9731i			0.449136i	−0.974458	+	0.224568i	$0.927903\pi$
$90$		0			0
$91$		−	46.4643i		−	0.510596i
$92$		0			0
$93$	48.0871			0.517066
$94$		0			0
$95$	−19.0342	+	187.392i	−0.200360	+	1.97255i
$96$		0			0
$97$		−	35.6440i		−	0.367464i	−0.982976	−	0.183732i	$-0.941182\pi$
$97$		−	35.6440i		−	0.367464i	0.982976	−	0.183732i	$-0.0588179\pi$
$98$		0			0
$99$	−32.4570			−0.327849
$100$		0			0
$101$	−6.89236			−0.0682412			−0.0341206	−	0.999418i	$-0.510863\pi$
$101$	−6.89236			−0.0682412			−0.0341206	+	0.999418i	$0.510863\pi$
$102$		0			0
$103$			101.969i			0.989992i	0.868895	+	0.494996i	$0.164831\pi$
$103$			101.969i			0.989992i	−0.868895	+	0.494996i	$0.835169\pi$
$104$		0			0
$105$			51.7614i			0.492966i
$106$		0			0
$107$		−	88.1128i		−	0.823484i	−0.911300	−	0.411742i	$-0.864920\pi$
$107$		−	88.1128i		−	0.823484i	0.911300	−	0.411742i	$-0.135080\pi$
$108$		0			0
$109$		−	11.4988i		−	0.105494i	−0.998608	−	0.0527469i	$-0.983202\pi$
$109$		−	11.4988i		−	0.105494i	0.998608	−	0.0527469i	$-0.0167977\pi$
$110$		0			0
$111$	89.3939			0.805350
$112$		0			0
$113$		−	126.730i		−	1.12151i	−0.827983	−	0.560753i	$-0.810512\pi$
$113$		−	126.730i		−	1.12151i	0.827983	−	0.560753i	$-0.189488\pi$
$114$		0			0
$115$	267.304			2.32438
$116$		0			0
$117$		−	46.2405i		−	0.395218i
$118$		0			0
$119$	−0.348303			−0.00292692
$120$		0			0
$121$	−3.94908			−0.0326370
$122$		0			0
$123$	−26.6294			−0.216499
$124$		0			0
$125$	478.599			3.82879
$126$		0			0
$127$		−	158.733i		−	1.24987i	−0.780678	−	0.624934i	$-0.785126\pi$
$127$		−	158.733i		−	1.24987i	0.780678	−	0.624934i	$-0.214874\pi$
$128$		0			0
$129$			86.9826i			0.674284i
$130$		0			0
$131$	142.597			1.08852			0.544262	−	0.838916i	$-0.316810\pi$
$131$	142.597			1.08852			0.544262	+	0.838916i	$0.316810\pi$
$132$		0			0
$133$	5.78796	−	56.9827i	0.0435185	−	0.428441i
$134$		0			0
$135$			51.5121i			0.381571i
$136$		0			0
$137$	166.791			1.21745			0.608727	−	0.793380i	$-0.291681\pi$
$137$	166.791			1.21745			0.608727	+	0.793380i	$0.291681\pi$
$138$		0			0
$139$	102.110			0.734606			0.367303	−	0.930101i	$-0.380281\pi$
$139$	102.110			0.734606			0.367303	+	0.930101i	$0.380281\pi$
$140$		0			0
$141$		−	43.3377i		−	0.307360i
$142$		0			0
$143$			166.759i			1.16614i
$144$		0			0
$145$		−	337.086i		−	2.32473i
$146$		0			0
$147$			69.1308i			0.470277i
$148$		0			0
$149$	−244.160			−1.63866			−0.819329	−	0.573323i	$-0.805654\pi$
$149$	−244.160			−1.63866			−0.819329	+	0.573323i	$0.805654\pi$
$150$		0			0
$151$		−	195.871i		−	1.29716i	−0.761146	−	0.648580i	$-0.775363\pi$
$151$		−	195.871i		−	1.29716i	0.761146	−	0.648580i	$-0.224637\pi$
$152$		0			0
$153$	−0.346625			−0.00226552
$154$		0			0
$155$			275.230i			1.77568i
$156$		0			0
$157$	117.369			0.747573			0.373787	−	0.927515i	$-0.378059\pi$
$157$	117.369			0.747573			0.373787	+	0.927515i	$0.378059\pi$
$158$		0			0
$159$	−146.978			−0.924389
$160$		0			0
$161$	−81.2823			−0.504859
$162$		0			0
$163$	−180.203			−1.10554			−0.552771	−	0.833333i	$-0.686430\pi$
$163$	−180.203			−1.10554			−0.552771	+	0.833333i	$0.686430\pi$
$164$		0			0
$165$		−	185.770i		−	1.12588i
$166$		0			0
$167$		−	113.141i		−	0.677491i	−0.940878	−	0.338745i	$-0.889997\pi$
$167$		−	113.141i		−	0.677491i	0.940878	−	0.338745i	$-0.110003\pi$
$168$		0			0
$169$	−68.5755			−0.405772
$170$		0			0
$171$	5.76008	−	56.7082i	0.0336847	−	0.331627i
$172$		0			0
$173$			298.411i			1.72492i	0.506127	+	0.862459i	$0.331077\pi$
$173$			298.411i			1.72492i	−0.506127	+	0.862459i	$0.668923\pi$
$174$		0			0
$175$	−220.897			−1.26227
$176$		0			0
$177$	104.052			0.587862
$178$		0			0
$179$		−	17.6491i		−	0.0985986i	−0.998784	−	0.0492993i	$-0.984301\pi$
$179$		−	17.6491i		−	0.0985986i	0.998784	−	0.0492993i	$-0.0156988\pi$
$180$		0			0
$181$			297.751i			1.64503i	0.568740	+	0.822517i	$0.307431\pi$
$181$			297.751i			1.64503i	−0.568740	+	0.822517i	$0.692569\pi$
$182$		0			0
$183$		−	28.7978i		−	0.157365i
$184$		0			0
$185$			511.651i			2.76568i
$186$		0			0
$187$	1.25005			0.00668474
$188$		0			0
$189$		−	15.6639i		−	0.0828778i
$190$		0			0
$191$	17.9862			0.0941688			0.0470844	−	0.998891i	$-0.485007\pi$
$191$	17.9862			0.0941688			0.0470844	+	0.998891i	$0.485007\pi$
$192$		0			0
$193$		−	104.090i		−	0.539327i	−0.962955	−	0.269664i	$-0.913087\pi$
$193$		−	104.090i		−	0.539327i	0.962955	−	0.269664i	$-0.0869125\pi$
$194$		0			0
$195$	264.660			1.35723
$196$		0			0
$197$	−196.215			−0.996017			−0.498008	−	0.867172i	$-0.665935\pi$
$197$	−196.215			−0.996017			−0.498008	+	0.867172i	$0.665935\pi$
$198$		0			0
$199$	−301.530			−1.51523			−0.757613	−	0.652704i	$-0.773634\pi$
$199$	−301.530			−1.51523			−0.757613	+	0.652704i	$0.773634\pi$
$200$		0			0
$201$	7.96453			0.0396245
$202$		0			0
$203$			102.502i			0.504936i
$204$		0			0
$205$		−	152.415i		−	0.743488i
$206$		0			0
$207$	−80.8908			−0.390777
$208$		0			0
$209$	−20.7728	+	204.509i	−0.0993914	+	0.978511i
$210$		0			0
$211$		−	153.409i		−	0.727055i	−0.931584	−	0.363527i	$-0.881572\pi$
$211$		−	153.409i		−	0.727055i	0.931584	−	0.363527i	$-0.118428\pi$
$212$		0			0
$213$	−128.130			−0.601549
$214$		0			0
$215$	−497.850			−2.31558
$216$		0			0
$217$		−	83.6925i		−	0.385680i
$218$		0			0
$219$			15.7397i			0.0718709i
$220$		0			0
$221$			1.78090i			0.00805837i
$222$		0			0
$223$			272.878i			1.22367i	0.790987	+	0.611834i	$0.209568\pi$
$223$			272.878i			1.22367i	−0.790987	+	0.611834i	$0.790432\pi$
$224$		0			0
$225$	−219.832			−0.977033
$226$		0			0
$227$		−	125.189i		−	0.551495i	−0.961230	−	0.275748i	$-0.911075\pi$
$227$		−	125.189i		−	0.551495i	0.961230	−	0.275748i	$-0.0889254\pi$
$228$		0			0
$229$	−274.350			−1.19803			−0.599017	−	0.800736i	$-0.704442\pi$
$229$	−274.350			−1.19803			−0.599017	+	0.800736i	$0.704442\pi$
$230$		0			0
$231$			56.4893i			0.244543i
$232$		0			0
$233$	−115.719			−0.496646			−0.248323	−	0.968677i	$-0.579879\pi$
$233$	−115.719			−0.496646			−0.248323	+	0.968677i	$0.579879\pi$
$234$		0			0
$235$	248.046			1.05552
$236$		0			0
$237$	−167.448			−0.706531
$238$		0			0
$239$	−67.2090			−0.281209			−0.140605	−	0.990066i	$-0.544905\pi$
$239$	−67.2090			−0.281209			−0.140605	+	0.990066i	$0.544905\pi$
$240$		0			0
$241$			124.189i			0.515307i	0.966237	+	0.257653i	$0.0829493\pi$
$241$			124.189i			0.515307i	−0.966237	+	0.257653i	$0.917051\pi$
$242$		0			0
$243$		−	15.5885i		−	0.0641500i
$244$		0			0
$245$	−395.674			−1.61500
$246$		0			0
$247$	−291.357	−	29.5943i	−1.17958	−	0.119815i
$248$		0			0
$249$		−	106.956i		−	0.429544i
$250$		0			0
$251$	12.1645			0.0484641			0.0242321	−	0.999706i	$-0.492286\pi$
$251$	12.1645			0.0484641			0.0242321	+	0.999706i	$0.492286\pi$
$252$		0			0
$253$	291.719			1.15304
$254$		0			0
$255$		−	1.98393i		−	0.00778012i
$256$		0			0
$257$			413.015i			1.60706i	0.595262	+	0.803532i	$0.297048\pi$
$257$			413.015i			1.60706i	−0.595262	+	0.803532i	$0.702952\pi$
$258$		0			0
$259$		−	155.584i		−	0.600711i
$260$		0			0
$261$			102.008i			0.390836i
$262$		0			0
$263$	−244.309			−0.928931			−0.464466	−	0.885591i	$-0.653754\pi$
$263$	−244.309			−0.928931			−0.464466	+	0.885591i	$0.653754\pi$
$264$		0			0
$265$		−	841.237i		−	3.17448i
$266$		0			0
$267$	69.2355			0.259309
$268$		0			0
$269$		−	221.977i		−	0.825193i	−0.910914	−	0.412596i	$-0.864622\pi$
$269$		−	221.977i		−	0.825193i	0.910914	−	0.412596i	$-0.135378\pi$
$270$		0			0
$271$	−464.627			−1.71449			−0.857245	−	0.514908i	$-0.827826\pi$
$271$	−464.627			−1.71449			−0.857245	+	0.514908i	$0.827826\pi$
$272$		0			0
$273$	−80.4785			−0.294793
$274$		0			0
$275$	792.790			2.88287
$276$		0			0
$277$	−338.509			−1.22206			−0.611028	−	0.791609i	$-0.709244\pi$
$277$	−338.509			−1.22206			−0.611028	+	0.791609i	$0.709244\pi$
$278$		0			0
$279$		−	83.2893i		−	0.298528i
$280$		0			0
$281$			171.878i			0.611666i	0.952085	+	0.305833i	$0.0989349\pi$
$281$			171.878i			0.611666i	−0.952085	+	0.305833i	$0.901065\pi$
$282$		0			0
$283$	394.594			1.39432			0.697162	−	0.716913i	$-0.254446\pi$
$283$	394.594			1.39432			0.697162	+	0.716913i	$0.254446\pi$
$284$		0			0
$285$	324.573	+	32.9682i	1.13885	+	0.115678i
$286$		0			0
$287$			46.3467i			0.161487i
$288$		0			0
$289$	−288.987			−0.999954
$290$		0			0
$291$	−61.7373			−0.212156
$292$		0			0
$293$		−	541.555i		−	1.84831i	−0.382016	−	0.924156i	$-0.624770\pi$
$293$		−	541.555i		−	1.84831i	0.382016	−	0.924156i	$-0.375230\pi$
$294$		0			0
$295$			595.546i			2.01880i
$296$		0			0
$297$			56.2172i			0.189284i
$298$		0			0
$299$			415.603i			1.38998i
$300$		0			0
$301$	151.387			0.502948
$302$		0			0
$303$			11.9379i			0.0393991i
$304$		0			0
$305$	164.826			0.540413
$306$		0			0
$307$		−	154.522i		−	0.503328i	−0.967815	−	0.251664i	$-0.919022\pi$
$307$		−	154.522i		−	0.503328i	0.967815	−	0.251664i	$-0.0809778\pi$
$308$		0			0
$309$	176.616			0.571572
$310$		0			0
$311$	200.544			0.644836			0.322418	−	0.946597i	$-0.395504\pi$
$311$	200.544			0.644836			0.322418	+	0.946597i	$0.395504\pi$
$312$		0			0
$313$	354.596			1.13289			0.566447	−	0.824098i	$-0.308317\pi$
$313$	354.596			1.13289			0.566447	+	0.824098i	$0.308317\pi$
$314$		0			0
$315$	89.6534			0.284614
$316$		0			0
$317$		−	172.282i		−	0.543477i	−0.962371	−	0.271738i	$-0.912401\pi$
$317$		−	172.282i		−	0.543477i	0.962371	−	0.271738i	$-0.0875985\pi$
$318$		0			0
$319$		−	367.876i		−	1.15322i
$320$		0			0
$321$	−152.616			−0.475439
$322$		0			0
$323$	−0.221843	+	2.18406i	−0.000686822	+	0.00676178i
$324$		0			0
$325$			1129.46i			3.47527i
$326$		0			0
$327$	−19.9166			−0.0609069
$328$		0			0
$329$	−75.4265			−0.229260
$330$		0			0
$331$			142.309i			0.429936i	0.976621	+	0.214968i	$0.0689648\pi$
$331$			142.309i			0.429936i	−0.976621	+	0.214968i	$0.931035\pi$
$332$		0			0
$333$		−	154.835i		−	0.464969i
$334$		0			0
$335$			45.5855i			0.136076i
$336$		0			0
$337$			38.4862i			0.114202i	0.998368	+	0.0571012i	$0.0181858\pi$
$337$			38.4862i			0.114202i	−0.998368	+	0.0571012i	$0.981814\pi$
$338$		0			0
$339$	−219.503			−0.647501
$340$		0			0
$341$			300.369i			0.880849i
$342$		0			0
$343$	268.029			0.781426
$344$		0			0
$345$		−	462.983i		−	1.34198i
$346$		0			0
$347$	−387.777			−1.11751			−0.558757	−	0.829331i	$-0.688722\pi$
$347$	−387.777			−1.11751			−0.558757	+	0.829331i	$0.688722\pi$
$348$		0			0
$349$	64.3080			0.184264			0.0921318	−	0.995747i	$-0.470632\pi$
$349$	64.3080			0.184264			0.0921318	+	0.995747i	$0.470632\pi$
$350$		0			0
$351$	−80.0908			−0.228179
$352$		0			0
$353$	−62.7329			−0.177713			−0.0888567	−	0.996044i	$-0.528321\pi$
$353$	−62.7329			−0.177713			−0.0888567	+	0.996044i	$0.528321\pi$
$354$		0			0
$355$		−	733.360i		−	2.06580i
$356$		0			0
$357$			0.603279i			0.00168986i
$358$		0			0
$359$	−609.306			−1.69723			−0.848616	−	0.529009i	$-0.822564\pi$
$359$	−609.306			−1.69723			−0.848616	+	0.529009i	$0.822564\pi$
$360$		0			0
$361$	−353.627	−	72.5876i	−0.979576	−	0.201074i
$362$		0			0
$363$			6.84001i			0.0188430i
$364$		0			0
$365$	−90.0873			−0.246815
$366$		0			0
$367$	−111.317			−0.303316			−0.151658	−	0.988433i	$-0.548461\pi$
$367$	−111.317			−0.303316			−0.151658	+	0.988433i	$0.548461\pi$
$368$		0			0
$369$			46.1235i			0.124996i
$370$		0			0
$371$			255.805i			0.689503i
$372$		0			0
$373$		−	202.144i		−	0.541940i	−0.962588	−	0.270970i	$-0.912656\pi$
$373$		−	202.144i		−	0.541940i	0.962588	−	0.270970i	$-0.0873445\pi$
$374$		0			0
$375$		−	828.958i		−	2.21055i
$376$		0			0
$377$	524.101			1.39019
$378$		0			0
$379$			490.422i			1.29399i	0.762495	+	0.646994i	$0.223974\pi$
$379$			490.422i			1.29399i	−0.762495	+	0.646994i	$0.776026\pi$
$380$		0			0
$381$	−274.934			−0.721611
$382$		0			0
$383$		−	36.2292i		−	0.0945933i	−0.998881	−	0.0472966i	$-0.984939\pi$
$383$		−	36.2292i		−	0.0945933i	0.998881	−	0.0472966i	$-0.0150606\pi$
$384$		0			0
$385$	−323.320			−0.839793
$386$		0			0
$387$	150.658			0.389298
$388$		0			0
$389$	185.177			0.476033			0.238016	−	0.971261i	$-0.423503\pi$
$389$	185.177			0.476033			0.238016	+	0.971261i	$0.423503\pi$
$390$		0			0
$391$	3.11542			0.00796783
$392$		0			0
$393$		−	246.984i		−	0.628459i
$394$		0			0
$395$		−	958.398i		−	2.42632i
$396$		0			0
$397$	−228.224			−0.574872			−0.287436	−	0.957800i	$-0.592803\pi$
$397$	−228.224			−0.574872			−0.287436	+	0.957800i	$0.592803\pi$
$398$		0			0
$399$	−98.6969	−	10.0250i	−0.247361	−	0.0251254i
$400$		0			0
$401$			72.1432i			0.179908i	0.995946	+	0.0899541i	$0.0286720\pi$
$401$			72.1432i			0.179908i	−0.995946	+	0.0899541i	$0.971328\pi$
$402$		0			0
$403$	−427.926			−1.06185
$404$		0			0
$405$	89.2215			0.220300
$406$		0			0
$407$			558.386i			1.37196i
$408$		0			0
$409$			400.431i			0.979048i	0.871990	+	0.489524i	$0.162829\pi$
$409$			400.431i			0.979048i	−0.871990	+	0.489524i	$0.837171\pi$
$410$		0			0
$411$		−	288.891i		−	0.702897i
$412$		0			0
$413$		−	181.095i		−	0.438487i
$414$		0			0
$415$	612.172			1.47511
$416$		0			0
$417$		−	176.860i		−	0.424125i
$418$		0			0
$419$	−456.141			−1.08864			−0.544321	−	0.838877i	$-0.683213\pi$
$419$	−456.141			−1.08864			−0.544321	+	0.838877i	$0.683213\pi$
$420$		0			0
$421$		−	163.425i		−	0.388182i	−0.980983	−	0.194091i	$-0.937824\pi$
$421$		−	163.425i		−	0.388182i	0.980983	−	0.194091i	$-0.0621758\pi$
$422$		0			0
$423$	−75.0631			−0.177454
$424$		0			0
$425$	8.46661			0.0199214
$426$		0			0
$427$	−50.1207			−0.117379
$428$		0			0
$429$	288.834			0.673274
$430$		0			0
$431$			223.091i			0.517613i	0.965929	+	0.258806i	$0.0833292\pi$
$431$			223.091i			0.517613i	−0.965929	+	0.258806i	$0.916671\pi$
$432$		0			0
$433$		−	60.0131i		−	0.138598i	−0.997596	−	0.0692991i	$-0.977924\pi$
$433$		−	60.0131i		−	0.138598i	0.997596	−	0.0692991i	$-0.0220763\pi$
$434$		0			0
$435$	−583.850			−1.34218
$436$		0			0
$437$	−51.7708	+	509.686i	−0.118469	+	1.16633i
$438$		0			0
$439$			183.424i			0.417823i	0.977935	+	0.208911i	$0.0669920\pi$
$439$			183.424i			0.417823i	−0.977935	+	0.208911i	$0.933008\pi$
$440$		0			0
$441$	119.738			0.271515
$442$		0			0
$443$	825.921			1.86438			0.932191	−	0.361967i	$-0.117895\pi$
$443$	825.921			1.86438			0.932191	+	0.361967i	$0.117895\pi$
$444$		0			0
$445$			396.274i			0.890503i
$446$		0			0
$447$			422.898i			0.946080i
$448$		0			0
$449$		−	559.499i		−	1.24610i	−0.782182	−	0.623050i	$-0.785893\pi$
$449$		−	559.499i		−	1.24610i	0.782182	−	0.623050i	$-0.214107\pi$
$450$		0			0
$451$		−	166.337i		−	0.368817i
$452$		0			0
$453$	−339.259			−0.748916
$454$		0			0
$455$		−	460.624i		−	1.01236i
$456$		0			0
$457$	511.991			1.12033			0.560166	−	0.828381i	$-0.310737\pi$
$457$	511.991			1.12033			0.560166	+	0.828381i	$0.310737\pi$
$458$		0			0
$459$			0.600372i			0.00130800i
$460$		0			0
$461$	−26.8317			−0.0582033			−0.0291016	−	0.999576i	$-0.509265\pi$
$461$	−26.8317			−0.0582033			−0.0291016	+	0.999576i	$0.509265\pi$
$462$		0			0
$463$	−431.887			−0.932802			−0.466401	−	0.884574i	$-0.654450\pi$
$463$	−431.887			−0.932802			−0.466401	+	0.884574i	$0.654450\pi$
$464$		0			0
$465$	476.712			1.02519
$466$		0			0
$467$	−64.2540			−0.137589			−0.0687944	−	0.997631i	$-0.521915\pi$
$467$	−64.2540			−0.137589			−0.0687944	+	0.997631i	$0.521915\pi$
$468$		0			0
$469$		−	13.8617i		−	0.0295559i
$470$		0			0
$471$		−	203.289i		−	0.431612i
$472$		0			0
$473$	−543.324			−1.14868
$474$		0			0
$475$	−140.695	+	1385.15i	−0.296200	+	2.91610i
$476$		0			0
$477$			254.573i			0.533696i
$478$		0			0
$479$	−640.901			−1.33800			−0.668999	−	0.743263i	$-0.733277\pi$
$479$	−640.901			−1.33800			−0.668999	+	0.743263i	$0.733277\pi$
$480$		0			0
$481$	−795.514			−1.65388
$482$		0			0
$483$			140.785i			0.291481i
$484$		0			0
$485$		−	353.357i		−	0.728571i
$486$		0			0
$487$			625.075i			1.28352i	0.766905	+	0.641760i	$0.221796\pi$
$487$			625.075i			1.28352i	−0.766905	+	0.641760i	$0.778204\pi$
$488$		0			0
$489$			312.121i			0.638285i
$490$		0			0
$491$	257.709			0.524867			0.262433	−	0.964950i	$-0.415475\pi$
$491$	257.709			0.524867			0.262433	+	0.964950i	$0.415475\pi$
$492$		0			0
$493$		−	3.92873i		−	0.00796904i
$494$		0			0
$495$	−321.763			−0.650026
$496$		0			0
$497$			223.002i			0.448696i
$498$		0			0
$499$	658.482			1.31960			0.659801	−	0.751440i	$-0.270640\pi$
$499$	658.482			1.31960			0.659801	+	0.751440i	$0.270640\pi$
$500$		0			0
$501$	−195.966			−0.391150
$502$		0			0
$503$	−236.146			−0.469474			−0.234737	−	0.972059i	$-0.575423\pi$
$503$	−236.146			−0.469474			−0.234737	+	0.972059i	$0.575423\pi$
$504$		0			0
$505$	−68.3274			−0.135302
$506$		0			0
$507$			118.776i			0.234273i
$508$		0			0
$509$		−	68.2300i		−	0.134047i	−0.997751	−	0.0670236i	$-0.978650\pi$
$509$		−	68.2300i		−	0.134047i	0.997751	−	0.0670236i	$-0.0213503\pi$
$510$		0			0
$511$	27.3940			0.0536086
$512$		0			0
$513$	−98.2215	−	9.97676i	−0.191465	−	0.0194479i
$514$		0			0
$515$			1010.87i			1.96286i
$516$		0			0
$517$	270.703			0.523603
$518$		0			0
$519$	516.863			0.995882
$520$		0			0
$521$		−	635.263i		−	1.21931i	−0.792665	−	0.609657i	$-0.791307\pi$
$521$		−	635.263i		−	1.21931i	0.792665	−	0.609657i	$-0.208693\pi$
$522$		0			0
$523$		−	472.488i		−	0.903419i	−0.892165	−	0.451710i	$-0.850814\pi$
$523$		−	472.488i		−	0.903419i	0.892165	−	0.451710i	$-0.149186\pi$
$524$		0			0
$525$			382.604i			0.728770i
$526$		0			0
$527$			3.20780i			0.00608690i
$528$		0			0
$529$	198.035			0.374358
$530$		0			0
$531$		−	180.223i		−	0.339403i
$532$		0			0
$533$	236.974			0.444605
$534$		0			0
$535$		−	873.506i		−	1.63272i
$536$		0			0
$537$	−30.5692			−0.0569259
$538$		0			0
$539$	−431.815			−0.801142
$540$		0			0
$541$	548.295			1.01348			0.506742	−	0.862098i	$-0.330850\pi$
$541$	548.295			1.01348			0.506742	+	0.862098i	$0.330850\pi$
$542$		0			0
$543$	515.720			0.949761
$544$		0			0
$545$		−	113.994i		−	0.209163i
$546$		0			0
$547$		−	369.232i		−	0.675012i	−0.941323	−	0.337506i	$-0.890417\pi$
$547$		−	369.232i		−	0.675012i	0.941323	−	0.337506i	$-0.109583\pi$
$548$		0			0
$549$	−49.8793			−0.0908548
$550$		0			0
$551$	642.745	+	65.2862i	1.16651	+	0.118487i
$552$		0			0
$553$			291.432i			0.527002i
$554$		0			0
$555$	886.206			1.59677
$556$		0			0
$557$	642.882			1.15419			0.577093	−	0.816678i	$-0.304187\pi$
$557$	642.882			1.15419			0.577093	+	0.816678i	$0.304187\pi$
$558$		0			0
$559$		−	774.056i		−	1.38472i
$560$		0			0
$561$		−	2.16514i		−	0.00385944i
$562$		0			0
$563$			267.879i			0.475807i	0.971289	+	0.237903i	$0.0764602\pi$
$563$			267.879i			0.475807i	−0.971289	+	0.237903i	$0.923540\pi$
$564$		0			0
$565$		−	1256.34i		−	2.22361i
$566$		0			0
$567$	−27.1307			−0.0478495
$568$		0			0
$569$			539.002i			0.947279i	0.880719	+	0.473640i	$0.157060\pi$
$569$			539.002i			0.947279i	−0.880719	+	0.473640i	$0.842940\pi$
$570$		0			0
$571$	136.244			0.238606			0.119303	−	0.992858i	$-0.461934\pi$
$571$	136.244			0.238606			0.119303	+	0.992858i	$0.461934\pi$
$572$		0			0
$573$		−	31.1531i		−	0.0543684i
$574$		0			0
$575$	1975.82			3.43622
$576$		0			0
$577$	331.980			0.575356			0.287678	−	0.957727i	$-0.407117\pi$
$577$	331.980			0.575356			0.287678	+	0.957727i	$0.407117\pi$
$578$		0			0
$579$	−180.289			−0.311381
$580$		0			0
$581$	−186.151			−0.320397
$582$		0			0
$583$		−	918.076i		−	1.57475i
$584$		0			0
$585$		−	458.405i		−	0.783598i
$586$		0			0
$587$	42.6626			0.0726791			0.0363395	−	0.999340i	$-0.488430\pi$
$587$	42.6626			0.0726791			0.0363395	+	0.999340i	$0.488430\pi$
$588$		0			0
$589$	−524.799	−	53.3060i	−0.891000	−	0.0905025i
$590$		0			0
$591$			339.855i			0.575050i
$592$		0			0
$593$	−298.058			−0.502628			−0.251314	−	0.967906i	$-0.580863\pi$
$593$	−298.058			−0.502628			−0.251314	+	0.967906i	$0.580863\pi$
$594$		0			0
$595$	−3.45290			−0.00580320
$596$		0			0
$597$			522.265i			0.874816i
$598$		0			0
$599$			690.206i			1.15226i	0.817357	+	0.576132i	$0.195439\pi$
$599$			690.206i			1.15226i	−0.817357	+	0.576132i	$0.804561\pi$
$600$		0			0
$601$			596.790i			0.992995i	0.868038	+	0.496497i	$0.165381\pi$
$601$			596.790i			0.992995i	−0.868038	+	0.496497i	$0.834619\pi$
$602$		0			0
$603$		−	13.7950i		−	0.0228772i
$604$		0			0
$605$	−39.1492			−0.0647094
$606$		0			0
$607$			334.973i			0.551851i	0.961179	+	0.275925i	$0.0889842\pi$
$607$			334.973i			0.551851i	−0.961179	+	0.275925i	$0.911016\pi$
$608$		0			0
$609$	177.539			0.291525
$610$		0			0
$611$			385.662i			0.631197i
$612$		0			0
$613$	−344.115			−0.561362			−0.280681	−	0.959801i	$-0.590560\pi$
$613$	−344.115			−0.561362			−0.280681	+	0.959801i	$0.590560\pi$
$614$		0			0
$615$	−263.991			−0.429253
$616$		0			0
$617$	−574.986			−0.931907			−0.465953	−	0.884809i	$-0.654289\pi$
$617$	−574.986			−0.931907			−0.465953	+	0.884809i	$0.654289\pi$
$618$		0			0
$619$	−127.502			−0.205981			−0.102990	−	0.994682i	$-0.532841\pi$
$619$	−127.502			−0.205981			−0.102990	+	0.994682i	$0.532841\pi$
$620$		0			0
$621$			140.107i			0.225615i
$622$		0			0
$623$		−	120.500i		−	0.193419i
$624$		0			0
$625$	2912.65			4.66025
$626$		0			0
$627$	354.220	+	35.9795i	0.564944	+	0.0573836i
$628$		0			0
$629$			5.96329i			0.00948059i
$630$		0			0
$631$	352.673			0.558912			0.279456	−	0.960159i	$-0.409846\pi$
$631$	352.673			0.558912			0.279456	+	0.960159i	$0.409846\pi$
$632$		0			0
$633$	−265.711			−0.419765
$634$		0			0
$635$		−	1573.60i		−	2.47811i
$636$		0			0
$637$		−	615.193i		−	0.965766i
$638$		0			0
$639$			221.928i			0.347305i
$640$		0			0
$641$			1133.50i			1.76833i	0.467171	+	0.884167i	$0.345273\pi$
$641$			1133.50i			1.76833i	−0.467171	+	0.884167i	$0.654727\pi$
$642$		0			0
$643$	−469.085			−0.729526			−0.364763	−	0.931100i	$-0.618850\pi$
$643$	−469.085			−0.729526			−0.364763	+	0.931100i	$0.618850\pi$
$644$		0			0
$645$			862.302i			1.33690i
$646$		0			0
$647$	203.611			0.314701			0.157350	−	0.987543i	$-0.449705\pi$
$647$	203.611			0.314701			0.157350	+	0.987543i	$0.449705\pi$
$648$		0			0
$649$			649.944i			1.00145i
$650$		0			0
$651$	−144.960			−0.222672
$652$		0			0
$653$	961.517			1.47246			0.736231	−	0.676731i	$-0.236604\pi$
$653$	961.517			1.47246			0.736231	+	0.676731i	$0.236604\pi$
$654$		0			0
$655$	1413.63			2.15822
$656$		0			0
$657$	27.2620			0.0414947
$658$		0			0
$659$			230.320i			0.349499i	0.984613	+	0.174749i	$0.0559115\pi$
$659$			230.320i			0.349499i	−0.984613	+	0.174749i	$0.944088\pi$
$660$		0			0
$661$			359.931i			0.544525i	0.962223	+	0.272262i	$0.0877719\pi$
$661$			359.931i			0.544525i	−0.962223	+	0.272262i	$0.912228\pi$
$662$		0			0
$663$	3.08461			0.00465250
$664$		0			0
$665$	57.3790	−	564.898i	0.0862842	−	0.849471i
$666$		0			0
$667$		−	916.836i		−	1.37457i
$668$		0			0
$669$	472.638			0.706484
$670$		0			0
$671$	179.881			0.268080
$672$		0			0
$673$		−	1124.17i		−	1.67039i	−0.549954	−	0.835195i	$-0.685355\pi$
$673$		−	1124.17i		−	1.67039i	0.549954	−	0.835195i	$-0.314645\pi$
$674$		0			0
$675$			380.761i			0.564090i
$676$		0			0
$677$		−	29.0597i		−	0.0429242i	−0.999770	−	0.0214621i	$-0.993168\pi$
$677$		−	29.0597i		−	0.0429242i	0.999770	−	0.0214621i	$-0.00683212\pi$
$678$		0			0
$679$			107.450i			0.158247i
$680$		0			0
$681$	−216.834			−0.318406
$682$		0			0
$683$		−	724.655i		−	1.06099i	−0.847689	−	0.530494i	$-0.822006\pi$
$683$		−	724.655i		−	1.06099i	0.847689	−	0.530494i	$-0.177994\pi$
$684$		0			0
$685$	1653.48			2.41385
$686$		0			0
$687$			475.188i			0.691686i
$688$		0			0
$689$	1307.95			1.89834
$690$		0			0
$691$	705.183			1.02053			0.510263	−	0.860018i	$-0.329548\pi$
$691$	705.183			1.02053			0.510263	+	0.860018i	$0.329548\pi$
$692$		0			0
$693$	97.8424			0.141187
$694$		0			0
$695$	1012.27			1.45650
$696$		0			0
$697$		−	1.77639i		−	0.00254863i
$698$		0			0
$699$			200.430i			0.286739i
$700$		0			0
$701$	−608.521			−0.868076			−0.434038	−	0.900895i	$-0.642912\pi$
$701$	−608.521			−0.868076			−0.434038	+	0.900895i	$0.642912\pi$
$702$		0			0
$703$	−975.600	−	99.0957i	−1.38777	−	0.140961i
$704$		0			0
$705$		−	429.629i		−	0.609402i
$706$		0			0
$707$	20.7772			0.0293878
$708$		0			0
$709$	−790.932			−1.11556			−0.557780	−	0.829989i	$-0.688347\pi$
$709$	−790.932			−1.11556			−0.557780	+	0.829989i	$0.688347\pi$
$710$		0			0
$711$			290.028i			0.407916i
$712$		0			0
$713$			748.593i			1.04992i
$714$		0			0
$715$			1653.16i			2.31211i
$716$		0			0
$717$			116.409i			0.162356i
$718$		0			0
$719$	−858.574			−1.19412			−0.597061	−	0.802196i	$-0.703665\pi$
$719$	−858.574			−1.19412			−0.597061	+	0.802196i	$0.703665\pi$
$720$		0			0
$721$		−	307.388i		−	0.426336i
$722$		0			0
$723$	215.102			0.297513
$724$		0			0
$725$		−	2491.64i		−	3.43674i
$726$		0			0
$727$	−1204.58			−1.65691			−0.828457	−	0.560053i	$-0.810781\pi$
$727$	−1204.58			−1.65691			−0.828457	+	0.560053i	$0.810781\pi$
$728$		0			0
$729$	−27.0000			−0.0370370
$730$		0			0
$731$	−5.80244			−0.00793767
$732$		0			0
$733$	−676.835			−0.923377			−0.461689	−	0.887042i	$-0.652756\pi$
$733$	−676.835			−0.923377			−0.461689	+	0.887042i	$0.652756\pi$
$734$		0			0
$735$			685.328i			0.932419i
$736$		0			0
$737$			49.7493i			0.0675024i
$738$		0			0
$739$	86.5075			0.117060			0.0585301	−	0.998286i	$-0.481359\pi$
$739$	86.5075			0.117060			0.0585301	+	0.998286i	$0.481359\pi$
$740$		0			0
$741$	−51.2589	+	504.645i	−0.0691753	+	0.681033i
$742$		0			0
$743$		−	1196.50i		−	1.61036i	−0.593030	−	0.805180i	$-0.702068\pi$
$743$		−	1196.50i		−	1.61036i	0.593030	−	0.805180i	$-0.297932\pi$
$744$		0			0
$745$	−2420.48			−3.24897
$746$		0			0
$747$	−185.254			−0.247997
$748$		0			0
$749$			265.618i			0.354630i
$750$		0			0
$751$		−	279.663i		−	0.372387i	−0.982513	−	0.186194i	$-0.940385\pi$
$751$		−	279.663i		−	0.372387i	0.982513	−	0.186194i	$-0.0596152\pi$
$752$		0			0
$753$		−	21.0695i		−	0.0279808i
$754$		0			0
$755$		−	1941.77i		−	2.57188i
$756$		0			0
$757$	234.363			0.309594			0.154797	−	0.987946i	$-0.450528\pi$
$757$	234.363			0.309594			0.154797	+	0.987946i	$0.450528\pi$
$758$		0			0
$759$		−	505.273i		−	0.665708i
$760$		0			0
$761$	201.790			0.265164			0.132582	−	0.991172i	$-0.457673\pi$
$761$	201.790			0.265164			0.132582	+	0.991172i	$0.457673\pi$
$762$		0			0
$763$			34.6635i			0.0454305i
$764$		0			0
$765$	−3.43627			−0.00449186
$766$		0			0
$767$	−925.954			−1.20724
$768$		0			0
$769$	−130.331			−0.169481			−0.0847406	−	0.996403i	$-0.527006\pi$
$769$	−130.331			−0.169481			−0.0847406	+	0.996403i	$0.527006\pi$
$770$		0			0
$771$	715.363			0.927838
$772$		0			0
$773$			590.469i			0.763867i	0.924190	+	0.381933i	$0.124742\pi$
$773$			590.469i			0.763867i	−0.924190	+	0.381933i	$0.875258\pi$
$774$		0			0
$775$			2034.41i			2.62505i
$776$		0			0
$777$	−269.480			−0.346821
$778$		0			0
$779$	290.620	+	29.5194i	0.373068	+	0.0378940i
$780$		0			0
$781$		−	800.346i		−	1.02477i
$782$		0			0
$783$	176.683			0.225649
$784$		0			0
$785$	1163.54			1.48221
$786$		0			0
$787$		−	571.324i		−	0.725952i	−0.931799	−	0.362976i	$-0.881761\pi$
$787$		−	571.324i		−	0.725952i	0.931799	−	0.362976i	$-0.118239\pi$
$788$		0			0
$789$			423.155i			0.536319i
$790$		0			0
$791$			382.030i			0.482972i
$792$		0			0
$793$			256.271i			0.323167i
$794$		0			0
$795$	−1457.07			−1.83279
$796$		0			0
$797$		−	935.941i		−	1.17433i	−0.809467	−	0.587165i	$-0.800244\pi$
$797$		−	935.941i		−	1.17433i	0.809467	−	0.587165i	$-0.199756\pi$
$798$		0			0
$799$	2.89098			0.00361824
$800$		0			0
$801$		−	119.919i		−	0.149712i
$802$		0			0
$803$	−98.3160			−0.122436
$804$		0			0
$805$	−805.792			−1.00098
$806$		0			0
$807$	−384.475			−0.476425
$808$		0			0
$809$	−1456.28			−1.80009			−0.900047	−	0.435792i	$-0.856468\pi$
$809$	−1456.28			−1.80009			−0.900047	+	0.435792i	$0.856468\pi$
$810$		0			0
$811$			1222.20i			1.50703i	0.657432	+	0.753514i	$0.271643\pi$
$811$			1222.20i			1.50703i	−0.657432	+	0.753514i	$0.728357\pi$
$812$		0			0
$813$			804.758i			0.989862i
$814$		0			0
$815$	−1786.45			−2.19196
$816$		0			0
$817$	96.4227	−	949.284i	0.118020	−	1.16191i
$818$		0			0
$819$			139.393i			0.170199i
$820$		0			0
$821$	306.106			0.372846			0.186423	−	0.982470i	$-0.440311\pi$
$821$	306.106			0.372846			0.186423	+	0.982470i	$0.440311\pi$
$822$		0			0
$823$	−217.902			−0.264766			−0.132383	−	0.991199i	$-0.542263\pi$
$823$	−217.902			−0.264766			−0.132383	+	0.991199i	$0.542263\pi$
$824$		0			0
$825$		−	1373.15i		−	1.66443i
$826$		0			0
$827$		−	20.2664i		−	0.0245059i	−0.999925	−	0.0122529i	$-0.996100\pi$
$827$		−	20.2664i		−	0.0245059i	0.999925	−	0.0122529i	$-0.00390033\pi$
$828$		0			0
$829$		−	533.462i		−	0.643501i	−0.946824	−	0.321751i	$-0.895729\pi$
$829$		−	533.462i		−	0.643501i	0.946824	−	0.321751i	$-0.104271\pi$
$830$		0			0
$831$			586.316i			0.705554i
$832$		0			0
$833$	−4.61158			−0.00553611
$834$		0			0
$835$		−	1121.62i		−	1.34326i
$836$		0			0
$837$	−144.261			−0.172355
$838$		0			0
$839$		−	1248.51i		−	1.48810i	−0.668126	−	0.744048i	$-0.732903\pi$
$839$		−	1248.51i		−	1.48810i	0.668126	−	0.744048i	$-0.267097\pi$
$840$		0			0
$841$	−315.186			−0.374775
$842$		0			0
$843$	297.702			0.353145
$844$		0			0
$845$	−679.823			−0.804525
$846$		0			0
$847$	11.9046			0.0140550
$848$		0			0
$849$		−	683.457i		−	0.805014i
$850$		0			0
$851$			1391.63i			1.63529i
$852$		0			0
$853$	1250.08			1.46551			0.732753	−	0.680495i	$-0.238235\pi$
$853$	1250.08			1.46551			0.732753	+	0.680495i	$0.238235\pi$
$854$		0			0
$855$	57.1026	−	562.177i	0.0667867	−	0.657517i
$856$		0			0
$857$		−	845.935i		−	0.987088i	−0.869721	−	0.493544i	$-0.835701\pi$
$857$		−	845.935i		−	0.987088i	0.869721	−	0.493544i	$-0.164299\pi$
$858$		0			0
$859$	829.883			0.966103			0.483052	−	0.875592i	$-0.339528\pi$
$859$	829.883			0.966103			0.483052	+	0.875592i	$0.339528\pi$
$860$		0			0
$861$	80.2749			0.0932345
$862$		0			0
$863$		−	963.692i		−	1.11668i	−0.829613	−	0.558338i	$-0.811439\pi$
$863$		−	963.692i		−	1.11668i	0.829613	−	0.558338i	$-0.188561\pi$
$864$		0			0
$865$			2958.30i			3.42000i
$866$		0			0
$867$			500.540i			0.577324i
$868$		0			0
$869$		−	1045.94i		−	1.20361i
$870$		0			0
$871$	−70.8761			−0.0813733
$872$		0			0
$873$			106.932i			0.122488i
$874$		0			0
$875$	−1442.75			−1.64885
$876$		0			0
$877$		−	145.023i		−	0.165363i	−0.996576	−	0.0826815i	$-0.973652\pi$
$877$		−	145.023i		−	0.165363i	0.996576	−	0.0826815i	$-0.0263484\pi$
$878$		0			0
$879$	−938.001			−1.06712
$880$		0			0
$881$	616.931			0.700262			0.350131	−	0.936701i	$-0.386137\pi$
$881$	616.931			0.700262			0.350131	+	0.936701i	$0.386137\pi$
$882$		0			0
$883$	−462.198			−0.523440			−0.261720	−	0.965144i	$-0.584290\pi$
$883$	−462.198			−0.523440			−0.261720	+	0.965144i	$0.584290\pi$
$884$		0			0
$885$	1031.52			1.16555
$886$		0			0
$887$		−	359.598i		−	0.405409i	−0.979240	−	0.202704i	$-0.935027\pi$
$887$		−	359.598i		−	0.405409i	0.979240	−	0.202704i	$-0.0649730\pi$
$888$		0			0
$889$			478.504i			0.538250i
$890$		0			0
$891$	97.3711			0.109283
$892$		0			0
$893$	−48.0411	+	472.966i	−0.0537974	+	0.529638i
$894$		0			0
$895$		−	174.965i		−	0.195491i
$896$		0			0
$897$	719.845			0.802503
$898$		0			0
$899$	944.022			1.05008
$900$		0			0
$901$		−	9.80461i		−	0.0108819i
$902$		0			0
$903$		−	262.211i		−	0.290377i
$904$		0			0
$905$			2951.76i			3.26161i
$906$		0			0
$907$			1629.31i			1.79637i	0.439614	+	0.898187i	$0.355115\pi$
$907$			1629.31i			1.79637i	−0.439614	+	0.898187i	$0.644885\pi$
$908$		0			0
$909$	20.6771			0.0227471
$910$		0			0
$911$			979.434i			1.07512i	0.843225	+	0.537560i	$0.180654\pi$
$911$			979.434i			1.07512i	−0.843225	+	0.537560i	$0.819346\pi$
$912$		0			0
$913$	668.088			0.731751
$914$		0			0
$915$		−	285.487i		−	0.312008i
$916$		0			0
$917$	−429.860			−0.468768
$918$		0			0
$919$	1229.87			1.33827			0.669133	−	0.743143i	$-0.266666\pi$
$919$	1229.87			1.33827			0.669133	+	0.743143i	$0.266666\pi$
$920$		0			0
$921$	−267.640			−0.290597
$922$		0			0
$923$	1140.23			1.23535
$924$		0			0
$925$			3781.97i			4.08861i
$926$		0			0
$927$		−	305.908i		−	0.329997i
$928$		0			0
$929$	−904.746			−0.973892			−0.486946	−	0.873432i	$-0.661889\pi$
$929$	−904.746			−0.973892			−0.486946	+	0.873432i	$0.661889\pi$
$930$		0			0
$931$	76.6334	−	754.459i	0.0823130	−	0.810374i
$932$		0			0
$933$		−	347.352i		−	0.372296i
$934$		0			0
$935$	12.3923			0.0132538
$936$		0			0
$937$	−754.845			−0.805598			−0.402799	−	0.915289i	$-0.631963\pi$
$937$	−754.845			−0.805598			−0.402799	+	0.915289i	$0.631963\pi$
$938$		0			0
$939$		−	614.178i		−	0.654077i
$940$		0			0
$941$		−	1256.04i		−	1.33479i	−0.744705	−	0.667394i	$-0.767410\pi$
$941$		−	1256.04i		−	1.33479i	0.744705	−	0.667394i	$-0.232590\pi$
$942$		0			0
$943$		−	414.551i		−	0.439609i
$944$		0			0
$945$		−	155.284i		−	0.164322i
$946$		0			0
$947$	391.891			0.413824			0.206912	−	0.978360i	$-0.433659\pi$
$947$	391.891			0.413824			0.206912	+	0.978360i	$0.433659\pi$
$948$		0			0
$949$		−	140.068i		−	0.147595i
$950$		0			0
$951$	−298.401			−0.313776
$952$		0			0
$953$			255.958i			0.268581i	0.990942	+	0.134291i	$0.0428755\pi$
$953$			255.958i			0.268581i	−0.990942	+	0.134291i	$0.957124\pi$
$954$		0			0
$955$	178.307			0.186709
$956$		0			0
$957$	−637.180			−0.665809
$958$		0			0
$959$	−502.795			−0.524291
$960$		0			0
$961$	190.209			0.197929
$962$		0			0
$963$			264.338i			0.274495i
$964$		0			0
$965$		−	1031.90i		−	1.06932i
$966$		0			0
$967$	1701.29			1.75935			0.879675	−	0.475575i	$-0.157760\pi$
$967$	1701.29			1.75935			0.879675	+	0.475575i	$0.157760\pi$
$968$		0			0
$969$	3.78289	+	0.384244i	0.00390392	+	0.000396537i
$970$		0			0
$971$		−	416.242i		−	0.428674i	−0.976760	−	0.214337i	$-0.931241\pi$
$971$		−	416.242i		−	0.428674i	0.976760	−	0.214337i	$-0.0687590\pi$
$972$		0			0
$973$	−307.813			−0.316355
$974$		0			0
$975$	1956.28			2.00645
$976$		0			0
$977$		−	950.474i		−	0.972850i	−0.873722	−	0.486425i	$-0.838301\pi$
$977$		−	950.474i		−	0.972850i	0.873722	−	0.486425i	$-0.161699\pi$
$978$		0			0
$979$			432.470i			0.441746i
$980$		0			0
$981$			34.4965i			0.0351646i
$982$		0			0
$983$		−	1052.29i		−	1.07049i	−0.844696	−	0.535246i	$-0.820219\pi$
$983$		−	1052.29i		−	1.07049i	0.844696	−	0.535246i	$-0.179781\pi$
$984$		0			0
$985$	−1945.18			−1.97480
$986$		0			0
$987$			130.642i			0.132363i
$988$		0			0
$989$	−1354.10			−1.36916
$990$		0			0
$991$		−	238.960i		−	0.241130i	−0.992705	−	0.120565i	$-0.961529\pi$
$991$		−	238.960i		−	0.241130i	0.992705	−	0.120565i	$-0.0384706\pi$
$992$		0			0
$993$	246.486			0.248224
$994$		0			0
$995$	−2989.22			−3.00424
$996$		0			0
$997$	35.0288			0.0351342			0.0175671	−	0.999846i	$-0.494408\pi$
$997$	35.0288			0.0351342			0.0175671	+	0.999846i	$0.494408\pi$
$998$		0			0
$999$	−268.182			−0.268450

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	912.3.o.d.721.4		8
3.2	odd	2		2736.3.o.n.721.2		8
4.3	odd	2		114.3.d.a.37.8	yes	8
12.11	even	2		342.3.d.b.37.1		8
19.18	odd	2	inner	912.3.o.d.721.8		8
57.56	even	2		2736.3.o.n.721.1		8
76.75	even	2		114.3.d.a.37.2	✓	8
228.227	odd	2		342.3.d.b.37.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.3.d.a.37.2	✓	8	76.75	even	2
114.3.d.a.37.8	yes	8	4.3	odd	2
342.3.d.b.37.1		8	12.11	even	2
342.3.d.b.37.5		8	228.227	odd	2
912.3.o.d.721.4		8	1.1	even	1	trivial
912.3.o.d.721.8		8	19.18	odd	2	inner
2736.3.o.n.721.1		8	57.56	even	2
2736.3.o.n.721.2		8	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	912.o (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.73205i q^{3} +9.91350 q^{5} -3.01452 q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q-1.73205i q^{3} +9.91350 q^{5} -3.01452 q^{7} -3.00000 q^{9} +10.8190 q^{11} +15.4135i q^{13} -17.1707i q^{15} +0.115542 q^{17} +(-1.92003 + 18.9027i) q^{19} +5.22130i q^{21} +26.9636 q^{23} +73.2775 q^{25} +5.19615i q^{27} -34.0027i q^{29} +27.7631i q^{31} -18.7391i q^{33} -29.8845 q^{35} +51.6116i q^{37} +26.6969 q^{39} -15.3745i q^{41} -50.2194 q^{43} -29.7405 q^{45} +25.0210 q^{47} -39.9127 q^{49} -0.200124i q^{51} -84.8577i q^{53} +107.254 q^{55} +(32.7405 + 3.32559i) q^{57} +60.0742i q^{59} +16.6264 q^{61} +9.04356 q^{63} +152.802i q^{65} +4.59832i q^{67} -46.7023i q^{69} -73.9759i q^{71} -9.08734 q^{73} -126.920i q^{75} -32.6141 q^{77} -96.6761i q^{79} +9.00000 q^{81} +61.7513 q^{83} +1.14542 q^{85} -58.8945 q^{87} +39.9731i q^{89} -46.4643i q^{91} +48.0871 q^{93} +(-19.0342 + 187.392i) q^{95} -35.6440i q^{97} -32.4570 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{5} + 12 q^{7} - 24 q^{9}+O(q^{10}) \) 8 * q + 4 * q^5 + 12 * q^7 - 24 * q^9	\( 8 q + 4 q^{5} + 12 q^{7} - 24 q^{9} - 4 q^{11} + 4 q^{17} + 36 q^{19} + 56 q^{23} + 140 q^{25} - 236 q^{35} + 96 q^{39} - 100 q^{43} - 12 q^{45} + 188 q^{47} - 36 q^{49} - 28 q^{55} + 36 q^{57} - 180 q^{61} - 36 q^{63} - 356 q^{73} + 68 q^{77} + 72 q^{81} - 136 q^{83} + 148 q^{85} + 144 q^{87} + 168 q^{93} + 140 q^{95} + 12 q^{99}+O(q^{100}) \) 8 * q + 4 * q^5 + 12 * q^7 - 24 * q^9 - 4 * q^11 + 4 * q^17 + 36 * q^19 + 56 * q^23 + 140 * q^25 - 236 * q^35 + 96 * q^39 - 100 * q^43 - 12 * q^45 + 188 * q^47 - 36 * q^49 - 28 * q^55 + 36 * q^57 - 180 * q^61 - 36 * q^63 - 356 * q^73 + 68 * q^77 + 72 * q^81 - 136 * q^83 + 148 * q^85 + 144 * q^87 + 168 * q^93 + 140 * q^95 + 12 * q^99

Introduction

L-functions

Modular forms

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Database

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