LMFDB - Embedded newform 69.2.c.a.68.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [69,2,Mod(68,69)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(69, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("69.68");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 69 = 3 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	69.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:	$0.550967773947$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.8869743.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{3} + 8 $ x^6 - 3*x^3 + 8
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			68.6
Root			$-0.261988 + 1.38973i$ of defining polynomial
Character	$\chi$	$=$	69.68
Dual form			69.2.c.a.68.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.77947i q^{2} +(1.60074 - 0.661546i) q^{3} -5.72545 q^{4} +(1.83875 + 4.44920i) q^{6} -10.3548i q^{8} +(2.12471 - 2.11792i) q^{9} +O(q^{10})$	$q+2.77947i q^{2} +(1.60074 - 0.661546i) q^{3} -5.72545 q^{4} +(1.83875 + 4.44920i) q^{6} -10.3548i q^{8} +(2.12471 - 2.11792i) q^{9} +(-9.16493 + 3.78765i) q^{12} -2.15352 q^{13} +17.3299 q^{16} +(5.88670 + 5.90557i) q^{18} +4.79583i q^{23} +(-6.85016 - 16.5753i) q^{24} -5.00000 q^{25} -5.98564i q^{26} +(2.00000 - 4.79583i) q^{27} -1.58966i q^{29} -5.29738 q^{31} +27.4583i q^{32} +(-12.1649 + 12.1261i) q^{36} +(-3.44722 + 1.42465i) q^{39} -9.52822i q^{41} -13.3299 q^{46} +7.14860i q^{47} +(27.7405 - 11.4645i) q^{48} +7.00000 q^{49} -13.8973i q^{50} +12.3299 q^{52} +(13.3299 + 5.55894i) q^{54} +4.41841 q^{58} +9.59166i q^{59} -14.7239i q^{62} -41.6597 q^{64} +(3.17267 + 7.67686i) q^{69} +15.0872i q^{71} +(-21.9306 - 22.0009i) q^{72} +17.0553 q^{73} +(-8.00368 + 3.30773i) q^{75} +(-3.95978 - 9.58143i) q^{78} +(0.0288070 - 8.99995i) q^{81} +26.4834 q^{82} +(-1.05163 - 2.54463i) q^{87} -27.4583i q^{92} +(-8.47970 + 3.50446i) q^{93} -19.8693 q^{94} +(18.1649 + 43.9535i) q^{96} +19.4563i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 12 q^{4} + 3 q^{6}+O(q^{10}) $ 6 * q - 12 * q^4 + 3 * q^6	$ 6 q - 12 q^{4} + 3 q^{6} - 15 q^{12} + 24 q^{16} + 21 q^{18} - 6 q^{24} - 30 q^{25} + 12 q^{27} - 33 q^{36} - 24 q^{39} + 69 q^{48} + 42 q^{49} - 6 q^{52} + 30 q^{58} - 90 q^{64} - 42 q^{72} - 51 q^{78} + 66 q^{82} + 48 q^{87} - 6 q^{93} - 78 q^{94} + 69 q^{96}+O(q^{100}) $ 6 * q - 12 * q^4 + 3 * q^6 - 15 * q^12 + 24 * q^16 + 21 * q^18 - 6 * q^24 - 30 * q^25 + 12 * q^27 - 33 * q^36 - 24 * q^39 + 69 * q^48 + 42 * q^49 - 6 * q^52 + 30 * q^58 - 90 * q^64 - 42 * q^72 - 51 * q^78 + 66 * q^82 + 48 * q^87 - 6 * q^93 - 78 * q^94 + 69 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/69\mathbb{Z}\right)^\times$.

$n$	$28$	$47$
$\chi(n)$	$-1$	$-1$

Coefficient data

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

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

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

$2$			2.77947i			1.96538i	0.185254	+	0.982691i	$0.440689\pi$
$2$			2.77947i			1.96538i	−0.185254	+	0.982691i	$0.559311\pi$
$3$	1.60074	−	0.661546i	0.924185	−	0.381944i
$3$	1.60074	−	0.661546i	0.924185	−	0.381944i
$4$	−5.72545			−2.86272
$5$		0			0			−	1.00000i	$-0.5\pi$
$5$		0			0				1.00000i	$0.5\pi$
$6$	1.83875	+	4.44920i	0.750666	+	1.81638i
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		−	10.3548i		−	3.66096i
$9$	2.12471	−	2.11792i	0.708238	−	0.705974i
$10$		0			0
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$	−9.16493	+	3.78765i	−2.64569	+	1.09340i
$13$	−2.15352			−0.597279			−0.298639	−	0.954366i	$-0.596533\pi$
$13$	−2.15352			−0.597279			−0.298639	+	0.954366i	$0.596533\pi$
$14$		0			0
$15$		0			0
$16$	17.3299			4.33247
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$	5.88670	+	5.90557i	1.38751	+	1.39196i
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			4.79583i			1.00000i
$23$			4.79583i			1.00000i
$24$	−6.85016	−	16.5753i	−1.39828	−	3.38341i
$25$	−5.00000			−1.00000
$26$		−	5.98564i		−	1.17388i
$27$	2.00000	−	4.79583i	0.384900	−	0.922958i
$28$		0			0
$29$		−	1.58966i		−	0.295192i	−0.989048	−	0.147596i	$-0.952846\pi$
$29$		−	1.58966i		−	0.295192i	0.989048	−	0.147596i	$-0.0471536\pi$
$30$		0			0
$31$	−5.29738			−0.951437			−0.475719	−	0.879598i	$-0.657812\pi$
$31$	−5.29738			−0.951437			−0.475719	+	0.879598i	$0.657812\pi$
$32$			27.4583i			4.85399i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−12.1649	+	12.1261i	−2.02749	+	2.02101i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$	−3.44722	+	1.42465i	−0.551996	+	0.228127i
$40$		0			0
$41$		−	9.52822i		−	1.48806i	−0.668148	−	0.744029i	$-0.732913\pi$
$41$		−	9.52822i		−	1.48806i	0.668148	−	0.744029i	$-0.267087\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$		0			0
$46$	−13.3299			−1.96538
$47$			7.14860i			1.04273i	0.853334	+	0.521365i	$0.174577\pi$
$47$			7.14860i			1.04273i	−0.853334	+	0.521365i	$0.825423\pi$
$48$	27.7405	−	11.4645i	4.00400	−	1.65476i
$49$	7.00000			1.00000
$50$		−	13.8973i		−	1.96538i
$51$		0			0
$52$	12.3299			1.70984
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$	13.3299	+	5.55894i	1.81396	+	0.756476i
$55$		0			0
$56$		0			0
$57$		0			0
$58$	4.41841			0.580166
$59$			9.59166i			1.24873i	0.781133	+	0.624364i	$0.214642\pi$
$59$			9.59166i			1.24873i	−0.781133	+	0.624364i	$0.785358\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		−	14.7239i		−	1.86994i
$63$		0			0
$64$	−41.6597			−5.20747
$65$		0			0
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$		0			0
$69$	3.17267	+	7.67686i	0.381944	+	0.924185i
$70$		0			0
$71$			15.0872i			1.79052i	0.445548	+	0.895258i	$0.353009\pi$
$71$			15.0872i			1.79052i	−0.445548	+	0.895258i	$0.646991\pi$
$72$	−21.9306	−	22.0009i	−2.58455	−	2.59283i
$73$	17.0553			1.99617			0.998087	−	0.0618285i	$-0.0196932\pi$
$73$	17.0553			1.99617			0.998087	+	0.0618285i	$0.0196932\pi$
$74$		0			0
$75$	−8.00368	+	3.30773i	−0.924185	+	0.381944i
$76$		0			0
$77$		0			0
$78$	−3.95978	−	9.58143i	−0.448357	−	1.08488i
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$		0			0
$81$	0.0288070	−	8.99995i	0.00320078	−	0.999995i
$82$	26.4834			2.92460
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−1.05163	−	2.54463i	−0.112747	−	0.272812i
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$		0			0
$92$		−	27.4583i		−	2.86272i
$93$	−8.47970	+	3.50446i	−0.879304	+	0.363396i
$94$	−19.8693			−2.04936
$95$		0			0
$96$	18.1649	+	43.9535i	1.85395	+	4.48598i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$			19.4563i			1.96538i
$99$		0			0
$100$	28.6272			2.86272
$101$		−	19.1833i		−	1.90881i	−0.298511	−	0.954406i	$-0.596490\pi$
$101$		−	19.1833i		−	1.90881i	0.298511	−	0.954406i	$-0.403510\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$			22.2992i			2.18662i
$105$		0			0
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$	−11.4509	+	27.4583i	−1.10186	+	2.64217i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$		0			0
$116$			9.10151i			0.845054i
$117$	−4.57561	+	4.56099i	−0.423015	+	0.421663i
$118$	−26.6597			−2.45423
$119$		0			0
$120$		0			0
$121$	−11.0000			−1.00000
$122$		0			0
$123$	−6.30336	−	15.2522i	−0.568355	−	1.37524i
$124$	30.3299			2.72370
$125$		0			0
$126$		0			0
$127$	13.9115			1.23444			0.617221	−	0.786790i	$-0.288258\pi$
$127$	13.9115			1.23444			0.617221	+	0.786790i	$0.288258\pi$
$128$		−	60.8754i		−	5.38067i
$129$		0			0
$130$		0			0
$131$		−	18.2665i		−	1.59595i	−0.602691	−	0.797975i	$-0.705905\pi$
$131$		−	18.2665i		−	1.59595i	0.602691	−	0.797975i	$-0.294095\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0			−	1.00000i	$-0.5\pi$
$137$		0			0				1.00000i	$0.5\pi$
$138$	−21.3376	+	8.81833i	−1.81638	+	0.750666i
$139$	0.990339			0.0839994			0.0419997	−	0.999118i	$-0.486627\pi$
$139$	0.990339			0.0839994			0.0419997	+	0.999118i	$0.486627\pi$
$140$		0			0
$141$	4.72913	+	11.4430i	0.398265	+	0.963676i
$142$	−41.9343			−3.51905
$143$		0			0
$144$	36.8210	−	36.7033i	3.06842	−	3.05861i
$145$		0			0
$146$			47.4047i			3.92324i
$147$	11.2052	−	4.63083i	0.924185	−	0.381944i
$148$		0			0
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$	−9.19374	−	22.2460i	−0.750666	−	1.81638i
$151$	−24.5062			−1.99429			−0.997144	−	0.0755288i	$-0.975936\pi$
$151$	−24.5062			−1.99429			−0.997144	+	0.0755288i	$0.975936\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	19.7369	−	8.15678i	1.58021	−	0.653065i
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	25.0151	+	0.0800681i	1.96537	+	0.00629074i
$163$	20.1992			1.58212			0.791061	−	0.611738i	$-0.209529\pi$
$163$	20.1992			1.58212			0.791061	+	0.611738i	$0.209529\pi$
$164$			54.5533i			4.25990i
$165$		0			0
$166$		0			0
$167$			9.59166i			0.742225i	0.928588	+	0.371113i	$0.121024\pi$
$167$			9.59166i			0.742225i	−0.928588	+	0.371113i	$0.878976\pi$
$168$		0			0
$169$	−8.36235			−0.643258
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	19.1833i		−	1.45848i	−0.684257	−	0.729241i	$-0.739873\pi$
$173$		−	19.1833i		−	1.45848i	0.684257	−	0.729241i	$-0.260127\pi$
$174$	7.07271	−	2.92298i	0.536181	−	0.221591i
$175$		0			0
$176$		0			0
$177$	6.34533	+	15.3537i	0.476944	+	1.15406i
$178$		0			0
$179$		−	26.2050i		−	1.95866i	−0.202279	−	0.979328i	$-0.564835\pi$
$179$		−	26.2050i		−	1.95866i	0.202279	−	0.979328i	$-0.435165\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$	49.6597			3.66096
$185$		0			0
$186$	−9.74054	−	23.5691i	−0.714211	−	1.72817i
$187$		0			0
$188$		−	40.9289i		−	2.98505i
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$	−66.6862	+	27.5598i	−4.81266	+	1.98896i
$193$	−8.44124			−0.607613			−0.303807	−	0.952734i	$-0.598258\pi$
$193$	−8.44124			−0.607613			−0.303807	+	0.952734i	$0.598258\pi$
$194$		0			0
$195$		0			0
$196$	−40.0781			−2.86272
$197$			23.8254i			1.69749i	0.528802	+	0.848745i	$0.322641\pi$
$197$			23.8254i			1.69749i	−0.528802	+	0.848745i	$0.677359\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$			51.7738i			3.66096i
$201$		0			0
$202$	53.3195			3.75154
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	10.1572	+	10.1898i	0.705974	+	0.708238i
$208$	−37.3202			−2.58769
$209$		0			0
$210$		0			0
$211$	−4.00000			−0.275371			−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000			−0.275371			−0.137686	+	0.990476i	$0.543966\pi$
$212$		0			0
$213$	9.98085	+	24.1506i	0.683877	+	1.65477i
$214$		0			0
$215$		0			0
$216$	−49.6597	−	20.7095i	−3.37892	−	1.40911i
$217$		0			0
$218$		0			0
$219$	27.3011	−	11.2829i	1.84483	−	0.762427i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	8.00000			0.535720			0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000			0.535720			0.267860	+	0.963458i	$0.413684\pi$
$224$		0			0
$225$	−10.6236	+	10.5896i	−0.708238	+	0.705974i
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$		0			0
$232$	−16.4606			−1.08069
$233$			6.34890i			0.415930i	0.978136	+	0.207965i	$0.0666840\pi$
$233$			6.34890i			0.415930i	−0.978136	+	0.207965i	$0.933316\pi$
$234$	−12.6771	−	12.7178i	−0.828730	−	0.831386i
$235$		0			0
$236$		−	54.9166i		−	3.57476i
$237$		0			0
$238$		0			0
$239$		−	0.789960i		−	0.0510982i	−0.999674	−	0.0255491i	$-0.991867\pi$
$239$		−	0.789960i		−	0.0510982i	0.999674	−	0.0255491i	$-0.00813342\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		−	30.5742i		−	1.96538i
$243$	−5.90778	−	14.4256i	−0.378984	−	0.925403i
$244$		0			0
$245$		0			0
$246$	42.3929	−	17.5200i	2.70287	−	1.11703i
$247$		0			0
$248$			54.8531i			3.48318i
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$			38.6665i			2.42615i
$255$		0			0
$256$	85.8817			5.36761
$257$			31.7640i			1.98138i	0.136130	+	0.990691i	$0.456534\pi$
$257$			31.7640i			1.98138i	−0.136130	+	0.990691i	$0.543466\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−3.36678	−	3.37757i	−0.208398	−	0.209066i
$262$	50.7711			3.13665
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	17.4668i		−	1.06497i	−0.846440	−	0.532484i	$-0.821259\pi$
$269$		−	17.4668i		−	1.06497i	0.846440	−	0.532484i	$-0.178741\pi$
$270$		0			0
$271$	−16.0000			−0.971931			−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000			−0.971931			−0.485965	+	0.873978i	$0.661532\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$	−18.1649	−	43.9535i	−1.09340	−	2.64569i
$277$	−27.6501			−1.66133			−0.830666	−	0.556771i	$-0.812040\pi$
$277$	−27.6501			−1.66133			−0.830666	+	0.556771i	$0.812040\pi$
$278$			2.75262i			0.165091i
$279$	−11.2554	+	11.2194i	−0.673843	+	0.671690i
$280$		0			0
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$	−31.8055	+	13.1445i	−1.89399	+	0.782742i
$283$		0			0		1.00000			$0$
$283$		0			0		−1.00000			$\pi$
$284$		−	86.3807i		−	5.12575i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	58.1545	+	58.3410i	3.42679	+	3.43777i
$289$	−17.0000			−1.00000
$290$		0			0
$291$		0			0
$292$	−97.6493			−5.71449
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$	12.8712	+	31.1444i	0.750666	+	1.81638i
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	10.3279i		−	0.597279i
$300$	45.8247	−	18.9383i	2.64569	−	1.09340i
$301$		0			0
$302$		−	68.1143i		−	3.91954i
$303$	−12.6907	−	30.7074i	−0.729059	−	1.76410i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	20.0000			1.14146			0.570730	−	0.821138i	$-0.306660\pi$
$307$	20.0000			1.14146			0.570730	+	0.821138i	$0.306660\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			23.0257i			1.30567i	0.757501	+	0.652834i	$0.226420\pi$
$311$			23.0257i			1.30567i	−0.757501	+	0.652834i	$0.773580\pi$
$312$	14.7520	+	35.6951i	0.835165	+	2.02084i
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	19.1833i		−	1.07744i	−0.842484	−	0.538721i	$-0.818908\pi$
$317$		−	19.1833i		−	1.07744i	0.842484	−	0.538721i	$-0.181092\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−0.164933	+	51.5288i	−0.00916294	+	2.86271i
$325$	10.7676			0.597279
$326$			56.1430i			3.10947i
$327$		0			0
$328$	−98.6625			−5.44772
$329$		0			0
$330$		0			0
$331$	−18.2185			−1.00138			−0.500690	−	0.865627i	$-0.666920\pi$
$331$	−18.2185			−1.00138			−0.500690	+	0.865627i	$0.666920\pi$
$332$		0			0
$333$		0			0
$334$	−26.6597			−1.45876
$335$		0			0
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$		−	23.2429i		−	1.26425i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	53.3195			2.86647
$347$			9.59166i			0.514907i	0.966291	+	0.257454i	$0.0828835\pi$
$347$			9.59166i			0.514907i	−0.966291	+	0.257454i	$0.917117\pi$
$348$	6.02107	+	14.5691i	0.322763	+	0.780987i
$349$	36.2641			1.94118			0.970588	−	0.240748i	$-0.0773927\pi$
$349$	36.2641			1.94118			0.970588	+	0.240748i	$0.0773927\pi$
$350$		0			0
$351$	−4.30704	+	10.3279i	−0.229893	+	0.551263i
$352$		0			0
$353$			15.8869i			0.845572i	0.906230	+	0.422786i	$0.138948\pi$
$353$			15.8869i			0.845572i	−0.906230	+	0.422786i	$0.861052\pi$
$354$	−42.6752	+	17.6367i	−2.26816	+	0.937377i
$355$		0			0
$356$		0			0
$357$		0			0
$358$	72.8361			3.84951
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	19.0000			1.00000
$362$		0			0
$363$	−17.6081	+	7.27701i	−0.924185	+	0.381944i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$			83.1111i			4.33247i
$369$	−20.1800	−	20.2447i	−1.05053	−	1.05390i
$370$		0			0
$371$		0			0
$372$	48.5501	−	20.0646i	2.51721	−	1.04030i
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$	74.0221			3.81740
$377$			3.42336i			0.176312i
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$	22.2686	−	9.20307i	1.14085	−	0.471488i
$382$		0			0
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$	−40.2719	−	97.4454i	−2.05512	−	4.97274i
$385$		0			0
$386$		−	23.4622i		−	1.19419i
$387$		0			0
$388$		0			0
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$		0			0
$392$		−	72.4834i		−	3.66096i
$393$	−12.0841	−	29.2398i	−0.609563	−	1.47495i
$394$	−66.2220			−3.33622
$395$		0			0
$396$		0			0
$397$	23.3430			1.17155			0.585777	−	0.810473i	$-0.300790\pi$
$397$	23.3430			1.17155			0.585777	+	0.810473i	$0.300790\pi$
$398$		0			0
$399$		0			0
$400$	−86.6493			−4.33247
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$	11.4080			0.568273
$404$			109.833i			5.46440i
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	4.13420			0.204423			0.102211	−	0.994763i	$-0.467408\pi$
$409$	4.13420			0.204423			0.102211	+	0.994763i	$0.467408\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	−28.3221	+	28.2316i	−1.39196	+	1.38751i
$415$		0			0
$416$		−	59.1320i		−	2.89918i
$417$	1.58527	−	0.655155i	0.0776311	−	0.0320831i
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		−	11.1179i		−	0.541210i
$423$	15.1402	+	15.1887i	0.736141	+	0.738501i
$424$		0			0
$425$		0			0
$426$	−67.1257	+	27.7415i	−3.25225	+	1.34408i
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$	34.6597	−	83.1111i	1.66757	−	3.99869i
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	31.3604	+	75.8825i	1.49846	+	3.62580i
$439$	33.1203			1.58075			0.790373	−	0.612626i	$-0.209887\pi$
$439$	33.1203			1.58075			0.790373	+	0.612626i	$0.209887\pi$
$440$		0			0
$441$	14.8730	−	14.8255i	0.708238	−	0.705974i
$442$		0			0
$443$		−	34.1436i		−	1.62221i	−0.584900	−	0.811105i	$-0.698866\pi$
$443$		−	34.1436i		−	1.62221i	0.584900	−	0.811105i	$-0.301134\pi$
$444$		0			0
$445$		0			0
$446$			22.2358i			1.05289i
$447$		0			0
$448$		0			0
$449$			38.3667i			1.81063i	0.424736	+	0.905317i	$0.360367\pi$
$449$			38.3667i			1.81063i	−0.424736	+	0.905317i	$0.639633\pi$
$450$	−29.4335	−	29.5279i	−1.38751	−	1.39196i
$451$		0			0
$452$		0			0
$453$	−39.2280	+	16.2120i	−1.84309	+	0.761706i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	42.8818i		−	1.99721i	−0.0528331	−	0.998603i	$-0.516825\pi$
$461$		−	42.8818i		−	1.99721i	0.0528331	−	0.998603i	$-0.483175\pi$
$462$		0			0
$463$	32.0000			1.48717			0.743583	−	0.668644i	$-0.233125\pi$
$463$	32.0000			1.48717			0.743583	+	0.668644i	$0.233125\pi$
$464$		−	27.5486i		−	1.27891i
$465$		0			0
$466$	−17.6466			−0.817461
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$	26.1974	−	26.1137i	1.21098	−	1.20711i
$469$		0			0
$470$		0			0
$471$		0			0
$472$	99.3195			4.57155
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	2.19567			0.100428
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	62.9799			2.86272
$485$		0			0
$486$	40.0955	−	16.4205i	1.81877	−	0.744848i
$487$	−43.7150			−1.98092			−0.990459	−	0.137808i	$-0.955994\pi$
$487$	−43.7150			−1.98092			−0.990459	+	0.137808i	$0.955994\pi$
$488$		0			0
$489$	32.3335	−	13.3627i	1.46217	−	0.604282i
$490$		0			0
$491$			40.5022i			1.82784i	0.405894	+	0.913920i	$0.366960\pi$
$491$			40.5022i			1.82784i	−0.405894	+	0.913920i	$0.633040\pi$
$492$	36.0896	+	87.3255i	1.62704	+	3.93694i
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−91.8028			−4.12207
$497$		0			0
$498$		0			0
$499$	−11.5851			−0.518620			−0.259310	−	0.965794i	$-0.583495\pi$
$499$	−11.5851			−0.518620			−0.259310	+	0.965794i	$0.583495\pi$
$500$		0			0
$501$	6.34533	+	15.3537i	0.283488	+	0.685954i
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−13.3859	+	5.53209i	−0.594490	+	0.245689i
$508$	−79.6493			−3.53387
$509$		−	27.0047i		−	1.19696i	−0.801136	−	0.598482i	$-0.795771\pi$
$509$		−	27.0047i		−	1.19696i	0.801136	−	0.598482i	$-0.204229\pi$
$510$		0			0
$511$		0			0
$512$			116.955i			5.16872i
$513$		0			0
$514$	−88.2870			−3.89417
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−12.6907	−	30.7074i	−0.557058	−	1.34791i
$520$		0			0
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$	9.38785	−	9.35785i	0.410895	−	0.409582i
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$			104.584i			4.56876i
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−23.0000			−1.00000
$530$		0			0
$531$	20.3144	+	20.3795i	0.881570	+	0.884396i
$532$		0			0
$533$			20.5192i			0.888785i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−17.3358	−	41.9473i	−0.748097	−	1.81016i
$538$	48.5484			2.09307
$539$		0			0
$540$		0			0
$541$	−40.5712			−1.74429			−0.872146	−	0.489246i	$-0.837272\pi$
$541$	−40.5712			−1.74429			−0.872146	+	0.489246i	$0.837272\pi$
$542$		−	44.4715i		−	1.91021i
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−30.7939			−1.31665			−0.658327	−	0.752732i	$-0.728735\pi$
$547$	−30.7939			−1.31665			−0.658327	+	0.752732i	$0.728735\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$	79.4921	−	32.8522i	3.38341	−	1.39828i
$553$		0			0
$554$		−	76.8525i		−	3.26515i
$555$		0			0
$556$	−5.67013			−0.240467
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$	−31.1841	−	31.2841i	−1.32013	−	1.32436i
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$	−27.0764	−	65.5164i	−1.14012	−	2.75874i
$565$		0			0
$566$		0			0
$567$		0			0
$568$	156.224			6.55501
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	23.9792i		−	1.00000i
$576$	−88.5150	+	88.2321i	−3.68812	+	3.67634i
$577$	−46.8589			−1.95076			−0.975381	−	0.220527i	$-0.929222\pi$
$577$	−46.8589			−1.95076			−0.975381	+	0.220527i	$0.929222\pi$
$578$		−	47.2510i		−	1.96538i
$579$	−13.5122	+	5.58427i	−0.561548	+	0.232074i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		−	176.604i		−	7.30792i
$585$		0			0
$586$		0			0
$587$			48.4408i			1.99937i	0.0251938	+	0.999683i	$0.491980\pi$
$587$			48.4408i			1.99937i	−0.0251938	+	0.999683i	$0.508020\pi$
$588$	−64.1545	+	26.5136i	−2.64569	+	1.09340i
$589$		0			0
$590$		0			0
$591$	15.7616	+	38.1382i	0.648346	+	1.56880i
$592$		0			0
$593$			38.3667i			1.57553i	0.615976	+	0.787765i	$0.288762\pi$
$593$			38.3667i			1.57553i	−0.615976	+	0.787765i	$0.711238\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$	28.7061			1.17388
$599$			9.59166i			0.391905i	0.980613	+	0.195952i	$0.0627798\pi$
$599$			9.59166i			0.391905i	−0.980613	+	0.195952i	$0.937220\pi$
$600$	34.2508	+	82.8763i	1.39828	+	3.38341i
$601$	42.5519			1.73573			0.867863	−	0.496803i	$-0.165493\pi$
$601$	42.5519			1.73573			0.867863	+	0.496803i	$0.165493\pi$
$602$		0			0
$603$		0			0
$604$	140.309			5.70909
$605$		0			0
$606$	85.3504	−	35.2733i	3.46712	−	1.43288i
$607$	−40.0000			−1.62355			−0.811775	−	0.583970i	$-0.801498\pi$
$607$	−40.0000			−1.62355			−0.811775	+	0.583970i	$0.801498\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	15.3946i		−	0.622801i
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$			55.5894i			2.24340i
$615$		0			0
$616$		0			0
$617$		0			0			−	1.00000i	$-0.5\pi$
$617$		0			0				1.00000i	$0.5\pi$
$618$		0			0
$619$		0			0		1.00000			$0$
$619$		0			0		−1.00000			$\pi$
$620$		0			0
$621$	23.0000	+	9.59166i	0.922958	+	0.384900i
$622$	−63.9993			−2.56614
$623$		0			0
$624$	−59.7398	+	24.6890i	−2.39151	+	0.988353i
$625$	25.0000			1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$	−6.40294	+	2.64619i	−0.254494	+	0.105176i
$634$	53.3195			2.11759
$635$		0			0
$636$		0			0
$637$	−15.0746			−0.597279
$638$		0			0
$639$	31.9534	+	32.0559i	1.26406	+	1.26811i
$640$		0			0
$641$		0			0			−	1.00000i	$-0.5\pi$
$641$		0			0				1.00000i	$0.5\pi$
$642$		0			0
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	8.72852i		−	0.343153i	−0.985171	−	0.171577i	$-0.945114\pi$
$647$		−	8.72852i		−	0.343153i	0.985171	−	0.171577i	$-0.0548861\pi$
$648$	−93.1925	−	0.298290i	−3.66095	−	0.0117179i
$649$		0			0
$650$			29.9282i			1.17388i
$651$		0			0
$652$	−115.649			−4.52918
$653$			14.2875i			0.559111i	0.960129	+	0.279556i	$0.0901872\pi$
$653$			14.2875i			0.559111i	−0.960129	+	0.279556i	$0.909813\pi$
$654$		0			0
$655$		0			0
$656$		−	165.123i		−	6.44696i
$657$	36.2376	−	36.1218i	1.41377	−	1.40925i
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		−	50.6377i		−	1.96809i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	7.62374			0.295192
$668$		−	54.9166i		−	2.12479i
$669$	12.8059	−	5.29237i	0.495104	−	0.204615i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	29.9764			1.15551			0.577753	−	0.816211i	$-0.303930\pi$
$673$	29.9764			1.15551			0.577753	+	0.816211i	$0.303930\pi$
$674$		0			0
$675$	−10.0000	+	23.9792i	−0.384900	+	0.922958i
$676$	47.8782			1.84147
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	2.38936i		−	0.0914263i	−0.998955	−	0.0457131i	$-0.985444\pi$
$683$		−	2.38936i		−	0.0914263i	0.998955	−	0.0457131i	$-0.0145560\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	44.0000			1.67384			0.836919	−	0.547326i	$-0.184354\pi$
$691$	44.0000			1.67384			0.836919	+	0.547326i	$0.184354\pi$
$692$			109.833i			4.17523i
$693$		0			0
$694$	−26.6597			−1.01199
$695$		0			0
$696$	−26.3490	+	10.8894i	−0.998757	+	0.412763i
$697$		0			0
$698$			100.795i			3.81515i
$699$	4.20009	+	10.1629i	0.158862	+	0.384397i
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$	−28.7061	−	11.9713i	−1.08344	−	0.451827i
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−44.1570			−1.66187
$707$		0			0
$708$	−36.3299	−	87.9069i	−1.36536	−	3.30375i
$709$		0			0		1.00000			$0$
$709$		0			0		−1.00000			$\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	25.4053i		−	0.951437i
$714$		0			0
$715$		0			0
$716$			150.036i			5.60709i
$717$	−0.522595	−	1.26452i	−0.0195167	−	0.0472243i
$718$		0			0
$719$		−	47.9583i		−	1.78854i	−0.447524	−	0.894272i	$-0.647694\pi$
$719$		−	47.9583i		−	1.78854i	0.447524	−	0.894272i	$-0.352306\pi$
$720$		0			0
$721$		0			0
$722$			52.8099i			1.96538i
$723$		0			0
$724$		0			0
$725$			7.94830i			0.295192i
$726$	−20.2262	−	48.9412i	−0.750666	−	1.81638i
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	−19.0000	−	19.1833i	−0.703704	−	0.710494i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		1.00000			$0$
$733$		0			0		−1.00000			$\pi$
$734$		0			0
$735$		0			0
$736$	−131.685			−4.85399
$737$		0			0
$738$	56.2696	−	56.0898i	2.07131	−	2.06469i
$739$	−37.4273			−1.37679			−0.688393	−	0.725338i	$-0.741684\pi$
$739$	−37.4273			−1.37679			−0.688393	+	0.725338i	$0.741684\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$	36.2879	+	87.8054i	1.33038	+	3.21910i
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$			123.884i			4.51759i
$753$		0			0
$754$	−9.51513			−0.346521
$755$		0			0
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	50.8204i		−	1.84224i	−0.389281	−	0.921119i	$-0.627276\pi$
$761$		−	50.8204i		−	1.84224i	0.389281	−	0.921119i	$-0.372724\pi$
$762$	25.5797	+	61.8948i	0.926653	+	2.24221i
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	20.6558i		−	0.745839i
$768$	137.474	−	56.8148i	4.96067	−	2.05013i
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$	21.0133	+	50.8457i	0.756777	+	1.83116i
$772$	48.3299			1.73943
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		0			0
$775$	26.4869			0.951437
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−7.62374	−	3.17932i	−0.272450	−	0.113620i
$784$	121.309			4.33247
$785$		0			0
$786$	81.2711	−	33.5874i	2.89885	−	1.19802i
$787$		0			0		1.00000			$0$
$787$		0			0		−1.00000			$\pi$
$788$		−	136.411i		−	4.85945i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$			64.8812i			2.30255i
$795$		0			0
$796$		0			0
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$		−	137.291i		−	4.85399i
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$			31.7082i			1.11687i
$807$	−11.5551	−	27.9597i	−0.406758	−	0.984228i
$808$	−198.639			−6.98809
$809$			38.3667i			1.34890i	0.738321	+	0.674450i	$0.235619\pi$
$809$			38.3667i			1.34890i	−0.738321	+	0.674450i	$0.764381\pi$
$810$		0			0
$811$	−50.0028			−1.75583			−0.877917	−	0.478812i	$-0.841067\pi$
$811$	−50.0028			−1.75583			−0.877917	+	0.478812i	$0.841067\pi$
$812$		0			0
$813$	−25.6118	+	10.5847i	−0.898244	+	0.371223i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$			11.4909i			0.401769i
$819$		0			0
$820$		0			0
$821$		−	19.1833i		−	0.669503i	−0.942306	−	0.334751i	$-0.891348\pi$
$821$		−	19.1833i		−	0.669503i	0.942306	−	0.334751i	$-0.108652\pi$
$822$		0			0
$823$	7.27806			0.253697			0.126849	−	0.991922i	$-0.459514\pi$
$823$	7.27806			0.253697			0.126849	+	0.991922i	$0.459514\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$	−58.1545	−	58.3410i	−2.02101	−	2.02749i
$829$	2.00000			0.0694629			0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000			0.0694629			0.0347314	+	0.999397i	$0.488942\pi$
$830$		0			0
$831$	−44.2605	+	18.2918i	−1.53538	+	0.634536i
$832$	89.7150			3.11031
$833$		0			0
$834$	1.82098	+	4.40621i	0.0630555	+	0.152575i
$835$		0			0
$836$		0			0
$837$	−10.5948	+	25.4053i	−0.366208	+	0.878137i
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	26.4730			0.912861
$842$		0			0
$843$		0			0
$844$	22.9018			0.788312
$845$		0			0
$846$	−42.2166	+	42.0817i	−1.45144	+	1.44680i
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$	−57.1449	−	138.273i	−1.95775	−	4.73715i
$853$	−10.0000			−0.342393			−0.171197	−	0.985237i	$-0.554763\pi$
$853$	−10.0000			−0.342393			−0.171197	+	0.985237i	$0.554763\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	19.0662i		−	0.651288i	−0.945492	−	0.325644i	$-0.894419\pi$
$857$		−	19.0662i		−	0.651288i	0.945492	−	0.325644i	$-0.105581\pi$
$858$		0			0
$859$	58.6168			1.99998			0.999990	−	0.00438140i	$-0.00139465\pi$
$859$	58.6168			1.99998			0.999990	+	0.00438140i	$0.00139465\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	43.6815i		−	1.48694i	−0.668771	−	0.743469i	$-0.733179\pi$
$863$		−	43.6815i		−	1.48694i	0.668771	−	0.743469i	$-0.266821\pi$
$864$	131.685	+	54.9166i	4.48003	+	1.86830i
$865$		0			0
$866$		0			0
$867$	−27.2125	+	11.2463i	−0.924185	+	0.381944i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$	−156.311	+	64.5996i	−5.28125	+	2.18262i
$877$	14.0000			0.472746			0.236373	−	0.971662i	$-0.424041\pi$
$877$	14.0000			0.472746			0.236373	+	0.971662i	$0.424041\pi$
$878$			92.0568i			3.10677i
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$	41.2069	+	41.3390i	1.38751	+	1.39196i
$883$	−52.0000			−1.74994			−0.874970	−	0.484178i	$-0.839119\pi$
$883$	−52.0000			−1.74994			−0.874970	+	0.484178i	$0.839119\pi$
$884$		0			0
$885$		0			0
$886$	94.9011			3.18826
$887$		−	59.5587i		−	1.99978i	−0.0146917	−	0.999892i	$-0.504677\pi$
$887$		−	59.5587i		−	1.99978i	0.0146917	−	0.999892i	$-0.495323\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$	−45.8036			−1.53362
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−6.83240	−	16.5323i	−0.228127	−	0.551996i
$898$	−106.639			−3.55859
$899$			8.42103i			0.280857i
$900$	60.8247	−	60.6303i	2.02749	−	2.02101i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$	−45.0607	−	109.033i	−1.49704	−	3.62238i
$907$		0			0		1.00000			$0$
$907$		0			0		−1.00000			$\pi$
$908$		0			0
$909$	−40.6288	−	40.7591i	−1.34757	−	1.35189i
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	32.0147	−	13.2309i	1.05492	−	0.435974i
$922$	119.189			3.92527
$923$		−	32.4905i		−	1.06944i
$924$		0			0
$925$		0			0
$926$			88.9430i			2.92285i
$927$		0			0
$928$	43.6493			1.43286
$929$			57.1790i			1.87598i	0.346658	+	0.937992i	$0.387317\pi$
$929$			57.1790i			1.87598i	−0.346658	+	0.937992i	$0.612683\pi$
$930$		0			0
$931$		0			0
$932$		−	36.3503i		−	1.19069i
$933$	15.2326	+	36.8581i	0.498692	+	1.20668i
$934$		0			0
$935$		0			0
$936$	47.2280	+	47.3794i	1.54369	+	1.54864i
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$	45.6957			1.48806
$944$			166.222i			5.41007i
$945$		0			0
$946$		0			0
$947$			24.6251i			0.800209i	0.916470	+	0.400104i	$0.131026\pi$
$947$			24.6251i			0.800209i	−0.916470	+	0.400104i	$0.868974\pi$
$948$		0			0
$949$	−36.7290			−1.19227
$950$		0			0
$951$	−12.6907	−	30.7074i	−0.411523	−	0.995757i
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$		0			0
$956$			4.52287i			0.146280i
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−2.93779			−0.0947673
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	52.3291			1.68279			0.841396	−	0.540420i	$-0.181735\pi$
$967$	52.3291			1.68279			0.841396	+	0.540420i	$0.181735\pi$
$968$			113.902i			3.66096i
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$	33.8247	+	82.5931i	1.08493	+	2.64917i
$973$		0			0
$974$		−	121.505i		−	3.89326i
$975$	17.2361	−	7.12327i	0.551996	−	0.228127i
$976$		0			0
$977$		0			0			−	1.00000i	$-0.5\pi$
$977$		0			0				1.00000i	$0.5\pi$
$978$	37.1412	+	89.8701i	1.18764	+	2.87373i
$979$		0			0
$980$		0			0
$981$		0			0
$982$	−112.575			−3.59240
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$	−157.933	+	65.2698i	−5.03471	+	2.08073i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	56.0000			1.77890			0.889449	−	0.457034i	$-0.151088\pi$
$991$	56.0000			1.77890			0.889449	+	0.457034i	$0.151088\pi$
$992$		−	145.457i		−	4.61826i
$993$	−29.1630	+	12.0524i	−0.925460	+	0.382471i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	26.0000			0.823428			0.411714	−	0.911313i	$-0.364930\pi$
$997$	26.0000			0.823428			0.411714	+	0.911313i	$0.364930\pi$
$998$		−	32.2004i		−	1.01929i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.2.c.a.68.6	yes	6
3.2	odd	2	inner	69.2.c.a.68.1	✓	6
4.3	odd	2		1104.2.m.a.689.2		6
12.11	even	2		1104.2.m.a.689.1		6
23.22	odd	2	CM	69.2.c.a.68.6	yes	6
69.68	even	2	inner	69.2.c.a.68.1	✓	6
92.91	even	2		1104.2.m.a.689.2		6
276.275	odd	2		1104.2.m.a.689.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.2.c.a.68.1	✓	6	3.2	odd	2	inner
69.2.c.a.68.1	✓	6	69.68	even	2	inner
69.2.c.a.68.6	yes	6	1.1	even	1	trivial
69.2.c.a.68.6	yes	6	23.22	odd	2	CM
1104.2.m.a.689.1		6	12.11	even	2
1104.2.m.a.689.1		6	276.275	odd	2
1104.2.m.a.689.2		6	4.3	odd	2
1104.2.m.a.689.2		6	92.91	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	69.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.77947i q^{2} +(1.60074 - 0.661546i) q^{3} -5.72545 q^{4} +(1.83875 + 4.44920i) q^{6} -10.3548i q^{8} +(2.12471 - 2.11792i) q^{9} +O(q^{10})\)	\(q+2.77947i q^{2} +(1.60074 - 0.661546i) q^{3} -5.72545 q^{4} +(1.83875 + 4.44920i) q^{6} -10.3548i q^{8} +(2.12471 - 2.11792i) q^{9} +(-9.16493 + 3.78765i) q^{12} -2.15352 q^{13} +17.3299 q^{16} +(5.88670 + 5.90557i) q^{18} +4.79583i q^{23} +(-6.85016 - 16.5753i) q^{24} -5.00000 q^{25} -5.98564i q^{26} +(2.00000 - 4.79583i) q^{27} -1.58966i q^{29} -5.29738 q^{31} +27.4583i q^{32} +(-12.1649 + 12.1261i) q^{36} +(-3.44722 + 1.42465i) q^{39} -9.52822i q^{41} -13.3299 q^{46} +7.14860i q^{47} +(27.7405 - 11.4645i) q^{48} +7.00000 q^{49} -13.8973i q^{50} +12.3299 q^{52} +(13.3299 + 5.55894i) q^{54} +4.41841 q^{58} +9.59166i q^{59} -14.7239i q^{62} -41.6597 q^{64} +(3.17267 + 7.67686i) q^{69} +15.0872i q^{71} +(-21.9306 - 22.0009i) q^{72} +17.0553 q^{73} +(-8.00368 + 3.30773i) q^{75} +(-3.95978 - 9.58143i) q^{78} +(0.0288070 - 8.99995i) q^{81} +26.4834 q^{82} +(-1.05163 - 2.54463i) q^{87} -27.4583i q^{92} +(-8.47970 + 3.50446i) q^{93} -19.8693 q^{94} +(18.1649 + 43.9535i) q^{96} +19.4563i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 12 q^{4} + 3 q^{6}+O(q^{10}) \) 6 * q - 12 * q^4 + 3 * q^6	\( 6 q - 12 q^{4} + 3 q^{6} - 15 q^{12} + 24 q^{16} + 21 q^{18} - 6 q^{24} - 30 q^{25} + 12 q^{27} - 33 q^{36} - 24 q^{39} + 69 q^{48} + 42 q^{49} - 6 q^{52} + 30 q^{58} - 90 q^{64} - 42 q^{72} - 51 q^{78} + 66 q^{82} + 48 q^{87} - 6 q^{93} - 78 q^{94} + 69 q^{96}+O(q^{100}) \) 6 * q - 12 * q^4 + 3 * q^6 - 15 * q^12 + 24 * q^16 + 21 * q^18 - 6 * q^24 - 30 * q^25 + 12 * q^27 - 33 * q^36 - 24 * q^39 + 69 * q^48 + 42 * q^49 - 6 * q^52 + 30 * q^58 - 90 * q^64 - 42 * q^72 - 51 * q^78 + 66 * q^82 + 48 * q^87 - 6 * q^93 - 78 * q^94 + 69 * q^96

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