LMFDB - Embedded newform 3240.2.f.k.649.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3240,2,Mod(649,3240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3240, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3240.649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3240 = 2^{3} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3240.f (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:	$25.8715302549$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} - x^{14} + 26 x^{13} - 44 x^{12} + 6 x^{11} + 225 x^{10} - 174 x^{9} + 102 x^{8} + \cdots + 390625 $ x^16 - 2x^15 - x^14 + 26x^13 - 44x^12 + 6x^11 + 225x^10 - 174x^9 + 102x^8 - 870x^7 + 5625x^6 + 750x^5 - 27500x^4 + 81250x^3 - 15625x^2 - 156250x + 390625
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 360)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.2
Root			$-2.23546 + 0.0521238i$ of defining polynomial
Character	$\chi$	$=$	3240.649
Dual form			3240.2.f.k.649.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.23546 + 0.0521238i) q^{5} -4.10221i q^{7} +O(q^{10})$	$q+(-2.23546 + 0.0521238i) q^{5} -4.10221i q^{7} -6.09464 q^{11} -5.13891i q^{13} +2.73647i q^{17} -1.80595 q^{19} +0.672542i q^{23} +(4.99457 - 0.233041i) q^{25} -3.50669 q^{29} -6.60133 q^{31} +(0.213823 + 9.17033i) q^{35} -4.44214i q^{37} +4.17846 q^{41} +4.63223i q^{43} +1.59647i q^{47} -9.82815 q^{49} -8.02220i q^{53} +(13.6243 - 0.317676i) q^{55} +5.30022 q^{59} +6.77883 q^{61} +(0.267860 + 11.4878i) q^{65} +11.3465i q^{67} +13.3686 q^{71} +6.50669i q^{73} +25.0015i q^{77} -2.99005 q^{79} +4.28444i q^{83} +(-0.142635 - 6.11728i) q^{85} +7.19878 q^{89} -21.0809 q^{91} +(4.03712 - 0.0941328i) q^{95} +3.85319i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 2 q^{5}+O(q^{10}) $ 16 * q + 2 * q^5	$ 16 q + 2 q^{5} - 16 q^{11} + 4 q^{19} + 6 q^{25} - 20 q^{29} + 12 q^{31} + 2 q^{35} + 8 q^{41} - 36 q^{49} + 10 q^{55} + 20 q^{61} - 10 q^{65} + 8 q^{71} - 4 q^{79} - 36 q^{85} - 48 q^{89} - 4 q^{91} + 32 q^{95}+O(q^{100}) $ 16 * q + 2 * q^5 - 16 * q^11 + 4 * q^19 + 6 * q^25 - 20 * q^29 + 12 * q^31 + 2 * q^35 + 8 * q^41 - 36 * q^49 + 10 * q^55 + 20 * q^61 - 10 * q^65 + 8 * q^71 - 4 * q^79 - 36 * q^85 - 48 * q^89 - 4 * q^91 + 32 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3240\mathbb{Z}\right)^\times$.

$n$	$1297$	$1621$	$2431$	$3161$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−2.23546	+	0.0521238i	−0.999728	+	0.0233105i
$5$	−2.23546	+	0.0521238i	−0.999728	+	0.0233105i
$6$		0			0
$7$		−	4.10221i		−	1.55049i	−0.631660	−	0.775245i	$-0.717626\pi$
$7$		−	4.10221i		−	1.55049i	0.631660	−	0.775245i	$-0.282374\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−6.09464			−1.83760			−0.918802	−	0.394719i	$-0.870842\pi$
$11$	−6.09464			−1.83760			−0.918802	+	0.394719i	$0.870842\pi$
$12$		0			0
$13$		−	5.13891i		−	1.42528i	−0.701531	−	0.712639i	$-0.747500\pi$
$13$		−	5.13891i		−	1.42528i	0.701531	−	0.712639i	$-0.252500\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			2.73647i			0.663692i	0.943334	+	0.331846i	$0.107671\pi$
$17$			2.73647i			0.663692i	−0.943334	+	0.331846i	$0.892329\pi$
$18$		0			0
$19$	−1.80595			−0.414313			−0.207156	−	0.978308i	$-0.566421\pi$
$19$	−1.80595			−0.414313			−0.207156	+	0.978308i	$0.566421\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			0.672542i			0.140235i	0.997539	+	0.0701174i	$0.0223374\pi$
$23$			0.672542i			0.140235i	−0.997539	+	0.0701174i	$0.977663\pi$
$24$		0			0
$25$	4.99457	−	0.233041i	0.998913	−	0.0466082i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−3.50669			−0.651176			−0.325588	−	0.945512i	$-0.605562\pi$
$29$	−3.50669			−0.651176			−0.325588	+	0.945512i	$0.605562\pi$
$30$		0			0
$31$	−6.60133			−1.18563			−0.592817	−	0.805337i	$-0.701984\pi$
$31$	−6.60133			−1.18563			−0.592817	+	0.805337i	$0.701984\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	0.213823	+	9.17033i	0.0361426	+	1.55007i
$36$		0			0
$37$		−	4.44214i		−	0.730283i	−0.930952	−	0.365141i	$-0.881021\pi$
$37$		−	4.44214i		−	0.730283i	0.930952	−	0.365141i	$-0.118979\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	4.17846			0.652566			0.326283	−	0.945272i	$-0.394204\pi$
$41$	4.17846			0.652566			0.326283	+	0.945272i	$0.394204\pi$
$42$		0			0
$43$			4.63223i			0.706408i	0.935546	+	0.353204i	$0.114908\pi$
$43$			4.63223i			0.706408i	−0.935546	+	0.353204i	$0.885092\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			1.59647i			0.232869i	0.993198	+	0.116434i	$0.0371465\pi$
$47$			1.59647i			0.232869i	−0.993198	+	0.116434i	$0.962854\pi$
$48$		0			0
$49$	−9.82815			−1.40402
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	8.02220i		−	1.10193i	−0.834527	−	0.550967i	$-0.814259\pi$
$53$		−	8.02220i		−	1.10193i	0.834527	−	0.550967i	$-0.185741\pi$
$54$		0			0
$55$	13.6243	−	0.317676i	1.83710	−	0.0428354i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	5.30022			0.690030			0.345015	−	0.938597i	$-0.387874\pi$
$59$	5.30022			0.690030			0.345015	+	0.938597i	$0.387874\pi$
$60$		0			0
$61$	6.77883			0.867940			0.433970	−	0.900927i	$-0.357112\pi$
$61$	6.77883			0.867940			0.433970	+	0.900927i	$0.357112\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	0.267860	+	11.4878i	0.0332239	+	1.42489i
$66$		0			0
$67$			11.3465i			1.38620i	0.720842	+	0.693100i	$0.243755\pi$
$67$			11.3465i			1.38620i	−0.720842	+	0.693100i	$0.756245\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	13.3686			1.58656			0.793280	−	0.608857i	$-0.208372\pi$
$71$	13.3686			1.58656			0.793280	+	0.608857i	$0.208372\pi$
$72$		0			0
$73$			6.50669i			0.761550i	0.924668	+	0.380775i	$0.124343\pi$
$73$			6.50669i			0.761550i	−0.924668	+	0.380775i	$0.875657\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			25.0015i			2.84919i
$78$		0			0
$79$	−2.99005			−0.336407			−0.168203	−	0.985752i	$-0.553797\pi$
$79$	−2.99005			−0.336407			−0.168203	+	0.985752i	$0.553797\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			4.28444i			0.470278i	0.971962	+	0.235139i	$0.0755546\pi$
$83$			4.28444i			0.470278i	−0.971962	+	0.235139i	$0.924445\pi$
$84$		0			0
$85$	−0.142635	−	6.11728i	−0.0154710	−	0.663512i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	7.19878			0.763070			0.381535	−	0.924354i	$-0.375396\pi$
$89$	7.19878			0.763070			0.381535	+	0.924354i	$0.375396\pi$
$90$		0			0
$91$	−21.0809			−2.20988
$92$		0			0
$93$		0			0
$94$		0			0
$95$	4.03712	−	0.0941328i	0.414200	−	0.00965782i
$96$		0			0
$97$			3.85319i			0.391232i	0.980681	+	0.195616i	$0.0626706\pi$
$97$			3.85319i			0.391232i	−0.980681	+	0.195616i	$0.937329\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−9.74966			−0.970128			−0.485064	−	0.874479i	$-0.661204\pi$
$101$	−9.74966			−0.970128			−0.485064	+	0.874479i	$0.661204\pi$
$102$		0			0
$103$			11.1389i			1.09755i	0.835970	+	0.548775i	$0.184906\pi$
$103$			11.1389i			1.09755i	−0.835970	+	0.548775i	$0.815094\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			1.19494i			0.115519i	0.998331	+	0.0577595i	$0.0183956\pi$
$107$			1.19494i			0.115519i	−0.998331	+	0.0577595i	$0.981604\pi$
$108$		0			0
$109$	9.05323			0.867143			0.433571	−	0.901119i	$-0.357253\pi$
$109$	9.05323			0.867143			0.433571	+	0.901119i	$0.357253\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	7.91795i		−	0.744858i	−0.928061	−	0.372429i	$-0.878525\pi$
$113$		−	7.91795i		−	0.744858i	0.928061	−	0.372429i	$-0.121475\pi$
$114$		0			0
$115$	−0.0350554	−	1.50344i	−0.00326894	−	0.140197i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	11.2256			1.02905
$120$		0			0
$121$	26.1447			2.37679
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−11.1530	+	0.781290i	−0.997555	+	0.0698807i
$126$		0			0
$127$		−	9.00959i		−	0.799471i	−0.916630	−	0.399736i	$-0.869102\pi$
$127$		−	9.00959i		−	0.799471i	0.916630	−	0.399736i	$-0.130898\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−2.53945			−0.221872			−0.110936	−	0.993828i	$-0.535385\pi$
$131$	−2.53945			−0.221872			−0.110936	+	0.993828i	$0.535385\pi$
$132$		0			0
$133$			7.40838i			0.642388i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			8.86656i			0.757522i	0.925495	+	0.378761i	$0.123650\pi$
$137$			8.86656i			0.757522i	−0.925495	+	0.378761i	$0.876350\pi$
$138$		0			0
$139$	−14.9946			−1.27183			−0.635913	−	0.771760i	$-0.719376\pi$
$139$	−14.9946			−1.27183			−0.635913	+	0.771760i	$0.719376\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			31.3198i			2.61910i
$144$		0			0
$145$	7.83906	−	0.182782i	0.650999	−	0.0151792i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−13.7963			−1.13023			−0.565117	−	0.825011i	$-0.691169\pi$
$149$	−13.7963			−1.13023			−0.565117	+	0.825011i	$0.691169\pi$
$150$		0			0
$151$	−2.60625			−0.212093			−0.106047	−	0.994361i	$-0.533819\pi$
$151$	−2.60625			−0.212093			−0.106047	+	0.994361i	$0.533819\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	14.7570	−	0.344086i	1.18531	−	0.0276377i
$156$		0			0
$157$		−	14.9509i		−	1.19321i	−0.802536	−	0.596604i	$-0.796516\pi$
$157$		−	14.9509i		−	1.19321i	0.802536	−	0.596604i	$-0.203484\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	2.75891			0.217433
$162$		0			0
$163$			9.49019i			0.743329i	0.928367	+	0.371664i	$0.121213\pi$
$163$			9.49019i			0.743329i	−0.928367	+	0.371664i	$0.878787\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	0.376417i		−	0.0291280i	−0.999894	−	0.0145640i	$-0.995364\pi$
$167$		−	0.376417i		−	0.0291280i	0.999894	−	0.0145640i	$-0.00463603\pi$
$168$		0			0
$169$	−13.4084			−1.03142
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	0.292513i		−	0.0222394i	−0.999938	−	0.0111197i	$-0.996460\pi$
$173$		−	0.292513i		−	0.0222394i	0.999938	−	0.0111197i	$-0.00353958\pi$
$174$		0			0
$175$	−0.955984	−	20.4888i	−0.0722656	−	1.54881i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−23.4931			−1.75596			−0.877978	−	0.478702i	$-0.841108\pi$
$179$	−23.4931			−1.75596			−0.877978	+	0.478702i	$0.841108\pi$
$180$		0			0
$181$	−2.37931			−0.176853			−0.0884263	−	0.996083i	$-0.528184\pi$
$181$	−2.37931			−0.176853			−0.0884263	+	0.996083i	$0.528184\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	0.231541	+	9.93022i	0.0170232	+	0.730084i
$186$		0			0
$187$		−	16.6778i		−	1.21960i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−11.3734			−0.822952			−0.411476	−	0.911421i	$-0.634987\pi$
$191$	−11.3734			−0.822952			−0.411476	+	0.911421i	$0.634987\pi$
$192$		0			0
$193$			21.6889i			1.56120i	0.625031	+	0.780600i	$0.285086\pi$
$193$			21.6889i			1.56120i	−0.625031	+	0.780600i	$0.714914\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			14.2214i			1.01323i	0.862171	+	0.506617i	$0.169104\pi$
$197$			14.2214i			1.01323i	−0.862171	+	0.506617i	$0.830896\pi$
$198$		0			0
$199$	−18.3820			−1.30306			−0.651532	−	0.758621i	$-0.725873\pi$
$199$	−18.3820			−1.30306			−0.651532	+	0.758621i	$0.725873\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			14.3852i			1.00964i
$204$		0			0
$205$	−9.34078	+	0.217797i	−0.652388	+	0.0152116i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	11.0066			0.761343
$210$		0			0
$211$	14.2337			0.979885			0.489943	−	0.871755i	$-0.337018\pi$
$211$	14.2337			0.979885			0.489943	+	0.871755i	$0.337018\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−0.241449	−	10.3552i	−0.0164667	−	0.706216i
$216$		0			0
$217$			27.0801i			1.83831i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	14.0625			0.945946
$222$		0			0
$223$		−	3.51246i		−	0.235212i	−0.993060	−	0.117606i	$-0.962478\pi$
$223$		−	3.51246i		−	0.235212i	0.993060	−	0.117606i	$-0.0375220\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			8.07805i			0.536159i	0.963397	+	0.268080i	$0.0863890\pi$
$227$			8.07805i			0.536159i	−0.963397	+	0.268080i	$0.913611\pi$
$228$		0			0
$229$	13.9437			0.921425			0.460713	−	0.887549i	$-0.347594\pi$
$229$	13.9437			0.921425			0.460713	+	0.887549i	$0.347594\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			5.09746i			0.333946i	0.985961	+	0.166973i	$0.0533992\pi$
$233$			5.09746i			0.333946i	−0.985961	+	0.166973i	$0.946601\pi$
$234$		0			0
$235$	−0.0832139	−	3.56884i	−0.00542828	−	0.232805i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−9.61864			−0.622178			−0.311089	−	0.950381i	$-0.600694\pi$
$239$	−9.61864			−0.622178			−0.311089	+	0.950381i	$0.600694\pi$
$240$		0			0
$241$	24.6395			1.58717			0.793586	−	0.608458i	$-0.208212\pi$
$241$	24.6395			1.58717			0.793586	+	0.608458i	$0.208212\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	21.9704	−	0.512280i	1.40364	−	0.0327284i
$246$		0			0
$247$			9.28061i			0.590511i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−16.2882			−1.02810			−0.514052	−	0.857759i	$-0.671856\pi$
$251$	−16.2882			−1.02810			−0.514052	+	0.857759i	$0.671856\pi$
$252$		0			0
$253$		−	4.09891i		−	0.257696i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	17.0770i		−	1.06524i	−0.846356	−	0.532618i	$-0.821208\pi$
$257$		−	17.0770i		−	1.06524i	0.846356	−	0.532618i	$-0.178792\pi$
$258$		0			0
$259$	−18.2226			−1.13230
$260$		0			0
$261$		0			0
$262$		0			0
$263$			17.6252i			1.08682i	0.839468	+	0.543409i	$0.182867\pi$
$263$			17.6252i			1.08682i	−0.839468	+	0.543409i	$0.817133\pi$
$264$		0			0
$265$	0.418147	+	17.9333i	0.0256866	+	1.10163i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−11.9751			−0.730132			−0.365066	−	0.930982i	$-0.618954\pi$
$269$	−11.9751			−0.730132			−0.365066	+	0.930982i	$0.618954\pi$
$270$		0			0
$271$	−1.08606			−0.0659734			−0.0329867	−	0.999456i	$-0.510502\pi$
$271$	−1.08606			−0.0659734			−0.0329867	+	0.999456i	$0.510502\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−30.4401	+	1.42030i	−1.83561	+	0.0856475i
$276$		0			0
$277$			15.8081i			0.949817i	0.880035	+	0.474908i	$0.157519\pi$
$277$			15.8081i			0.949817i	−0.880035	+	0.474908i	$0.842481\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	1.46295			0.0872724			0.0436362	−	0.999047i	$-0.486106\pi$
$281$	1.46295			0.0872724			0.0436362	+	0.999047i	$0.486106\pi$
$282$		0			0
$283$		−	14.0122i		−	0.832937i	−0.909150	−	0.416468i	$-0.863268\pi$
$283$		−	14.0122i		−	0.832937i	0.909150	−	0.416468i	$-0.136732\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	17.1409i		−	1.01180i
$288$		0			0
$289$	9.51172			0.559513
$290$		0			0
$291$		0			0
$292$		0			0
$293$			19.7020i			1.15100i	0.817800	+	0.575502i	$0.195193\pi$
$293$			19.7020i			1.15100i	−0.817800	+	0.575502i	$0.804807\pi$
$294$		0			0
$295$	−11.8484	+	0.276267i	−0.689842	+	0.0160849i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	3.45614			0.199874
$300$		0			0
$301$	19.0024			1.09528
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−15.1538	+	0.353338i	−0.867705	+	0.0202321i
$306$		0			0
$307$		−	21.0537i		−	1.20160i	−0.799400	−	0.600799i	$-0.794849\pi$
$307$		−	21.0537i		−	1.20160i	0.799400	−	0.600799i	$-0.205151\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−0.523747			−0.0296990			−0.0148495	−	0.999890i	$-0.504727\pi$
$311$	−0.523747			−0.0296990			−0.0148495	+	0.999890i	$0.504727\pi$
$312$		0			0
$313$			9.12311i			0.515669i	0.966189	+	0.257834i	$0.0830089\pi$
$313$			9.12311i			0.515669i	−0.966189	+	0.257834i	$0.916991\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	15.6256i		−	0.877620i	−0.898580	−	0.438810i	$-0.855400\pi$
$317$		−	15.6256i		−	0.877620i	0.898580	−	0.438810i	$-0.144600\pi$
$318$		0			0
$319$	21.3720			1.19660
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	4.94193i		−	0.274976i
$324$		0			0
$325$	−1.19758	−	25.6666i	−0.0664297	−	1.42373i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	6.54905			0.361061
$330$		0			0
$331$	−26.3384			−1.44769			−0.723845	−	0.689962i	$-0.757627\pi$
$331$	−26.3384			−1.44769			−0.723845	+	0.689962i	$0.757627\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−0.591424	−	25.3647i	−0.0323129	−	1.38582i
$336$		0			0
$337$		−	27.2424i		−	1.48399i	−0.670406	−	0.741994i	$-0.733880\pi$
$337$		−	27.2424i		−	1.48399i	0.670406	−	0.741994i	$-0.266120\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	40.2328			2.17873
$342$		0			0
$343$			11.6017i			0.626431i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	16.3695i		−	0.878761i	−0.898301	−	0.439380i	$-0.855198\pi$
$347$		−	16.3695i		−	0.878761i	0.898301	−	0.439380i	$-0.144802\pi$
$348$		0			0
$349$	18.2945			0.979282			0.489641	−	0.871924i	$-0.337128\pi$
$349$	18.2945			0.979282			0.489641	+	0.871924i	$0.337128\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			20.3978i			1.08567i	0.839840	+	0.542833i	$0.182648\pi$
$353$			20.3978i			1.08567i	−0.839840	+	0.542833i	$0.817352\pi$
$354$		0			0
$355$	−29.8850	+	0.696821i	−1.58613	+	0.0369834i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	9.33006			0.492422			0.246211	−	0.969216i	$-0.420814\pi$
$359$	9.33006			0.492422			0.246211	+	0.969216i	$0.420814\pi$
$360$		0			0
$361$	−15.7386			−0.828345
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−0.339153	−	14.5454i	−0.0177521	−	0.761343i
$366$		0			0
$367$			9.16288i			0.478299i	0.970983	+	0.239149i	$0.0768685\pi$
$367$			9.16288i			0.478299i	−0.970983	+	0.239149i	$0.923131\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−32.9088			−1.70854
$372$		0			0
$373$			24.3601i			1.26132i	0.776060	+	0.630659i	$0.217215\pi$
$373$			24.3601i			1.26132i	−0.776060	+	0.630659i	$0.782785\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			18.0206i			0.928107i
$378$		0			0
$379$	7.84198			0.402815			0.201408	−	0.979508i	$-0.435448\pi$
$379$	7.84198			0.402815			0.201408	+	0.979508i	$0.435448\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			30.8152i			1.57458i	0.616580	+	0.787292i	$0.288518\pi$
$383$			30.8152i			1.57458i	−0.616580	+	0.787292i	$0.711482\pi$
$384$		0			0
$385$	−1.30317	−	55.8899i	−0.0664159	−	2.84841i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	4.74040			0.240348			0.120174	−	0.992753i	$-0.461655\pi$
$389$	4.74040			0.240348			0.120174	+	0.992753i	$0.461655\pi$
$390$		0			0
$391$	−1.84039			−0.0930727
$392$		0			0
$393$		0			0
$394$		0			0
$395$	6.68413	−	0.155853i	0.336315	−	0.00784179i
$396$		0			0
$397$		−	36.8417i		−	1.84903i	−0.381146	−	0.924515i	$-0.624471\pi$
$397$		−	36.8417i		−	1.84903i	0.381146	−	0.924515i	$-0.375529\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−7.40497			−0.369787			−0.184893	−	0.982759i	$-0.559194\pi$
$401$	−7.40497			−0.369787			−0.184893	+	0.982759i	$0.559194\pi$
$402$		0			0
$403$			33.9237i			1.68986i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			27.0732i			1.34197i
$408$		0			0
$409$	0.608016			0.0300645			0.0150322	−	0.999887i	$-0.495215\pi$
$409$	0.608016			0.0300645			0.0150322	+	0.999887i	$0.495215\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	21.7426i		−	1.06988i
$414$		0			0
$415$	−0.223321	−	9.57769i	−0.0109624	−	0.470150i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	12.3668			0.604157			0.302078	−	0.953283i	$-0.402320\pi$
$419$	12.3668			0.604157			0.302078	+	0.953283i	$0.402320\pi$
$420$		0			0
$421$	12.9958			0.633375			0.316688	−	0.948530i	$-0.397429\pi$
$421$	12.9958			0.633375			0.316688	+	0.948530i	$0.397429\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	0.637711	+	13.6675i	0.0309335	+	0.662971i
$426$		0			0
$427$		−	27.8082i		−	1.34573i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−4.47990			−0.215789			−0.107895	−	0.994162i	$-0.534411\pi$
$431$	−4.47990			−0.215789			−0.107895	+	0.994162i	$0.534411\pi$
$432$		0			0
$433$			16.5313i			0.794443i	0.917723	+	0.397222i	$0.130026\pi$
$433$			16.5313i			0.794443i	−0.917723	+	0.397222i	$0.869974\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	1.21458i		−	0.0581011i
$438$		0			0
$439$	23.3853			1.11612			0.558060	−	0.829801i	$-0.311546\pi$
$439$	23.3853			1.11612			0.558060	+	0.829801i	$0.311546\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	20.6024i		−	0.978850i	−0.872045	−	0.489425i	$-0.837207\pi$
$443$		−	20.6024i		−	0.978850i	0.872045	−	0.489425i	$-0.162793\pi$
$444$		0			0
$445$	−16.0926	+	0.375228i	−0.762862	+	0.0177875i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−13.1837			−0.622177			−0.311088	−	0.950381i	$-0.600694\pi$
$449$	−13.1837			−0.622177			−0.311088	+	0.950381i	$0.600694\pi$
$450$		0			0
$451$	−25.4662			−1.19916
$452$		0			0
$453$		0			0
$454$		0			0
$455$	47.1256	−	1.09882i	2.20928	−	0.0515133i
$456$		0			0
$457$		−	36.7063i		−	1.71705i	−0.512772	−	0.858525i	$-0.671381\pi$
$457$		−	36.7063i		−	1.71705i	0.512772	−	0.858525i	$-0.328619\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	15.3704			0.715869			0.357934	−	0.933747i	$-0.383481\pi$
$461$	15.3704			0.715869			0.357934	+	0.933747i	$0.383481\pi$
$462$		0			0
$463$		−	0.813669i		−	0.0378144i	−0.999821	−	0.0189072i	$-0.993981\pi$
$463$		−	0.813669i		−	0.0378144i	0.999821	−	0.0189072i	$-0.00601871\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	15.2103i		−	0.703850i	−0.936028	−	0.351925i	$-0.885527\pi$
$467$		−	15.2103i		−	0.703850i	0.936028	−	0.351925i	$-0.114473\pi$
$468$		0			0
$469$	46.5459			2.14929
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	28.2318i		−	1.29810i
$474$		0			0
$475$	−9.01993	+	0.420860i	−0.413863	+	0.0193104i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	7.40975			0.338560			0.169280	−	0.985568i	$-0.445856\pi$
$479$	7.40975			0.338560			0.169280	+	0.985568i	$0.445856\pi$
$480$		0			0
$481$	−22.8278			−1.04086
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−0.200843	−	8.61365i	−0.00911979	−	0.391126i
$486$		0			0
$487$			42.0092i			1.90362i	0.306692	+	0.951809i	$0.400778\pi$
$487$			42.0092i			1.90362i	−0.306692	+	0.951809i	$0.599222\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	7.68009			0.346597			0.173299	−	0.984869i	$-0.444557\pi$
$491$	7.68009			0.346597			0.173299	+	0.984869i	$0.444557\pi$
$492$		0			0
$493$		−	9.59596i		−	0.432180i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	54.8408i		−	2.45995i
$498$		0			0
$499$	−39.6725			−1.77599			−0.887993	−	0.459856i	$-0.847901\pi$
$499$	−39.6725			−1.77599			−0.887993	+	0.459856i	$0.847901\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	26.0215i		−	1.16024i	−0.814530	−	0.580121i	$-0.803005\pi$
$503$		−	26.0215i		−	1.16024i	0.814530	−	0.580121i	$-0.196995\pi$
$504$		0			0
$505$	21.7950	−	0.508189i	0.969864	−	0.0226141i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	14.1306			0.626328			0.313164	−	0.949699i	$-0.398611\pi$
$509$	14.1306			0.626328			0.313164	+	0.949699i	$0.398611\pi$
$510$		0			0
$511$	26.6918			1.18078
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−0.580602	−	24.9006i	−0.0255844	−	1.09725i
$516$		0			0
$517$		−	9.72990i		−	0.427921i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	13.0285			0.570788			0.285394	−	0.958410i	$-0.407876\pi$
$521$	13.0285			0.570788			0.285394	+	0.958410i	$0.407876\pi$
$522$		0			0
$523$			21.9922i			0.961651i	0.876816	+	0.480826i	$0.159663\pi$
$523$			21.9922i			0.961651i	−0.876816	+	0.480826i	$0.840337\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	18.0644i		−	0.786896i
$528$		0			0
$529$	22.5477			0.980334
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	21.4727i		−	0.930088i
$534$		0			0
$535$	−0.0622846	−	2.67123i	−0.00269280	−	0.115488i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	59.8991			2.58003
$540$		0			0
$541$	−24.7006			−1.06196			−0.530981	−	0.847384i	$-0.678177\pi$
$541$	−24.7006			−1.06196			−0.530981	+	0.847384i	$0.678177\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−20.2381	+	0.471889i	−0.866907	+	0.0202135i
$546$		0			0
$547$		−	11.3012i		−	0.483204i	−0.970375	−	0.241602i	$-0.922327\pi$
$547$		−	11.3012i		−	0.483204i	0.970375	−	0.241602i	$-0.0776729\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	6.33290			0.269790
$552$		0			0
$553$			12.2658i			0.521595i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			25.0264i			1.06040i	0.847872	+	0.530201i	$0.177884\pi$
$557$			25.0264i			1.06040i	−0.847872	+	0.530201i	$0.822116\pi$
$558$		0			0
$559$	23.8046			1.00683
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	27.3852i		−	1.15415i	−0.816692	−	0.577074i	$-0.804194\pi$
$563$		−	27.3852i		−	1.15415i	0.816692	−	0.577074i	$-0.195806\pi$
$564$		0			0
$565$	0.412713	+	17.7003i	0.0173630	+	0.744656i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−36.2360			−1.51909			−0.759545	−	0.650455i	$-0.774578\pi$
$569$	−36.2360			−1.51909			−0.759545	+	0.650455i	$0.774578\pi$
$570$		0			0
$571$	−11.7997			−0.493802			−0.246901	−	0.969041i	$-0.579412\pi$
$571$	−11.7997			−0.493802			−0.246901	+	0.969041i	$0.579412\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	0.156730	+	3.35906i	0.00653610	+	0.140082i
$576$		0			0
$577$		−	8.24270i		−	0.343148i	−0.985171	−	0.171574i	$-0.945115\pi$
$577$		−	8.24270i		−	0.343148i	0.985171	−	0.171574i	$-0.0548853\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	17.5757			0.729162
$582$		0			0
$583$			48.8924i			2.02492i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			43.1630i			1.78153i	0.454466	+	0.890764i	$0.349830\pi$
$587$			43.1630i			1.78153i	−0.454466	+	0.890764i	$0.650170\pi$
$588$		0			0
$589$	11.9217			0.491223
$590$		0			0
$591$		0			0
$592$		0			0
$593$			41.9976i			1.72464i	0.506367	+	0.862318i	$0.330988\pi$
$593$			41.9976i			1.72464i	−0.506367	+	0.862318i	$0.669012\pi$
$594$		0			0
$595$	−25.0944	+	0.585120i	−1.02877	+	0.0239876i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	1.69018			0.0690588			0.0345294	−	0.999404i	$-0.489007\pi$
$599$	1.69018			0.0690588			0.0345294	+	0.999404i	$0.489007\pi$
$600$		0			0
$601$	−28.2253			−1.15133			−0.575666	−	0.817685i	$-0.695257\pi$
$601$	−28.2253			−1.15133			−0.575666	+	0.817685i	$0.695257\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−58.4454	+	1.36276i	−2.37614	+	0.0554040i
$606$		0			0
$607$			15.1250i			0.613905i	0.951725	+	0.306952i	$0.0993092\pi$
$607$			15.1250i			0.613905i	−0.951725	+	0.306952i	$0.900691\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	8.20411			0.331903
$612$		0			0
$613$			33.1010i			1.33694i	0.743741	+	0.668468i	$0.233050\pi$
$613$			33.1010i			1.33694i	−0.743741	+	0.668468i	$0.766950\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	45.6077i		−	1.83610i	−0.396468	−	0.918048i	$-0.629764\pi$
$617$		−	45.6077i		−	1.83610i	0.396468	−	0.918048i	$-0.370236\pi$
$618$		0			0
$619$	−1.63919			−0.0658846			−0.0329423	−	0.999457i	$-0.510488\pi$
$619$	−1.63919			−0.0658846			−0.0329423	+	0.999457i	$0.510488\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	29.5309i		−	1.18313i
$624$		0			0
$625$	24.8914	−	2.32788i	0.995655	−	0.0931152i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	12.1558			0.484683
$630$		0			0
$631$	28.4741			1.13354			0.566769	−	0.823877i	$-0.308193\pi$
$631$	28.4741			1.13354			0.566769	+	0.823877i	$0.308193\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	0.469613	+	20.1406i	0.0186360	+	0.799254i
$636$		0			0
$637$			50.5060i			2.00112i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−42.9228			−1.69535			−0.847674	−	0.530517i	$-0.821998\pi$
$641$	−42.9228			−1.69535			−0.847674	+	0.530517i	$0.821998\pi$
$642$		0			0
$643$			50.4673i			1.99024i	0.0986876	+	0.995118i	$0.468536\pi$
$643$			50.4673i			1.99024i	−0.0986876	+	0.995118i	$0.531464\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	13.8898i		−	0.546064i	−0.962005	−	0.273032i	$-0.911974\pi$
$647$		−	13.8898i		−	0.546064i	0.962005	−	0.273032i	$-0.0880265\pi$
$648$		0			0
$649$	−32.3029			−1.26800
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	11.0874i		−	0.433884i	−0.976184	−	0.216942i	$-0.930392\pi$
$653$		−	11.0874i		−	0.433884i	0.976184	−	0.216942i	$-0.0696083\pi$
$654$		0			0
$655$	5.67683	−	0.132365i	0.221812	−	0.00517194i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−4.65546			−0.181351			−0.0906755	−	0.995880i	$-0.528903\pi$
$659$	−4.65546			−0.181351			−0.0906755	+	0.995880i	$0.528903\pi$
$660$		0			0
$661$	−42.1111			−1.63793			−0.818966	−	0.573842i	$-0.805452\pi$
$661$	−42.1111			−1.63793			−0.818966	+	0.573842i	$0.805452\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−0.386153	−	16.5611i	−0.0149744	−	0.642214i
$666$		0			0
$667$		−	2.35840i		−	0.0913175i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−41.3146			−1.59493
$672$		0			0
$673$			9.11131i			0.351215i	0.984460	+	0.175608i	$0.0561890\pi$
$673$			9.11131i			0.351215i	−0.984460	+	0.175608i	$0.943811\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	16.3963i		−	0.630160i	−0.949065	−	0.315080i	$-0.897969\pi$
$677$		−	16.3963i		−	0.630160i	0.949065	−	0.315080i	$-0.102031\pi$
$678$		0			0
$679$	15.8066			0.606601
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	31.3665i		−	1.20021i	−0.799923	−	0.600103i	$-0.795126\pi$
$683$		−	31.3665i		−	1.20021i	0.799923	−	0.600103i	$-0.204874\pi$
$684$		0			0
$685$	−0.462159	−	19.8209i	−0.0176582	−	0.757316i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−41.2254			−1.57056
$690$		0			0
$691$	−3.12177			−0.118758			−0.0593788	−	0.998236i	$-0.518912\pi$
$691$	−3.12177			−0.118758			−0.0593788	+	0.998236i	$0.518912\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	33.5199	−	0.781576i	1.27148	−	0.0296469i
$696$		0			0
$697$			11.4342i			0.433103i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−36.9206			−1.39447			−0.697236	−	0.716841i	$-0.745587\pi$
$701$	−36.9206			−1.39447			−0.697236	+	0.716841i	$0.745587\pi$
$702$		0			0
$703$			8.02227i			0.302566i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			39.9952i			1.50417i
$708$		0			0
$709$	33.3089			1.25094			0.625471	−	0.780248i	$-0.284907\pi$
$709$	33.3089			1.25094			0.625471	+	0.780248i	$0.284907\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	4.43968i		−	0.166267i
$714$		0			0
$715$	−1.63251	−	70.0143i	−0.0610523	−	2.61839i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−40.6734			−1.51686			−0.758431	−	0.651753i	$-0.774034\pi$
$719$	−40.6734			−1.51686			−0.758431	+	0.651753i	$0.774034\pi$
$720$		0			0
$721$	45.6942			1.70174
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−17.5144	+	0.817203i	−0.650468	+	0.0303502i
$726$		0			0
$727$		−	33.3980i		−	1.23866i	−0.785129	−	0.619332i	$-0.787403\pi$
$727$		−	33.3980i		−	1.23866i	0.785129	−	0.619332i	$-0.212597\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−12.6760			−0.468837
$732$		0			0
$733$			53.0590i			1.95978i	0.199542	+	0.979889i	$0.436054\pi$
$733$			53.0590i			1.95978i	−0.199542	+	0.979889i	$0.563946\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	69.1531i		−	2.54729i
$738$		0			0
$739$	9.18128			0.337739			0.168869	−	0.985638i	$-0.445988\pi$
$739$	9.18128			0.337739			0.168869	+	0.985638i	$0.445988\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	36.3147i		−	1.33226i	−0.745838	−	0.666128i	$-0.767951\pi$
$743$		−	36.3147i		−	1.33226i	0.745838	−	0.666128i	$-0.232049\pi$
$744$		0			0
$745$	30.8410	−	0.719113i	1.12993	−	0.0263463i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	4.90189			0.179111
$750$		0			0
$751$	−14.5971			−0.532657			−0.266328	−	0.963882i	$-0.585811\pi$
$751$	−14.5971			−0.532657			−0.266328	+	0.963882i	$0.585811\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	5.82616	−	0.135847i	0.212036	−	0.00494399i
$756$		0			0
$757$			44.8778i			1.63111i	0.578677	+	0.815556i	$0.303569\pi$
$757$			44.8778i			1.63111i	−0.578677	+	0.815556i	$0.696431\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	44.8976			1.62754			0.813769	−	0.581188i	$-0.197412\pi$
$761$	44.8976			1.62754			0.813769	+	0.581188i	$0.197412\pi$
$762$		0			0
$763$		−	37.1383i		−	1.34450i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	27.2374i		−	0.983484i
$768$		0			0
$769$	16.8636			0.608118			0.304059	−	0.952653i	$-0.401658\pi$
$769$	16.8636			0.608118			0.304059	+	0.952653i	$0.401658\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	41.8399i		−	1.50488i	−0.658663	−	0.752438i	$-0.728878\pi$
$773$		−	41.8399i		−	1.50488i	0.658663	−	0.752438i	$-0.271122\pi$
$774$		0			0
$775$	−32.9708	+	1.53838i	−1.18435	+	0.0552603i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−7.54608			−0.270366
$780$		0			0
$781$	−81.4768			−2.91547
$782$		0			0
$783$		0			0
$784$		0			0
$785$	0.779295	+	33.4221i	0.0278142	+	1.19288i
$786$		0			0
$787$			32.5358i			1.15978i	0.814696	+	0.579888i	$0.196904\pi$
$787$			32.5358i			1.15978i	−0.814696	+	0.579888i	$0.803096\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−32.4811			−1.15490
$792$		0			0
$793$		−	34.8358i		−	1.23706i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			39.1264i			1.38593i	0.720971	+	0.692965i	$0.243696\pi$
$797$			39.1264i			1.38593i	−0.720971	+	0.692965i	$0.756304\pi$
$798$		0			0
$799$	−4.36869			−0.154553
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	39.6559i		−	1.39943i
$804$		0			0
$805$	−6.16744	+	0.143805i	−0.217374	+	0.00506845i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−31.5260			−1.10840			−0.554198	−	0.832385i	$-0.686975\pi$
$809$	−31.5260			−1.10840			−0.554198	+	0.832385i	$0.686975\pi$
$810$		0			0
$811$	−6.54430			−0.229801			−0.114901	−	0.993377i	$-0.536655\pi$
$811$	−6.54430			−0.229801			−0.114901	+	0.993377i	$0.536655\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−0.494664	−	21.2149i	−0.0173273	−	0.743127i
$816$		0			0
$817$		−	8.36556i		−	0.292674i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	53.3810			1.86301			0.931505	−	0.363728i	$-0.118496\pi$
$821$	53.3810			1.86301			0.931505	+	0.363728i	$0.118496\pi$
$822$		0			0
$823$		−	37.7588i		−	1.31619i	−0.752935	−	0.658095i	$-0.771362\pi$
$823$		−	37.7588i		−	1.31619i	0.752935	−	0.658095i	$-0.228638\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	6.15055i		−	0.213875i	−0.994266	−	0.106938i	$-0.965895\pi$
$827$		−	6.15055i		−	0.213875i	0.994266	−	0.106938i	$-0.0341045\pi$
$828$		0			0
$829$	−43.1456			−1.49851			−0.749255	−	0.662282i	$-0.769588\pi$
$829$	−43.1456			−1.49851			−0.749255	+	0.662282i	$0.769588\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	26.8945i		−	0.931838i
$834$		0			0
$835$	0.0196202	+	0.841464i	0.000678987	+	0.0291201i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−35.7855			−1.23545			−0.617726	−	0.786394i	$-0.711946\pi$
$839$	−35.7855			−1.23545			−0.617726	+	0.786394i	$0.711946\pi$
$840$		0			0
$841$	−16.7031			−0.575970
$842$		0			0
$843$		0			0
$844$		0			0
$845$	29.9740	−	0.698898i	1.03114	−	0.0240428i
$846$		0			0
$847$		−	107.251i		−	3.68519i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	2.98753			0.102411
$852$		0			0
$853$			15.1868i			0.519987i	0.965610	+	0.259993i	$0.0837204\pi$
$853$			15.1868i			0.519987i	−0.965610	+	0.259993i	$0.916280\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			1.06162i			0.0362643i	0.999836	+	0.0181321i	$0.00577196\pi$
$857$			1.06162i			0.0362643i	−0.999836	+	0.0181321i	$0.994228\pi$
$858$		0			0
$859$	−1.20807			−0.0412188			−0.0206094	−	0.999788i	$-0.506561\pi$
$859$	−1.20807			−0.0412188			−0.0206094	+	0.999788i	$0.506561\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	13.2305i		−	0.450373i	−0.974316	−	0.225186i	$-0.927701\pi$
$863$		−	13.2305i		−	0.450373i	0.974316	−	0.225186i	$-0.0722991\pi$
$864$		0			0
$865$	0.0152469	+	0.653901i	0.000518409	+	0.0222333i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	18.2233			0.618182
$870$		0			0
$871$	58.3088			1.97572
$872$		0			0
$873$		0			0
$874$		0			0
$875$	3.20502	+	45.7520i	0.108349	+	1.54670i
$876$		0			0
$877$		−	40.8886i		−	1.38071i	−0.723471	−	0.690355i	$-0.757455\pi$
$877$		−	40.8886i		−	1.38071i	0.723471	−	0.690355i	$-0.242545\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	23.2103			0.781975			0.390988	−	0.920396i	$-0.372133\pi$
$881$	23.2103			0.781975			0.390988	+	0.920396i	$0.372133\pi$
$882$		0			0
$883$		−	20.1285i		−	0.677377i	−0.940899	−	0.338689i	$-0.890017\pi$
$883$		−	20.1285i		−	0.677377i	0.940899	−	0.338689i	$-0.109983\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	1.09260i		−	0.0366858i	−0.999832	−	0.0183429i	$-0.994161\pi$
$887$		−	1.09260i		−	0.0366858i	0.999832	−	0.0183429i	$-0.00583905\pi$
$888$		0			0
$889$	−36.9592			−1.23957
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	2.88314i		−	0.0964805i
$894$		0			0
$895$	52.5178	−	1.22455i	1.75548	−	0.0409321i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	23.1488			0.772056
$900$		0			0
$901$	21.9525			0.731345
$902$		0			0
$903$		0			0
$904$		0			0
$905$	5.31885	−	0.124019i	0.176805	−	0.00412252i
$906$		0			0
$907$			4.15255i			0.137883i	0.997621	+	0.0689416i	$0.0219622\pi$
$907$			4.15255i			0.137883i	−0.997621	+	0.0689416i	$0.978038\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	28.0748			0.930160			0.465080	−	0.885269i	$-0.346026\pi$
$911$	28.0748			0.930160			0.465080	+	0.885269i	$0.346026\pi$
$912$		0			0
$913$		−	26.1121i		−	0.864185i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			10.4173i			0.344011i
$918$		0			0
$919$	18.0993			0.597041			0.298520	−	0.954403i	$-0.403507\pi$
$919$	18.0993			0.597041			0.298520	+	0.954403i	$0.403507\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	68.7000i		−	2.26129i
$924$		0			0
$925$	−1.03520	−	22.1865i	−0.0340372	−	0.729489i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	29.0265			0.952328			0.476164	−	0.879357i	$-0.342027\pi$
$929$	29.0265			0.952328			0.476164	+	0.879357i	$0.342027\pi$
$930$		0			0
$931$	17.7491			0.581704
$932$		0			0
$933$		0			0
$934$		0			0
$935$	0.869311	+	37.2826i	0.0284295	+	1.21927i
$936$		0			0
$937$		−	26.8353i		−	0.876672i	−0.898811	−	0.438336i	$-0.855568\pi$
$937$		−	26.8353i		−	0.876672i	0.898811	−	0.438336i	$-0.144432\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−41.5393			−1.35414			−0.677071	−	0.735918i	$-0.736751\pi$
$941$	−41.5393			−1.35414			−0.677071	+	0.735918i	$0.736751\pi$
$942$		0			0
$943$			2.81019i			0.0915124i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	43.9716i		−	1.42889i	−0.699694	−	0.714443i	$-0.746680\pi$
$947$		−	43.9716i		−	1.42889i	0.699694	−	0.714443i	$-0.253320\pi$
$948$		0			0
$949$	33.4373			1.08542
$950$		0			0
$951$		0			0
$952$		0			0
$953$			19.3790i			0.627749i	0.949465	+	0.313874i	$0.101627\pi$
$953$			19.3790i			0.627749i	−0.949465	+	0.313874i	$0.898373\pi$
$954$		0			0
$955$	25.4248	−	0.592825i	0.822728	−	0.0191834i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	36.3725			1.17453
$960$		0			0
$961$	12.5776			0.405729
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−1.13051	−	48.4846i	−0.0363923	−	1.56078i
$966$		0			0
$967$		−	1.89520i		−	0.0609456i	−0.999536	−	0.0304728i	$-0.990299\pi$
$967$		−	1.89520i		−	0.0609456i	0.999536	−	0.0304728i	$-0.00970130\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−36.8043			−1.18111			−0.590553	−	0.806999i	$-0.701090\pi$
$971$	−36.8043			−1.18111			−0.590553	+	0.806999i	$0.701090\pi$
$972$		0			0
$973$			61.5111i			1.97196i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			39.5452i			1.26516i	0.774493	+	0.632582i	$0.218005\pi$
$977$			39.5452i			1.26516i	−0.774493	+	0.632582i	$0.781995\pi$
$978$		0			0
$979$	−43.8740			−1.40222
$980$		0			0
$981$		0			0
$982$		0			0
$983$			15.5070i			0.494597i	0.968939	+	0.247298i	$0.0795428\pi$
$983$			15.5070i			0.494597i	−0.968939	+	0.247298i	$0.920457\pi$
$984$		0			0
$985$	−0.741274	−	31.7914i	−0.0236190	−	1.01296i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−3.11537			−0.0990629
$990$		0			0
$991$	23.4814			0.745910			0.372955	−	0.927849i	$-0.378345\pi$
$991$	23.4814			0.745910			0.372955	+	0.927849i	$0.378345\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	41.0922	−	0.958137i	1.30271	−	0.0303750i
$996$		0			0
$997$		−	27.1601i		−	0.860169i	−0.902789	−	0.430084i	$-0.858484\pi$
$997$		−	27.1601i		−	0.860169i	0.902789	−	0.430084i	$-0.141516\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3240.2.f.k.649.2		16
3.2	odd	2		3240.2.f.i.649.15		16
5.4	even	2	inner	3240.2.f.k.649.1		16
9.2	odd	6		1080.2.bi.b.1009.6		32
9.4	even	3		360.2.bi.b.169.8	yes	32
9.5	odd	6		1080.2.bi.b.289.5		32
9.7	even	3		360.2.bi.b.49.9	yes	32
15.14	odd	2		3240.2.f.i.649.16		16
36.7	odd	6		720.2.by.f.49.8		32
36.11	even	6		2160.2.by.f.1009.6		32
36.23	even	6		2160.2.by.f.289.5		32
36.31	odd	6		720.2.by.f.529.9		32
45.4	even	6		360.2.bi.b.169.9	yes	32
45.14	odd	6		1080.2.bi.b.289.6		32
45.29	odd	6		1080.2.bi.b.1009.5		32
45.34	even	6		360.2.bi.b.49.8	✓	32
180.59	even	6		2160.2.by.f.289.6		32
180.79	odd	6		720.2.by.f.49.9		32
180.119	even	6		2160.2.by.f.1009.5		32
180.139	odd	6		720.2.by.f.529.8		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.2.bi.b.49.8	✓	32	45.34	even	6
360.2.bi.b.49.9	yes	32	9.7	even	3
360.2.bi.b.169.8	yes	32	9.4	even	3
360.2.bi.b.169.9	yes	32	45.4	even	6
720.2.by.f.49.8		32	36.7	odd	6
720.2.by.f.49.9		32	180.79	odd	6
720.2.by.f.529.8		32	180.139	odd	6
720.2.by.f.529.9		32	36.31	odd	6
1080.2.bi.b.289.5		32	9.5	odd	6
1080.2.bi.b.289.6		32	45.14	odd	6
1080.2.bi.b.1009.5		32	45.29	odd	6
1080.2.bi.b.1009.6		32	9.2	odd	6
2160.2.by.f.289.5		32	36.23	even	6
2160.2.by.f.289.6		32	180.59	even	6
2160.2.by.f.1009.5		32	180.119	even	6
2160.2.by.f.1009.6		32	36.11	even	6
3240.2.f.i.649.15		16	3.2	odd	2
3240.2.f.i.649.16		16	15.14	odd	2
3240.2.f.k.649.1		16	5.4	even	2	inner
3240.2.f.k.649.2		16	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 3240 = 2^{3} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3240.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.23546 + 0.0521238i) q^{5} -4.10221i q^{7} +O(q^{10})\)	\(q+(-2.23546 + 0.0521238i) q^{5} -4.10221i q^{7} -6.09464 q^{11} -5.13891i q^{13} +2.73647i q^{17} -1.80595 q^{19} +0.672542i q^{23} +(4.99457 - 0.233041i) q^{25} -3.50669 q^{29} -6.60133 q^{31} +(0.213823 + 9.17033i) q^{35} -4.44214i q^{37} +4.17846 q^{41} +4.63223i q^{43} +1.59647i q^{47} -9.82815 q^{49} -8.02220i q^{53} +(13.6243 - 0.317676i) q^{55} +5.30022 q^{59} +6.77883 q^{61} +(0.267860 + 11.4878i) q^{65} +11.3465i q^{67} +13.3686 q^{71} +6.50669i q^{73} +25.0015i q^{77} -2.99005 q^{79} +4.28444i q^{83} +(-0.142635 - 6.11728i) q^{85} +7.19878 q^{89} -21.0809 q^{91} +(4.03712 - 0.0941328i) q^{95} +3.85319i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 2 q^{5}+O(q^{10}) \) 16 * q + 2 * q^5	\( 16 q + 2 q^{5} - 16 q^{11} + 4 q^{19} + 6 q^{25} - 20 q^{29} + 12 q^{31} + 2 q^{35} + 8 q^{41} - 36 q^{49} + 10 q^{55} + 20 q^{61} - 10 q^{65} + 8 q^{71} - 4 q^{79} - 36 q^{85} - 48 q^{89} - 4 q^{91} + 32 q^{95}+O(q^{100}) \) 16 * q + 2 * q^5 - 16 * q^11 + 4 * q^19 + 6 * q^25 - 20 * q^29 + 12 * q^31 + 2 * q^35 + 8 * q^41 - 36 * q^49 + 10 * q^55 + 20 * q^61 - 10 * q^65 + 8 * q^71 - 4 * q^79 - 36 * q^85 - 48 * q^89 - 4 * q^91 + 32 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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