LMFDB - Embedded newform 4368.2.h.a.337.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4368,2,Mod(337,4368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4368, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4368.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4368.h (of order $2$, degree $1$, not 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:	$34.8786556029$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1092)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	4368.337
Dual form			4368.2.h.a.337.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +4.00000i q^{5} -1.00000i q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +4.00000i q^{5} -1.00000i q^{7} +1.00000 q^{9} +(-3.00000 + 2.00000i) q^{13} -4.00000i q^{15} -2.00000 q^{17} +1.00000i q^{21} -4.00000 q^{23} -11.0000 q^{25} -1.00000 q^{27} +2.00000 q^{29} -8.00000i q^{31} +4.00000 q^{35} -8.00000i q^{37} +(3.00000 - 2.00000i) q^{39} -4.00000i q^{41} +8.00000 q^{43} +4.00000i q^{45} -1.00000 q^{49} +2.00000 q^{51} +6.00000 q^{53} -8.00000i q^{59} -10.0000 q^{61} -1.00000i q^{63} +(-8.00000 - 12.0000i) q^{65} -8.00000i q^{67} +4.00000 q^{69} +4.00000i q^{73} +11.0000 q^{75} -12.0000 q^{79} +1.00000 q^{81} +8.00000i q^{83} -8.00000i q^{85} -2.00000 q^{87} +12.0000i q^{89} +(2.00000 + 3.00000i) q^{91} +8.00000i q^{93} +12.0000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 2 * q^9	$ 2 q - 2 q^{3} + 2 q^{9} - 6 q^{13} - 4 q^{17} - 8 q^{23} - 22 q^{25} - 2 q^{27} + 4 q^{29} + 8 q^{35} + 6 q^{39} + 16 q^{43} - 2 q^{49} + 4 q^{51} + 12 q^{53} - 20 q^{61} - 16 q^{65} + 8 q^{69} + 22 q^{75} - 24 q^{79} + 2 q^{81} - 4 q^{87} + 4 q^{91}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^9 - 6 * q^13 - 4 * q^17 - 8 * q^23 - 22 * q^25 - 2 * q^27 + 4 * q^29 + 8 * q^35 + 6 * q^39 + 16 * q^43 - 2 * q^49 + 4 * q^51 + 12 * q^53 - 20 * q^61 - 16 * q^65 + 8 * q^69 + 22 * q^75 - 24 * q^79 + 2 * q^81 - 4 * q^87 + 4 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4368\mathbb{Z}\right)^\times$.

$n$	$1093$	$1249$	$1457$	$2017$	$3823$
$\chi(n)$	$1$	$1$	$1$	$-1$	$1$

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
$3$	−1.00000			−0.577350
$3$	−1.00000			−0.577350
$4$		0			0
$5$			4.00000i			1.78885i	0.447214	+	0.894427i	$0.352416\pi$
$5$			4.00000i			1.78885i	−0.447214	+	0.894427i	$0.647584\pi$
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$	−3.00000	+	2.00000i	−0.832050	+	0.554700i
$13$	−3.00000	+	2.00000i	−0.832050	+	0.554700i
$14$		0			0
$15$		−	4.00000i		−	1.03280i
$16$		0			0
$17$	−2.00000			−0.485071			−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000			−0.485071			−0.242536	+	0.970143i	$0.577979\pi$
$18$		0			0
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$		0			0
$21$			1.00000i			0.218218i
$22$		0			0
$23$	−4.00000			−0.834058			−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000			−0.834058			−0.417029	+	0.908893i	$0.636929\pi$
$24$		0			0
$25$	−11.0000			−2.20000
$26$		0			0
$27$	−1.00000			−0.192450
$28$		0			0
$29$	2.00000			0.371391			0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000			0.371391			0.185695	+	0.982607i	$0.440546\pi$
$30$		0			0
$31$		−	8.00000i		−	1.43684i	−0.695608	−	0.718421i	$-0.744865\pi$
$31$		−	8.00000i		−	1.43684i	0.695608	−	0.718421i	$-0.255135\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	4.00000			0.676123
$36$		0			0
$37$		−	8.00000i		−	1.31519i	−0.753371	−	0.657596i	$-0.771573\pi$
$37$		−	8.00000i		−	1.31519i	0.753371	−	0.657596i	$-0.228427\pi$
$38$		0			0
$39$	3.00000	−	2.00000i	0.480384	−	0.320256i
$40$		0			0
$41$		−	4.00000i		−	0.624695i	−0.949968	−	0.312348i	$-0.898885\pi$
$41$		−	4.00000i		−	0.624695i	0.949968	−	0.312348i	$-0.101115\pi$
$42$		0			0
$43$	8.00000			1.21999			0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000			1.21999			0.609994	+	0.792406i	$0.291172\pi$
$44$		0			0
$45$			4.00000i			0.596285i
$46$		0			0
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	2.00000			0.280056
$52$		0			0
$53$	6.00000			0.824163			0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000			0.824163			0.412082	+	0.911147i	$0.364802\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	8.00000i		−	1.04151i	−0.853706	−	0.520756i	$-0.825650\pi$
$59$		−	8.00000i		−	1.04151i	0.853706	−	0.520756i	$-0.174350\pi$
$60$		0			0
$61$	−10.0000			−1.28037			−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000			−1.28037			−0.640184	+	0.768221i	$0.721142\pi$
$62$		0			0
$63$		−	1.00000i		−	0.125988i
$64$		0			0
$65$	−8.00000	−	12.0000i	−0.992278	−	1.48842i
$66$		0			0
$67$		−	8.00000i		−	0.977356i	−0.872464	−	0.488678i	$-0.837479\pi$
$67$		−	8.00000i		−	0.977356i	0.872464	−	0.488678i	$-0.162521\pi$
$68$		0			0
$69$	4.00000			0.481543
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$			4.00000i			0.468165i	0.972217	+	0.234082i	$0.0752085\pi$
$73$			4.00000i			0.468165i	−0.972217	+	0.234082i	$0.924791\pi$
$74$		0			0
$75$	11.0000			1.27017
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−12.0000			−1.35011			−0.675053	−	0.737769i	$-0.735879\pi$
$79$	−12.0000			−1.35011			−0.675053	+	0.737769i	$0.735879\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			8.00000i			0.878114i	0.898459	+	0.439057i	$0.144687\pi$
$83$			8.00000i			0.878114i	−0.898459	+	0.439057i	$0.855313\pi$
$84$		0			0
$85$		−	8.00000i		−	0.867722i
$86$		0			0
$87$	−2.00000			−0.214423
$88$		0			0
$89$			12.0000i			1.27200i	0.771690	+	0.635999i	$0.219412\pi$
$89$			12.0000i			1.27200i	−0.771690	+	0.635999i	$0.780588\pi$
$90$		0			0
$91$	2.00000	+	3.00000i	0.209657	+	0.314485i
$92$		0			0
$93$			8.00000i			0.829561i
$94$		0			0
$95$		0			0
$96$		0			0
$97$			12.0000i			1.21842i	0.793011	+	0.609208i	$0.208512\pi$
$97$			12.0000i			1.21842i	−0.793011	+	0.609208i	$0.791488\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	14.0000			1.39305			0.696526	−	0.717532i	$-0.254728\pi$
$101$	14.0000			1.39305			0.696526	+	0.717532i	$0.254728\pi$
$102$		0			0
$103$	−8.00000			−0.788263			−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000			−0.788263			−0.394132	+	0.919054i	$0.628955\pi$
$104$		0			0
$105$	−4.00000			−0.390360
$106$		0			0
$107$	−8.00000			−0.773389			−0.386695	−	0.922208i	$-0.626383\pi$
$107$	−8.00000			−0.773389			−0.386695	+	0.922208i	$0.626383\pi$
$108$		0			0
$109$			8.00000i			0.766261i	0.923694	+	0.383131i	$0.125154\pi$
$109$			8.00000i			0.766261i	−0.923694	+	0.383131i	$0.874846\pi$
$110$		0			0
$111$			8.00000i			0.759326i
$112$		0			0
$113$	14.0000			1.31701			0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000			1.31701			0.658505	+	0.752577i	$0.271189\pi$
$114$		0			0
$115$		−	16.0000i		−	1.49201i
$116$		0			0
$117$	−3.00000	+	2.00000i	−0.277350	+	0.184900i
$118$		0			0
$119$			2.00000i			0.183340i
$120$		0			0
$121$	11.0000			1.00000
$122$		0			0
$123$			4.00000i			0.360668i
$124$		0			0
$125$		−	24.0000i		−	2.14663i
$126$		0			0
$127$	−20.0000			−1.77471			−0.887357	−	0.461084i	$-0.847461\pi$
$127$	−20.0000			−1.77471			−0.887357	+	0.461084i	$0.847461\pi$
$128$		0			0
$129$	−8.00000			−0.704361
$130$		0			0
$131$	−4.00000			−0.349482			−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000			−0.349482			−0.174741	+	0.984614i	$0.555909\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		−	4.00000i		−	0.344265i
$136$		0			0
$137$		−	16.0000i		−	1.36697i	−0.729964	−	0.683486i	$-0.760463\pi$
$137$		−	16.0000i		−	1.36697i	0.729964	−	0.683486i	$-0.239537\pi$
$138$		0			0
$139$	20.0000			1.69638			0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000			1.69638			0.848189	+	0.529694i	$0.177693\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$			8.00000i			0.664364i
$146$		0			0
$147$	1.00000			0.0824786
$148$		0			0
$149$		−	8.00000i		−	0.655386i	−0.944784	−	0.327693i	$-0.893729\pi$
$149$		−	8.00000i		−	0.655386i	0.944784	−	0.327693i	$-0.106271\pi$
$150$		0			0
$151$		−	8.00000i		−	0.651031i	−0.945537	−	0.325515i	$-0.894462\pi$
$151$		−	8.00000i		−	0.651031i	0.945537	−	0.325515i	$-0.105538\pi$
$152$		0			0
$153$	−2.00000			−0.161690
$154$		0			0
$155$	32.0000			2.57030
$156$		0			0
$157$	10.0000			0.798087			0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000			0.798087			0.399043	+	0.916932i	$0.369342\pi$
$158$		0			0
$159$	−6.00000			−0.475831
$160$		0			0
$161$			4.00000i			0.315244i
$162$		0			0
$163$		−	8.00000i		−	0.626608i	−0.949653	−	0.313304i	$-0.898564\pi$
$163$		−	8.00000i		−	0.626608i	0.949653	−	0.313304i	$-0.101436\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	5.00000	−	12.0000i	0.384615	−	0.923077i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	6.00000			0.456172			0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000			0.456172			0.228086	+	0.973641i	$0.426753\pi$
$174$		0			0
$175$			11.0000i			0.831522i
$176$		0			0
$177$			8.00000i			0.601317i
$178$		0			0
$179$	24.0000			1.79384			0.896922	−	0.442189i	$-0.145798\pi$
$179$	24.0000			1.79384			0.896922	+	0.442189i	$0.145798\pi$
$180$		0			0
$181$	−14.0000			−1.04061			−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000			−1.04061			−0.520306	+	0.853980i	$0.674182\pi$
$182$		0			0
$183$	10.0000			0.739221
$184$		0			0
$185$	32.0000			2.35269
$186$		0			0
$187$		0			0
$188$		0			0
$189$			1.00000i			0.0727393i
$190$		0			0
$191$	12.0000			0.868290			0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000			0.868290			0.434145	+	0.900843i	$0.357051\pi$
$192$		0			0
$193$		−	16.0000i		−	1.15171i	−0.817554	−	0.575853i	$-0.804670\pi$
$193$		−	16.0000i		−	1.15171i	0.817554	−	0.575853i	$-0.195330\pi$
$194$		0			0
$195$	8.00000	+	12.0000i	0.572892	+	0.859338i
$196$		0			0
$197$		−	8.00000i		−	0.569976i	−0.958531	−	0.284988i	$-0.908010\pi$
$197$		−	8.00000i		−	0.569976i	0.958531	−	0.284988i	$-0.0919897\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$			8.00000i			0.564276i
$202$		0			0
$203$		−	2.00000i		−	0.140372i
$204$		0			0
$205$	16.0000			1.11749
$206$		0			0
$207$	−4.00000			−0.278019
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0			−	1.00000i	$-0.5\pi$
$211$		0			0				1.00000i	$0.5\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			32.0000i			2.18238i
$216$		0			0
$217$	−8.00000			−0.543075
$218$		0			0
$219$		−	4.00000i		−	0.270295i
$220$		0			0
$221$	6.00000	−	4.00000i	0.403604	−	0.269069i
$222$		0			0
$223$		−	8.00000i		−	0.535720i	−0.963458	−	0.267860i	$-0.913684\pi$
$223$		−	8.00000i		−	0.535720i	0.963458	−	0.267860i	$-0.0863164\pi$
$224$		0			0
$225$	−11.0000			−0.733333
$226$		0			0
$227$			8.00000i			0.530979i	0.964114	+	0.265489i	$0.0855335\pi$
$227$			8.00000i			0.530979i	−0.964114	+	0.265489i	$0.914466\pi$
$228$		0			0
$229$		−	28.0000i		−	1.85029i	−0.379611	−	0.925146i	$-0.623942\pi$
$229$		−	28.0000i		−	1.85029i	0.379611	−	0.925146i	$-0.376058\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−6.00000			−0.393073			−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000			−0.393073			−0.196537	+	0.980497i	$0.562969\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	12.0000			0.779484
$238$		0			0
$239$		−	16.0000i		−	1.03495i	−0.855697	−	0.517477i	$-0.826871\pi$
$239$		−	16.0000i		−	1.03495i	0.855697	−	0.517477i	$-0.173129\pi$
$240$		0			0
$241$			4.00000i			0.257663i	0.991667	+	0.128831i	$0.0411226\pi$
$241$			4.00000i			0.257663i	−0.991667	+	0.128831i	$0.958877\pi$
$242$		0			0
$243$	−1.00000			−0.0641500
$244$		0			0
$245$		−	4.00000i		−	0.255551i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		−	8.00000i		−	0.506979i
$250$		0			0
$251$	−20.0000			−1.26239			−0.631194	−	0.775625i	$-0.717435\pi$
$251$	−20.0000			−1.26239			−0.631194	+	0.775625i	$0.717435\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$			8.00000i			0.500979i
$256$		0			0
$257$	18.0000			1.12281			0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000			1.12281			0.561405	+	0.827541i	$0.310261\pi$
$258$		0			0
$259$	−8.00000			−0.497096
$260$		0			0
$261$	2.00000			0.123797
$262$		0			0
$263$	−12.0000			−0.739952			−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000			−0.739952			−0.369976	+	0.929041i	$0.620634\pi$
$264$		0			0
$265$			24.0000i			1.47431i
$266$		0			0
$267$		−	12.0000i		−	0.734388i
$268$		0			0
$269$	26.0000			1.58525			0.792624	−	0.609711i	$-0.208714\pi$
$269$	26.0000			1.58525			0.792624	+	0.609711i	$0.208714\pi$
$270$		0			0
$271$			8.00000i			0.485965i	0.970031	+	0.242983i	$0.0781258\pi$
$271$			8.00000i			0.485965i	−0.970031	+	0.242983i	$0.921874\pi$
$272$		0			0
$273$	−2.00000	−	3.00000i	−0.121046	−	0.181568i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−26.0000			−1.56219			−0.781094	−	0.624413i	$-0.785338\pi$
$277$	−26.0000			−1.56219			−0.781094	+	0.624413i	$0.785338\pi$
$278$		0			0
$279$		−	8.00000i		−	0.478947i
$280$		0			0
$281$		−	32.0000i		−	1.90896i	−0.298275	−	0.954480i	$-0.596411\pi$
$281$		−	32.0000i		−	1.90896i	0.298275	−	0.954480i	$-0.403589\pi$
$282$		0			0
$283$	−4.00000			−0.237775			−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000			−0.237775			−0.118888	+	0.992908i	$0.537933\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−4.00000			−0.236113
$288$		0			0
$289$	−13.0000			−0.764706
$290$		0			0
$291$		−	12.0000i		−	0.703452i
$292$		0			0
$293$		−	4.00000i		−	0.233682i	−0.993151	−	0.116841i	$-0.962723\pi$
$293$		−	4.00000i		−	0.233682i	0.993151	−	0.116841i	$-0.0372769\pi$
$294$		0			0
$295$	32.0000			1.86311
$296$		0			0
$297$		0			0
$298$		0			0
$299$	12.0000	−	8.00000i	0.693978	−	0.462652i
$300$		0			0
$301$		−	8.00000i		−	0.461112i
$302$		0			0
$303$	−14.0000			−0.804279
$304$		0			0
$305$		−	40.0000i		−	2.29039i
$306$		0			0
$307$			16.0000i			0.913168i	0.889680	+	0.456584i	$0.150927\pi$
$307$			16.0000i			0.913168i	−0.889680	+	0.456584i	$0.849073\pi$
$308$		0			0
$309$	8.00000			0.455104
$310$		0			0
$311$	−32.0000			−1.81455			−0.907277	−	0.420534i	$-0.861843\pi$
$311$	−32.0000			−1.81455			−0.907277	+	0.420534i	$0.861843\pi$
$312$		0			0
$313$	22.0000			1.24351			0.621757	−	0.783210i	$-0.286419\pi$
$313$	22.0000			1.24351			0.621757	+	0.783210i	$0.286419\pi$
$314$		0			0
$315$	4.00000			0.225374
$316$		0			0
$317$			24.0000i			1.34797i	0.738743	+	0.673987i	$0.235420\pi$
$317$			24.0000i			1.34797i	−0.738743	+	0.673987i	$0.764580\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	8.00000			0.446516
$322$		0			0
$323$		0			0
$324$		0			0
$325$	33.0000	−	22.0000i	1.83051	−	1.22034i
$326$		0			0
$327$		−	8.00000i		−	0.442401i
$328$		0			0
$329$		0			0
$330$		0			0
$331$			8.00000i			0.439720i	0.975531	+	0.219860i	$0.0705600\pi$
$331$			8.00000i			0.439720i	−0.975531	+	0.219860i	$0.929440\pi$
$332$		0			0
$333$		−	8.00000i		−	0.438397i
$334$		0			0
$335$	32.0000			1.74835
$336$		0			0
$337$	−2.00000			−0.108947			−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000			−0.108947			−0.0544735	+	0.998515i	$0.517348\pi$
$338$		0			0
$339$	−14.0000			−0.760376
$340$		0			0
$341$		0			0
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$			16.0000i			0.861411i
$346$		0			0
$347$	−24.0000			−1.28839			−0.644194	−	0.764862i	$-0.722807\pi$
$347$	−24.0000			−1.28839			−0.644194	+	0.764862i	$0.722807\pi$
$348$		0			0
$349$			28.0000i			1.49881i	0.662114	+	0.749403i	$0.269659\pi$
$349$			28.0000i			1.49881i	−0.662114	+	0.749403i	$0.730341\pi$
$350$		0			0
$351$	3.00000	−	2.00000i	0.160128	−	0.106752i
$352$		0			0
$353$			4.00000i			0.212899i	0.994318	+	0.106449i	$0.0339482\pi$
$353$			4.00000i			0.212899i	−0.994318	+	0.106449i	$0.966052\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		−	2.00000i		−	0.105851i
$358$		0			0
$359$		−	32.0000i		−	1.68890i	−0.535638	−	0.844448i	$-0.679929\pi$
$359$		−	32.0000i		−	1.68890i	0.535638	−	0.844448i	$-0.320071\pi$
$360$		0			0
$361$	19.0000			1.00000
$362$		0			0
$363$	−11.0000			−0.577350
$364$		0			0
$365$	−16.0000			−0.837478
$366$		0			0
$367$	16.0000			0.835193			0.417597	−	0.908633i	$-0.362873\pi$
$367$	16.0000			0.835193			0.417597	+	0.908633i	$0.362873\pi$
$368$		0			0
$369$		−	4.00000i		−	0.208232i
$370$		0			0
$371$		−	6.00000i		−	0.311504i
$372$		0			0
$373$	−6.00000			−0.310668			−0.155334	−	0.987862i	$-0.549645\pi$
$373$	−6.00000			−0.310668			−0.155334	+	0.987862i	$0.549645\pi$
$374$		0			0
$375$			24.0000i			1.23935i
$376$		0			0
$377$	−6.00000	+	4.00000i	−0.309016	+	0.206010i
$378$		0			0
$379$		−	8.00000i		−	0.410932i	−0.978664	−	0.205466i	$-0.934129\pi$
$379$		−	8.00000i		−	0.410932i	0.978664	−	0.205466i	$-0.0658711\pi$
$380$		0			0
$381$	20.0000			1.02463
$382$		0			0
$383$		−	16.0000i		−	0.817562i	−0.912633	−	0.408781i	$-0.865954\pi$
$383$		−	16.0000i		−	0.817562i	0.912633	−	0.408781i	$-0.134046\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	8.00000			0.406663
$388$		0			0
$389$	−6.00000			−0.304212			−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000			−0.304212			−0.152106	+	0.988364i	$0.548606\pi$
$390$		0			0
$391$	8.00000			0.404577
$392$		0			0
$393$	4.00000			0.201773
$394$		0			0
$395$		−	48.0000i		−	2.41514i
$396$		0			0
$397$		−	20.0000i		−	1.00377i	−0.864934	−	0.501886i	$-0.832640\pi$
$397$		−	20.0000i		−	1.00377i	0.864934	−	0.501886i	$-0.167360\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$	16.0000	+	24.0000i	0.797017	+	1.19553i
$404$		0			0
$405$			4.00000i			0.198762i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		−	4.00000i		−	0.197787i	−0.995098	−	0.0988936i	$-0.968470\pi$
$409$		−	4.00000i		−	0.197787i	0.995098	−	0.0988936i	$-0.0315304\pi$
$410$		0			0
$411$			16.0000i			0.789222i
$412$		0			0
$413$	−8.00000			−0.393654
$414$		0			0
$415$	−32.0000			−1.57082
$416$		0			0
$417$	−20.0000			−0.979404
$418$		0			0
$419$	20.0000			0.977064			0.488532	−	0.872546i	$-0.337533\pi$
$419$	20.0000			0.977064			0.488532	+	0.872546i	$0.337533\pi$
$420$		0			0
$421$		−	32.0000i		−	1.55958i	−0.626038	−	0.779792i	$-0.715325\pi$
$421$		−	32.0000i		−	1.55958i	0.626038	−	0.779792i	$-0.284675\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	22.0000			1.06716
$426$		0			0
$427$			10.0000i			0.483934i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	32.0000i		−	1.54139i	−0.637207	−	0.770693i	$-0.719910\pi$
$431$		−	32.0000i		−	1.54139i	0.637207	−	0.770693i	$-0.280090\pi$
$432$		0			0
$433$	−14.0000			−0.672797			−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000			−0.672797			−0.336399	+	0.941720i	$0.609209\pi$
$434$		0			0
$435$		−	8.00000i		−	0.383571i
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	−1.00000			−0.0476190
$442$		0			0
$443$	16.0000			0.760183			0.380091	−	0.924949i	$-0.375893\pi$
$443$	16.0000			0.760183			0.380091	+	0.924949i	$0.375893\pi$
$444$		0			0
$445$	−48.0000			−2.27542
$446$		0			0
$447$			8.00000i			0.378387i
$448$		0			0
$449$		−	24.0000i		−	1.13263i	−0.824189	−	0.566315i	$-0.808369\pi$
$449$		−	24.0000i		−	1.13263i	0.824189	−	0.566315i	$-0.191631\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$			8.00000i			0.375873i
$454$		0			0
$455$	−12.0000	+	8.00000i	−0.562569	+	0.375046i
$456$		0			0
$457$			32.0000i			1.49690i	0.663193	+	0.748448i	$0.269201\pi$
$457$			32.0000i			1.49690i	−0.663193	+	0.748448i	$0.730799\pi$
$458$		0			0
$459$	2.00000			0.0933520
$460$		0			0
$461$		−	36.0000i		−	1.67669i	−0.545142	−	0.838344i	$-0.683524\pi$
$461$		−	36.0000i		−	1.67669i	0.545142	−	0.838344i	$-0.316476\pi$
$462$		0			0
$463$			24.0000i			1.11537i	0.830051	+	0.557687i	$0.188311\pi$
$463$			24.0000i			1.11537i	−0.830051	+	0.557687i	$0.811689\pi$
$464$		0			0
$465$	−32.0000			−1.48396
$466$		0			0
$467$	−36.0000			−1.66588			−0.832941	−	0.553362i	$-0.813345\pi$
$467$	−36.0000			−1.66588			−0.832941	+	0.553362i	$0.813345\pi$
$468$		0			0
$469$	−8.00000			−0.369406
$470$		0			0
$471$	−10.0000			−0.460776
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	6.00000			0.274721
$478$		0			0
$479$		0			0		1.00000			$0$
$479$		0			0		−1.00000			$\pi$
$480$		0			0
$481$	16.0000	+	24.0000i	0.729537	+	1.09431i
$482$		0			0
$483$		−	4.00000i		−	0.182006i
$484$		0			0
$485$	−48.0000			−2.17957
$486$		0			0
$487$			24.0000i			1.08754i	0.839233	+	0.543772i	$0.183004\pi$
$487$			24.0000i			1.08754i	−0.839233	+	0.543772i	$0.816996\pi$
$488$		0			0
$489$			8.00000i			0.361773i
$490$		0			0
$491$	40.0000			1.80517			0.902587	−	0.430507i	$-0.141665\pi$
$491$	40.0000			1.80517			0.902587	+	0.430507i	$0.141665\pi$
$492$		0			0
$493$	−4.00000			−0.180151
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		−	40.0000i		−	1.79065i	−0.445418	−	0.895323i	$-0.646945\pi$
$499$		−	40.0000i		−	1.79065i	0.445418	−	0.895323i	$-0.353055\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$			56.0000i			2.49197i
$506$		0			0
$507$	−5.00000	+	12.0000i	−0.222058	+	0.532939i
$508$		0			0
$509$			36.0000i			1.59567i	0.602875	+	0.797836i	$0.294022\pi$
$509$			36.0000i			1.59567i	−0.602875	+	0.797836i	$0.705978\pi$
$510$		0			0
$511$	4.00000			0.176950
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	32.0000i		−	1.41009i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−6.00000			−0.263371
$520$		0			0
$521$	−42.0000			−1.84005			−0.920027	−	0.391856i	$-0.871833\pi$
$521$	−42.0000			−1.84005			−0.920027	+	0.391856i	$0.871833\pi$
$522$		0			0
$523$	20.0000			0.874539			0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000			0.874539			0.437269	+	0.899331i	$0.355946\pi$
$524$		0			0
$525$		−	11.0000i		−	0.480079i
$526$		0			0
$527$			16.0000i			0.696971i
$528$		0			0
$529$	−7.00000			−0.304348
$530$		0			0
$531$		−	8.00000i		−	0.347170i
$532$		0			0
$533$	8.00000	+	12.0000i	0.346518	+	0.519778i
$534$		0			0
$535$		−	32.0000i		−	1.38348i
$536$		0			0
$537$	−24.0000			−1.03568
$538$		0			0
$539$		0			0
$540$		0			0
$541$		−	16.0000i		−	0.687894i	−0.938989	−	0.343947i	$-0.888236\pi$
$541$		−	16.0000i		−	0.687894i	0.938989	−	0.343947i	$-0.111764\pi$
$542$		0			0
$543$	14.0000			0.600798
$544$		0			0
$545$	−32.0000			−1.37073
$546$		0			0
$547$	8.00000			0.342055			0.171028	−	0.985266i	$-0.445291\pi$
$547$	8.00000			0.342055			0.171028	+	0.985266i	$0.445291\pi$
$548$		0			0
$549$	−10.0000			−0.426790
$550$		0			0
$551$		0			0
$552$		0			0
$553$			12.0000i			0.510292i
$554$		0			0
$555$	−32.0000			−1.35832
$556$		0			0
$557$			24.0000i			1.01691i	0.861088	+	0.508456i	$0.169784\pi$
$557$			24.0000i			1.01691i	−0.861088	+	0.508456i	$0.830216\pi$
$558$		0			0
$559$	−24.0000	+	16.0000i	−1.01509	+	0.676728i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−20.0000			−0.842900			−0.421450	−	0.906852i	$-0.638479\pi$
$563$	−20.0000			−0.842900			−0.421450	+	0.906852i	$0.638479\pi$
$564$		0			0
$565$			56.0000i			2.35594i
$566$		0			0
$567$		−	1.00000i		−	0.0419961i
$568$		0			0
$569$	10.0000			0.419222			0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000			0.419222			0.209611	+	0.977785i	$0.432780\pi$
$570$		0			0
$571$	−32.0000			−1.33916			−0.669579	−	0.742741i	$-0.733526\pi$
$571$	−32.0000			−1.33916			−0.669579	+	0.742741i	$0.733526\pi$
$572$		0			0
$573$	−12.0000			−0.501307
$574$		0			0
$575$	44.0000			1.83493
$576$		0			0
$577$		−	20.0000i		−	0.832611i	−0.909225	−	0.416305i	$-0.863325\pi$
$577$		−	20.0000i		−	0.832611i	0.909225	−	0.416305i	$-0.136675\pi$
$578$		0			0
$579$			16.0000i			0.664937i
$580$		0			0
$581$	8.00000			0.331896
$582$		0			0
$583$		0			0
$584$		0			0
$585$	−8.00000	−	12.0000i	−0.330759	−	0.496139i
$586$		0			0
$587$		−	8.00000i		−	0.330195i	−0.986277	−	0.165098i	$-0.947206\pi$
$587$		−	8.00000i		−	0.330195i	0.986277	−	0.165098i	$-0.0527939\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$			8.00000i			0.329076i
$592$		0			0
$593$		−	44.0000i		−	1.80686i	−0.428732	−	0.903432i	$-0.641040\pi$
$593$		−	44.0000i		−	1.80686i	0.428732	−	0.903432i	$-0.358960\pi$
$594$		0			0
$595$	−8.00000			−0.327968
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−12.0000			−0.490307			−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000			−0.490307			−0.245153	+	0.969484i	$0.578838\pi$
$600$		0			0
$601$	22.0000			0.897399			0.448699	−	0.893683i	$-0.351887\pi$
$601$	22.0000			0.897399			0.448699	+	0.893683i	$0.351887\pi$
$602$		0			0
$603$		−	8.00000i		−	0.325785i
$604$		0			0
$605$			44.0000i			1.78885i
$606$		0			0
$607$	8.00000			0.324710			0.162355	−	0.986732i	$-0.448091\pi$
$607$	8.00000			0.324710			0.162355	+	0.986732i	$0.448091\pi$
$608$		0			0
$609$			2.00000i			0.0810441i
$610$		0			0
$611$		0			0
$612$		0			0
$613$			8.00000i			0.323117i	0.986863	+	0.161558i	$0.0516520\pi$
$613$			8.00000i			0.323117i	−0.986863	+	0.161558i	$0.948348\pi$
$614$		0			0
$615$	−16.0000			−0.645182
$616$		0			0
$617$			16.0000i			0.644136i	0.946717	+	0.322068i	$0.104378\pi$
$617$			16.0000i			0.644136i	−0.946717	+	0.322068i	$0.895622\pi$
$618$		0			0
$619$		−	32.0000i		−	1.28619i	−0.765787	−	0.643094i	$-0.777650\pi$
$619$		−	32.0000i		−	1.28619i	0.765787	−	0.643094i	$-0.222350\pi$
$620$		0			0
$621$	4.00000			0.160514
$622$		0			0
$623$	12.0000			0.480770
$624$		0			0
$625$	41.0000			1.64000
$626$		0			0
$627$		0			0
$628$		0			0
$629$			16.0000i			0.637962i
$630$		0			0
$631$			40.0000i			1.59237i	0.605050	+	0.796187i	$0.293153\pi$
$631$			40.0000i			1.59237i	−0.605050	+	0.796187i	$0.706847\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	80.0000i		−	3.17470i
$636$		0			0
$637$	3.00000	−	2.00000i	0.118864	−	0.0792429i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−18.0000			−0.710957			−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000			−0.710957			−0.355479	+	0.934684i	$0.615682\pi$
$642$		0			0
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$		0			0
$645$		−	32.0000i		−	1.26000i
$646$		0			0
$647$	−40.0000			−1.57256			−0.786281	−	0.617869i	$-0.787996\pi$
$647$	−40.0000			−1.57256			−0.786281	+	0.617869i	$0.787996\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	8.00000			0.313545
$652$		0			0
$653$	−14.0000			−0.547862			−0.273931	−	0.961749i	$-0.588324\pi$
$653$	−14.0000			−0.547862			−0.273931	+	0.961749i	$0.588324\pi$
$654$		0			0
$655$		−	16.0000i		−	0.625172i
$656$		0			0
$657$			4.00000i			0.156055i
$658$		0			0
$659$	40.0000			1.55818			0.779089	−	0.626913i	$-0.215682\pi$
$659$	40.0000			1.55818			0.779089	+	0.626913i	$0.215682\pi$
$660$		0			0
$661$		−	20.0000i		−	0.777910i	−0.921257	−	0.388955i	$-0.872836\pi$
$661$		−	20.0000i		−	0.777910i	0.921257	−	0.388955i	$-0.127164\pi$
$662$		0			0
$663$	−6.00000	+	4.00000i	−0.233021	+	0.155347i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−8.00000			−0.309761
$668$		0			0
$669$			8.00000i			0.309298i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−18.0000			−0.693849			−0.346925	−	0.937893i	$-0.612774\pi$
$673$	−18.0000			−0.693849			−0.346925	+	0.937893i	$0.612774\pi$
$674$		0			0
$675$	11.0000			0.423390
$676$		0			0
$677$	−18.0000			−0.691796			−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000			−0.691796			−0.345898	+	0.938272i	$0.612426\pi$
$678$		0			0
$679$	12.0000			0.460518
$680$		0			0
$681$		−	8.00000i		−	0.306561i
$682$		0			0
$683$		−	48.0000i		−	1.83667i	−0.395805	−	0.918334i	$-0.629534\pi$
$683$		−	48.0000i		−	1.83667i	0.395805	−	0.918334i	$-0.370466\pi$
$684$		0			0
$685$	64.0000			2.44531
$686$		0			0
$687$			28.0000i			1.06827i
$688$		0			0
$689$	−18.0000	+	12.0000i	−0.685745	+	0.457164i
$690$		0			0
$691$		−	16.0000i		−	0.608669i	−0.952565	−	0.304334i	$-0.901566\pi$
$691$		−	16.0000i		−	0.608669i	0.952565	−	0.304334i	$-0.0984340\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			80.0000i			3.03457i
$696$		0			0
$697$			8.00000i			0.303022i
$698$		0			0
$699$	6.00000			0.226941
$700$		0			0
$701$	34.0000			1.28416			0.642081	−	0.766637i	$-0.278071\pi$
$701$	34.0000			1.28416			0.642081	+	0.766637i	$0.278071\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	14.0000i		−	0.526524i
$708$		0			0
$709$			24.0000i			0.901339i	0.892691	+	0.450669i	$0.148815\pi$
$709$			24.0000i			0.901339i	−0.892691	+	0.450669i	$0.851185\pi$
$710$		0			0
$711$	−12.0000			−0.450035
$712$		0			0
$713$			32.0000i			1.19841i
$714$		0			0
$715$		0			0
$716$		0			0
$717$			16.0000i			0.597531i
$718$		0			0
$719$	16.0000			0.596699			0.298350	−	0.954457i	$-0.403564\pi$
$719$	16.0000			0.596699			0.298350	+	0.954457i	$0.403564\pi$
$720$		0			0
$721$			8.00000i			0.297936i
$722$		0			0
$723$		−	4.00000i		−	0.148762i
$724$		0			0
$725$	−22.0000			−0.817059
$726$		0			0
$727$	−8.00000			−0.296704			−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000			−0.296704			−0.148352	+	0.988935i	$0.547397\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	−16.0000			−0.591781
$732$		0			0
$733$			20.0000i			0.738717i	0.929287	+	0.369358i	$0.120423\pi$
$733$			20.0000i			0.738717i	−0.929287	+	0.369358i	$0.879577\pi$
$734$		0			0
$735$			4.00000i			0.147542i
$736$		0			0
$737$		0			0
$738$		0			0
$739$			40.0000i			1.47142i	0.677295	+	0.735712i	$0.263152\pi$
$739$			40.0000i			1.47142i	−0.677295	+	0.735712i	$0.736848\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			16.0000i			0.586983i	0.955962	+	0.293492i	$0.0948173\pi$
$743$			16.0000i			0.586983i	−0.955962	+	0.293492i	$0.905183\pi$
$744$		0			0
$745$	32.0000			1.17239
$746$		0			0
$747$			8.00000i			0.292705i
$748$		0			0
$749$			8.00000i			0.292314i
$750$		0			0
$751$	−20.0000			−0.729810			−0.364905	−	0.931045i	$-0.618899\pi$
$751$	−20.0000			−0.729810			−0.364905	+	0.931045i	$0.618899\pi$
$752$		0			0
$753$	20.0000			0.728841
$754$		0			0
$755$	32.0000			1.16460
$756$		0			0
$757$	−10.0000			−0.363456			−0.181728	−	0.983349i	$-0.558169\pi$
$757$	−10.0000			−0.363456			−0.181728	+	0.983349i	$0.558169\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	12.0000i		−	0.435000i	−0.976060	−	0.217500i	$-0.930210\pi$
$761$		−	12.0000i		−	0.435000i	0.976060	−	0.217500i	$-0.0697902\pi$
$762$		0			0
$763$	8.00000			0.289619
$764$		0			0
$765$		−	8.00000i		−	0.289241i
$766$		0			0
$767$	16.0000	+	24.0000i	0.577727	+	0.866590i
$768$		0			0
$769$		−	12.0000i		−	0.432731i	−0.976312	−	0.216366i	$-0.930580\pi$
$769$		−	12.0000i		−	0.432731i	0.976312	−	0.216366i	$-0.0694203\pi$
$770$		0			0
$771$	−18.0000			−0.648254
$772$		0			0
$773$		−	12.0000i		−	0.431610i	−0.976436	−	0.215805i	$-0.930762\pi$
$773$		−	12.0000i		−	0.431610i	0.976436	−	0.215805i	$-0.0692376\pi$
$774$		0			0
$775$			88.0000i			3.16105i
$776$		0			0
$777$	8.00000			0.286998
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−2.00000			−0.0714742
$784$		0			0
$785$			40.0000i			1.42766i
$786$		0			0
$787$		−	16.0000i		−	0.570338i	−0.958477	−	0.285169i	$-0.907950\pi$
$787$		−	16.0000i		−	0.570338i	0.958477	−	0.285169i	$-0.0920498\pi$
$788$		0			0
$789$	12.0000			0.427211
$790$		0			0
$791$		−	14.0000i		−	0.497783i
$792$		0			0
$793$	30.0000	−	20.0000i	1.06533	−	0.710221i
$794$		0			0
$795$		−	24.0000i		−	0.851192i
$796$		0			0
$797$	10.0000			0.354218			0.177109	−	0.984191i	$-0.443325\pi$
$797$	10.0000			0.354218			0.177109	+	0.984191i	$0.443325\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$			12.0000i			0.423999i
$802$		0			0
$803$		0			0
$804$		0			0
$805$	−16.0000			−0.563926
$806$		0			0
$807$	−26.0000			−0.915243
$808$		0			0
$809$	−26.0000			−0.914111			−0.457056	−	0.889438i	$-0.651096\pi$
$809$	−26.0000			−0.914111			−0.457056	+	0.889438i	$0.651096\pi$
$810$		0			0
$811$			16.0000i			0.561836i	0.959732	+	0.280918i	$0.0906389\pi$
$811$			16.0000i			0.561836i	−0.959732	+	0.280918i	$0.909361\pi$
$812$		0			0
$813$		−	8.00000i		−	0.280572i
$814$		0			0
$815$	32.0000			1.12091
$816$		0			0
$817$		0			0
$818$		0			0
$819$	2.00000	+	3.00000i	0.0698857	+	0.104828i
$820$		0			0
$821$		−	24.0000i		−	0.837606i	−0.908077	−	0.418803i	$-0.862450\pi$
$821$		−	24.0000i		−	0.837606i	0.908077	−	0.418803i	$-0.137550\pi$
$822$		0			0
$823$	−12.0000			−0.418294			−0.209147	−	0.977884i	$-0.567069\pi$
$823$	−12.0000			−0.418294			−0.209147	+	0.977884i	$0.567069\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			48.0000i			1.66912i	0.550914	+	0.834562i	$0.314279\pi$
$827$			48.0000i			1.66912i	−0.550914	+	0.834562i	$0.685721\pi$
$828$		0			0
$829$	−42.0000			−1.45872			−0.729360	−	0.684130i	$-0.760182\pi$
$829$	−42.0000			−1.45872			−0.729360	+	0.684130i	$0.760182\pi$
$830$		0			0
$831$	26.0000			0.901930
$832$		0			0
$833$	2.00000			0.0692959
$834$		0			0
$835$		0			0
$836$		0			0
$837$			8.00000i			0.276520i
$838$		0			0
$839$		−	16.0000i		−	0.552381i	−0.961103	−	0.276191i	$-0.910928\pi$
$839$		−	16.0000i		−	0.552381i	0.961103	−	0.276191i	$-0.0890721\pi$
$840$		0			0
$841$	−25.0000			−0.862069
$842$		0			0
$843$			32.0000i			1.10214i
$844$		0			0
$845$	48.0000	+	20.0000i	1.65125	+	0.688021i
$846$		0			0
$847$		−	11.0000i		−	0.377964i
$848$		0			0
$849$	4.00000			0.137280
$850$		0			0
$851$			32.0000i			1.09695i
$852$		0			0
$853$		−	44.0000i		−	1.50653i	−0.657716	−	0.753266i	$-0.728477\pi$
$853$		−	44.0000i		−	1.50653i	0.657716	−	0.753266i	$-0.271523\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	26.0000			0.888143			0.444072	−	0.895991i	$-0.353534\pi$
$857$	26.0000			0.888143			0.444072	+	0.895991i	$0.353534\pi$
$858$		0			0
$859$	44.0000			1.50126			0.750630	−	0.660722i	$-0.229750\pi$
$859$	44.0000			1.50126			0.750630	+	0.660722i	$0.229750\pi$
$860$		0			0
$861$	4.00000			0.136320
$862$		0			0
$863$			16.0000i			0.544646i	0.962206	+	0.272323i	$0.0877920\pi$
$863$			16.0000i			0.544646i	−0.962206	+	0.272323i	$0.912208\pi$
$864$		0			0
$865$			24.0000i			0.816024i
$866$		0			0
$867$	13.0000			0.441503
$868$		0			0
$869$		0			0
$870$		0			0
$871$	16.0000	+	24.0000i	0.542139	+	0.813209i
$872$		0			0
$873$			12.0000i			0.406138i
$874$		0			0
$875$	−24.0000			−0.811348
$876$		0			0
$877$		−	8.00000i		−	0.270141i	−0.990836	−	0.135070i	$-0.956874\pi$
$877$		−	8.00000i		−	0.270141i	0.990836	−	0.135070i	$-0.0431261\pi$
$878$		0			0
$879$			4.00000i			0.134917i
$880$		0			0
$881$	−30.0000			−1.01073			−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000			−1.01073			−0.505363	+	0.862907i	$0.668641\pi$
$882$		0			0
$883$	−48.0000			−1.61533			−0.807664	−	0.589643i	$-0.799269\pi$
$883$	−48.0000			−1.61533			−0.807664	+	0.589643i	$0.799269\pi$
$884$		0			0
$885$	−32.0000			−1.07567
$886$		0			0
$887$	48.0000			1.61168			0.805841	−	0.592132i	$-0.201714\pi$
$887$	48.0000			1.61168			0.805841	+	0.592132i	$0.201714\pi$
$888$		0			0
$889$			20.0000i			0.670778i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$			96.0000i			3.20893i
$896$		0			0
$897$	−12.0000	+	8.00000i	−0.400668	+	0.267112i
$898$		0			0
$899$		−	16.0000i		−	0.533630i
$900$		0			0
$901$	−12.0000			−0.399778
$902$		0			0
$903$			8.00000i			0.266223i
$904$		0			0
$905$		−	56.0000i		−	1.86150i
$906$		0			0
$907$	8.00000			0.265636			0.132818	−	0.991140i	$-0.457597\pi$
$907$	8.00000			0.265636			0.132818	+	0.991140i	$0.457597\pi$
$908$		0			0
$909$	14.0000			0.464351
$910$		0			0
$911$	−36.0000			−1.19273			−0.596367	−	0.802712i	$-0.703390\pi$
$911$	−36.0000			−1.19273			−0.596367	+	0.802712i	$0.703390\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$			40.0000i			1.32236i
$916$		0			0
$917$			4.00000i			0.132092i
$918$		0			0
$919$	12.0000			0.395843			0.197922	−	0.980218i	$-0.436581\pi$
$919$	12.0000			0.395843			0.197922	+	0.980218i	$0.436581\pi$
$920$		0			0
$921$		−	16.0000i		−	0.527218i
$922$		0			0
$923$		0			0
$924$		0			0
$925$			88.0000i			2.89342i
$926$		0			0
$927$	−8.00000			−0.262754
$928$		0			0
$929$			60.0000i			1.96854i	0.176682	+	0.984268i	$0.443464\pi$
$929$			60.0000i			1.96854i	−0.176682	+	0.984268i	$0.556536\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	32.0000			1.04763
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−38.0000			−1.24141			−0.620703	−	0.784046i	$-0.713153\pi$
$937$	−38.0000			−1.24141			−0.620703	+	0.784046i	$0.713153\pi$
$938$		0			0
$939$	−22.0000			−0.717943
$940$		0			0
$941$			28.0000i			0.912774i	0.889781	+	0.456387i	$0.150857\pi$
$941$			28.0000i			0.912774i	−0.889781	+	0.456387i	$0.849143\pi$
$942$		0			0
$943$			16.0000i			0.521032i
$944$		0			0
$945$	−4.00000			−0.130120
$946$		0			0
$947$		−	16.0000i		−	0.519930i	−0.965618	−	0.259965i	$-0.916289\pi$
$947$		−	16.0000i		−	0.519930i	0.965618	−	0.259965i	$-0.0837111\pi$
$948$		0			0
$949$	−8.00000	−	12.0000i	−0.259691	−	0.389536i
$950$		0			0
$951$		−	24.0000i		−	0.778253i
$952$		0			0
$953$	42.0000			1.36051			0.680257	−	0.732974i	$-0.261868\pi$
$953$	42.0000			1.36051			0.680257	+	0.732974i	$0.261868\pi$
$954$		0			0
$955$			48.0000i			1.55324i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−16.0000			−0.516667
$960$		0			0
$961$	−33.0000			−1.06452
$962$		0			0
$963$	−8.00000			−0.257796
$964$		0			0
$965$	64.0000			2.06023
$966$		0			0
$967$			56.0000i			1.80084i	0.435023	+	0.900419i	$0.356740\pi$
$967$			56.0000i			1.80084i	−0.435023	+	0.900419i	$0.643260\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−60.0000			−1.92549			−0.962746	−	0.270408i	$-0.912841\pi$
$971$	−60.0000			−1.92549			−0.962746	+	0.270408i	$0.912841\pi$
$972$		0			0
$973$		−	20.0000i		−	0.641171i
$974$		0			0
$975$	−33.0000	+	22.0000i	−1.05685	+	0.704564i
$976$		0			0
$977$		−	48.0000i		−	1.53566i	−0.640656	−	0.767828i	$-0.721338\pi$
$977$		−	48.0000i		−	1.53566i	0.640656	−	0.767828i	$-0.278662\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$			8.00000i			0.255420i
$982$		0			0
$983$		0			0		1.00000			$0$
$983$		0			0		−1.00000			$\pi$
$984$		0			0
$985$	32.0000			1.01960
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−32.0000			−1.01754
$990$		0			0
$991$	−4.00000			−0.127064			−0.0635321	−	0.997980i	$-0.520237\pi$
$991$	−4.00000			−0.127064			−0.0635321	+	0.997980i	$0.520237\pi$
$992$		0			0
$993$		−	8.00000i		−	0.253872i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−18.0000			−0.570066			−0.285033	−	0.958518i	$-0.592005\pi$
$997$	−18.0000			−0.570066			−0.285033	+	0.958518i	$0.592005\pi$
$998$		0			0
$999$			8.00000i			0.253109i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4368.2.h.a.337.2		2
4.3	odd	2		1092.2.e.b.337.2	yes	2
12.11	even	2		3276.2.e.b.2521.1		2
13.12	even	2	inner	4368.2.h.a.337.1		2
28.27	even	2		7644.2.e.e.4705.1		2
52.51	odd	2		1092.2.e.b.337.1	✓	2
156.155	even	2		3276.2.e.b.2521.2		2
364.363	even	2		7644.2.e.e.4705.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1092.2.e.b.337.1	✓	2	52.51	odd	2
1092.2.e.b.337.2	yes	2	4.3	odd	2
3276.2.e.b.2521.1		2	12.11	even	2
3276.2.e.b.2521.2		2	156.155	even	2
4368.2.h.a.337.1		2	13.12	even	2	inner
4368.2.h.a.337.2		2	1.1	even	1	trivial
7644.2.e.e.4705.1		2	28.27	even	2
7644.2.e.e.4705.2		2	364.363	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4368.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +4.00000i q^{5} -1.00000i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +4.00000i q^{5} -1.00000i q^{7} +1.00000 q^{9} +(-3.00000 + 2.00000i) q^{13} -4.00000i q^{15} -2.00000 q^{17} +1.00000i q^{21} -4.00000 q^{23} -11.0000 q^{25} -1.00000 q^{27} +2.00000 q^{29} -8.00000i q^{31} +4.00000 q^{35} -8.00000i q^{37} +(3.00000 - 2.00000i) q^{39} -4.00000i q^{41} +8.00000 q^{43} +4.00000i q^{45} -1.00000 q^{49} +2.00000 q^{51} +6.00000 q^{53} -8.00000i q^{59} -10.0000 q^{61} -1.00000i q^{63} +(-8.00000 - 12.0000i) q^{65} -8.00000i q^{67} +4.00000 q^{69} +4.00000i q^{73} +11.0000 q^{75} -12.0000 q^{79} +1.00000 q^{81} +8.00000i q^{83} -8.00000i q^{85} -2.00000 q^{87} +12.0000i q^{89} +(2.00000 + 3.00000i) q^{91} +8.00000i q^{93} +12.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 2 * q^9	\( 2 q - 2 q^{3} + 2 q^{9} - 6 q^{13} - 4 q^{17} - 8 q^{23} - 22 q^{25} - 2 q^{27} + 4 q^{29} + 8 q^{35} + 6 q^{39} + 16 q^{43} - 2 q^{49} + 4 q^{51} + 12 q^{53} - 20 q^{61} - 16 q^{65} + 8 q^{69} + 22 q^{75} - 24 q^{79} + 2 q^{81} - 4 q^{87} + 4 q^{91}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^9 - 6 * q^13 - 4 * q^17 - 8 * q^23 - 22 * q^25 - 2 * q^27 + 4 * q^29 + 8 * q^35 + 6 * q^39 + 16 * q^43 - 2 * q^49 + 4 * q^51 + 12 * q^53 - 20 * q^61 - 16 * q^65 + 8 * q^69 + 22 * q^75 - 24 * q^79 + 2 * q^81 - 4 * q^87 + 4 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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