LMFDB - Embedded newform 69.4.c.b.68.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [69,4,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, 4, names="a")

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

chi := DirichletCharacter("69.68");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 69 = 3 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
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:	$4.07113179040$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-23})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 5x^{2} - 6x + 36 $ x^4 - x^3 - 5x^2 - 6x + 36
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			68.4
Root			$-1.82666 + 1.63197i$ of defining polynomial
Character	$\chi$	$=$	69.68
Dual form			69.4.c.b.68.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.99599i q^{2} +(-5.15331 + 0.665865i) q^{3} -16.9599 q^{4} +(-3.32666 - 25.7459i) q^{6} -44.7638i q^{8} +(26.1132 - 6.86282i) q^{9} +O(q^{10})$	$q+4.99599i q^{2} +(-5.15331 + 0.665865i) q^{3} -16.9599 q^{4} +(-3.32666 - 25.7459i) q^{6} -44.7638i q^{8} +(26.1132 - 6.86282i) q^{9} +(87.3998 - 11.2930i) q^{12} -86.8397 q^{13} +87.9599 q^{16} +(34.2866 + 130.462i) q^{18} -110.304i q^{23} +(29.8066 + 230.682i) q^{24} -125.000 q^{25} -433.851i q^{26} +(-130.000 + 52.7541i) q^{27} +311.361i q^{29} -147.080 q^{31} +81.3370i q^{32} +(-442.879 + 116.393i) q^{36} +(447.512 - 57.8235i) q^{39} +522.393i q^{41} +551.079 q^{46} -279.751i q^{47} +(-453.285 + 58.5694i) q^{48} +343.000 q^{49} -624.499i q^{50} +1472.80 q^{52} +(-263.559 - 649.479i) q^{54} -1555.56 q^{58} +815.291i q^{59} -734.811i q^{62} +297.321 q^{64} +(73.4477 + 568.432i) q^{69} -908.142i q^{71} +(-307.206 - 1168.93i) q^{72} -812.359 q^{73} +(644.164 - 83.2331i) q^{75} +(288.886 + 2235.77i) q^{78} +(634.803 - 358.421i) q^{81} -2609.87 q^{82} +(-207.324 - 1604.54i) q^{87} +1870.75i q^{92} +(757.950 - 97.9355i) q^{93} +1397.64 q^{94} +(-54.1595 - 419.155i) q^{96} +1713.63i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} - 18 q^{4} - 5 q^{6} + 38 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 - 18 * q^4 - 5 * q^6 + 38 * q^9	$ 4 q - 4 q^{3} - 18 q^{4} - 5 q^{6} + 38 q^{9} + 225 q^{12} - 148 q^{13} + 302 q^{16} + 79 q^{18} + 86 q^{24} - 500 q^{25} - 520 q^{27} - 688 q^{31} - 999 q^{36} + 976 q^{39} + 1058 q^{46} - 509 q^{48} + 1372 q^{49} + 3150 q^{52} - 506 q^{54} - 3182 q^{58} + 1588 q^{64} + 1058 q^{69} - 448 q^{72} - 2452 q^{73} + 500 q^{75} + 599 q^{78} + 14 q^{81} - 5306 q^{82} + 1048 q^{87} + 274 q^{93} + 2650 q^{94} + 207 q^{96}+O(q^{100}) $ 4 * q - 4 * q^3 - 18 * q^4 - 5 * q^6 + 38 * q^9 + 225 * q^12 - 148 * q^13 + 302 * q^16 + 79 * q^18 + 86 * q^24 - 500 * q^25 - 520 * q^27 - 688 * q^31 - 999 * q^36 + 976 * q^39 + 1058 * q^46 - 509 * q^48 + 1372 * q^49 + 3150 * q^52 - 506 * q^54 - 3182 * q^58 + 1588 * q^64 + 1058 * q^69 - 448 * q^72 - 2452 * q^73 + 500 * q^75 + 599 * q^78 + 14 * q^81 - 5306 * q^82 + 1048 * q^87 + 274 * q^93 + 2650 * q^94 + 207 * 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 $

$2$			4.99599i			1.76635i	0.469044	+	0.883175i	$0.344599\pi$
$2$			4.99599i			1.76635i	−0.469044	+	0.883175i	$0.655401\pi$
$3$	−5.15331	+	0.665865i	−0.991755	+	0.128146i
$3$	−5.15331	+	0.665865i	−0.991755	+	0.128146i
$4$	−16.9599			−2.11999
$5$		0			0			−	1.00000i	$-0.5\pi$
$5$		0			0				1.00000i	$0.5\pi$
$6$	−3.32666	−	25.7459i	−0.226350	−	1.75179i
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		−	44.7638i		−	1.97830i
$9$	26.1132	−	6.86282i	0.967157	−	0.254179i
$10$		0			0
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$	87.3998	−	11.2930i	2.10251	−	0.271668i
$13$	−86.8397			−1.85269			−0.926347	−	0.376672i	$-0.877068\pi$
$13$	−86.8397			−1.85269			−0.926347	+	0.376672i	$0.877068\pi$
$14$		0			0
$15$		0			0
$16$	87.9599			1.37437
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$	34.2866	+	130.462i	0.448968	+	1.70834i
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	110.304i		−	1.00000i
$23$		−	110.304i		−	1.00000i
$24$	29.8066	+	230.682i	0.253510	+	1.96199i
$25$	−125.000			−1.00000
$26$		−	433.851i		−	3.27250i
$27$	−130.000	+	52.7541i	−0.926612	+	0.376020i
$28$		0			0
$29$			311.361i			1.99373i	0.0791039	+	0.996866i	$0.474794\pi$
$29$			311.361i			1.99373i	−0.0791039	+	0.996866i	$0.525206\pi$
$30$		0			0
$31$	−147.080			−0.852141			−0.426070	−	0.904690i	$-0.640102\pi$
$31$	−147.080			−0.852141			−0.426070	+	0.904690i	$0.640102\pi$
$32$			81.3370i			0.449328i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−442.879	+	116.393i	−2.05037	+	0.538856i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$	447.512	−	57.8235i	1.83742	−	0.237415i
$40$		0			0
$41$			522.393i			1.98986i	0.100583	+	0.994929i	$0.467929\pi$
$41$			522.393i			1.98986i	−0.100583	+	0.994929i	$0.532071\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$		0			0
$46$	551.079			1.76635
$47$		−	279.751i		−	0.868212i	−0.900862	−	0.434106i	$-0.857064\pi$
$47$		−	279.751i		−	0.868212i	0.900862	−	0.434106i	$-0.142936\pi$
$48$	−453.285	+	58.5694i	−1.36304	+	0.176120i
$49$	343.000			1.00000
$50$		−	624.499i		−	1.76635i
$51$		0			0
$52$	1472.80			3.92769
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$	−263.559	−	649.479i	−0.664183	−	1.63672i
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−1555.56			−3.52163
$59$			815.291i			1.79902i	0.436905	+	0.899508i	$0.356075\pi$
$59$			815.291i			1.79902i	−0.436905	+	0.899508i	$0.643925\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		−	734.811i		−	1.50518i
$63$		0			0
$64$	297.321			0.580704
$65$		0			0
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$		0			0
$69$	73.4477	+	568.432i	0.128146	+	0.991755i
$70$		0			0
$71$		−	908.142i		−	1.51798i	−0.651102	−	0.758990i	$-0.725693\pi$
$71$		−	908.142i		−	1.51798i	0.651102	−	0.758990i	$-0.274307\pi$
$72$	−307.206	−	1168.93i	−0.502841	−	1.91333i
$73$	−812.359			−1.30246			−0.651229	−	0.758881i	$-0.725746\pi$
$73$	−812.359			−1.30246			−0.651229	+	0.758881i	$0.725746\pi$
$74$		0			0
$75$	644.164	−	83.2331i	0.991755	−	0.128146i
$76$		0			0
$77$		0			0
$78$	288.886	+	2235.77i	0.419358	+	3.24552i
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$		0			0
$81$	634.803	−	358.421i	0.870787	−	0.491661i
$82$	−2609.87			−3.51478
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−207.324	−	1604.54i	−0.255488	−	1.97730i
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$		0			0
$92$			1870.75i			2.11999i
$93$	757.950	−	97.9355i	0.845115	−	0.109198i
$94$	1397.64			1.53357
$95$		0			0
$96$	−54.1595	−	419.155i	−0.0575794	−	0.445623i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$			1713.63i			1.76635i
$99$		0			0
$100$	2119.99			2.11999
$101$			1246.92i			1.22844i	0.789133	+	0.614222i	$0.210530\pi$
$101$			1246.92i			1.22844i	−0.789133	+	0.614222i	$0.789470\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$			3887.27i			3.66518i
$105$		0			0
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$	2204.79	−	894.707i	1.96441	−	0.797159i
$109$		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
$116$		−	5280.66i		−	4.22670i
$117$	−2267.67	+	595.965i	−1.79185	+	0.470915i
$118$	−4073.19			−3.17769
$119$		0			0
$120$		0			0
$121$	−1331.00			−1.00000
$122$		0			0
$123$	−347.843	−	2692.06i	−0.254992	−	1.97345i
$124$	2494.47			1.80653
$125$		0			0
$126$		0			0
$127$	282.285			0.197234			0.0986172	−	0.995125i	$-0.468558\pi$
$127$	282.285			0.197234			0.0986172	+	0.995125i	$0.468558\pi$
$128$			2136.11i			1.47505i
$129$		0			0
$130$		0			0
$131$		−	1879.48i		−	1.25352i	−0.779213	−	0.626759i	$-0.784381\pi$
$131$		−	1879.48i		−	1.25352i	0.779213	−	0.626759i	$-0.215619\pi$
$132$		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	$0.5\pi$
$138$	−2839.88	+	366.944i	−1.75179	+	0.226350i
$139$	−2609.95			−1.59261			−0.796305	−	0.604896i	$-0.793215\pi$
$139$	−2609.95			−1.59261			−0.796305	+	0.604896i	$0.793215\pi$
$140$		0			0
$141$	186.277	+	1441.65i	0.111258	+	0.861054i
$142$	4537.07			2.68128
$143$		0			0
$144$	2296.92	−	603.653i	1.32924	−	0.349336i
$145$		0			0
$146$		−	4058.54i		−	2.30060i
$147$	−1767.59	+	228.392i	−0.991755	+	0.128146i
$148$		0			0
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$	415.832	+	3218.24i	0.226350	+	1.75179i
$151$	1085.32			0.584917			0.292458	−	0.956278i	$-0.405527\pi$
$151$	1085.32			0.584917			0.292458	+	0.956278i	$0.405527\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−7589.78	+	980.684i	−3.89531	+	0.503317i
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	1790.67	+	3171.47i	0.868446	+	1.53811i
$163$	4132.34			1.98571			0.992853	−	0.119344i	$-0.0380790\pi$
$163$	4132.34			1.98571			0.992853	+	0.119344i	$0.0380790\pi$
$164$		−	8859.76i		−	4.21848i
$165$		0			0
$166$		0			0
$167$			3922.99i			1.81778i	0.417030	+	0.908892i	$0.363071\pi$
$167$			3922.99i			1.81778i	−0.417030	+	0.908892i	$0.636929\pi$
$168$		0			0
$169$	5344.14			2.43247
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	2896.68i		−	1.27301i	−0.771273	−	0.636505i	$-0.780380\pi$
$173$		−	2896.68i		−	1.27301i	0.771273	−	0.636505i	$-0.219620\pi$
$174$	8016.27	−	1035.79i	3.49260	−	0.451282i
$175$		0			0
$176$		0			0
$177$	−542.874	−	4201.45i	−0.230536	−	1.78418i
$178$		0			0
$179$			418.343i			0.174684i	0.996178	+	0.0873419i	$0.0278373\pi$
$179$			418.343i			0.174684i	−0.996178	+	0.0873419i	$0.972163\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$	−4937.63			−1.97830
$185$		0			0
$186$	489.285	+	3786.71i	0.192882	+	1.49277i
$187$		0			0
$188$			4744.57i			1.84060i
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$	−1532.19	+	197.975i	−0.575916	+	0.0744148i
$193$	648.026			0.241689			0.120844	−	0.992671i	$-0.461440\pi$
$193$	648.026			0.241689			0.120844	+	0.992671i	$0.461440\pi$
$194$		0			0
$195$		0			0
$196$	−5817.26			−2.11999
$197$			4486.71i			1.62267i	0.584585	+	0.811333i	$0.301257\pi$
$197$			4486.71i			1.62267i	−0.584585	+	0.811333i	$0.698743\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$			5595.47i			1.97830i
$201$		0			0
$202$	−6229.58			−2.16986
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−756.997	−	2880.40i	−0.254179	−	0.967157i
$208$	−7638.42			−2.54629
$209$		0			0
$210$		0			0
$211$	2468.00			0.805233			0.402616	−	0.915369i	$-0.368101\pi$
$211$	2468.00			0.805233			0.402616	+	0.915369i	$0.368101\pi$
$212$		0			0
$213$	604.700	+	4679.94i	0.194523	+	1.50547i
$214$		0			0
$215$		0			0
$216$	2361.47	+	5819.29i	0.743880	+	1.83311i
$217$		0			0
$218$		0			0
$219$	4186.34	−	540.921i	1.29172	−	0.166904i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−4840.00			−1.45341			−0.726705	−	0.686950i	$-0.758949\pi$
$223$	−4840.00			−1.45341			−0.726705	+	0.686950i	$0.758949\pi$
$224$		0			0
$225$	−3264.16	+	857.852i	−0.967157	+	0.254179i
$226$		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
$232$	13937.7			3.94420
$233$			2892.13i			0.813174i	0.913612	+	0.406587i	$0.133281\pi$
$233$			2892.13i			0.813174i	−0.913612	+	0.406587i	$0.866719\pi$
$234$	−2977.44	−	11329.2i	−0.831800	−	3.16503i
$235$		0			0
$236$		−	13827.3i		−	3.81390i
$237$		0			0
$238$		0			0
$239$		−	6097.92i		−	1.65038i	−0.564852	−	0.825192i	$-0.691067\pi$
$239$		−	6097.92i		−	1.65038i	0.564852	−	0.825192i	$-0.308933\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		−	6649.67i		−	1.76635i
$243$	−3032.68	+	2269.75i	−0.800603	+	0.599195i
$244$		0			0
$245$		0			0
$246$	13449.5	−	1737.82i	3.48581	−	0.450405i
$247$		0			0
$248$			6583.86i			1.68579i
$249$		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
$254$			1410.29i			0.348385i
$255$		0			0
$256$	−8293.41			−2.02476
$257$			936.820i			0.227382i	0.993516	+	0.113691i	$0.0362674\pi$
$257$			936.820i			0.227382i	−0.993516	+	0.113691i	$0.963733\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	2136.81	+	8130.64i	0.506764	+	1.92825i
$262$	9389.86			2.21415
$263$		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
$269$			5251.15i			1.19022i	0.803646	+	0.595108i	$0.202891\pi$
$269$			5251.15i			1.19022i	−0.803646	+	0.595108i	$0.797109\pi$
$270$		0			0
$271$	8912.00			1.99766			0.998829	−	0.0483752i	$-0.0154043\pi$
$271$	8912.00			1.99766			0.998829	+	0.0483752i	$0.0154043\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$	−1245.67	−	9640.56i	−0.271668	−	2.10251i
$277$	−8743.84			−1.89663			−0.948315	−	0.317331i	$-0.897213\pi$
$277$	−8743.84			−1.89663			−0.948315	+	0.317331i	$0.897213\pi$
$278$		−	13039.3i		−	2.81311i
$279$	−3840.74	+	1009.38i	−0.824154	+	0.216596i
$280$		0			0
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$	−7202.46	+	930.637i	−1.52092	+	0.196520i
$283$		0			0		1.00000			$0$
$283$		0			0		−1.00000			$\pi$
$284$			15402.0i			3.21811i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	558.201	+	2123.97i	0.114209	+	0.434571i
$289$	−4913.00			−1.00000
$290$		0			0
$291$		0			0
$292$	13777.6			2.76120
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$	−1141.04	−	8830.85i	−0.226350	−	1.75179i
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$			9578.78i			1.85269i
$300$	−10925.0	+	1411.63i	−2.10251	+	0.271668i
$301$		0			0
$302$			5422.27i			1.03317i
$303$	−830.278	−	6425.75i	−0.157420	−	1.21832i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−10420.0			−1.93714			−0.968568	−	0.248749i	$-0.919981\pi$
$307$	−10420.0			−1.93714			−0.968568	+	0.248749i	$0.919981\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	9709.01i		−	1.77025i	−0.465354	−	0.885125i	$-0.654073\pi$
$311$		−	9709.01i		−	1.77025i	0.465354	−	0.885125i	$-0.345927\pi$
$312$	−2588.40	−	20032.3i	−0.469677	−	3.63496i
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	11183.9i		−	1.98154i	−0.135542	−	0.990772i	$-0.543277\pi$
$317$		−	11183.9i		−	1.98154i	0.135542	−	0.990772i	$-0.456723\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−10766.2	+	6078.80i	−1.84606	+	1.04232i
$325$	10855.0			1.85269
$326$			20645.2i			3.50745i
$327$		0			0
$328$	23384.3			3.93653
$329$		0			0
$330$		0			0
$331$	−5997.08			−0.995859			−0.497930	−	0.867218i	$-0.665906\pi$
$331$	−5997.08			−0.995859			−0.497930	+	0.867218i	$0.665906\pi$
$332$		0			0
$333$		0			0
$334$	−19599.2			−3.21084
$335$		0			0
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$			26699.3i			4.29660i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	14471.8			2.24858
$347$			9102.49i			1.40821i	0.710098	+	0.704103i	$0.248650\pi$
$347$			9102.49i			1.40821i	−0.710098	+	0.704103i	$0.751350\pi$
$348$	3516.21	+	27212.9i	0.541633	+	4.19185i
$349$	−12386.8			−1.89986			−0.949929	−	0.312467i	$-0.898845\pi$
$349$	−12386.8			−1.89986			−0.949929	+	0.312467i	$0.898845\pi$
$350$		0			0
$351$	11289.2	−	4581.16i	1.71673	−	0.696650i
$352$		0			0
$353$		−	9374.23i		−	1.41343i	−0.707499	−	0.706714i	$-0.750177\pi$
$353$		−	9374.23i		−	1.41343i	0.707499	−	0.706714i	$-0.249823\pi$
$354$	20990.4	−	2712.19i	3.15149	−	0.407208i
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−2090.04			−0.308553
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	6859.00			1.00000
$362$		0			0
$363$	6859.06	−	886.266i	0.991755	−	0.128146i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$		−	9702.34i		−	1.37437i
$369$	3585.09	+	13641.4i	0.505779	+	1.92451i
$370$		0			0
$371$		0			0
$372$	−12854.8	+	1660.98i	−1.79164	+	0.231499i
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$	−12522.7			−1.71758
$377$		−	27038.5i		−	3.69378i
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$	−1454.70	+	187.964i	−0.195608	+	0.0252748i
$382$		0			0
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$	−1422.36	−	11008.0i	−0.189022	−	1.46289i
$385$		0			0
$386$			3237.53i			0.426907i
$387$		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
$392$		−	15354.0i		−	1.97830i
$393$	1251.48	+	9685.54i	0.160633	+	1.24318i
$394$	−22415.6			−2.86620
$395$		0			0
$396$		0			0
$397$	11677.7			1.47629			0.738145	−	0.674642i	$-0.235702\pi$
$397$	11677.7			1.47629			0.738145	+	0.674642i	$0.235702\pi$
$398$		0			0
$399$		0			0
$400$	−10995.0			−1.37437
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$	12772.4			1.57876
$404$		−	21147.6i		−	2.60429i
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−11155.1			−1.34862			−0.674308	−	0.738451i	$-0.735558\pi$
$409$	−11155.1			−1.34862			−0.674308	+	0.738451i	$0.735558\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	14390.4	−	3781.95i	1.70834	−	0.448968i
$415$		0			0
$416$		−	7063.28i		−	0.832466i
$417$	13449.9	−	1737.87i	1.57948	−	0.204086i
$418$		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			$\pi$
$422$			12330.1i			1.42232i
$423$	−1919.88	−	7305.22i	−0.220681	−	0.839697i
$424$		0			0
$425$		0			0
$426$	−23380.9	+	3021.08i	−2.65918	+	0.343595i
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$	−11434.8	+	4640.25i	−1.27351	+	0.516792i
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	2702.44	+	20914.9i	0.294812	+	2.28163i
$439$	−10984.0			−1.19416			−0.597080	−	0.802182i	$-0.703672\pi$
$439$	−10984.0			−1.19416			−0.597080	+	0.802182i	$0.703672\pi$
$440$		0			0
$441$	8956.84	−	2353.95i	0.967157	−	0.254179i
$442$		0			0
$443$			18198.2i			1.95174i	0.218352	+	0.975870i	$0.429932\pi$
$443$			18198.2i			1.95174i	−0.218352	+	0.975870i	$0.570068\pi$
$444$		0			0
$445$		0			0
$446$		−	24180.6i		−	2.56723i
$447$		0			0
$448$		0			0
$449$		−	4795.83i		−	0.504074i	−0.967718	−	0.252037i	$-0.918900\pi$
$449$		−	4795.83i		−	0.504074i	0.967718	−	0.252037i	$-0.0811005\pi$
$450$	−4285.82	−	16307.7i	−0.448968	−	1.70834i
$451$		0			0
$452$		0			0
$453$	−5593.01	+	722.679i	−0.580094	+	0.0749546i
$454$		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
$461$		−	7066.71i		−	0.713947i	−0.934115	−	0.356973i	$-0.883809\pi$
$461$		−	7066.71i		−	0.713947i	0.934115	−	0.356973i	$-0.116191\pi$
$462$		0			0
$463$	−11680.0			−1.17239			−0.586194	−	0.810171i	$-0.699374\pi$
$463$	−11680.0			−1.17239			−0.586194	+	0.810171i	$0.699374\pi$
$464$			27387.3i			2.74013i
$465$		0			0
$466$	−14449.0			−1.43635
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$	38459.5	−	10107.5i	3.79870	−	0.998336i
$469$		0			0
$470$		0			0
$471$		0			0
$472$	36495.5			3.55899
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	30465.2			2.91516
$479$		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
$484$	22573.7			2.11999
$485$		0			0
$486$	−11339.6	−	15151.2i	−1.05839	−	1.41414i
$487$	17336.9			1.61316			0.806580	−	0.591125i	$-0.201316\pi$
$487$	17336.9			1.61316			0.806580	+	0.591125i	$0.201316\pi$
$488$		0			0
$489$	−21295.3	+	2751.58i	−1.96933	+	0.254460i
$490$		0			0
$491$			21296.6i			1.95744i	0.205206	+	0.978719i	$0.434214\pi$
$491$			21296.6i			1.95744i	−0.205206	+	0.978719i	$0.565786\pi$
$492$	5899.40	+	45657.1i	0.540581	+	4.18370i
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−12937.2			−1.17116
$497$		0			0
$498$		0			0
$499$	5737.17			0.514691			0.257346	−	0.966319i	$-0.417152\pi$
$499$	5737.17			0.514691			0.257346	+	0.966319i	$0.417152\pi$
$500$		0			0
$501$	−2612.18	−	20216.4i	−0.232941	−	1.80280i
$502$		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
$507$	−27540.0	+	3558.48i	−2.41242	+	0.311711i
$508$	−4787.54			−0.418135
$509$			3876.14i			0.337538i	0.985656	+	0.168769i	$0.0539792\pi$
$509$			3876.14i			0.337538i	−0.985656	+	0.168769i	$0.946021\pi$
$510$		0			0
$511$		0			0
$512$		−	24344.9i		−	2.10138i
$513$		0			0
$514$	−4680.34			−0.401636
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	1928.80	+	14927.5i	0.163131	+	1.26251i
$520$		0			0
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$	−40620.6	+	10675.5i	−3.40597	+	0.895123i
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$			31875.8i			2.65745i
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−12167.0			−1.00000
$530$		0			0
$531$	5595.20	+	21289.9i	0.457271	+	1.73993i
$532$		0			0
$533$		−	45364.5i		−	3.68659i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−278.560	−	2155.85i	−0.0223850	−	0.173244i
$538$	−26234.7			−2.10234
$539$		0			0
$540$		0			0
$541$	22247.0			1.76797			0.883985	−	0.467515i	$-0.154851\pi$
$541$	22247.0			1.76797			0.883985	+	0.467515i	$0.154851\pi$
$542$			44524.3i			3.52856i
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−22901.7			−1.79014			−0.895068	−	0.445930i	$-0.852873\pi$
$547$	−22901.7			−1.79014			−0.895068	+	0.445930i	$0.852873\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$	25445.1	−	3287.79i	1.96199	−	0.253510i
$553$		0			0
$554$		−	43684.2i		−	3.35011i
$555$		0			0
$556$	44264.5			3.37632
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$	−5042.88	−	19188.3i	−0.382584	−	1.45575i
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$	−3159.24	−	24450.2i	−0.235865	−	1.82543i
$565$		0			0
$566$		0			0
$567$		0			0
$568$	−40651.8			−3.00302
$569$		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
$575$			13788.0i			1.00000i
$576$	7764.00	−	2040.46i	0.561632	−	0.147603i
$577$	−3963.24			−0.285948			−0.142974	−	0.989726i	$-0.545667\pi$
$577$	−3963.24			−0.285948			−0.142974	+	0.989726i	$0.545667\pi$
$578$		−	24545.3i		−	1.76635i
$579$	−3339.48	+	431.498i	−0.239696	+	0.0309714i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$			36364.2i			2.57665i
$585$		0			0
$586$		0			0
$587$			12321.1i			0.866344i	0.901311	+	0.433172i	$0.142606\pi$
$587$			12321.1i			0.866344i	−0.901311	+	0.433172i	$0.857394\pi$
$588$	29978.1	−	3873.51i	2.10251	−	0.271668i
$589$		0			0
$590$		0			0
$591$	−2987.55	−	23121.4i	−0.207938	−	1.60929i
$592$		0			0
$593$			11778.6i			0.815662i	0.913057	+	0.407831i	$0.133715\pi$
$593$			11778.6i			0.815662i	−0.913057	+	0.407831i	$0.866285\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$	−47855.5			−3.27250
$599$			16353.8i			1.11552i	0.830002	+	0.557761i	$0.188339\pi$
$599$			16353.8i			1.11552i	−0.830002	+	0.557761i	$0.811661\pi$
$600$	−3725.83	−	28835.2i	−0.253510	−	1.96199i
$601$	25354.9			1.72088			0.860441	−	0.509550i	$-0.170188\pi$
$601$	25354.9			1.72088			0.860441	+	0.509550i	$0.170188\pi$
$602$		0			0
$603$		0			0
$604$	−18407.0			−1.24002
$605$		0			0
$606$	32103.0	−	4148.06i	2.15197	−	0.278059i
$607$	8840.00			0.591111			0.295556	−	0.955326i	$-0.404495\pi$
$607$	8840.00			0.591111			0.295556	+	0.955326i	$0.404495\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			24293.5i			1.60853i
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$		−	52058.2i		−	3.42166i
$615$		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
$621$	5819.00	+	14339.5i	0.376020	+	0.926612i
$622$	48506.2			3.12688
$623$		0			0
$624$	39363.2	−	5086.16i	2.52530	−	0.326297i
$625$	15625.0			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
$633$	−12718.4	+	1643.35i	−0.798594	+	0.103187i
$634$	55874.6			3.50010
$635$		0			0
$636$		0			0
$637$	−29786.0			−1.85269
$638$		0			0
$639$	−6232.41	−	23714.5i	−0.385838	−	1.46813i
$640$		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
$647$		−	16636.7i		−	1.01091i	−0.862854	−	0.505454i	$-0.831325\pi$
$647$		−	16636.7i		−	1.01091i	0.862854	−	0.505454i	$-0.168675\pi$
$648$	−16044.3	−	28416.2i	−0.972652	−	1.72268i
$649$		0			0
$650$			54231.3i			3.27250i
$651$		0			0
$652$	−70084.3			−4.20968
$653$			6538.77i			0.391856i	0.980618	+	0.195928i	$0.0627719\pi$
$653$			6538.77i			0.391856i	−0.980618	+	0.195928i	$0.937228\pi$
$654$		0			0
$655$		0			0
$656$			45949.7i			2.73481i
$657$	−21213.3	+	5575.07i	−1.25968	+	0.331057i
$658$		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			$\pi$
$662$		−	29961.4i		−	1.75904i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	34344.4			1.99373
$668$		−	66533.7i		−	3.85369i
$669$	24942.0	−	3222.79i	1.44143	−	0.186248i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	8519.60			0.487974			0.243987	−	0.969778i	$-0.421545\pi$
$673$	8519.60			0.487974			0.243987	+	0.969778i	$0.421545\pi$
$674$		0			0
$675$	16250.0	−	6594.27i	0.926612	−	0.376020i
$676$	−90636.3			−5.15682
$677$		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
$683$			33067.2i			1.85253i	0.376867	+	0.926267i	$0.377001\pi$
$683$			33067.2i			1.85253i	−0.376867	+	0.926267i	$0.622999\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−6028.00			−0.331861			−0.165930	−	0.986137i	$-0.553063\pi$
$691$	−6028.00			−0.331861			−0.165930	+	0.986137i	$0.553063\pi$
$692$			49127.5i			2.69877i
$693$		0			0
$694$	−45476.0			−2.48738
$695$		0			0
$696$	−71825.2	+	9280.61i	−3.91168	+	0.505432i
$697$		0			0
$698$		−	61884.4i		−	3.35581i
$699$	−1925.77	−	14904.0i	−0.104205	−	0.806470i
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$	22887.4	+	56400.6i	1.23053	+	3.03234i
$703$		0			0
$704$		0			0
$705$		0			0
$706$	46833.6			2.49661
$707$		0			0
$708$	9207.11	+	71256.3i	0.488735	+	3.78245i
$709$		0			0		1.00000			$0$
$709$		0			0		−1.00000			$\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			16223.5i			0.852141i
$714$		0			0
$715$		0			0
$716$		−	7095.07i		−	0.370328i
$717$	4060.39	+	31424.5i	0.211490	+	1.63678i
$718$		0			0
$719$			6858.04i			0.355719i	0.984056	+	0.177859i	$0.0569172\pi$
$719$			6858.04i			0.355719i	−0.984056	+	0.177859i	$0.943083\pi$
$720$		0			0
$721$		0			0
$722$			34267.5i			1.76635i
$723$		0			0
$724$		0			0
$725$		−	38920.1i		−	1.99373i
$726$	4427.78	+	34267.8i	0.226350	+	1.75179i
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	14117.0	−	13716.1i	0.717218	−	0.696849i
$730$		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
$736$	8971.81			0.449328
$737$		0			0
$738$	−68152.3	+	17911.1i	−3.39935	+	0.893383i
$739$	7328.32			0.364786			0.182393	−	0.983226i	$-0.441616\pi$
$739$	7328.32			0.364786			0.182393	+	0.983226i	$0.441616\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$	−4383.96	−	33928.7i	−0.216027	−	1.67189i
$745$		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			$\pi$
$752$		−	24606.9i		−	1.19325i
$753$		0			0
$754$	135084.			6.52450
$755$		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
$761$			26268.3i			1.25128i	0.780111	+	0.625641i	$0.215163\pi$
$761$			26268.3i			1.25128i	−0.780111	+	0.625641i	$0.784837\pi$
$762$	−939.066	−	7267.69i	−0.0446441	−	0.345513i
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	70799.7i		−	3.33302i
$768$	42738.5	−	5522.29i	2.00806	−	0.259464i
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$	−623.796	−	4827.72i	−0.0291381	−	0.225507i
$772$	−10990.5			−0.512378
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		0			0
$775$	18385.0			0.852141
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−16425.6	−	40476.9i	−0.749683	−	1.84742i
$784$	30170.3			1.37437
$785$		0			0
$786$	−48388.9	+	6252.38i	−2.19590	+	0.283734i
$787$		0			0		1.00000			$0$
$787$		0			0		−1.00000			$\pi$
$788$		−	76094.4i		−	3.44004i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$			58341.7i			2.60764i
$795$		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
$800$		−	10167.1i		−	0.449328i
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$			63810.8i			2.78864i
$807$	−3496.56	−	27060.8i	−0.152521	−	1.18040i
$808$	55816.7			2.43023
$809$			36640.2i			1.59234i	0.605076	+	0.796168i	$0.293143\pi$
$809$			36640.2i			1.59234i	−0.605076	+	0.796168i	$0.706857\pi$
$810$		0			0
$811$	−38214.7			−1.65462			−0.827312	−	0.561743i	$-0.810131\pi$
$811$	−38214.7			−1.65462			−0.827312	+	0.561743i	$0.810131\pi$
$812$		0			0
$813$	−45926.3	+	5934.19i	−1.98119	+	0.255991i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		−	55730.7i		−	2.38213i
$819$		0			0
$820$		0			0
$821$		−	40189.1i		−	1.70841i	−0.519933	−	0.854207i	$-0.674043\pi$
$821$		−	40189.1i		−	1.70841i	0.519933	−	0.854207i	$-0.325957\pi$
$822$		0			0
$823$	−29161.0			−1.23510			−0.617550	−	0.786532i	$-0.711875\pi$
$823$	−29161.0			−1.23510			−0.617550	+	0.786532i	$0.711875\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$	12838.6	+	48851.4i	0.538856	+	2.05037i
$829$	−4966.00			−0.208053			−0.104027	−	0.994575i	$-0.533173\pi$
$829$	−4966.00			−0.208053			−0.104027	+	0.994575i	$0.533173\pi$
$830$		0			0
$831$	45059.7	−	5822.22i	1.88099	−	0.243045i
$832$	−25819.2			−1.07587
$833$		0			0
$834$	8682.39	+	67195.4i	0.360488	+	2.78991i
$835$		0			0
$836$		0			0
$837$	19120.4	−	7759.09i	0.789604	−	0.320422i
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	−72556.6			−2.97497
$842$		0			0
$843$		0			0
$844$	−41857.1			−1.70709
$845$		0			0
$846$	36496.8	−	9591.73i	1.48320	−	0.389799i
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$	−10255.7	−	79371.4i	−0.412387	−	3.19157i
$853$	24590.0			0.987041			0.493520	−	0.869734i	$-0.335710\pi$
$853$	24590.0			0.987041			0.493520	+	0.869734i	$0.335710\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			44702.2i			1.78179i	0.454205	+	0.890897i	$0.349923\pi$
$857$			44702.2i			1.78179i	−0.454205	+	0.890897i	$0.650077\pi$
$858$		0			0
$859$	−25747.2			−1.02268			−0.511340	−	0.859379i	$-0.670851\pi$
$859$	−25747.2			−1.02268			−0.511340	+	0.859379i	$0.670851\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	20690.6i		−	0.816125i	−0.912954	−	0.408063i	$-0.866205\pi$
$863$		−	20690.6i		−	0.816125i	0.912954	−	0.408063i	$-0.133795\pi$
$864$	−4290.86	−	10573.8i	−0.168956	−	0.416352i
$865$		0			0
$866$		0			0
$867$	25318.2	−	3271.39i	0.991755	−	0.128146i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$	−71000.0	+	9173.99i	−2.73844	+	0.353836i
$877$	−34090.0			−1.31259			−0.656293	−	0.754506i	$-0.727876\pi$
$877$	−34090.0			−1.31259			−0.656293	+	0.754506i	$0.727876\pi$
$878$		−	54875.8i		−	2.10930i
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$	11760.3	+	44748.3i	0.448968	+	1.70834i
$883$	−2860.00			−0.109000			−0.0544998	−	0.998514i	$-0.517356\pi$
$883$	−2860.00			−0.109000			−0.0544998	+	0.998514i	$0.517356\pi$
$884$		0			0
$885$		0			0
$886$	−90917.9			−3.44746
$887$		−	28407.6i		−	1.07535i	−0.843153	−	0.537673i	$-0.819303\pi$
$887$		−	28407.6i		−	1.07535i	0.843153	−	0.537673i	$-0.180697\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$	82086.1			3.08122
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−6378.18	−	49362.5i	−0.237415	−	1.83742i
$898$	23959.9			0.890371
$899$		−	45795.0i		−	1.69894i
$900$	55359.9	−	14549.1i	2.05037	−	0.538856i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$	−3610.50	−	27942.6i	−0.132396	−	1.02465i
$907$		0			0		1.00000			$0$
$907$		0			0		−1.00000			$\pi$
$908$		0			0
$909$	8557.36	+	32561.0i	0.312244	+	1.18810i
$910$		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
$921$	53697.5	−	6938.31i	1.92117	−	0.248236i
$922$	35305.2			1.26108
$923$			78862.8i			2.81235i
$924$		0			0
$925$		0			0
$926$		−	58353.2i		−	2.07085i
$927$		0			0
$928$	−25325.2			−0.895839
$929$			56624.7i			1.99978i	0.0148179	+	0.999890i	$0.495283\pi$
$929$			56624.7i			1.99978i	−0.0148179	+	0.999890i	$0.504717\pi$
$930$		0			0
$931$		0			0
$932$		−	49050.3i		−	1.72392i
$933$	6464.89	+	50033.6i	0.226850	+	1.75565i
$934$		0			0
$935$		0			0
$936$	26677.7	+	101509.i	0.931610	+	3.54480i
$937$		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
$943$	57622.2			1.98986
$944$			71713.0i			2.47252i
$945$		0			0
$946$		0			0
$947$		−	44151.8i		−	1.51504i	−0.652814	−	0.757518i	$-0.726412\pi$
$947$		−	44151.8i		−	1.51504i	0.652814	−	0.757518i	$-0.273588\pi$
$948$		0			0
$949$	70545.0			2.41305
$950$		0			0
$951$	7446.95	+	57634.0i	0.253926	+	1.96521i
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$		0			0
$956$			103420.i			3.49880i
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−8158.44			−0.273856
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−47303.2			−1.57308			−0.786540	−	0.617539i	$-0.788130\pi$
$967$	−47303.2			−1.57308			−0.786540	+	0.617539i	$0.788130\pi$
$968$			59580.6i			1.97830i
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$	51434.1	−	38494.8i	1.69727	−	1.27029i
$973$		0			0
$974$			86614.9i			2.84941i
$975$	−55939.0	+	7227.94i	−1.83742	+	0.237415i
$976$		0			0
$977$		0			0			−	1.00000i	$-0.5\pi$
$977$		0			0				1.00000i	$0.5\pi$
$978$	−13746.9	−	106391.i	−0.449465	−	3.47853i
$979$		0			0
$980$		0			0
$981$		0			0
$982$	−106398.			−3.45752
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$	−120507.	+	15570.8i	−3.90407	+	0.504450i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	9128.00			0.292594			0.146297	−	0.989241i	$-0.453265\pi$
$991$	9128.00			0.292594			0.146297	+	0.989241i	$0.453265\pi$
$992$		−	11963.1i		−	0.382891i
$993$	30904.8	−	3993.25i	0.987649	−	0.127615i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−60190.0			−1.91197			−0.955986	−	0.293412i	$-0.905209\pi$
$997$	−60190.0			−1.91197			−0.955986	+	0.293412i	$0.905209\pi$
$998$			28662.9i			0.909125i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.4.c.b.68.4	yes	4
3.2	odd	2	inner	69.4.c.b.68.1	✓	4
23.22	odd	2	CM	69.4.c.b.68.4	yes	4
69.68	even	2	inner	69.4.c.b.68.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.4.c.b.68.1	✓	4	3.2	odd	2	inner
69.4.c.b.68.1	✓	4	69.68	even	2	inner
69.4.c.b.68.4	yes	4	1.1	even	1	trivial
69.4.c.b.68.4	yes	4	23.22	odd	2	CM

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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	69.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+4.99599i q^{2} +(-5.15331 + 0.665865i) q^{3} -16.9599 q^{4} +(-3.32666 - 25.7459i) q^{6} -44.7638i q^{8} +(26.1132 - 6.86282i) q^{9} +O(q^{10})\)	\(q+4.99599i q^{2} +(-5.15331 + 0.665865i) q^{3} -16.9599 q^{4} +(-3.32666 - 25.7459i) q^{6} -44.7638i q^{8} +(26.1132 - 6.86282i) q^{9} +(87.3998 - 11.2930i) q^{12} -86.8397 q^{13} +87.9599 q^{16} +(34.2866 + 130.462i) q^{18} -110.304i q^{23} +(29.8066 + 230.682i) q^{24} -125.000 q^{25} -433.851i q^{26} +(-130.000 + 52.7541i) q^{27} +311.361i q^{29} -147.080 q^{31} +81.3370i q^{32} +(-442.879 + 116.393i) q^{36} +(447.512 - 57.8235i) q^{39} +522.393i q^{41} +551.079 q^{46} -279.751i q^{47} +(-453.285 + 58.5694i) q^{48} +343.000 q^{49} -624.499i q^{50} +1472.80 q^{52} +(-263.559 - 649.479i) q^{54} -1555.56 q^{58} +815.291i q^{59} -734.811i q^{62} +297.321 q^{64} +(73.4477 + 568.432i) q^{69} -908.142i q^{71} +(-307.206 - 1168.93i) q^{72} -812.359 q^{73} +(644.164 - 83.2331i) q^{75} +(288.886 + 2235.77i) q^{78} +(634.803 - 358.421i) q^{81} -2609.87 q^{82} +(-207.324 - 1604.54i) q^{87} +1870.75i q^{92} +(757.950 - 97.9355i) q^{93} +1397.64 q^{94} +(-54.1595 - 419.155i) q^{96} +1713.63i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} - 18 q^{4} - 5 q^{6} + 38 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 - 18 * q^4 - 5 * q^6 + 38 * q^9	\( 4 q - 4 q^{3} - 18 q^{4} - 5 q^{6} + 38 q^{9} + 225 q^{12} - 148 q^{13} + 302 q^{16} + 79 q^{18} + 86 q^{24} - 500 q^{25} - 520 q^{27} - 688 q^{31} - 999 q^{36} + 976 q^{39} + 1058 q^{46} - 509 q^{48} + 1372 q^{49} + 3150 q^{52} - 506 q^{54} - 3182 q^{58} + 1588 q^{64} + 1058 q^{69} - 448 q^{72} - 2452 q^{73} + 500 q^{75} + 599 q^{78} + 14 q^{81} - 5306 q^{82} + 1048 q^{87} + 274 q^{93} + 2650 q^{94} + 207 q^{96}+O(q^{100}) \) 4 * q - 4 * q^3 - 18 * q^4 - 5 * q^6 + 38 * q^9 + 225 * q^12 - 148 * q^13 + 302 * q^16 + 79 * q^18 + 86 * q^24 - 500 * q^25 - 520 * q^27 - 688 * q^31 - 999 * q^36 + 976 * q^39 + 1058 * q^46 - 509 * q^48 + 1372 * q^49 + 3150 * q^52 - 506 * q^54 - 3182 * q^58 + 1588 * q^64 + 1058 * q^69 - 448 * q^72 - 2452 * q^73 + 500 * q^75 + 599 * q^78 + 14 * q^81 - 5306 * q^82 + 1048 * q^87 + 274 * q^93 + 2650 * q^94 + 207 * q^96

Introduction

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