LMFDB - Embedded newform 4368.2.h.q.337.4

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			337.4
Root			$-1.29051i$ of defining polynomial
Character	$\chi$	$=$	4368.337
Dual form			4368.2.h.q.337.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.33457i q^{5} -1.00000i q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.33457i q^{5} -1.00000i q^{7} +1.00000 q^{9} -4.88434i q^{11} +(3.59383 - 0.290514i) q^{13} -1.33457i q^{15} +4.58103 q^{17} -2.96874i q^{19} -1.00000i q^{21} +6.13080 q^{23} +3.21892 q^{25} +1.00000 q^{27} -7.63789 q^{29} +5.63789i q^{31} -4.88434i q^{33} -1.33457 q^{35} -9.76869i q^{37} +(3.59383 - 0.290514i) q^{39} +3.69669i q^{41} -9.63789 q^{43} -1.33457i q^{45} +5.27206i q^{47} -1.00000 q^{49} +4.58103 q^{51} +10.7374 q^{53} -6.51851 q^{55} -2.96874i q^{57} +10.3033i q^{59} -11.1877 q^{61} -1.00000i q^{63} +(-0.387712 - 4.79623i) q^{65} -5.50709i q^{67} +6.13080 q^{69} +4.88434i q^{71} +4.45023i q^{73} +3.21892 q^{75} -4.88434 q^{77} +3.54977 q^{79} +1.00000 q^{81} -5.27206i q^{83} -6.11372i q^{85} -7.63789 q^{87} -9.94120i q^{89} +(-0.290514 - 3.59383i) q^{91} +5.63789i q^{93} -3.96200 q^{95} -18.2189i q^{97} -4.88434i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$		−	1.33457i		−	0.596839i	−0.954435	−	0.298420i	$-0.903541\pi$
$5$		−	1.33457i		−	0.596839i	0.954435	−	0.298420i	$-0.0964595\pi$
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$		−	4.88434i		−	1.47268i	−0.676609	−	0.736342i	$-0.736551\pi$
$11$		−	4.88434i		−	1.47268i	0.676609	−	0.736342i	$-0.263449\pi$
$12$		0			0
$13$	3.59383	−	0.290514i	0.996749	−	0.0805742i
$13$	3.59383	−	0.290514i	0.996749	−	0.0805742i
$14$		0			0
$15$		−	1.33457i		−	0.344585i
$16$		0			0
$17$	4.58103			1.11106			0.555531	−	0.831496i	$-0.312515\pi$
$17$	4.58103			1.11106			0.555531	+	0.831496i	$0.312515\pi$
$18$		0			0
$19$		−	2.96874i		−	0.681076i	−0.940231	−	0.340538i	$-0.889391\pi$
$19$		−	2.96874i		−	0.681076i	0.940231	−	0.340538i	$-0.110609\pi$
$20$		0			0
$21$		−	1.00000i		−	0.218218i
$22$		0			0
$23$	6.13080			1.27836			0.639180	−	0.769057i	$-0.279274\pi$
$23$	6.13080			1.27836			0.639180	+	0.769057i	$0.279274\pi$
$24$		0			0
$25$	3.21892			0.643783
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$	−7.63789			−1.41832			−0.709160	−	0.705048i	$-0.750926\pi$
$29$	−7.63789			−1.41832			−0.709160	+	0.705048i	$0.750926\pi$
$30$		0			0
$31$			5.63789i			1.01259i	0.862359	+	0.506297i	$0.168986\pi$
$31$			5.63789i			1.01259i	−0.862359	+	0.506297i	$0.831014\pi$
$32$		0			0
$33$		−	4.88434i		−	0.850255i
$34$		0			0
$35$	−1.33457			−0.225584
$36$		0			0
$37$		−	9.76869i		−	1.60596i	−0.596005	−	0.802981i	$-0.703246\pi$
$37$		−	9.76869i		−	1.60596i	0.596005	−	0.802981i	$-0.296754\pi$
$38$		0			0
$39$	3.59383	−	0.290514i	0.575473	−	0.0465195i
$40$		0			0
$41$			3.69669i			0.577325i	0.957431	+	0.288663i	$0.0932106\pi$
$41$			3.69669i			0.577325i	−0.957431	+	0.288663i	$0.906789\pi$
$42$		0			0
$43$	−9.63789			−1.46976			−0.734882	−	0.678195i	$-0.762762\pi$
$43$	−9.63789			−1.46976			−0.734882	+	0.678195i	$0.762762\pi$
$44$		0			0
$45$		−	1.33457i		−	0.198946i
$46$		0			0
$47$			5.27206i			0.769008i	0.923123	+	0.384504i	$0.125628\pi$
$47$			5.27206i			0.769008i	−0.923123	+	0.384504i	$0.874372\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	4.58103			0.641472
$52$		0			0
$53$	10.7374			1.47490			0.737449	−	0.675402i	$-0.236030\pi$
$53$	10.7374			1.47490			0.737449	+	0.675402i	$0.236030\pi$
$54$		0			0
$55$	−6.51851			−0.878956
$56$		0			0
$57$		−	2.96874i		−	0.393219i
$58$		0			0
$59$			10.3033i			1.34138i	0.741739	+	0.670689i	$0.234001\pi$
$59$			10.3033i			1.34138i	−0.741739	+	0.670689i	$0.765999\pi$
$60$		0			0
$61$	−11.1877			−1.43243			−0.716216	−	0.697878i	$-0.754128\pi$
$61$	−11.1877			−1.43243			−0.716216	+	0.697878i	$0.754128\pi$
$62$		0			0
$63$		−	1.00000i		−	0.125988i
$64$		0			0
$65$	−0.387712	−	4.79623i	−0.0480898	−	0.594899i
$66$		0			0
$67$		−	5.50709i		−	0.672798i	−0.941720	−	0.336399i	$-0.890791\pi$
$67$		−	5.50709i		−	0.672798i	0.941720	−	0.336399i	$-0.109209\pi$
$68$		0			0
$69$	6.13080			0.738061
$70$		0			0
$71$			4.88434i			0.579665i	0.957077	+	0.289832i	$0.0935996\pi$
$71$			4.88434i			0.579665i	−0.957077	+	0.289832i	$0.906400\pi$
$72$		0			0
$73$			4.45023i			0.520860i	0.965493	+	0.260430i	$0.0838643\pi$
$73$			4.45023i			0.520860i	−0.965493	+	0.260430i	$0.916136\pi$
$74$		0			0
$75$	3.21892			0.371688
$76$		0			0
$77$	−4.88434			−0.556622
$78$		0			0
$79$	3.54977			0.399380			0.199690	−	0.979859i	$-0.436006\pi$
$79$	3.54977			0.399380			0.199690	+	0.979859i	$0.436006\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	5.27206i		−	0.578683i	−0.957226	−	0.289342i	$-0.906564\pi$
$83$		−	5.27206i		−	0.578683i	0.957226	−	0.289342i	$-0.0934364\pi$
$84$		0			0
$85$		−	6.11372i		−	0.663126i
$86$		0			0
$87$	−7.63789			−0.818867
$88$		0			0
$89$		−	9.94120i		−	1.05377i	−0.849938	−	0.526883i	$-0.823361\pi$
$89$		−	9.94120i		−	1.05377i	0.849938	−	0.526883i	$-0.176639\pi$
$90$		0			0
$91$	−0.290514	−	3.59383i	−0.0304542	−	0.376736i
$92$		0			0
$93$			5.63789i			0.584622i
$94$		0			0
$95$	−3.96200			−0.406493
$96$		0			0
$97$		−	18.2189i		−	1.84985i	−0.380149	−	0.924925i	$-0.624127\pi$
$97$		−	18.2189i		−	1.84985i	0.380149	−	0.924925i	$-0.375873\pi$
$98$		0			0
$99$		−	4.88434i		−	0.490895i
$100$		0			0
$101$	−10.6947			−1.06417			−0.532083	−	0.846692i	$-0.678591\pi$
$101$	−10.6947			−1.06417			−0.532083	+	0.846692i	$0.678591\pi$
$102$		0			0
$103$	−1.48149			−0.145975			−0.0729877	−	0.997333i	$-0.523253\pi$
$103$	−1.48149			−0.145975			−0.0729877	+	0.997333i	$0.523253\pi$
$104$		0			0
$105$	−1.33457			−0.130241
$106$		0			0
$107$	−4.49291			−0.434346			−0.217173	−	0.976133i	$-0.569684\pi$
$107$	−4.49291			−0.434346			−0.217173	+	0.976133i	$0.569684\pi$
$108$		0			0
$109$			7.83120i			0.750093i	0.927006	+	0.375047i	$0.122373\pi$
$109$			7.83120i			0.750093i	−0.927006	+	0.375047i	$0.877627\pi$
$110$		0			0
$111$		−	9.76869i		−	0.927203i
$112$		0			0
$113$	7.63789			0.718512			0.359256	−	0.933239i	$-0.383031\pi$
$113$	7.63789			0.718512			0.359256	+	0.933239i	$0.383031\pi$
$114$		0			0
$115$		−	8.18200i		−	0.762975i
$116$		0			0
$117$	3.59383	−	0.290514i	0.332250	−	0.0268581i
$118$		0			0
$119$		−	4.58103i		−	0.419942i
$120$		0			0
$121$	−12.8568			−1.16880
$122$		0			0
$123$			3.69669i			0.333319i
$124$		0			0
$125$		−	10.9687i		−	0.981074i
$126$		0			0
$127$	16.4122			1.45635			0.728175	−	0.685391i	$-0.240369\pi$
$127$	16.4122			1.45635			0.728175	+	0.685391i	$0.240369\pi$
$128$		0			0
$129$	−9.63789			−0.848569
$130$		0			0
$131$	8.43783			0.737217			0.368608	−	0.929585i	$-0.379834\pi$
$131$	8.43783			0.737217			0.368608	+	0.929585i	$0.379834\pi$
$132$		0			0
$133$	−2.96874			−0.257423
$134$		0			0
$135$		−	1.33457i		−	0.114862i
$136$		0			0
$137$		−	16.8037i		−	1.43563i	−0.696232	−	0.717817i	$-0.745141\pi$
$137$		−	16.8037i		−	1.43563i	0.696232	−	0.717817i	$-0.254859\pi$
$138$		0			0
$139$	−2.51851			−0.213617			−0.106809	−	0.994280i	$-0.534063\pi$
$139$	−2.51851			−0.213617			−0.106809	+	0.994280i	$0.534063\pi$
$140$		0			0
$141$			5.27206i			0.443987i
$142$		0			0
$143$	−1.41897	−	17.5535i	−0.118660	−	1.46790i
$144$		0			0
$145$			10.1933i			0.846509i
$146$		0			0
$147$	−1.00000			−0.0824786
$148$		0			0
$149$			11.6416i			0.953717i	0.878980	+	0.476859i	$0.158225\pi$
$149$			11.6416i			0.953717i	−0.878980	+	0.476859i	$0.841775\pi$
$150$		0			0
$151$		−	15.5374i		−	1.26441i	−0.774800	−	0.632207i	$-0.782149\pi$
$151$		−	15.5374i		−	1.26441i	0.774800	−	0.632207i	$-0.217851\pi$
$152$		0			0
$153$	4.58103			0.370354
$154$		0			0
$155$	7.52417			0.604356
$156$		0			0
$157$	−7.16206			−0.571594			−0.285797	−	0.958290i	$-0.592258\pi$
$157$	−7.16206			−0.571594			−0.285797	+	0.958290i	$0.592258\pi$
$158$		0			0
$159$	10.7374			0.851533
$160$		0			0
$161$		−	6.13080i		−	0.483175i
$162$		0			0
$163$		−	7.70617i		−	0.603594i	−0.953372	−	0.301797i	$-0.902414\pi$
$163$		−	7.70617i		−	0.603594i	0.953372	−	0.301797i	$-0.0975864\pi$
$164$		0			0
$165$	−6.51851			−0.507465
$166$		0			0
$167$			12.5478i			0.970980i	0.874242	+	0.485490i	$0.161359\pi$
$167$			12.5478i			0.970980i	−0.874242	+	0.485490i	$0.838641\pi$
$168$		0			0
$169$	12.8312	−	2.08812i	0.987016	−	0.160624i
$170$		0			0
$171$		−	2.96874i		−	0.227025i
$172$		0			0
$173$	14.5185			1.10382			0.551911	−	0.833903i	$-0.313899\pi$
$173$	14.5185			1.10382			0.551911	+	0.833903i	$0.313899\pi$
$174$		0			0
$175$		−	3.21892i		−	0.243327i
$176$		0			0
$177$			10.3033i			0.774444i
$178$		0			0
$179$	−16.0683			−1.20100			−0.600500	−	0.799625i	$-0.705032\pi$
$179$	−16.0683			−1.20100			−0.600500	+	0.799625i	$0.705032\pi$
$180$		0			0
$181$	17.2758			1.28410			0.642049	−	0.766664i	$-0.278085\pi$
$181$	17.2758			1.28410			0.642049	+	0.766664i	$0.278085\pi$
$182$		0			0
$183$	−11.1877			−0.827015
$184$		0			0
$185$	−13.0370			−0.958501
$186$		0			0
$187$		−	22.3753i		−	1.63624i
$188$		0			0
$189$		−	1.00000i		−	0.0727393i
$190$		0			0
$191$	4.66915			0.337848			0.168924	−	0.985629i	$-0.445971\pi$
$191$	4.66915			0.337848			0.168924	+	0.985629i	$0.445971\pi$
$192$		0			0
$193$		−	2.66915i		−	0.192129i	−0.995375	−	0.0960647i	$-0.969374\pi$
$193$		−	2.66915i		−	0.192129i	0.995375	−	0.0960647i	$-0.0306256\pi$
$194$		0			0
$195$	−0.387712	−	4.79623i	−0.0277647	−	0.343465i
$196$		0			0
$197$			13.6967i			0.975848i	0.872886	+	0.487924i	$0.162246\pi$
$197$			13.6967i			0.975848i	−0.872886	+	0.487924i	$0.837754\pi$
$198$		0			0
$199$	5.93748			0.420897			0.210448	−	0.977605i	$-0.432508\pi$
$199$	5.93748			0.420897			0.210448	+	0.977605i	$0.432508\pi$
$200$		0			0
$201$		−	5.50709i		−	0.388440i
$202$		0			0
$203$			7.63789i			0.536075i
$204$		0			0
$205$	4.93350			0.344570
$206$		0			0
$207$	6.13080			0.426120
$208$		0			0
$209$	−14.5003			−1.00301
$210$		0			0
$211$	−18.6493			−1.28387			−0.641936	−	0.766758i	$-0.721868\pi$
$211$	−18.6493			−1.28387			−0.641936	+	0.766758i	$0.721868\pi$
$212$		0			0
$213$			4.88434i			0.334670i
$214$		0			0
$215$			12.8625i			0.877213i
$216$		0			0
$217$	5.63789			0.382725
$218$		0			0
$219$			4.45023i			0.300719i
$220$		0			0
$221$	16.4634	−	1.33085i	1.10745	−	0.0895230i
$222$		0			0
$223$		−	15.4066i		−	1.03170i	−0.856679	−	0.515850i	$-0.827476\pi$
$223$		−	15.4066i		−	1.03170i	0.856679	−	0.515850i	$-0.172524\pi$
$224$		0			0
$225$	3.21892			0.214594
$226$		0			0
$227$			7.89576i			0.524060i	0.965060	+	0.262030i	$0.0843920\pi$
$227$			7.89576i			0.524060i	−0.965060	+	0.262030i	$0.915608\pi$
$228$		0			0
$229$		−	9.01142i		−	0.595492i	−0.954645	−	0.297746i	$-0.903765\pi$
$229$		−	9.01142i		−	0.595492i	0.954645	−	0.297746i	$-0.0962348\pi$
$230$		0			0
$231$	−4.88434			−0.321366
$232$		0			0
$233$	−10.7374			−0.703432			−0.351716	−	0.936107i	$-0.614402\pi$
$233$	−10.7374			−0.703432			−0.351716	+	0.936107i	$0.614402\pi$
$234$		0			0
$235$	7.03594			0.458974
$236$		0			0
$237$	3.54977			0.230582
$238$		0			0
$239$			15.9839i			1.03391i	0.856012	+	0.516956i	$0.172935\pi$
$239$			15.9839i			1.03391i	−0.856012	+	0.516956i	$0.827065\pi$
$240$		0			0
$241$			21.6635i			1.39547i	0.716357	+	0.697734i	$0.245808\pi$
$241$			21.6635i			1.39547i	−0.716357	+	0.697734i	$0.754192\pi$
$242$		0			0
$243$	1.00000			0.0641500
$244$		0			0
$245$			1.33457i			0.0852627i
$246$		0			0
$247$	−0.862462	−	10.6691i	−0.0548771	−	0.678861i
$248$		0			0
$249$		−	5.27206i		−	0.334103i
$250$		0			0
$251$	3.87496			0.244586			0.122293	−	0.992494i	$-0.460975\pi$
$251$	3.87496			0.244586			0.122293	+	0.992494i	$0.460975\pi$
$252$		0			0
$253$		−	29.9449i		−	1.88262i
$254$		0			0
$255$		−	6.11372i		−	0.382856i
$256$		0			0
$257$	16.8938			1.05381			0.526904	−	0.849925i	$-0.323353\pi$
$257$	16.8938			1.05381			0.526904	+	0.849925i	$0.323353\pi$
$258$		0			0
$259$	−9.76869			−0.606997
$260$		0			0
$261$	−7.63789			−0.472773
$262$		0			0
$263$	−9.40657			−0.580034			−0.290017	−	0.957021i	$-0.593661\pi$
$263$	−9.40657			−0.580034			−0.290017	+	0.957021i	$0.593661\pi$
$264$		0			0
$265$		−	14.3299i		−	0.880277i
$266$		0			0
$267$		−	9.94120i		−	0.608392i
$268$		0			0
$269$	−17.6181			−1.07419			−0.537096	−	0.843521i	$-0.680479\pi$
$269$	−17.6181			−1.07419			−0.537096	+	0.843521i	$0.680479\pi$
$270$		0			0
$271$			23.1070i			1.40365i	0.712350	+	0.701824i	$0.247631\pi$
$271$			23.1070i			1.40365i	−0.712350	+	0.701824i	$0.752369\pi$
$272$		0			0
$273$	−0.290514	−	3.59383i	−0.0175827	−	0.217508i
$274$		0			0
$275$		−	15.7223i		−	0.948089i
$276$		0			0
$277$	−20.0757			−1.20623			−0.603116	−	0.797653i	$-0.706075\pi$
$277$	−20.0757			−1.20623			−0.603116	+	0.797653i	$0.706075\pi$
$278$		0			0
$279$			5.63789i			0.337531i
$280$		0			0
$281$		−	11.2038i		−	0.668361i	−0.942509	−	0.334181i	$-0.891540\pi$
$281$		−	11.2038i		−	0.668361i	0.942509	−	0.334181i	$-0.108460\pi$
$282$		0			0
$283$	1.33829			0.0795532			0.0397766	−	0.999209i	$-0.487335\pi$
$283$	1.33829			0.0795532			0.0397766	+	0.999209i	$0.487335\pi$
$284$		0			0
$285$	−3.96200			−0.234689
$286$		0			0
$287$	3.69669			0.218208
$288$		0			0
$289$	3.98582			0.234460
$290$		0			0
$291$		−	18.2189i		−	1.06801i
$292$		0			0
$293$			8.60291i			0.502587i	0.967911	+	0.251294i	$0.0808560\pi$
$293$			8.60291i			0.502587i	−0.967911	+	0.251294i	$0.919144\pi$
$294$		0			0
$295$	13.7505			0.800586
$296$		0			0
$297$		−	4.88434i		−	0.283418i
$298$		0			0
$299$	22.0330	−	1.78108i	1.27420	−	0.103003i
$300$		0			0
$301$			9.63789i			0.555519i
$302$		0			0
$303$	−10.6947			−0.614397
$304$		0			0
$305$			14.9307i			0.854932i
$306$		0			0
$307$		−	7.57537i		−	0.432349i	−0.976355	−	0.216175i	$-0.930642\pi$
$307$		−	7.57537i		−	0.432349i	0.976355	−	0.216175i	$-0.0693580\pi$
$308$		0			0
$309$	−1.48149			−0.0842790
$310$		0			0
$311$	−6.11372			−0.346677			−0.173339	−	0.984862i	$-0.555455\pi$
$311$	−6.11372			−0.346677			−0.173339	+	0.984862i	$0.555455\pi$
$312$		0			0
$313$	−7.31269			−0.413338			−0.206669	−	0.978411i	$-0.566262\pi$
$313$	−7.31269			−0.413338			−0.206669	+	0.978411i	$0.566262\pi$
$314$		0			0
$315$	−1.33457			−0.0751947
$316$		0			0
$317$		−	1.70412i		−	0.0957131i	−0.998854	−	0.0478565i	$-0.984761\pi$
$317$		−	1.70412i		−	0.0957131i	0.998854	−	0.0478565i	$-0.0152390\pi$
$318$		0			0
$319$			37.3061i			2.08874i
$320$		0			0
$321$	−4.49291			−0.250770
$322$		0			0
$323$		−	13.5999i		−	0.756718i
$324$		0			0
$325$	11.5682	−	0.935141i	0.641690	−	0.0518723i
$326$		0			0
$327$			7.83120i			0.433067i
$328$		0			0
$329$	5.27206			0.290658
$330$		0			0
$331$		−	16.5232i		−	0.908197i	−0.890952	−	0.454098i	$-0.849961\pi$
$331$		−	16.5232i		−	0.908197i	0.890952	−	0.454098i	$-0.150039\pi$
$332$		0			0
$333$		−	9.76869i		−	0.535321i
$334$		0			0
$335$	−7.34961			−0.401552
$336$		0			0
$337$	12.3043			0.670257			0.335128	−	0.942172i	$-0.391220\pi$
$337$	12.3043			0.670257			0.335128	+	0.942172i	$0.391220\pi$
$338$		0			0
$339$	7.63789			0.414833
$340$		0			0
$341$	27.5374			1.49123
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$		−	8.18200i		−	0.440504i
$346$		0			0
$347$	−9.88240			−0.530515			−0.265258	−	0.964178i	$-0.585457\pi$
$347$	−9.88240			−0.530515			−0.265258	+	0.964178i	$0.585457\pi$
$348$		0			0
$349$			29.1497i			1.56035i	0.625564	+	0.780173i	$0.284869\pi$
$349$			29.1497i			1.56035i	−0.625564	+	0.780173i	$0.715131\pi$
$350$		0			0
$351$	3.59383	−	0.290514i	0.191824	−	0.0155065i
$352$		0			0
$353$			8.90994i			0.474228i	0.971482	+	0.237114i	$0.0762016\pi$
$353$			8.90994i			0.474228i	−0.971482	+	0.237114i	$0.923798\pi$
$354$		0			0
$355$	6.51851			0.345967
$356$		0			0
$357$		−	4.58103i		−	0.242454i
$358$		0			0
$359$			29.8776i			1.57688i	0.615112	+	0.788440i	$0.289111\pi$
$359$			29.8776i			1.57688i	−0.615112	+	0.788440i	$0.710889\pi$
$360$		0			0
$361$	10.1866			0.536136
$362$		0			0
$363$	−12.8568			−0.674807
$364$		0			0
$365$	5.93916			0.310870
$366$		0			0
$367$	5.91932			0.308986			0.154493	−	0.987994i	$-0.450626\pi$
$367$	5.91932			0.308986			0.154493	+	0.987994i	$0.450626\pi$
$368$		0			0
$369$			3.69669i			0.192442i
$370$		0			0
$371$		−	10.7374i		−	0.557459i
$372$		0			0
$373$	7.35645			0.380903			0.190451	−	0.981697i	$-0.439005\pi$
$373$	7.35645			0.380903			0.190451	+	0.981697i	$0.439005\pi$
$374$		0			0
$375$		−	10.9687i		−	0.566423i
$376$		0			0
$377$	−27.4493	+	2.21892i	−1.41371	+	0.114280i
$378$		0			0
$379$		−	12.0000i		−	0.616399i	−0.951322	−	0.308199i	$-0.900274\pi$
$379$		−	12.0000i		−	0.616399i	0.951322	−	0.308199i	$-0.0997264\pi$
$380$		0			0
$381$	16.4122			0.840824
$382$		0			0
$383$		−	15.2891i		−	0.781238i	−0.920552	−	0.390619i	$-0.872261\pi$
$383$		−	15.2891i		−	0.781238i	0.920552	−	0.390619i	$-0.127739\pi$
$384$		0			0
$385$			6.51851i			0.332214i
$386$		0			0
$387$	−9.63789			−0.489921
$388$		0			0
$389$	−7.51453			−0.381002			−0.190501	−	0.981687i	$-0.561011\pi$
$389$	−7.51453			−0.381002			−0.190501	+	0.981687i	$0.561011\pi$
$390$		0			0
$391$	28.0854			1.42034
$392$		0			0
$393$	8.43783			0.425632
$394$		0			0
$395$		−	4.73743i		−	0.238366i
$396$		0			0
$397$			10.3951i			0.521718i	0.965377	+	0.260859i	$0.0840057\pi$
$397$			10.3951i			0.521718i	−0.965377	+	0.260859i	$0.915994\pi$
$398$		0			0
$399$	−2.96874			−0.148623
$400$		0			0
$401$			13.4029i			0.669307i	0.942341	+	0.334653i	$0.108619\pi$
$401$			13.4029i			0.669307i	−0.942341	+	0.334653i	$0.891381\pi$
$402$		0			0
$403$	1.63789	+	20.2616i	0.0815890	+	1.00930i
$404$		0			0
$405$		−	1.33457i		−	0.0663155i
$406$		0			0
$407$	−47.7136			−2.36508
$408$		0			0
$409$		−	34.7421i		−	1.71789i	−0.512071	−	0.858943i	$-0.671121\pi$
$409$		−	34.7421i		−	1.71789i	0.512071	−	0.858943i	$-0.328879\pi$
$410$		0			0
$411$		−	16.8037i		−	0.828864i
$412$		0			0
$413$	10.3033			0.506993
$414$		0			0
$415$	−7.03594			−0.345381
$416$		0			0
$417$	−2.51851			−0.123332
$418$		0			0
$419$	−3.01418			−0.147252			−0.0736261	−	0.997286i	$-0.523457\pi$
$419$	−3.01418			−0.147252			−0.0736261	+	0.997286i	$0.523457\pi$
$420$		0			0
$421$			32.1366i			1.56624i	0.621871	+	0.783120i	$0.286373\pi$
$421$			32.1366i			1.56624i	−0.621871	+	0.783120i	$0.713627\pi$
$422$		0			0
$423$			5.27206i			0.256336i
$424$		0			0
$425$	14.7459			0.715283
$426$		0			0
$427$			11.1877i			0.541409i
$428$		0			0
$429$	−1.41897	−	17.5535i	−0.0685086	−	0.847490i
$430$		0			0
$431$		−	9.00938i		−	0.433966i	−0.976175	−	0.216983i	$-0.930378\pi$
$431$		−	9.00938i		−	0.433966i	0.976175	−	0.216983i	$-0.0696217\pi$
$432$		0			0
$433$	−38.6397			−1.85690			−0.928452	−	0.371453i	$-0.878860\pi$
$433$	−38.6397			−1.85690			−0.928452	+	0.371453i	$0.878860\pi$
$434$		0			0
$435$			10.1933i			0.488732i
$436$		0			0
$437$		−	18.2008i		−	0.870660i
$438$		0			0
$439$	−1.48149			−0.0707076			−0.0353538	−	0.999375i	$-0.511256\pi$
$439$	−1.48149			−0.0707076			−0.0353538	+	0.999375i	$0.511256\pi$
$440$		0			0
$441$	−1.00000			−0.0476190
$442$		0			0
$443$	12.0683			0.573381			0.286691	−	0.958023i	$-0.407445\pi$
$443$	12.0683			0.573381			0.286691	+	0.958023i	$0.407445\pi$
$444$		0			0
$445$	−13.2673			−0.628928
$446$		0			0
$447$			11.6416i			0.550629i
$448$		0			0
$449$			24.0794i			1.13638i	0.822898	+	0.568189i	$0.192356\pi$
$449$			24.0794i			1.13638i	−0.822898	+	0.568189i	$0.807644\pi$
$450$		0			0
$451$	18.0559			0.850218
$452$		0			0
$453$		−	15.5374i		−	0.730009i
$454$		0			0
$455$	−4.79623	+	0.387712i	−0.224851	+	0.0181762i
$456$		0			0
$457$		−	32.9933i		−	1.54336i	−0.636011	−	0.771680i	$-0.719417\pi$
$457$		−	32.9933i		−	1.54336i	0.636011	−	0.771680i	$-0.280583\pi$
$458$		0			0
$459$	4.58103			0.213824
$460$		0			0
$461$			13.4654i			0.627145i	0.949564	+	0.313572i	$0.101526\pi$
$461$			13.4654i			0.627145i	−0.949564	+	0.313572i	$0.898474\pi$
$462$		0			0
$463$			22.3679i			1.03952i	0.854311	+	0.519762i	$0.173979\pi$
$463$			22.3679i			1.03952i	−0.854311	+	0.519762i	$0.826021\pi$
$464$		0			0
$465$	7.52417			0.348925
$466$		0			0
$467$	−16.5629			−0.766438			−0.383219	−	0.923658i	$-0.625185\pi$
$467$	−16.5629			−0.766438			−0.383219	+	0.923658i	$0.625185\pi$
$468$		0			0
$469$	−5.50709			−0.254294
$470$		0			0
$471$	−7.16206			−0.330010
$472$		0			0
$473$			47.0747i			2.16450i
$474$		0			0
$475$		−	9.55613i		−	0.438465i
$476$		0			0
$477$	10.7374			0.491633
$478$		0			0
$479$			10.1403i			0.463321i	0.972797	+	0.231661i	$0.0744159\pi$
$479$			10.1403i			0.463321i	−0.972797	+	0.231661i	$0.925584\pi$
$480$		0			0
$481$	−2.83794	−	35.1070i	−0.129399	−	1.60074i
$482$		0			0
$483$		−	6.13080i		−	0.278961i
$484$		0			0
$485$	−24.3145			−1.10406
$486$		0			0
$487$		−	11.6624i		−	0.528474i	−0.964458	−	0.264237i	$-0.914880\pi$
$487$		−	11.6624i		−	0.528474i	0.964458	−	0.264237i	$-0.0851201\pi$
$488$		0			0
$489$		−	7.70617i		−	0.348485i
$490$		0			0
$491$	24.8454			1.12126			0.560628	−	0.828068i	$-0.310560\pi$
$491$	24.8454			1.12126			0.560628	+	0.828068i	$0.310560\pi$
$492$		0			0
$493$	−34.9894			−1.57584
$494$		0			0
$495$	−6.51851			−0.292985
$496$		0			0
$497$	4.88434			0.219093
$498$		0			0
$499$		−	7.40081i		−	0.331306i	−0.986184	−	0.165653i	$-0.947027\pi$
$499$		−	7.40081i		−	0.331306i	0.986184	−	0.165653i	$-0.0529731\pi$
$500$		0			0
$501$			12.5478i			0.560596i
$502$		0			0
$503$	35.8898			1.60025			0.800124	−	0.599834i	$-0.204767\pi$
$503$	35.8898			1.60025			0.800124	+	0.599834i	$0.204767\pi$
$504$		0			0
$505$			14.2729i			0.635136i
$506$		0			0
$507$	12.8312	−	2.08812i	0.569854	−	0.0927365i
$508$		0			0
$509$		−	3.99628i		−	0.177132i	−0.996070	−	0.0885660i	$-0.971772\pi$
$509$		−	3.99628i		−	0.177132i	0.996070	−	0.0885660i	$-0.0282284\pi$
$510$		0			0
$511$	4.45023			0.196867
$512$		0			0
$513$		−	2.96874i		−	0.131073i
$514$		0			0
$515$			1.97716i			0.0871239i
$516$		0			0
$517$	25.7505			1.13251
$518$		0			0
$519$	14.5185			0.637292
$520$		0			0
$521$	−28.5562			−1.25107			−0.625536	−	0.780196i	$-0.715119\pi$
$521$	−28.5562			−1.25107			−0.625536	+	0.780196i	$0.715119\pi$
$522$		0			0
$523$	23.5555			1.03001			0.515006	−	0.857187i	$-0.327790\pi$
$523$	23.5555			1.03001			0.515006	+	0.857187i	$0.327790\pi$
$524$		0			0
$525$		−	3.21892i		−	0.140485i
$526$		0			0
$527$			25.8273i			1.12506i
$528$		0			0
$529$	14.5867			0.634204
$530$		0			0
$531$			10.3033i			0.447126i
$532$		0			0
$533$	1.07394	+	13.2853i	0.0465175	+	0.575448i
$534$		0			0
$535$			5.99612i			0.259235i
$536$		0			0
$537$	−16.0683			−0.693397
$538$		0			0
$539$			4.88434i			0.210384i
$540$		0			0
$541$		−	6.32411i		−	0.271895i	−0.990716	−	0.135947i	$-0.956592\pi$
$541$		−	6.32411i		−	0.271895i	0.990716	−	0.135947i	$-0.0434078\pi$
$542$		0			0
$543$	17.2758			0.741374
$544$		0			0
$545$	10.4513			0.447685
$546$		0			0
$547$	−17.8141			−0.761677			−0.380838	−	0.924642i	$-0.624365\pi$
$547$	−17.8141			−0.761677			−0.380838	+	0.924642i	$0.624365\pi$
$548$		0			0
$549$	−11.1877			−0.477478
$550$		0			0
$551$			22.6749i			0.965984i
$552$		0			0
$553$		−	3.54977i		−	0.150952i
$554$		0			0
$555$	−13.0370			−0.553391
$556$		0			0
$557$			2.04915i			0.0868255i	0.999057	+	0.0434127i	$0.0138230\pi$
$557$			2.04915i			0.0868255i	−0.999057	+	0.0434127i	$0.986177\pi$
$558$		0			0
$559$	−34.6369	+	2.79994i	−1.46499	+	0.118425i
$560$		0			0
$561$		−	22.3753i		−	0.944686i
$562$		0			0
$563$	29.5999			1.24749			0.623743	−	0.781629i	$-0.285611\pi$
$563$	29.5999			1.24749			0.623743	+	0.781629i	$0.285611\pi$
$564$		0			0
$565$		−	10.1933i		−	0.428836i
$566$		0			0
$567$		−	1.00000i		−	0.0419961i
$568$		0			0
$569$	7.63789			0.320197			0.160098	−	0.987101i	$-0.448819\pi$
$569$	7.63789			0.320197			0.160098	+	0.987101i	$0.448819\pi$
$570$		0			0
$571$	11.9364			0.499523			0.249761	−	0.968307i	$-0.419648\pi$
$571$	11.9364			0.499523			0.249761	+	0.968307i	$0.419648\pi$
$572$		0			0
$573$	4.66915			0.195056
$574$		0			0
$575$	19.7345			0.822986
$576$		0			0
$577$		−	47.8320i		−	1.99127i	−0.0933200	−	0.995636i	$-0.529748\pi$
$577$		−	47.8320i		−	1.99127i	0.0933200	−	0.995636i	$-0.470252\pi$
$578$		0			0
$579$		−	2.66915i		−	0.110926i
$580$		0			0
$581$	−5.27206			−0.218722
$582$		0			0
$583$		−	52.4453i		−	2.17206i
$584$		0			0
$585$	−0.387712	−	4.79623i	−0.0160299	−	0.198300i
$586$		0			0
$587$		−	12.2350i		−	0.504994i	−0.967598	−	0.252497i	$-0.918748\pi$
$587$		−	12.2350i		−	0.504994i	0.967598	−	0.252497i	$-0.0812518\pi$
$588$		0			0
$589$	16.7374			0.689654
$590$		0			0
$591$			13.6967i			0.563406i
$592$		0			0
$593$			33.5714i			1.37861i	0.724471	+	0.689305i	$0.242084\pi$
$593$			33.5714i			1.37861i	−0.724471	+	0.689305i	$0.757916\pi$
$594$		0			0
$595$	−6.11372			−0.250638
$596$		0			0
$597$	5.93748			0.243005
$598$		0			0
$599$	−20.4549			−0.835765			−0.417883	−	0.908501i	$-0.637228\pi$
$599$	−20.4549			−0.835765			−0.417883	+	0.908501i	$0.637228\pi$
$600$		0			0
$601$	25.3014			1.03206			0.516032	−	0.856569i	$-0.327408\pi$
$601$	25.3014			1.03206			0.516032	+	0.856569i	$0.327408\pi$
$602$		0			0
$603$		−	5.50709i		−	0.224266i
$604$		0			0
$605$			17.1583i			0.697586i
$606$		0			0
$607$	16.0330			0.650761			0.325380	−	0.945583i	$-0.394508\pi$
$607$	16.0330			0.650761			0.325380	+	0.945583i	$0.394508\pi$
$608$		0			0
$609$			7.63789i			0.309503i
$610$		0			0
$611$	1.53161	+	18.9469i	0.0619622	+	0.766508i
$612$		0			0
$613$		−	39.9752i		−	1.61458i	−0.590153	−	0.807292i	$-0.700933\pi$
$613$		−	39.9752i		−	1.61458i	0.590153	−	0.807292i	$-0.299067\pi$
$614$		0			0
$615$	4.93350			0.198938
$616$		0			0
$617$		−	1.05782i		−	0.0425863i	−0.999773	−	0.0212932i	$-0.993222\pi$
$617$		−	1.05782i		−	0.0425863i	0.999773	−	0.0212932i	$-0.00677834\pi$
$618$		0			0
$619$			41.8199i			1.68088i	0.541902	+	0.840442i	$0.317704\pi$
$619$			41.8199i			1.68088i	−0.541902	+	0.840442i	$0.682296\pi$
$620$		0			0
$621$	6.13080			0.246020
$622$		0			0
$623$	−9.94120			−0.398286
$624$		0			0
$625$	1.45599			0.0582397
$626$		0			0
$627$	−14.5003			−0.579088
$628$		0			0
$629$		−	44.7506i		−	1.78432i
$630$		0			0
$631$			20.0928i			0.799882i	0.916541	+	0.399941i	$0.130969\pi$
$631$			20.0928i			0.799882i	−0.916541	+	0.399941i	$0.869031\pi$
$632$		0			0
$633$	−18.6493			−0.741243
$634$		0			0
$635$		−	21.9033i		−	0.869207i
$636$		0			0
$637$	−3.59383	+	0.290514i	−0.142393	+	0.0115106i
$638$		0			0
$639$			4.88434i			0.193222i
$640$		0			0
$641$	12.2008			0.481901			0.240950	−	0.970537i	$-0.422541\pi$
$641$	12.2008			0.481901			0.240950	+	0.970537i	$0.422541\pi$
$642$		0			0
$643$		−	0.0511985i		−	0.00201907i	−0.999999	−	0.00100954i	$-0.999679\pi$
$643$		−	0.0511985i		−	0.00201907i	0.999999	−	0.00100954i	$-0.000321345\pi$
$644$		0			0
$645$			12.8625i			0.506459i
$646$		0			0
$647$	−6.63691			−0.260924			−0.130462	−	0.991453i	$-0.541646\pi$
$647$	−6.63691			−0.260924			−0.130462	+	0.991453i	$0.541646\pi$
$648$		0			0
$649$	50.3249			1.97543
$650$		0			0
$651$	5.63789			0.220966
$652$		0			0
$653$	14.1250			0.552755			0.276378	−	0.961049i	$-0.410866\pi$
$653$	14.1250			0.552755			0.276378	+	0.961049i	$0.410866\pi$
$654$		0			0
$655$		−	11.2609i		−	0.440000i
$656$		0			0
$657$			4.45023i			0.173620i
$658$		0			0
$659$	9.95456			0.387775			0.193887	−	0.981024i	$-0.437890\pi$
$659$	9.95456			0.387775			0.193887	+	0.981024i	$0.437890\pi$
$660$		0			0
$661$		−	11.0359i		−	0.429248i	−0.976697	−	0.214624i	$-0.931147\pi$
$661$		−	11.0359i		−	0.429248i	0.976697	−	0.214624i	$-0.0688527\pi$
$662$		0			0
$663$	16.4634	−	1.33085i	0.639387	−	0.0516861i
$664$		0			0
$665$			3.96200i			0.153640i
$666$		0			0
$667$	−46.8263			−1.81312
$668$		0			0
$669$		−	15.4066i		−	0.595652i
$670$		0			0
$671$			54.6443i			2.10952i
$672$		0			0
$673$	−9.64921			−0.371950			−0.185975	−	0.982555i	$-0.559544\pi$
$673$	−9.64921			−0.371950			−0.185975	+	0.982555i	$0.559544\pi$
$674$		0			0
$675$	3.21892			0.123896
$676$		0			0
$677$	27.8320			1.06967			0.534835	−	0.844956i	$-0.320374\pi$
$677$	27.8320			1.06967			0.534835	+	0.844956i	$0.320374\pi$
$678$		0			0
$679$	−18.2189			−0.699178
$680$		0			0
$681$			7.89576i			0.302566i
$682$		0			0
$683$			43.8151i			1.67654i	0.545257	+	0.838269i	$0.316432\pi$
$683$			43.8151i			1.67654i	−0.545257	+	0.838269i	$0.683568\pi$
$684$		0			0
$685$	−22.4257			−0.856842
$686$		0			0
$687$		−	9.01142i		−	0.343807i
$688$		0			0
$689$	38.5885	−	3.11938i	1.47010	−	0.118839i
$690$		0			0
$691$		−	41.6057i		−	1.58275i	−0.611329	−	0.791377i	$-0.709365\pi$
$691$		−	41.6057i		−	1.58275i	0.611329	−	0.791377i	$-0.290635\pi$
$692$		0			0
$693$	−4.88434			−0.185541
$694$		0			0
$695$			3.36114i			0.127495i
$696$		0			0
$697$			16.9346i			0.641445i
$698$		0			0
$699$	−10.7374			−0.406127
$700$		0			0
$701$	−35.4765			−1.33993			−0.669965	−	0.742393i	$-0.733691\pi$
$701$	−35.4765			−1.33993			−0.669965	+	0.742393i	$0.733691\pi$
$702$		0			0
$703$	−29.0007			−1.09378
$704$		0			0
$705$	7.03594			0.264989
$706$		0			0
$707$			10.6947i			0.402217i
$708$		0			0
$709$			17.8937i			0.672013i	0.941860	+	0.336006i	$0.109076\pi$
$709$			17.8937i			0.672013i	−0.941860	+	0.336006i	$0.890924\pi$
$710$		0			0
$711$	3.54977			0.133127
$712$		0			0
$713$			34.5647i			1.29446i
$714$		0			0
$715$	−23.4264	+	1.89372i	−0.876098	+	0.0708211i
$716$		0			0
$717$			15.9839i			0.596929i
$718$		0			0
$719$	34.1137			1.27223			0.636113	−	0.771596i	$-0.280541\pi$
$719$	34.1137			1.27223			0.636113	+	0.771596i	$0.280541\pi$
$720$		0			0
$721$			1.48149i			0.0551735i
$722$		0			0
$723$			21.6635i			0.805674i
$724$		0			0
$725$	−24.5857			−0.913090
$726$		0			0
$727$	0.599191			0.0222228			0.0111114	−	0.999938i	$-0.496463\pi$
$727$	0.599191			0.0222228			0.0111114	+	0.999938i	$0.496463\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	−44.1514			−1.63300
$732$		0			0
$733$			41.7125i			1.54069i	0.637629	+	0.770344i	$0.279915\pi$
$733$			41.7125i			1.54069i	−0.637629	+	0.770344i	$0.720085\pi$
$734$		0			0
$735$			1.33457i			0.0492265i
$736$		0			0
$737$	−26.8985			−0.990819
$738$		0			0
$739$			19.4446i			0.715280i	0.933860	+	0.357640i	$0.116419\pi$
$739$			19.4446i			0.715280i	−0.933860	+	0.357640i	$0.883581\pi$
$740$		0			0
$741$	−0.862462	−	10.6691i	−0.0316833	−	0.391941i
$742$		0			0
$743$			9.44721i			0.346584i	0.984870	+	0.173292i	$0.0554405\pi$
$743$			9.44721i			0.346584i	−0.984870	+	0.173292i	$0.944559\pi$
$744$		0			0
$745$	15.5366			0.569216
$746$		0			0
$747$		−	5.27206i		−	0.192894i
$748$		0			0
$749$			4.49291i			0.164167i
$750$		0			0
$751$	8.91366			0.325264			0.162632	−	0.986687i	$-0.448002\pi$
$751$	8.91366			0.325264			0.162632	+	0.986687i	$0.448002\pi$
$752$		0			0
$753$	3.87496			0.141212
$754$		0			0
$755$	−20.7358			−0.754651
$756$		0			0
$757$	−43.8416			−1.59345			−0.796726	−	0.604341i	$-0.793437\pi$
$757$	−43.8416			−1.59345			−0.796726	+	0.604341i	$0.793437\pi$
$758$		0			0
$759$		−	29.9449i		−	1.08693i
$760$		0			0
$761$			11.2283i			0.407025i	0.979072	+	0.203513i	$0.0652358\pi$
$761$			11.2283i			0.407025i	−0.979072	+	0.203513i	$0.934764\pi$
$762$		0			0
$763$	7.83120			0.283509
$764$		0			0
$765$		−	6.11372i		−	0.221042i
$766$		0			0
$767$	2.99326	+	37.0283i	0.108080	+	1.33702i
$768$		0			0
$769$			13.1860i			0.475499i	0.971327	+	0.237749i	$0.0764097\pi$
$769$			13.1860i			0.475499i	−0.971327	+	0.237749i	$0.923590\pi$
$770$		0			0
$771$	16.8938			0.608416
$772$		0			0
$773$			25.6094i			0.921105i	0.887632	+	0.460552i	$0.152349\pi$
$773$			25.6094i			0.921105i	−0.887632	+	0.460552i	$0.847651\pi$
$774$		0			0
$775$			18.1479i			0.651891i
$776$		0			0
$777$	−9.76869			−0.350450
$778$		0			0
$779$	10.9745			0.393202
$780$		0			0
$781$	23.8568			0.853663
$782$		0			0
$783$	−7.63789			−0.272956
$784$		0			0
$785$			9.55829i			0.341150i
$786$		0			0
$787$		−	47.8047i		−	1.70405i	−0.523497	−	0.852027i	$-0.675373\pi$
$787$		−	47.8047i		−	1.70405i	0.523497	−	0.852027i	$-0.324627\pi$
$788$		0			0
$789$	−9.40657			−0.334883
$790$		0			0
$791$		−	7.63789i		−	0.271572i
$792$		0			0
$793$	−40.2065	+	3.25017i	−1.42778	+	0.115417i
$794$		0			0
$795$		−	14.3299i		−	0.508228i
$796$		0			0
$797$	23.1326			0.819398			0.409699	−	0.912221i	$-0.365634\pi$
$797$	23.1326			0.819398			0.409699	+	0.912221i	$0.365634\pi$
$798$		0			0
$799$			24.1514i			0.854416i
$800$		0			0
$801$		−	9.94120i		−	0.351255i
$802$		0			0
$803$	21.7364			0.767063
$804$		0			0
$805$	−8.18200			−0.288377
$806$		0			0
$807$	−17.6181			−0.620185
$808$		0			0
$809$	−40.0757			−1.40899			−0.704494	−	0.709710i	$-0.748826\pi$
$809$	−40.0757			−1.40899			−0.704494	+	0.709710i	$0.748826\pi$
$810$		0			0
$811$		−	23.5374i		−	0.826509i	−0.910616	−	0.413254i	$-0.864392\pi$
$811$		−	23.5374i		−	0.826509i	0.910616	−	0.413254i	$-0.135608\pi$
$812$		0			0
$813$			23.1070i			0.810397i
$814$		0			0
$815$	−10.2844			−0.360248
$816$		0			0
$817$			28.6124i			1.00102i
$818$		0			0
$819$	−0.290514	−	3.59383i	−0.0101514	−	0.125579i
$820$		0			0
$821$			3.71760i			0.129745i	0.997894	+	0.0648726i	$0.0206641\pi$
$821$			3.71760i			0.129745i	−0.997894	+	0.0648726i	$0.979336\pi$
$822$		0			0
$823$	−39.0747			−1.36206			−0.681030	−	0.732256i	$-0.738468\pi$
$823$	−39.0747			−1.36206			−0.681030	+	0.732256i	$0.738468\pi$
$824$		0			0
$825$		−	15.7223i		−	0.547380i
$826$		0			0
$827$			3.55349i			0.123567i	0.998090	+	0.0617834i	$0.0196788\pi$
$827$			3.55349i			0.123567i	−0.998090	+	0.0617834i	$0.980321\pi$
$828$		0			0
$829$	−53.3107			−1.85156			−0.925779	−	0.378065i	$-0.876590\pi$
$829$	−53.3107			−1.85156			−0.925779	+	0.378065i	$0.876590\pi$
$830$		0			0
$831$	−20.0757			−0.696419
$832$		0			0
$833$	−4.58103			−0.158723
$834$		0			0
$835$	16.7460			0.579519
$836$		0			0
$837$			5.63789i			0.194874i
$838$		0			0
$839$		−	19.7344i		−	0.681307i	−0.940189	−	0.340654i	$-0.889352\pi$
$839$		−	19.7344i		−	0.681307i	0.940189	−	0.340654i	$-0.110648\pi$
$840$		0			0
$841$	29.3373			1.01163
$842$		0			0
$843$		−	11.2038i		−	0.385878i
$844$		0			0
$845$	−2.78674	−	17.1242i	−0.0958669	−	0.589090i
$846$		0			0
$847$			12.8568i			0.441765i
$848$		0			0
$849$	1.33829			0.0459300
$850$		0			0
$851$		−	59.8898i		−	2.05300i
$852$		0			0
$853$			11.8188i			0.404668i	0.979317	+	0.202334i	$0.0648527\pi$
$853$			11.8188i			0.404668i	−0.979317	+	0.202334i	$0.935147\pi$
$854$		0			0
$855$	−3.96200			−0.135498
$856$		0			0
$857$	−6.76858			−0.231210			−0.115605	−	0.993295i	$-0.536881\pi$
$857$	−6.76858			−0.231210			−0.115605	+	0.993295i	$0.536881\pi$
$858$		0			0
$859$	54.6881			1.86593			0.932967	−	0.359962i	$-0.117210\pi$
$859$	54.6881			1.86593			0.932967	+	0.359962i	$0.117210\pi$
$860$		0			0
$861$	3.69669			0.125983
$862$		0			0
$863$		−	6.17144i		−	0.210078i	−0.994468	−	0.105039i	$-0.966503\pi$
$863$		−	6.17144i		−	0.210078i	0.994468	−	0.105039i	$-0.0334968\pi$
$864$		0			0
$865$		−	19.3760i		−	0.658804i
$866$		0			0
$867$	3.98582			0.135366
$868$		0			0
$869$		−	17.3383i		−	0.588161i
$870$		0			0
$871$	−1.59989	−	19.7915i	−0.0542101	−	0.670610i
$872$		0			0
$873$		−	18.2189i		−	0.616617i
$874$		0			0
$875$	−10.9687			−0.370811
$876$		0			0
$877$		−	6.22388i		−	0.210165i	−0.994463	−	0.105083i	$-0.966489\pi$
$877$		−	6.22388i		−	0.210165i	0.994463	−	0.105083i	$-0.0335107\pi$
$878$		0			0
$879$			8.60291i			0.290169i
$880$		0			0
$881$	28.5810			0.962919			0.481460	−	0.876468i	$-0.340107\pi$
$881$	28.5810			0.962919			0.481460	+	0.876468i	$0.340107\pi$
$882$		0			0
$883$	49.3391			1.66039			0.830196	−	0.557471i	$-0.188228\pi$
$883$	49.3391			1.66039			0.830196	+	0.557471i	$0.188228\pi$
$884$		0			0
$885$	13.7505			0.462219
$886$		0			0
$887$	−4.90046			−0.164541			−0.0822707	−	0.996610i	$-0.526217\pi$
$887$	−4.90046			−0.164541			−0.0822707	+	0.996610i	$0.526217\pi$
$888$		0			0
$889$		−	16.4122i		−	0.550449i
$890$		0			0
$891$		−	4.88434i		−	0.163632i
$892$		0			0
$893$	15.6514			0.523753
$894$		0			0
$895$			21.4443i			0.716804i
$896$		0			0
$897$	22.0330	−	1.78108i	0.735662	−	0.0594687i
$898$		0			0
$899$		−	43.0615i		−	1.43618i
$900$		0			0
$901$	49.1885			1.63871
$902$		0			0
$903$			9.63789i			0.320729i
$904$		0			0
$905$		−	23.0558i		−	0.766400i
$906$		0			0
$907$	25.5021			0.846784			0.423392	−	0.905947i	$-0.360839\pi$
$907$	25.5021			0.846784			0.423392	+	0.905947i	$0.360839\pi$
$908$		0			0
$909$	−10.6947			−0.354722
$910$		0			0
$911$	−37.2452			−1.23399			−0.616994	−	0.786967i	$-0.711650\pi$
$911$	−37.2452			−1.23399			−0.616994	+	0.786967i	$0.711650\pi$
$912$		0			0
$913$	−25.7505			−0.852218
$914$		0			0
$915$			14.9307i			0.493595i
$916$		0			0
$917$		−	8.43783i		−	0.278642i
$918$		0			0
$919$	−23.0739			−0.761139			−0.380570	−	0.924752i	$-0.624272\pi$
$919$	−23.0739			−0.761139			−0.380570	+	0.924752i	$0.624272\pi$
$920$		0			0
$921$		−	7.57537i		−	0.249617i
$922$		0			0
$923$	1.41897	+	17.5535i	0.0467060	+	0.577780i
$924$		0			0
$925$		−	31.4446i		−	1.03389i
$926$		0			0
$927$	−1.48149			−0.0486585
$928$		0			0
$929$			41.0085i			1.34545i	0.739895	+	0.672723i	$0.234875\pi$
$929$			41.0085i			1.34545i	−0.739895	+	0.672723i	$0.765125\pi$
$930$		0			0
$931$			2.96874i			0.0972966i
$932$		0			0
$933$	−6.11372			−0.200154
$934$		0			0
$935$	−29.8615			−0.976575
$936$		0			0
$937$	8.17348			0.267016			0.133508	−	0.991048i	$-0.457376\pi$
$937$	8.17348			0.267016			0.133508	+	0.991048i	$0.457376\pi$
$938$		0			0
$939$	−7.31269			−0.238641
$940$		0			0
$941$			33.1732i			1.08142i	0.841210	+	0.540708i	$0.181844\pi$
$941$			33.1732i			1.08142i	−0.841210	+	0.540708i	$0.818156\pi$
$942$		0			0
$943$			22.6636i			0.738030i
$944$		0			0
$945$	−1.33457			−0.0434137
$946$		0			0
$947$			8.32148i			0.270412i	0.990818	+	0.135206i	$0.0431696\pi$
$947$			8.32148i			0.270412i	−0.990818	+	0.135206i	$0.956830\pi$
$948$		0			0
$949$	1.29286	+	15.9934i	0.0419679	+	0.519167i
$950$		0			0
$951$		−	1.70412i		−	0.0552600i
$952$		0			0
$953$	6.39047			0.207008			0.103504	−	0.994629i	$-0.466995\pi$
$953$	6.39047			0.207008			0.103504	+	0.994629i	$0.466995\pi$
$954$		0			0
$955$		−	6.23131i		−	0.201641i
$956$		0			0
$957$			37.3061i			1.20593i
$958$		0			0
$959$	−16.8037			−0.542619
$960$		0			0
$961$	−0.785767			−0.0253473
$962$		0			0
$963$	−4.49291			−0.144782
$964$		0			0
$965$	−3.56217			−0.114670
$966$		0			0
$967$			12.4304i			0.399735i	0.979823	+	0.199867i	$0.0640511\pi$
$967$			12.4304i			0.399735i	−0.979823	+	0.199867i	$0.935949\pi$
$968$		0			0
$969$		−	13.5999i		−	0.436891i
$970$		0			0
$971$	−0.462630			−0.0148465			−0.00742325	−	0.999972i	$-0.502363\pi$
$971$	−0.462630			−0.0148465			−0.00742325	+	0.999972i	$0.502363\pi$
$972$		0			0
$973$			2.51851i			0.0807398i
$974$		0			0
$975$	11.5682	−	0.935141i	0.370480	−	0.0299485i
$976$		0			0
$977$			10.6087i			0.339401i	0.985496	+	0.169701i	$0.0542801\pi$
$977$			10.6087i			0.339401i	−0.985496	+	0.169701i	$0.945720\pi$
$978$		0			0
$979$	−48.5562			−1.55186
$980$		0			0
$981$			7.83120i			0.250031i
$982$		0			0
$983$			22.9042i			0.730530i	0.930904	+	0.365265i	$0.119022\pi$
$983$			22.9042i			0.730530i	−0.930904	+	0.365265i	$0.880978\pi$
$984$		0			0
$985$	18.2792			0.582425
$986$		0			0
$987$	5.27206			0.167811
$988$		0			0
$989$	−59.0879			−1.87889
$990$		0			0
$991$	12.3013			0.390763			0.195381	−	0.980727i	$-0.437406\pi$
$991$	12.3013			0.390763			0.195381	+	0.980727i	$0.437406\pi$
$992$		0			0
$993$		−	16.5232i		−	0.524348i
$994$		0			0
$995$		−	7.92400i		−	0.251208i
$996$		0			0
$997$	53.3875			1.69080			0.845400	−	0.534134i	$-0.179362\pi$
$997$	53.3875			1.69080			0.845400	+	0.534134i	$0.179362\pi$
$998$		0			0
$999$		−	9.76869i		−	0.309068i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4368.2.h.q.337.4		8
4.3	odd	2		273.2.c.c.64.6	yes	8
12.11	even	2		819.2.c.d.64.3		8
13.12	even	2	inner	4368.2.h.q.337.5		8
28.27	even	2		1911.2.c.l.883.6		8
52.31	even	4		3549.2.a.x.1.3		4
52.47	even	4		3549.2.a.v.1.2		4
52.51	odd	2		273.2.c.c.64.3	✓	8
156.155	even	2		819.2.c.d.64.6		8
364.363	even	2		1911.2.c.l.883.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.c.c.64.3	✓	8	52.51	odd	2
273.2.c.c.64.6	yes	8	4.3	odd	2
819.2.c.d.64.3		8	12.11	even	2
819.2.c.d.64.6		8	156.155	even	2
1911.2.c.l.883.3		8	364.363	even	2
1911.2.c.l.883.6		8	28.27	even	2
3549.2.a.v.1.2		4	52.47	even	4
3549.2.a.x.1.3		4	52.31	even	4
4368.2.h.q.337.4		8	1.1	even	1	trivial
4368.2.h.q.337.5		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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.33457i q^{5} -1.00000i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.33457i q^{5} -1.00000i q^{7} +1.00000 q^{9} -4.88434i q^{11} +(3.59383 - 0.290514i) q^{13} -1.33457i q^{15} +4.58103 q^{17} -2.96874i q^{19} -1.00000i q^{21} +6.13080 q^{23} +3.21892 q^{25} +1.00000 q^{27} -7.63789 q^{29} +5.63789i q^{31} -4.88434i q^{33} -1.33457 q^{35} -9.76869i q^{37} +(3.59383 - 0.290514i) q^{39} +3.69669i q^{41} -9.63789 q^{43} -1.33457i q^{45} +5.27206i q^{47} -1.00000 q^{49} +4.58103 q^{51} +10.7374 q^{53} -6.51851 q^{55} -2.96874i q^{57} +10.3033i q^{59} -11.1877 q^{61} -1.00000i q^{63} +(-0.387712 - 4.79623i) q^{65} -5.50709i q^{67} +6.13080 q^{69} +4.88434i q^{71} +4.45023i q^{73} +3.21892 q^{75} -4.88434 q^{77} +3.54977 q^{79} +1.00000 q^{81} -5.27206i q^{83} -6.11372i q^{85} -7.63789 q^{87} -9.94120i q^{89} +(-0.290514 - 3.59383i) q^{91} +5.63789i q^{93} -3.96200 q^{95} -18.2189i q^{97} -4.88434i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{3} + 8 q^{9}+O(q^{10}) \) 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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