LMFDB - Embedded newform 370.2.g.b.327.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [370,2,Mod(43,370)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(370, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([3, 3]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("370.43");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 370 = 2 \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	370.g (of order $4$, degree $2$, 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:	$2.95446487479$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			327.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	370.327
Dual form			370.2.g.b.43.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000i q^{2} -1.00000 q^{4} +(-2.00000 + 1.00000i) q^{5} +(-2.00000 - 2.00000i) q^{7} -1.00000i q^{8} -3.00000i q^{9} +(-1.00000 - 2.00000i) q^{10} -4.00000i q^{13} +(2.00000 - 2.00000i) q^{14} +1.00000 q^{16} -2.00000 q^{17} +3.00000 q^{18} +(2.00000 + 2.00000i) q^{19} +(2.00000 - 1.00000i) q^{20} -4.00000i q^{23} +(3.00000 - 4.00000i) q^{25} +4.00000 q^{26} +(2.00000 + 2.00000i) q^{28} +(-7.00000 + 7.00000i) q^{29} +(-4.00000 - 4.00000i) q^{31} +1.00000i q^{32} -2.00000i q^{34} +(6.00000 + 2.00000i) q^{35} +3.00000i q^{36} +(-6.00000 - 1.00000i) q^{37} +(-2.00000 + 2.00000i) q^{38} +(1.00000 + 2.00000i) q^{40} -4.00000i q^{43} +(3.00000 + 6.00000i) q^{45} +4.00000 q^{46} +(-2.00000 - 2.00000i) q^{47} +1.00000i q^{49} +(4.00000 + 3.00000i) q^{50} +4.00000i q^{52} +(-1.00000 + 1.00000i) q^{53} +(-2.00000 + 2.00000i) q^{56} +(-7.00000 - 7.00000i) q^{58} +(2.00000 + 2.00000i) q^{59} +(1.00000 + 1.00000i) q^{61} +(4.00000 - 4.00000i) q^{62} +(-6.00000 + 6.00000i) q^{63} -1.00000 q^{64} +(4.00000 + 8.00000i) q^{65} +2.00000 q^{68} +(-2.00000 + 6.00000i) q^{70} +12.0000 q^{71} -3.00000 q^{72} +(-5.00000 - 5.00000i) q^{73} +(1.00000 - 6.00000i) q^{74} +(-2.00000 - 2.00000i) q^{76} +(12.0000 + 12.0000i) q^{79} +(-2.00000 + 1.00000i) q^{80} -9.00000 q^{81} +(4.00000 - 4.00000i) q^{83} +(4.00000 - 2.00000i) q^{85} +4.00000 q^{86} +(-7.00000 + 7.00000i) q^{89} +(-6.00000 + 3.00000i) q^{90} +(-8.00000 + 8.00000i) q^{91} +4.00000i q^{92} +(2.00000 - 2.00000i) q^{94} +(-6.00000 - 2.00000i) q^{95} -12.0000 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 4 q^{5} - 4 q^{7} - 2 q^{10} + 4 q^{14} + 2 q^{16} - 4 q^{17} + 6 q^{18} + 4 q^{19} + 4 q^{20} + 6 q^{25} + 8 q^{26} + 4 q^{28} - 14 q^{29} - 8 q^{31} + 12 q^{35} - 12 q^{37} - 4 q^{38}+ \cdots - 2 q^{98}+O(q^{100}) $ 2 * q - 2 * q^4 - 4 * q^5 - 4 * q^7 - 2 * q^10 + 4 * q^14 + 2 * q^16 - 4 * q^17 + 6 * q^18 + 4 * q^19 + 4 * q^20 + 6 * q^25 + 8 * q^26 + 4 * q^28 - 14 * q^29 - 8 * q^31 + 12 * q^35 - 12 * q^37 - 4 * q^38 + 2 * q^40 + 6 * q^45 + 8 * q^46 - 4 * q^47 + 8 * q^50 - 2 * q^53 - 4 * q^56 - 14 * q^58 + 4 * q^59 + 2 * q^61 + 8 * q^62 - 12 * q^63 - 2 * q^64 + 8 * q^65 + 4 * q^68 - 4 * q^70 + 24 * q^71 - 6 * q^72 - 10 * q^73 + 2 * q^74 - 4 * q^76 + 24 * q^79 - 4 * q^80 - 18 * q^81 + 8 * q^83 + 8 * q^85 + 8 * q^86 - 14 * q^89 - 12 * q^90 - 16 * q^91 + 4 * q^94 - 12 * q^95 - 24 * q^97 - 2 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/370\mathbb{Z}\right)^\times$.

$n$	$261$	$297$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$e\left(\frac{1}{4}\right)$

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 $

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$3$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$4$	−1.00000			−0.500000
$5$	−2.00000	+	1.00000i	−0.894427	+	0.447214i
$5$	−2.00000	+	1.00000i	−0.894427	+	0.447214i
$6$		0			0
$7$	−2.00000	−	2.00000i	−0.755929	−	0.755929i	0.219650	−	0.975579i	$-0.429509\pi$
$7$	−2.00000	−	2.00000i	−0.755929	−	0.755929i	−0.975579	+	0.219650i	$0.929509\pi$
$8$		−	1.00000i		−	0.353553i
$9$		−	3.00000i		−	1.00000i
$10$	−1.00000	−	2.00000i	−0.316228	−	0.632456i
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$		−	4.00000i		−	1.10940i	−0.832050	−	0.554700i	$-0.812833\pi$
$13$		−	4.00000i		−	1.10940i	0.832050	−	0.554700i	$-0.187167\pi$
$14$	2.00000	−	2.00000i	0.534522	−	0.534522i
$15$		0			0
$16$	1.00000			0.250000
$17$	−2.00000			−0.485071			−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000			−0.485071			−0.242536	+	0.970143i	$0.577979\pi$
$18$	3.00000			0.707107
$19$	2.00000	+	2.00000i	0.458831	+	0.458831i	0.898272	−	0.439440i	$-0.144823\pi$
$19$	2.00000	+	2.00000i	0.458831	+	0.458831i	−0.439440	+	0.898272i	$0.644823\pi$
$20$	2.00000	−	1.00000i	0.447214	−	0.223607i
$21$		0			0
$22$		0			0
$23$		−	4.00000i		−	0.834058i	−0.908893	−	0.417029i	$-0.863071\pi$
$23$		−	4.00000i		−	0.834058i	0.908893	−	0.417029i	$-0.136929\pi$
$24$		0			0
$25$	3.00000	−	4.00000i	0.600000	−	0.800000i
$26$	4.00000			0.784465
$27$		0			0
$28$	2.00000	+	2.00000i	0.377964	+	0.377964i
$29$	−7.00000	+	7.00000i	−1.29987	+	1.29987i	−0.371391	+	0.928477i	$0.621119\pi$
$29$	−7.00000	+	7.00000i	−1.29987	+	1.29987i	−0.928477	+	0.371391i	$0.878881\pi$
$30$		0			0
$31$	−4.00000	−	4.00000i	−0.718421	−	0.718421i	0.249861	−	0.968282i	$-0.419615\pi$
$31$	−4.00000	−	4.00000i	−0.718421	−	0.718421i	−0.968282	+	0.249861i	$0.919615\pi$
$32$			1.00000i			0.176777i
$33$		0			0
$34$		−	2.00000i		−	0.342997i
$35$	6.00000	+	2.00000i	1.01419	+	0.338062i
$36$			3.00000i			0.500000i
$37$	−6.00000	−	1.00000i	−0.986394	−	0.164399i
$37$	−6.00000	−	1.00000i	−0.986394	−	0.164399i
$38$	−2.00000	+	2.00000i	−0.324443	+	0.324443i
$39$		0			0
$40$	1.00000	+	2.00000i	0.158114	+	0.316228i
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$		−	4.00000i		−	0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$		−	4.00000i		−	0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$		0			0
$45$	3.00000	+	6.00000i	0.447214	+	0.894427i
$46$	4.00000			0.589768
$47$	−2.00000	−	2.00000i	−0.291730	−	0.291730i	0.546033	−	0.837763i	$-0.316137\pi$
$47$	−2.00000	−	2.00000i	−0.291730	−	0.291730i	−0.837763	+	0.546033i	$0.816137\pi$
$48$		0			0
$49$			1.00000i			0.142857i
$50$	4.00000	+	3.00000i	0.565685	+	0.424264i
$51$		0			0
$52$			4.00000i			0.554700i
$53$	−1.00000	+	1.00000i	−0.137361	+	0.137361i	−0.772444	−	0.635083i	$-0.780966\pi$
$53$	−1.00000	+	1.00000i	−0.137361	+	0.137361i	0.635083	+	0.772444i	$0.280966\pi$
$54$		0			0
$55$		0			0
$56$	−2.00000	+	2.00000i	−0.267261	+	0.267261i
$57$		0			0
$58$	−7.00000	−	7.00000i	−0.919145	−	0.919145i
$59$	2.00000	+	2.00000i	0.260378	+	0.260378i	0.825208	−	0.564830i	$-0.191058\pi$
$59$	2.00000	+	2.00000i	0.260378	+	0.260378i	−0.564830	+	0.825208i	$0.691058\pi$
$60$		0			0
$61$	1.00000	+	1.00000i	0.128037	+	0.128037i	0.768221	−	0.640184i	$-0.221142\pi$
$61$	1.00000	+	1.00000i	0.128037	+	0.128037i	−0.640184	+	0.768221i	$0.721142\pi$
$62$	4.00000	−	4.00000i	0.508001	−	0.508001i
$63$	−6.00000	+	6.00000i	−0.755929	+	0.755929i
$64$	−1.00000			−0.125000
$65$	4.00000	+	8.00000i	0.496139	+	0.992278i
$66$		0			0
$67$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$67$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$68$	2.00000			0.242536
$69$		0			0
$70$	−2.00000	+	6.00000i	−0.239046	+	0.717137i
$71$	12.0000			1.42414			0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000			1.42414			0.712069	+	0.702109i	$0.247758\pi$
$72$	−3.00000			−0.353553
$73$	−5.00000	−	5.00000i	−0.585206	−	0.585206i	0.351123	−	0.936329i	$-0.385800\pi$
$73$	−5.00000	−	5.00000i	−0.585206	−	0.585206i	−0.936329	+	0.351123i	$0.885800\pi$
$74$	1.00000	−	6.00000i	0.116248	−	0.697486i
$75$		0			0
$76$	−2.00000	−	2.00000i	−0.229416	−	0.229416i
$77$		0			0
$78$		0			0
$79$	12.0000	+	12.0000i	1.35011	+	1.35011i	0.885537	+	0.464568i	$0.153790\pi$
$79$	12.0000	+	12.0000i	1.35011	+	1.35011i	0.464568	+	0.885537i	$0.346210\pi$
$80$	−2.00000	+	1.00000i	−0.223607	+	0.111803i
$81$	−9.00000			−1.00000
$82$		0			0
$83$	4.00000	−	4.00000i	0.439057	−	0.439057i	−0.452638	−	0.891695i	$-0.649517\pi$
$83$	4.00000	−	4.00000i	0.439057	−	0.439057i	0.891695	+	0.452638i	$0.149517\pi$
$84$		0			0
$85$	4.00000	−	2.00000i	0.433861	−	0.216930i
$86$	4.00000			0.431331
$87$		0			0
$88$		0			0
$89$	−7.00000	+	7.00000i	−0.741999	+	0.741999i	−0.972962	−	0.230964i	$-0.925812\pi$
$89$	−7.00000	+	7.00000i	−0.741999	+	0.741999i	0.230964	+	0.972962i	$0.425812\pi$
$90$	−6.00000	+	3.00000i	−0.632456	+	0.316228i
$91$	−8.00000	+	8.00000i	−0.838628	+	0.838628i
$92$			4.00000i			0.417029i
$93$		0			0
$94$	2.00000	−	2.00000i	0.206284	−	0.206284i
$95$	−6.00000	−	2.00000i	−0.615587	−	0.205196i
$96$		0			0
$97$	−12.0000			−1.21842			−0.609208	−	0.793011i	$-0.708512\pi$
$97$	−12.0000			−1.21842			−0.609208	+	0.793011i	$0.708512\pi$
$98$	−1.00000			−0.101015
$99$		0			0
$100$	−3.00000	+	4.00000i	−0.300000	+	0.400000i
$101$		−	10.0000i		−	0.995037i	−0.867453	−	0.497519i	$-0.834245\pi$
$101$		−	10.0000i		−	0.995037i	0.867453	−	0.497519i	$-0.165755\pi$
$102$		0			0
$103$	−16.0000			−1.57653			−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000			−1.57653			−0.788263	+	0.615338i	$0.789020\pi$
$104$	−4.00000			−0.392232
$105$		0			0
$106$	−1.00000	−	1.00000i	−0.0971286	−	0.0971286i
$107$	8.00000	+	8.00000i	0.773389	+	0.773389i	0.978697	−	0.205308i	$-0.0658197\pi$
$107$	8.00000	+	8.00000i	0.773389	+	0.773389i	−0.205308	+	0.978697i	$0.565820\pi$
$108$		0			0
$109$	−3.00000	−	3.00000i	−0.287348	−	0.287348i	0.548683	−	0.836031i	$-0.315129\pi$
$109$	−3.00000	−	3.00000i	−0.287348	−	0.287348i	−0.836031	+	0.548683i	$0.815129\pi$
$110$		0			0
$111$		0			0
$112$	−2.00000	−	2.00000i	−0.188982	−	0.188982i
$113$	14.0000			1.31701			0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000			1.31701			0.658505	+	0.752577i	$0.271189\pi$
$114$		0			0
$115$	4.00000	+	8.00000i	0.373002	+	0.746004i
$116$	7.00000	−	7.00000i	0.649934	−	0.649934i
$117$	−12.0000			−1.10940
$118$	−2.00000	+	2.00000i	−0.184115	+	0.184115i
$119$	4.00000	+	4.00000i	0.366679	+	0.366679i
$120$		0			0
$121$	11.0000			1.00000
$122$	−1.00000	+	1.00000i	−0.0905357	+	0.0905357i
$123$		0			0
$124$	4.00000	+	4.00000i	0.359211	+	0.359211i
$125$	−2.00000	+	11.0000i	−0.178885	+	0.983870i
$126$	−6.00000	−	6.00000i	−0.534522	−	0.534522i
$127$	−2.00000	−	2.00000i	−0.177471	−	0.177471i	0.612781	−	0.790253i	$-0.290051\pi$
$127$	−2.00000	−	2.00000i	−0.177471	−	0.177471i	−0.790253	+	0.612781i	$0.790051\pi$
$128$		−	1.00000i		−	0.0883883i
$129$		0			0
$130$	−8.00000	+	4.00000i	−0.701646	+	0.350823i
$131$	6.00000	+	6.00000i	0.524222	+	0.524222i	0.918844	−	0.394621i	$-0.129124\pi$
$131$	6.00000	+	6.00000i	0.524222	+	0.524222i	−0.394621	+	0.918844i	$0.629124\pi$
$132$		0			0
$133$		−	8.00000i		−	0.693688i
$134$		0			0
$135$		0			0
$136$			2.00000i			0.171499i
$137$	−7.00000	−	7.00000i	−0.598050	−	0.598050i	0.341743	−	0.939793i	$-0.388983\pi$
$137$	−7.00000	−	7.00000i	−0.598050	−	0.598050i	−0.939793	+	0.341743i	$0.888983\pi$
$138$		0			0
$139$	20.0000			1.69638			0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000			1.69638			0.848189	+	0.529694i	$0.177693\pi$
$140$	−6.00000	−	2.00000i	−0.507093	−	0.169031i
$141$		0			0
$142$			12.0000i			1.00702i
$143$		0			0
$144$		−	3.00000i		−	0.250000i
$145$	7.00000	−	21.0000i	0.581318	−	1.74396i
$146$	5.00000	−	5.00000i	0.413803	−	0.413803i
$147$		0			0
$148$	6.00000	+	1.00000i	0.493197	+	0.0821995i
$149$		−	6.00000i		−	0.491539i	−0.969328	−	0.245770i	$-0.920959\pi$
$149$		−	6.00000i		−	0.491539i	0.969328	−	0.245770i	$-0.0790407\pi$
$150$		0			0
$151$		−	20.0000i		−	1.62758i	−0.581161	−	0.813788i	$-0.697401\pi$
$151$		−	20.0000i		−	1.62758i	0.581161	−	0.813788i	$-0.302599\pi$
$152$	2.00000	−	2.00000i	0.162221	−	0.162221i
$153$			6.00000i			0.485071i
$154$		0			0
$155$	12.0000	+	4.00000i	0.963863	+	0.321288i
$156$		0			0
$157$	−7.00000	−	7.00000i	−0.558661	−	0.558661i	0.370265	−	0.928926i	$-0.379267\pi$
$157$	−7.00000	−	7.00000i	−0.558661	−	0.558661i	−0.928926	+	0.370265i	$0.879267\pi$
$158$	−12.0000	+	12.0000i	−0.954669	+	0.954669i
$159$		0			0
$160$	−1.00000	−	2.00000i	−0.0790569	−	0.158114i
$161$	−8.00000	+	8.00000i	−0.630488	+	0.630488i
$162$		−	9.00000i		−	0.707107i
$163$	24.0000			1.87983			0.939913	−	0.341415i	$-0.110906\pi$
$163$	24.0000			1.87983			0.939913	+	0.341415i	$0.110906\pi$
$164$		0			0
$165$		0			0
$166$	4.00000	+	4.00000i	0.310460	+	0.310460i
$167$	8.00000			0.619059			0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000			0.619059			0.309529	+	0.950890i	$0.399829\pi$
$168$		0			0
$169$	−3.00000			−0.230769
$170$	2.00000	+	4.00000i	0.153393	+	0.306786i
$171$	6.00000	−	6.00000i	0.458831	−	0.458831i
$172$			4.00000i			0.304997i
$173$	5.00000	+	5.00000i	0.380143	+	0.380143i	0.871154	−	0.491011i	$-0.163372\pi$
$173$	5.00000	+	5.00000i	0.380143	+	0.380143i	−0.491011	+	0.871154i	$0.663372\pi$
$174$		0			0
$175$	−14.0000	+	2.00000i	−1.05830	+	0.151186i
$176$		0			0
$177$		0			0
$178$	−7.00000	−	7.00000i	−0.524672	−	0.524672i
$179$	−2.00000	+	2.00000i	−0.149487	+	0.149487i	−0.777889	−	0.628402i	$-0.783709\pi$
$179$	−2.00000	+	2.00000i	−0.149487	+	0.149487i	0.628402	+	0.777889i	$0.283709\pi$
$180$	−3.00000	−	6.00000i	−0.223607	−	0.447214i
$181$	−8.00000			−0.594635			−0.297318	−	0.954779i	$-0.596092\pi$
$181$	−8.00000			−0.594635			−0.297318	+	0.954779i	$0.596092\pi$
$182$	−8.00000	−	8.00000i	−0.592999	−	0.592999i
$183$		0			0
$184$	−4.00000			−0.294884
$185$	13.0000	−	4.00000i	0.955779	−	0.294086i
$186$		0			0
$187$		0			0
$188$	2.00000	+	2.00000i	0.145865	+	0.145865i
$189$		0			0
$190$	2.00000	−	6.00000i	0.145095	−	0.435286i
$191$	16.0000	−	16.0000i	1.15772	−	1.15772i	0.172754	−	0.984965i	$-0.444733\pi$
$191$	16.0000	−	16.0000i	1.15772	−	1.15772i	0.984965	−	0.172754i	$-0.0552667\pi$
$192$		0			0
$193$		−	14.0000i		−	1.00774i	−0.863779	−	0.503871i	$-0.831909\pi$
$193$		−	14.0000i		−	1.00774i	0.863779	−	0.503871i	$-0.168091\pi$
$194$		−	12.0000i		−	0.861550i
$195$		0			0
$196$		−	1.00000i		−	0.0714286i
$197$	3.00000	+	3.00000i	0.213741	+	0.213741i	0.805855	−	0.592113i	$-0.201706\pi$
$197$	3.00000	+	3.00000i	0.213741	+	0.213741i	−0.592113	+	0.805855i	$0.701706\pi$
$198$		0			0
$199$	8.00000	−	8.00000i	0.567105	−	0.567105i	−0.364211	−	0.931316i	$-0.618661\pi$
$199$	8.00000	−	8.00000i	0.567105	−	0.567105i	0.931316	+	0.364211i	$0.118661\pi$
$200$	−4.00000	−	3.00000i	−0.282843	−	0.212132i
$201$		0			0
$202$	10.0000			0.703598
$203$	28.0000			1.96521
$204$		0			0
$205$		0			0
$206$		−	16.0000i		−	1.11477i
$207$	−12.0000			−0.834058
$208$		−	4.00000i		−	0.277350i
$209$		0			0
$210$		0			0
$211$	−8.00000			−0.550743			−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000			−0.550743			−0.275371	+	0.961338i	$0.588801\pi$
$212$	1.00000	−	1.00000i	0.0686803	−	0.0686803i
$213$		0			0
$214$	−8.00000	+	8.00000i	−0.546869	+	0.546869i
$215$	4.00000	+	8.00000i	0.272798	+	0.545595i
$216$		0			0
$217$			16.0000i			1.08615i
$218$	3.00000	−	3.00000i	0.203186	−	0.203186i
$219$		0			0
$220$		0			0
$221$			8.00000i			0.538138i
$222$		0			0
$223$	−6.00000	+	6.00000i	−0.401790	+	0.401790i	−0.878863	−	0.477074i	$-0.841698\pi$
$223$	−6.00000	+	6.00000i	−0.401790	+	0.401790i	0.477074	+	0.878863i	$0.341698\pi$
$224$	2.00000	−	2.00000i	0.133631	−	0.133631i
$225$	−12.0000	−	9.00000i	−0.800000	−	0.600000i
$226$			14.0000i			0.931266i
$227$	−12.0000			−0.796468			−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000			−0.796468			−0.398234	+	0.917284i	$0.630377\pi$
$228$		0			0
$229$		−	16.0000i		−	1.05731i	−0.848837	−	0.528655i	$-0.822697\pi$
$229$		−	16.0000i		−	1.05731i	0.848837	−	0.528655i	$-0.177303\pi$
$230$	−8.00000	+	4.00000i	−0.527504	+	0.263752i
$231$		0			0
$232$	7.00000	+	7.00000i	0.459573	+	0.459573i
$233$	−5.00000	−	5.00000i	−0.327561	−	0.327561i	0.524097	−	0.851658i	$-0.324403\pi$
$233$	−5.00000	−	5.00000i	−0.327561	−	0.327561i	−0.851658	+	0.524097i	$0.824403\pi$
$234$		−	12.0000i		−	0.784465i
$235$	6.00000	+	2.00000i	0.391397	+	0.130466i
$236$	−2.00000	−	2.00000i	−0.130189	−	0.130189i
$237$		0			0
$238$	−4.00000	+	4.00000i	−0.259281	+	0.259281i
$239$	−8.00000	−	8.00000i	−0.517477	−	0.517477i	0.399330	−	0.916807i	$-0.369243\pi$
$239$	−8.00000	−	8.00000i	−0.517477	−	0.517477i	−0.916807	+	0.399330i	$0.869243\pi$
$240$		0			0
$241$	−19.0000	+	19.0000i	−1.22390	+	1.22390i	−0.257663	+	0.966235i	$0.582952\pi$
$241$	−19.0000	+	19.0000i	−1.22390	+	1.22390i	−0.966235	+	0.257663i	$0.917048\pi$
$242$			11.0000i			0.707107i
$243$		0			0
$244$	−1.00000	−	1.00000i	−0.0640184	−	0.0640184i
$245$	−1.00000	−	2.00000i	−0.0638877	−	0.127775i
$246$		0			0
$247$	8.00000	−	8.00000i	0.509028	−	0.509028i
$248$	−4.00000	+	4.00000i	−0.254000	+	0.254000i
$249$		0			0
$250$	−11.0000	−	2.00000i	−0.695701	−	0.126491i
$251$	−14.0000	−	14.0000i	−0.883672	−	0.883672i	0.110234	−	0.993906i	$-0.464840\pi$
$251$	−14.0000	−	14.0000i	−0.883672	−	0.883672i	−0.993906	+	0.110234i	$0.964840\pi$
$252$	6.00000	−	6.00000i	0.377964	−	0.377964i
$253$		0			0
$254$	2.00000	−	2.00000i	0.125491	−	0.125491i
$255$		0			0
$256$	1.00000			0.0625000
$257$	18.0000			1.12281			0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000			1.12281			0.561405	+	0.827541i	$0.310261\pi$
$258$		0			0
$259$	10.0000	+	14.0000i	0.621370	+	0.869918i
$260$	−4.00000	−	8.00000i	−0.248069	−	0.496139i
$261$	21.0000	+	21.0000i	1.29987	+	1.29987i
$262$	−6.00000	+	6.00000i	−0.370681	+	0.370681i
$263$	−10.0000	−	10.0000i	−0.616626	−	0.616626i	0.328038	−	0.944664i	$-0.393613\pi$
$263$	−10.0000	−	10.0000i	−0.616626	−	0.616626i	−0.944664	+	0.328038i	$0.893613\pi$
$264$		0			0
$265$	1.00000	−	3.00000i	0.0614295	−	0.184289i
$266$	8.00000			0.490511
$267$		0			0
$268$		0			0
$269$			14.0000i			0.853595i	0.904347	+	0.426798i	$0.140358\pi$
$269$			14.0000i			0.853595i	−0.904347	+	0.426798i	$0.859642\pi$
$270$		0			0
$271$	−28.0000			−1.70088			−0.850439	−	0.526073i	$-0.823664\pi$
$271$	−28.0000			−1.70088			−0.850439	+	0.526073i	$0.823664\pi$
$272$	−2.00000			−0.121268
$273$		0			0
$274$	7.00000	−	7.00000i	0.422885	−	0.422885i
$275$		0			0
$276$		0			0
$277$		−	12.0000i		−	0.721010i	−0.932757	−	0.360505i	$-0.882604\pi$
$277$		−	12.0000i		−	0.721010i	0.932757	−	0.360505i	$-0.117396\pi$
$278$			20.0000i			1.19952i
$279$	−12.0000	+	12.0000i	−0.718421	+	0.718421i
$280$	2.00000	−	6.00000i	0.119523	−	0.358569i
$281$	1.00000	−	1.00000i	0.0596550	−	0.0596550i	−0.676650	−	0.736305i	$-0.736569\pi$
$281$	1.00000	−	1.00000i	0.0596550	−	0.0596550i	0.736305	+	0.676650i	$0.236569\pi$
$282$		0			0
$283$	24.0000			1.42665			0.713326	−	0.700832i	$-0.247188\pi$
$283$	24.0000			1.42665			0.713326	+	0.700832i	$0.247188\pi$
$284$	−12.0000			−0.712069
$285$		0			0
$286$		0			0
$287$		0			0
$288$	3.00000			0.176777
$289$	−13.0000			−0.764706
$290$	21.0000	+	7.00000i	1.23316	+	0.411054i
$291$		0			0
$292$	5.00000	+	5.00000i	0.292603	+	0.292603i
$293$	9.00000	−	9.00000i	0.525786	−	0.525786i	−0.393527	−	0.919313i	$-0.628745\pi$
$293$	9.00000	−	9.00000i	0.525786	−	0.525786i	0.919313	+	0.393527i	$0.128745\pi$
$294$		0			0
$295$	−6.00000	−	2.00000i	−0.349334	−	0.116445i
$296$	−1.00000	+	6.00000i	−0.0581238	+	0.348743i
$297$		0			0
$298$	6.00000			0.347571
$299$	−16.0000			−0.925304
$300$		0			0
$301$	−8.00000	+	8.00000i	−0.461112	+	0.461112i
$302$	20.0000			1.15087
$303$		0			0
$304$	2.00000	+	2.00000i	0.114708	+	0.114708i
$305$	−3.00000	−	1.00000i	−0.171780	−	0.0572598i
$306$	−6.00000			−0.342997
$307$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$307$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$308$		0			0
$309$		0			0
$310$	−4.00000	+	12.0000i	−0.227185	+	0.681554i
$311$	−4.00000	−	4.00000i	−0.226819	−	0.226819i	0.584543	−	0.811363i	$-0.301274\pi$
$311$	−4.00000	−	4.00000i	−0.226819	−	0.226819i	−0.811363	+	0.584543i	$0.801274\pi$
$312$		0			0
$313$			26.0000i			1.46961i	0.678280	+	0.734803i	$0.262726\pi$
$313$			26.0000i			1.46961i	−0.678280	+	0.734803i	$0.737274\pi$
$314$	7.00000	−	7.00000i	0.395033	−	0.395033i
$315$	6.00000	−	18.0000i	0.338062	−	1.01419i
$316$	−12.0000	−	12.0000i	−0.675053	−	0.675053i
$317$	−5.00000	+	5.00000i	−0.280828	+	0.280828i	−0.833439	−	0.552611i	$-0.813631\pi$
$317$	−5.00000	+	5.00000i	−0.280828	+	0.280828i	0.552611	+	0.833439i	$0.313631\pi$
$318$		0			0
$319$		0			0
$320$	2.00000	−	1.00000i	0.111803	−	0.0559017i
$321$		0			0
$322$	−8.00000	−	8.00000i	−0.445823	−	0.445823i
$323$	−4.00000	−	4.00000i	−0.222566	−	0.222566i
$324$	9.00000			0.500000
$325$	−16.0000	−	12.0000i	−0.887520	−	0.665640i
$326$			24.0000i			1.32924i
$327$		0			0
$328$		0			0
$329$			8.00000i			0.441054i
$330$		0			0
$331$	6.00000	−	6.00000i	0.329790	−	0.329790i	−0.522717	−	0.852506i	$-0.675081\pi$
$331$	6.00000	−	6.00000i	0.329790	−	0.329790i	0.852506	+	0.522717i	$0.175081\pi$
$332$	−4.00000	+	4.00000i	−0.219529	+	0.219529i
$333$	−3.00000	+	18.0000i	−0.164399	+	0.986394i
$334$			8.00000i			0.437741i
$335$		0			0
$336$		0			0
$337$	15.0000	−	15.0000i	0.817102	−	0.817102i	−0.168585	−	0.985687i	$-0.553920\pi$
$337$	15.0000	−	15.0000i	0.817102	−	0.817102i	0.985687	+	0.168585i	$0.0539198\pi$
$338$		−	3.00000i		−	0.163178i
$339$		0			0
$340$	−4.00000	+	2.00000i	−0.216930	+	0.108465i
$341$		0			0
$342$	6.00000	+	6.00000i	0.324443	+	0.324443i
$343$	−12.0000	+	12.0000i	−0.647939	+	0.647939i
$344$	−4.00000			−0.215666
$345$		0			0
$346$	−5.00000	+	5.00000i	−0.268802	+	0.268802i
$347$		−	12.0000i		−	0.644194i	−0.946707	−	0.322097i	$-0.895612\pi$
$347$		−	12.0000i		−	0.644194i	0.946707	−	0.322097i	$-0.104388\pi$
$348$		0			0
$349$			14.0000i			0.749403i	0.927146	+	0.374701i	$0.122255\pi$
$349$			14.0000i			0.749403i	−0.927146	+	0.374701i	$0.877745\pi$
$350$	−2.00000	−	14.0000i	−0.106904	−	0.748331i
$351$		0			0
$352$		0			0
$353$	−36.0000			−1.91609			−0.958043	−	0.286623i	$-0.907467\pi$
$353$	−36.0000			−1.91609			−0.958043	+	0.286623i	$0.907467\pi$
$354$		0			0
$355$	−24.0000	+	12.0000i	−1.27379	+	0.636894i
$356$	7.00000	−	7.00000i	0.370999	−	0.370999i
$357$		0			0
$358$	−2.00000	−	2.00000i	−0.105703	−	0.105703i
$359$			24.0000i			1.26667i	0.773877	+	0.633336i	$0.218315\pi$
$359$			24.0000i			1.26667i	−0.773877	+	0.633336i	$0.781685\pi$
$360$	6.00000	−	3.00000i	0.316228	−	0.158114i
$361$		−	11.0000i		−	0.578947i
$362$		−	8.00000i		−	0.420471i
$363$		0			0
$364$	8.00000	−	8.00000i	0.419314	−	0.419314i
$365$	15.0000	+	5.00000i	0.785136	+	0.261712i
$366$		0			0
$367$	18.0000	+	18.0000i	0.939592	+	0.939592i	0.998277	−	0.0586842i	$-0.0186905\pi$
$367$	18.0000	+	18.0000i	0.939592	+	0.939592i	−0.0586842	+	0.998277i	$0.518691\pi$
$368$		−	4.00000i		−	0.208514i
$369$		0			0
$370$	4.00000	+	13.0000i	0.207950	+	0.675838i
$371$	4.00000			0.207670
$372$		0			0
$373$	−25.0000	−	25.0000i	−1.29445	−	1.29445i	−0.932005	−	0.362446i	$-0.881942\pi$
$373$	−25.0000	−	25.0000i	−1.29445	−	1.29445i	−0.362446	−	0.932005i	$-0.618058\pi$
$374$		0			0
$375$		0			0
$376$	−2.00000	+	2.00000i	−0.103142	+	0.103142i
$377$	28.0000	+	28.0000i	1.44207	+	1.44207i
$378$		0			0
$379$			4.00000i			0.205466i	0.994709	+	0.102733i	$0.0327588\pi$
$379$			4.00000i			0.205466i	−0.994709	+	0.102733i	$0.967241\pi$
$380$	6.00000	+	2.00000i	0.307794	+	0.102598i
$381$		0			0
$382$	16.0000	+	16.0000i	0.818631	+	0.818631i
$383$			36.0000i			1.83951i	0.392488	+	0.919757i	$0.371614\pi$
$383$			36.0000i			1.83951i	−0.392488	+	0.919757i	$0.628386\pi$
$384$		0			0
$385$		0			0
$386$	14.0000			0.712581
$387$	−12.0000			−0.609994
$388$	12.0000			0.609208
$389$	17.0000	+	17.0000i	0.861934	+	0.861934i	0.991563	−	0.129628i	$-0.0413785\pi$
$389$	17.0000	+	17.0000i	0.861934	+	0.861934i	−0.129628	+	0.991563i	$0.541378\pi$
$390$		0			0
$391$			8.00000i			0.404577i
$392$	1.00000			0.0505076
$393$		0			0
$394$	−3.00000	+	3.00000i	−0.151138	+	0.151138i
$395$	−36.0000	−	12.0000i	−1.81136	−	0.603786i
$396$		0			0
$397$	25.0000	−	25.0000i	1.25471	−	1.25471i	0.301131	−	0.953583i	$-0.402636\pi$
$397$	25.0000	−	25.0000i	1.25471	−	1.25471i	0.953583	−	0.301131i	$-0.0973643\pi$
$398$	8.00000	+	8.00000i	0.401004	+	0.401004i
$399$		0			0
$400$	3.00000	−	4.00000i	0.150000	−	0.200000i
$401$	1.00000	+	1.00000i	0.0499376	+	0.0499376i	0.731635	−	0.681697i	$-0.238758\pi$
$401$	1.00000	+	1.00000i	0.0499376	+	0.0499376i	−0.681697	+	0.731635i	$0.738758\pi$
$402$		0			0
$403$	−16.0000	+	16.0000i	−0.797017	+	0.797017i
$404$			10.0000i			0.497519i
$405$	18.0000	−	9.00000i	0.894427	−	0.447214i
$406$			28.0000i			1.38962i
$407$		0			0
$408$		0			0
$409$	13.0000	−	13.0000i	0.642809	−	0.642809i	−0.308436	−	0.951245i	$-0.599806\pi$
$409$	13.0000	−	13.0000i	0.642809	−	0.642809i	0.951245	+	0.308436i	$0.0998057\pi$
$410$		0			0
$411$		0			0
$412$	16.0000			0.788263
$413$		−	8.00000i		−	0.393654i
$414$		−	12.0000i		−	0.589768i
$415$	−4.00000	+	12.0000i	−0.196352	+	0.589057i
$416$	4.00000			0.196116
$417$		0			0
$418$		0			0
$419$		−	36.0000i		−	1.75872i	−0.476162	−	0.879358i	$-0.657972\pi$
$419$		−	36.0000i		−	1.75872i	0.476162	−	0.879358i	$-0.342028\pi$
$420$		0			0
$421$	11.0000	+	11.0000i	0.536107	+	0.536107i	0.922383	−	0.386276i	$-0.126239\pi$
$421$	11.0000	+	11.0000i	0.536107	+	0.536107i	−0.386276	+	0.922383i	$0.626239\pi$
$422$		−	8.00000i		−	0.389434i
$423$	−6.00000	+	6.00000i	−0.291730	+	0.291730i
$424$	1.00000	+	1.00000i	0.0485643	+	0.0485643i
$425$	−6.00000	+	8.00000i	−0.291043	+	0.388057i
$426$		0			0
$427$		−	4.00000i		−	0.193574i
$428$	−8.00000	−	8.00000i	−0.386695	−	0.386695i
$429$		0			0
$430$	−8.00000	+	4.00000i	−0.385794	+	0.192897i
$431$	−4.00000	−	4.00000i	−0.192673	−	0.192673i	0.604177	−	0.796850i	$-0.293502\pi$
$431$	−4.00000	−	4.00000i	−0.192673	−	0.192673i	−0.796850	+	0.604177i	$0.793502\pi$
$432$		0			0
$433$	−21.0000	+	21.0000i	−1.00920	+	1.00920i	−0.00923827	+	0.999957i	$0.502941\pi$
$433$	−21.0000	+	21.0000i	−1.00920	+	1.00920i	−0.999957	+	0.00923827i	$0.997059\pi$
$434$	−16.0000			−0.768025
$435$		0			0
$436$	3.00000	+	3.00000i	0.143674	+	0.143674i
$437$	8.00000	−	8.00000i	0.382692	−	0.382692i
$438$		0			0
$439$	8.00000	−	8.00000i	0.381819	−	0.381819i	−0.489938	−	0.871757i	$-0.662981\pi$
$439$	8.00000	−	8.00000i	0.381819	−	0.381819i	0.871757	+	0.489938i	$0.162981\pi$
$440$		0			0
$441$	3.00000			0.142857
$442$	−8.00000			−0.380521
$443$	20.0000	+	20.0000i	0.950229	+	0.950229i	0.998819	−	0.0485901i	$-0.0154728\pi$
$443$	20.0000	+	20.0000i	0.950229	+	0.950229i	−0.0485901	+	0.998819i	$0.515473\pi$
$444$		0			0
$445$	7.00000	−	21.0000i	0.331832	−	0.995495i
$446$	−6.00000	−	6.00000i	−0.284108	−	0.284108i
$447$		0			0
$448$	2.00000	+	2.00000i	0.0944911	+	0.0944911i
$449$	−23.0000	−	23.0000i	−1.08544	−	1.08544i	−0.995992	−	0.0894454i	$-0.971491\pi$
$449$	−23.0000	−	23.0000i	−1.08544	−	1.08544i	−0.0894454	−	0.995992i	$-0.528509\pi$
$450$	9.00000	−	12.0000i	0.424264	−	0.565685i
$451$		0			0
$452$	−14.0000			−0.658505
$453$		0			0
$454$		−	12.0000i		−	0.563188i
$455$	8.00000	−	24.0000i	0.375046	−	1.12514i
$456$		0			0
$457$	28.0000			1.30978			0.654892	−	0.755722i	$-0.272714\pi$
$457$	28.0000			1.30978			0.654892	+	0.755722i	$0.272714\pi$
$458$	16.0000			0.747631
$459$		0			0
$460$	−4.00000	−	8.00000i	−0.186501	−	0.373002i
$461$	−19.0000	+	19.0000i	−0.884918	+	0.884918i	−0.994030	−	0.109111i	$-0.965200\pi$
$461$	−19.0000	+	19.0000i	−0.884918	+	0.884918i	0.109111	+	0.994030i	$0.465200\pi$
$462$		0			0
$463$		−	4.00000i		−	0.185896i	−0.995671	−	0.0929479i	$-0.970371\pi$
$463$		−	4.00000i		−	0.185896i	0.995671	−	0.0929479i	$-0.0296290\pi$
$464$	−7.00000	+	7.00000i	−0.324967	+	0.324967i
$465$		0			0
$466$	5.00000	−	5.00000i	0.231621	−	0.231621i
$467$	8.00000			0.370196			0.185098	−	0.982720i	$-0.440740\pi$
$467$	8.00000			0.370196			0.185098	+	0.982720i	$0.440740\pi$
$468$	12.0000			0.554700
$469$		0			0
$470$	−2.00000	+	6.00000i	−0.0922531	+	0.276759i
$471$		0			0
$472$	2.00000	−	2.00000i	0.0920575	−	0.0920575i
$473$		0			0
$474$		0			0
$475$	14.0000	−	2.00000i	0.642364	−	0.0917663i
$476$	−4.00000	−	4.00000i	−0.183340	−	0.183340i
$477$	3.00000	+	3.00000i	0.137361	+	0.137361i
$478$	8.00000	−	8.00000i	0.365911	−	0.365911i
$479$	−28.0000	−	28.0000i	−1.27935	−	1.27935i	−0.941030	−	0.338322i	$-0.890141\pi$
$479$	−28.0000	−	28.0000i	−1.27935	−	1.27935i	−0.338322	−	0.941030i	$-0.609859\pi$
$480$		0			0
$481$	−4.00000	+	24.0000i	−0.182384	+	1.09431i
$482$	−19.0000	−	19.0000i	−0.865426	−	0.865426i
$483$		0			0
$484$	−11.0000			−0.500000
$485$	24.0000	−	12.0000i	1.08978	−	0.544892i
$486$		0			0
$487$	−32.0000			−1.45006			−0.725029	−	0.688718i	$-0.758174\pi$
$487$	−32.0000			−1.45006			−0.725029	+	0.688718i	$0.758174\pi$
$488$	1.00000	−	1.00000i	0.0452679	−	0.0452679i
$489$		0			0
$490$	2.00000	−	1.00000i	0.0903508	−	0.0451754i
$491$	−8.00000			−0.361035			−0.180517	−	0.983572i	$-0.557777\pi$
$491$	−8.00000			−0.361035			−0.180517	+	0.983572i	$0.557777\pi$
$492$		0			0
$493$	14.0000	−	14.0000i	0.630528	−	0.630528i
$494$	8.00000	+	8.00000i	0.359937	+	0.359937i
$495$		0			0
$496$	−4.00000	−	4.00000i	−0.179605	−	0.179605i
$497$	−24.0000	−	24.0000i	−1.07655	−	1.07655i
$498$		0			0
$499$	−2.00000	+	2.00000i	−0.0895323	+	0.0895323i	−0.750454	−	0.660922i	$-0.770165\pi$
$499$	−2.00000	+	2.00000i	−0.0895323	+	0.0895323i	0.660922	+	0.750454i	$0.270165\pi$
$500$	2.00000	−	11.0000i	0.0894427	−	0.491935i
$501$		0			0
$502$	14.0000	−	14.0000i	0.624851	−	0.624851i
$503$			16.0000i			0.713405i	0.934218	+	0.356702i	$0.116099\pi$
$503$			16.0000i			0.713405i	−0.934218	+	0.356702i	$0.883901\pi$
$504$	6.00000	+	6.00000i	0.267261	+	0.267261i
$505$	10.0000	+	20.0000i	0.444994	+	0.889988i
$506$		0			0
$507$		0			0
$508$	2.00000	+	2.00000i	0.0887357	+	0.0887357i
$509$	−30.0000			−1.32973			−0.664863	−	0.746965i	$-0.731510\pi$
$509$	−30.0000			−1.32973			−0.664863	+	0.746965i	$0.731510\pi$
$510$		0			0
$511$			20.0000i			0.884748i
$512$			1.00000i			0.0441942i
$513$		0			0
$514$			18.0000i			0.793946i
$515$	32.0000	−	16.0000i	1.41009	−	0.705044i
$516$		0			0
$517$		0			0
$518$	−14.0000	+	10.0000i	−0.615125	+	0.439375i
$519$		0			0
$520$	8.00000	−	4.00000i	0.350823	−	0.175412i
$521$			10.0000i			0.438108i	0.975713	+	0.219054i	$0.0702971\pi$
$521$			10.0000i			0.438108i	−0.975713	+	0.219054i	$0.929703\pi$
$522$	−21.0000	+	21.0000i	−0.919145	+	0.919145i
$523$		−	24.0000i		−	1.04945i	−0.851273	−	0.524723i	$-0.824169\pi$
$523$		−	24.0000i		−	1.04945i	0.851273	−	0.524723i	$-0.175831\pi$
$524$	−6.00000	−	6.00000i	−0.262111	−	0.262111i
$525$		0			0
$526$	10.0000	−	10.0000i	0.436021	−	0.436021i
$527$	8.00000	+	8.00000i	0.348485	+	0.348485i
$528$		0			0
$529$	7.00000			0.304348
$530$	3.00000	+	1.00000i	0.130312	+	0.0434372i
$531$	6.00000	−	6.00000i	0.260378	−	0.260378i
$532$			8.00000i			0.346844i
$533$		0			0
$534$		0			0
$535$	−24.0000	−	8.00000i	−1.03761	−	0.345870i
$536$		0			0
$537$		0			0
$538$	−14.0000			−0.603583
$539$		0			0
$540$		0			0
$541$	11.0000	−	11.0000i	0.472927	−	0.472927i	−0.429934	−	0.902861i	$-0.641463\pi$
$541$	11.0000	−	11.0000i	0.472927	−	0.472927i	0.902861	+	0.429934i	$0.141463\pi$
$542$		−	28.0000i		−	1.20270i
$543$		0			0
$544$		−	2.00000i		−	0.0857493i
$545$	9.00000	+	3.00000i	0.385518	+	0.128506i
$546$		0			0
$547$			8.00000i			0.342055i	0.985266	+	0.171028i	$0.0547087\pi$
$547$			8.00000i			0.342055i	−0.985266	+	0.171028i	$0.945291\pi$
$548$	7.00000	+	7.00000i	0.299025	+	0.299025i
$549$	3.00000	−	3.00000i	0.128037	−	0.128037i
$550$		0			0
$551$	−28.0000			−1.19284
$552$		0			0
$553$		−	48.0000i		−	2.04117i
$554$	12.0000			0.509831
$555$		0			0
$556$	−20.0000			−0.848189
$557$		−	12.0000i		−	0.508456i	−0.967144	−	0.254228i	$-0.918179\pi$
$557$		−	12.0000i		−	0.508456i	0.967144	−	0.254228i	$-0.0818214\pi$
$558$	−12.0000	−	12.0000i	−0.508001	−	0.508001i
$559$	−16.0000			−0.676728
$560$	6.00000	+	2.00000i	0.253546	+	0.0845154i
$561$		0			0
$562$	1.00000	+	1.00000i	0.0421825	+	0.0421825i
$563$		−	24.0000i		−	1.01148i	−0.862686	−	0.505740i	$-0.831220\pi$
$563$		−	24.0000i		−	1.01148i	0.862686	−	0.505740i	$-0.168780\pi$
$564$		0			0
$565$	−28.0000	+	14.0000i	−1.17797	+	0.588984i
$566$			24.0000i			1.00880i
$567$	18.0000	+	18.0000i	0.755929	+	0.755929i
$568$		−	12.0000i		−	0.503509i
$569$	3.00000	−	3.00000i	0.125767	−	0.125767i	−0.641422	−	0.767188i	$-0.721655\pi$
$569$	3.00000	−	3.00000i	0.125767	−	0.125767i	0.767188	+	0.641422i	$0.221655\pi$
$570$		0			0
$571$	12.0000			0.502184			0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000			0.502184			0.251092	+	0.967963i	$0.419210\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−16.0000	−	12.0000i	−0.667246	−	0.500435i
$576$			3.00000i			0.125000i
$577$	18.0000			0.749350			0.374675	−	0.927156i	$-0.377754\pi$
$577$	18.0000			0.749350			0.374675	+	0.927156i	$0.377754\pi$
$578$		−	13.0000i		−	0.540729i
$579$		0			0
$580$	−7.00000	+	21.0000i	−0.290659	+	0.871978i
$581$	−16.0000			−0.663792
$582$		0			0
$583$		0			0
$584$	−5.00000	+	5.00000i	−0.206901	+	0.206901i
$585$	24.0000	−	12.0000i	0.992278	−	0.496139i
$586$	9.00000	+	9.00000i	0.371787	+	0.371787i
$587$			28.0000i			1.15568i	0.816149	+	0.577842i	$0.196105\pi$
$587$			28.0000i			1.15568i	−0.816149	+	0.577842i	$0.803895\pi$
$588$		0			0
$589$		−	16.0000i		−	0.659269i
$590$	2.00000	−	6.00000i	0.0823387	−	0.247016i
$591$		0			0
$592$	−6.00000	−	1.00000i	−0.246598	−	0.0410997i
$593$	−1.00000	+	1.00000i	−0.0410651	+	0.0410651i	−0.727341	−	0.686276i	$-0.759244\pi$
$593$	−1.00000	+	1.00000i	−0.0410651	+	0.0410651i	0.686276	+	0.727341i	$0.259244\pi$
$594$		0			0
$595$	−12.0000	−	4.00000i	−0.491952	−	0.163984i
$596$			6.00000i			0.245770i
$597$		0			0
$598$		−	16.0000i		−	0.654289i
$599$		−	36.0000i		−	1.47092i	−0.677568	−	0.735460i	$-0.736966\pi$
$599$		−	36.0000i		−	1.47092i	0.677568	−	0.735460i	$-0.263034\pi$
$600$		0			0
$601$	32.0000			1.30531			0.652654	−	0.757656i	$-0.273656\pi$
$601$	32.0000			1.30531			0.652654	+	0.757656i	$0.273656\pi$
$602$	−8.00000	−	8.00000i	−0.326056	−	0.326056i
$603$		0			0
$604$			20.0000i			0.813788i
$605$	−22.0000	+	11.0000i	−0.894427	+	0.447214i
$606$		0			0
$607$		−	12.0000i		−	0.487065i	−0.969893	−	0.243532i	$-0.921694\pi$
$607$		−	12.0000i		−	0.487065i	0.969893	−	0.243532i	$-0.0783062\pi$
$608$	−2.00000	+	2.00000i	−0.0811107	+	0.0811107i
$609$		0			0
$610$	1.00000	−	3.00000i	0.0404888	−	0.121466i
$611$	−8.00000	+	8.00000i	−0.323645	+	0.323645i
$612$		−	6.00000i		−	0.242536i
$613$	−5.00000	−	5.00000i	−0.201948	−	0.201948i	0.598886	−	0.800834i	$-0.295610\pi$
$613$	−5.00000	−	5.00000i	−0.201948	−	0.201948i	−0.800834	+	0.598886i	$0.795610\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	15.0000	−	15.0000i	0.603877	−	0.603877i	−0.337462	−	0.941339i	$-0.609568\pi$
$617$	15.0000	−	15.0000i	0.603877	−	0.603877i	0.941339	+	0.337462i	$0.109568\pi$
$618$		0			0
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$	−12.0000	−	4.00000i	−0.481932	−	0.160644i
$621$		0			0
$622$	4.00000	−	4.00000i	0.160385	−	0.160385i
$623$	28.0000			1.12180
$624$		0			0
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$	−26.0000			−1.03917
$627$		0			0
$628$	7.00000	+	7.00000i	0.279330	+	0.279330i
$629$	12.0000	+	2.00000i	0.478471	+	0.0797452i
$630$	18.0000	+	6.00000i	0.717137	+	0.239046i
$631$	16.0000	+	16.0000i	0.636950	+	0.636950i	0.949802	−	0.312852i	$-0.101284\pi$
$631$	16.0000	+	16.0000i	0.636950	+	0.636950i	−0.312852	+	0.949802i	$0.601284\pi$
$632$	12.0000	−	12.0000i	0.477334	−	0.477334i
$633$		0			0
$634$	−5.00000	−	5.00000i	−0.198575	−	0.198575i
$635$	6.00000	+	2.00000i	0.238103	+	0.0793676i
$636$		0			0
$637$	4.00000			0.158486
$638$		0			0
$639$		−	36.0000i		−	1.42414i
$640$	1.00000	+	2.00000i	0.0395285	+	0.0790569i
$641$	−48.0000			−1.89589			−0.947943	−	0.318440i	$-0.896841\pi$
$641$	−48.0000			−1.89589			−0.947943	+	0.318440i	$0.896841\pi$
$642$		0			0
$643$	44.0000			1.73519			0.867595	−	0.497271i	$-0.165665\pi$
$643$	44.0000			1.73519			0.867595	+	0.497271i	$0.165665\pi$
$644$	8.00000	−	8.00000i	0.315244	−	0.315244i
$645$		0			0
$646$	4.00000	−	4.00000i	0.157378	−	0.157378i
$647$		−	32.0000i		−	1.25805i	−0.777385	−	0.629025i	$-0.783454\pi$
$647$		−	32.0000i		−	1.25805i	0.777385	−	0.629025i	$-0.216546\pi$
$648$			9.00000i			0.353553i
$649$		0			0
$650$	12.0000	−	16.0000i	0.470679	−	0.627572i
$651$		0			0
$652$	−24.0000			−0.939913
$653$	4.00000			0.156532			0.0782660	−	0.996933i	$-0.475062\pi$
$653$	4.00000			0.156532			0.0782660	+	0.996933i	$0.475062\pi$
$654$		0			0
$655$	−18.0000	−	6.00000i	−0.703318	−	0.234439i
$656$		0			0
$657$	−15.0000	+	15.0000i	−0.585206	+	0.585206i
$658$	−8.00000			−0.311872
$659$	−20.0000			−0.779089			−0.389545	−	0.921008i	$-0.627368\pi$
$659$	−20.0000			−0.779089			−0.389545	+	0.921008i	$0.627368\pi$
$660$		0			0
$661$	21.0000	+	21.0000i	0.816805	+	0.816805i	0.985644	−	0.168838i	$-0.0540016\pi$
$661$	21.0000	+	21.0000i	0.816805	+	0.816805i	−0.168838	+	0.985644i	$0.554002\pi$
$662$	6.00000	+	6.00000i	0.233197	+	0.233197i
$663$		0			0
$664$	−4.00000	−	4.00000i	−0.155230	−	0.155230i
$665$	8.00000	+	16.0000i	0.310227	+	0.620453i
$666$	−18.0000	−	3.00000i	−0.697486	−	0.116248i
$667$	28.0000	+	28.0000i	1.08416	+	1.08416i
$668$	−8.00000			−0.309529
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	9.00000	−	9.00000i	0.346925	−	0.346925i	−0.512038	−	0.858963i	$-0.671109\pi$
$673$	9.00000	−	9.00000i	0.346925	−	0.346925i	0.858963	+	0.512038i	$0.171109\pi$
$674$	15.0000	+	15.0000i	0.577778	+	0.577778i
$675$		0			0
$676$	3.00000			0.115385
$677$	−25.0000	+	25.0000i	−0.960828	+	0.960828i	−0.999261	−	0.0384331i	$-0.987763\pi$
$677$	−25.0000	+	25.0000i	−0.960828	+	0.960828i	0.0384331	+	0.999261i	$0.487763\pi$
$678$		0			0
$679$	24.0000	+	24.0000i	0.921035	+	0.921035i
$680$	−2.00000	−	4.00000i	−0.0766965	−	0.153393i
$681$		0			0
$682$		0			0
$683$			36.0000i			1.37750i	0.724998	+	0.688751i	$0.241841\pi$
$683$			36.0000i			1.37750i	−0.724998	+	0.688751i	$0.758159\pi$
$684$	−6.00000	+	6.00000i	−0.229416	+	0.229416i
$685$	21.0000	+	7.00000i	0.802369	+	0.267456i
$686$	−12.0000	−	12.0000i	−0.458162	−	0.458162i
$687$		0			0
$688$		−	4.00000i		−	0.152499i
$689$	4.00000	+	4.00000i	0.152388	+	0.152388i
$690$		0			0
$691$		−	20.0000i		−	0.760836i	−0.924815	−	0.380418i	$-0.875780\pi$
$691$		−	20.0000i		−	0.760836i	0.924815	−	0.380418i	$-0.124220\pi$
$692$	−5.00000	−	5.00000i	−0.190071	−	0.190071i
$693$		0			0
$694$	12.0000			0.455514
$695$	−40.0000	+	20.0000i	−1.51729	+	0.758643i
$696$		0			0
$697$		0			0
$698$	−14.0000			−0.529908
$699$		0			0
$700$	14.0000	−	2.00000i	0.529150	−	0.0755929i
$701$	31.0000	−	31.0000i	1.17085	−	1.17085i	0.188847	−	0.982006i	$-0.439525\pi$
$701$	31.0000	−	31.0000i	1.17085	−	1.17085i	0.982006	−	0.188847i	$-0.0604752\pi$
$702$		0			0
$703$	−10.0000	−	14.0000i	−0.377157	−	0.528020i
$704$		0			0
$705$		0			0
$706$		−	36.0000i		−	1.35488i
$707$	−20.0000	+	20.0000i	−0.752177	+	0.752177i
$708$		0			0
$709$	27.0000	+	27.0000i	1.01401	+	1.01401i	0.999901	+	0.0141058i	$0.00449016\pi$
$709$	27.0000	+	27.0000i	1.01401	+	1.01401i	0.0141058	+	0.999901i	$0.495510\pi$
$710$	−12.0000	−	24.0000i	−0.450352	−	0.900704i
$711$	36.0000	−	36.0000i	1.35011	−	1.35011i
$712$	7.00000	+	7.00000i	0.262336	+	0.262336i
$713$	−16.0000	+	16.0000i	−0.599205	+	0.599205i
$714$		0			0
$715$		0			0
$716$	2.00000	−	2.00000i	0.0747435	−	0.0747435i
$717$		0			0
$718$	−24.0000			−0.895672
$719$			4.00000i			0.149175i	0.997214	+	0.0745874i	$0.0237640\pi$
$719$			4.00000i			0.149175i	−0.997214	+	0.0745874i	$0.976236\pi$
$720$	3.00000	+	6.00000i	0.111803	+	0.223607i
$721$	32.0000	+	32.0000i	1.19174	+	1.19174i
$722$	11.0000			0.409378
$723$		0			0
$724$	8.00000			0.297318
$725$	7.00000	+	49.0000i	0.259973	+	1.81981i
$726$		0			0
$727$			28.0000i			1.03846i	0.854634	+	0.519231i	$0.173782\pi$
$727$			28.0000i			1.03846i	−0.854634	+	0.519231i	$0.826218\pi$
$728$	8.00000	+	8.00000i	0.296500	+	0.296500i
$729$			27.0000i			1.00000i
$730$	−5.00000	+	15.0000i	−0.185058	+	0.555175i
$731$			8.00000i			0.295891i
$732$		0			0
$733$	15.0000	+	15.0000i	0.554038	+	0.554038i	0.927604	−	0.373566i	$-0.121865\pi$
$733$	15.0000	+	15.0000i	0.554038	+	0.554038i	−0.373566	+	0.927604i	$0.621865\pi$
$734$	−18.0000	+	18.0000i	−0.664392	+	0.664392i
$735$		0			0
$736$	4.00000			0.147442
$737$		0			0
$738$		0			0
$739$	20.0000			0.735712			0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000			0.735712			0.367856	+	0.929883i	$0.380092\pi$
$740$	−13.0000	+	4.00000i	−0.477890	+	0.147043i
$741$		0			0
$742$			4.00000i			0.146845i
$743$	10.0000	+	10.0000i	0.366864	+	0.366864i	0.866332	−	0.499468i	$-0.166471\pi$
$743$	10.0000	+	10.0000i	0.366864	+	0.366864i	−0.499468	+	0.866332i	$0.666471\pi$
$744$		0			0
$745$	6.00000	+	12.0000i	0.219823	+	0.439646i
$746$	25.0000	−	25.0000i	0.915315	−	0.915315i
$747$	−12.0000	−	12.0000i	−0.439057	−	0.439057i
$748$		0			0
$749$		−	32.0000i		−	1.16925i
$750$		0			0
$751$		−	40.0000i		−	1.45962i	−0.683650	−	0.729810i	$-0.739608\pi$
$751$		−	40.0000i		−	1.45962i	0.683650	−	0.729810i	$-0.260392\pi$
$752$	−2.00000	−	2.00000i	−0.0729325	−	0.0729325i
$753$		0			0
$754$	−28.0000	+	28.0000i	−1.01970	+	1.01970i
$755$	20.0000	+	40.0000i	0.727875	+	1.45575i
$756$		0			0
$757$	28.0000			1.01768			0.508839	−	0.860862i	$-0.330075\pi$
$757$	28.0000			1.01768			0.508839	+	0.860862i	$0.330075\pi$
$758$	−4.00000			−0.145287
$759$		0			0
$760$	−2.00000	+	6.00000i	−0.0725476	+	0.217643i
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$			12.0000i			0.434429i
$764$	−16.0000	+	16.0000i	−0.578860	+	0.578860i
$765$	−6.00000	−	12.0000i	−0.216930	−	0.433861i
$766$	−36.0000			−1.30073
$767$	8.00000	−	8.00000i	0.288863	−	0.288863i
$768$		0			0
$769$	3.00000	−	3.00000i	0.108183	−	0.108183i	−0.650943	−	0.759126i	$-0.725627\pi$
$769$	3.00000	−	3.00000i	0.108183	−	0.108183i	0.759126	+	0.650943i	$0.225627\pi$
$770$		0			0
$771$		0			0
$772$			14.0000i			0.503871i
$773$	−21.0000	+	21.0000i	−0.755318	+	0.755318i	−0.975466	−	0.220149i	$-0.929346\pi$
$773$	−21.0000	+	21.0000i	−0.755318	+	0.755318i	0.220149	+	0.975466i	$0.429346\pi$
$774$		−	12.0000i		−	0.431331i
$775$	−28.0000	+	4.00000i	−1.00579	+	0.143684i
$776$			12.0000i			0.430775i
$777$		0			0
$778$	−17.0000	+	17.0000i	−0.609480	+	0.609480i
$779$		0			0
$780$		0			0
$781$		0			0
$782$	−8.00000			−0.286079
$783$		0			0
$784$			1.00000i			0.0357143i
$785$	21.0000	+	7.00000i	0.749522	+	0.249841i

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:	\( N \)	\(=\)	\( 370 = 2 \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	370.g (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +(-2.00000 + 1.00000i) q^{5} +(-2.00000 - 2.00000i) q^{7} -1.00000i q^{8} -3.00000i q^{9} +(-1.00000 - 2.00000i) q^{10} -4.00000i q^{13} +(2.00000 - 2.00000i) q^{14} +1.00000 q^{16} -2.00000 q^{17} +3.00000 q^{18} +(2.00000 + 2.00000i) q^{19} +(2.00000 - 1.00000i) q^{20} -4.00000i q^{23} +(3.00000 - 4.00000i) q^{25} +4.00000 q^{26} +(2.00000 + 2.00000i) q^{28} +(-7.00000 + 7.00000i) q^{29} +(-4.00000 - 4.00000i) q^{31} +1.00000i q^{32} -2.00000i q^{34} +(6.00000 + 2.00000i) q^{35} +3.00000i q^{36} +(-6.00000 - 1.00000i) q^{37} +(-2.00000 + 2.00000i) q^{38} +(1.00000 + 2.00000i) q^{40} -4.00000i q^{43} +(3.00000 + 6.00000i) q^{45} +4.00000 q^{46} +(-2.00000 - 2.00000i) q^{47} +1.00000i q^{49} +(4.00000 + 3.00000i) q^{50} +4.00000i q^{52} +(-1.00000 + 1.00000i) q^{53} +(-2.00000 + 2.00000i) q^{56} +(-7.00000 - 7.00000i) q^{58} +(2.00000 + 2.00000i) q^{59} +(1.00000 + 1.00000i) q^{61} +(4.00000 - 4.00000i) q^{62} +(-6.00000 + 6.00000i) q^{63} -1.00000 q^{64} +(4.00000 + 8.00000i) q^{65} +2.00000 q^{68} +(-2.00000 + 6.00000i) q^{70} +12.0000 q^{71} -3.00000 q^{72} +(-5.00000 - 5.00000i) q^{73} +(1.00000 - 6.00000i) q^{74} +(-2.00000 - 2.00000i) q^{76} +(12.0000 + 12.0000i) q^{79} +(-2.00000 + 1.00000i) q^{80} -9.00000 q^{81} +(4.00000 - 4.00000i) q^{83} +(4.00000 - 2.00000i) q^{85} +4.00000 q^{86} +(-7.00000 + 7.00000i) q^{89} +(-6.00000 + 3.00000i) q^{90} +(-8.00000 + 8.00000i) q^{91} +4.00000i q^{92} +(2.00000 - 2.00000i) q^{94} +(-6.00000 - 2.00000i) q^{95} -12.0000 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 4 q^{5} - 4 q^{7} - 2 q^{10} + 4 q^{14} + 2 q^{16} - 4 q^{17} + 6 q^{18} + 4 q^{19} + 4 q^{20} + 6 q^{25} + 8 q^{26} + 4 q^{28} - 14 q^{29} - 8 q^{31} + 12 q^{35} - 12 q^{37} - 4 q^{38}+ \cdots - 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^4 - 4 * q^5 - 4 * q^7 - 2 * q^10 + 4 * q^14 + 2 * q^16 - 4 * q^17 + 6 * q^18 + 4 * q^19 + 4 * q^20 + 6 * q^25 + 8 * q^26 + 4 * q^28 - 14 * q^29 - 8 * q^31 + 12 * q^35 - 12 * q^37 - 4 * q^38 + 2 * q^40 + 6 * q^45 + 8 * q^46 - 4 * q^47 + 8 * q^50 - 2 * q^53 - 4 * q^56 - 14 * q^58 + 4 * q^59 + 2 * q^61 + 8 * q^62 - 12 * q^63 - 2 * q^64 + 8 * q^65 + 4 * q^68 - 4 * q^70 + 24 * q^71 - 6 * q^72 - 10 * q^73 + 2 * q^74 - 4 * q^76 + 24 * q^79 - 4 * q^80 - 18 * q^81 + 8 * q^83 + 8 * q^85 + 8 * q^86 - 14 * q^89 - 12 * q^90 - 16 * q^91 + 4 * q^94 - 12 * q^95 - 24 * q^97 - 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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