LMFDB - Embedded newform 4368.2.h.r.337.1

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:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 20x^{8} + 138x^{6} + 364x^{4} + 249x^{2} + 36 $ x^10 + 20x^8 + 138x^6 + 364x^4 + 249x^2 + 36
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 2184)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.1
Root			$-2.28392i$ of defining polynomial
Character	$\chi$	$=$	4368.337
Dual form			4368.2.h.r.337.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -2.89917i q^{5} -1.00000i q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -2.89917i q^{5} -1.00000i q^{7} +1.00000 q^{9} -4.62147i q^{11} +(-2.28392 + 2.78993i) q^{13} +2.89917i q^{15} +2.68288 q^{17} -6.97302i q^{19} +1.00000i q^{21} -5.74252 q^{23} -3.40518 q^{25} -1.00000 q^{27} -9.63568 q^{29} -4.95280i q^{31} +4.62147i q^{33} -2.89917 q^{35} -3.11546i q^{37} +(2.28392 - 2.78993i) q^{39} +9.43961i q^{41} +2.54044 q^{43} -2.89917i q^{45} +9.87937i q^{47} -1.00000 q^{49} -2.68288 q^{51} +12.0885 q^{53} -13.3984 q^{55} +6.97302i q^{57} -13.4520i q^{59} -6.69532 q^{61} -1.00000i q^{63} +(8.08848 + 6.62147i) q^{65} -5.56826i q^{67} +5.74252 q^{69} -5.50601i q^{71} -6.20352i q^{73} +3.40518 q^{75} -4.62147 q^{77} -2.65548 q^{79} +1.00000 q^{81} +15.9482i q^{83} -7.77812i q^{85} +9.63568 q^{87} -2.24470i q^{89} +(2.78993 + 2.28392i) q^{91} +4.95280i q^{93} -20.2160 q^{95} -14.5731i q^{97} -4.62147i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 10 q^{3} + 10 q^{9}+O(q^{10}) $ 10 * q - 10 * q^3 + 10 * q^9	$ 10 q - 10 q^{3} + 10 q^{9} - 4 q^{13} + 10 q^{17} - 8 q^{23} - 2 q^{25} - 10 q^{27} - 44 q^{29} + 4 q^{39} - 20 q^{43} - 10 q^{49} - 10 q^{51} + 10 q^{53} + 2 q^{55} + 18 q^{61} - 30 q^{65} + 8 q^{69} + 2 q^{75} - 2 q^{77} - 2 q^{79} + 10 q^{81} + 44 q^{87} + 6 q^{91} - 44 q^{95}+O(q^{100}) $ 10 * q - 10 * q^3 + 10 * q^9 - 4 * q^13 + 10 * q^17 - 8 * q^23 - 2 * q^25 - 10 * q^27 - 44 * q^29 + 4 * q^39 - 20 * q^43 - 10 * q^49 - 10 * q^51 + 10 * q^53 + 2 * q^55 + 18 * q^61 - 30 * q^65 + 8 * q^69 + 2 * q^75 - 2 * q^77 - 2 * q^79 + 10 * q^81 + 44 * q^87 + 6 * q^91 - 44 * q^95

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
$5$		−	2.89917i		−	1.29655i	−0.761407	−	0.648274i	$-0.775491\pi$
$5$		−	2.89917i		−	1.29655i	0.761407	−	0.648274i	$-0.224509\pi$
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$		−	4.62147i		−	1.39343i	−0.717350	−	0.696713i	$-0.754645\pi$
$11$		−	4.62147i		−	1.39343i	0.717350	−	0.696713i	$-0.245355\pi$
$12$		0			0
$13$	−2.28392	+	2.78993i	−0.633445	+	0.773787i
$13$	−2.28392	+	2.78993i	−0.633445	+	0.773787i
$14$		0			0
$15$			2.89917i			0.748562i
$16$		0			0
$17$	2.68288			0.650694			0.325347	−	0.945595i	$-0.394519\pi$
$17$	2.68288			0.650694			0.325347	+	0.945595i	$0.394519\pi$
$18$		0			0
$19$		−	6.97302i		−	1.59972i	−0.600186	−	0.799860i	$-0.704907\pi$
$19$		−	6.97302i		−	1.59972i	0.600186	−	0.799860i	$-0.295093\pi$
$20$		0			0
$21$			1.00000i			0.218218i
$22$		0			0
$23$	−5.74252			−1.19740			−0.598699	−	0.800974i	$-0.704316\pi$
$23$	−5.74252			−1.19740			−0.598699	+	0.800974i	$0.704316\pi$
$24$		0			0
$25$	−3.40518			−0.681036
$26$		0			0
$27$	−1.00000			−0.192450
$28$		0			0
$29$	−9.63568			−1.78930			−0.894650	−	0.446768i	$-0.852575\pi$
$29$	−9.63568			−1.78930			−0.894650	+	0.446768i	$0.852575\pi$
$30$		0			0
$31$		−	4.95280i		−	0.889549i	−0.895643	−	0.444774i	$-0.853284\pi$
$31$		−	4.95280i		−	0.889549i	0.895643	−	0.444774i	$-0.146716\pi$
$32$		0			0
$33$			4.62147i			0.804495i
$34$		0			0
$35$	−2.89917			−0.490049
$36$		0			0
$37$		−	3.11546i		−	0.512178i	−0.966653	−	0.256089i	$-0.917566\pi$
$37$		−	3.11546i		−	0.512178i	0.966653	−	0.256089i	$-0.0824341\pi$
$38$		0			0
$39$	2.28392	−	2.78993i	0.365720	−	0.446746i
$40$		0			0
$41$			9.43961i			1.47422i	0.675773	+	0.737109i	$0.263810\pi$
$41$			9.43961i			1.47422i	−0.675773	+	0.737109i	$0.736190\pi$
$42$		0			0
$43$	2.54044			0.387413			0.193707	−	0.981060i	$-0.437949\pi$
$43$	2.54044			0.387413			0.193707	+	0.981060i	$0.437949\pi$
$44$		0			0
$45$		−	2.89917i		−	0.432183i
$46$		0			0
$47$			9.87937i			1.44105i	0.693427	+	0.720527i	$0.256100\pi$
$47$			9.87937i			1.44105i	−0.693427	+	0.720527i	$0.743900\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−2.68288			−0.375678
$52$		0			0
$53$	12.0885			1.66048			0.830240	−	0.557406i	$-0.188203\pi$
$53$	12.0885			1.66048			0.830240	+	0.557406i	$0.188203\pi$
$54$		0			0
$55$	−13.3984			−1.80664
$56$		0			0
$57$			6.97302i			0.923599i
$58$		0			0
$59$		−	13.4520i		−	1.75131i	−0.482940	−	0.875654i	$-0.660431\pi$
$59$		−	13.4520i		−	1.75131i	0.482940	−	0.875654i	$-0.339569\pi$
$60$		0			0
$61$	−6.69532			−0.857248			−0.428624	−	0.903483i	$-0.641001\pi$
$61$	−6.69532			−0.857248			−0.428624	+	0.903483i	$0.641001\pi$
$62$		0			0
$63$		−	1.00000i		−	0.125988i
$64$		0			0
$65$	8.08848	+	6.62147i	1.00325	+	0.821292i
$66$		0			0
$67$		−	5.56826i		−	0.680271i	−0.940376	−	0.340136i	$-0.889527\pi$
$67$		−	5.56826i		−	0.680271i	0.940376	−	0.340136i	$-0.110473\pi$
$68$		0			0
$69$	5.74252			0.691318
$70$		0			0
$71$		−	5.50601i		−	0.653443i	−0.945121	−	0.326722i	$-0.894056\pi$
$71$		−	5.50601i		−	0.653443i	0.945121	−	0.326722i	$-0.105944\pi$
$72$		0			0
$73$		−	6.20352i		−	0.726067i	−0.931776	−	0.363033i	$-0.881741\pi$
$73$		−	6.20352i		−	0.726067i	0.931776	−	0.363033i	$-0.118259\pi$
$74$		0			0
$75$	3.40518			0.393196
$76$		0			0
$77$	−4.62147			−0.526665
$78$		0			0
$79$	−2.65548			−0.298764			−0.149382	−	0.988780i	$-0.547728\pi$
$79$	−2.65548			−0.298764			−0.149382	+	0.988780i	$0.547728\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			15.9482i			1.75055i	0.483630	+	0.875273i	$0.339318\pi$
$83$			15.9482i			1.75055i	−0.483630	+	0.875273i	$0.660682\pi$
$84$		0			0
$85$		−	7.77812i		−	0.843655i
$86$		0			0
$87$	9.63568			1.03305
$88$		0			0
$89$		−	2.24470i		−	0.237938i	−0.992898	−	0.118969i	$-0.962041\pi$
$89$		−	2.24470i		−	0.237938i	0.992898	−	0.118969i	$-0.0379589\pi$
$90$		0			0
$91$	2.78993	+	2.28392i	0.292464	+	0.239420i
$92$		0			0
$93$			4.95280i			0.513581i
$94$		0			0
$95$	−20.2160			−2.07411
$96$		0			0
$97$		−	14.5731i		−	1.47967i	−0.672786	−	0.739837i	$-0.734903\pi$
$97$		−	14.5731i		−	1.47967i	0.672786	−	0.739837i	$-0.265097\pi$
$98$		0			0
$99$		−	4.62147i		−	0.464475i
$100$		0			0
$101$	14.2309			1.41603			0.708015	−	0.706198i	$-0.249591\pi$
$101$	14.2309			1.41603			0.708015	+	0.706198i	$0.249591\pi$
$102$		0			0
$103$	−15.5889			−1.53602			−0.768010	−	0.640438i	$-0.778753\pi$
$103$	−15.5889			−1.53602			−0.768010	+	0.640438i	$0.778753\pi$
$104$		0			0
$105$	2.89917			0.282930
$106$		0			0
$107$	−0.297322			−0.0287432			−0.0143716	−	0.999897i	$-0.504575\pi$
$107$	−0.297322			−0.0287432			−0.0143716	+	0.999897i	$0.504575\pi$
$108$		0			0
$109$			11.9258i			1.14229i	0.820851	+	0.571143i	$0.193500\pi$
$109$			11.9258i			1.14229i	−0.820851	+	0.571143i	$0.806500\pi$
$110$		0			0
$111$			3.11546i			0.295706i
$112$		0			0
$113$	11.1049			1.04466			0.522329	−	0.852744i	$-0.325063\pi$
$113$	11.1049			1.04466			0.522329	+	0.852744i	$0.325063\pi$
$114$		0			0
$115$			16.6485i			1.55248i
$116$		0			0
$117$	−2.28392	+	2.78993i	−0.211148	+	0.257929i
$118$		0			0
$119$		−	2.68288i		−	0.245939i
$120$		0			0
$121$	−10.3580			−0.941634
$122$		0			0
$123$		−	9.43961i		−	0.851141i
$124$		0			0
$125$		−	4.62366i		−	0.413552i
$126$		0			0
$127$	13.6091			1.20761			0.603807	−	0.797131i	$-0.293650\pi$
$127$	13.6091			1.20761			0.603807	+	0.797131i	$0.293650\pi$
$128$		0			0
$129$	−2.54044			−0.223673
$130$		0			0
$131$	−1.35756			−0.118611			−0.0593054	−	0.998240i	$-0.518889\pi$
$131$	−1.35756			−0.118611			−0.0593054	+	0.998240i	$0.518889\pi$
$132$		0			0
$133$	−6.97302			−0.604637
$134$		0			0
$135$			2.89917i			0.249521i
$136$		0			0
$137$			19.0285i			1.62571i	0.582463	+	0.812857i	$0.302089\pi$
$137$			19.0285i			1.62571i	−0.582463	+	0.812857i	$0.697911\pi$
$138$		0			0
$139$	−9.02446			−0.765445			−0.382722	−	0.923863i	$-0.625013\pi$
$139$	−9.02446			−0.765445			−0.382722	+	0.923863i	$0.625013\pi$
$140$		0			0
$141$		−	9.87937i		−	0.831993i
$142$		0			0
$143$	12.8936	+	10.5551i	1.07821	+	0.882659i
$144$		0			0
$145$			27.9354i			2.31991i
$146$		0			0
$147$	1.00000			0.0824786
$148$		0			0
$149$			18.5877i			1.52277i	0.648302	+	0.761383i	$0.275479\pi$
$149$			18.5877i			1.52277i	−0.648302	+	0.761383i	$0.724521\pi$
$150$		0			0
$151$			14.1679i			1.15297i	0.817108	+	0.576485i	$0.195576\pi$
$151$			14.1679i			1.15297i	−0.817108	+	0.576485i	$0.804424\pi$
$152$		0			0
$153$	2.68288			0.216898
$154$		0			0
$155$	−14.3590			−1.15334
$156$		0			0
$157$	6.62892			0.529045			0.264523	−	0.964380i	$-0.414786\pi$
$157$	6.62892			0.529045			0.264523	+	0.964380i	$0.414786\pi$
$158$		0			0
$159$	−12.0885			−0.958679
$160$		0			0
$161$			5.74252i			0.452574i
$162$		0			0
$163$		−	6.89358i		−	0.539946i	−0.962868	−	0.269973i	$-0.912985\pi$
$163$		−	6.89358i		−	0.539946i	0.962868	−	0.269973i	$-0.0870149\pi$
$164$		0			0
$165$	13.3984			1.04307
$166$		0			0
$167$			22.8336i			1.76692i	0.468509	+	0.883459i	$0.344791\pi$
$167$			22.8336i			1.76692i	−0.468509	+	0.883459i	$0.655209\pi$
$168$		0			0
$169$	−2.56742	−	12.7440i	−0.197494	−	0.980304i
$170$		0			0
$171$		−	6.97302i		−	0.533240i
$172$		0			0
$173$	−13.6612			−1.03864			−0.519319	−	0.854580i	$-0.673814\pi$
$173$	−13.6612			−1.03864			−0.519319	+	0.854580i	$0.673814\pi$
$174$		0			0
$175$			3.40518i			0.257407i
$176$		0			0
$177$			13.4520i			1.01112i
$178$		0			0
$179$	−16.1001			−1.20338			−0.601688	−	0.798731i	$-0.705505\pi$
$179$	−16.1001			−1.20338			−0.601688	+	0.798731i	$0.705505\pi$
$180$		0			0
$181$	−11.0172			−0.818905			−0.409452	−	0.912332i	$-0.634280\pi$
$181$	−11.0172			−0.818905			−0.409452	+	0.912332i	$0.634280\pi$
$182$		0			0
$183$	6.69532			0.494932
$184$		0			0
$185$	−9.03224			−0.664064
$186$		0			0
$187$		−	12.3988i		−	0.906693i
$188$		0			0
$189$			1.00000i			0.0727393i
$190$		0			0
$191$	4.67128			0.338002			0.169001	−	0.985616i	$-0.445946\pi$
$191$	4.67128			0.338002			0.169001	+	0.985616i	$0.445946\pi$
$192$		0			0
$193$			2.24210i			0.161390i	0.996739	+	0.0806951i	$0.0257140\pi$
$193$			2.24210i			0.161390i	−0.996739	+	0.0806951i	$0.974286\pi$
$194$		0			0
$195$	−8.08848	−	6.62147i	−0.579228	−	0.474173i
$196$		0			0
$197$			2.14463i			0.152798i	0.997077	+	0.0763992i	$0.0243423\pi$
$197$			2.14463i			0.152798i	−0.997077	+	0.0763992i	$0.975658\pi$
$198$		0			0
$199$	19.1447			1.35713			0.678566	−	0.734539i	$-0.262602\pi$
$199$	19.1447			1.35713			0.678566	+	0.734539i	$0.262602\pi$
$200$		0			0
$201$			5.56826i			0.392755i
$202$		0			0
$203$			9.63568i			0.676292i
$204$		0			0
$205$	27.3670			1.91139
$206$		0			0
$207$	−5.74252			−0.399133
$208$		0			0
$209$	−32.2256			−2.22909
$210$		0			0
$211$	17.9734			1.23734			0.618671	−	0.785650i	$-0.287671\pi$
$211$	17.9734			1.23734			0.618671	+	0.785650i	$0.287671\pi$
$212$		0			0
$213$			5.50601i			0.377266i
$214$		0			0
$215$		−	7.36516i		−	0.502300i
$216$		0			0
$217$	−4.95280			−0.336218
$218$		0			0
$219$			6.20352i			0.419195i
$220$		0			0
$221$	−6.12748	+	7.48504i	−0.412179	+	0.503498i
$222$		0			0
$223$			4.61546i			0.309074i	0.987987	+	0.154537i	$0.0493886\pi$
$223$			4.61546i			0.309074i	−0.987987	+	0.154537i	$0.950611\pi$
$224$		0			0
$225$	−3.40518			−0.227012
$226$		0			0
$227$		−	20.8606i		−	1.38457i	−0.721626	−	0.692283i	$-0.756605\pi$
$227$		−	20.8606i		−	1.38457i	0.721626	−	0.692283i	$-0.243395\pi$
$228$		0			0
$229$		−	7.49664i		−	0.495392i	−0.968838	−	0.247696i	$-0.920327\pi$
$229$		−	7.49664i		−	0.495392i	0.968838	−	0.247696i	$-0.0796734\pi$
$230$		0			0
$231$	4.62147			0.304070
$232$		0			0
$233$	−13.7092			−0.898119			−0.449060	−	0.893502i	$-0.648241\pi$
$233$	−13.7092			−0.898119			−0.449060	+	0.893502i	$0.648241\pi$
$234$		0			0
$235$	28.6420			1.86840
$236$		0			0
$237$	2.65548			0.172492
$238$		0			0
$239$		−	5.18069i		−	0.335111i	−0.985863	−	0.167555i	$-0.946413\pi$
$239$		−	5.18069i		−	0.335111i	0.985863	−	0.167555i	$-0.0535873\pi$
$240$		0			0
$241$		−	16.3310i		−	1.05197i	−0.850493	−	0.525986i	$-0.823696\pi$
$241$		−	16.3310i		−	1.05197i	0.850493	−	0.525986i	$-0.176304\pi$
$242$		0			0
$243$	−1.00000			−0.0641500
$244$		0			0
$245$			2.89917i			0.185221i
$246$		0			0
$247$	19.4542	+	15.9258i	1.23784	+	1.01334i
$248$		0			0
$249$		−	15.9482i		−	1.01068i
$250$		0			0
$251$	−19.7638			−1.24748			−0.623739	−	0.781633i	$-0.714387\pi$
$251$	−19.7638			−1.24748			−0.623739	+	0.781633i	$0.714387\pi$
$252$		0			0
$253$			26.5389i			1.66849i
$254$		0			0
$255$			7.77812i			0.487085i
$256$		0			0
$257$	−4.25496			−0.265417			−0.132709	−	0.991155i	$-0.542367\pi$
$257$	−4.25496			−0.265417			−0.132709	+	0.991155i	$0.542367\pi$
$258$		0			0
$259$	−3.11546			−0.193585
$260$		0			0
$261$	−9.63568			−0.596433
$262$		0			0
$263$	2.13084			0.131393			0.0656966	−	0.997840i	$-0.479073\pi$
$263$	2.13084			0.131393			0.0656966	+	0.997840i	$0.479073\pi$
$264$		0			0
$265$		−	35.0465i		−	2.15289i
$266$		0			0
$267$			2.24470i			0.137374i
$268$		0			0
$269$	−22.2010			−1.35362			−0.676809	−	0.736158i	$-0.736638\pi$
$269$	−22.2010			−1.35362			−0.676809	+	0.736158i	$0.736638\pi$
$270$		0			0
$271$			1.74353i			0.105912i	0.998597	+	0.0529560i	$0.0168643\pi$
$271$			1.74353i			0.105912i	−0.998597	+	0.0529560i	$0.983136\pi$
$272$		0			0
$273$	−2.78993	−	2.28392i	−0.168854	−	0.138229i
$274$		0			0
$275$			15.7369i			0.948972i
$276$		0			0
$277$	20.8988			1.25569			0.627845	−	0.778339i	$-0.283937\pi$
$277$	20.8988			1.25569			0.627845	+	0.778339i	$0.283937\pi$
$278$		0			0
$279$		−	4.95280i		−	0.296516i
$280$		0			0
$281$			11.3988i			0.679993i	0.940427	+	0.339996i	$0.110426\pi$
$281$			11.3988i			0.679993i	−0.940427	+	0.339996i	$0.889574\pi$
$282$		0			0
$283$	−22.5014			−1.33757			−0.668786	−	0.743455i	$-0.733186\pi$
$283$	−22.5014			−1.33757			−0.668786	+	0.743455i	$0.733186\pi$
$284$		0			0
$285$	20.2160			1.19749
$286$		0			0
$287$	9.43961			0.557202
$288$		0			0
$289$	−9.80216			−0.576598
$290$		0			0
$291$			14.5731i			0.854290i
$292$		0			0
$293$			32.5005i			1.89870i	0.314221	+	0.949350i	$0.398257\pi$
$293$			32.5005i			1.89870i	−0.314221	+	0.949350i	$0.601743\pi$
$294$		0			0
$295$	−38.9998			−2.27065
$296$		0			0
$297$			4.62147i			0.268165i
$298$		0			0
$299$	13.1155	−	16.0212i	0.758487	−	0.926532i
$300$		0			0
$301$		−	2.54044i		−	0.146428i
$302$		0			0
$303$	−14.2309			−0.817545
$304$		0			0
$305$			19.4109i			1.11146i
$306$		0			0
$307$		−	1.22332i		−	0.0698183i	−0.999390	−	0.0349092i	$-0.988886\pi$
$307$		−	1.22332i		−	0.0698183i	0.999390	−	0.0349092i	$-0.0111142\pi$
$308$		0			0
$309$	15.5889			0.886821
$310$		0			0
$311$	−30.2010			−1.71254			−0.856271	−	0.516527i	$-0.827224\pi$
$311$	−30.2010			−1.71254			−0.856271	+	0.516527i	$0.827224\pi$
$312$		0			0
$313$	−28.8632			−1.63145			−0.815723	−	0.578442i	$-0.803661\pi$
$313$	−28.8632			−1.63145			−0.815723	+	0.578442i	$0.803661\pi$
$314$		0			0
$315$	−2.89917			−0.163350
$316$		0			0
$317$		−	8.42800i		−	0.473364i	−0.971587	−	0.236682i	$-0.923940\pi$
$317$		−	8.42800i		−	0.473364i	0.971587	−	0.236682i	$-0.0760599\pi$
$318$		0			0
$319$			44.5310i			2.49326i
$320$		0			0
$321$	0.297322			0.0165949
$322$		0			0
$323$		−	18.7078i		−	1.04093i
$324$		0			0
$325$	7.77716	−	9.50021i	0.431399	−	0.526977i
$326$		0			0
$327$		−	11.9258i		−	0.659499i
$328$		0			0
$329$	9.87937			0.544667
$330$		0			0
$331$		−	1.97596i		−	0.108609i	−0.998524	−	0.0543043i	$-0.982706\pi$
$331$		−	1.97596i		−	0.108609i	0.998524	−	0.0543043i	$-0.0172941\pi$
$332$		0			0
$333$		−	3.11546i		−	0.170726i
$334$		0			0
$335$	−16.1433			−0.882004
$336$		0			0
$337$	−16.5245			−0.900145			−0.450072	−	0.892992i	$-0.648602\pi$
$337$	−16.5245			−0.900145			−0.450072	+	0.892992i	$0.648602\pi$
$338$		0			0
$339$	−11.1049			−0.603134
$340$		0			0
$341$	−22.8892			−1.23952
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$		−	16.6485i		−	0.896327i
$346$		0			0
$347$	12.3946			0.665377			0.332688	−	0.943037i	$-0.392044\pi$
$347$	12.3946			0.665377			0.332688	+	0.943037i	$0.392044\pi$
$348$		0			0
$349$		−	13.0453i		−	0.698298i	−0.937067	−	0.349149i	$-0.886471\pi$
$349$		−	13.0453i		−	0.698298i	0.937067	−	0.349149i	$-0.113529\pi$
$350$		0			0
$351$	2.28392	−	2.78993i	0.121907	−	0.148915i
$352$		0			0
$353$		−	17.7769i		−	0.946171i	−0.881016	−	0.473086i	$-0.843140\pi$
$353$		−	17.7769i		−	0.946171i	0.881016	−	0.473086i	$-0.156860\pi$
$354$		0			0
$355$	−15.9629			−0.847220
$356$		0			0
$357$			2.68288i			0.141993i
$358$		0			0
$359$			7.60041i			0.401135i	0.979680	+	0.200567i	$0.0642785\pi$
$359$			7.60041i			0.401135i	−0.979680	+	0.200567i	$0.935721\pi$
$360$		0			0
$361$	−29.6230			−1.55910
$362$		0			0
$363$	10.3580			0.543653
$364$		0			0
$365$	−17.9850			−0.941380
$366$		0			0
$367$	−10.2313			−0.534071			−0.267036	−	0.963687i	$-0.586044\pi$
$367$	−10.2313			−0.534071			−0.267036	+	0.963687i	$0.586044\pi$
$368$		0			0
$369$			9.43961i			0.491406i
$370$		0			0
$371$		−	12.0885i		−	0.627602i
$372$		0			0
$373$	9.00042			0.466024			0.233012	−	0.972474i	$-0.425142\pi$
$373$	9.00042			0.466024			0.233012	+	0.972474i	$0.425142\pi$
$374$		0			0
$375$			4.62366i			0.238765i
$376$		0			0
$377$	22.0071	−	26.8829i	1.13342	−	1.38454i
$378$		0			0
$379$			25.6207i			1.31605i	0.752997	+	0.658024i	$0.228607\pi$
$379$			25.6207i			1.31605i	−0.752997	+	0.658024i	$0.771393\pi$
$380$		0			0
$381$	−13.6091			−0.697216
$382$		0			0
$383$			18.8808i			0.964764i	0.875961	+	0.482382i	$0.160228\pi$
$383$			18.8808i			0.964764i	−0.875961	+	0.482382i	$0.839772\pi$
$384$		0			0
$385$			13.3984i			0.682847i
$386$		0			0
$387$	2.54044			0.129138
$388$		0			0
$389$	17.6216			0.893448			0.446724	−	0.894672i	$-0.352591\pi$
$389$	17.6216			0.893448			0.446724	+	0.894672i	$0.352591\pi$
$390$		0			0
$391$	−15.4065			−0.779139
$392$		0			0
$393$	1.35756			0.0684800
$394$		0			0
$395$			7.69867i			0.387362i
$396$		0			0
$397$		−	10.2478i		−	0.514321i	−0.966369	−	0.257161i	$-0.917213\pi$
$397$		−	10.2478i		−	0.514321i	0.966369	−	0.257161i	$-0.0827869\pi$
$398$		0			0
$399$	6.97302			0.349088
$400$		0			0
$401$			23.3576i			1.16643i	0.812320	+	0.583213i	$0.198205\pi$
$401$			23.3576i			1.16643i	−0.812320	+	0.583213i	$0.801795\pi$
$402$		0			0
$403$	13.8180	+	11.3118i	0.688322	+	0.563481i
$404$		0			0
$405$		−	2.89917i		−	0.144061i
$406$		0			0
$407$	−14.3980			−0.713682
$408$		0			0
$409$			7.69209i			0.380349i	0.981750	+	0.190175i	$0.0609054\pi$
$409$			7.69209i			0.380349i	−0.981750	+	0.190175i	$0.939095\pi$
$410$		0			0
$411$		−	19.0285i		−	0.938607i
$412$		0			0
$413$	−13.4520			−0.661932
$414$		0			0
$415$	46.2366			2.26966
$416$		0			0
$417$	9.02446			0.441930
$418$		0			0
$419$	−7.92144			−0.386988			−0.193494	−	0.981101i	$-0.561982\pi$
$419$	−7.92144			−0.386988			−0.193494	+	0.981101i	$0.561982\pi$
$420$		0			0
$421$			2.78632i			0.135797i	0.997692	+	0.0678984i	$0.0216294\pi$
$421$			2.78632i			0.135797i	−0.997692	+	0.0678984i	$0.978371\pi$
$422$		0			0
$423$			9.87937i			0.480351i
$424$		0			0
$425$	−9.13568			−0.443146
$426$		0			0
$427$			6.69532i			0.324009i
$428$		0			0
$429$	−12.8936	−	10.5551i	−0.622508	−	0.509603i
$430$		0			0
$431$			0.584021i			0.0281313i	0.999901	+	0.0140656i	$0.00447738\pi$
$431$			0.584021i			0.0281313i	−0.999901	+	0.0140656i	$0.995523\pi$
$432$		0			0
$433$	25.8632			1.24291			0.621454	−	0.783451i	$-0.286542\pi$
$433$	25.8632			1.24291			0.621454	+	0.783451i	$0.286542\pi$
$434$		0			0
$435$		−	27.9354i		−	1.33940i
$436$		0			0
$437$			40.0427i			1.91550i
$438$		0			0
$439$	−7.96734			−0.380260			−0.190130	−	0.981759i	$-0.560891\pi$
$439$	−7.96734			−0.380260			−0.190130	+	0.981759i	$0.560891\pi$
$440$		0			0
$441$	−1.00000			−0.0476190
$442$		0			0
$443$	−23.5783			−1.12024			−0.560120	−	0.828412i	$-0.689245\pi$
$443$	−23.5783			−1.12024			−0.560120	+	0.828412i	$0.689245\pi$
$444$		0			0
$445$	−6.50778			−0.308498
$446$		0			0
$447$		−	18.5877i		−	0.879169i
$448$		0			0
$449$		−	8.64551i		−	0.408007i	−0.978970	−	0.204003i	$-0.934605\pi$
$449$		−	8.64551i		−	0.408007i	0.978970	−	0.204003i	$-0.0653954\pi$
$450$		0			0
$451$	43.6248			2.05421
$452$		0			0
$453$		−	14.1679i		−	0.665667i
$454$		0			0
$455$	6.62147	−	8.08848i	0.310419	−	0.379194i
$456$		0			0
$457$			17.5683i			0.821809i	0.911678	+	0.410904i	$0.134787\pi$
$457$			17.5683i			0.821809i	−0.911678	+	0.410904i	$0.865213\pi$
$458$		0			0
$459$	−2.68288			−0.125226
$460$		0			0
$461$		−	38.9337i		−	1.81332i	−0.421860	−	0.906661i	$-0.638622\pi$
$461$		−	38.9337i		−	1.81332i	0.421860	−	0.906661i	$-0.361378\pi$
$462$		0			0
$463$		−	26.4273i		−	1.22818i	−0.789237	−	0.614089i	$-0.789523\pi$
$463$		−	26.4273i		−	1.22818i	0.789237	−	0.614089i	$-0.210477\pi$
$464$		0			0
$465$	14.3590			0.665882
$466$		0			0
$467$	28.1140			1.30096			0.650479	−	0.759524i	$-0.274568\pi$
$467$	28.1140			1.30096			0.650479	+	0.759524i	$0.274568\pi$
$468$		0			0
$469$	−5.56826			−0.257118
$470$		0			0
$471$	−6.62892			−0.305444
$472$		0			0
$473$		−	11.7406i		−	0.539831i
$474$		0			0
$475$			23.7444i			1.08947i
$476$		0			0
$477$	12.0885			0.553493
$478$		0			0
$479$			13.6139i			0.622033i	0.950404	+	0.311017i	$0.100670\pi$
$479$			13.6139i			0.622033i	−0.950404	+	0.311017i	$0.899330\pi$
$480$		0			0
$481$	8.69191	+	7.11546i	0.396317	+	0.324437i
$482$		0			0
$483$		−	5.74252i		−	0.261294i
$484$		0			0
$485$	−42.2499			−1.91847
$486$		0			0
$487$		−	10.8411i		−	0.491258i	−0.969364	−	0.245629i	$-0.921005\pi$
$487$		−	10.8411i		−	0.491258i	0.969364	−	0.245629i	$-0.0789945\pi$
$488$		0			0
$489$			6.89358i			0.311738i
$490$		0			0
$491$	31.8497			1.43736			0.718679	−	0.695342i	$-0.244747\pi$
$491$	31.8497			1.43736			0.718679	+	0.695342i	$0.244747\pi$
$492$		0			0
$493$	−25.8513			−1.16429
$494$		0			0
$495$	−13.3984			−0.602214
$496$		0			0
$497$	−5.50601			−0.246978
$498$		0			0
$499$			30.4560i			1.36340i	0.731634	+	0.681698i	$0.238758\pi$
$499$			30.4560i			1.36340i	−0.731634	+	0.681698i	$0.761242\pi$
$500$		0			0
$501$		−	22.8336i		−	1.02013i
$502$		0			0
$503$	−40.1470			−1.79007			−0.895034	−	0.445998i	$-0.852849\pi$
$503$	−40.1470			−1.79007			−0.895034	+	0.445998i	$0.852849\pi$
$504$		0			0
$505$		−	41.2578i		−	1.83595i
$506$		0			0
$507$	2.56742	+	12.7440i	0.114023	+	0.565979i
$508$		0			0
$509$		−	0.0192885i		−	0.000854947i	−1.00000		0.000427473i	$-0.999864\pi$
$509$		−	0.0192885i		−	0.000854947i	1.00000		0.000427473i	$-0.000136069\pi$
$510$		0			0
$511$	−6.20352			−0.274427
$512$		0			0
$513$			6.97302i			0.307866i
$514$		0			0
$515$			45.1948i			1.99152i
$516$		0			0
$517$	45.6572			2.00800
$518$		0			0
$519$	13.6612			0.599658
$520$		0			0
$521$	29.7490			1.30333			0.651664	−	0.758507i	$-0.274071\pi$
$521$	29.7490			1.30333			0.651664	+	0.758507i	$0.274071\pi$
$522$		0			0
$523$	33.1062			1.44763			0.723816	−	0.689993i	$-0.242386\pi$
$523$	33.1062			1.44763			0.723816	+	0.689993i	$0.242386\pi$
$524$		0			0
$525$		−	3.40518i		−	0.148614i
$526$		0			0
$527$		−	13.2878i		−	0.578824i
$528$		0			0
$529$	9.97655			0.433763
$530$		0			0
$531$		−	13.4520i		−	0.583769i
$532$		0			0
$533$	−26.3358	−	21.5593i	−1.14073	−	0.933837i
$534$		0			0
$535$			0.861985i			0.0372669i
$536$		0			0
$537$	16.1001			0.694770
$538$		0			0
$539$			4.62147i			0.199061i
$540$		0			0
$541$		−	22.2496i		−	0.956586i	−0.878200	−	0.478293i	$-0.841256\pi$
$541$		−	22.2496i		−	0.956586i	0.878200	−	0.478293i	$-0.158744\pi$
$542$		0			0
$543$	11.0172			0.472795
$544$		0			0
$545$	34.5750			1.48103
$546$		0			0
$547$	13.9364			0.595878			0.297939	−	0.954585i	$-0.403701\pi$
$547$	13.9364			0.595878			0.297939	+	0.954585i	$0.403701\pi$
$548$		0			0
$549$	−6.69532			−0.285749
$550$		0			0
$551$			67.1897i			2.86238i
$552$		0			0
$553$			2.65548i			0.112922i
$554$		0			0
$555$	9.03224			0.383397
$556$		0			0
$557$		−	27.4529i		−	1.16322i	−0.813469	−	0.581608i	$-0.802424\pi$
$557$		−	27.4529i		−	1.16322i	0.813469	−	0.581608i	$-0.197576\pi$
$558$		0			0
$559$	−5.80216	+	7.08764i	−0.245405	+	0.299775i
$560$		0			0
$561$			12.3988i			0.523479i
$562$		0			0
$563$	−19.0474			−0.802754			−0.401377	−	0.915913i	$-0.631468\pi$
$563$	−19.0474			−0.802754			−0.401377	+	0.915913i	$0.631468\pi$
$564$		0			0
$565$		−	32.1949i		−	1.35445i
$566$		0			0
$567$		−	1.00000i		−	0.0419961i
$568$		0			0
$569$	23.3754			0.979947			0.489974	−	0.871737i	$-0.337006\pi$
$569$	23.3754			0.979947			0.489974	+	0.871737i	$0.337006\pi$
$570$		0			0
$571$	−7.45902			−0.312150			−0.156075	−	0.987745i	$-0.549884\pi$
$571$	−7.45902			−0.312150			−0.156075	+	0.987745i	$0.549884\pi$
$572$		0			0
$573$	−4.67128			−0.195145
$574$		0			0
$575$	19.5543			0.815471
$576$		0			0
$577$		−	17.6203i		−	0.733543i	−0.930311	−	0.366771i	$-0.880463\pi$
$577$		−	17.6203i		−	0.733543i	0.930311	−	0.366771i	$-0.119537\pi$
$578$		0			0
$579$		−	2.24210i		−	0.0931787i
$580$		0			0
$581$	15.9482			0.661644
$582$		0			0
$583$		−	55.8665i		−	2.31376i
$584$		0			0
$585$	8.08848	+	6.62147i	0.334417	+	0.273764i
$586$		0			0
$587$		−	9.18064i		−	0.378926i	−0.981888	−	0.189463i	$-0.939325\pi$
$587$		−	9.18064i		−	0.378926i	0.981888	−	0.189463i	$-0.0606747\pi$
$588$		0			0
$589$	−34.5360			−1.42303
$590$		0			0
$591$		−	2.14463i		−	0.0882182i
$592$		0			0
$593$			2.31069i			0.0948889i	0.998874	+	0.0474444i	$0.0151077\pi$
$593$			2.31069i			0.0948889i	−0.998874	+	0.0474444i	$0.984892\pi$
$594$		0			0
$595$	−7.77812			−0.318872
$596$		0			0
$597$	−19.1447			−0.783541
$598$		0			0
$599$	−2.61652			−0.106908			−0.0534540	−	0.998570i	$-0.517023\pi$
$599$	−2.61652			−0.106908			−0.0534540	+	0.998570i	$0.517023\pi$
$600$		0			0
$601$	−41.4118			−1.68922			−0.844612	−	0.535379i	$-0.820169\pi$
$601$	−41.4118			−1.68922			−0.844612	+	0.535379i	$0.820169\pi$
$602$		0			0
$603$		−	5.56826i		−	0.226757i
$604$		0			0
$605$			30.0295i			1.22087i
$606$		0			0
$607$	46.6407			1.89309			0.946544	−	0.322576i	$-0.104549\pi$
$607$	46.6407			1.89309			0.946544	+	0.322576i	$0.104549\pi$
$608$		0			0
$609$		−	9.63568i		−	0.390457i
$610$		0			0
$611$	−27.5627	−	22.5637i	−1.11507	−	0.912829i
$612$		0			0
$613$		−	10.8493i		−	0.438200i	−0.975702	−	0.219100i	$-0.929688\pi$
$613$		−	10.8493i		−	0.438200i	0.975702	−	0.219100i	$-0.0703120\pi$
$614$		0			0
$615$	−27.3670			−1.10354
$616$		0			0
$617$		−	40.6648i		−	1.63710i	−0.574434	−	0.818551i	$-0.694778\pi$
$617$		−	40.6648i		−	1.63710i	0.574434	−	0.818551i	$-0.305222\pi$
$618$		0			0
$619$		−	6.38479i		−	0.256626i	−0.991734	−	0.128313i	$-0.959044\pi$
$619$		−	6.38479i		−	0.256626i	0.991734	−	0.128313i	$-0.0409563\pi$
$620$		0			0
$621$	5.74252			0.230439
$622$		0			0
$623$	−2.24470			−0.0899322
$624$		0			0
$625$	−30.4307			−1.21723
$626$		0			0
$627$	32.2256			1.28697
$628$		0			0
$629$		−	8.35840i		−	0.333271i
$630$		0			0
$631$		−	24.2760i		−	0.966413i	−0.875506	−	0.483207i	$-0.839472\pi$
$631$		−	24.2760i		−	0.966413i	0.875506	−	0.483207i	$-0.160528\pi$
$632$		0			0
$633$	−17.9734			−0.714380
$634$		0			0
$635$		−	39.4551i		−	1.56573i
$636$		0			0
$637$	2.28392	−	2.78993i	0.0904922	−	0.110541i
$638$		0			0
$639$		−	5.50601i		−	0.217814i
$640$		0			0
$641$	−0.470628			−0.0185887			−0.00929434	−	0.999957i	$-0.502959\pi$
$641$	−0.470628			−0.0185887			−0.00929434	+	0.999957i	$0.502959\pi$
$642$		0			0
$643$		−	9.99467i		−	0.394151i	−0.980388	−	0.197076i	$-0.936856\pi$
$643$		−	9.99467i		−	0.394151i	0.980388	−	0.197076i	$-0.0631445\pi$
$644$		0			0
$645$			7.36516i			0.290003i
$646$		0			0
$647$	0.325322			0.0127897			0.00639487	−	0.999980i	$-0.497964\pi$
$647$	0.325322			0.0127897			0.00639487	+	0.999980i	$0.497964\pi$
$648$		0			0
$649$	−62.1682			−2.44032
$650$		0			0
$651$	4.95280			0.194115
$652$		0			0
$653$	38.9175			1.52296			0.761480	−	0.648189i	$-0.224473\pi$
$653$	38.9175			1.52296			0.761480	+	0.648189i	$0.224473\pi$
$654$		0			0
$655$			3.93581i			0.153785i
$656$		0			0
$657$		−	6.20352i		−	0.242022i
$658$		0			0
$659$	14.0057			0.545584			0.272792	−	0.962073i	$-0.412053\pi$
$659$	14.0057			0.545584			0.272792	+	0.962073i	$0.412053\pi$
$660$		0			0
$661$			2.23659i			0.0869933i	0.999054	+	0.0434966i	$0.0138498\pi$
$661$			2.23659i			0.0869933i	−0.999054	+	0.0434966i	$0.986150\pi$
$662$		0			0
$663$	6.12748	−	7.48504i	0.237972	−	0.290695i
$664$		0			0
$665$			20.2160i			0.783941i
$666$		0			0
$667$	55.3331			2.14251
$668$		0			0
$669$		−	4.61546i		−	0.178444i
$670$		0			0
$671$			30.9422i			1.19451i
$672$		0			0
$673$	10.2527			0.395212			0.197606	−	0.980282i	$-0.436683\pi$
$673$	10.2527			0.395212			0.197606	+	0.980282i	$0.436683\pi$
$674$		0			0
$675$	3.40518			0.131065
$676$		0			0
$677$	−4.09525			−0.157393			−0.0786965	−	0.996899i	$-0.525076\pi$
$677$	−4.09525			−0.157393			−0.0786965	+	0.996899i	$0.525076\pi$
$678$		0			0
$679$	−14.5731			−0.559264
$680$		0			0
$681$			20.8606i			0.799379i
$682$		0			0
$683$			10.3082i			0.394431i	0.980360	+	0.197216i	$0.0631899\pi$
$683$			10.3082i			0.394431i	−0.980360	+	0.197216i	$0.936810\pi$
$684$		0			0
$685$	55.1668			2.10782
$686$		0			0
$687$			7.49664i			0.286015i
$688$		0			0
$689$	−27.6091	+	33.7260i	−1.05182	+	1.28486i
$690$		0			0
$691$		−	14.3297i		−	0.545129i	−0.962138	−	0.272565i	$-0.912128\pi$
$691$		−	14.3297i		−	0.545129i	0.962138	−	0.272565i	$-0.0878718\pi$
$692$		0			0
$693$	−4.62147			−0.175555
$694$		0			0
$695$			26.1634i			0.992436i
$696$		0			0
$697$			25.3253i			0.959265i
$698$		0			0
$699$	13.7092			0.518529
$700$		0			0
$701$	−22.0413			−0.832489			−0.416245	−	0.909253i	$-0.636654\pi$
$701$	−22.0413			−0.832489			−0.416245	+	0.909253i	$0.636654\pi$
$702$		0			0
$703$	−21.7242			−0.819342
$704$		0			0
$705$	−28.6420			−1.07872
$706$		0			0
$707$		−	14.2309i		−	0.535209i
$708$		0			0
$709$			9.51346i			0.357285i	0.983914	+	0.178643i	$0.0571706\pi$
$709$			9.51346i			0.357285i	−0.983914	+	0.178643i	$0.942829\pi$
$710$		0			0
$711$	−2.65548			−0.0995881
$712$		0			0
$713$			28.4415i			1.06514i
$714$		0			0
$715$	30.6009	−	37.3806i	1.14441	−	1.39796i
$716$		0			0
$717$			5.18069i			0.193476i
$718$		0			0
$719$	−51.7985			−1.93176			−0.965879	−	0.258993i	$-0.916609\pi$
$719$	−51.7985			−1.93176			−0.965879	+	0.258993i	$0.916609\pi$
$720$		0			0
$721$			15.5889i			0.580561i
$722$		0			0
$723$			16.3310i			0.607356i
$724$		0			0
$725$	32.8112			1.21858
$726$		0			0
$727$	−16.1708			−0.599741			−0.299871	−	0.953980i	$-0.596944\pi$
$727$	−16.1708			−0.599741			−0.299871	+	0.953980i	$0.596944\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	6.81569			0.252087
$732$		0			0
$733$			5.61504i			0.207396i	0.994609	+	0.103698i	$0.0330676\pi$
$733$			5.61504i			0.207396i	−0.994609	+	0.103698i	$0.966932\pi$
$734$		0			0
$735$		−	2.89917i		−	0.106937i
$736$		0			0
$737$	−25.7335			−0.947907
$738$		0			0
$739$		−	40.5847i		−	1.49293i	−0.665424	−	0.746465i	$-0.731749\pi$
$739$		−	40.5847i		−	1.49293i	0.665424	−	0.746465i	$-0.268251\pi$
$740$		0			0
$741$	−19.4542	−	15.9258i	−0.714669	−	0.585049i
$742$		0			0
$743$		−	39.1777i		−	1.43729i	−0.695377	−	0.718645i	$-0.744763\pi$
$743$		−	39.1777i		−	1.43729i	0.695377	−	0.718645i	$-0.255237\pi$
$744$		0			0
$745$	53.8890			1.97434
$746$		0			0
$747$			15.9482i			0.583515i
$748$		0			0
$749$			0.297322i			0.0108639i
$750$		0			0
$751$	−25.8180			−0.942111			−0.471055	−	0.882104i	$-0.656127\pi$
$751$	−25.8180			−0.942111			−0.471055	+	0.882104i	$0.656127\pi$
$752$		0			0
$753$	19.7638			0.720231
$754$		0			0
$755$	41.0752			1.49488
$756$		0			0
$757$	14.0013			0.508886			0.254443	−	0.967088i	$-0.418108\pi$
$757$	14.0013			0.508886			0.254443	+	0.967088i	$0.418108\pi$
$758$		0			0
$759$		−	26.5389i		−	0.963300i
$760$		0			0
$761$		−	27.0131i		−	0.979225i	−0.871940	−	0.489613i	$-0.837138\pi$
$761$		−	27.0131i		−	0.979225i	0.871940	−	0.489613i	$-0.162862\pi$
$762$		0			0
$763$	11.9258			0.431744
$764$		0			0
$765$		−	7.77812i		−	0.281218i
$766$		0			0
$767$	37.5303	+	30.7234i	1.35514	+	1.10936i
$768$		0			0
$769$		−	22.2844i		−	0.803596i	−0.915728	−	0.401798i	$-0.868385\pi$
$769$		−	22.2844i		−	0.803596i	0.915728	−	0.401798i	$-0.131615\pi$
$770$		0			0
$771$	4.25496			0.153239
$772$		0			0
$773$		−	25.5175i		−	0.917801i	−0.888488	−	0.458900i	$-0.848244\pi$
$773$		−	25.5175i		−	0.917801i	0.888488	−	0.458900i	$-0.151756\pi$
$774$		0			0
$775$			16.8652i			0.605814i
$776$		0			0
$777$	3.11546			0.111766
$778$		0			0
$779$	65.8226			2.35834
$780$		0			0
$781$	−25.4459			−0.910524
$782$		0			0
$783$	9.63568			0.344351
$784$		0			0
$785$		−	19.2183i		−	0.685932i
$786$		0			0
$787$			25.9999i			0.926797i	0.886150	+	0.463399i	$0.153370\pi$
$787$			25.9999i			0.926797i	−0.886150	+	0.463399i	$0.846630\pi$
$788$		0			0
$789$	−2.13084			−0.0758599
$790$		0			0
$791$		−	11.1049i		−	0.394844i
$792$		0			0
$793$	15.2916	−	18.6795i	0.543020	−	0.663327i
$794$		0			0
$795$			35.0465i			1.24297i
$796$		0			0
$797$	−43.8149			−1.55200			−0.776002	−	0.630731i	$-0.782755\pi$
$797$	−43.8149			−1.55200			−0.776002	+	0.630731i	$0.782755\pi$
$798$		0			0
$799$			26.5051i			0.937685i
$800$		0			0
$801$		−	2.24470i		−	0.0793127i
$802$		0			0
$803$	−28.6694			−1.01172
$804$		0			0
$805$	16.6485			0.586784
$806$		0			0
$807$	22.2010			0.781512
$808$		0			0
$809$	29.0278			1.02056			0.510282	−	0.860007i	$-0.329541\pi$
$809$	29.0278			1.02056			0.510282	+	0.860007i	$0.329541\pi$
$810$		0			0
$811$		−	19.5181i		−	0.685374i	−0.939450	−	0.342687i	$-0.888663\pi$
$811$		−	19.5181i		−	0.685374i	0.939450	−	0.342687i	$-0.111337\pi$
$812$		0			0
$813$		−	1.74353i		−	0.0611484i
$814$		0			0
$815$	−19.9856			−0.700066
$816$		0			0
$817$		−	17.7145i		−	0.619753i
$818$		0			0
$819$	2.78993	+	2.28392i	0.0974880	+	0.0798066i
$820$		0			0
$821$			36.7002i			1.28085i	0.768022	+	0.640423i	$0.221241\pi$
$821$			36.7002i			1.28085i	−0.768022	+	0.640423i	$0.778759\pi$
$822$		0			0
$823$	26.4859			0.923240			0.461620	−	0.887078i	$-0.347268\pi$
$823$	26.4859			0.923240			0.461620	+	0.887078i	$0.347268\pi$
$824$		0			0
$825$		−	15.7369i		−	0.547890i
$826$		0			0
$827$			1.85389i			0.0644660i	0.999480	+	0.0322330i	$0.0102619\pi$
$827$			1.85389i			0.0644660i	−0.999480	+	0.0322330i	$0.989738\pi$
$828$		0			0
$829$	39.8166			1.38289			0.691444	−	0.722430i	$-0.256975\pi$
$829$	39.8166			1.38289			0.691444	+	0.722430i	$0.256975\pi$
$830$		0			0
$831$	−20.8988			−0.724973
$832$		0			0
$833$	−2.68288			−0.0929562
$834$		0			0
$835$	66.1985			2.29089
$836$		0			0
$837$			4.95280i			0.171194i
$838$		0			0
$839$			13.0743i			0.451374i	0.974200	+	0.225687i	$0.0724627\pi$
$839$			13.0743i			0.451374i	−0.974200	+	0.225687i	$0.927537\pi$
$840$		0			0
$841$	63.8463			2.20159
$842$		0			0
$843$		−	11.3988i		−	0.392594i
$844$		0			0
$845$	−36.9469	+	7.44338i	−1.27101	+	0.256060i
$846$		0			0
$847$			10.3580i			0.355904i
$848$		0			0
$849$	22.5014			0.772247
$850$		0			0
$851$			17.8906i			0.613282i
$852$		0			0
$853$			29.1938i			0.999577i	0.866148	+	0.499788i	$0.166589\pi$
$853$			29.1938i			0.999577i	−0.866148	+	0.499788i	$0.833411\pi$
$854$		0			0
$855$	−20.2160			−0.691371
$856$		0			0
$857$	−28.6687			−0.979305			−0.489652	−	0.871918i	$-0.662876\pi$
$857$	−28.6687			−0.979305			−0.489652	+	0.871918i	$0.662876\pi$
$858$		0			0
$859$	−8.51664			−0.290584			−0.145292	−	0.989389i	$-0.546412\pi$
$859$	−8.51664			−0.290584			−0.145292	+	0.989389i	$0.546412\pi$
$860$		0			0
$861$	−9.43961			−0.321701
$862$		0			0
$863$		−	26.2888i		−	0.894881i	−0.894314	−	0.447440i	$-0.852336\pi$
$863$		−	26.2888i		−	0.894881i	0.894314	−	0.447440i	$-0.147664\pi$
$864$		0			0
$865$			39.6060i			1.34664i
$866$		0			0
$867$	9.80216			0.332899
$868$		0			0
$869$			12.2722i			0.416306i
$870$		0			0
$871$	15.5351	+	12.7175i	0.526385	+	0.430915i
$872$		0			0
$873$		−	14.5731i		−	0.493225i
$874$		0			0
$875$	−4.62366			−0.156308
$876$		0			0
$877$		−	9.38012i		−	0.316744i	−0.987380	−	0.158372i	$-0.949375\pi$
$877$		−	9.38012i		−	0.316744i	0.987380	−	0.158372i	$-0.0506245\pi$
$878$		0			0
$879$		−	32.5005i		−	1.09621i
$880$		0			0
$881$	29.8202			1.00467			0.502334	−	0.864673i	$-0.332475\pi$
$881$	29.8202			1.00467			0.502334	+	0.864673i	$0.332475\pi$
$882$		0			0
$883$	−33.9848			−1.14368			−0.571839	−	0.820366i	$-0.693770\pi$
$883$	−33.9848			−1.14368			−0.571839	+	0.820366i	$0.693770\pi$
$884$		0			0
$885$	38.9998			1.31096
$886$		0			0
$887$	−20.6776			−0.694284			−0.347142	−	0.937813i	$-0.612848\pi$
$887$	−20.6776			−0.694284			−0.347142	+	0.937813i	$0.612848\pi$
$888$		0			0
$889$		−	13.6091i		−	0.456435i
$890$		0			0
$891$		−	4.62147i		−	0.154825i
$892$		0			0
$893$	68.8890			2.30528
$894$		0			0
$895$			46.6768i			1.56023i
$896$		0			0
$897$	−13.1155	+	16.0212i	−0.437912	+	0.534933i
$898$		0			0
$899$			47.7236i			1.59167i
$900$		0			0
$901$	32.4319			1.08046
$902$		0			0
$903$			2.54044i			0.0845405i
$904$		0			0
$905$			31.9408i			1.06175i
$906$		0			0
$907$	−20.2425			−0.672140			−0.336070	−	0.941837i	$-0.609098\pi$
$907$	−20.2425			−0.672140			−0.336070	+	0.941837i	$0.609098\pi$
$908$		0			0
$909$	14.2309			0.472010
$910$		0			0
$911$	11.4972			0.380921			0.190460	−	0.981695i	$-0.439002\pi$
$911$	11.4972			0.380921			0.190460	+	0.981695i	$0.439002\pi$
$912$		0			0
$913$	73.7042			2.43925
$914$		0			0
$915$		−	19.4109i		−	0.641703i
$916$		0			0
$917$			1.35756i			0.0448307i
$918$		0			0
$919$	−38.8708			−1.28223			−0.641115	−	0.767445i	$-0.721528\pi$
$919$	−38.8708			−1.28223			−0.641115	+	0.767445i	$0.721528\pi$
$920$		0			0
$921$			1.22332i			0.0403096i
$922$		0			0
$923$	15.3614	+	12.5753i	0.505626	+	0.413921i
$924$		0			0
$925$			10.6087i			0.348812i
$926$		0			0
$927$	−15.5889			−0.512006
$928$		0			0
$929$		−	18.2283i		−	0.598052i	−0.954245	−	0.299026i	$-0.903338\pi$
$929$		−	18.2283i		−	0.598052i	0.954245	−	0.299026i	$-0.0966617\pi$
$930$		0			0
$931$			6.97302i			0.228531i
$932$		0			0
$933$	30.2010			0.988736
$934$		0			0
$935$	−35.9463			−1.17557
$936$		0			0
$937$	33.8738			1.10661			0.553304	−	0.832980i	$-0.313367\pi$
$937$	33.8738			1.10661			0.553304	+	0.832980i	$0.313367\pi$
$938$		0			0
$939$	28.8632			0.941916
$940$		0			0
$941$		−	15.4920i		−	0.505025i	−0.967594	−	0.252513i	$-0.918743\pi$
$941$		−	15.4920i		−	0.505025i	0.967594	−	0.252513i	$-0.0812569\pi$
$942$		0			0
$943$		−	54.2071i		−	1.76523i
$944$		0			0
$945$	2.89917			0.0943100
$946$		0			0
$947$			7.01891i			0.228084i	0.993476	+	0.114042i	$0.0363798\pi$
$947$			7.01891i			0.228084i	−0.993476	+	0.114042i	$0.963620\pi$
$948$		0			0
$949$	17.3074	+	14.1683i	0.561821	+	0.459924i
$950$		0			0
$951$			8.42800i			0.273297i
$952$		0			0
$953$	7.16649			0.232145			0.116073	−	0.993241i	$-0.462969\pi$
$953$	7.16649			0.232145			0.116073	+	0.993241i	$0.462969\pi$
$954$		0			0
$955$		−	13.5428i		−	0.438236i
$956$		0			0
$957$		−	44.5310i		−	1.43948i
$958$		0			0
$959$	19.0285			0.614462
$960$		0			0
$961$	6.46979			0.208703
$962$		0			0
$963$	−0.297322			−0.00958105
$964$		0			0
$965$	6.50024			0.209250
$966$		0			0
$967$			44.4812i			1.43042i	0.698910	+	0.715210i	$0.253669\pi$
$967$			44.4812i			1.43042i	−0.698910	+	0.715210i	$0.746331\pi$
$968$		0			0
$969$			18.7078i			0.600980i
$970$		0			0
$971$	7.81119			0.250673			0.125337	−	0.992114i	$-0.459999\pi$
$971$	7.81119			0.250673			0.125337	+	0.992114i	$0.459999\pi$
$972$		0			0
$973$			9.02446i			0.289311i
$974$		0			0
$975$	−7.77716	+	9.50021i	−0.249068	+	0.304250i
$976$		0			0
$977$		−	40.3274i		−	1.29019i	−0.764103	−	0.645095i	$-0.776818\pi$
$977$		−	40.3274i		−	1.29019i	0.764103	−	0.645095i	$-0.223182\pi$
$978$		0			0
$979$	−10.3738			−0.331549
$980$		0			0
$981$			11.9258i			0.380762i
$982$		0			0
$983$			30.2102i			0.963556i	0.876293	+	0.481778i	$0.160009\pi$
$983$			30.2102i			0.963556i	−0.876293	+	0.481778i	$0.839991\pi$
$984$		0			0
$985$	6.21764			0.198110
$986$		0			0
$987$	−9.87937			−0.314464
$988$		0			0
$989$	−14.5885			−0.463888
$990$		0			0
$991$	27.4302			0.871350			0.435675	−	0.900104i	$-0.356510\pi$
$991$	27.4302			0.871350			0.435675	+	0.900104i	$0.356510\pi$
$992$		0			0
$993$			1.97596i			0.0627052i
$994$		0			0
$995$		−	55.5038i		−	1.75959i
$996$		0			0
$997$	−2.67074			−0.0845831			−0.0422916	−	0.999105i	$-0.513466\pi$
$997$	−2.67074			−0.0845831			−0.0422916	+	0.999105i	$0.513466\pi$
$998$		0			0
$999$			3.11546i			0.0985688i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4368.2.h.r.337.1		10
4.3	odd	2		2184.2.h.f.337.1	✓	10
13.12	even	2	inner	4368.2.h.r.337.10		10
52.51	odd	2		2184.2.h.f.337.10	yes	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2184.2.h.f.337.1	✓	10	4.3	odd	2
2184.2.h.f.337.10	yes	10	52.51	odd	2
4368.2.h.r.337.1		10	1.1	even	1	trivial
4368.2.h.r.337.10		10	13.12	even	2	inner

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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -2.89917i q^{5} -1.00000i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -2.89917i q^{5} -1.00000i q^{7} +1.00000 q^{9} -4.62147i q^{11} +(-2.28392 + 2.78993i) q^{13} +2.89917i q^{15} +2.68288 q^{17} -6.97302i q^{19} +1.00000i q^{21} -5.74252 q^{23} -3.40518 q^{25} -1.00000 q^{27} -9.63568 q^{29} -4.95280i q^{31} +4.62147i q^{33} -2.89917 q^{35} -3.11546i q^{37} +(2.28392 - 2.78993i) q^{39} +9.43961i q^{41} +2.54044 q^{43} -2.89917i q^{45} +9.87937i q^{47} -1.00000 q^{49} -2.68288 q^{51} +12.0885 q^{53} -13.3984 q^{55} +6.97302i q^{57} -13.4520i q^{59} -6.69532 q^{61} -1.00000i q^{63} +(8.08848 + 6.62147i) q^{65} -5.56826i q^{67} +5.74252 q^{69} -5.50601i q^{71} -6.20352i q^{73} +3.40518 q^{75} -4.62147 q^{77} -2.65548 q^{79} +1.00000 q^{81} +15.9482i q^{83} -7.77812i q^{85} +9.63568 q^{87} -2.24470i q^{89} +(2.78993 + 2.28392i) q^{91} +4.95280i q^{93} -20.2160 q^{95} -14.5731i q^{97} -4.62147i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{3} + 10 q^{9}+O(q^{10}) \) 10 * q - 10 * q^3 + 10 * q^9	\( 10 q - 10 q^{3} + 10 q^{9} - 4 q^{13} + 10 q^{17} - 8 q^{23} - 2 q^{25} - 10 q^{27} - 44 q^{29} + 4 q^{39} - 20 q^{43} - 10 q^{49} - 10 q^{51} + 10 q^{53} + 2 q^{55} + 18 q^{61} - 30 q^{65} + 8 q^{69} + 2 q^{75} - 2 q^{77} - 2 q^{79} + 10 q^{81} + 44 q^{87} + 6 q^{91} - 44 q^{95}+O(q^{100}) \) 10 * q - 10 * q^3 + 10 * q^9 - 4 * q^13 + 10 * q^17 - 8 * q^23 - 2 * q^25 - 10 * q^27 - 44 * q^29 + 4 * q^39 - 20 * q^43 - 10 * q^49 - 10 * q^51 + 10 * q^53 + 2 * q^55 + 18 * q^61 - 30 * q^65 + 8 * q^69 + 2 * q^75 - 2 * q^77 - 2 * q^79 + 10 * q^81 + 44 * q^87 + 6 * q^91 - 44 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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