LMFDB - Embedded newform 4368.2.h.q.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:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 15x^{6} + 67x^{4} + 77x^{2} + 4 $ x^8 + 15x^6 + 67x^4 + 77*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 273)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.1
Root			$-2.60520i$ of defining polynomial
Character	$\chi$	$=$	4368.337
Dual form			4368.2.h.q.337.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -3.78706i q^{5} +1.00000i q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -3.78706i q^{5} +1.00000i q^{7} +1.00000 q^{9} -2.55475i q^{11} +(0.0504439 - 3.60520i) q^{13} -3.78706i q^{15} -3.21040 q^{17} +8.44270i q^{19} +1.00000i q^{21} -3.97809 q^{23} -9.34181 q^{25} +1.00000 q^{27} -2.86858 q^{29} -0.868584i q^{31} -2.55475i q^{33} +3.78706 q^{35} -5.10951i q^{37} +(0.0504439 - 3.60520i) q^{39} -3.34436i q^{41} -4.86858 q^{43} -3.78706i q^{45} -11.0983i q^{47} -1.00000 q^{49} -3.21040 q^{51} +1.33319 q^{53} -9.67501 q^{55} +8.44270i q^{57} -10.6556i q^{59} -4.10089 q^{61} +1.00000i q^{63} +(-13.6531 - 0.191034i) q^{65} +10.8467i q^{67} -3.97809 q^{69} +2.55475i q^{71} -6.76770i q^{73} -9.34181 q^{75} +2.55475 q^{77} +1.23230 q^{79} +1.00000 q^{81} +11.0983i q^{83} +12.1580i q^{85} -2.86858 q^{87} +5.52423i q^{89} +(3.60520 + 0.0504439i) q^{91} -0.868584i q^{93} +31.9730 q^{95} +5.65819i q^{97} -2.55475i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{3} + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^3 + 8 * q^9	$ 8 q + 8 q^{3} + 8 q^{9} - 6 q^{13} + 20 q^{17} + 6 q^{23} - 34 q^{25} + 8 q^{27} - 18 q^{29} + 6 q^{35} - 6 q^{39} - 34 q^{43} - 8 q^{49} + 20 q^{51} - 10 q^{53} - 16 q^{55} - 20 q^{61} - 10 q^{65} + 6 q^{69} - 34 q^{75} + 4 q^{77} + 2 q^{79} + 8 q^{81} - 18 q^{87} + 6 q^{91} + 78 q^{95}+O(q^{100}) $ 8 * q + 8 * q^3 + 8 * q^9 - 6 * q^13 + 20 * q^17 + 6 * q^23 - 34 * q^25 + 8 * q^27 - 18 * q^29 + 6 * q^35 - 6 * q^39 - 34 * q^43 - 8 * q^49 + 20 * q^51 - 10 * q^53 - 16 * q^55 - 20 * q^61 - 10 * q^65 + 6 * q^69 - 34 * q^75 + 4 * q^77 + 2 * q^79 + 8 * q^81 - 18 * q^87 + 6 * q^91 + 78 * 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
$5$		−	3.78706i		−	1.69362i	−0.531892	−	0.846812i	$-0.678519\pi$
$5$		−	3.78706i		−	1.69362i	0.531892	−	0.846812i	$-0.321481\pi$
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$		−	2.55475i		−	0.770287i	−0.922857	−	0.385144i	$-0.874152\pi$
$11$		−	2.55475i		−	0.770287i	0.922857	−	0.385144i	$-0.125848\pi$
$12$		0			0
$13$	0.0504439	−	3.60520i	0.0139906	−	0.999902i
$13$	0.0504439	−	3.60520i	0.0139906	−	0.999902i
$14$		0			0
$15$		−	3.78706i		−	0.977814i
$16$		0			0
$17$	−3.21040			−0.778636			−0.389318	−	0.921103i	$-0.627289\pi$
$17$	−3.21040			−0.778636			−0.389318	+	0.921103i	$0.627289\pi$
$18$		0			0
$19$			8.44270i			1.93689i	0.249230	+	0.968444i	$0.419822\pi$
$19$			8.44270i			1.93689i	−0.249230	+	0.968444i	$0.580178\pi$
$20$		0			0
$21$			1.00000i			0.218218i
$22$		0			0
$23$	−3.97809			−0.829490			−0.414745	−	0.909938i	$-0.636129\pi$
$23$	−3.97809			−0.829490			−0.414745	+	0.909938i	$0.636129\pi$
$24$		0			0
$25$	−9.34181			−1.86836
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$	−2.86858			−0.532683			−0.266341	−	0.963879i	$-0.585815\pi$
$29$	−2.86858			−0.532683			−0.266341	+	0.963879i	$0.585815\pi$
$30$		0			0
$31$		−	0.868584i		−	0.156002i	−0.996953	−	0.0780011i	$-0.975146\pi$
$31$		−	0.868584i		−	0.156002i	0.996953	−	0.0780011i	$-0.0248538\pi$
$32$		0			0
$33$		−	2.55475i		−	0.444726i
$34$		0			0
$35$	3.78706			0.640130
$36$		0			0
$37$		−	5.10951i		−	0.839998i	−0.907525	−	0.419999i	$-0.862030\pi$
$37$		−	5.10951i		−	0.839998i	0.907525	−	0.419999i	$-0.137970\pi$
$38$		0			0
$39$	0.0504439	−	3.60520i	0.00807749	−	0.577294i
$40$		0			0
$41$		−	3.34436i		−	0.522301i	−0.965298	−	0.261150i	$-0.915898\pi$
$41$		−	3.34436i		−	0.522301i	0.965298	−	0.261150i	$-0.0841019\pi$
$42$		0			0
$43$	−4.86858			−0.742452			−0.371226	−	0.928543i	$-0.621062\pi$
$43$	−4.86858			−0.742452			−0.371226	+	0.928543i	$0.621062\pi$
$44$		0			0
$45$		−	3.78706i		−	0.564541i
$46$		0			0
$47$		−	11.0983i		−	1.61886i	−0.587217	−	0.809430i	$-0.699776\pi$
$47$		−	11.0983i		−	1.61886i	0.587217	−	0.809430i	$-0.300224\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−3.21040			−0.449545
$52$		0			0
$53$	1.33319			0.183128			0.0915640	−	0.995799i	$-0.470813\pi$
$53$	1.33319			0.183128			0.0915640	+	0.995799i	$0.470813\pi$
$54$		0			0
$55$	−9.67501			−1.30458
$56$		0			0
$57$			8.44270i			1.11826i
$58$		0			0
$59$		−	10.6556i		−	1.38725i	−0.720338	−	0.693623i	$-0.756013\pi$
$59$		−	10.6556i		−	1.38725i	0.720338	−	0.693623i	$-0.243987\pi$
$60$		0			0
$61$	−4.10089			−0.525065			−0.262532	−	0.964923i	$-0.584558\pi$
$61$	−4.10089			−0.525065			−0.262532	+	0.964923i	$0.584558\pi$
$62$		0			0
$63$			1.00000i			0.125988i
$64$		0			0
$65$	−13.6531	−	0.191034i	−1.69346	−	0.0236949i
$66$		0			0
$67$			10.8467i			1.32513i	0.749003	+	0.662566i	$0.230533\pi$
$67$			10.8467i			1.32513i	−0.749003	+	0.662566i	$0.769467\pi$
$68$		0			0
$69$	−3.97809			−0.478906
$70$		0			0
$71$			2.55475i			0.303194i	0.988442	+	0.151597i	$0.0484415\pi$
$71$			2.55475i			0.303194i	−0.988442	+	0.151597i	$0.951558\pi$
$72$		0			0
$73$		−	6.76770i		−	0.792099i	−0.918229	−	0.396049i	$-0.870381\pi$
$73$		−	6.76770i		−	0.792099i	0.918229	−	0.396049i	$-0.129619\pi$
$74$		0			0
$75$	−9.34181			−1.07870
$76$		0			0
$77$	2.55475			0.291141
$78$		0			0
$79$	1.23230			0.138645			0.0693225	−	0.997594i	$-0.477916\pi$
$79$	1.23230			0.138645			0.0693225	+	0.997594i	$0.477916\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			11.0983i			1.21820i	0.793093	+	0.609101i	$0.208469\pi$
$83$			11.0983i			1.21820i	−0.793093	+	0.609101i	$0.791531\pi$
$84$		0			0
$85$			12.1580i			1.31872i
$86$		0			0
$87$	−2.86858			−0.307544
$88$		0			0
$89$			5.52423i			0.585567i	0.956179	+	0.292783i	$0.0945815\pi$
$89$			5.52423i			0.585567i	−0.956179	+	0.292783i	$0.905419\pi$
$90$		0			0
$91$	3.60520	+	0.0504439i	0.377927	+	0.00528796i
$92$		0			0
$93$		−	0.868584i		−	0.0900679i
$94$		0			0
$95$	31.9730			3.28036
$96$		0			0
$97$			5.65819i			0.574502i	0.957855	+	0.287251i	$0.0927414\pi$
$97$			5.65819i			0.574502i	−0.957855	+	0.287251i	$0.907259\pi$
$98$		0			0
$99$		−	2.55475i		−	0.256762i
$100$		0			0
$101$	−8.94756			−0.890316			−0.445158	−	0.895452i	$-0.646852\pi$
$101$	−8.94756			−0.890316			−0.445158	+	0.895452i	$0.646852\pi$
$102$		0			0
$103$	1.67501			0.165043			0.0825216	−	0.996589i	$-0.473703\pi$
$103$	1.67501			0.165043			0.0825216	+	0.996589i	$0.473703\pi$
$104$		0			0
$105$	3.78706			0.369579
$106$		0			0
$107$	0.846676			0.0818513			0.0409256	−	0.999162i	$-0.486969\pi$
$107$	0.846676			0.0818513			0.0409256	+	0.999162i	$0.486969\pi$
$108$		0			0
$109$			17.9949i			1.72360i	0.507248	+	0.861800i	$0.330663\pi$
$109$			17.9949i			1.72360i	−0.507248	+	0.861800i	$0.669337\pi$
$110$		0			0
$111$		−	5.10951i		−	0.484973i
$112$		0			0
$113$	2.86858			0.269854			0.134927	−	0.990856i	$-0.456920\pi$
$113$	2.86858			0.269854			0.134927	+	0.990856i	$0.456920\pi$
$114$		0			0
$115$			15.0653i			1.40484i
$116$		0			0
$117$	0.0504439	−	3.60520i	0.00466354	−	0.333301i
$118$		0			0
$119$		−	3.21040i		−	0.294297i
$120$		0			0
$121$	4.47323			0.406657
$122$		0			0
$123$		−	3.34436i		−	0.301551i
$124$		0			0
$125$			16.4427i			1.47068i
$126$		0			0
$127$	−17.2053			−1.52672			−0.763362	−	0.645971i	$-0.776453\pi$
$127$	−17.2053			−1.52672			−0.763362	+	0.645971i	$0.776453\pi$
$128$		0			0
$129$	−4.86858			−0.428655
$130$		0			0
$131$	−16.6836			−1.45766			−0.728828	−	0.684697i	$-0.759934\pi$
$131$	−16.6836			−1.45766			−0.728828	+	0.684697i	$0.759934\pi$
$132$		0			0
$133$	−8.44270			−0.732075
$134$		0			0
$135$		−	3.78706i		−	0.325938i
$136$		0			0
$137$		−	18.9134i		−	1.61588i	−0.589265	−	0.807940i	$-0.700583\pi$
$137$		−	18.9134i		−	1.61588i	0.589265	−	0.807940i	$-0.299417\pi$
$138$		0			0
$139$	−5.67501			−0.481348			−0.240674	−	0.970606i	$-0.577368\pi$
$139$	−5.67501			−0.481348			−0.240674	+	0.970606i	$0.577368\pi$
$140$		0			0
$141$		−	11.0983i		−	0.934649i
$142$		0			0
$143$	−9.21040	−	0.128872i	−0.770212	−	0.0107768i
$144$		0			0
$145$			10.8635i			0.902164i
$146$		0			0
$147$	−1.00000			−0.0824786
$148$		0			0
$149$			8.49259i			0.695740i	0.937543	+	0.347870i	$0.113095\pi$
$149$			8.49259i			0.695740i	−0.937543	+	0.347870i	$0.886905\pi$
$150$		0			0
$151$		−	14.2190i		−	1.15713i	−0.815637	−	0.578564i	$-0.803613\pi$
$151$		−	14.2190i		−	1.15713i	0.815637	−	0.578564i	$-0.196387\pi$
$152$		0			0
$153$	−3.21040			−0.259545
$154$		0			0
$155$	−3.28938			−0.264209
$156$		0			0
$157$	8.42079			0.672052			0.336026	−	0.941853i	$-0.390917\pi$
$157$	8.42079			0.672052			0.336026	+	0.941853i	$0.390917\pi$
$158$		0			0
$159$	1.33319			0.105729
$160$		0			0
$161$		−	3.97809i		−	0.313518i
$162$		0			0
$163$			3.77589i			0.295751i	0.989006	+	0.147875i	$0.0472435\pi$
$163$			3.77589i			0.295751i	−0.989006	+	0.147875i	$0.952757\pi$
$164$		0			0
$165$	−9.67501			−0.753198
$166$		0			0
$167$		−	8.83551i		−	0.683712i	−0.939752	−	0.341856i	$-0.888944\pi$
$167$		−	8.83551i		−	0.683712i	0.939752	−	0.341856i	$-0.111056\pi$
$168$		0			0
$169$	−12.9949	−	0.363721i	−0.999609	−	0.0279785i
$170$		0			0
$171$			8.44270i			0.645629i
$172$		0			0
$173$	17.6750			1.34381			0.671903	−	0.740639i	$-0.265477\pi$
$173$	17.6750			1.34381			0.671903	+	0.740639i	$0.265477\pi$
$174$		0			0
$175$		−	9.34181i		−	0.706175i
$176$		0			0
$177$		−	10.6556i		−	0.800927i
$178$		0			0
$179$	−16.9073			−1.26371			−0.631856	−	0.775086i	$-0.717707\pi$
$179$	−16.9073			−1.26371			−0.631856	+	0.775086i	$0.717707\pi$
$180$		0			0
$181$	7.73717			0.575099			0.287550	−	0.957766i	$-0.407159\pi$
$181$	7.73717			0.575099			0.287550	+	0.957766i	$0.407159\pi$
$182$		0			0
$183$	−4.10089			−0.303146
$184$		0			0
$185$	−19.3500			−1.42264
$186$		0			0
$187$			8.20178i			0.599773i
$188$		0			0
$189$			1.00000i			0.0727393i
$190$		0			0
$191$	−5.57412			−0.403329			−0.201664	−	0.979455i	$-0.564635\pi$
$191$	−5.57412			−0.403329			−0.201664	+	0.979455i	$0.564635\pi$
$192$		0			0
$193$		−	7.57412i		−	0.545197i	−0.962128	−	0.272598i	$-0.912117\pi$
$193$		−	7.57412i		−	0.545197i	0.962128	−	0.272598i	$-0.0878830\pi$
$194$		0			0
$195$	−13.6531	−	0.191034i	−0.977719	−	0.0136802i
$196$		0			0
$197$		−	13.3444i		−	0.950746i	−0.879784	−	0.475373i	$-0.842313\pi$
$197$		−	13.3444i		−	0.950746i	0.879784	−	0.475373i	$-0.157687\pi$
$198$		0			0
$199$	16.8854			1.19697			0.598487	−	0.801132i	$-0.295769\pi$
$199$	16.8854			1.19697			0.598487	+	0.801132i	$0.295769\pi$
$200$		0			0
$201$			10.8467i			0.765066i
$202$		0			0
$203$		−	2.86858i		−	0.201335i
$204$		0			0
$205$	−12.6653			−0.884581
$206$		0			0
$207$	−3.97809			−0.276497
$208$		0			0
$209$	21.5690			1.49196
$210$		0			0
$211$	−11.6969			−0.805249			−0.402624	−	0.915365i	$-0.631902\pi$
$211$	−11.6969			−0.805249			−0.402624	+	0.915365i	$0.631902\pi$
$212$		0			0
$213$			2.55475i			0.175049i
$214$		0			0
$215$			18.4376i			1.25743i
$216$		0			0
$217$	0.868584			0.0589633
$218$		0			0
$219$		−	6.76770i		−	0.457319i
$220$		0			0
$221$	−0.161945	+	11.5741i	−0.0108936	+	0.778559i
$222$		0			0
$223$		−	4.24093i		−	0.283993i	−0.989867	−	0.141997i	$-0.954648\pi$
$223$		−	4.24093i		−	0.283993i	0.989867	−	0.141997i	$-0.0453522\pi$
$224$		0			0
$225$	−9.34181			−0.622788
$226$		0			0
$227$			1.72643i			0.114587i	0.998357	+	0.0572934i	$0.0182471\pi$
$227$			1.72643i			0.114587i	−0.998357	+	0.0572934i	$0.981753\pi$
$228$		0			0
$229$			6.82833i			0.451229i	0.974217	+	0.225614i	$0.0724389\pi$
$229$			6.82833i			0.451229i	−0.974217	+	0.225614i	$0.927561\pi$
$230$		0			0
$231$	2.55475			0.168091
$232$		0			0
$233$	−1.33319			−0.0873403			−0.0436702	−	0.999046i	$-0.513905\pi$
$233$	−1.33319			−0.0873403			−0.0436702	+	0.999046i	$0.513905\pi$
$234$		0			0
$235$	−42.0301			−2.74174
$236$		0			0
$237$	1.23230			0.0800468
$238$		0			0
$239$		−	3.90985i		−	0.252907i	−0.991973	−	0.126454i	$-0.959640\pi$
$239$		−	3.90985i		−	0.252907i	0.991973	−	0.126454i	$-0.0403595\pi$
$240$		0			0
$241$		−	25.3903i		−	1.63553i	−0.575552	−	0.817765i	$-0.695213\pi$
$241$		−	25.3903i		−	1.63553i	0.575552	−	0.817765i	$-0.304787\pi$
$242$		0			0
$243$	1.00000			0.0641500
$244$		0			0
$245$			3.78706i			0.241946i
$246$		0			0
$247$	30.4376	+	0.425883i	1.93670	+	0.0270983i
$248$		0			0
$249$			11.0983i			0.703329i
$250$		0			0
$251$	25.7708			1.62664			0.813319	−	0.581818i	$-0.197658\pi$
$251$	25.7708			1.62664			0.813319	+	0.581818i	$0.197658\pi$
$252$		0			0
$253$			10.1631i			0.638945i
$254$		0			0
$255$			12.1580i			0.761361i
$256$		0			0
$257$	5.87678			0.366584			0.183292	−	0.983059i	$-0.441325\pi$
$257$	5.87678			0.366584			0.183292	+	0.983059i	$0.441325\pi$
$258$		0			0
$259$	5.10951			0.317489
$260$		0			0
$261$	−2.86858			−0.177561
$262$		0			0
$263$	10.2409			0.631483			0.315741	−	0.948845i	$-0.397747\pi$
$263$	10.2409			0.631483			0.315741	+	0.948845i	$0.397747\pi$
$264$		0			0
$265$		−	5.04888i		−	0.310150i
$266$		0			0
$267$			5.52423i			0.338077i
$268$		0			0
$269$	−16.1396			−0.984050			−0.492025	−	0.870581i	$-0.663743\pi$
$269$	−16.1396			−0.984050			−0.492025	+	0.870581i	$0.663743\pi$
$270$		0			0
$271$			12.2577i			0.744605i	0.928111	+	0.372302i	$0.121432\pi$
$271$			12.2577i			0.744605i	−0.928111	+	0.372302i	$0.878568\pi$
$272$		0			0
$273$	3.60520	+	0.0504439i	0.218197	+	0.00305300i
$274$		0			0
$275$			23.8660i			1.43918i
$276$		0			0
$277$	9.81504			0.589729			0.294864	−	0.955539i	$-0.404726\pi$
$277$	9.81504			0.589729			0.294864	+	0.955539i	$0.404726\pi$
$278$		0			0
$279$		−	0.868584i		−	0.0520007i
$280$		0			0
$281$			16.1910i			0.965876i	0.875655	+	0.482938i	$0.160430\pi$
$281$			16.1910i			0.965876i	−0.875655	+	0.482938i	$0.839570\pi$
$282$		0			0
$283$	−19.1482			−1.13824			−0.569122	−	0.822253i	$-0.692717\pi$
$283$	−19.1482			−1.13824			−0.569122	+	0.822253i	$0.692717\pi$
$284$		0			0
$285$	31.9730			1.89392
$286$		0			0
$287$	3.34436			0.197411
$288$		0			0
$289$	−6.69335			−0.393727
$290$		0			0
$291$			5.65819i			0.331689i
$292$		0			0
$293$		−	24.6725i		−	1.44138i	−0.693257	−	0.720690i	$-0.743825\pi$
$293$		−	24.6725i		−	1.44138i	0.693257	−	0.720690i	$-0.256175\pi$
$294$		0			0
$295$	−40.3535			−2.34947
$296$		0			0
$297$		−	2.55475i		−	0.148242i
$298$		0			0
$299$	−0.200670	+	14.3418i	−0.0116051	+	0.829408i
$300$		0			0
$301$		−	4.86858i		−	0.280620i
$302$		0			0
$303$	−8.94756			−0.514024
$304$		0			0
$305$			15.5303i			0.889263i
$306$		0			0
$307$			13.7540i			0.784981i	0.919756	+	0.392491i	$0.128386\pi$
$307$			13.7540i			0.784981i	−0.919756	+	0.392491i	$0.871614\pi$
$308$		0			0
$309$	1.67501			0.0952877
$310$		0			0
$311$	−12.1580			−0.689415			−0.344707	−	0.938710i	$-0.612022\pi$
$311$	−12.1580			−0.689415			−0.344707	+	0.938710i	$0.612022\pi$
$312$		0			0
$313$	21.6699			1.22486			0.612428	−	0.790526i	$-0.290193\pi$
$313$	21.6699			1.22486			0.612428	+	0.790526i	$0.290193\pi$
$314$		0			0
$315$	3.78706			0.213377
$316$		0			0
$317$		−	29.3780i		−	1.65003i	−0.565109	−	0.825016i	$-0.691166\pi$
$317$		−	29.3780i		−	1.65003i	0.565109	−	0.825016i	$-0.308834\pi$
$318$		0			0
$319$			7.32853i			0.410319i
$320$		0			0
$321$	0.846676			0.0472569
$322$		0			0
$323$		−	27.1044i		−	1.50813i
$324$		0			0
$325$	−0.471237	+	33.6791i	−0.0261396	+	1.86818i
$326$		0			0
$327$			17.9949i			0.995121i
$328$		0			0
$329$	11.0983			0.611871
$330$		0			0
$331$		−	23.9124i		−	1.31434i	−0.753741	−	0.657171i	$-0.771753\pi$
$331$		−	23.9124i		−	1.31434i	0.753741	−	0.657171i	$-0.228247\pi$
$332$		0			0
$333$		−	5.10951i		−	0.279999i
$334$		0			0
$335$	41.0770			2.24428
$336$		0			0
$337$	−15.5706			−0.848182			−0.424091	−	0.905620i	$-0.639406\pi$
$337$	−15.5706			−0.848182			−0.424091	+	0.905620i	$0.639406\pi$
$338$		0			0
$339$	2.86858			0.155800
$340$		0			0
$341$	−2.21902			−0.120167
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$			15.0653i			0.811087i
$346$		0			0
$347$	−1.04845			−0.0562838			−0.0281419	−	0.999604i	$-0.508959\pi$
$347$	−1.04845			−0.0562838			−0.0281419	+	0.999604i	$0.508959\pi$
$348$		0			0
$349$			13.8721i			0.742557i	0.928521	+	0.371279i	$0.121081\pi$
$349$			13.8721i			0.742557i	−0.928521	+	0.371279i	$0.878919\pi$
$350$		0			0
$351$	0.0504439	−	3.60520i	0.00269250	−	0.192431i
$352$		0			0
$353$		−	9.96693i		−	0.530486i	−0.964182	−	0.265243i	$-0.914548\pi$
$353$		−	9.96693i		−	0.530486i	0.964182	−	0.265243i	$-0.0854522\pi$
$354$		0			0
$355$	9.67501			0.513496
$356$		0			0
$357$		−	3.21040i		−	0.169912i
$358$		0			0
$359$			18.9705i			1.00122i	0.865672	+	0.500611i	$0.166891\pi$
$359$			18.9705i			1.00122i	−0.865672	+	0.500611i	$0.833109\pi$
$360$		0			0
$361$	−52.2792			−2.75154
$362$		0			0
$363$	4.47323			0.234784
$364$		0			0
$365$	−25.6297			−1.34152
$366$		0			0
$367$	−22.3586			−1.16711			−0.583556	−	0.812073i	$-0.698339\pi$
$367$	−22.3586			−1.16711			−0.583556	+	0.812073i	$0.698339\pi$
$368$		0			0
$369$		−	3.34436i		−	0.174100i
$370$		0			0
$371$			1.33319i			0.0692159i
$372$		0			0
$373$	26.0958			1.35119			0.675595	−	0.737273i	$-0.263887\pi$
$373$	26.0958			1.35119			0.675595	+	0.737273i	$0.263887\pi$
$374$		0			0
$375$			16.4427i			0.849097i
$376$		0			0
$377$	−0.144703	+	10.3418i	−0.00745256	+	0.532630i
$378$		0			0
$379$			12.0000i			0.616399i	0.951322	+	0.308199i	$0.0997264\pi$
$379$			12.0000i			0.616399i	−0.951322	+	0.308199i	$0.900274\pi$
$380$		0			0
$381$	−17.2053			−0.881455
$382$		0			0
$383$			4.96229i			0.253561i	0.991931	+	0.126781i	$0.0404644\pi$
$383$			4.96229i			0.253561i	−0.991931	+	0.126781i	$0.959536\pi$
$384$		0			0
$385$		−	9.67501i		−	0.493084i
$386$		0			0
$387$	−4.86858			−0.247484
$388$		0			0
$389$	17.8757			0.906333			0.453166	−	0.891426i	$-0.350294\pi$
$389$	17.8757			0.906333			0.453166	+	0.891426i	$0.350294\pi$
$390$		0			0
$391$	12.7713			0.645870
$392$		0			0
$393$	−16.6836			−0.841578
$394$		0			0
$395$		−	4.66681i		−	0.234813i
$396$		0			0
$397$			7.06925i			0.354796i	0.984139	+	0.177398i	$0.0567679\pi$
$397$			7.06925i			0.354796i	−0.984139	+	0.177398i	$0.943232\pi$
$398$		0			0
$399$	−8.44270			−0.422664
$400$		0			0
$401$		−	9.12025i		−	0.455444i	−0.973726	−	0.227722i	$-0.926872\pi$
$401$		−	9.12025i		−	0.455444i	0.973726	−	0.227722i	$-0.0731277\pi$
$402$		0			0
$403$	−3.13142	−	0.0438147i	−0.155987	−	0.00218257i
$404$		0			0
$405$		−	3.78706i		−	0.188180i
$406$		0			0
$407$	−13.0535			−0.647040
$408$		0			0
$409$		−	18.2542i		−	0.902611i	−0.892369	−	0.451306i	$-0.850958\pi$
$409$		−	18.2542i		−	0.902611i	0.892369	−	0.451306i	$-0.149042\pi$
$410$		0			0
$411$		−	18.9134i		−	0.932929i
$412$		0			0
$413$	10.6556			0.524330
$414$		0			0
$415$	42.0301			2.06318
$416$		0			0
$417$	−5.67501			−0.277906
$418$		0			0
$419$	−13.6934			−0.668964			−0.334482	−	0.942402i	$-0.608561\pi$
$419$	−13.6934			−0.668964			−0.334482	+	0.942402i	$0.608561\pi$
$420$		0			0
$421$		−	33.8146i		−	1.64802i	−0.566573	−	0.824012i	$-0.691731\pi$
$421$		−	33.8146i		−	1.64802i	0.566573	−	0.824012i	$-0.308269\pi$
$422$		0			0
$423$		−	11.0983i		−	0.539620i
$424$		0			0
$425$	29.9909			1.45477
$426$		0			0
$427$		−	4.10089i		−	0.198456i
$428$		0			0
$429$	−9.21040	−	0.128872i	−0.444682	−	0.00622199i
$430$		0			0
$431$		−	20.3256i		−	0.979048i	−0.871990	−	0.489524i	$-0.837171\pi$
$431$		−	20.3256i		−	0.979048i	0.871990	−	0.489524i	$-0.162829\pi$
$432$		0			0
$433$	−17.1106			−0.822284			−0.411142	−	0.911571i	$-0.634870\pi$
$433$	−17.1106			−0.822284			−0.411142	+	0.911571i	$0.634870\pi$
$434$		0			0
$435$			10.8635i			0.520865i
$436$		0			0
$437$		−	33.5858i		−	1.60663i
$438$		0			0
$439$	1.67501			0.0799436			0.0399718	−	0.999201i	$-0.487273\pi$
$439$	1.67501			0.0799436			0.0399718	+	0.999201i	$0.487273\pi$
$440$		0			0
$441$	−1.00000			−0.0476190
$442$		0			0
$443$	12.9073			0.613245			0.306622	−	0.951831i	$-0.400801\pi$
$443$	12.9073			0.613245			0.306622	+	0.951831i	$0.400801\pi$
$444$		0			0
$445$	20.9206			0.991730
$446$		0			0
$447$			8.49259i			0.401686i
$448$		0			0
$449$			21.1762i			0.999368i	0.866208	+	0.499684i	$0.166550\pi$
$449$			21.1762i			0.999368i	−0.866208	+	0.499684i	$0.833450\pi$
$450$		0			0
$451$	−8.54401			−0.402322
$452$		0			0
$453$		−	14.2190i		−	0.668068i
$454$		0			0
$455$	0.191034	−	13.6531i	0.00895581	−	0.640067i
$456$		0			0
$457$		−	8.41570i		−	0.393670i	−0.980437	−	0.196835i	$-0.936934\pi$
$457$		−	8.41570i		−	0.393670i	0.980437	−	0.196835i	$-0.0630663\pi$
$458$		0			0
$459$	−3.21040			−0.149848
$460$		0			0
$461$			1.76515i			0.0822113i	0.999155	+	0.0411056i	$0.0130880\pi$
$461$			1.76515i			0.0822113i	−0.999155	+	0.0411056i	$0.986912\pi$
$462$		0			0
$463$		−	38.9241i		−	1.80896i	−0.426519	−	0.904479i	$-0.640260\pi$
$463$		−	38.9241i		−	1.80896i	0.426519	−	0.904479i	$-0.359740\pi$
$464$		0			0
$465$	−3.28938			−0.152541
$466$		0			0
$467$	30.4544			1.40926			0.704631	−	0.709573i	$-0.251112\pi$
$467$	30.4544			1.40926			0.704631	+	0.709573i	$0.251112\pi$
$468$		0			0
$469$	−10.8467			−0.500853
$470$		0			0
$471$	8.42079			0.388010
$472$		0			0
$473$			12.4380i			0.571902i
$474$		0			0
$475$		−	78.8701i		−	3.61881i
$476$		0			0
$477$	1.33319			0.0610427
$478$		0			0
$479$			3.54656i			0.162046i	0.996712	+	0.0810232i	$0.0258188\pi$
$479$			3.54656i			0.162046i	−0.996712	+	0.0810232i	$0.974181\pi$
$480$		0			0
$481$	−18.4208	−	0.257744i	−0.839916	−	0.0117521i
$482$		0			0
$483$		−	3.97809i		−	0.181009i
$484$		0			0
$485$	21.4279			0.972990
$486$		0			0
$487$		−	39.9898i		−	1.81211i	−0.423158	−	0.906056i	$-0.639078\pi$
$487$		−	39.9898i		−	1.81211i	0.423158	−	0.906056i	$-0.360922\pi$
$488$		0			0
$489$			3.77589i			0.170752i
$490$		0			0
$491$	9.69844			0.437685			0.218842	−	0.975760i	$-0.429772\pi$
$491$	9.69844			0.437685			0.218842	+	0.975760i	$0.429772\pi$
$492$		0			0
$493$	9.20929			0.414766
$494$		0			0
$495$	−9.67501			−0.434859
$496$		0			0
$497$	−2.55475			−0.114596
$498$		0			0
$499$		−	24.0336i		−	1.07589i	−0.842979	−	0.537947i	$-0.819200\pi$
$499$		−	24.0336i		−	1.07589i	0.842979	−	0.537947i	$-0.180800\pi$
$500$		0			0
$501$		−	8.83551i		−	0.394741i
$502$		0			0
$503$	−3.67390			−0.163811			−0.0819056	−	0.996640i	$-0.526101\pi$
$503$	−3.67390			−0.163811			−0.0819056	+	0.996640i	$0.526101\pi$
$504$		0			0
$505$			33.8849i			1.50786i
$506$		0			0
$507$	−12.9949	−	0.363721i	−0.577124	−	0.0161534i
$508$		0			0
$509$			19.3612i			0.858169i	0.903264	+	0.429085i	$0.141164\pi$
$509$			19.3612i			0.858169i	−0.903264	+	0.429085i	$0.858836\pi$
$510$		0			0
$511$	6.76770			0.299385
$512$		0			0
$513$			8.44270i			0.372754i
$514$		0			0
$515$		−	6.34334i		−	0.279521i
$516$		0			0
$517$	−28.3535			−1.24699
$518$		0			0
$519$	17.6750			0.775847
$520$		0			0
$521$	34.1130			1.49452			0.747260	−	0.664532i	$-0.231369\pi$
$521$	34.1130			1.49452			0.747260	+	0.664532i	$0.231369\pi$
$522$		0			0
$523$	33.0250			1.44408			0.722042	−	0.691850i	$-0.243204\pi$
$523$	33.0250			1.44408			0.722042	+	0.691850i	$0.243204\pi$
$524$		0			0
$525$		−	9.34181i		−	0.407710i
$526$		0			0
$527$			2.78850i			0.121469i
$528$		0			0
$529$	−7.17478			−0.311947
$530$		0			0
$531$		−	10.6556i		−	0.462415i
$532$		0			0
$533$	−12.0571	−	0.168702i	−0.522250	−	0.00730731i
$534$		0			0
$535$		−	3.20641i		−	0.138625i
$536$		0			0
$537$	−16.9073			−0.729604
$538$		0			0
$539$			2.55475i			0.110041i
$540$		0			0
$541$		−	24.8416i		−	1.06802i	−0.845477	−	0.534012i	$-0.820684\pi$
$541$		−	24.8416i		−	1.06802i	0.845477	−	0.534012i	$-0.179316\pi$
$542$		0			0
$543$	7.73717			0.332034
$544$		0			0
$545$	68.1478			2.91913
$546$		0			0
$547$	−8.14114			−0.348090			−0.174045	−	0.984738i	$-0.555684\pi$
$547$	−8.14114			−0.348090			−0.174045	+	0.984738i	$0.555684\pi$
$548$		0			0
$549$	−4.10089			−0.175022
$550$		0			0
$551$		−	24.2186i		−	1.03175i
$552$		0			0
$553$			1.23230i			0.0524029i
$554$		0			0
$555$	−19.3500			−0.821362
$556$		0			0
$557$			8.11052i			0.343654i	0.985127	+	0.171827i	$0.0549670\pi$
$557$			8.11052i			0.343654i	−0.985127	+	0.171827i	$0.945033\pi$
$558$		0			0
$559$	−0.245590	+	17.5522i	−0.0103874	+	0.742379i
$560$		0			0
$561$			8.20178i			0.346279i
$562$		0			0
$563$	−11.1044			−0.467995			−0.233998	−	0.972237i	$-0.575181\pi$
$563$	−11.1044			−0.467995			−0.233998	+	0.972237i	$0.575181\pi$
$564$		0			0
$565$		−	10.8635i		−	0.457031i
$566$		0			0
$567$			1.00000i			0.0419961i
$568$		0			0
$569$	2.86858			0.120257			0.0601286	−	0.998191i	$-0.480849\pi$
$569$	2.86858			0.120257			0.0601286	+	0.998191i	$0.480849\pi$
$570$		0			0
$571$	−32.4947			−1.35986			−0.679930	−	0.733277i	$-0.737990\pi$
$571$	−32.4947			−1.35986			−0.679930	+	0.733277i	$0.737990\pi$
$572$		0			0
$573$	−5.57412			−0.232862
$574$		0			0
$575$	37.1626			1.54979
$576$		0			0
$577$		−	24.3759i		−	1.01478i	−0.861716	−	0.507390i	$-0.830610\pi$
$577$		−	24.3759i		−	1.01478i	0.861716	−	0.507390i	$-0.169390\pi$
$578$		0			0
$579$		−	7.57412i		−	0.314770i
$580$		0			0
$581$	−11.0983			−0.460437
$582$		0			0
$583$		−	3.40598i		−	0.141061i
$584$		0			0
$585$	−13.6531	−	0.191034i	−0.564486	−	0.00789828i
$586$		0			0
$587$			11.7483i			0.484906i	0.970163	+	0.242453i	$0.0779520\pi$
$587$			11.7483i			0.484906i	−0.970163	+	0.242453i	$0.922048\pi$
$588$		0			0
$589$	7.33319			0.302159
$590$		0			0
$591$		−	13.3444i		−	0.548914i
$592$		0			0
$593$			46.6459i			1.91552i	0.287574	+	0.957759i	$0.407151\pi$
$593$			46.6459i			1.91552i	−0.287574	+	0.957759i	$0.592849\pi$
$594$		0			0
$595$	−12.1580			−0.498428
$596$		0			0
$597$	16.8854			0.691073
$598$		0			0
$599$	20.8197			0.850669			0.425335	−	0.905036i	$-0.360156\pi$
$599$	20.8197			0.850669			0.425335	+	0.905036i	$0.360156\pi$
$600$		0			0
$601$	24.2588			0.989539			0.494770	−	0.869024i	$-0.335252\pi$
$601$	24.2588			0.989539			0.494770	+	0.869024i	$0.335252\pi$
$602$		0			0
$603$			10.8467i			0.441711i
$604$		0			0
$605$		−	16.9404i		−	0.688724i
$606$		0			0
$607$	−6.20067			−0.251677			−0.125839	−	0.992051i	$-0.540162\pi$
$607$	−6.20067			−0.251677			−0.125839	+	0.992051i	$0.540162\pi$
$608$		0			0
$609$		−	2.86858i		−	0.116241i
$610$		0			0
$611$	−40.0117	−	0.559844i	−1.61870	−	0.0226489i
$612$		0			0
$613$		−	14.9026i		−	0.601912i	−0.953638	−	0.300956i	$-0.902694\pi$
$613$		−	14.9026i		−	0.601912i	0.953638	−	0.300956i	$-0.0973058\pi$
$614$		0			0
$615$	−12.6653			−0.510713
$616$		0			0
$617$		−	24.1472i		−	0.972130i	−0.873923	−	0.486065i	$-0.838432\pi$
$617$		−	24.1472i		−	0.972130i	0.873923	−	0.486065i	$-0.161568\pi$
$618$		0			0
$619$		−	43.9339i		−	1.76585i	−0.469513	−	0.882925i	$-0.655571\pi$
$619$		−	43.9339i		−	1.76585i	0.469513	−	0.882925i	$-0.344429\pi$
$620$		0			0
$621$	−3.97809			−0.159635
$622$		0			0
$623$	−5.52423			−0.221323
$624$		0			0
$625$	15.5604			0.622416
$626$		0			0
$627$	21.5690			0.861384
$628$		0			0
$629$			16.4036i			0.654052i
$630$		0			0
$631$			25.9511i			1.03310i	0.856258	+	0.516548i	$0.172783\pi$
$631$			25.9511i			1.03310i	−0.856258	+	0.516548i	$0.827217\pi$
$632$		0			0
$633$	−11.6969			−0.464911
$634$		0			0
$635$			65.1575i			2.58570i
$636$		0			0
$637$	−0.0504439	+	3.60520i	−0.00199866	+	0.142843i
$638$		0			0
$639$			2.55475i			0.101065i
$640$		0			0
$641$	−39.5858			−1.56355			−0.781773	−	0.623563i	$-0.785685\pi$
$641$	−39.5858			−1.56355			−0.781773	+	0.623563i	$0.785685\pi$
$642$		0			0
$643$			17.0434i			0.672125i	0.941840	+	0.336062i	$0.109095\pi$
$643$			17.0434i			0.672125i	−0.941840	+	0.336062i	$0.890905\pi$
$644$		0			0
$645$			18.4376i			0.725980i
$646$		0			0
$647$	27.7544			1.09114			0.545569	−	0.838066i	$-0.316313\pi$
$647$	27.7544			1.09114			0.545569	+	0.838066i	$0.316313\pi$
$648$		0			0
$649$	−27.2225			−1.06858
$650$		0			0
$651$	0.868584			0.0340425
$652$		0			0
$653$	−7.77080			−0.304095			−0.152048	−	0.988373i	$-0.548587\pi$
$653$	−7.77080			−0.304095			−0.152048	+	0.988373i	$0.548587\pi$
$654$		0			0
$655$			63.1819i			2.46872i
$656$		0			0
$657$		−	6.76770i		−	0.264033i
$658$		0			0
$659$	4.74935			0.185008			0.0925042	−	0.995712i	$-0.470513\pi$
$659$	4.74935			0.185008			0.0925042	+	0.995712i	$0.470513\pi$
$660$		0			0
$661$		−	38.0301i		−	1.47920i	−0.673047	−	0.739599i	$-0.735015\pi$
$661$		−	38.0301i		−	1.47920i	0.673047	−	0.739599i	$-0.264985\pi$
$662$		0			0
$663$	−0.161945	+	11.5741i	−0.00628942	+	0.449501i
$664$		0			0
$665$			31.9730i			1.23986i
$666$		0			0
$667$	11.4115			0.441855
$668$		0			0
$669$		−	4.24093i		−	0.163964i
$670$		0			0
$671$			10.4768i			0.404451i
$672$		0			0
$673$	23.0602			0.888905			0.444452	−	0.895803i	$-0.353398\pi$
$673$	23.0602			0.888905			0.444452	+	0.895803i	$0.353398\pi$
$674$		0			0
$675$	−9.34181			−0.359567
$676$		0			0
$677$	−44.3759			−1.70550			−0.852752	−	0.522317i	$-0.825068\pi$
$677$	−44.3759			−1.70550			−0.852752	+	0.522317i	$0.825068\pi$
$678$		0			0
$679$	−5.65819			−0.217141
$680$		0			0
$681$			1.72643i			0.0661568i
$682$		0			0
$683$		−	5.91494i		−	0.226329i	−0.993576	−	0.113165i	$-0.963901\pi$
$683$		−	5.91494i		−	0.226329i	0.993576	−	0.113165i	$-0.0360987\pi$
$684$		0			0
$685$	−71.6261			−2.73669
$686$		0			0
$687$			6.82833i			0.260517i
$688$		0			0
$689$	0.0672514	−	4.80642i	0.00256207	−	0.183110i
$690$		0			0
$691$			12.6883i			0.482685i	0.970440	+	0.241343i	$0.0775878\pi$
$691$			12.6883i			0.482685i	−0.970440	+	0.241343i	$0.922412\pi$
$692$		0			0
$693$	2.55475			0.0970471
$694$		0			0
$695$			21.4916i			0.815222i
$696$		0			0
$697$			10.7367i			0.406682i
$698$		0			0
$699$	−1.33319			−0.0504260
$700$		0			0
$701$	25.8487			0.976291			0.488146	−	0.872762i	$-0.337674\pi$
$701$	25.8487			0.976291			0.488146	+	0.872762i	$0.337674\pi$
$702$		0			0
$703$	43.1381			1.62698
$704$		0			0
$705$	−42.0301			−1.58294
$706$		0			0
$707$		−	8.94756i		−	0.336508i
$708$		0			0
$709$			18.8803i			0.709065i	0.935044	+	0.354533i	$0.115360\pi$
$709$			18.8803i			0.709065i	−0.935044	+	0.354533i	$0.884640\pi$
$710$		0			0
$711$	1.23230			0.0462150
$712$		0			0
$713$			3.45531i			0.129402i
$714$		0			0
$715$	−0.488045	+	34.8803i	−0.0182518	+	1.30445i
$716$		0			0
$717$		−	3.90985i		−	0.146016i
$718$		0			0
$719$	40.1580			1.49764			0.748820	−	0.662774i	$-0.230621\pi$
$719$	40.1580			1.49764			0.748820	+	0.662774i	$0.230621\pi$
$720$		0			0
$721$			1.67501i			0.0623804i
$722$		0			0
$723$		−	25.3903i		−	0.944274i
$724$		0			0
$725$	26.7978			0.995244
$726$		0			0
$727$	32.0336			1.18806			0.594031	−	0.804442i	$-0.297536\pi$
$727$	32.0336			1.18806			0.594031	+	0.804442i	$0.297536\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	15.6301			0.578100
$732$		0			0
$733$			48.3265i			1.78498i	0.451066	+	0.892491i	$0.351044\pi$
$733$			48.3265i			1.78498i	−0.451066	+	0.892491i	$0.648956\pi$
$734$		0			0
$735$			3.78706i			0.139688i
$736$		0			0
$737$	27.7106			1.02073
$738$		0			0
$739$		−	35.7321i		−	1.31443i	−0.753705	−	0.657213i	$-0.771735\pi$
$739$		−	35.7321i		−	1.31443i	0.753705	−	0.657213i	$-0.228265\pi$
$740$		0			0
$741$	30.4376	+	0.425883i	1.11815	+	0.0156452i
$742$		0			0
$743$			45.0092i			1.65123i	0.564236	+	0.825613i	$0.309171\pi$
$743$			45.0092i			1.65123i	−0.564236	+	0.825613i	$0.690829\pi$
$744$		0			0
$745$	32.1619			1.17832
$746$		0			0
$747$			11.0983i			0.406067i
$748$		0			0
$749$			0.846676i			0.0309369i
$750$		0			0
$751$	−5.39425			−0.196839			−0.0984195	−	0.995145i	$-0.531379\pi$
$751$	−5.39425			−0.196839			−0.0984195	+	0.995145i	$0.531379\pi$
$752$		0			0
$753$	25.7708			0.939140
$754$		0			0
$755$	−53.8483			−1.95974
$756$		0			0
$757$	13.7896			0.501191			0.250595	−	0.968092i	$-0.419374\pi$
$757$	13.7896			0.501191			0.250595	+	0.968092i	$0.419374\pi$
$758$		0			0
$759$			10.1631i			0.368895i
$760$		0			0
$761$			30.6674i			1.11169i	0.831286	+	0.555846i	$0.187605\pi$
$761$			30.6674i			1.11169i	−0.831286	+	0.555846i	$0.812395\pi$
$762$		0			0
$763$	−17.9949			−0.651460
$764$		0			0
$765$			12.1580i			0.439572i
$766$		0			0
$767$	−38.4157	−	0.537512i	−1.38711	−	0.0194084i
$768$		0			0
$769$		−	48.6160i		−	1.75314i	−0.481278	−	0.876568i	$-0.659827\pi$
$769$		−	48.6160i		−	1.75314i	0.481278	−	0.876568i	$-0.340173\pi$
$770$		0			0
$771$	5.87678			0.211647
$772$		0			0
$773$			18.6729i			0.671617i	0.941930	+	0.335808i	$0.109009\pi$
$773$			18.6729i			0.671617i	−0.941930	+	0.335808i	$0.890991\pi$
$774$		0			0
$775$			8.11415i			0.291469i
$776$		0			0
$777$	5.10951			0.183303
$778$		0			0
$779$	28.2354			1.01164
$780$		0			0
$781$	6.52677			0.233546
$782$		0			0
$783$	−2.86858			−0.102515
$784$		0			0
$785$		−	31.8900i		−	1.13820i
$786$		0			0
$787$			9.61751i			0.342827i	0.985199	+	0.171414i	$0.0548334\pi$
$787$			9.61751i			0.342827i	−0.985199	+	0.171414i	$0.945167\pi$
$788$		0			0
$789$	10.2409			0.364587
$790$		0			0
$791$			2.86858i			0.101995i
$792$		0			0
$793$	−0.206865	+	14.7845i	−0.00734598	+	0.525013i
$794$		0			0
$795$		−	5.04888i		−	0.179065i
$796$		0			0
$797$	−3.73606			−0.132338			−0.0661691	−	0.997808i	$-0.521078\pi$
$797$	−3.73606			−0.132338			−0.0661691	+	0.997808i	$0.521078\pi$
$798$		0			0
$799$			35.6301i			1.26050i
$800$		0			0
$801$			5.52423i			0.195189i
$802$		0			0
$803$	−17.2898			−0.610144
$804$		0			0
$805$	−15.0653			−0.530981
$806$		0			0
$807$	−16.1396			−0.568141
$808$		0			0
$809$	−10.1850			−0.358084			−0.179042	−	0.983841i	$-0.557300\pi$
$809$	−10.1850			−0.358084			−0.179042	+	0.983841i	$0.557300\pi$
$810$		0			0
$811$		−	6.21902i		−	0.218379i	−0.994021	−	0.109190i	$-0.965174\pi$
$811$		−	6.21902i		−	0.218379i	0.994021	−	0.109190i	$-0.0348256\pi$
$812$		0			0
$813$			12.2577i			0.429898i
$814$		0			0
$815$	14.2995			0.500891
$816$		0			0
$817$		−	41.1040i		−	1.43805i
$818$		0			0
$819$	3.60520	+	0.0504439i	0.125976	+	0.00176265i
$820$		0			0
$821$		−	55.4534i		−	1.93534i	−0.252224	−	0.967669i	$-0.581162\pi$
$821$		−	55.4534i		−	1.93534i	0.252224	−	0.967669i	$-0.418838\pi$
$822$		0			0
$823$	20.4380			0.712425			0.356213	−	0.934405i	$-0.384068\pi$
$823$	20.4380			0.712425			0.356213	+	0.934405i	$0.384068\pi$
$824$		0			0
$825$			23.8660i			0.830909i
$826$		0			0
$827$			14.1289i			0.491309i	0.969357	+	0.245655i	$0.0790029\pi$
$827$			14.1289i			0.491309i	−0.969357	+	0.245655i	$0.920997\pi$
$828$		0			0
$829$	34.9159			1.21268			0.606340	−	0.795206i	$-0.292637\pi$
$829$	34.9159			1.21268			0.606340	+	0.795206i	$0.292637\pi$
$830$		0			0
$831$	9.81504			0.340480
$832$		0			0
$833$	3.21040			0.111234
$834$		0			0
$835$	−33.4606			−1.15795
$836$		0			0
$837$		−	0.868584i		−	0.0300226i
$838$		0			0
$839$		−	46.4437i		−	1.60341i	−0.597717	−	0.801707i	$-0.703925\pi$
$839$		−	46.4437i		−	1.60341i	0.597717	−	0.801707i	$-0.296075\pi$
$840$		0			0
$841$	−20.7712			−0.716249
$842$		0			0
$843$			16.1910i			0.557649i
$844$		0			0
$845$	−1.37743	+	49.2125i	−0.0473851	+	1.69296i
$846$		0			0
$847$			4.47323i			0.153702i
$848$		0			0
$849$	−19.1482			−0.657166
$850$		0			0
$851$			20.3261i			0.696770i
$852$		0			0
$853$			41.4462i			1.41909i	0.704659	+	0.709546i	$0.251100\pi$
$853$			41.4462i			1.41909i	−0.704659	+	0.709546i	$0.748900\pi$
$854$		0			0
$855$	31.9730			1.09345
$856$		0			0
$857$	33.8666			1.15686			0.578431	−	0.815732i	$-0.303665\pi$
$857$	33.8666			1.15686			0.578431	+	0.815732i	$0.303665\pi$
$858$		0			0
$859$	37.2890			1.27228			0.636141	−	0.771573i	$-0.280530\pi$
$859$	37.2890			1.27228			0.636141	+	0.771573i	$0.280530\pi$
$860$		0			0
$861$	3.34436			0.113975
$862$		0			0
$863$		−	38.7464i		−	1.31894i	−0.751730	−	0.659471i	$-0.770781\pi$
$863$		−	38.7464i		−	1.31894i	0.751730	−	0.659471i	$-0.229219\pi$
$864$		0			0
$865$		−	66.9363i		−	2.27590i
$866$		0			0
$867$	−6.69335			−0.227318
$868$		0			0
$869$		−	3.14823i		−	0.106797i
$870$		0			0
$871$	39.1044	+	0.547149i	1.32500	+	0.0185394i
$872$		0			0
$873$			5.65819i			0.191501i
$874$		0			0
$875$	−16.4427			−0.555865
$876$		0			0
$877$			51.8319i			1.75024i	0.483908	+	0.875119i	$0.339217\pi$
$877$			51.8319i			1.75024i	−0.483908	+	0.875119i	$0.660783\pi$
$878$		0			0
$879$		−	24.6725i		−	0.832181i
$880$		0			0
$881$	20.7896			0.700420			0.350210	−	0.936671i	$-0.386110\pi$
$881$	20.7896			0.700420			0.350210	+	0.936671i	$0.386110\pi$
$882$		0			0
$883$	−17.5292			−0.589904			−0.294952	−	0.955512i	$-0.595304\pi$
$883$	−17.5292			−0.589904			−0.294952	+	0.955512i	$0.595304\pi$
$884$		0			0
$885$	−40.3535			−1.35647
$886$		0			0
$887$	−9.53539			−0.320167			−0.160084	−	0.987103i	$-0.551176\pi$
$887$	−9.53539			−0.320167			−0.160084	+	0.987103i	$0.551176\pi$
$888$		0			0
$889$		−	17.2053i		−	0.577047i
$890$		0			0
$891$		−	2.55475i		−	0.0855875i
$892$		0			0
$893$	93.7000			3.13555
$894$		0			0
$895$			64.0290i			2.14025i
$896$		0			0
$897$	−0.200670	+	14.3418i	−0.00670019	+	0.478859i
$898$		0			0
$899$			2.49160i			0.0830997i
$900$		0			0
$901$	−4.28008			−0.142590
$902$		0			0
$903$		−	4.86858i		−	0.162016i
$904$		0			0
$905$		−	29.3011i		−	0.974002i
$906$		0			0
$907$	−27.3270			−0.907378			−0.453689	−	0.891160i	$-0.649892\pi$
$907$	−27.3270			−0.907378			−0.453689	+	0.891160i	$0.649892\pi$
$908$		0			0
$909$	−8.94756			−0.296772
$910$		0			0
$911$	38.9582			1.29074			0.645371	−	0.763869i	$-0.276703\pi$
$911$	38.9582			1.29074			0.645371	+	0.763869i	$0.276703\pi$
$912$		0			0
$913$	28.3535			0.938365
$914$		0			0
$915$			15.5303i			0.513416i
$916$		0			0
$917$		−	16.6836i		−	0.550942i
$918$		0			0
$919$	−9.94293			−0.327987			−0.163993	−	0.986461i	$-0.552438\pi$
$919$	−9.94293			−0.327987			−0.163993	+	0.986461i	$0.552438\pi$
$920$		0			0
$921$			13.7540i			0.453209i
$922$		0			0
$923$	9.21040	+	0.128872i	0.303164	+	0.00424187i
$924$		0			0
$925$			47.7321i			1.56942i
$926$		0			0
$927$	1.67501			0.0550144
$928$		0			0
$929$		−	7.80854i		−	0.256190i	−0.991762	−	0.128095i	$-0.959114\pi$
$929$		−	7.80854i		−	0.256190i	0.991762	−	0.128095i	$-0.0408862\pi$
$930$		0			0
$931$		−	8.44270i		−	0.276698i
$932$		0			0
$933$	−12.1580			−0.398034
$934$		0			0
$935$	31.0606			1.01579
$936$		0			0
$937$	−9.59246			−0.313372			−0.156686	−	0.987648i	$-0.550081\pi$
$937$	−9.59246			−0.313372			−0.156686	+	0.987648i	$0.550081\pi$
$938$		0			0
$939$	21.6699			0.707171
$940$		0			0
$941$			28.5043i			0.929214i	0.885517	+	0.464607i	$0.153804\pi$
$941$			28.5043i			0.929214i	−0.885517	+	0.464607i	$0.846196\pi$
$942$		0			0
$943$			13.3042i			0.433243i
$944$		0			0
$945$	3.78706			0.123193
$946$		0			0
$947$		−	47.8997i		−	1.55653i	−0.627936	−	0.778265i	$-0.716100\pi$
$947$		−	47.8997i		−	1.55653i	0.627936	−	0.778265i	$-0.283900\pi$
$948$		0			0
$949$	−24.3989	−	0.341389i	−0.792021	−	0.0110820i
$950$		0			0
$951$		−	29.3780i		−	0.952647i
$952$		0			0
$953$	32.5181			1.05337			0.526683	−	0.850062i	$-0.323436\pi$
$953$	32.5181			1.05337			0.526683	+	0.850062i	$0.323436\pi$
$954$		0			0
$955$			21.1095i			0.683088i
$956$		0			0
$957$			7.32853i			0.236898i
$958$		0			0
$959$	18.9134			0.610745
$960$		0			0
$961$	30.2456			0.975663
$962$		0			0
$963$	0.846676			0.0272838
$964$		0			0
$965$	−28.6836			−0.923359
$966$		0			0
$967$		−	18.0387i		−	0.580086i	−0.957014	−	0.290043i	$-0.906330\pi$
$967$		−	18.0387i		−	0.580086i	0.957014	−	0.290043i	$-0.0936697\pi$
$968$		0			0
$969$		−	27.1044i		−	0.870719i
$970$		0			0
$971$	−30.2190			−0.969774			−0.484887	−	0.874577i	$-0.661139\pi$
$971$	−30.2190			−0.969774			−0.484887	+	0.874577i	$0.661139\pi$
$972$		0			0
$973$		−	5.67501i		−	0.181932i
$974$		0			0
$975$	−0.471237	+	33.6791i	−0.0150917	+	1.07859i
$976$		0			0
$977$		−	38.4652i		−	1.23061i	−0.788289	−	0.615305i	$-0.789033\pi$
$977$		−	38.4652i		−	1.23061i	0.788289	−	0.615305i	$-0.210967\pi$
$978$		0			0
$979$	14.1130			0.451055
$980$		0			0
$981$			17.9949i			0.574533i
$982$		0			0
$983$		−	12.1742i		−	0.388297i	−0.980972	−	0.194149i	$-0.937806\pi$
$983$		−	12.1742i		−	0.388297i	0.980972	−	0.194149i	$-0.0621944\pi$
$984$		0			0
$985$	−50.5359			−1.61021
$986$		0			0
$987$	11.0983			0.353264
$988$		0			0
$989$	19.3677			0.615856
$990$		0			0
$991$	−14.4982			−0.460552			−0.230276	−	0.973125i	$-0.573963\pi$
$991$	−14.4982			−0.460552			−0.230276	+	0.973125i	$0.573963\pi$
$992$		0			0
$993$		−	23.9124i		−	0.758836i
$994$		0			0
$995$		−	63.9460i		−	2.02722i
$996$		0			0
$997$	−9.35086			−0.296145			−0.148072	−	0.988977i	$-0.547307\pi$
$997$	−9.35086			−0.296145			−0.148072	+	0.988977i	$0.547307\pi$
$998$		0			0
$999$		−	5.10951i		−	0.161658i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4368.2.h.q.337.1		8
4.3	odd	2		273.2.c.c.64.8	yes	8
12.11	even	2		819.2.c.d.64.1		8
13.12	even	2	inner	4368.2.h.q.337.8		8
28.27	even	2		1911.2.c.l.883.8		8
52.31	even	4		3549.2.a.v.1.4		4
52.47	even	4		3549.2.a.x.1.1		4
52.51	odd	2		273.2.c.c.64.1	✓	8
156.155	even	2		819.2.c.d.64.8		8
364.363	even	2		1911.2.c.l.883.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.c.c.64.1	✓	8	52.51	odd	2
273.2.c.c.64.8	yes	8	4.3	odd	2
819.2.c.d.64.1		8	12.11	even	2
819.2.c.d.64.8		8	156.155	even	2
1911.2.c.l.883.1		8	364.363	even	2
1911.2.c.l.883.8		8	28.27	even	2
3549.2.a.v.1.4		4	52.31	even	4
3549.2.a.x.1.1		4	52.47	even	4
4368.2.h.q.337.1		8	1.1	even	1	trivial
4368.2.h.q.337.8		8	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -3.78706i q^{5} +1.00000i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -3.78706i q^{5} +1.00000i q^{7} +1.00000 q^{9} -2.55475i q^{11} +(0.0504439 - 3.60520i) q^{13} -3.78706i q^{15} -3.21040 q^{17} +8.44270i q^{19} +1.00000i q^{21} -3.97809 q^{23} -9.34181 q^{25} +1.00000 q^{27} -2.86858 q^{29} -0.868584i q^{31} -2.55475i q^{33} +3.78706 q^{35} -5.10951i q^{37} +(0.0504439 - 3.60520i) q^{39} -3.34436i q^{41} -4.86858 q^{43} -3.78706i q^{45} -11.0983i q^{47} -1.00000 q^{49} -3.21040 q^{51} +1.33319 q^{53} -9.67501 q^{55} +8.44270i q^{57} -10.6556i q^{59} -4.10089 q^{61} +1.00000i q^{63} +(-13.6531 - 0.191034i) q^{65} +10.8467i q^{67} -3.97809 q^{69} +2.55475i q^{71} -6.76770i q^{73} -9.34181 q^{75} +2.55475 q^{77} +1.23230 q^{79} +1.00000 q^{81} +11.0983i q^{83} +12.1580i q^{85} -2.86858 q^{87} +5.52423i q^{89} +(3.60520 + 0.0504439i) q^{91} -0.868584i q^{93} +31.9730 q^{95} +5.65819i q^{97} -2.55475i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{3} + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^3 + 8 * q^9	\( 8 q + 8 q^{3} + 8 q^{9} - 6 q^{13} + 20 q^{17} + 6 q^{23} - 34 q^{25} + 8 q^{27} - 18 q^{29} + 6 q^{35} - 6 q^{39} - 34 q^{43} - 8 q^{49} + 20 q^{51} - 10 q^{53} - 16 q^{55} - 20 q^{61} - 10 q^{65} + 6 q^{69} - 34 q^{75} + 4 q^{77} + 2 q^{79} + 8 q^{81} - 18 q^{87} + 6 q^{91} + 78 q^{95}+O(q^{100}) \) 8 * q + 8 * q^3 + 8 * q^9 - 6 * q^13 + 20 * q^17 + 6 * q^23 - 34 * q^25 + 8 * q^27 - 18 * q^29 + 6 * q^35 - 6 * q^39 - 34 * q^43 - 8 * q^49 + 20 * q^51 - 10 * q^53 - 16 * q^55 - 20 * q^61 - 10 * q^65 + 6 * q^69 - 34 * q^75 + 4 * q^77 + 2 * q^79 + 8 * q^81 - 18 * q^87 + 6 * q^91 + 78 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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