LMFDB - Embedded newform 5520.2.be.b.1471.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5520,2,Mod(1471,5520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5520, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("5520.1471");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5520.be (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:	$44.0774219157$
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} - 4 x^{15} + 8 x^{14} + 28 x^{13} + 373 x^{12} - 920 x^{11} + 1088 x^{10} - 168 x^{9} + 16460 x^{8} + \cdots + 64 $ x^16 - 4x^15 + 8x^14 + 28x^13 + 373x^12 - 920x^11 + 1088x^10 - 168x^9 + 16460x^8 - 45408x^7 + 62624x^6 - 18048x^5 + 2160x^4 - 1664x^3 + 6272x^2 - 896*x + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1471.9
Root			$1.49885 - 1.49885i$ of defining polynomial
Character	$\chi$	$=$	5520.1471
Dual form			5520.2.be.b.1471.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +1.00000i q^{5} -4.84428 q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +1.00000i q^{5} -4.84428 q^{7} -1.00000 q^{9} -3.09487 q^{11} -4.53183 q^{13} -1.00000 q^{15} +1.04944i q^{17} -3.05893 q^{19} -4.84428i q^{21} +(-4.79485 - 0.0971617i) q^{23} -1.00000 q^{25} -1.00000i q^{27} -2.44565 q^{29} -3.08441i q^{31} -3.09487i q^{33} -4.84428i q^{35} +3.78545i q^{37} -4.53183i q^{39} +4.04608 q^{41} -2.53119 q^{43} -1.00000i q^{45} +3.83718i q^{47} +16.4671 q^{49} -1.04944 q^{51} +8.14281i q^{53} -3.09487i q^{55} -3.05893i q^{57} -4.64825i q^{59} -11.3650i q^{61} +4.84428 q^{63} -4.53183i q^{65} -3.77906 q^{67} +(0.0971617 - 4.79485i) q^{69} +13.2737i q^{71} +2.53642 q^{73} -1.00000i q^{75} +14.9924 q^{77} -1.04304 q^{79} +1.00000 q^{81} +4.24974 q^{83} -1.04944 q^{85} -2.44565i q^{87} +3.89654i q^{89} +21.9535 q^{91} +3.08441 q^{93} -3.05893i q^{95} -5.72909i q^{97} +3.09487 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 8 q^{7} - 16 q^{9}+O(q^{10}) $ 16 * q + 8 * q^7 - 16 * q^9	$ 16 q + 8 q^{7} - 16 q^{9} - 8 q^{11} + 8 q^{13} - 16 q^{15} - 12 q^{23} - 16 q^{25} - 4 q^{29} + 4 q^{41} + 20 q^{49} + 4 q^{51} - 8 q^{63} + 16 q^{67} + 40 q^{73} + 24 q^{77} - 32 q^{79} + 16 q^{81} + 4 q^{85} + 48 q^{91} + 8 q^{99}+O(q^{100}) $ 16 * q + 8 * q^7 - 16 * q^9 - 8 * q^11 + 8 * q^13 - 16 * q^15 - 12 * q^23 - 16 * q^25 - 4 * q^29 + 4 * q^41 + 20 * q^49 + 4 * q^51 - 8 * q^63 + 16 * q^67 + 40 * q^73 + 24 * q^77 - 32 * q^79 + 16 * q^81 + 4 * q^85 + 48 * q^91 + 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5520\mathbb{Z}\right)^\times$.

$n$	$1201$	$1381$	$1841$	$4417$	$4831$
$\chi(n)$	$-1$	$1$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$		0			0
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$	−4.84428			−1.83097			−0.915484	−	0.402355i	$-0.868192\pi$
$7$	−4.84428			−1.83097			−0.915484	+	0.402355i	$0.868192\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−3.09487			−0.933138			−0.466569	−	0.884485i	$-0.654510\pi$
$11$	−3.09487			−0.933138			−0.466569	+	0.884485i	$0.654510\pi$
$12$		0			0
$13$	−4.53183			−1.25690			−0.628452	−	0.777848i	$-0.716311\pi$
$13$	−4.53183			−1.25690			−0.628452	+	0.777848i	$0.716311\pi$
$14$		0			0
$15$	−1.00000			−0.258199
$16$		0			0
$17$			1.04944i			0.254526i	0.991869	+	0.127263i	$0.0406192\pi$
$17$			1.04944i			0.254526i	−0.991869	+	0.127263i	$0.959381\pi$
$18$		0			0
$19$	−3.05893			−0.701766			−0.350883	−	0.936419i	$-0.614119\pi$
$19$	−3.05893			−0.701766			−0.350883	+	0.936419i	$0.614119\pi$
$20$		0			0
$21$		−	4.84428i		−	1.05711i
$22$		0			0
$23$	−4.79485	−	0.0971617i	−0.999795	−	0.0202596i
$23$	−4.79485	−	0.0971617i	−0.999795	−	0.0202596i
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	−2.44565			−0.454146			−0.227073	−	0.973878i	$-0.572916\pi$
$29$	−2.44565			−0.454146			−0.227073	+	0.973878i	$0.572916\pi$
$30$		0			0
$31$		−	3.08441i		−	0.553977i	−0.960873	−	0.276988i	$-0.910664\pi$
$31$		−	3.08441i		−	0.553977i	0.960873	−	0.276988i	$-0.0893363\pi$
$32$		0			0
$33$		−	3.09487i		−	0.538748i
$34$		0			0
$35$		−	4.84428i		−	0.818833i
$36$		0			0
$37$			3.78545i			0.622325i	0.950357	+	0.311162i	$0.100718\pi$
$37$			3.78545i			0.622325i	−0.950357	+	0.311162i	$0.899282\pi$
$38$		0			0
$39$		−	4.53183i		−	0.725674i
$40$		0			0
$41$	4.04608			0.631891			0.315946	−	0.948777i	$-0.397678\pi$
$41$	4.04608			0.631891			0.315946	+	0.948777i	$0.397678\pi$
$42$		0			0
$43$	−2.53119			−0.386002			−0.193001	−	0.981199i	$-0.561822\pi$
$43$	−2.53119			−0.386002			−0.193001	+	0.981199i	$0.561822\pi$
$44$		0			0
$45$		−	1.00000i		−	0.149071i
$46$		0			0
$47$			3.83718i			0.559711i	0.960042	+	0.279855i	$0.0902865\pi$
$47$			3.83718i			0.559711i	−0.960042	+	0.279855i	$0.909713\pi$
$48$		0			0
$49$	16.4671			2.35244
$50$		0			0
$51$	−1.04944			−0.146950
$52$		0			0
$53$			8.14281i			1.11850i	0.828999	+	0.559251i	$0.188911\pi$
$53$			8.14281i			1.11850i	−0.828999	+	0.559251i	$0.811089\pi$
$54$		0			0
$55$		−	3.09487i		−	0.417312i
$56$		0			0
$57$		−	3.05893i		−	0.405165i
$58$		0			0
$59$		−	4.64825i		−	0.605151i	−0.953125	−	0.302575i	$-0.902154\pi$
$59$		−	4.64825i		−	0.605151i	0.953125	−	0.302575i	$-0.0978464\pi$
$60$		0			0
$61$		−	11.3650i		−	1.45514i	−0.686033	−	0.727570i	$-0.740649\pi$
$61$		−	11.3650i		−	1.45514i	0.686033	−	0.727570i	$-0.259351\pi$
$62$		0			0
$63$	4.84428			0.610322
$64$		0			0
$65$		−	4.53183i		−	0.562105i
$66$		0			0
$67$	−3.77906			−0.461686			−0.230843	−	0.972991i	$-0.574148\pi$
$67$	−3.77906			−0.461686			−0.230843	+	0.972991i	$0.574148\pi$
$68$		0			0
$69$	0.0971617	−	4.79485i	0.0116969	−	0.577232i
$70$		0			0
$71$			13.2737i			1.57530i	0.616122	+	0.787651i	$0.288703\pi$
$71$			13.2737i			1.57530i	−0.616122	+	0.787651i	$0.711297\pi$
$72$		0			0
$73$	2.53642			0.296865			0.148433	−	0.988923i	$-0.452577\pi$
$73$	2.53642			0.296865			0.148433	+	0.988923i	$0.452577\pi$
$74$		0			0
$75$		−	1.00000i		−	0.115470i
$76$		0			0
$77$	14.9924			1.70855
$78$		0			0
$79$	−1.04304			−0.117351			−0.0586754	−	0.998277i	$-0.518688\pi$
$79$	−1.04304			−0.117351			−0.0586754	+	0.998277i	$0.518688\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	4.24974			0.466470			0.233235	−	0.972420i	$-0.425069\pi$
$83$	4.24974			0.466470			0.233235	+	0.972420i	$0.425069\pi$
$84$		0			0
$85$	−1.04944			−0.113827
$86$		0			0
$87$		−	2.44565i		−	0.262202i
$88$		0			0
$89$			3.89654i			0.413033i	0.978443	+	0.206516i	$0.0662127\pi$
$89$			3.89654i			0.413033i	−0.978443	+	0.206516i	$0.933787\pi$
$90$		0			0
$91$	21.9535			2.30135
$92$		0			0
$93$	3.08441			0.319839
$94$		0			0
$95$		−	3.05893i		−	0.313839i
$96$		0			0
$97$		−	5.72909i		−	0.581701i	−0.956768	−	0.290851i	$-0.906062\pi$
$97$		−	5.72909i		−	0.581701i	0.956768	−	0.290851i	$-0.0939383\pi$
$98$		0			0
$99$	3.09487			0.311046
$100$		0			0
$101$	3.97419			0.395447			0.197724	−	0.980258i	$-0.436645\pi$
$101$	3.97419			0.395447			0.197724	+	0.980258i	$0.436645\pi$
$102$		0			0
$103$	−13.3249			−1.31294			−0.656471	−	0.754351i	$-0.727952\pi$
$103$	−13.3249			−1.31294			−0.656471	+	0.754351i	$0.727952\pi$
$104$		0			0
$105$	4.84428			0.472754
$106$		0			0
$107$	11.4779			1.10961			0.554806	−	0.831980i	$-0.312793\pi$
$107$	11.4779			1.10961			0.554806	+	0.831980i	$0.312793\pi$
$108$		0			0
$109$			13.3604i			1.27970i	0.768501	+	0.639848i	$0.221003\pi$
$109$			13.3604i			1.27970i	−0.768501	+	0.639848i	$0.778997\pi$
$110$		0			0
$111$	−3.78545			−0.359299
$112$		0			0
$113$			12.7468i			1.19912i	0.800331	+	0.599559i	$0.204657\pi$
$113$			12.7468i			1.19912i	−0.800331	+	0.599559i	$0.795343\pi$
$114$		0			0
$115$	0.0971617	−	4.79485i	0.00906038	−	0.447122i
$116$		0			0
$117$	4.53183			0.418968
$118$		0			0
$119$		−	5.08377i		−	0.466028i
$120$		0			0
$121$	−1.42178			−0.129253
$122$		0			0
$123$			4.04608i			0.364823i
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$		−	1.34087i		−	0.118983i	−0.998229	−	0.0594913i	$-0.981052\pi$
$127$		−	1.34087i		−	0.118983i	0.998229	−	0.0594913i	$-0.0189479\pi$
$128$		0			0
$129$		−	2.53119i		−	0.222859i
$130$		0			0
$131$		−	12.8225i		−	1.12030i	−0.828390	−	0.560152i	$-0.810743\pi$
$131$		−	12.8225i		−	1.12030i	0.828390	−	0.560152i	$-0.189257\pi$
$132$		0			0
$133$	14.8183			1.28491
$134$		0			0
$135$	1.00000			0.0860663
$136$		0			0
$137$			3.67251i			0.313763i	0.987617	+	0.156882i	$0.0501441\pi$
$137$			3.67251i			0.313763i	−0.987617	+	0.156882i	$0.949856\pi$
$138$		0			0
$139$		−	11.5659i		−	0.981009i	−0.871439	−	0.490504i	$-0.836813\pi$
$139$		−	11.5659i		−	0.981009i	0.871439	−	0.490504i	$-0.163187\pi$
$140$		0			0
$141$	−3.83718			−0.323149
$142$		0			0
$143$	14.0254			1.17287
$144$		0			0
$145$		−	2.44565i		−	0.203100i
$146$		0			0
$147$			16.4671i			1.35818i
$148$		0			0
$149$			12.2359i			1.00240i	0.865330	+	0.501202i	$0.167109\pi$
$149$			12.2359i			1.00240i	−0.865330	+	0.501202i	$0.832891\pi$
$150$		0			0
$151$			11.3471i			0.923415i	0.887032	+	0.461708i	$0.152763\pi$
$151$			11.3471i			0.923415i	−0.887032	+	0.461708i	$0.847237\pi$
$152$		0			0
$153$		−	1.04944i		−	0.0848419i
$154$		0			0
$155$	3.08441			0.247746
$156$		0			0
$157$		−	10.3681i		−	0.827461i	−0.910399	−	0.413730i	$-0.864226\pi$
$157$		−	10.3681i		−	0.827461i	0.910399	−	0.413730i	$-0.135774\pi$
$158$		0			0
$159$	−8.14281			−0.645767
$160$		0			0
$161$	23.2276	+	0.470679i	1.83059	+	0.0370947i
$162$		0			0
$163$			11.7920i			0.923623i	0.886978	+	0.461811i	$0.152800\pi$
$163$			11.7920i			0.923623i	−0.886978	+	0.461811i	$0.847200\pi$
$164$		0			0
$165$	3.09487			0.240935
$166$		0			0
$167$		−	15.3234i		−	1.18576i	−0.805290	−	0.592881i	$-0.797990\pi$
$167$		−	15.3234i		−	1.18576i	0.805290	−	0.592881i	$-0.202010\pi$
$168$		0			0
$169$	7.53751			0.579808
$170$		0			0
$171$	3.05893			0.233922
$172$		0			0
$173$	3.12803			0.237820			0.118910	−	0.992905i	$-0.462060\pi$
$173$	3.12803			0.237820			0.118910	+	0.992905i	$0.462060\pi$
$174$		0			0
$175$	4.84428			0.366193
$176$		0			0
$177$	4.64825			0.349384
$178$		0			0
$179$		−	25.9800i		−	1.94184i	−0.239404	−	0.970920i	$-0.576952\pi$
$179$		−	25.9800i		−	1.94184i	0.239404	−	0.970920i	$-0.423048\pi$
$180$		0			0
$181$		−	15.3758i		−	1.14288i	−0.820645	−	0.571439i	$-0.806385\pi$
$181$		−	15.3758i		−	1.14288i	0.820645	−	0.571439i	$-0.193615\pi$
$182$		0			0
$183$	11.3650			0.840126
$184$		0			0
$185$	−3.78545			−0.278312
$186$		0			0
$187$		−	3.24787i		−	0.237508i
$188$		0			0
$189$			4.84428i			0.352370i
$190$		0			0
$191$	−0.659707			−0.0477347			−0.0238674	−	0.999715i	$-0.507598\pi$
$191$	−0.659707			−0.0477347			−0.0238674	+	0.999715i	$0.507598\pi$
$192$		0			0
$193$	13.2828			0.956120			0.478060	−	0.878327i	$-0.341340\pi$
$193$	13.2828			0.956120			0.478060	+	0.878327i	$0.341340\pi$
$194$		0			0
$195$	4.53183			0.324531
$196$		0			0
$197$	−1.33165			−0.0948761			−0.0474380	−	0.998874i	$-0.515106\pi$
$197$	−1.33165			−0.0948761			−0.0474380	+	0.998874i	$0.515106\pi$
$198$		0			0
$199$	3.42710			0.242941			0.121470	−	0.992595i	$-0.461239\pi$
$199$	3.42710			0.242941			0.121470	+	0.992595i	$0.461239\pi$
$200$		0			0
$201$		−	3.77906i		−	0.266554i
$202$		0			0
$203$	11.8474			0.831527
$204$		0			0
$205$			4.04608i			0.282590i
$206$		0			0
$207$	4.79485	+	0.0971617i	0.333265	+	0.00675321i
$208$		0			0
$209$	9.46698			0.654845
$210$		0			0
$211$			22.7762i			1.56798i	0.620774	+	0.783989i	$0.286818\pi$
$211$			22.7762i			1.56798i	−0.620774	+	0.783989i	$0.713182\pi$
$212$		0			0
$213$	−13.2737			−0.909501
$214$		0			0
$215$		−	2.53119i		−	0.172626i
$216$		0			0
$217$			14.9418i			1.01431i
$218$		0			0
$219$			2.53642i			0.171395i
$220$		0			0
$221$		−	4.75587i		−	0.319914i
$222$		0			0
$223$			12.5342i			0.839353i	0.907674	+	0.419677i	$0.137856\pi$
$223$			12.5342i			0.839353i	−0.907674	+	0.419677i	$0.862144\pi$
$224$		0			0
$225$	1.00000			0.0666667
$226$		0			0
$227$	−24.4186			−1.62072			−0.810359	−	0.585933i	$-0.800728\pi$
$227$	−24.4186			−1.62072			−0.810359	+	0.585933i	$0.800728\pi$
$228$		0			0
$229$		−	25.3713i		−	1.67658i	−0.545225	−	0.838290i	$-0.683556\pi$
$229$		−	25.3713i		−	1.67658i	0.545225	−	0.838290i	$-0.316444\pi$
$230$		0			0
$231$			14.9924i			0.986429i
$232$		0			0
$233$	0.675614			0.0442609			0.0221305	−	0.999755i	$-0.492955\pi$
$233$	0.675614			0.0442609			0.0221305	+	0.999755i	$0.492955\pi$
$234$		0			0
$235$	−3.83718			−0.250310
$236$		0			0
$237$		−	1.04304i		−	0.0677526i
$238$		0			0
$239$		−	9.80756i		−	0.634398i	−0.948359	−	0.317199i	$-0.897258\pi$
$239$		−	9.80756i		−	0.634398i	0.948359	−	0.317199i	$-0.102742\pi$
$240$		0			0
$241$			10.2922i			0.662978i	0.943459	+	0.331489i	$0.107551\pi$
$241$			10.2922i			0.662978i	−0.943459	+	0.331489i	$0.892449\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$			16.4671i			1.05204i
$246$		0			0
$247$	13.8625			0.882053
$248$		0			0
$249$			4.24974i			0.269316i
$250$		0			0
$251$	8.41799			0.531339			0.265669	−	0.964064i	$-0.414407\pi$
$251$	8.41799			0.531339			0.265669	+	0.964064i	$0.414407\pi$
$252$		0			0
$253$	14.8394	+	0.300703i	0.932947	+	0.0189050i
$254$		0			0
$255$		−	1.04944i		−	0.0657182i
$256$		0			0
$257$	4.93222			0.307663			0.153832	−	0.988097i	$-0.450839\pi$
$257$	4.93222			0.307663			0.153832	+	0.988097i	$0.450839\pi$
$258$		0			0
$259$		−	18.3378i		−	1.13946i
$260$		0			0
$261$	2.44565			0.151382
$262$		0			0
$263$	−12.3726			−0.762924			−0.381462	−	0.924384i	$-0.624579\pi$
$263$	−12.3726			−0.762924			−0.381462	+	0.924384i	$0.624579\pi$
$264$		0			0
$265$	−8.14281			−0.500209
$266$		0			0
$267$	−3.89654			−0.238465
$268$		0			0
$269$	−29.2075			−1.78081			−0.890406	−	0.455166i	$-0.849580\pi$
$269$	−29.2075			−1.78081			−0.890406	+	0.455166i	$0.849580\pi$
$270$		0			0
$271$			16.1560i			0.981409i	0.871326	+	0.490704i	$0.163260\pi$
$271$			16.1560i			0.981409i	−0.871326	+	0.490704i	$0.836740\pi$
$272$		0			0
$273$			21.9535i			1.32869i
$274$		0			0
$275$	3.09487			0.186628
$276$		0			0
$277$	−24.8559			−1.49345			−0.746723	−	0.665135i	$-0.768374\pi$
$277$	−24.8559			−1.49345			−0.746723	+	0.665135i	$0.768374\pi$
$278$		0			0
$279$			3.08441i			0.184659i
$280$		0			0
$281$		−	18.4409i		−	1.10009i	−0.835134	−	0.550046i	$-0.814610\pi$
$281$		−	18.4409i		−	1.10009i	0.835134	−	0.550046i	$-0.185390\pi$
$282$		0			0
$283$	−0.823367			−0.0489441			−0.0244720	−	0.999701i	$-0.507790\pi$
$283$	−0.823367			−0.0489441			−0.0244720	+	0.999701i	$0.507790\pi$
$284$		0			0
$285$	3.05893			0.181195
$286$		0			0
$287$	−19.6004			−1.15697
$288$		0			0
$289$	15.8987			0.935217
$290$		0			0
$291$	5.72909			0.335845
$292$		0			0
$293$		−	6.91225i		−	0.403818i	−0.979404	−	0.201909i	$-0.935285\pi$
$293$		−	6.91225i		−	0.403818i	0.979404	−	0.201909i	$-0.0647145\pi$
$294$		0			0
$295$	4.64825			0.270632
$296$		0			0
$297$			3.09487i			0.179583i
$298$		0			0
$299$	21.7294	+	0.440321i	1.25665	+	0.0254644i
$300$		0			0
$301$	12.2618			0.706758
$302$		0			0
$303$			3.97419i			0.228312i
$304$		0			0
$305$	11.3650			0.650759
$306$		0			0
$307$			30.2075i			1.72403i	0.506879	+	0.862017i	$0.330799\pi$
$307$			30.2075i			1.72403i	−0.506879	+	0.862017i	$0.669201\pi$
$308$		0			0
$309$		−	13.3249i		−	0.758028i
$310$		0			0
$311$			16.6281i			0.942896i	0.881894	+	0.471448i	$0.156268\pi$
$311$			16.6281i			0.942896i	−0.881894	+	0.471448i	$0.843732\pi$
$312$		0			0
$313$			31.9251i			1.80451i	0.431202	+	0.902255i	$0.358090\pi$
$313$			31.9251i			1.80451i	−0.431202	+	0.902255i	$0.641910\pi$
$314$		0			0
$315$			4.84428i			0.272944i
$316$		0			0
$317$	21.2873			1.19561			0.597806	−	0.801641i	$-0.296039\pi$
$317$	21.2873			1.19561			0.597806	+	0.801641i	$0.296039\pi$
$318$		0			0
$319$	7.56898			0.423781
$320$		0			0
$321$			11.4779i			0.640635i
$322$		0			0
$323$		−	3.21015i		−	0.178617i
$324$		0			0
$325$	4.53183			0.251381
$326$		0			0
$327$	−13.3604			−0.738833
$328$		0			0
$329$		−	18.5884i		−	1.02481i
$330$		0			0
$331$		−	19.0651i		−	1.04791i	−0.851745	−	0.523956i	$-0.824456\pi$
$331$		−	19.0651i		−	1.04791i	0.851745	−	0.523956i	$-0.175544\pi$
$332$		0			0
$333$		−	3.78545i		−	0.207442i
$334$		0			0
$335$		−	3.77906i		−	0.206472i
$336$		0			0
$337$		−	18.3374i		−	0.998901i	−0.866342	−	0.499451i	$-0.833535\pi$
$337$		−	18.3374i		−	0.998901i	0.866342	−	0.499451i	$-0.166465\pi$
$338$		0			0
$339$	−12.7468			−0.692311
$340$		0			0
$341$			9.54585i			0.516937i
$342$		0			0
$343$	−45.8612			−2.47627
$344$		0			0
$345$	4.79485	+	0.0971617i	0.258146	+	0.00523101i
$346$		0			0
$347$		−	13.6634i		−	0.733489i	−0.930322	−	0.366744i	$-0.880472\pi$
$347$		−	13.6634i		−	0.733489i	0.930322	−	0.366744i	$-0.119528\pi$
$348$		0			0
$349$	31.1112			1.66534			0.832671	−	0.553768i	$-0.186810\pi$
$349$	31.1112			1.66534			0.832671	+	0.553768i	$0.186810\pi$
$350$		0			0
$351$			4.53183i			0.241891i
$352$		0			0
$353$	31.5128			1.67726			0.838630	−	0.544702i	$-0.183357\pi$
$353$	31.5128			1.67726			0.838630	+	0.544702i	$0.183357\pi$
$354$		0			0
$355$	−13.2737			−0.704496
$356$		0			0
$357$	5.08377			0.269061
$358$		0			0
$359$	−34.1464			−1.80218			−0.901088	−	0.433636i	$-0.857230\pi$
$359$	−34.1464			−1.80218			−0.901088	+	0.433636i	$0.857230\pi$
$360$		0			0
$361$	−9.64296			−0.507524
$362$		0			0
$363$		−	1.42178i		−	0.0746242i
$364$		0			0
$365$			2.53642i			0.132762i
$366$		0			0
$367$	−9.17921			−0.479151			−0.239576	−	0.970878i	$-0.577008\pi$
$367$	−9.17921			−0.479151			−0.239576	+	0.970878i	$0.577008\pi$
$368$		0			0
$369$	−4.04608			−0.210630
$370$		0			0
$371$		−	39.4461i		−	2.04794i
$372$		0			0
$373$		−	15.7943i		−	0.817795i	−0.912580	−	0.408898i	$-0.865913\pi$
$373$		−	15.7943i		−	0.817795i	0.912580	−	0.408898i	$-0.134087\pi$
$374$		0			0
$375$	1.00000			0.0516398
$376$		0			0
$377$	11.0833			0.570819
$378$		0			0
$379$	11.0170			0.565905			0.282952	−	0.959134i	$-0.408686\pi$
$379$	11.0170			0.565905			0.282952	+	0.959134i	$0.408686\pi$
$380$		0			0
$381$	1.34087			0.0686947
$382$		0			0
$383$	−0.892024			−0.0455803			−0.0227901	−	0.999740i	$-0.507255\pi$
$383$	−0.892024			−0.0455803			−0.0227901	+	0.999740i	$0.507255\pi$
$384$		0			0
$385$			14.9924i			0.764085i
$386$		0			0
$387$	2.53119			0.128667
$388$		0			0
$389$		−	0.970666i		−	0.0492147i	−0.999697	−	0.0246074i	$-0.992166\pi$
$389$		−	0.970666i		−	0.0492147i	0.999697	−	0.0246074i	$-0.00783356\pi$
$390$		0			0
$391$	0.101965	−	5.03189i	0.00515659	−	0.254473i
$392$		0			0
$393$	12.8225			0.646808
$394$		0			0
$395$		−	1.04304i		−	0.0524809i
$396$		0			0
$397$	−9.57904			−0.480758			−0.240379	−	0.970679i	$-0.577272\pi$
$397$	−9.57904			−0.480758			−0.240379	+	0.970679i	$0.577272\pi$
$398$		0			0
$399$			14.8183i			0.741844i
$400$		0			0
$401$		−	20.6070i		−	1.02906i	−0.857471	−	0.514532i	$-0.827966\pi$
$401$		−	20.6070i		−	1.02906i	0.857471	−	0.514532i	$-0.172034\pi$
$402$		0			0
$403$			13.9780i			0.696296i
$404$		0			0
$405$			1.00000i			0.0496904i
$406$		0			0
$407$		−	11.7155i		−	0.580715i
$408$		0			0
$409$	15.4878			0.765822			0.382911	−	0.923785i	$-0.374922\pi$
$409$	15.4878			0.765822			0.382911	+	0.923785i	$0.374922\pi$
$410$		0			0
$411$	−3.67251			−0.181151
$412$		0			0
$413$			22.5174i			1.10801i
$414$		0			0
$415$			4.24974i			0.208612i
$416$		0			0
$417$	11.5659			0.566386
$418$		0			0
$419$	−35.3366			−1.72631			−0.863154	−	0.504941i	$-0.831514\pi$
$419$	−35.3366			−1.72631			−0.863154	+	0.504941i	$0.831514\pi$
$420$		0			0
$421$			9.54836i			0.465359i	0.972554	+	0.232679i	$0.0747492\pi$
$421$			9.54836i			0.465359i	−0.972554	+	0.232679i	$0.925251\pi$
$422$		0			0
$423$		−	3.83718i		−	0.186570i
$424$		0			0
$425$		−	1.04944i		−	0.0509051i
$426$		0			0
$427$			55.0553i			2.66431i
$428$		0			0
$429$			14.0254i			0.677154i
$430$		0			0
$431$	24.9390			1.20127			0.600634	−	0.799524i	$-0.294915\pi$
$431$	24.9390			1.20127			0.600634	+	0.799524i	$0.294915\pi$
$432$		0			0
$433$			38.3664i			1.84377i	0.387464	+	0.921885i	$0.373351\pi$
$433$			38.3664i			1.84377i	−0.387464	+	0.921885i	$0.626649\pi$
$434$		0			0
$435$	2.44565			0.117260
$436$		0			0
$437$	14.6671	+	0.297211i	0.701622	+	0.0142175i
$438$		0			0
$439$			2.96764i			0.141638i	0.997489	+	0.0708189i	$0.0225612\pi$
$439$			2.96764i			0.141638i	−0.997489	+	0.0708189i	$0.977439\pi$
$440$		0			0
$441$	−16.4671			−0.784147
$442$		0			0
$443$		−	0.100101i		−	0.00475595i	−0.999997	−	0.00237797i	$-0.999243\pi$
$443$		−	0.100101i		−	0.00475595i	0.999997	−	0.00237797i	$-0.000756933\pi$
$444$		0			0
$445$	−3.89654			−0.184714
$446$		0			0
$447$	−12.2359			−0.578739
$448$		0			0
$449$	−32.2186			−1.52049			−0.760245	−	0.649637i	$-0.774921\pi$
$449$	−32.2186			−1.52049			−0.760245	+	0.649637i	$0.774921\pi$
$450$		0			0
$451$	−12.5221			−0.589642
$452$		0			0
$453$	−11.3471			−0.533134
$454$		0			0
$455$			21.9535i			1.02920i
$456$		0			0
$457$		−	17.8778i		−	0.836288i	−0.908381	−	0.418144i	$-0.862681\pi$
$457$		−	17.8778i		−	0.836288i	0.908381	−	0.418144i	$-0.137319\pi$
$458$		0			0
$459$	1.04944			0.0489835
$460$		0			0
$461$	5.34372			0.248882			0.124441	−	0.992227i	$-0.460286\pi$
$461$	5.34372			0.248882			0.124441	+	0.992227i	$0.460286\pi$
$462$		0			0
$463$			25.9996i			1.20830i	0.796870	+	0.604151i	$0.206488\pi$
$463$			25.9996i			1.20830i	−0.796870	+	0.604151i	$0.793512\pi$
$464$		0			0
$465$			3.08441i			0.143036i
$466$		0			0
$467$	3.57869			0.165602			0.0828010	−	0.996566i	$-0.473613\pi$
$467$	3.57869			0.165602			0.0828010	+	0.996566i	$0.473613\pi$
$468$		0			0
$469$	18.3068			0.845331
$470$		0			0
$471$	10.3681			0.477735
$472$		0			0
$473$	7.83370			0.360194
$474$		0			0
$475$	3.05893			0.140353
$476$		0			0
$477$		−	8.14281i		−	0.372834i
$478$		0			0
$479$	29.8270			1.36283			0.681416	−	0.731896i	$-0.261365\pi$
$479$	29.8270			1.36283			0.681416	+	0.731896i	$0.261365\pi$
$480$		0			0
$481$		−	17.1550i		−	0.782203i
$482$		0			0
$483$	−0.470679	+	23.2276i	−0.0214166	+	1.05689i
$484$		0			0
$485$	5.72909			0.260145
$486$		0			0
$487$		−	36.6321i		−	1.65996i	−0.557793	−	0.829980i	$-0.688352\pi$
$487$		−	36.6321i		−	1.65996i	0.557793	−	0.829980i	$-0.311648\pi$
$488$		0			0
$489$	−11.7920			−0.533254
$490$		0			0
$491$		−	28.1982i		−	1.27257i	−0.771455	−	0.636284i	$-0.780471\pi$
$491$		−	28.1982i		−	1.27257i	0.771455	−	0.636284i	$-0.219529\pi$
$492$		0			0
$493$		−	2.56656i		−	0.115592i
$494$		0			0
$495$			3.09487i			0.139104i
$496$		0			0
$497$		−	64.3017i		−	2.88432i
$498$		0			0
$499$			25.4668i			1.14005i	0.821627	+	0.570025i	$0.193066\pi$
$499$			25.4668i			1.14005i	−0.821627	+	0.570025i	$0.806934\pi$
$500$		0			0
$501$	15.3234			0.684600
$502$		0			0
$503$	21.4532			0.956552			0.478276	−	0.878210i	$-0.341262\pi$
$503$	21.4532			0.956552			0.478276	+	0.878210i	$0.341262\pi$
$504$		0			0
$505$			3.97419i			0.176849i
$506$		0			0
$507$			7.53751i			0.334752i
$508$		0			0
$509$	9.58966			0.425054			0.212527	−	0.977155i	$-0.431831\pi$
$509$	9.58966			0.425054			0.212527	+	0.977155i	$0.431831\pi$
$510$		0			0
$511$	−12.2871			−0.543550
$512$		0			0
$513$			3.05893i			0.135055i
$514$		0			0
$515$		−	13.3249i		−	0.587166i
$516$		0			0
$517$		−	11.8756i		−	0.522287i
$518$		0			0
$519$			3.12803i			0.137305i
$520$		0			0
$521$		−	13.0956i		−	0.573727i	−0.957971	−	0.286863i	$-0.907387\pi$
$521$		−	13.0956i		−	0.573727i	0.957971	−	0.286863i	$-0.0926126\pi$
$522$		0			0
$523$	−6.40732			−0.280173			−0.140086	−	0.990139i	$-0.544738\pi$
$523$	−6.40732			−0.280173			−0.140086	+	0.990139i	$0.544738\pi$
$524$		0			0
$525$			4.84428i			0.211422i
$526$		0			0
$527$	3.23689			0.141001
$528$		0			0
$529$	22.9811	+	0.931751i	0.999179	+	0.0405109i
$530$		0			0
$531$			4.64825i			0.201717i
$532$		0			0
$533$	−18.3361			−0.794227
$534$		0			0
$535$			11.4779i			0.496233i
$536$		0			0
$537$	25.9800			1.12112
$538$		0			0
$539$	−50.9635			−2.19515
$540$		0			0
$541$	−30.4405			−1.30874			−0.654370	−	0.756174i	$-0.727066\pi$
$541$	−30.4405			−1.30874			−0.654370	+	0.756174i	$0.727066\pi$
$542$		0			0
$543$	15.3758			0.659841
$544$		0			0
$545$	−13.3604			−0.572298
$546$		0			0
$547$		−	28.9045i		−	1.23587i	−0.786230	−	0.617934i	$-0.787970\pi$
$547$		−	28.9045i		−	1.23587i	0.786230	−	0.617934i	$-0.212030\pi$
$548$		0			0
$549$			11.3650i			0.485047i
$550$		0			0
$551$	7.48108			0.318705
$552$		0			0
$553$	5.05277			0.214866
$554$		0			0
$555$		−	3.78545i		−	0.160684i
$556$		0			0
$557$		−	15.2748i		−	0.647212i	−0.946192	−	0.323606i	$-0.895105\pi$
$557$		−	15.2748i		−	0.647212i	0.946192	−	0.323606i	$-0.104895\pi$
$558$		0			0
$559$	11.4709			0.485168
$560$		0			0
$561$	3.24787			0.137125
$562$		0			0
$563$	−2.19626			−0.0925612			−0.0462806	−	0.998928i	$-0.514737\pi$
$563$	−2.19626			−0.0925612			−0.0462806	+	0.998928i	$0.514737\pi$
$564$		0			0
$565$	−12.7468			−0.536262
$566$		0			0
$567$	−4.84428			−0.203441
$568$		0			0
$569$		−	27.6847i		−	1.16060i	−0.814401	−	0.580302i	$-0.802934\pi$
$569$		−	27.6847i		−	1.16060i	0.814401	−	0.580302i	$-0.197066\pi$
$570$		0			0
$571$	28.6755			1.20003			0.600016	−	0.799988i	$-0.295161\pi$
$571$	28.6755			1.20003			0.600016	+	0.799988i	$0.295161\pi$
$572$		0			0
$573$		−	0.659707i		−	0.0275597i
$574$		0			0
$575$	4.79485	+	0.0971617i	0.199959	+	0.00405192i
$576$		0			0
$577$	−2.26705			−0.0943785			−0.0471892	−	0.998886i	$-0.515026\pi$
$577$	−2.26705			−0.0943785			−0.0471892	+	0.998886i	$0.515026\pi$
$578$		0			0
$579$			13.2828i			0.552016i
$580$		0			0
$581$	−20.5869			−0.854090
$582$		0			0
$583$		−	25.2009i		−	1.04372i
$584$		0			0
$585$			4.53183i			0.187368i
$586$		0			0
$587$		−	10.1170i		−	0.417574i	−0.977961	−	0.208787i	$-0.933048\pi$
$587$		−	10.1170i		−	0.417574i	0.977961	−	0.208787i	$-0.0669515\pi$
$588$		0			0
$589$			9.43499i			0.388762i
$590$		0			0
$591$		−	1.33165i		−	0.0547767i
$592$		0			0
$593$	32.3649			1.32907			0.664533	−	0.747259i	$-0.268630\pi$
$593$	32.3649			1.32907			0.664533	+	0.747259i	$0.268630\pi$
$594$		0			0
$595$	5.08377			0.208414
$596$		0			0
$597$			3.42710i			0.140262i
$598$		0			0
$599$			3.27236i			0.133705i	0.997763	+	0.0668526i	$0.0212957\pi$
$599$			3.27236i			0.133705i	−0.997763	+	0.0668526i	$0.978704\pi$
$600$		0			0
$601$	35.2903			1.43952			0.719761	−	0.694222i	$-0.244251\pi$
$601$	35.2903			1.43952			0.719761	+	0.694222i	$0.244251\pi$
$602$		0			0
$603$	3.77906			0.153895
$604$		0			0
$605$		−	1.42178i		−	0.0578036i
$606$		0			0
$607$			5.72385i			0.232324i	0.993230	+	0.116162i	$0.0370591\pi$
$607$			5.72385i			0.232324i	−0.993230	+	0.116162i	$0.962941\pi$
$608$		0			0
$609$			11.8474i			0.480082i
$610$		0			0
$611$		−	17.3895i		−	0.703503i
$612$		0			0
$613$			11.1031i			0.448449i	0.974538	+	0.224224i	$0.0719848\pi$
$613$			11.1031i			0.448449i	−0.974538	+	0.224224i	$0.928015\pi$
$614$		0			0
$615$	−4.04608			−0.163154
$616$		0			0
$617$		−	0.915087i		−	0.0368400i	−0.999830	−	0.0184200i	$-0.994136\pi$
$617$		−	0.915087i		−	0.0368400i	0.999830	−	0.0184200i	$-0.00586360\pi$
$618$		0			0
$619$	32.2320			1.29552			0.647758	−	0.761847i	$-0.275707\pi$
$619$	32.2320			1.29552			0.647758	+	0.761847i	$0.275707\pi$
$620$		0			0
$621$	−0.0971617	+	4.79485i	−0.00389896	+	0.192411i
$622$		0			0
$623$		−	18.8760i		−	0.756249i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$			9.46698i			0.378075i
$628$		0			0
$629$	−3.97259			−0.158398
$630$		0			0
$631$	−0.724543			−0.0288436			−0.0144218	−	0.999896i	$-0.504591\pi$
$631$	−0.724543			−0.0288436			−0.0144218	+	0.999896i	$0.504591\pi$
$632$		0			0
$633$	−22.7762			−0.905273
$634$		0			0
$635$	1.34087			0.0532107
$636$		0			0
$637$	−74.6261			−2.95679
$638$		0			0
$639$		−	13.2737i		−	0.525100i
$640$		0			0
$641$			9.49638i			0.375084i	0.982257	+	0.187542i	$0.0600521\pi$
$641$			9.49638i			0.375084i	−0.982257	+	0.187542i	$0.939948\pi$
$642$		0			0
$643$	7.33887			0.289417			0.144708	−	0.989474i	$-0.453776\pi$
$643$	7.33887			0.289417			0.144708	+	0.989474i	$0.453776\pi$
$644$		0			0
$645$	2.53119			0.0996654
$646$		0			0
$647$		−	8.89013i		−	0.349507i	−0.984612	−	0.174753i	$-0.944087\pi$
$647$		−	8.89013i		−	0.349507i	0.984612	−	0.174753i	$-0.0559128\pi$
$648$		0			0
$649$			14.3857i			0.564689i
$650$		0			0
$651$	−14.9418			−0.585614
$652$		0			0
$653$	−42.3035			−1.65546			−0.827732	−	0.561124i	$-0.810369\pi$
$653$	−42.3035			−1.65546			−0.827732	+	0.561124i	$0.810369\pi$
$654$		0			0
$655$	12.8225			0.501015
$656$		0			0
$657$	−2.53642			−0.0989550
$658$		0			0
$659$	2.34063			0.0911781			0.0455891	−	0.998960i	$-0.485484\pi$
$659$	2.34063			0.0911781			0.0455891	+	0.998960i	$0.485484\pi$
$660$		0			0
$661$		−	6.00716i		−	0.233652i	−0.993152	−	0.116826i	$-0.962728\pi$
$661$		−	6.00716i		−	0.233652i	0.993152	−	0.116826i	$-0.0372719\pi$
$662$		0			0
$663$	4.75587			0.184703
$664$		0			0
$665$			14.8183i			0.574630i
$666$		0			0
$667$	11.7265	+	0.237624i	0.454053	+	0.00920083i
$668$		0			0
$669$	−12.5342			−0.484601
$670$		0			0
$671$			35.1732i			1.35785i
$672$		0			0
$673$	−39.2984			−1.51484			−0.757420	−	0.652928i	$-0.773540\pi$
$673$	−39.2984			−1.51484			−0.757420	+	0.652928i	$0.773540\pi$
$674$		0			0
$675$			1.00000i			0.0384900i
$676$		0			0
$677$			10.1936i			0.391772i	0.980627	+	0.195886i	$0.0627583\pi$
$677$			10.1936i			0.391772i	−0.980627	+	0.195886i	$0.937242\pi$
$678$		0			0
$679$			27.7533i			1.06508i
$680$		0			0
$681$		−	24.4186i		−	0.935722i
$682$		0			0
$683$		−	4.94854i		−	0.189350i	−0.995508	−	0.0946752i	$-0.969819\pi$
$683$		−	4.94854i		−	0.189350i	0.995508	−	0.0946752i	$-0.0301813\pi$
$684$		0			0
$685$	−3.67251			−0.140319
$686$		0			0
$687$	25.3713			0.967974
$688$		0			0
$689$		−	36.9019i		−	1.40585i
$690$		0			0
$691$		−	30.0509i		−	1.14319i	−0.820536	−	0.571595i	$-0.806325\pi$
$691$		−	30.0509i		−	1.14319i	0.820536	−	0.571595i	$-0.193675\pi$
$692$		0			0
$693$	−14.9924			−0.569515
$694$		0			0
$695$	11.5659			0.438720
$696$		0			0
$697$			4.24610i			0.160833i
$698$		0			0
$699$			0.675614i			0.0255541i
$700$		0			0
$701$		−	18.9951i		−	0.717434i	−0.933446	−	0.358717i	$-0.883214\pi$
$701$		−	18.9951i		−	0.717434i	0.933446	−	0.358717i	$-0.116786\pi$
$702$		0			0
$703$		−	11.5794i		−	0.436727i
$704$		0			0
$705$		−	3.83718i		−	0.144517i
$706$		0			0
$707$	−19.2521			−0.724051
$708$		0			0
$709$		−	10.3208i		−	0.387605i	−0.981041	−	0.193802i	$-0.937918\pi$
$709$		−	10.3208i		−	0.387605i	0.981041	−	0.193802i	$-0.0620820\pi$
$710$		0			0
$711$	1.04304			0.0391170
$712$		0			0
$713$	−0.299687	+	14.7893i	−0.0112234	+	0.553863i
$714$		0			0
$715$			14.0254i			0.524521i
$716$		0			0
$717$	9.80756			0.366270
$718$		0			0
$719$			28.4693i			1.06172i	0.847458	+	0.530862i	$0.178132\pi$
$719$			28.4693i			1.06172i	−0.847458	+	0.530862i	$0.821868\pi$
$720$		0			0
$721$	64.5497			2.40395
$722$		0			0
$723$	−10.2922			−0.382770
$724$		0			0
$725$	2.44565			0.0908293
$726$		0			0
$727$	3.62136			0.134309			0.0671543	−	0.997743i	$-0.478608\pi$
$727$	3.62136			0.134309			0.0671543	+	0.997743i	$0.478608\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$		−	2.65632i		−	0.0982475i
$732$		0			0
$733$			3.39563i			0.125420i	0.998032	+	0.0627102i	$0.0199744\pi$
$733$			3.39563i			0.125420i	−0.998032	+	0.0627102i	$0.980026\pi$
$734$		0			0
$735$	−16.4671			−0.607397
$736$		0			0
$737$	11.6957			0.430817
$738$		0			0
$739$			45.6318i			1.67859i	0.543675	+	0.839296i	$0.317032\pi$
$739$			45.6318i			1.67859i	−0.543675	+	0.839296i	$0.682968\pi$
$740$		0			0
$741$			13.8625i			0.509253i
$742$		0			0
$743$	8.29853			0.304444			0.152222	−	0.988346i	$-0.451357\pi$
$743$	8.29853			0.304444			0.152222	+	0.988346i	$0.451357\pi$
$744$		0			0
$745$	−12.2359			−0.448289
$746$		0			0
$747$	−4.24974			−0.155490
$748$		0			0
$749$	−55.6023			−2.03166
$750$		0			0
$751$	39.1817			1.42976			0.714880	−	0.699247i	$-0.246481\pi$
$751$	39.1817			1.42976			0.714880	+	0.699247i	$0.246481\pi$
$752$		0			0
$753$			8.41799i			0.306769i
$754$		0			0
$755$	−11.3471			−0.412964
$756$		0			0
$757$			37.3641i			1.35802i	0.734128	+	0.679011i	$0.237591\pi$
$757$			37.3641i			1.35802i	−0.734128	+	0.679011i	$0.762409\pi$
$758$		0			0
$759$	−0.300703	+	14.8394i	−0.0109148	+	0.538637i
$760$		0			0
$761$	13.2385			0.479896			0.239948	−	0.970786i	$-0.422870\pi$
$761$	13.2385			0.479896			0.239948	+	0.970786i	$0.422870\pi$
$762$		0			0
$763$		−	64.7217i		−	2.34308i
$764$		0			0
$765$	1.04944			0.0379424
$766$		0			0
$767$			21.0651i			0.760616i
$768$		0			0
$769$		−	33.6263i		−	1.21260i	−0.795237	−	0.606298i	$-0.792654\pi$
$769$		−	33.6263i		−	1.21260i	0.795237	−	0.606298i	$-0.207346\pi$
$770$		0			0
$771$			4.93222i			0.177630i
$772$		0			0
$773$			17.5764i			0.632177i	0.948730	+	0.316089i	$0.102370\pi$
$773$			17.5764i			0.632177i	−0.948730	+	0.316089i	$0.897630\pi$
$774$		0			0
$775$			3.08441i			0.110795i
$776$		0			0
$777$	18.3378			0.657865
$778$		0			0
$779$	−12.3767			−0.443440
$780$		0			0
$781$		−	41.0804i		−	1.46997i
$782$		0			0
$783$			2.44565i			0.0874005i
$784$		0			0
$785$	10.3681			0.370052
$786$		0			0
$787$	−34.3182			−1.22331			−0.611656	−	0.791124i	$-0.709496\pi$
$787$	−34.3182			−1.22331			−0.611656	+	0.791124i	$0.709496\pi$
$788$		0			0
$789$		−	12.3726i		−	0.440475i
$790$		0			0
$791$		−	61.7491i		−	2.19555i
$792$		0			0
$793$			51.5043i			1.82897i
$794$		0			0
$795$		−	8.14281i		−	0.288796i
$796$		0			0
$797$		−	0.808358i		−	0.0286335i	−0.999898	−	0.0143168i	$-0.995443\pi$
$797$		−	0.808358i		−	0.0286335i	0.999898	−	0.0143168i	$-0.00455732\pi$
$798$		0			0
$799$	−4.02688			−0.142461
$800$		0			0
$801$		−	3.89654i		−	0.137678i
$802$		0			0
$803$	−7.84988			−0.277016
$804$		0			0
$805$	−0.470679	+	23.2276i	−0.0165892	+	0.818665i
$806$		0			0
$807$		−	29.2075i		−	1.02815i
$808$		0			0
$809$	10.4177			0.366268			0.183134	−	0.983088i	$-0.441376\pi$
$809$	10.4177			0.366268			0.183134	+	0.983088i	$0.441376\pi$
$810$		0			0
$811$		−	21.0907i		−	0.740593i	−0.928914	−	0.370297i	$-0.879256\pi$
$811$		−	21.0907i		−	0.740593i	0.928914	−	0.370297i	$-0.120744\pi$
$812$		0			0
$813$	−16.1560			−0.566617
$814$		0			0
$815$	−11.7920			−0.413057
$816$		0			0
$817$	7.74272			0.270883
$818$		0			0
$819$	−21.9535			−0.767117
$820$		0			0
$821$	−11.8269			−0.412761			−0.206380	−	0.978472i	$-0.566168\pi$
$821$	−11.8269			−0.412761			−0.206380	+	0.978472i	$0.566168\pi$
$822$		0			0
$823$			13.4121i			0.467517i	0.972295	+	0.233758i	$0.0751025\pi$
$823$			13.4121i			0.467517i	−0.972295	+	0.233758i	$0.924898\pi$
$824$		0			0
$825$			3.09487i			0.107750i
$826$		0			0
$827$	44.8208			1.55857			0.779286	−	0.626668i	$-0.215582\pi$
$827$	44.8208			1.55857			0.779286	+	0.626668i	$0.215582\pi$
$828$		0			0
$829$	−39.5431			−1.37339			−0.686694	−	0.726947i	$-0.740939\pi$
$829$	−39.5431			−1.37339			−0.686694	+	0.726947i	$0.740939\pi$
$830$		0			0
$831$		−	24.8559i		−	0.862241i
$832$		0			0
$833$			17.2812i			0.598756i
$834$		0			0
$835$	15.3234			0.530289
$836$		0			0
$837$	−3.08441			−0.106613
$838$		0			0
$839$	52.1962			1.80201			0.901007	−	0.433805i	$-0.142829\pi$
$839$	52.1962			1.80201			0.901007	+	0.433805i	$0.142829\pi$
$840$		0			0
$841$	−23.0188			−0.793751
$842$		0			0
$843$	18.4409			0.635138
$844$		0			0
$845$			7.53751i			0.259298i
$846$		0			0
$847$	6.88751			0.236658
$848$		0			0
$849$		−	0.823367i		−	0.0282579i
$850$		0			0
$851$	0.367801	−	18.1507i	0.0126081	−	0.622197i
$852$		0			0
$853$	57.1046			1.95522			0.977612	−	0.210415i	$-0.0674815\pi$
$853$	57.1046			1.95522			0.977612	+	0.210415i	$0.0674815\pi$
$854$		0			0
$855$			3.05893i			0.104613i
$856$		0			0
$857$	−33.0004			−1.12727			−0.563636	−	0.826024i	$-0.690598\pi$
$857$	−33.0004			−1.12727			−0.563636	+	0.826024i	$0.690598\pi$
$858$		0			0
$859$			46.3157i			1.58027i	0.612932	+	0.790135i	$0.289990\pi$
$859$			46.3157i			1.58027i	−0.612932	+	0.790135i	$0.710010\pi$
$860$		0			0
$861$		−	19.6004i		−	0.667978i
$862$		0			0
$863$		−	28.8658i		−	0.982604i	−0.870989	−	0.491302i	$-0.836521\pi$
$863$		−	28.8658i		−	0.982604i	0.870989	−	0.491302i	$-0.163479\pi$
$864$		0			0
$865$			3.12803i			0.106356i
$866$		0			0
$867$			15.8987i			0.539948i
$868$		0			0
$869$	3.22806			0.109505
$870$		0			0
$871$	17.1261			0.580295
$872$		0			0
$873$			5.72909i			0.193900i
$874$		0			0
$875$			4.84428i			0.163767i
$876$		0			0
$877$	−25.1171			−0.848145			−0.424072	−	0.905628i	$-0.639400\pi$
$877$	−25.1171			−0.848145			−0.424072	+	0.905628i	$0.639400\pi$
$878$		0			0
$879$	6.91225			0.233144
$880$		0			0
$881$		−	48.8946i		−	1.64730i	−0.567098	−	0.823650i	$-0.691934\pi$
$881$		−	48.8946i		−	1.64730i	0.567098	−	0.823650i	$-0.308066\pi$
$882$		0			0
$883$			39.3348i			1.32372i	0.749627	+	0.661860i	$0.230233\pi$
$883$			39.3348i			1.32372i	−0.749627	+	0.661860i	$0.769767\pi$
$884$		0			0
$885$			4.64825i			0.156249i
$886$		0			0
$887$		−	56.2201i		−	1.88769i	−0.330393	−	0.943843i	$-0.607181\pi$
$887$		−	56.2201i		−	1.88769i	0.330393	−	0.943843i	$-0.392819\pi$
$888$		0			0
$889$			6.49554i			0.217853i
$890$		0			0
$891$	−3.09487			−0.103682
$892$		0			0
$893$		−	11.7377i		−	0.392786i
$894$		0			0
$895$	25.9800			0.868417
$896$		0			0
$897$	−0.440321	+	21.7294i	−0.0147019	+	0.725525i
$898$		0			0
$899$			7.54340i			0.251587i
$900$		0			0
$901$	−8.54536			−0.284687
$902$		0			0
$903$			12.2618i			0.408047i
$904$		0			0
$905$	15.3758			0.511110
$906$		0			0
$907$	−37.5550			−1.24699			−0.623496	−	0.781826i	$-0.714288\pi$
$907$	−37.5550			−1.24699			−0.623496	+	0.781826i	$0.714288\pi$
$908$		0			0
$909$	−3.97419			−0.131816
$910$		0			0
$911$	−10.2173			−0.338513			−0.169256	−	0.985572i	$-0.554137\pi$
$911$	−10.2173			−0.338513			−0.169256	+	0.985572i	$0.554137\pi$
$912$		0			0
$913$	−13.1524			−0.435281
$914$		0			0
$915$			11.3650i			0.375716i
$916$		0			0
$917$			62.1157i			2.05124i
$918$		0			0
$919$	35.4425			1.16914			0.584569	−	0.811344i	$-0.301263\pi$
$919$	35.4425			1.16914			0.584569	+	0.811344i	$0.301263\pi$
$920$		0			0
$921$	−30.2075			−0.995372
$922$		0			0
$923$		−	60.1543i		−	1.98000i
$924$		0			0
$925$		−	3.78545i		−	0.124465i
$926$		0			0
$927$	13.3249			0.437648
$928$		0			0
$929$	27.3376			0.896918			0.448459	−	0.893803i	$-0.351973\pi$
$929$	27.3376			0.896918			0.448459	+	0.893803i	$0.351973\pi$
$930$		0			0
$931$	−50.3716			−1.65086
$932$		0			0
$933$	−16.6281			−0.544381
$934$		0			0
$935$	3.24787			0.106217
$936$		0			0
$937$			28.1261i			0.918838i	0.888220	+	0.459419i	$0.151942\pi$
$937$			28.1261i			0.918838i	−0.888220	+	0.459419i	$0.848058\pi$
$938$		0			0
$939$	−31.9251			−1.04183
$940$		0			0
$941$		−	49.1173i		−	1.60118i	−0.599214	−	0.800589i	$-0.704520\pi$
$941$		−	49.1173i		−	1.60118i	0.599214	−	0.800589i	$-0.295480\pi$
$942$		0			0
$943$	−19.4003	−	0.393124i	−0.631762	−	0.0128019i
$944$		0			0
$945$	−4.84428			−0.157585
$946$		0			0
$947$		−	1.22418i		−	0.0397804i	−0.999802	−	0.0198902i	$-0.993668\pi$
$947$		−	1.22418i		−	0.0397804i	0.999802	−	0.0198902i	$-0.00633166\pi$
$948$		0			0
$949$	−11.4946			−0.373131
$950$		0			0
$951$			21.2873i			0.690287i
$952$		0			0
$953$			31.8279i			1.03101i	0.856887	+	0.515504i	$0.172395\pi$
$953$			31.8279i			1.03101i	−0.856887	+	0.515504i	$0.827605\pi$
$954$		0			0
$955$		−	0.659707i		−	0.0213476i
$956$		0			0
$957$			7.56898i			0.244670i
$958$		0			0
$959$		−	17.7907i		−	0.574490i
$960$		0			0
$961$	21.4864			0.693110
$962$		0			0
$963$	−11.4779			−0.369871
$964$		0			0
$965$			13.2828i			0.427590i
$966$		0			0
$967$		−	3.01006i		−	0.0967971i	−0.998828	−	0.0483985i	$-0.984588\pi$
$967$		−	3.01006i		−	0.0967971i	0.998828	−	0.0483985i	$-0.0154118\pi$
$968$		0			0
$969$	3.21015			0.103125
$970$		0			0
$971$	0.278352			0.00893273			0.00446636	−	0.999990i	$-0.498578\pi$
$971$	0.278352			0.00893273			0.00446636	+	0.999990i	$0.498578\pi$
$972$		0			0
$973$			56.0286i			1.79619i
$974$		0			0
$975$			4.53183i			0.145135i
$976$		0			0
$977$		−	44.5047i		−	1.42383i	−0.702264	−	0.711916i	$-0.747827\pi$
$977$		−	44.5047i		−	1.42383i	0.702264	−	0.711916i	$-0.252173\pi$
$978$		0			0
$979$		−	12.0593i		−	0.385417i
$980$		0			0
$981$		−	13.3604i		−	0.426566i
$982$		0			0
$983$	−7.02437			−0.224043			−0.112021	−	0.993706i	$-0.535733\pi$
$983$	−7.02437			−0.224043			−0.112021	+	0.993706i	$0.535733\pi$
$984$		0			0
$985$		−	1.33165i		−	0.0424299i
$986$		0			0
$987$	18.5884			0.591675
$988$		0			0
$989$	12.1367	+	0.245934i	0.385923	+	0.00782026i
$990$		0			0
$991$			51.5501i			1.63754i	0.574119	+	0.818772i	$0.305345\pi$
$991$			51.5501i			1.63754i	−0.574119	+	0.818772i	$0.694655\pi$
$992$		0			0
$993$	19.0651			0.605012
$994$		0			0
$995$			3.42710i			0.108646i
$996$		0			0
$997$	−13.3647			−0.423264			−0.211632	−	0.977349i	$-0.567878\pi$
$997$	−13.3647			−0.423264			−0.211632	+	0.977349i	$0.567878\pi$
$998$		0			0
$999$	3.78545			0.119766

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5520.2.be.b.1471.9	yes	16
4.3	odd	2		5520.2.be.a.1471.8	✓	16
23.22	odd	2		5520.2.be.a.1471.16	yes	16
92.91	even	2	inner	5520.2.be.b.1471.1	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5520.2.be.a.1471.8	✓	16	4.3	odd	2
5520.2.be.a.1471.16	yes	16	23.22	odd	2
5520.2.be.b.1471.1	yes	16	92.91	even	2	inner
5520.2.be.b.1471.9	yes	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:	\( N \)	\(=\)	\( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5520.be (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +1.00000i q^{5} -4.84428 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +1.00000i q^{5} -4.84428 q^{7} -1.00000 q^{9} -3.09487 q^{11} -4.53183 q^{13} -1.00000 q^{15} +1.04944i q^{17} -3.05893 q^{19} -4.84428i q^{21} +(-4.79485 - 0.0971617i) q^{23} -1.00000 q^{25} -1.00000i q^{27} -2.44565 q^{29} -3.08441i q^{31} -3.09487i q^{33} -4.84428i q^{35} +3.78545i q^{37} -4.53183i q^{39} +4.04608 q^{41} -2.53119 q^{43} -1.00000i q^{45} +3.83718i q^{47} +16.4671 q^{49} -1.04944 q^{51} +8.14281i q^{53} -3.09487i q^{55} -3.05893i q^{57} -4.64825i q^{59} -11.3650i q^{61} +4.84428 q^{63} -4.53183i q^{65} -3.77906 q^{67} +(0.0971617 - 4.79485i) q^{69} +13.2737i q^{71} +2.53642 q^{73} -1.00000i q^{75} +14.9924 q^{77} -1.04304 q^{79} +1.00000 q^{81} +4.24974 q^{83} -1.04944 q^{85} -2.44565i q^{87} +3.89654i q^{89} +21.9535 q^{91} +3.08441 q^{93} -3.05893i q^{95} -5.72909i q^{97} +3.09487 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{7} - 16 q^{9}+O(q^{10}) \) 16 * q + 8 * q^7 - 16 * q^9	\( 16 q + 8 q^{7} - 16 q^{9} - 8 q^{11} + 8 q^{13} - 16 q^{15} - 12 q^{23} - 16 q^{25} - 4 q^{29} + 4 q^{41} + 20 q^{49} + 4 q^{51} - 8 q^{63} + 16 q^{67} + 40 q^{73} + 24 q^{77} - 32 q^{79} + 16 q^{81} + 4 q^{85} + 48 q^{91} + 8 q^{99}+O(q^{100}) \) 16 * q + 8 * q^7 - 16 * q^9 - 8 * q^11 + 8 * q^13 - 16 * q^15 - 12 * q^23 - 16 * q^25 - 4 * q^29 + 4 * q^41 + 20 * q^49 + 4 * q^51 - 8 * q^63 + 16 * q^67 + 40 * q^73 + 24 * q^77 - 32 * q^79 + 16 * q^81 + 4 * q^85 + 48 * q^91 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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