LMFDB - Embedded newform 1840.2.e.b.369.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,2,Mod(369,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))

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

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

chi := DirichletCharacter("1840.369");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1840.e (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:	$14.6924739719$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 115)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			369.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1840.369
Dual form			1840.2.e.b.369.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{3} +(2.00000 + 1.00000i) q^{5} -1.00000i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+2.00000i q^{3} +(2.00000 + 1.00000i) q^{5} -1.00000i q^{7} -1.00000 q^{9} -2.00000i q^{13} +(-2.00000 + 4.00000i) q^{15} -5.00000i q^{17} +8.00000 q^{19} +2.00000 q^{21} -1.00000i q^{23} +(3.00000 + 4.00000i) q^{25} +4.00000i q^{27} +5.00000 q^{29} +5.00000 q^{31} +(1.00000 - 2.00000i) q^{35} +7.00000i q^{37} +4.00000 q^{39} -7.00000 q^{41} +4.00000i q^{43} +(-2.00000 - 1.00000i) q^{45} +2.00000i q^{47} +6.00000 q^{49} +10.0000 q^{51} +1.00000i q^{53} +16.0000i q^{57} +3.00000 q^{59} -6.00000 q^{61} +1.00000i q^{63} +(2.00000 - 4.00000i) q^{65} -13.0000i q^{67} +2.00000 q^{69} -13.0000 q^{71} -8.00000i q^{73} +(-8.00000 + 6.00000i) q^{75} -14.0000 q^{79} -11.0000 q^{81} -3.00000i q^{83} +(5.00000 - 10.0000i) q^{85} +10.0000i q^{87} +14.0000 q^{89} -2.00000 q^{91} +10.0000i q^{93} +(16.0000 + 8.00000i) q^{95} +14.0000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{5} - 2 q^{9}+O(q^{10}) $ 2 * q + 4 * q^5 - 2 * q^9	$ 2 q + 4 q^{5} - 2 q^{9} - 4 q^{15} + 16 q^{19} + 4 q^{21} + 6 q^{25} + 10 q^{29} + 10 q^{31} + 2 q^{35} + 8 q^{39} - 14 q^{41} - 4 q^{45} + 12 q^{49} + 20 q^{51} + 6 q^{59} - 12 q^{61} + 4 q^{65} + 4 q^{69} - 26 q^{71} - 16 q^{75} - 28 q^{79} - 22 q^{81} + 10 q^{85} + 28 q^{89} - 4 q^{91} + 32 q^{95}+O(q^{100}) $ 2 * q + 4 * q^5 - 2 * q^9 - 4 * q^15 + 16 * q^19 + 4 * q^21 + 6 * q^25 + 10 * q^29 + 10 * q^31 + 2 * q^35 + 8 * q^39 - 14 * q^41 - 4 * q^45 + 12 * q^49 + 20 * q^51 + 6 * q^59 - 12 * q^61 + 4 * q^65 + 4 * q^69 - 26 * q^71 - 16 * q^75 - 28 * q^79 - 22 * q^81 + 10 * q^85 + 28 * q^89 - 4 * q^91 + 32 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times$.

$n$	$737$	$1151$	$1201$	$1381$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$3$			2.00000i			1.15470i	0.816497	+	0.577350i	$0.195913\pi$
$3$			2.00000i			1.15470i	−0.816497	+	0.577350i	$0.804087\pi$
$4$		0			0
$5$	2.00000	+	1.00000i	0.894427	+	0.447214i
$5$	2.00000	+	1.00000i	0.894427	+	0.447214i
$6$		0			0
$7$		−	1.00000i		−	0.377964i	−0.981981	−	0.188982i	$-0.939481\pi$
$7$		−	1.00000i		−	0.377964i	0.981981	−	0.188982i	$-0.0605189\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$		0			0
$13$		−	2.00000i		−	0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$		−	2.00000i		−	0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$		0			0
$15$	−2.00000	+	4.00000i	−0.516398	+	1.03280i
$16$		0			0
$17$		−	5.00000i		−	1.21268i	−0.795206	−	0.606339i	$-0.792637\pi$
$17$		−	5.00000i		−	1.21268i	0.795206	−	0.606339i	$-0.207363\pi$
$18$		0			0
$19$	8.00000			1.83533			0.917663	−	0.397360i	$-0.130073\pi$
$19$	8.00000			1.83533			0.917663	+	0.397360i	$0.130073\pi$
$20$		0			0
$21$	2.00000			0.436436
$22$		0			0
$23$		−	1.00000i		−	0.208514i
$23$		−	1.00000i		−	0.208514i
$24$		0			0
$25$	3.00000	+	4.00000i	0.600000	+	0.800000i
$26$		0			0
$27$			4.00000i			0.769800i
$28$		0			0
$29$	5.00000			0.928477			0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000			0.928477			0.464238	+	0.885710i	$0.346328\pi$
$30$		0			0
$31$	5.00000			0.898027			0.449013	−	0.893525i	$-0.351776\pi$
$31$	5.00000			0.898027			0.449013	+	0.893525i	$0.351776\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.00000	−	2.00000i	0.169031	−	0.338062i
$36$		0			0
$37$			7.00000i			1.15079i	0.817875	+	0.575396i	$0.195152\pi$
$37$			7.00000i			1.15079i	−0.817875	+	0.575396i	$0.804848\pi$
$38$		0			0
$39$	4.00000			0.640513
$40$		0			0
$41$	−7.00000			−1.09322			−0.546608	−	0.837389i	$-0.684081\pi$
$41$	−7.00000			−1.09322			−0.546608	+	0.837389i	$0.684081\pi$
$42$		0			0
$43$			4.00000i			0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$			4.00000i			0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$		0			0
$45$	−2.00000	−	1.00000i	−0.298142	−	0.149071i
$46$		0			0
$47$			2.00000i			0.291730i	0.989305	+	0.145865i	$0.0465965\pi$
$47$			2.00000i			0.291730i	−0.989305	+	0.145865i	$0.953403\pi$
$48$		0			0
$49$	6.00000			0.857143
$50$		0			0
$51$	10.0000			1.40028
$52$		0			0
$53$			1.00000i			0.137361i	0.997639	+	0.0686803i	$0.0218788\pi$
$53$			1.00000i			0.137361i	−0.997639	+	0.0686803i	$0.978121\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$			16.0000i			2.11925i
$58$		0			0
$59$	3.00000			0.390567			0.195283	−	0.980747i	$-0.437437\pi$
$59$	3.00000			0.390567			0.195283	+	0.980747i	$0.437437\pi$
$60$		0			0
$61$	−6.00000			−0.768221			−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000			−0.768221			−0.384111	+	0.923287i	$0.625492\pi$
$62$		0			0
$63$			1.00000i			0.125988i
$64$		0			0
$65$	2.00000	−	4.00000i	0.248069	−	0.496139i
$66$		0			0
$67$		−	13.0000i		−	1.58820i	−0.607785	−	0.794101i	$-0.707942\pi$
$67$		−	13.0000i		−	1.58820i	0.607785	−	0.794101i	$-0.292058\pi$
$68$		0			0
$69$	2.00000			0.240772
$70$		0			0
$71$	−13.0000			−1.54282			−0.771408	−	0.636341i	$-0.780447\pi$
$71$	−13.0000			−1.54282			−0.771408	+	0.636341i	$0.780447\pi$
$72$		0			0
$73$		−	8.00000i		−	0.936329i	−0.883641	−	0.468165i	$-0.844915\pi$
$73$		−	8.00000i		−	0.936329i	0.883641	−	0.468165i	$-0.155085\pi$
$74$		0			0
$75$	−8.00000	+	6.00000i	−0.923760	+	0.692820i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−14.0000			−1.57512			−0.787562	−	0.616236i	$-0.788657\pi$
$79$	−14.0000			−1.57512			−0.787562	+	0.616236i	$0.788657\pi$
$80$		0			0
$81$	−11.0000			−1.22222
$82$		0			0
$83$		−	3.00000i		−	0.329293i	−0.986353	−	0.164646i	$-0.947352\pi$
$83$		−	3.00000i		−	0.329293i	0.986353	−	0.164646i	$-0.0526483\pi$
$84$		0			0
$85$	5.00000	−	10.0000i	0.542326	−	1.08465i
$86$		0			0
$87$			10.0000i			1.07211i
$88$		0			0
$89$	14.0000			1.48400			0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000			1.48400			0.741999	+	0.670402i	$0.233878\pi$
$90$		0			0
$91$	−2.00000			−0.209657
$92$		0			0
$93$			10.0000i			1.03695i
$94$		0			0
$95$	16.0000	+	8.00000i	1.64157	+	0.820783i
$96$		0			0
$97$			14.0000i			1.42148i	0.703452	+	0.710742i	$0.251641\pi$
$97$			14.0000i			1.42148i	−0.703452	+	0.710742i	$0.748359\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	15.0000			1.49256			0.746278	−	0.665635i	$-0.231839\pi$
$101$	15.0000			1.49256			0.746278	+	0.665635i	$0.231839\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$	4.00000	+	2.00000i	0.390360	+	0.195180i
$106$		0			0
$107$		−	9.00000i		−	0.870063i	−0.900415	−	0.435031i	$-0.856737\pi$
$107$		−	9.00000i		−	0.870063i	0.900415	−	0.435031i	$-0.143263\pi$
$108$		0			0
$109$	−18.0000			−1.72409			−0.862044	−	0.506834i	$-0.830816\pi$
$109$	−18.0000			−1.72409			−0.862044	+	0.506834i	$0.830816\pi$
$110$		0			0
$111$	−14.0000			−1.32882
$112$		0			0
$113$			1.00000i			0.0940721i	0.998893	+	0.0470360i	$0.0149776\pi$
$113$			1.00000i			0.0940721i	−0.998893	+	0.0470360i	$0.985022\pi$
$114$		0			0
$115$	1.00000	−	2.00000i	0.0932505	−	0.186501i
$116$		0			0
$117$			2.00000i			0.184900i
$118$		0			0
$119$	−5.00000			−0.458349
$120$		0			0
$121$	−11.0000			−1.00000
$122$		0			0
$123$		−	14.0000i		−	1.26234i
$124$		0			0
$125$	2.00000	+	11.0000i	0.178885	+	0.983870i
$126$		0			0
$127$			20.0000i			1.77471i	0.461084	+	0.887357i	$0.347461\pi$
$127$			20.0000i			1.77471i	−0.461084	+	0.887357i	$0.652539\pi$
$128$		0			0
$129$	−8.00000			−0.704361
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$		−	8.00000i		−	0.693688i
$134$		0			0
$135$	−4.00000	+	8.00000i	−0.344265	+	0.688530i
$136$		0			0
$137$			6.00000i			0.512615i	0.966595	+	0.256307i	$0.0825059\pi$
$137$			6.00000i			0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$		0			0
$139$	9.00000			0.763370			0.381685	−	0.924292i	$-0.375344\pi$
$139$	9.00000			0.763370			0.381685	+	0.924292i	$0.375344\pi$
$140$		0			0
$141$	−4.00000			−0.336861
$142$		0			0
$143$		0			0
$144$		0			0
$145$	10.0000	+	5.00000i	0.830455	+	0.415227i
$146$		0			0
$147$			12.0000i			0.989743i
$148$		0			0
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$		0			0
$151$	−8.00000			−0.651031			−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000			−0.651031			−0.325515	+	0.945537i	$0.605538\pi$
$152$		0			0
$153$			5.00000i			0.404226i
$154$		0			0
$155$	10.0000	+	5.00000i	0.803219	+	0.401610i
$156$		0			0
$157$		−	3.00000i		−	0.239426i	−0.992809	−	0.119713i	$-0.961803\pi$
$157$		−	3.00000i		−	0.239426i	0.992809	−	0.119713i	$-0.0381975\pi$
$158$		0			0
$159$	−2.00000			−0.158610
$160$		0			0
$161$	−1.00000			−0.0788110
$162$		0			0
$163$			24.0000i			1.87983i	0.341415	+	0.939913i	$0.389094\pi$
$163$			24.0000i			1.87983i	−0.341415	+	0.939913i	$0.610906\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	16.0000i		−	1.23812i	−0.785345	−	0.619059i	$-0.787514\pi$
$167$		−	16.0000i		−	1.23812i	0.785345	−	0.619059i	$-0.212486\pi$
$168$		0			0
$169$	9.00000			0.692308
$170$		0			0
$171$	−8.00000			−0.611775
$172$		0			0
$173$		−	6.00000i		−	0.456172i	−0.973641	−	0.228086i	$-0.926753\pi$
$173$		−	6.00000i		−	0.456172i	0.973641	−	0.228086i	$-0.0732467\pi$
$174$		0			0
$175$	4.00000	−	3.00000i	0.302372	−	0.226779i
$176$		0			0
$177$			6.00000i			0.450988i
$178$		0			0
$179$	−4.00000			−0.298974			−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000			−0.298974			−0.149487	+	0.988764i	$0.547762\pi$
$180$		0			0
$181$	−14.0000			−1.04061			−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000			−1.04061			−0.520306	+	0.853980i	$0.674182\pi$
$182$		0			0
$183$		−	12.0000i		−	0.887066i
$184$		0			0
$185$	−7.00000	+	14.0000i	−0.514650	+	1.02930i
$186$		0			0
$187$		0			0
$188$		0			0
$189$	4.00000			0.290957
$190$		0			0
$191$	8.00000			0.578860			0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000			0.578860			0.289430	+	0.957199i	$0.406534\pi$
$192$		0			0
$193$			12.0000i			0.863779i	0.901927	+	0.431889i	$0.142153\pi$
$193$			12.0000i			0.863779i	−0.901927	+	0.431889i	$0.857847\pi$
$194$		0			0
$195$	8.00000	+	4.00000i	0.572892	+	0.286446i
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	2.00000			0.141776			0.0708881	−	0.997484i	$-0.477417\pi$
$199$	2.00000			0.141776			0.0708881	+	0.997484i	$0.477417\pi$
$200$		0			0
$201$	26.0000			1.83390
$202$		0			0
$203$		−	5.00000i		−	0.350931i
$204$		0			0
$205$	−14.0000	−	7.00000i	−0.977802	−	0.488901i
$206$		0			0
$207$			1.00000i			0.0695048i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	9.00000			0.619586			0.309793	−	0.950804i	$-0.399740\pi$
$211$	9.00000			0.619586			0.309793	+	0.950804i	$0.399740\pi$
$212$		0			0
$213$		−	26.0000i		−	1.78149i
$214$		0			0
$215$	−4.00000	+	8.00000i	−0.272798	+	0.545595i
$216$		0			0
$217$		−	5.00000i		−	0.339422i
$218$		0			0
$219$	16.0000			1.08118
$220$		0			0
$221$	−10.0000			−0.672673
$222$		0			0
$223$		−	14.0000i		−	0.937509i	−0.883328	−	0.468755i	$-0.844703\pi$
$223$		−	14.0000i		−	0.937509i	0.883328	−	0.468755i	$-0.155297\pi$
$224$		0			0
$225$	−3.00000	−	4.00000i	−0.200000	−	0.266667i
$226$		0			0
$227$		0			0		1.00000			$0$
$227$		0			0		−1.00000			$\pi$
$228$		0			0
$229$	−4.00000			−0.264327			−0.132164	−	0.991228i	$-0.542192\pi$
$229$	−4.00000			−0.264327			−0.132164	+	0.991228i	$0.542192\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	6.00000i		−	0.393073i	−0.980497	−	0.196537i	$-0.937031\pi$
$233$		−	6.00000i		−	0.393073i	0.980497	−	0.196537i	$-0.0629694\pi$
$234$		0			0
$235$	−2.00000	+	4.00000i	−0.130466	+	0.260931i
$236$		0			0
$237$		−	28.0000i		−	1.81880i
$238$		0			0
$239$	−1.00000			−0.0646846			−0.0323423	−	0.999477i	$-0.510297\pi$
$239$	−1.00000			−0.0646846			−0.0323423	+	0.999477i	$0.510297\pi$
$240$		0			0
$241$	6.00000			0.386494			0.193247	−	0.981150i	$-0.438098\pi$
$241$	6.00000			0.386494			0.193247	+	0.981150i	$0.438098\pi$
$242$		0			0
$243$		−	10.0000i		−	0.641500i
$244$		0			0
$245$	12.0000	+	6.00000i	0.766652	+	0.383326i
$246$		0			0
$247$		−	16.0000i		−	1.01806i
$248$		0			0
$249$	6.00000			0.380235
$250$		0			0
$251$	−18.0000			−1.13615			−0.568075	−	0.822977i	$-0.692312\pi$
$251$	−18.0000			−1.13615			−0.568075	+	0.822977i	$0.692312\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	20.0000	+	10.0000i	1.25245	+	0.626224i
$256$		0			0
$257$			22.0000i			1.37232i	0.727450	+	0.686161i	$0.240706\pi$
$257$			22.0000i			1.37232i	−0.727450	+	0.686161i	$0.759294\pi$
$258$		0			0
$259$	7.00000			0.434959
$260$		0			0
$261$	−5.00000			−0.309492
$262$		0			0
$263$		−	13.0000i		−	0.801614i	−0.916162	−	0.400807i	$-0.868730\pi$
$263$		−	13.0000i		−	0.801614i	0.916162	−	0.400807i	$-0.131270\pi$
$264$		0			0
$265$	−1.00000	+	2.00000i	−0.0614295	+	0.122859i
$266$		0			0
$267$			28.0000i			1.71357i
$268$		0			0
$269$	15.0000			0.914566			0.457283	−	0.889321i	$-0.348823\pi$
$269$	15.0000			0.914566			0.457283	+	0.889321i	$0.348823\pi$
$270$		0			0
$271$	15.0000			0.911185			0.455593	−	0.890188i	$-0.349427\pi$
$271$	15.0000			0.911185			0.455593	+	0.890188i	$0.349427\pi$
$272$		0			0
$273$		−	4.00000i		−	0.242091i
$274$		0			0
$275$		0			0
$276$		0			0
$277$		−	26.0000i		−	1.56219i	−0.624413	−	0.781094i	$-0.714662\pi$
$277$		−	26.0000i		−	1.56219i	0.624413	−	0.781094i	$-0.285338\pi$
$278$		0			0
$279$	−5.00000			−0.299342
$280$		0			0
$281$	−12.0000			−0.715860			−0.357930	−	0.933748i	$-0.616517\pi$
$281$	−12.0000			−0.715860			−0.357930	+	0.933748i	$0.616517\pi$
$282$		0			0
$283$			11.0000i			0.653882i	0.945045	+	0.326941i	$0.106018\pi$
$283$			11.0000i			0.653882i	−0.945045	+	0.326941i	$0.893982\pi$
$284$		0			0
$285$	−16.0000	+	32.0000i	−0.947758	+	1.89552i
$286$		0			0
$287$			7.00000i			0.413197i
$288$		0			0
$289$	−8.00000			−0.470588
$290$		0			0
$291$	−28.0000			−1.64139
$292$		0			0
$293$		−	29.0000i		−	1.69420i	−0.531435	−	0.847099i	$-0.678347\pi$
$293$		−	29.0000i		−	1.69420i	0.531435	−	0.847099i	$-0.321653\pi$
$294$		0			0
$295$	6.00000	+	3.00000i	0.349334	+	0.174667i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−2.00000			−0.115663
$300$		0			0
$301$	4.00000			0.230556
$302$		0			0
$303$			30.0000i			1.72345i
$304$		0			0
$305$	−12.0000	−	6.00000i	−0.687118	−	0.343559i
$306$		0			0
$307$		−	14.0000i		−	0.799022i	−0.916728	−	0.399511i	$-0.869180\pi$
$307$		−	14.0000i		−	0.799022i	0.916728	−	0.399511i	$-0.130820\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−4.00000			−0.226819			−0.113410	−	0.993548i	$-0.536177\pi$
$311$	−4.00000			−0.226819			−0.113410	+	0.993548i	$0.536177\pi$
$312$		0			0
$313$		−	21.0000i		−	1.18699i	−0.804838	−	0.593495i	$-0.797748\pi$
$313$		−	21.0000i		−	1.18699i	0.804838	−	0.593495i	$-0.202252\pi$
$314$		0			0
$315$	−1.00000	+	2.00000i	−0.0563436	+	0.112687i
$316$		0			0
$317$		−	24.0000i		−	1.34797i	−0.738743	−	0.673987i	$-0.764580\pi$
$317$		−	24.0000i		−	1.34797i	0.738743	−	0.673987i	$-0.235420\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	18.0000			1.00466
$322$		0			0
$323$		−	40.0000i		−	2.22566i
$324$		0			0
$325$	8.00000	−	6.00000i	0.443760	−	0.332820i
$326$		0			0
$327$		−	36.0000i		−	1.99080i
$328$		0			0
$329$	2.00000			0.110264
$330$		0			0
$331$	5.00000			0.274825			0.137412	−	0.990514i	$-0.456121\pi$
$331$	5.00000			0.274825			0.137412	+	0.990514i	$0.456121\pi$
$332$		0			0
$333$		−	7.00000i		−	0.383598i
$334$		0			0
$335$	13.0000	−	26.0000i	0.710266	−	1.42053i
$336$		0			0
$337$			26.0000i			1.41631i	0.706057	+	0.708155i	$0.250472\pi$
$337$			26.0000i			1.41631i	−0.706057	+	0.708155i	$0.749528\pi$
$338$		0			0
$339$	−2.00000			−0.108625
$340$		0			0
$341$		0			0
$342$		0			0
$343$		−	13.0000i		−	0.701934i
$344$		0			0
$345$	4.00000	+	2.00000i	0.215353	+	0.107676i
$346$		0			0
$347$		−	8.00000i		−	0.429463i	−0.976673	−	0.214731i	$-0.931112\pi$
$347$		−	8.00000i		−	0.429463i	0.976673	−	0.214731i	$-0.0688876\pi$
$348$		0			0
$349$	−23.0000			−1.23116			−0.615581	−	0.788074i	$-0.711079\pi$
$349$	−23.0000			−1.23116			−0.615581	+	0.788074i	$0.711079\pi$
$350$		0			0
$351$	8.00000			0.427008
$352$		0			0
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$	−26.0000	−	13.0000i	−1.37994	−	0.689968i
$356$		0			0
$357$		−	10.0000i		−	0.529256i
$358$		0			0
$359$	−6.00000			−0.316668			−0.158334	−	0.987386i	$-0.550612\pi$
$359$	−6.00000			−0.316668			−0.158334	+	0.987386i	$0.550612\pi$
$360$		0			0
$361$	45.0000			2.36842
$362$		0			0
$363$		−	22.0000i		−	1.15470i
$364$		0			0
$365$	8.00000	−	16.0000i	0.418739	−	0.837478i
$366$		0			0
$367$			13.0000i			0.678594i	0.940679	+	0.339297i	$0.110189\pi$
$367$			13.0000i			0.678594i	−0.940679	+	0.339297i	$0.889811\pi$
$368$		0			0
$369$	7.00000			0.364405
$370$		0			0
$371$	1.00000			0.0519174
$372$		0			0
$373$			6.00000i			0.310668i	0.987862	+	0.155334i	$0.0496454\pi$
$373$			6.00000i			0.310668i	−0.987862	+	0.155334i	$0.950355\pi$
$374$		0			0
$375$	−22.0000	+	4.00000i	−1.13608	+	0.206559i
$376$		0			0
$377$		−	10.0000i		−	0.515026i
$378$		0			0
$379$	6.00000			0.308199			0.154100	−	0.988055i	$-0.450752\pi$
$379$	6.00000			0.308199			0.154100	+	0.988055i	$0.450752\pi$
$380$		0			0
$381$	−40.0000			−2.04926
$382$		0			0
$383$			3.00000i			0.153293i	0.997058	+	0.0766464i	$0.0244213\pi$
$383$			3.00000i			0.153293i	−0.997058	+	0.0766464i	$0.975579\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		−	4.00000i		−	0.203331i
$388$		0			0
$389$	−22.0000			−1.11544			−0.557722	−	0.830028i	$-0.688325\pi$
$389$	−22.0000			−1.11544			−0.557722	+	0.830028i	$0.688325\pi$
$390$		0			0
$391$	−5.00000			−0.252861
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−28.0000	−	14.0000i	−1.40883	−	0.704416i
$396$		0			0
$397$			16.0000i			0.803017i	0.915855	+	0.401508i	$0.131514\pi$
$397$			16.0000i			0.803017i	−0.915855	+	0.401508i	$0.868486\pi$
$398$		0			0
$399$	16.0000			0.801002
$400$		0			0
$401$	2.00000			0.0998752			0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000			0.0998752			0.0499376	+	0.998752i	$0.484098\pi$
$402$		0			0
$403$		−	10.0000i		−	0.498135i
$404$		0			0
$405$	−22.0000	−	11.0000i	−1.09319	−	0.546594i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	19.0000			0.939490			0.469745	−	0.882802i	$-0.344346\pi$
$409$	19.0000			0.939490			0.469745	+	0.882802i	$0.344346\pi$
$410$		0			0
$411$	−12.0000			−0.591916
$412$		0			0
$413$		−	3.00000i		−	0.147620i
$414$		0			0
$415$	3.00000	−	6.00000i	0.147264	−	0.294528i
$416$		0			0
$417$			18.0000i			0.881464i
$418$		0			0
$419$	−2.00000			−0.0977064			−0.0488532	−	0.998806i	$-0.515557\pi$
$419$	−2.00000			−0.0977064			−0.0488532	+	0.998806i	$0.515557\pi$
$420$		0			0
$421$	10.0000			0.487370			0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000			0.487370			0.243685	+	0.969854i	$0.421644\pi$
$422$		0			0
$423$		−	2.00000i		−	0.0972433i
$424$		0			0
$425$	20.0000	−	15.0000i	0.970143	−	0.727607i
$426$		0			0
$427$			6.00000i			0.290360i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	12.0000			0.578020			0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000			0.578020			0.289010	+	0.957326i	$0.406674\pi$
$432$		0			0
$433$		−	19.0000i		−	0.913082i	−0.889702	−	0.456541i	$-0.849088\pi$
$433$		−	19.0000i		−	0.913082i	0.889702	−	0.456541i	$-0.150912\pi$
$434$		0			0
$435$	−10.0000	+	20.0000i	−0.479463	+	0.958927i
$436$		0			0
$437$		−	8.00000i		−	0.382692i
$438$		0			0
$439$	−28.0000			−1.33637			−0.668184	−	0.743996i	$-0.732928\pi$
$439$	−28.0000			−1.33637			−0.668184	+	0.743996i	$0.732928\pi$
$440$		0			0
$441$	−6.00000			−0.285714
$442$		0			0
$443$		−	24.0000i		−	1.14027i	−0.821549	−	0.570137i	$-0.806890\pi$
$443$		−	24.0000i		−	1.14027i	0.821549	−	0.570137i	$-0.193110\pi$
$444$		0			0
$445$	28.0000	+	14.0000i	1.32733	+	0.663664i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−15.0000			−0.707894			−0.353947	−	0.935266i	$-0.615161\pi$
$449$	−15.0000			−0.707894			−0.353947	+	0.935266i	$0.615161\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		−	16.0000i		−	0.751746i
$454$		0			0
$455$	−4.00000	−	2.00000i	−0.187523	−	0.0937614i
$456$		0			0
$457$			1.00000i			0.0467780i	0.999726	+	0.0233890i	$0.00744563\pi$
$457$			1.00000i			0.0467780i	−0.999726	+	0.0233890i	$0.992554\pi$
$458$		0			0
$459$	20.0000			0.933520
$460$		0			0
$461$	−18.0000			−0.838344			−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000			−0.838344			−0.419172	+	0.907907i	$0.637680\pi$
$462$		0			0
$463$		−	4.00000i		−	0.185896i	−0.995671	−	0.0929479i	$-0.970371\pi$
$463$		−	4.00000i		−	0.185896i	0.995671	−	0.0929479i	$-0.0296290\pi$
$464$		0			0
$465$	−10.0000	+	20.0000i	−0.463739	+	0.927478i
$466$		0			0
$467$			27.0000i			1.24941i	0.780860	+	0.624705i	$0.214781\pi$
$467$			27.0000i			1.24941i	−0.780860	+	0.624705i	$0.785219\pi$
$468$		0			0
$469$	−13.0000			−0.600284
$470$		0			0
$471$	6.00000			0.276465
$472$		0			0
$473$		0			0
$474$		0			0
$475$	24.0000	+	32.0000i	1.10120	+	1.46826i
$476$		0			0
$477$		−	1.00000i		−	0.0457869i
$478$		0			0
$479$	−28.0000			−1.27935			−0.639676	−	0.768644i	$-0.720932\pi$
$479$	−28.0000			−1.27935			−0.639676	+	0.768644i	$0.720932\pi$
$480$		0			0
$481$	14.0000			0.638345
$482$		0			0
$483$		−	2.00000i		−	0.0910032i
$484$		0			0
$485$	−14.0000	+	28.0000i	−0.635707	+	1.27141i
$486$		0			0
$487$			32.0000i			1.45006i	0.688718	+	0.725029i	$0.258174\pi$
$487$			32.0000i			1.45006i	−0.688718	+	0.725029i	$0.741826\pi$
$488$		0			0
$489$	−48.0000			−2.17064
$490$		0			0
$491$	31.0000			1.39901			0.699505	−	0.714628i	$-0.253404\pi$
$491$	31.0000			1.39901			0.699505	+	0.714628i	$0.253404\pi$
$492$		0			0
$493$		−	25.0000i		−	1.12594i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			13.0000i			0.583130i
$498$		0			0
$499$	−11.0000			−0.492428			−0.246214	−	0.969216i	$-0.579187\pi$
$499$	−11.0000			−0.492428			−0.246214	+	0.969216i	$0.579187\pi$
$500$		0			0
$501$	32.0000			1.42965
$502$		0			0
$503$		−	9.00000i		−	0.401290i	−0.979664	−	0.200645i	$-0.935696\pi$
$503$		−	9.00000i		−	0.401290i	0.979664	−	0.200645i	$-0.0643038\pi$
$504$		0			0
$505$	30.0000	+	15.0000i	1.33498	+	0.667491i
$506$		0			0
$507$			18.0000i			0.799408i
$508$		0			0
$509$	−26.0000			−1.15243			−0.576215	−	0.817298i	$-0.695471\pi$
$509$	−26.0000			−1.15243			−0.576215	+	0.817298i	$0.695471\pi$
$510$		0			0
$511$	−8.00000			−0.353899
$512$		0			0
$513$			32.0000i			1.41283i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	12.0000			0.526742
$520$		0			0
$521$	−20.0000			−0.876216			−0.438108	−	0.898922i	$-0.644351\pi$
$521$	−20.0000			−0.876216			−0.438108	+	0.898922i	$0.644351\pi$
$522$		0			0
$523$		−	28.0000i		−	1.22435i	−0.790721	−	0.612177i	$-0.790294\pi$
$523$		−	28.0000i		−	1.22435i	0.790721	−	0.612177i	$-0.209706\pi$
$524$		0			0
$525$	6.00000	+	8.00000i	0.261861	+	0.349149i
$526$		0			0
$527$		−	25.0000i		−	1.08902i
$528$		0			0
$529$	−1.00000			−0.0434783
$530$		0			0
$531$	−3.00000			−0.130189
$532$		0			0
$533$			14.0000i			0.606407i
$534$		0			0
$535$	9.00000	−	18.0000i	0.389104	−	0.778208i
$536$		0			0
$537$		−	8.00000i		−	0.345225i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	14.0000			0.601907			0.300954	−	0.953639i	$-0.402695\pi$
$541$	14.0000			0.601907			0.300954	+	0.953639i	$0.402695\pi$
$542$		0			0
$543$		−	28.0000i		−	1.20160i
$544$		0			0
$545$	−36.0000	−	18.0000i	−1.54207	−	0.771035i
$546$		0			0
$547$			4.00000i			0.171028i	0.996337	+	0.0855138i	$0.0272532\pi$
$547$			4.00000i			0.171028i	−0.996337	+	0.0855138i	$0.972747\pi$
$548$		0			0
$549$	6.00000			0.256074
$550$		0			0
$551$	40.0000			1.70406
$552$		0			0
$553$			14.0000i			0.595341i
$554$		0			0
$555$	−28.0000	−	14.0000i	−1.18853	−	0.594267i
$556$		0			0
$557$		−	33.0000i		−	1.39825i	−0.714997	−	0.699127i	$-0.753572\pi$
$557$		−	33.0000i		−	1.39825i	0.714997	−	0.699127i	$-0.246428\pi$
$558$		0			0
$559$	8.00000			0.338364
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	25.0000i		−	1.05362i	−0.849982	−	0.526812i	$-0.823387\pi$
$563$		−	25.0000i		−	1.05362i	0.849982	−	0.526812i	$-0.176613\pi$
$564$		0			0
$565$	−1.00000	+	2.00000i	−0.0420703	+	0.0841406i
$566$		0			0
$567$			11.0000i			0.461957i
$568$		0			0
$569$	12.0000			0.503066			0.251533	−	0.967849i	$-0.419065\pi$
$569$	12.0000			0.503066			0.251533	+	0.967849i	$0.419065\pi$
$570$		0			0
$571$	−18.0000			−0.753277			−0.376638	−	0.926360i	$-0.622920\pi$
$571$	−18.0000			−0.753277			−0.376638	+	0.926360i	$0.622920\pi$
$572$		0			0
$573$			16.0000i			0.668410i
$574$		0			0
$575$	4.00000	−	3.00000i	0.166812	−	0.125109i
$576$		0			0
$577$			22.0000i			0.915872i	0.888985	+	0.457936i	$0.151411\pi$
$577$			22.0000i			0.915872i	−0.888985	+	0.457936i	$0.848589\pi$
$578$		0			0
$579$	−24.0000			−0.997406
$580$		0			0
$581$	−3.00000			−0.124461
$582$		0			0
$583$		0			0
$584$		0			0
$585$	−2.00000	+	4.00000i	−0.0826898	+	0.165380i
$586$		0			0
$587$			12.0000i			0.495293i	0.968850	+	0.247647i	$0.0796572\pi$
$587$			12.0000i			0.495293i	−0.968850	+	0.247647i	$0.920343\pi$
$588$		0			0
$589$	40.0000			1.64817
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	30.0000i		−	1.23195i	−0.787765	−	0.615976i	$-0.788762\pi$
$593$		−	30.0000i		−	1.23195i	0.787765	−	0.615976i	$-0.211238\pi$
$594$		0			0
$595$	−10.0000	−	5.00000i	−0.409960	−	0.204980i
$596$		0			0
$597$			4.00000i			0.163709i
$598$		0			0
$599$	−48.0000			−1.96123			−0.980613	−	0.195952i	$-0.937220\pi$
$599$	−48.0000			−1.96123			−0.980613	+	0.195952i	$0.937220\pi$
$600$		0			0
$601$	−25.0000			−1.01977			−0.509886	−	0.860242i	$-0.670312\pi$
$601$	−25.0000			−1.01977			−0.509886	+	0.860242i	$0.670312\pi$
$602$		0			0
$603$			13.0000i			0.529401i
$604$		0			0
$605$	−22.0000	−	11.0000i	−0.894427	−	0.447214i
$606$		0			0
$607$			38.0000i			1.54237i	0.636610	+	0.771186i	$0.280336\pi$
$607$			38.0000i			1.54237i	−0.636610	+	0.771186i	$0.719664\pi$
$608$		0			0
$609$	10.0000			0.405220
$610$		0			0
$611$	4.00000			0.161823
$612$		0			0
$613$		−	6.00000i		−	0.242338i	−0.992632	−	0.121169i	$-0.961336\pi$
$613$		−	6.00000i		−	0.242338i	0.992632	−	0.121169i	$-0.0386643\pi$
$614$		0			0
$615$	14.0000	−	28.0000i	0.564534	−	1.12907i
$616$		0			0
$617$		−	47.0000i		−	1.89215i	−0.323949	−	0.946074i	$-0.605011\pi$
$617$		−	47.0000i		−	1.89215i	0.323949	−	0.946074i	$-0.394989\pi$
$618$		0			0
$619$	−10.0000			−0.401934			−0.200967	−	0.979598i	$-0.564408\pi$
$619$	−10.0000			−0.401934			−0.200967	+	0.979598i	$0.564408\pi$
$620$		0			0
$621$	4.00000			0.160514
$622$		0			0
$623$		−	14.0000i		−	0.560898i
$624$		0			0
$625$	−7.00000	+	24.0000i	−0.280000	+	0.960000i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	35.0000			1.39554
$630$		0			0
$631$	20.0000			0.796187			0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000			0.796187			0.398094	+	0.917345i	$0.369672\pi$
$632$		0			0
$633$			18.0000i			0.715436i
$634$		0			0
$635$	−20.0000	+	40.0000i	−0.793676	+	1.58735i
$636$		0			0
$637$		−	12.0000i		−	0.475457i
$638$		0			0
$639$	13.0000			0.514272
$640$		0			0
$641$	−24.0000			−0.947943			−0.473972	−	0.880540i	$-0.657180\pi$
$641$	−24.0000			−0.947943			−0.473972	+	0.880540i	$0.657180\pi$
$642$		0			0
$643$			37.0000i			1.45914i	0.683907	+	0.729569i	$0.260279\pi$
$643$			37.0000i			1.45914i	−0.683907	+	0.729569i	$0.739721\pi$
$644$		0			0
$645$	−16.0000	−	8.00000i	−0.629999	−	0.315000i
$646$		0			0
$647$			18.0000i			0.707653i	0.935311	+	0.353827i	$0.115120\pi$
$647$			18.0000i			0.707653i	−0.935311	+	0.353827i	$0.884880\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	10.0000			0.391931
$652$		0			0
$653$		−	2.00000i		−	0.0782660i	−0.999234	−	0.0391330i	$-0.987540\pi$
$653$		−	2.00000i		−	0.0782660i	0.999234	−	0.0391330i	$-0.0124596\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$			8.00000i			0.312110i
$658$		0			0
$659$	22.0000			0.856998			0.428499	−	0.903542i	$-0.359042\pi$
$659$	22.0000			0.856998			0.428499	+	0.903542i	$0.359042\pi$
$660$		0			0
$661$	−16.0000			−0.622328			−0.311164	−	0.950356i	$-0.600719\pi$
$661$	−16.0000			−0.622328			−0.311164	+	0.950356i	$0.600719\pi$
$662$		0			0
$663$		−	20.0000i		−	0.776736i
$664$		0			0
$665$	8.00000	−	16.0000i	0.310227	−	0.620453i
$666$		0			0
$667$		−	5.00000i		−	0.193601i
$668$		0			0
$669$	28.0000			1.08254
$670$		0			0
$671$		0			0
$672$		0			0
$673$		−	24.0000i		−	0.925132i	−0.886585	−	0.462566i	$-0.846929\pi$
$673$		−	24.0000i		−	0.925132i	0.886585	−	0.462566i	$-0.153071\pi$
$674$		0			0
$675$	−16.0000	+	12.0000i	−0.615840	+	0.461880i
$676$		0			0
$677$		−	3.00000i		−	0.115299i	−0.998337	−	0.0576497i	$-0.981639\pi$
$677$		−	3.00000i		−	0.115299i	0.998337	−	0.0576497i	$-0.0183606\pi$
$678$		0			0
$679$	14.0000			0.537271
$680$		0			0
$681$		0			0
$682$		0			0
$683$			46.0000i			1.76014i	0.474843	+	0.880071i	$0.342505\pi$
$683$			46.0000i			1.76014i	−0.474843	+	0.880071i	$0.657495\pi$
$684$		0			0
$685$	−6.00000	+	12.0000i	−0.229248	+	0.458496i
$686$		0			0
$687$		−	8.00000i		−	0.305219i
$688$		0			0
$689$	2.00000			0.0761939
$690$		0			0
$691$	28.0000			1.06517			0.532585	−	0.846376i	$-0.321221\pi$
$691$	28.0000			1.06517			0.532585	+	0.846376i	$0.321221\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	18.0000	+	9.00000i	0.682779	+	0.341389i
$696$		0			0
$697$			35.0000i			1.32572i
$698$		0			0
$699$	12.0000			0.453882
$700$		0			0
$701$	−28.0000			−1.05755			−0.528773	−	0.848763i	$-0.677348\pi$
$701$	−28.0000			−1.05755			−0.528773	+	0.848763i	$0.677348\pi$
$702$		0			0
$703$			56.0000i			2.11208i
$704$		0			0
$705$	−8.00000	−	4.00000i	−0.301297	−	0.150649i
$706$		0			0
$707$		−	15.0000i		−	0.564133i
$708$		0			0
$709$	−4.00000			−0.150223			−0.0751116	−	0.997175i	$-0.523931\pi$
$709$	−4.00000			−0.150223			−0.0751116	+	0.997175i	$0.523931\pi$
$710$		0			0
$711$	14.0000			0.525041
$712$		0			0
$713$		−	5.00000i		−	0.187251i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		−	2.00000i		−	0.0746914i
$718$		0			0
$719$	−45.0000			−1.67822			−0.839108	−	0.543964i	$-0.816923\pi$
$719$	−45.0000			−1.67822			−0.839108	+	0.543964i	$0.816923\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$			12.0000i			0.446285i
$724$		0			0
$725$	15.0000	+	20.0000i	0.557086	+	0.742781i
$726$		0			0
$727$		−	27.0000i		−	1.00137i	−0.865628	−	0.500687i	$-0.833081\pi$
$727$		−	27.0000i		−	1.00137i	0.865628	−	0.500687i	$-0.166919\pi$
$728$		0			0
$729$	−13.0000			−0.481481
$730$		0			0
$731$	20.0000			0.739727
$732$		0			0
$733$		−	7.00000i		−	0.258551i	−0.991609	−	0.129275i	$-0.958735\pi$
$733$		−	7.00000i		−	0.258551i	0.991609	−	0.129275i	$-0.0412651\pi$
$734$		0			0
$735$	−12.0000	+	24.0000i	−0.442627	+	0.885253i
$736$		0			0
$737$		0			0
$738$		0			0
$739$	7.00000			0.257499			0.128750	−	0.991677i	$-0.458904\pi$
$739$	7.00000			0.257499			0.128750	+	0.991677i	$0.458904\pi$
$740$		0			0
$741$	32.0000			1.17555
$742$		0			0
$743$		−	32.0000i		−	1.17397i	−0.809599	−	0.586983i	$-0.800316\pi$
$743$		−	32.0000i		−	1.17397i	0.809599	−	0.586983i	$-0.199684\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$			3.00000i			0.109764i
$748$		0			0
$749$	−9.00000			−0.328853
$750$		0			0
$751$	−14.0000			−0.510867			−0.255434	−	0.966827i	$-0.582218\pi$
$751$	−14.0000			−0.510867			−0.255434	+	0.966827i	$0.582218\pi$
$752$		0			0
$753$		−	36.0000i		−	1.31191i
$754$		0			0
$755$	−16.0000	−	8.00000i	−0.582300	−	0.291150i
$756$		0			0
$757$		−	35.0000i		−	1.27210i	−0.771649	−	0.636048i	$-0.780568\pi$
$757$		−	35.0000i		−	1.27210i	0.771649	−	0.636048i	$-0.219432\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	25.0000			0.906249			0.453125	−	0.891447i	$-0.350309\pi$
$761$	25.0000			0.906249			0.453125	+	0.891447i	$0.350309\pi$
$762$		0			0
$763$			18.0000i			0.651644i
$764$		0			0
$765$	−5.00000	+	10.0000i	−0.180775	+	0.361551i
$766$		0			0
$767$		−	6.00000i		−	0.216647i
$768$		0			0
$769$	−10.0000			−0.360609			−0.180305	−	0.983611i	$-0.557708\pi$
$769$	−10.0000			−0.360609			−0.180305	+	0.983611i	$0.557708\pi$
$770$		0			0
$771$	−44.0000			−1.58462
$772$		0			0
$773$			42.0000i			1.51064i	0.655359	+	0.755318i	$0.272517\pi$
$773$			42.0000i			1.51064i	−0.655359	+	0.755318i	$0.727483\pi$
$774$		0			0
$775$	15.0000	+	20.0000i	0.538816	+	0.718421i
$776$		0			0
$777$			14.0000i			0.502247i
$778$		0			0
$779$	−56.0000			−2.00641
$780$		0			0
$781$		0			0
$782$		0			0
$783$			20.0000i			0.714742i
$784$		0			0
$785$	3.00000	−	6.00000i	0.107075	−	0.214149i
$786$		0			0
$787$		−	17.0000i		−	0.605985i	−0.952993	−	0.302992i	$-0.902014\pi$
$787$		−	17.0000i		−	0.605985i	0.952993	−	0.302992i	$-0.0979856\pi$
$788$		0			0
$789$	26.0000			0.925625
$790$		0			0
$791$	1.00000			0.0355559
$792$		0			0
$793$			12.0000i			0.426132i
$794$		0			0
$795$	−4.00000	−	2.00000i	−0.141865	−	0.0709327i
$796$		0			0
$797$		−	15.0000i		−	0.531327i	−0.964066	−	0.265664i	$-0.914409\pi$
$797$		−	15.0000i		−	0.531327i	0.964066	−	0.265664i	$-0.0855911\pi$
$798$		0			0
$799$	10.0000			0.353775
$800$		0			0
$801$	−14.0000			−0.494666
$802$		0			0
$803$		0			0
$804$		0			0
$805$	−2.00000	−	1.00000i	−0.0704907	−	0.0352454i
$806$		0			0
$807$			30.0000i			1.05605i
$808$		0			0
$809$	25.0000			0.878953			0.439477	−	0.898254i	$-0.355164\pi$
$809$	25.0000			0.878953			0.439477	+	0.898254i	$0.355164\pi$
$810$		0			0
$811$	−31.0000			−1.08856			−0.544279	−	0.838905i	$-0.683197\pi$
$811$	−31.0000			−1.08856			−0.544279	+	0.838905i	$0.683197\pi$
$812$		0			0
$813$			30.0000i			1.05215i
$814$		0			0
$815$	−24.0000	+	48.0000i	−0.840683	+	1.68137i
$816$		0			0
$817$			32.0000i			1.11954i
$818$		0			0
$819$	2.00000			0.0698857
$820$		0			0
$821$	30.0000			1.04701			0.523504	−	0.852023i	$-0.324625\pi$
$821$	30.0000			1.04701			0.523504	+	0.852023i	$0.324625\pi$
$822$		0			0
$823$		−	34.0000i		−	1.18517i	−0.805510	−	0.592583i	$-0.798108\pi$
$823$		−	34.0000i		−	1.18517i	0.805510	−	0.592583i	$-0.201892\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			9.00000i			0.312961i	0.987681	+	0.156480i	$0.0500148\pi$
$827$			9.00000i			0.312961i	−0.987681	+	0.156480i	$0.949985\pi$
$828$		0			0
$829$	−37.0000			−1.28506			−0.642532	−	0.766259i	$-0.722116\pi$
$829$	−37.0000			−1.28506			−0.642532	+	0.766259i	$0.722116\pi$
$830$		0			0
$831$	52.0000			1.80386
$832$		0			0
$833$		−	30.0000i		−	1.03944i
$834$		0			0
$835$	16.0000	−	32.0000i	0.553703	−	1.10741i
$836$		0			0
$837$			20.0000i			0.691301i
$838$		0			0
$839$	14.0000			0.483334			0.241667	−	0.970359i	$-0.422306\pi$
$839$	14.0000			0.483334			0.241667	+	0.970359i	$0.422306\pi$
$840$		0			0
$841$	−4.00000			−0.137931
$842$		0			0
$843$		−	24.0000i		−	0.826604i
$844$		0			0
$845$	18.0000	+	9.00000i	0.619219	+	0.309609i
$846$		0			0
$847$			11.0000i			0.377964i
$848$		0			0
$849$	−22.0000			−0.755038
$850$		0			0
$851$	7.00000			0.239957
$852$		0			0
$853$		−	8.00000i		−	0.273915i	−0.990577	−	0.136957i	$-0.956268\pi$
$853$		−	8.00000i		−	0.273915i	0.990577	−	0.136957i	$-0.0437323\pi$
$854$		0			0
$855$	−16.0000	−	8.00000i	−0.547188	−	0.273594i
$856$		0			0
$857$			24.0000i			0.819824i	0.912125	+	0.409912i	$0.134441\pi$
$857$			24.0000i			0.819824i	−0.912125	+	0.409912i	$0.865559\pi$
$858$		0			0
$859$	43.0000			1.46714			0.733571	−	0.679613i	$-0.237852\pi$
$859$	43.0000			1.46714			0.733571	+	0.679613i	$0.237852\pi$
$860$		0			0
$861$	−14.0000			−0.477119
$862$		0			0
$863$		−	18.0000i		−	0.612727i	−0.951915	−	0.306364i	$-0.900888\pi$
$863$		−	18.0000i		−	0.612727i	0.951915	−	0.306364i	$-0.0991123\pi$
$864$		0			0
$865$	6.00000	−	12.0000i	0.204006	−	0.408012i
$866$		0			0
$867$		−	16.0000i		−	0.543388i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−26.0000			−0.880976
$872$		0			0
$873$		−	14.0000i		−	0.473828i
$874$		0			0
$875$	11.0000	−	2.00000i	0.371868	−	0.0676123i
$876$		0			0
$877$			8.00000i			0.270141i	0.990836	+	0.135070i	$0.0431261\pi$
$877$			8.00000i			0.270141i	−0.990836	+	0.135070i	$0.956874\pi$
$878$		0			0
$879$	58.0000			1.95629
$880$		0			0
$881$	54.0000			1.81931			0.909653	−	0.415369i	$-0.136347\pi$
$881$	54.0000			1.81931			0.909653	+	0.415369i	$0.136347\pi$
$882$		0			0
$883$		−	20.0000i		−	0.673054i	−0.941674	−	0.336527i	$-0.890748\pi$
$883$		−	20.0000i		−	0.673054i	0.941674	−	0.336527i	$-0.109252\pi$
$884$		0			0
$885$	−6.00000	+	12.0000i	−0.201688	+	0.403376i
$886$		0			0
$887$		−	42.0000i		−	1.41022i	−0.709097	−	0.705111i	$-0.750897\pi$
$887$		−	42.0000i		−	1.41022i	0.709097	−	0.705111i	$-0.249103\pi$
$888$		0			0
$889$	20.0000			0.670778
$890$		0			0
$891$		0			0
$892$		0			0
$893$			16.0000i			0.535420i
$894$		0			0
$895$	−8.00000	−	4.00000i	−0.267411	−	0.133705i
$896$		0			0
$897$		−	4.00000i		−	0.133556i
$898$		0			0
$899$	25.0000			0.833797
$900$		0			0
$901$	5.00000			0.166574
$902$		0			0
$903$			8.00000i			0.266223i
$904$		0			0
$905$	−28.0000	−	14.0000i	−0.930751	−	0.465376i
$906$		0			0
$907$		−	9.00000i		−	0.298840i	−0.988774	−	0.149420i	$-0.952259\pi$
$907$		−	9.00000i		−	0.298840i	0.988774	−	0.149420i	$-0.0477407\pi$
$908$		0			0
$909$	−15.0000			−0.497519
$910$		0			0
$911$	12.0000			0.397578			0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000			0.397578			0.198789	+	0.980042i	$0.436299\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$	12.0000	−	24.0000i	0.396708	−	0.793416i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	4.00000			0.131948			0.0659739	−	0.997821i	$-0.478985\pi$
$919$	4.00000			0.131948			0.0659739	+	0.997821i	$0.478985\pi$
$920$		0			0
$921$	28.0000			0.922631
$922$		0			0
$923$			26.0000i			0.855800i
$924$		0			0
$925$	−28.0000	+	21.0000i	−0.920634	+	0.690476i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−35.0000			−1.14831			−0.574156	−	0.818746i	$-0.694670\pi$
$929$	−35.0000			−1.14831			−0.574156	+	0.818746i	$0.694670\pi$
$930$		0			0
$931$	48.0000			1.57314
$932$		0			0
$933$		−	8.00000i		−	0.261908i
$934$		0			0
$935$		0			0
$936$		0			0
$937$			54.0000i			1.76410i	0.471153	+	0.882052i	$0.343838\pi$
$937$			54.0000i			1.76410i	−0.471153	+	0.882052i	$0.656162\pi$
$938$		0			0
$939$	42.0000			1.37062
$940$		0			0
$941$	2.00000			0.0651981			0.0325991	−	0.999469i	$-0.489622\pi$
$941$	2.00000			0.0651981			0.0325991	+	0.999469i	$0.489622\pi$
$942$		0			0
$943$			7.00000i			0.227951i
$944$		0			0
$945$	8.00000	+	4.00000i	0.260240	+	0.130120i
$946$		0			0
$947$		−	14.0000i		−	0.454939i	−0.973785	−	0.227469i	$-0.926955\pi$
$947$		−	14.0000i		−	0.454939i	0.973785	−	0.227469i	$-0.0730452\pi$
$948$		0			0
$949$	−16.0000			−0.519382
$950$		0			0
$951$	48.0000			1.55651
$952$		0			0
$953$		−	54.0000i		−	1.74923i	−0.484817	−	0.874616i	$-0.661114\pi$
$953$		−	54.0000i		−	1.74923i	0.484817	−	0.874616i	$-0.338886\pi$
$954$		0			0
$955$	16.0000	+	8.00000i	0.517748	+	0.258874i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	6.00000			0.193750
$960$		0			0
$961$	−6.00000			−0.193548
$962$		0			0
$963$			9.00000i			0.290021i
$964$		0			0
$965$	−12.0000	+	24.0000i	−0.386294	+	0.772587i
$966$		0			0
$967$			18.0000i			0.578841i	0.957202	+	0.289420i	$0.0934626\pi$
$967$			18.0000i			0.578841i	−0.957202	+	0.289420i	$0.906537\pi$
$968$		0			0
$969$	80.0000			2.56997
$970$		0			0
$971$	6.00000			0.192549			0.0962746	−	0.995355i	$-0.469307\pi$
$971$	6.00000			0.192549			0.0962746	+	0.995355i	$0.469307\pi$
$972$		0			0
$973$		−	9.00000i		−	0.288527i
$974$		0			0
$975$	12.0000	+	16.0000i	0.384308	+	0.512410i
$976$		0			0
$977$		−	39.0000i		−	1.24772i	−0.781536	−	0.623860i	$-0.785563\pi$
$977$		−	39.0000i		−	1.24772i	0.781536	−	0.623860i	$-0.214437\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	18.0000			0.574696
$982$		0			0
$983$			21.0000i			0.669796i	0.942254	+	0.334898i	$0.108702\pi$
$983$			21.0000i			0.669796i	−0.942254	+	0.334898i	$0.891298\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$			4.00000i			0.127321i
$988$		0			0
$989$	4.00000			0.127193
$990$		0			0
$991$	−19.0000			−0.603555			−0.301777	−	0.953378i	$-0.597580\pi$
$991$	−19.0000			−0.603555			−0.301777	+	0.953378i	$0.597580\pi$
$992$		0			0
$993$			10.0000i			0.317340i
$994$		0			0
$995$	4.00000	+	2.00000i	0.126809	+	0.0634043i
$996$		0			0
$997$			20.0000i			0.633406i	0.948525	+	0.316703i	$0.102576\pi$
$997$			20.0000i			0.633406i	−0.948525	+	0.316703i	$0.897424\pi$
$998$		0			0
$999$	−28.0000			−0.885881

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.2.e.b.369.2		2
4.3	odd	2		115.2.b.a.24.2	yes	2
5.2	odd	4		9200.2.a.bg.1.1		1
5.3	odd	4		9200.2.a.g.1.1		1
5.4	even	2	inner	1840.2.e.b.369.1		2
12.11	even	2		1035.2.b.a.829.1		2
20.3	even	4		575.2.a.e.1.1		1
20.7	even	4		575.2.a.a.1.1		1
20.19	odd	2		115.2.b.a.24.1	✓	2
60.23	odd	4		5175.2.a.a.1.1		1
60.47	odd	4		5175.2.a.z.1.1		1
60.59	even	2		1035.2.b.a.829.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.2.b.a.24.1	✓	2	20.19	odd	2
115.2.b.a.24.2	yes	2	4.3	odd	2
575.2.a.a.1.1		1	20.7	even	4
575.2.a.e.1.1		1	20.3	even	4
1035.2.b.a.829.1		2	12.11	even	2
1035.2.b.a.829.2		2	60.59	even	2
1840.2.e.b.369.1		2	5.4	even	2	inner
1840.2.e.b.369.2		2	1.1	even	1	trivial
5175.2.a.a.1.1		1	60.23	odd	4
5175.2.a.z.1.1		1	60.47	odd	4
9200.2.a.g.1.1		1	5.3	odd	4
9200.2.a.bg.1.1		1	5.2	odd	4

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:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1840.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{3} +(2.00000 + 1.00000i) q^{5} -1.00000i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+2.00000i q^{3} +(2.00000 + 1.00000i) q^{5} -1.00000i q^{7} -1.00000 q^{9} -2.00000i q^{13} +(-2.00000 + 4.00000i) q^{15} -5.00000i q^{17} +8.00000 q^{19} +2.00000 q^{21} -1.00000i q^{23} +(3.00000 + 4.00000i) q^{25} +4.00000i q^{27} +5.00000 q^{29} +5.00000 q^{31} +(1.00000 - 2.00000i) q^{35} +7.00000i q^{37} +4.00000 q^{39} -7.00000 q^{41} +4.00000i q^{43} +(-2.00000 - 1.00000i) q^{45} +2.00000i q^{47} +6.00000 q^{49} +10.0000 q^{51} +1.00000i q^{53} +16.0000i q^{57} +3.00000 q^{59} -6.00000 q^{61} +1.00000i q^{63} +(2.00000 - 4.00000i) q^{65} -13.0000i q^{67} +2.00000 q^{69} -13.0000 q^{71} -8.00000i q^{73} +(-8.00000 + 6.00000i) q^{75} -14.0000 q^{79} -11.0000 q^{81} -3.00000i q^{83} +(5.00000 - 10.0000i) q^{85} +10.0000i q^{87} +14.0000 q^{89} -2.00000 q^{91} +10.0000i q^{93} +(16.0000 + 8.00000i) q^{95} +14.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{5} - 2 q^{9}+O(q^{10}) \) 2 * q + 4 * q^5 - 2 * q^9	\( 2 q + 4 q^{5} - 2 q^{9} - 4 q^{15} + 16 q^{19} + 4 q^{21} + 6 q^{25} + 10 q^{29} + 10 q^{31} + 2 q^{35} + 8 q^{39} - 14 q^{41} - 4 q^{45} + 12 q^{49} + 20 q^{51} + 6 q^{59} - 12 q^{61} + 4 q^{65} + 4 q^{69} - 26 q^{71} - 16 q^{75} - 28 q^{79} - 22 q^{81} + 10 q^{85} + 28 q^{89} - 4 q^{91} + 32 q^{95}+O(q^{100}) \) 2 * q + 4 * q^5 - 2 * q^9 - 4 * q^15 + 16 * q^19 + 4 * q^21 + 6 * q^25 + 10 * q^29 + 10 * q^31 + 2 * q^35 + 8 * q^39 - 14 * q^41 - 4 * q^45 + 12 * q^49 + 20 * q^51 + 6 * q^59 - 12 * q^61 + 4 * q^65 + 4 * q^69 - 26 * q^71 - 16 * q^75 - 28 * q^79 - 22 * q^81 + 10 * q^85 + 28 * q^89 - 4 * q^91 + 32 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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