LMFDB - Embedded newform 1344.3.f.g.769.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1344,3,Mod(769,1344)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1344, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1344.769"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1344.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,-4,0,-24,0,16,0,0,0,24,0,0,0,0,0,-12,0,96,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)

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

Self dual:	no
Analytic conductor:	$36.6213475300$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.551252791296.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 17x^{6} + 26x^{5} + 270x^{4} - 302x^{3} - 1007x^{2} - 3502x + 10609 $ x^8 - 2x^7 - 17x^6 + 26x^5 + 270x^4 - 302x^3 - 1007x^2 - 3502*x + 10609
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			769.7
Root			$-3.47991 - 1.76998i$ of defining polynomial
Character	$\chi$	$=$	1344.769
Dual form			1344.3.f.g.769.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{3} +2.55678i q^{5} +(6.95983 - 0.748863i) q^{7} -3.00000 q^{9} +5.82252 q^{11} +21.8397i q^{13} -4.42847 q^{15} -16.7166i q^{17} -1.37203i q^{19} +(1.29707 + 12.0548i) q^{21} +10.2911 q^{23} +18.4629 q^{25} -5.19615i q^{27} +32.1825 q^{29} +35.1222i q^{31} +10.0849i q^{33} +(1.91468 + 17.7947i) q^{35} -47.2786 q^{37} -37.8275 q^{39} -12.2290i q^{41} -20.4805 q^{43} -7.67033i q^{45} +86.4736i q^{47} +(47.8784 - 10.4239i) q^{49} +28.9541 q^{51} +51.3355 q^{53} +14.8869i q^{55} +2.37643 q^{57} -41.5283i q^{59} -103.227i q^{61} +(-20.8795 + 2.24659i) q^{63} -55.8393 q^{65} -25.5411 q^{67} +17.8247i q^{69} +74.8267 q^{71} -6.02572i q^{73} +31.9787i q^{75} +(40.5238 - 4.36027i) q^{77} -20.8784 q^{79} +9.00000 q^{81} +141.066i q^{83} +42.7407 q^{85} +55.7417i q^{87} +137.025i q^{89} +(16.3550 + 152.001i) q^{91} -60.8335 q^{93} +3.50798 q^{95} -56.0358i q^{97} -17.4676 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{7} - 24 q^{9} + 16 q^{11} + 24 q^{15} - 12 q^{21} + 96 q^{23} + 64 q^{29} + 120 q^{35} + 40 q^{37} - 48 q^{39} - 136 q^{43} + 112 q^{49} + 24 q^{51} - 112 q^{53} - 72 q^{57} + 12 q^{63} - 208 q^{65}+ \cdots - 48 q^{99}+O(q^{100}) $ 8 * q - 4 * q^7 - 24 * q^9 + 16 * q^11 + 24 * q^15 - 12 * q^21 + 96 * q^23 + 64 * q^29 + 120 * q^35 + 40 * q^37 - 48 * q^39 - 136 * q^43 + 112 * q^49 + 24 * q^51 - 112 * q^53 - 72 * q^57 + 12 * q^63 - 208 * q^65 + 8 * q^67 + 16 * q^71 - 288 * q^77 + 104 * q^79 + 72 * q^81 + 424 * q^85 + 192 * q^91 - 144 * q^93 - 240 * q^95 - 48 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times$.

$n$	$127$	$449$	$577$	$1093$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$5$			2.55678i			0.511356i	0.966762	+	0.255678i	$0.0822986\pi$
$5$			2.55678i			0.511356i	−0.966762	+	0.255678i	$0.917701\pi$
$6$		0			0
$7$	6.95983	−	0.748863i	0.994261	−	0.106980i
$7$	6.95983	−	0.748863i	0.994261	−	0.106980i
$8$		0			0
$9$	−3.00000			−0.333333
$10$		0			0
$11$	5.82252			0.529320			0.264660	−	0.964342i	$-0.414740\pi$
$11$	5.82252			0.529320			0.264660	+	0.964342i	$0.414740\pi$
$12$		0			0
$13$			21.8397i			1.67998i	0.542603	+	0.839989i	$0.317439\pi$
$13$			21.8397i			1.67998i	−0.542603	+	0.839989i	$0.682561\pi$
$14$		0			0
$15$	−4.42847			−0.295231
$16$		0			0
$17$		−	16.7166i		−	0.983332i	−0.870784	−	0.491666i	$-0.836388\pi$
$17$		−	16.7166i		−	0.983332i	0.870784	−	0.491666i	$-0.163612\pi$
$18$		0			0
$19$		−	1.37203i		−	0.0722121i	−0.999348	−	0.0361060i	$-0.988505\pi$
$19$		−	1.37203i		−	0.0722121i	0.999348	−	0.0361060i	$-0.0114954\pi$
$20$		0			0
$21$	1.29707	+	12.0548i	0.0617652	+	0.574037i
$22$		0			0
$23$	10.2911			0.447439			0.223720	−	0.974654i	$-0.428180\pi$
$23$	10.2911			0.447439			0.223720	+	0.974654i	$0.428180\pi$
$24$		0			0
$25$	18.4629			0.738515
$26$		0			0
$27$		−	5.19615i		−	0.192450i
$28$		0			0
$29$	32.1825			1.10974			0.554870	−	0.831937i	$-0.312768\pi$
$29$	32.1825			1.10974			0.554870	+	0.831937i	$0.312768\pi$
$30$		0			0
$31$			35.1222i			1.13298i	0.824070	+	0.566488i	$0.191698\pi$
$31$			35.1222i			1.13298i	−0.824070	+	0.566488i	$0.808302\pi$
$32$		0			0
$33$			10.0849i			0.305603i
$34$		0			0
$35$	1.91468	+	17.7947i	0.0547050	+	0.508421i
$36$		0			0
$37$	−47.2786			−1.27780			−0.638900	−	0.769290i	$-0.720610\pi$
$37$	−47.2786			−1.27780			−0.638900	+	0.769290i	$0.720610\pi$
$38$		0			0
$39$	−37.8275			−0.969936
$40$		0			0
$41$		−	12.2290i		−	0.298269i	−0.988817	−	0.149135i	$-0.952351\pi$
$41$		−	12.2290i		−	0.298269i	0.988817	−	0.149135i	$-0.0476488\pi$
$42$		0			0
$43$	−20.4805			−0.476291			−0.238146	−	0.971229i	$-0.576540\pi$
$43$	−20.4805			−0.476291			−0.238146	+	0.971229i	$0.576540\pi$
$44$		0			0
$45$		−	7.67033i		−	0.170452i
$46$		0			0
$47$			86.4736i			1.83986i	0.392077	+	0.919932i	$0.371757\pi$
$47$			86.4736i			1.83986i	−0.392077	+	0.919932i	$0.628243\pi$
$48$		0			0
$49$	47.8784	−	10.4239i	0.977110	−	0.212733i
$50$		0			0
$51$	28.9541			0.567727
$52$		0			0
$53$	51.3355			0.968594			0.484297	−	0.874904i	$-0.339075\pi$
$53$	51.3355			0.968594			0.484297	+	0.874904i	$0.339075\pi$
$54$		0			0
$55$			14.8869i			0.270671i
$56$		0			0
$57$	2.37643			0.0416917
$58$		0			0
$59$		−	41.5283i		−	0.703870i	−0.936025	−	0.351935i	$-0.885524\pi$
$59$		−	41.5283i		−	0.703870i	0.936025	−	0.351935i	$-0.114476\pi$
$60$		0			0
$61$		−	103.227i		−	1.69224i	−0.532991	−	0.846121i	$-0.678932\pi$
$61$		−	103.227i		−	1.69224i	0.532991	−	0.846121i	$-0.321068\pi$
$62$		0			0
$63$	−20.8795	+	2.24659i	−0.331420	+	0.0356601i
$64$		0			0
$65$	−55.8393			−0.859066
$66$		0			0
$67$	−25.5411			−0.381210			−0.190605	−	0.981667i	$-0.561045\pi$
$67$	−25.5411			−0.381210			−0.190605	+	0.981667i	$0.561045\pi$
$68$		0			0
$69$			17.8247i			0.258329i
$70$		0			0
$71$	74.8267			1.05390			0.526948	−	0.849897i	$-0.323336\pi$
$71$	74.8267			1.05390			0.526948	+	0.849897i	$0.323336\pi$
$72$		0			0
$73$		−	6.02572i		−	0.0825441i	−0.999148	−	0.0412720i	$-0.986859\pi$
$73$		−	6.02572i		−	0.0825441i	0.999148	−	0.0412720i	$-0.0131410\pi$
$74$		0			0
$75$			31.9787i			0.426382i
$76$		0			0
$77$	40.5238	−	4.36027i	0.526282	−	0.0566269i
$78$		0			0
$79$	−20.8784			−0.264284			−0.132142	−	0.991231i	$-0.542185\pi$
$79$	−20.8784			−0.264284			−0.132142	+	0.991231i	$0.542185\pi$
$80$		0			0
$81$	9.00000			0.111111
$82$		0			0
$83$			141.066i			1.69959i	0.527112	+	0.849796i	$0.323275\pi$
$83$			141.066i			1.69959i	−0.527112	+	0.849796i	$0.676725\pi$
$84$		0			0
$85$	42.7407			0.502832
$86$		0			0
$87$			55.7417i			0.640709i
$88$		0			0
$89$			137.025i			1.53961i	0.638281	+	0.769804i	$0.279646\pi$
$89$			137.025i			1.53961i	−0.638281	+	0.769804i	$0.720354\pi$
$90$		0			0
$91$	16.3550	+	152.001i	0.179725	+	1.67034i
$92$		0			0
$93$	−60.8335			−0.654124
$94$		0			0
$95$	3.50798			0.0369261
$96$		0			0
$97$		−	56.0358i		−	0.577689i	−0.957376	−	0.288844i	$-0.906729\pi$
$97$		−	56.0358i		−	0.577689i	0.957376	−	0.288844i	$-0.0932710\pi$
$98$		0			0
$99$	−17.4676			−0.176440
$100$		0			0
$101$			64.2057i			0.635700i	0.948141	+	0.317850i	$0.102961\pi$
$101$			64.2057i			0.635700i	−0.948141	+	0.317850i	$0.897039\pi$
$102$		0			0
$103$			34.8932i			0.338768i	0.985550	+	0.169384i	$0.0541779\pi$
$103$			34.8932i			0.338768i	−0.985550	+	0.169384i	$0.945822\pi$
$104$		0			0
$105$	−30.8214	+	3.31632i	−0.293537	+	0.0315840i
$106$		0			0
$107$	−118.803			−1.11031			−0.555156	−	0.831746i	$-0.687342\pi$
$107$	−118.803			−1.11031			−0.555156	+	0.831746i	$0.687342\pi$
$108$		0			0
$109$	−123.572			−1.13369			−0.566845	−	0.823824i	$-0.691836\pi$
$109$	−123.572			−1.13369			−0.566845	+	0.823824i	$0.691836\pi$
$110$		0			0
$111$		−	81.8889i		−	0.737738i
$112$		0			0
$113$	40.3527			0.357103			0.178552	−	0.983931i	$-0.442859\pi$
$113$	40.3527			0.357103			0.178552	+	0.983931i	$0.442859\pi$
$114$		0			0
$115$			26.3121i			0.228801i
$116$		0			0
$117$		−	65.5192i		−	0.559993i
$118$		0			0
$119$	−12.5185	−	116.345i	−0.105197	−	0.977689i
$120$		0			0
$121$	−87.0982			−0.719820
$122$		0			0
$123$	21.1813			0.172206
$124$		0			0
$125$			111.125i			0.889000i
$126$		0			0
$127$	−187.867			−1.47927			−0.739634	−	0.673009i	$-0.765001\pi$
$127$	−187.867			−1.47927			−0.739634	+	0.673009i	$0.765001\pi$
$128$		0			0
$129$		−	35.4733i		−	0.274987i
$130$		0			0
$131$			149.663i			1.14247i	0.820787	+	0.571235i	$0.193535\pi$
$131$			149.663i			1.14247i	−0.820787	+	0.571235i	$0.806465\pi$
$132$		0			0
$133$	−1.02746	−	9.54909i	−0.00772528	−	0.0717977i
$134$		0			0
$135$	13.2854			0.0984104
$136$		0			0
$137$	32.2395			0.235325			0.117662	−	0.993054i	$-0.462460\pi$
$137$	32.2395			0.235325			0.117662	+	0.993054i	$0.462460\pi$
$138$		0			0
$139$			102.864i			0.740026i	0.929027	+	0.370013i	$0.120647\pi$
$139$			102.864i			0.740026i	−0.929027	+	0.370013i	$0.879353\pi$
$140$		0			0
$141$	−149.777			−1.06225
$142$		0			0
$143$			127.162i			0.889246i
$144$		0			0
$145$			82.2834i			0.567472i
$146$		0			0
$147$	18.0547	+	82.9278i	0.122821	+	0.564135i
$148$		0			0
$149$	184.202			1.23626			0.618129	−	0.786077i	$-0.287891\pi$
$149$	184.202			1.23626			0.618129	+	0.786077i	$0.287891\pi$
$150$		0			0
$151$	−19.6611			−0.130206			−0.0651030	−	0.997879i	$-0.520738\pi$
$151$	−19.6611			−0.130206			−0.0651030	+	0.997879i	$0.520738\pi$
$152$		0			0
$153$			50.1499i			0.327777i
$154$		0			0
$155$	−89.7998			−0.579353
$156$		0			0
$157$		−	6.07390i		−	0.0386872i	−0.999813	−	0.0193436i	$-0.993842\pi$
$157$		−	6.07390i		−	0.0386872i	0.999813	−	0.0193436i	$-0.00615765\pi$
$158$		0			0
$159$			88.9157i			0.559218i
$160$		0			0
$161$	71.6243	−	7.70662i	0.444871	−	0.0478672i
$162$		0			0
$163$	218.020			1.33754			0.668772	−	0.743467i	$-0.266820\pi$
$163$	218.020			1.33754			0.668772	+	0.743467i	$0.266820\pi$
$164$		0			0
$165$	−25.7849			−0.156272
$166$		0			0
$167$			250.335i			1.49901i	0.661997	+	0.749506i	$0.269709\pi$
$167$			250.335i			1.49901i	−0.661997	+	0.749506i	$0.730291\pi$
$168$		0			0
$169$	−307.973			−1.82233
$170$		0			0
$171$			4.11609i			0.0240707i
$172$		0			0
$173$		−	326.475i		−	1.88714i	−0.331174	−	0.943570i	$-0.607445\pi$
$173$		−	326.475i		−	1.88714i	0.331174	−	0.943570i	$-0.392555\pi$
$174$		0			0
$175$	128.499	−	13.8262i	0.734277	−	0.0790067i
$176$		0			0
$177$	71.9291			0.406379
$178$		0			0
$179$	−190.382			−1.06359			−0.531793	−	0.846874i	$-0.678482\pi$
$179$	−190.382			−1.06359			−0.531793	+	0.846874i	$0.678482\pi$
$180$		0			0
$181$			199.662i			1.10310i	0.834141	+	0.551552i	$0.185964\pi$
$181$			199.662i			1.10310i	−0.834141	+	0.551552i	$0.814036\pi$
$182$		0			0
$183$	178.794			0.977016
$184$		0			0
$185$		−	120.881i		−	0.653410i
$186$		0			0
$187$		−	97.3330i		−	0.520497i
$188$		0			0
$189$	−3.89121	−	36.1643i	−0.0205884	−	0.191346i
$190$		0			0
$191$	161.628			0.846222			0.423111	−	0.906078i	$-0.360938\pi$
$191$	161.628			0.846222			0.423111	+	0.906078i	$0.360938\pi$
$192$		0			0
$193$	−1.25133			−0.00648355			−0.00324178	−	0.999995i	$-0.501032\pi$
$193$	−1.25133			−0.00648355			−0.00324178	+	0.999995i	$0.501032\pi$
$194$		0			0
$195$		−	96.7165i		−	0.495982i
$196$		0			0
$197$	−262.759			−1.33380			−0.666901	−	0.745146i	$-0.732380\pi$
$197$	−262.759			−1.33380			−0.666901	+	0.745146i	$0.732380\pi$
$198$		0			0
$199$			343.362i			1.72544i	0.505685	+	0.862718i	$0.331240\pi$
$199$			343.362i			1.72544i	−0.505685	+	0.862718i	$0.668760\pi$
$200$		0			0
$201$		−	44.2384i		−	0.220092i
$202$		0			0
$203$	223.984	−	24.1002i	1.10337	−	0.118720i
$204$		0			0
$205$	31.2669			0.152522
$206$		0			0
$207$	−30.8733			−0.149146
$208$		0			0
$209$		−	7.98867i		−	0.0382233i
$210$		0			0
$211$	297.884			1.41177			0.705886	−	0.708325i	$-0.250549\pi$
$211$	297.884			1.41177			0.705886	+	0.708325i	$0.250549\pi$
$212$		0			0
$213$			129.604i			0.608468i
$214$		0			0
$215$		−	52.3641i		−	0.243554i
$216$		0			0
$217$	26.3017	+	244.445i	0.121206	+	1.12647i
$218$		0			0
$219$	10.4369			0.0476569
$220$		0			0
$221$	365.087			1.65198
$222$		0			0
$223$			99.6525i			0.446872i	0.974719	+	0.223436i	$0.0717274\pi$
$223$			99.6525i			0.446872i	−0.974719	+	0.223436i	$0.928273\pi$
$224$		0			0
$225$	−55.3887			−0.246172
$226$		0			0
$227$		−	334.346i		−	1.47289i	−0.676498	−	0.736444i	$-0.736503\pi$
$227$		−	334.346i		−	1.47289i	0.676498	−	0.736444i	$-0.263497\pi$
$228$		0			0
$229$		−	49.7313i		−	0.217167i	−0.994087	−	0.108584i	$-0.965368\pi$
$229$		−	49.7313i		−	0.217167i	0.994087	−	0.108584i	$-0.0346316\pi$
$230$		0			0
$231$	7.55221	+	70.1892i	0.0326936	+	0.303849i
$232$		0			0
$233$	406.511			1.74468			0.872340	−	0.488899i	$-0.162601\pi$
$233$	406.511			1.74468			0.872340	+	0.488899i	$0.162601\pi$
$234$		0			0
$235$	−221.094			−0.940825
$236$		0			0
$237$		−	36.1625i		−	0.152584i
$238$		0			0
$239$	−302.533			−1.26583			−0.632914	−	0.774222i	$-0.718141\pi$
$239$	−302.533			−1.26583			−0.632914	+	0.774222i	$0.718141\pi$
$240$		0			0
$241$		−	391.822i		−	1.62582i	−0.582392	−	0.812908i	$-0.697883\pi$
$241$		−	391.822i		−	1.62582i	0.582392	−	0.812908i	$-0.302117\pi$
$242$		0			0
$243$			15.5885i			0.0641500i
$244$		0			0
$245$	26.6516	+	122.414i	0.108782	+	0.499651i
$246$		0			0
$247$	29.9647			0.121315
$248$		0			0
$249$	−244.334			−0.981260
$250$		0			0
$251$			33.7831i			0.134594i	0.997733	+	0.0672971i	$0.0214375\pi$
$251$			33.7831i			0.134594i	−0.997733	+	0.0672971i	$0.978562\pi$
$252$		0			0
$253$	59.9202			0.236839
$254$		0			0
$255$			74.0291i			0.290310i
$256$		0			0
$257$		−	357.247i		−	1.39007i	−0.718977	−	0.695033i	$-0.755390\pi$
$257$		−	357.247i		−	1.39007i	0.718977	−	0.695033i	$-0.244610\pi$
$258$		0			0
$259$	−329.051	+	35.4052i	−1.27047	+	0.136700i
$260$		0			0
$261$	−96.5474			−0.369913
$262$		0			0
$263$	−374.711			−1.42476			−0.712379	−	0.701795i	$-0.752382\pi$
$263$	−374.711			−1.42476			−0.712379	+	0.701795i	$0.752382\pi$
$264$		0			0
$265$			131.253i			0.495296i
$266$		0			0
$267$	−237.334			−0.888893
$268$		0			0
$269$			69.3103i			0.257659i	0.991667	+	0.128830i	$0.0411220\pi$
$269$			69.3103i			0.257659i	−0.991667	+	0.128830i	$0.958878\pi$
$270$		0			0
$271$		−	454.972i		−	1.67886i	−0.543466	−	0.839431i	$-0.682888\pi$
$271$		−	454.972i		−	1.67886i	0.543466	−	0.839431i	$-0.317112\pi$
$272$		0			0
$273$	−263.273	+	28.3276i	−0.964370	+	0.103764i
$274$		0			0
$275$	107.501			0.390911
$276$		0			0
$277$	−141.878			−0.512195			−0.256098	−	0.966651i	$-0.582437\pi$
$277$	−141.878			−0.512195			−0.256098	+	0.966651i	$0.582437\pi$
$278$		0			0
$279$		−	105.367i		−	0.377658i
$280$		0			0
$281$	477.922			1.70079			0.850395	−	0.526144i	$-0.176363\pi$
$281$	477.922			1.70079			0.850395	+	0.526144i	$0.176363\pi$
$282$		0			0
$283$			37.6953i			0.133199i	0.997780	+	0.0665994i	$0.0212150\pi$
$283$			37.6953i			0.133199i	−0.997780	+	0.0665994i	$0.978785\pi$
$284$		0			0
$285$			6.07599i			0.0213193i
$286$		0			0
$287$	−9.15788	−	85.1120i	−0.0319090	−	0.296558i
$288$		0			0
$289$	9.55390			0.0330585
$290$		0			0
$291$	97.0569			0.333529
$292$		0			0
$293$			71.7594i			0.244913i	0.992474	+	0.122456i	$0.0390771\pi$
$293$			71.7594i			0.244913i	−0.992474	+	0.122456i	$0.960923\pi$
$294$		0			0
$295$	106.179			0.359928
$296$		0			0
$297$		−	30.2547i		−	0.101868i
$298$		0			0
$299$			224.755i			0.751688i
$300$		0			0
$301$	−142.541	+	15.3371i	−0.473558	+	0.0509538i
$302$		0			0
$303$	−111.207			−0.367021
$304$		0			0
$305$	263.928			0.865337
$306$		0			0
$307$		−	414.109i		−	1.34889i	−0.738325	−	0.674445i	$-0.764383\pi$
$307$		−	414.109i		−	1.34889i	0.738325	−	0.674445i	$-0.235617\pi$
$308$		0			0
$309$	−60.4367			−0.195588
$310$		0			0
$311$		−	241.225i		−	0.775644i	−0.921734	−	0.387822i	$-0.873228\pi$
$311$		−	241.225i		−	0.775644i	0.921734	−	0.387822i	$-0.126772\pi$
$312$		0			0
$313$		−	197.478i		−	0.630919i	−0.948939	−	0.315460i	$-0.897841\pi$
$313$		−	197.478i		−	0.630919i	0.948939	−	0.315460i	$-0.102159\pi$
$314$		0			0
$315$	−5.74403	−	53.3842i	−0.0182350	−	0.169474i
$316$		0			0
$317$	446.669			1.40905			0.704524	−	0.709680i	$-0.251160\pi$
$317$	446.669			1.40905			0.704524	+	0.709680i	$0.251160\pi$
$318$		0			0
$319$	187.383			0.587408
$320$		0			0
$321$		−	205.774i		−	0.641039i
$322$		0			0
$323$	−22.9357			−0.0710085
$324$		0			0
$325$			403.224i			1.24069i
$326$		0			0
$327$		−	214.033i		−	0.654536i
$328$		0			0
$329$	64.7569	+	601.842i	0.196829	+	1.82931i
$330$		0			0
$331$	−129.870			−0.392355			−0.196178	−	0.980568i	$-0.562853\pi$
$331$	−129.870			−0.392355			−0.196178	+	0.980568i	$0.562853\pi$
$332$		0			0
$333$	141.836			0.425933
$334$		0			0
$335$		−	65.3029i		−	0.194934i
$336$		0			0
$337$	−368.682			−1.09401			−0.547006	−	0.837128i	$-0.684233\pi$
$337$	−368.682			−1.09401			−0.547006	+	0.837128i	$0.684233\pi$
$338$		0			0
$339$			69.8929i			0.206174i
$340$		0			0
$341$			204.500i			0.599707i
$342$		0			0
$343$	325.419	−	108.403i	0.948745	−	0.316044i
$344$		0			0
$345$	−45.5738			−0.132098
$346$		0			0
$347$	489.785			1.41148			0.705742	−	0.708468i	$-0.250614\pi$
$347$	489.785			1.41148			0.705742	+	0.708468i	$0.250614\pi$
$348$		0			0
$349$		−	288.064i		−	0.825400i	−0.910867	−	0.412700i	$-0.864586\pi$
$349$		−	288.064i		−	0.825400i	0.910867	−	0.412700i	$-0.135414\pi$
$350$		0			0
$351$	113.483			0.323312
$352$		0			0
$353$		−	97.2115i		−	0.275387i	−0.990475	−	0.137693i	$-0.956031\pi$
$353$		−	97.2115i		−	0.275387i	0.990475	−	0.137693i	$-0.0439688\pi$
$354$		0			0
$355$			191.315i			0.538916i
$356$		0			0
$357$	201.515	−	21.6826i	0.564469	−	0.0607357i
$358$		0			0
$359$	284.701			0.793039			0.396520	−	0.918026i	$-0.370218\pi$
$359$	284.701			0.793039			0.396520	+	0.918026i	$0.370218\pi$
$360$		0			0
$361$	359.118			0.994785
$362$		0			0
$363$		−	150.859i		−	0.415588i
$364$		0			0
$365$	15.4064			0.0422094
$366$		0			0
$367$		−	51.7192i		−	0.140924i	−0.997514	−	0.0704622i	$-0.977553\pi$
$367$		−	51.7192i		−	0.140924i	0.997514	−	0.0704622i	$-0.0224474\pi$
$368$		0			0
$369$			36.6871i			0.0994231i
$370$		0			0
$371$	357.286	−	38.4432i	0.963035	−	0.103621i
$372$		0			0
$373$	−597.420			−1.60166			−0.800832	−	0.598890i	$-0.795609\pi$
$373$	−597.420			−1.60166			−0.800832	+	0.598890i	$0.795609\pi$
$374$		0			0
$375$	−192.474			−0.513264
$376$		0			0
$377$			702.856i			1.86434i
$378$		0			0
$379$	71.2607			0.188023			0.0940115	−	0.995571i	$-0.470031\pi$
$379$	71.2607			0.188023			0.0940115	+	0.995571i	$0.470031\pi$
$380$		0			0
$381$		−	325.395i		−	0.854056i
$382$		0			0
$383$		−	363.128i		−	0.948116i	−0.880494	−	0.474058i	$-0.842789\pi$
$383$		−	363.128i		−	0.948116i	0.880494	−	0.474058i	$-0.157211\pi$
$384$		0			0
$385$	11.1482	+	103.610i	0.0289565	+	0.269118i
$386$		0			0
$387$	61.4415			0.158764
$388$		0			0
$389$	236.454			0.607850			0.303925	−	0.952696i	$-0.401703\pi$
$389$	236.454			0.607850			0.303925	+	0.952696i	$0.401703\pi$
$390$		0			0
$391$		−	172.033i		−	0.439981i
$392$		0			0
$393$	−259.225			−0.659605
$394$		0			0
$395$		−	53.3815i		−	0.135143i
$396$		0			0
$397$		−	460.991i		−	1.16119i	−0.814194	−	0.580593i	$-0.802821\pi$
$397$		−	460.991i		−	1.16119i	0.814194	−	0.580593i	$-0.197179\pi$
$398$		0			0
$399$	16.5395	−	1.77962i	0.0414524	−	0.00446019i
$400$		0			0
$401$	−295.902			−0.737909			−0.368955	−	0.929447i	$-0.620284\pi$
$401$	−295.902			−0.737909			−0.368955	+	0.929447i	$0.620284\pi$
$402$		0			0
$403$	−767.060			−1.90337
$404$		0			0
$405$			23.0110i			0.0568173i
$406$		0			0
$407$	−275.281			−0.676365
$408$		0			0
$409$			547.907i			1.33962i	0.742530	+	0.669812i	$0.233625\pi$
$409$			547.907i			1.33962i	−0.742530	+	0.669812i	$0.766375\pi$
$410$		0			0
$411$			55.8404i			0.135865i
$412$		0			0
$413$	−31.0990	−	289.030i	−0.0753003	−	0.699830i
$414$		0			0
$415$	−360.675			−0.869096
$416$		0			0
$417$	−178.165			−0.427254
$418$		0			0
$419$		−	352.472i		−	0.841223i	−0.907241	−	0.420611i	$-0.861816\pi$
$419$		−	352.472i		−	0.841223i	0.907241	−	0.420611i	$-0.138184\pi$
$420$		0			0
$421$	679.902			1.61497			0.807485	−	0.589888i	$-0.200828\pi$
$421$	679.902			1.61497			0.807485	+	0.589888i	$0.200828\pi$
$422$		0			0
$423$		−	259.421i		−	0.613288i
$424$		0			0
$425$		−	308.637i		−	0.726206i
$426$		0			0
$427$	−77.3027	−	718.440i	−0.181037	−	1.68253i
$428$		0			0
$429$	−220.251			−0.513407
$430$		0			0
$431$	529.775			1.22918			0.614589	−	0.788848i	$-0.289322\pi$
$431$	529.775			1.22918			0.614589	+	0.788848i	$0.289322\pi$
$432$		0			0
$433$			398.400i			0.920093i	0.887895	+	0.460047i	$0.152167\pi$
$433$			398.400i			0.920093i	−0.887895	+	0.460047i	$0.847833\pi$
$434$		0			0
$435$	−142.519			−0.327630
$436$		0			0
$437$		−	14.1197i		−	0.0323105i
$438$		0			0
$439$		−	372.685i		−	0.848940i	−0.905442	−	0.424470i	$-0.860460\pi$
$439$		−	372.685i		−	0.848940i	0.905442	−	0.424470i	$-0.139540\pi$
$440$		0			0
$441$	−143.635	+	31.2717i	−0.325703	+	0.0709110i
$442$		0			0
$443$	24.4995			0.0553036			0.0276518	−	0.999618i	$-0.491197\pi$
$443$	24.4995			0.0553036			0.0276518	+	0.999618i	$0.491197\pi$
$444$		0			0
$445$	−350.343			−0.787287
$446$		0			0
$447$			319.048i			0.713754i
$448$		0			0
$449$	891.739			1.98606			0.993028	−	0.117879i	$-0.0376094\pi$
$449$	891.739			1.98606			0.993028	+	0.117879i	$0.0376094\pi$
$450$		0			0
$451$		−	71.2039i		−	0.157880i
$452$		0			0
$453$		−	34.0540i		−	0.0751745i
$454$		0			0
$455$	−388.632	+	41.8160i	−0.854136	+	0.0919033i
$456$		0			0
$457$	−457.868			−1.00190			−0.500950	−	0.865476i	$-0.667016\pi$
$457$	−457.868			−1.00190			−0.500950	+	0.865476i	$0.667016\pi$
$458$		0			0
$459$	−86.8622			−0.189242
$460$		0			0
$461$			37.6805i			0.0817364i	0.999165	+	0.0408682i	$0.0130124\pi$
$461$			37.6805i			0.0817364i	−0.999165	+	0.0408682i	$0.986988\pi$
$462$		0			0
$463$	116.909			0.252504			0.126252	−	0.991998i	$-0.459705\pi$
$463$	116.909			0.252504			0.126252	+	0.991998i	$0.459705\pi$
$464$		0			0
$465$		−	155.538i		−	0.334490i
$466$		0			0
$467$		−	348.521i		−	0.746297i	−0.927772	−	0.373148i	$-0.878278\pi$
$467$		−	348.521i		−	0.746297i	0.927772	−	0.373148i	$-0.121722\pi$
$468$		0			0
$469$	−177.762	+	19.1268i	−0.379022	+	0.0407820i
$470$		0			0
$471$	10.5203			0.0223361
$472$		0			0
$473$	−119.248			−0.252110
$474$		0			0
$475$		−	25.3316i		−	0.0533297i
$476$		0			0
$477$	−154.006			−0.322865
$478$		0			0
$479$		−	361.489i		−	0.754674i	−0.926076	−	0.377337i	$-0.876840\pi$
$479$		−	361.489i		−	0.754674i	0.926076	−	0.377337i	$-0.123160\pi$
$480$		0			0
$481$		−	1032.55i		−	2.14668i
$482$		0			0
$483$	13.3483	+	124.057i	0.0276362	+	0.256847i
$484$		0			0
$485$	143.271			0.295404
$486$		0			0
$487$	−278.730			−0.572341			−0.286170	−	0.958179i	$-0.592382\pi$
$487$	−278.730			−0.572341			−0.286170	+	0.958179i	$0.592382\pi$
$488$		0			0
$489$			377.621i			0.772232i
$490$		0			0
$491$	114.355			0.232902			0.116451	−	0.993196i	$-0.462848\pi$
$491$	114.355			0.232902			0.116451	+	0.993196i	$0.462848\pi$
$492$		0			0
$493$		−	537.983i		−	1.09124i
$494$		0			0
$495$		−	44.6607i		−	0.0902236i
$496$		0			0
$497$	520.781	−	56.0349i	1.04785	−	0.112746i
$498$		0			0
$499$	731.571			1.46607			0.733037	−	0.680189i	$-0.238102\pi$
$499$	731.571			1.46607			0.733037	+	0.680189i	$0.238102\pi$
$500$		0			0
$501$	−433.593			−0.865455
$502$		0			0
$503$		−	178.420i		−	0.354712i	−0.984147	−	0.177356i	$-0.943246\pi$
$503$		−	178.420i		−	0.354712i	0.984147	−	0.177356i	$-0.0567544\pi$
$504$		0			0
$505$	−164.160			−0.325069
$506$		0			0
$507$		−	533.425i		−	1.05212i
$508$		0			0
$509$		−	581.588i		−	1.14261i	−0.820738	−	0.571305i	$-0.806437\pi$
$509$		−	581.588i		−	1.14261i	0.820738	−	0.571305i	$-0.193563\pi$
$510$		0			0
$511$	−4.51244	−	41.9380i	−0.00883060	−	0.0820704i
$512$		0			0
$513$	−7.12928			−0.0138972
$514$		0			0
$515$	−89.2140			−0.173231
$516$		0			0
$517$			503.495i			0.973878i
$518$		0			0
$519$	565.471			1.08954
$520$		0			0
$521$		−	223.945i		−	0.429836i	−0.976632	−	0.214918i	$-0.931052\pi$
$521$		−	223.945i		−	0.429836i	0.976632	−	0.214918i	$-0.0689485\pi$
$522$		0			0
$523$			77.0909i			0.147401i	0.997280	+	0.0737006i	$0.0234809\pi$
$523$			77.0909i			0.147401i	−0.997280	+	0.0737006i	$0.976519\pi$
$524$		0			0
$525$	23.9476	+	222.566i	0.0456145	+	0.423935i
$526$		0			0
$527$	587.126			1.11409
$528$		0			0
$529$	−423.093			−0.799798
$530$		0			0
$531$			124.585i			0.234623i
$532$		0			0
$533$	267.079			0.501086
$534$		0			0
$535$		−	303.754i		−	0.567765i
$536$		0			0
$537$		−	329.751i		−	0.614062i
$538$		0			0
$539$	278.773	−	60.6935i	0.517204	−	0.112604i
$540$		0			0
$541$	−11.6427			−0.0215208			−0.0107604	−	0.999942i	$-0.503425\pi$
$541$	−11.6427			−0.0215208			−0.0107604	+	0.999942i	$0.503425\pi$
$542$		0			0
$543$	−345.824			−0.636877
$544$		0			0
$545$		−	315.947i		−	0.579719i
$546$		0			0
$547$	140.549			0.256945			0.128473	−	0.991713i	$-0.458993\pi$
$547$	140.549			0.256945			0.128473	+	0.991713i	$0.458993\pi$
$548$		0			0
$549$			309.680i			0.564080i
$550$		0			0
$551$		−	44.1553i		−	0.0801366i
$552$		0			0
$553$	−145.310	+	15.6351i	−0.262767	+	0.0282732i
$554$		0			0
$555$	209.372			0.377246
$556$		0			0
$557$	−214.834			−0.385698			−0.192849	−	0.981228i	$-0.561773\pi$
$557$	−214.834			−0.385698			−0.192849	+	0.981228i	$0.561773\pi$
$558$		0			0
$559$		−	447.289i		−	0.800159i
$560$		0			0
$561$	168.586			0.300509
$562$		0			0
$563$			338.814i			0.601801i	0.953655	+	0.300901i	$0.0972873\pi$
$563$			338.814i			0.601801i	−0.953655	+	0.300901i	$0.902713\pi$
$564$		0			0
$565$			103.173i			0.182607i
$566$		0			0
$567$	62.6385	−	6.73977i	0.110473	−	0.0118867i
$568$		0			0
$569$	−497.478			−0.874302			−0.437151	−	0.899388i	$-0.644012\pi$
$569$	−497.478			−0.874302			−0.437151	+	0.899388i	$0.644012\pi$
$570$		0			0
$571$	614.030			1.07536			0.537679	−	0.843149i	$-0.319301\pi$
$571$	614.030			1.07536			0.537679	+	0.843149i	$0.319301\pi$
$572$		0			0
$573$			279.949i			0.488567i
$574$		0			0
$575$	190.003			0.330441
$576$		0			0
$577$		−	922.638i		−	1.59903i	−0.600649	−	0.799513i	$-0.705091\pi$
$577$		−	922.638i		−	1.59903i	0.600649	−	0.799513i	$-0.294909\pi$
$578$		0			0
$579$		−	2.16736i		−	0.00374328i
$580$		0			0
$581$	105.639	+	981.796i	0.181823	+	1.68984i
$582$		0			0
$583$	298.902			0.512696
$584$		0			0
$585$	167.518			0.286355
$586$		0			0
$587$		−	522.573i		−	0.890244i	−0.895470	−	0.445122i	$-0.853160\pi$
$587$		−	522.573i		−	0.890244i	0.895470	−	0.445122i	$-0.146840\pi$
$588$		0			0
$589$	48.1888			0.0818145
$590$		0			0
$591$		−	455.112i		−	0.770071i
$592$		0			0
$593$		−	675.966i		−	1.13991i	−0.821676	−	0.569955i	$-0.806961\pi$
$593$		−	675.966i		−	1.13991i	0.821676	−	0.569955i	$-0.193039\pi$
$594$		0			0
$595$	297.468	−	32.0070i	0.499947	−	0.0537932i
$596$		0			0
$597$	−594.720			−0.996181
$598$		0			0
$599$	18.2174			0.0304131			0.0152065	−	0.999884i	$-0.495159\pi$
$599$	18.2174			0.0304131			0.0152065	+	0.999884i	$0.495159\pi$
$600$		0			0
$601$			467.784i			0.778342i	0.921165	+	0.389171i	$0.127239\pi$
$601$			467.784i			0.778342i	−0.921165	+	0.389171i	$0.872761\pi$
$602$		0			0
$603$	76.6232			0.127070
$604$		0			0
$605$		−	222.691i		−	0.368084i
$606$		0			0
$607$			979.229i			1.61323i	0.591079	+	0.806614i	$0.298702\pi$
$607$			979.229i			1.61323i	−0.591079	+	0.806614i	$0.701298\pi$
$608$		0			0
$609$	41.7429	+	387.952i	0.0685433	+	0.637032i
$610$		0			0
$611$	−1888.56			−3.09093
$612$		0			0
$613$	−423.249			−0.690455			−0.345227	−	0.938519i	$-0.612198\pi$
$613$	−423.249			−0.690455			−0.345227	+	0.938519i	$0.612198\pi$
$614$		0			0
$615$			54.1559i			0.0880584i
$616$		0			0
$617$	11.3273			0.0183586			0.00917932	−	0.999958i	$-0.497078\pi$
$617$	11.3273			0.0183586			0.00917932	+	0.999958i	$0.497078\pi$
$618$		0			0
$619$		−	2.10827i		−	0.00340592i	−0.999999	−	0.00170296i	$-0.999458\pi$
$619$		−	2.10827i		−	0.00340592i	0.999999	−	0.00170296i	$-0.000542070\pi$
$620$		0			0
$621$		−	53.4741i		−	0.0861097i
$622$		0			0
$623$	102.613	+	953.671i	0.164708	+	1.53077i
$624$		0			0
$625$	177.450			0.283921
$626$		0			0
$627$	13.8368			0.0220682
$628$		0			0
$629$			790.339i			1.25650i
$630$		0			0
$631$	−646.524			−1.02460			−0.512301	−	0.858806i	$-0.671207\pi$
$631$	−646.524			−1.02460			−0.512301	+	0.858806i	$0.671207\pi$
$632$		0			0
$633$			515.950i			0.815087i
$634$		0			0
$635$		−	480.334i		−	0.756432i
$636$		0			0
$637$	227.655	+	1045.65i	0.357387	+	1.64152i
$638$		0			0
$639$	−224.480			−0.351299
$640$		0			0
$641$	−302.562			−0.472015			−0.236008	−	0.971751i	$-0.575839\pi$
$641$	−302.562			−0.472015			−0.236008	+	0.971751i	$0.575839\pi$
$642$		0			0
$643$			302.605i			0.470615i	0.971921	+	0.235307i	$0.0756097\pi$
$643$			302.605i			0.470615i	−0.971921	+	0.235307i	$0.924390\pi$
$644$		0			0
$645$	90.6973			0.140616
$646$		0			0
$647$		−	0.0187510i		−	2.89815e-5i	−1.00000		1.44908e-5i	$-0.999995\pi$
$647$		−	0.0187510i		−	2.89815e-5i	1.00000		1.44908e-5i	$-4.61255e-6\pi$
$648$		0			0
$649$		−	241.799i		−	0.372572i
$650$		0			0
$651$	−423.391	+	45.5559i	−0.650370	+	0.0699784i
$652$		0			0
$653$	809.123			1.23909			0.619543	−	0.784963i	$-0.287318\pi$
$653$	809.123			1.23909			0.619543	+	0.784963i	$0.287318\pi$
$654$		0			0
$655$	−382.656			−0.584208
$656$		0			0
$657$			18.0772i			0.0275147i
$658$		0			0
$659$	−854.384			−1.29649			−0.648243	−	0.761434i	$-0.724496\pi$
$659$	−854.384			−1.29649			−0.648243	+	0.761434i	$0.724496\pi$
$660$		0			0
$661$		−	520.330i		−	0.787186i	−0.919285	−	0.393593i	$-0.871232\pi$
$661$		−	520.330i		−	0.787186i	0.919285	−	0.393593i	$-0.128768\pi$
$662$		0			0
$663$			632.349i			0.953769i
$664$		0			0
$665$	24.4149	−	2.62699i	0.0367141	−	0.00395037i
$666$		0			0
$667$	331.193			0.496541
$668$		0			0
$669$	−172.603			−0.258002
$670$		0			0
$671$		−	601.040i		−	0.895738i
$672$		0			0
$673$	−140.121			−0.208203			−0.104101	−	0.994567i	$-0.533197\pi$
$673$	−140.121			−0.208203			−0.104101	+	0.994567i	$0.533197\pi$
$674$		0			0
$675$		−	95.9360i		−	0.142127i
$676$		0			0
$677$		−	134.613i		−	0.198837i	−0.995046	−	0.0994184i	$-0.968302\pi$
$677$		−	134.613i		−	0.198837i	0.995046	−	0.0994184i	$-0.0316982\pi$
$678$		0			0
$679$	−41.9631	−	390.000i	−0.0618014	−	0.574373i
$680$		0			0
$681$	579.104			0.850373
$682$		0			0
$683$	−99.1586			−0.145181			−0.0725905	−	0.997362i	$-0.523127\pi$
$683$	−99.1586			−0.145181			−0.0725905	+	0.997362i	$0.523127\pi$
$684$		0			0
$685$			82.4292i			0.120335i
$686$		0			0
$687$	86.1372			0.125382
$688$		0			0
$689$			1121.15i			1.62722i
$690$		0			0
$691$		−	935.002i		−	1.35311i	−0.736390	−	0.676557i	$-0.763471\pi$
$691$		−	935.002i		−	1.35311i	0.736390	−	0.676557i	$-0.236529\pi$
$692$		0			0
$693$	−121.571	+	13.0808i	−0.175427	+	0.0188756i
$694$		0			0
$695$	−262.999			−0.378416
$696$		0			0
$697$	−204.428			−0.293298
$698$		0			0
$699$			704.097i			1.00729i
$700$		0			0
$701$	−18.2160			−0.0259857			−0.0129928	−	0.999916i	$-0.504136\pi$
$701$	−18.2160			−0.0259857			−0.0129928	+	0.999916i	$0.504136\pi$
$702$		0			0
$703$			64.8676i			0.0922726i
$704$		0			0
$705$		−	382.946i		−	0.543186i
$706$		0			0
$707$	48.0812	+	446.860i	0.0680074	+	0.632051i
$708$		0			0
$709$	290.529			0.409773			0.204887	−	0.978786i	$-0.434317\pi$
$709$	290.529			0.409773			0.204887	+	0.978786i	$0.434317\pi$
$710$		0			0
$711$	62.6352			0.0880946
$712$		0			0
$713$			361.446i			0.506938i
$714$		0			0
$715$	−325.126			−0.454721
$716$		0			0
$717$		−	524.002i		−	0.730826i
$718$		0			0
$719$			529.734i			0.736765i	0.929674	+	0.368383i	$0.120088\pi$
$719$			529.734i			0.736765i	−0.929674	+	0.368383i	$0.879912\pi$
$720$		0			0
$721$	26.1302	+	242.850i	0.0362416	+	0.336824i
$722$		0			0
$723$	678.655			0.938665
$724$		0			0
$725$	594.181			0.819560
$726$		0			0
$727$			858.179i			1.18044i	0.807243	+	0.590219i	$0.200959\pi$
$727$			858.179i			1.18044i	−0.807243	+	0.590219i	$0.799041\pi$
$728$		0			0
$729$	−27.0000			−0.0370370
$730$		0			0
$731$			342.365i			0.468352i
$732$		0			0
$733$		−	989.920i		−	1.35050i	−0.737587	−	0.675252i	$-0.764035\pi$
$733$		−	989.920i		−	1.35050i	0.737587	−	0.675252i	$-0.235965\pi$
$734$		0			0
$735$	−212.028	+	46.1620i	−0.288474	+	0.0628054i
$736$		0			0
$737$	−148.714			−0.201782
$738$		0			0
$739$	1042.74			1.41102			0.705511	−	0.708699i	$-0.250718\pi$
$739$	1042.74			1.41102			0.705511	+	0.708699i	$0.250718\pi$
$740$		0			0
$741$			51.9005i			0.0700411i
$742$		0			0
$743$	−1225.64			−1.64959			−0.824794	−	0.565434i	$-0.808709\pi$
$743$	−1225.64			−1.64959			−0.824794	+	0.565434i	$0.808709\pi$
$744$		0			0
$745$			470.965i			0.632167i
$746$		0			0
$747$		−	423.198i		−	0.566530i
$748$		0			0
$749$	−826.852	+	88.9675i	−1.10394	+	0.118782i
$750$		0			0
$751$	−742.516			−0.988703			−0.494351	−	0.869262i	$-0.664594\pi$
$751$	−742.516			−0.988703			−0.494351	+	0.869262i	$0.664594\pi$
$752$		0			0
$753$	−58.5141			−0.0777080
$754$		0			0
$755$		−	50.2691i		−	0.0665816i
$756$		0			0
$757$	−449.076			−0.593232			−0.296616	−	0.954997i	$-0.595858\pi$
$757$	−449.076			−0.593232			−0.296616	+	0.954997i	$0.595858\pi$
$758$		0			0
$759$			103.785i			0.136739i
$760$		0			0
$761$			504.360i			0.662759i	0.943498	+	0.331380i	$0.107514\pi$
$761$			504.360i			0.662759i	−0.943498	+	0.331380i	$0.892486\pi$
$762$		0			0
$763$	−860.042	+	92.5387i	−1.12718	+	0.121283i
$764$		0			0
$765$	−128.222			−0.167611
$766$		0			0
$767$	906.967			1.18249
$768$		0			0
$769$			152.033i			0.197702i	0.995102	+	0.0988508i	$0.0315167\pi$
$769$			152.033i			0.197702i	−0.995102	+	0.0988508i	$0.968483\pi$
$770$		0			0
$771$	618.770			0.802555
$772$		0			0
$773$		−	1084.50i		−	1.40297i	−0.712685	−	0.701485i	$-0.752521\pi$
$773$		−	1084.50i		−	1.40297i	0.712685	−	0.701485i	$-0.247479\pi$
$774$		0			0
$775$			648.458i			0.836720i
$776$		0			0
$777$	−61.3236	−	569.933i	−0.0789235	−	0.733504i
$778$		0			0
$779$	−16.7786			−0.0215387
$780$		0			0
$781$	435.680			0.557849
$782$		0			0
$783$		−	167.225i		−	0.213570i
$784$		0			0
$785$	15.5296			0.0197829
$786$		0			0
$787$			1037.74i			1.31860i	0.751882	+	0.659298i	$0.229146\pi$
$787$			1037.74i			1.31860i	−0.751882	+	0.659298i	$0.770854\pi$
$788$		0			0
$789$		−	649.019i		−	0.822585i
$790$		0			0
$791$	280.848	−	30.2186i	0.355054	−	0.0382031i
$792$		0			0
$793$	2254.44			2.84293
$794$		0			0
$795$	−227.338			−0.285959
$796$		0			0
$797$			409.579i			0.513901i	0.966425	+	0.256950i	$0.0827177\pi$
$797$			409.579i			0.513901i	−0.966425	+	0.256950i	$0.917282\pi$
$798$		0			0
$799$	1445.55			1.80920
$800$		0			0
$801$		−	411.075i		−	0.513202i
$802$		0			0
$803$		−	35.0849i		−	0.0436923i
$804$		0			0
$805$	19.7041	+	183.127i	0.0244772	+	0.227487i
$806$		0			0
$807$	−120.049			−0.148760
$808$		0			0
$809$	−555.816			−0.687041			−0.343520	−	0.939145i	$-0.611619\pi$
$809$	−555.816			−0.687041			−0.343520	+	0.939145i	$0.611619\pi$
$810$		0			0
$811$			1234.27i			1.52191i	0.648805	+	0.760955i	$0.275269\pi$
$811$			1234.27i			1.52191i	−0.648805	+	0.760955i	$0.724731\pi$
$812$		0			0
$813$	788.034			0.969292
$814$		0			0
$815$			557.428i			0.683961i
$816$		0			0
$817$			28.0999i			0.0343940i
$818$		0			0
$819$	−49.0649	−	456.002i	−0.0599083	−	0.556779i
$820$		0			0
$821$	679.883			0.828116			0.414058	−	0.910250i	$-0.364111\pi$
$821$	679.883			0.828116			0.414058	+	0.910250i	$0.364111\pi$
$822$		0			0
$823$	928.586			1.12829			0.564147	−	0.825674i	$-0.309205\pi$
$823$	928.586			1.12829			0.564147	+	0.825674i	$0.309205\pi$
$824$		0			0
$825$			186.196i			0.225693i
$826$		0			0
$827$	1321.98			1.59852			0.799262	−	0.600982i	$-0.205224\pi$
$827$	1321.98			1.59852			0.799262	+	0.600982i	$0.205224\pi$
$828$		0			0
$829$		−	1036.65i		−	1.25048i	−0.780433	−	0.625240i	$-0.785001\pi$
$829$		−	1036.65i		−	1.25048i	0.780433	−	0.625240i	$-0.214999\pi$
$830$		0			0
$831$		−	245.740i		−	0.295716i
$832$		0			0
$833$	−174.253	−	800.366i	−0.209187	−	0.960824i
$834$		0			0
$835$	−640.051			−0.766528
$836$		0			0
$837$	182.500			0.218041
$838$		0			0
$839$		−	653.282i		−	0.778644i	−0.921102	−	0.389322i	$-0.872709\pi$
$839$		−	653.282i		−	0.778644i	0.921102	−	0.389322i	$-0.127291\pi$
$840$		0			0
$841$	194.711			0.231523
$842$		0			0
$843$			827.786i			0.981952i
$844$		0			0
$845$		−	787.419i		−	0.931857i
$846$		0			0
$847$	−606.189	+	65.2246i	−0.715689	+	0.0770067i
$848$		0			0
$849$	−65.2901			−0.0769024
$850$		0			0
$851$	−486.549			−0.571738
$852$		0			0
$853$			1227.98i			1.43961i	0.694178	+	0.719803i	$0.255768\pi$
$853$			1227.98i			1.43961i	−0.694178	+	0.719803i	$0.744232\pi$
$854$		0			0
$855$	−10.5239			−0.0123087
$856$		0			0
$857$		−	378.900i		−	0.442123i	−0.975260	−	0.221062i	$-0.929048\pi$
$857$		−	378.900i		−	0.442123i	0.975260	−	0.221062i	$-0.0709522\pi$
$858$		0			0
$859$			322.806i			0.375793i	0.982189	+	0.187897i	$0.0601670\pi$
$859$			322.806i			0.375793i	−0.982189	+	0.187897i	$0.939833\pi$
$860$		0			0
$861$	147.418	−	15.8619i	0.171218	−	0.0184227i
$862$		0			0
$863$	183.339			0.212444			0.106222	−	0.994342i	$-0.466125\pi$
$863$	183.339			0.212444			0.106222	+	0.994342i	$0.466125\pi$
$864$		0			0
$865$	834.724			0.964999
$866$		0			0
$867$			16.5478i			0.0190863i
$868$		0			0
$869$	−121.565			−0.139891
$870$		0			0
$871$		−	557.810i		−	0.640425i
$872$		0			0
$873$			168.107i			0.192563i
$874$		0			0
$875$	83.2174	+	773.411i	0.0951056	+	0.883898i
$876$		0			0
$877$	−88.5528			−0.100972			−0.0504862	−	0.998725i	$-0.516077\pi$
$877$	−88.5528			−0.100972			−0.0504862	+	0.998725i	$0.516077\pi$
$878$		0			0
$879$	−124.291			−0.141400
$880$		0			0
$881$			993.847i			1.12809i	0.825744	+	0.564045i	$0.190755\pi$
$881$			993.847i			1.12809i	−0.825744	+	0.564045i	$0.809245\pi$
$882$		0			0
$883$	−264.028			−0.299012			−0.149506	−	0.988761i	$-0.547768\pi$
$883$	−264.028			−0.299012			−0.149506	+	0.988761i	$0.547768\pi$
$884$		0			0
$885$			183.907i			0.207804i
$886$		0			0
$887$			1093.22i			1.23249i	0.787553	+	0.616247i	$0.211348\pi$
$887$			1093.22i			1.23249i	−0.787553	+	0.616247i	$0.788652\pi$
$888$		0			0
$889$	−1307.52	+	140.687i	−1.47078	+	0.158253i
$890$		0			0
$891$	52.4027			0.0588134
$892$		0			0
$893$	118.644			0.132860
$894$		0			0
$895$		−	486.764i		−	0.543871i
$896$		0			0
$897$	−389.287			−0.433987
$898$		0			0
$899$			1130.32i			1.25731i
$900$		0			0
$901$		−	858.157i		−	0.952449i
$902$		0			0
$903$	−26.5646	−	246.888i	−0.0294182	−	0.273409i
$904$		0			0
$905$	−510.491			−0.564078
$906$		0			0
$907$	−587.162			−0.647367			−0.323684	−	0.946165i	$-0.604921\pi$
$907$	−587.162			−0.647367			−0.323684	+	0.946165i	$0.604921\pi$
$908$		0			0
$909$		−	192.617i		−	0.211900i
$910$		0			0
$911$	32.9303			0.0361475			0.0180737	−	0.999837i	$-0.494247\pi$
$911$	32.9303			0.0361475			0.0180737	+	0.999837i	$0.494247\pi$
$912$		0			0
$913$			821.360i			0.899628i
$914$		0			0
$915$			457.136i			0.499603i
$916$		0			0
$917$	112.077	+	1041.63i	0.122222	+	1.13591i
$918$		0			0
$919$	−90.0432			−0.0979796			−0.0489898	−	0.998799i	$-0.515600\pi$
$919$	−90.0432			−0.0979796			−0.0489898	+	0.998799i	$0.515600\pi$
$920$		0			0
$921$	717.258			0.778782
$922$		0			0
$923$			1634.19i			1.77052i
$924$		0			0
$925$	−872.899			−0.943675
$926$		0			0
$927$		−	104.679i		−	0.112923i
$928$		0			0
$929$		−	1754.02i		−	1.88807i	−0.329842	−	0.944036i	$-0.606995\pi$
$929$		−	1754.02i		−	1.88807i	0.329842	−	0.944036i	$-0.393005\pi$
$930$		0			0
$931$	−14.3019	−	65.6906i	−0.0153619	−	0.0705592i
$932$		0			0
$933$	417.814			0.447818
$934$		0			0
$935$	248.859			0.266159
$936$		0			0
$937$		−	126.768i		−	0.135291i	−0.997709	−	0.0676454i	$-0.978451\pi$
$937$		−	126.768i		−	0.135291i	0.997709	−	0.0676454i	$-0.0215487\pi$
$938$		0			0
$939$	342.041			0.364261
$940$		0			0
$941$			9.81118i			0.0104263i	0.999986	+	0.00521316i	$0.00165941\pi$
$941$			9.81118i			0.0104263i	−0.999986	+	0.00521316i	$0.998341\pi$
$942$		0			0
$943$		−	125.850i		−	0.133457i
$944$		0			0
$945$	92.4642	−	9.94895i	0.0978457	−	0.0105280i
$946$		0			0
$947$	409.685			0.432614			0.216307	−	0.976325i	$-0.430599\pi$
$947$	409.685			0.432614			0.216307	+	0.976325i	$0.430599\pi$
$948$		0			0
$949$	131.600			0.138672
$950$		0			0
$951$			773.653i			0.813515i
$952$		0			0
$953$	−1074.46			−1.12745			−0.563725	−	0.825962i	$-0.690632\pi$
$953$	−1074.46			−1.12745			−0.563725	+	0.825962i	$0.690632\pi$
$954$		0			0
$955$			413.248i			0.432720i
$956$		0			0
$957$			324.557i			0.339140i
$958$		0			0
$959$	224.381	−	24.1430i	0.233974	−	0.0251751i
$960$		0			0
$961$	−272.571			−0.283633
$962$		0			0
$963$	356.410			0.370104
$964$		0			0
$965$		−	3.19936i		−	0.00331540i
$966$		0			0
$967$	1124.26			1.16263			0.581316	−	0.813678i	$-0.302538\pi$
$967$	1124.26			1.16263			0.581316	+	0.813678i	$0.302538\pi$
$968$		0			0
$969$		−	39.7259i		−	0.0409968i
$970$		0			0
$971$		−	214.012i		−	0.220403i	−0.993909	−	0.110202i	$-0.964850\pi$
$971$		−	214.012i		−	0.220403i	0.993909	−	0.110202i	$-0.0351497\pi$
$972$		0			0
$973$	77.0308	+	715.913i	0.0791683	+	0.735779i
$974$		0			0
$975$	−698.405			−0.716313
$976$		0			0
$977$	750.558			0.768227			0.384114	−	0.923286i	$-0.374507\pi$
$977$	750.558			0.768227			0.384114	+	0.923286i	$0.374507\pi$
$978$		0			0
$979$			797.831i			0.814945i
$980$		0			0
$981$	370.717			0.377897
$982$		0			0
$983$			117.208i			0.119235i	0.998221	+	0.0596176i	$0.0189881\pi$
$983$			117.208i			0.119235i	−0.998221	+	0.0596176i	$0.981012\pi$
$984$		0			0
$985$		−	671.817i		−	0.682048i
$986$		0			0
$987$	−1042.42	+	112.162i	−1.05615	+	0.113640i
$988$		0			0
$989$	−210.767			−0.213111
$990$		0			0
$991$	−1466.43			−1.47975			−0.739875	−	0.672744i	$-0.765115\pi$
$991$	−1466.43			−1.47975			−0.739875	+	0.672744i	$0.765115\pi$
$992$		0			0
$993$		−	224.941i		−	0.226527i
$994$		0			0
$995$	−877.900			−0.882311
$996$		0			0
$997$			485.333i			0.486793i	0.969927	+	0.243397i	$0.0782616\pi$
$997$			485.333i			0.486793i	−0.969927	+	0.243397i	$0.921738\pi$
$998$		0			0
$999$			245.667i			0.245913i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.3.f.g.769.7		8
4.3	odd	2		1344.3.f.h.769.3		8
7.6	odd	2	inner	1344.3.f.g.769.2		8
8.3	odd	2		336.3.f.d.97.6		8
8.5	even	2		168.3.f.a.97.2	✓	8
24.5	odd	2		504.3.f.c.433.5		8
24.11	even	2		1008.3.f.j.433.5		8
28.27	even	2		1344.3.f.h.769.6		8
56.5	odd	6		1176.3.z.a.913.3		8
56.13	odd	2		168.3.f.a.97.7	yes	8
56.27	even	2		336.3.f.d.97.3		8
56.37	even	6		1176.3.z.f.913.2		8
56.45	odd	6		1176.3.z.f.313.2		8
56.53	even	6		1176.3.z.a.313.3		8
168.83	odd	2		1008.3.f.j.433.4		8
168.125	even	2		504.3.f.c.433.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.3.f.a.97.2	✓	8	8.5	even	2
168.3.f.a.97.7	yes	8	56.13	odd	2
336.3.f.d.97.3		8	56.27	even	2
336.3.f.d.97.6		8	8.3	odd	2
504.3.f.c.433.4		8	168.125	even	2
504.3.f.c.433.5		8	24.5	odd	2
1008.3.f.j.433.4		8	168.83	odd	2
1008.3.f.j.433.5		8	24.11	even	2
1176.3.z.a.313.3		8	56.53	even	6
1176.3.z.a.913.3		8	56.5	odd	6
1176.3.z.f.313.2		8	56.45	odd	6
1176.3.z.f.913.2		8	56.37	even	6
1344.3.f.g.769.2		8	7.6	odd	2	inner
1344.3.f.g.769.7		8	1.1	even	1	trivial
1344.3.f.h.769.3		8	4.3	odd	2
1344.3.f.h.769.6		8	28.27	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1344.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.73205i q^{3} +2.55678i q^{5} +(6.95983 - 0.748863i) q^{7} -3.00000 q^{9} +5.82252 q^{11} +21.8397i q^{13} -4.42847 q^{15} -16.7166i q^{17} -1.37203i q^{19} +(1.29707 + 12.0548i) q^{21} +10.2911 q^{23} +18.4629 q^{25} -5.19615i q^{27} +32.1825 q^{29} +35.1222i q^{31} +10.0849i q^{33} +(1.91468 + 17.7947i) q^{35} -47.2786 q^{37} -37.8275 q^{39} -12.2290i q^{41} -20.4805 q^{43} -7.67033i q^{45} +86.4736i q^{47} +(47.8784 - 10.4239i) q^{49} +28.9541 q^{51} +51.3355 q^{53} +14.8869i q^{55} +2.37643 q^{57} -41.5283i q^{59} -103.227i q^{61} +(-20.8795 + 2.24659i) q^{63} -55.8393 q^{65} -25.5411 q^{67} +17.8247i q^{69} +74.8267 q^{71} -6.02572i q^{73} +31.9787i q^{75} +(40.5238 - 4.36027i) q^{77} -20.8784 q^{79} +9.00000 q^{81} +141.066i q^{83} +42.7407 q^{85} +55.7417i q^{87} +137.025i q^{89} +(16.3550 + 152.001i) q^{91} -60.8335 q^{93} +3.50798 q^{95} -56.0358i q^{97} -17.4676 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{7} - 24 q^{9} + 16 q^{11} + 24 q^{15} - 12 q^{21} + 96 q^{23} + 64 q^{29} + 120 q^{35} + 40 q^{37} - 48 q^{39} - 136 q^{43} + 112 q^{49} + 24 q^{51} - 112 q^{53} - 72 q^{57} + 12 q^{63} - 208 q^{65}+ \cdots - 48 q^{99}+O(q^{100}) \) 8 * q - 4 * q^7 - 24 * q^9 + 16 * q^11 + 24 * q^15 - 12 * q^21 + 96 * q^23 + 64 * q^29 + 120 * q^35 + 40 * q^37 - 48 * q^39 - 136 * q^43 + 112 * q^49 + 24 * q^51 - 112 * q^53 - 72 * q^57 + 12 * q^63 - 208 * q^65 + 8 * q^67 + 16 * q^71 - 288 * q^77 + 104 * q^79 + 72 * q^81 + 424 * q^85 + 192 * q^91 - 144 * q^93 - 240 * q^95 - 48 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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