LMFDB - Embedded newform 1840.2.i.c.1471.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,2,Mod(1471,1840)]

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([1, 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("1840.1471");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1840.i (of order $2$, degree $1$, 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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 8x^{14} + 44x^{12} + 162x^{10} + 404x^{8} - 84x^{6} - 79x^{4} - 4x^{2} + 16 $ x^16 + 8x^14 + 44x^12 + 162x^10 + 404x^8 - 84x^6 - 79x^4 - 4*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1471.12
Root			$-1.47123 - 1.54824i$ of defining polynomial
Character	$\chi$	$=$	1840.1471
Dual form			1840.2.i.c.1471.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.21040i q^{3} +1.00000i q^{5} -4.59480 q^{7} +1.53493 q^{9} +O(q^{10})$	$q+1.21040i q^{3} +1.00000i q^{5} -4.59480 q^{7} +1.53493 q^{9} +3.99480 q^{11} -0.942747 q^{13} -1.21040 q^{15} -4.91920i q^{17} -8.05890 q^{19} -5.56155i q^{21} +(-1.82081 - 4.43674i) q^{23} -1.00000 q^{25} +5.48909i q^{27} -8.30039 q^{29} +5.27450i q^{31} +4.83531i q^{33} -4.59480i q^{35} +3.06985i q^{37} -1.14110i q^{39} +9.21958 q^{41} -7.24432 q^{43} +1.53493i q^{45} -12.4495i q^{47} +14.1122 q^{49} +5.95421 q^{51} -3.64543i q^{53} +3.99480i q^{55} -9.75451i q^{57} -3.46410i q^{59} -11.9278i q^{61} -7.05267 q^{63} -0.942747i q^{65} +6.02351 q^{67} +(5.37024 - 2.20391i) q^{69} -8.00001i q^{71} -4.90117 q^{73} -1.21040i q^{75} -18.3553 q^{77} +1.22081 q^{79} -2.03923 q^{81} -7.10572 q^{83} +4.91920 q^{85} -10.0468i q^{87} -4.19296i q^{89} +4.33173 q^{91} -6.38427 q^{93} -8.05890i q^{95} -3.27376i q^{97} +6.13172 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 8 q^{13} - 16 q^{25} - 84 q^{29} + 24 q^{41} + 36 q^{49} - 12 q^{69} + 68 q^{73} - 48 q^{77} + 32 q^{81} + 4 q^{85} - 52 q^{93}+O(q^{100}) $ 16 * q - 8 * q^13 - 16 * q^25 - 84 * q^29 + 24 * q^41 + 36 * q^49 - 12 * q^69 + 68 * q^73 - 48 * q^77 + 32 * q^81 + 4 * q^85 - 52 * q^93

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
$3$			1.21040i			0.698826i	0.936969	+	0.349413i	$0.113619\pi$
$3$			1.21040i			0.698826i	−0.936969	+	0.349413i	$0.886381\pi$
$4$		0			0
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$	−4.59480			−1.73667			−0.868335	−	0.495978i	$-0.834810\pi$
$7$	−4.59480			−1.73667			−0.868335	+	0.495978i	$0.834810\pi$
$8$		0			0
$9$	1.53493			0.511642
$10$		0			0
$11$	3.99480			1.20448			0.602239	−	0.798316i	$-0.294276\pi$
$11$	3.99480			1.20448			0.602239	+	0.798316i	$0.294276\pi$
$12$		0			0
$13$	−0.942747			−0.261471			−0.130735	−	0.991417i	$-0.541734\pi$
$13$	−0.942747			−0.261471			−0.130735	+	0.991417i	$0.541734\pi$
$14$		0			0
$15$	−1.21040			−0.312525
$16$		0			0
$17$		−	4.91920i		−	1.19308i	−0.802583	−	0.596540i	$-0.796542\pi$
$17$		−	4.91920i		−	1.19308i	0.802583	−	0.596540i	$-0.203458\pi$
$18$		0			0
$19$	−8.05890			−1.84884			−0.924419	−	0.381378i	$-0.875449\pi$
$19$	−8.05890			−1.84884			−0.924419	+	0.381378i	$0.875449\pi$
$20$		0			0
$21$		−	5.56155i		−	1.21363i
$22$		0			0
$23$	−1.82081	−	4.43674i	−0.379665	−	0.925124i
$23$	−1.82081	−	4.43674i	−0.379665	−	0.925124i
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$			5.48909i			1.05637i
$28$		0			0
$29$	−8.30039			−1.54134			−0.770672	−	0.637232i	$-0.780079\pi$
$29$	−8.30039			−1.54134			−0.770672	+	0.637232i	$0.780079\pi$
$30$		0			0
$31$			5.27450i			0.947328i	0.880706	+	0.473664i	$0.157069\pi$
$31$			5.27450i			0.947328i	−0.880706	+	0.473664i	$0.842931\pi$
$32$		0			0
$33$			4.83531i			0.841720i
$34$		0			0
$35$		−	4.59480i		−	0.776662i
$36$		0			0
$37$			3.06985i			0.504680i	0.967639	+	0.252340i	$0.0812002\pi$
$37$			3.06985i			0.504680i	−0.967639	+	0.252340i	$0.918800\pi$
$38$		0			0
$39$		−	1.14110i		−	0.182723i
$40$		0			0
$41$	9.21958			1.43986			0.719929	−	0.694048i	$-0.244175\pi$
$41$	9.21958			1.43986			0.719929	+	0.694048i	$0.244175\pi$
$42$		0			0
$43$	−7.24432			−1.10475			−0.552374	−	0.833596i	$-0.686278\pi$
$43$	−7.24432			−1.10475			−0.552374	+	0.833596i	$0.686278\pi$
$44$		0			0
$45$			1.53493i			0.228813i
$46$		0			0
$47$		−	12.4495i		−	1.81595i	−0.419026	−	0.907974i	$-0.637628\pi$
$47$		−	12.4495i		−	1.81595i	0.419026	−	0.907974i	$-0.362372\pi$
$48$		0			0
$49$	14.1122			2.01602
$50$		0			0
$51$	5.95421			0.833756
$52$		0			0
$53$		−	3.64543i		−	0.500739i	−0.968150	−	0.250369i	$-0.919448\pi$
$53$		−	3.64543i		−	0.500739i	0.968150	−	0.250369i	$-0.0805521\pi$
$54$		0			0
$55$			3.99480i			0.538659i
$56$		0			0
$57$		−	9.75451i		−	1.29202i
$58$		0			0
$59$		−	3.46410i		−	0.450988i	−0.974245	−	0.225494i	$-0.927600\pi$
$59$		−	3.46410i		−	0.450988i	0.974245	−	0.225494i	$-0.0723995\pi$
$60$		0			0
$61$		−	11.9278i		−	1.52720i	−0.645691	−	0.763599i	$-0.723431\pi$
$61$		−	11.9278i		−	1.52720i	0.645691	−	0.763599i	$-0.276569\pi$
$62$		0			0
$63$	−7.05267			−0.888553
$64$		0			0
$65$		−	0.942747i		−	0.116933i
$66$		0			0
$67$	6.02351			0.735888			0.367944	−	0.929848i	$-0.380062\pi$
$67$	6.02351			0.735888			0.367944	+	0.929848i	$0.380062\pi$
$68$		0			0
$69$	5.37024	−	2.20391i	0.646501	−	0.265320i
$70$		0			0
$71$		−	8.00001i		−	0.949426i	−0.880141	−	0.474713i	$-0.842552\pi$
$71$		−	8.00001i		−	0.949426i	0.880141	−	0.474713i	$-0.157448\pi$
$72$		0			0
$73$	−4.90117			−0.573638			−0.286819	−	0.957985i	$-0.592598\pi$
$73$	−4.90117			−0.573638			−0.286819	+	0.957985i	$0.592598\pi$
$74$		0			0
$75$		−	1.21040i		−	0.139765i
$76$		0			0
$77$	−18.3553			−2.09178
$78$		0			0
$79$	1.22081			0.137352			0.0686760	−	0.997639i	$-0.478123\pi$
$79$	1.22081			0.137352			0.0686760	+	0.997639i	$0.478123\pi$
$80$		0			0
$81$	−2.03923			−0.226581
$82$		0			0
$83$	−7.10572			−0.779954			−0.389977	−	0.920825i	$-0.627517\pi$
$83$	−7.10572			−0.779954			−0.389977	+	0.920825i	$0.627517\pi$
$84$		0			0
$85$	4.91920			0.533562
$86$		0			0
$87$		−	10.0468i		−	1.07713i
$88$		0			0
$89$		−	4.19296i		−	0.444453i	−0.974995	−	0.222226i	$-0.928668\pi$
$89$		−	4.19296i		−	0.444453i	0.974995	−	0.222226i	$-0.0713324\pi$
$90$		0			0
$91$	4.33173			0.454089
$92$		0			0
$93$	−6.38427			−0.662018
$94$		0			0
$95$		−	8.05890i		−	0.826825i
$96$		0			0
$97$		−	3.27376i		−	0.332400i	−0.986092	−	0.166200i	$-0.946850\pi$
$97$		−	3.27376i		−	0.332400i	0.986092	−	0.166200i	$-0.0531498\pi$
$98$		0			0
$99$	6.13172			0.616261
$100$		0			0
$101$	7.83839			0.779949			0.389974	−	0.920826i	$-0.372484\pi$
$101$	7.83839			0.779949			0.389974	+	0.920826i	$0.372484\pi$
$102$		0			0
$103$	−13.0069			−1.28161			−0.640803	−	0.767705i	$-0.721398\pi$
$103$	−13.0069			−1.28161			−0.640803	+	0.767705i	$0.721398\pi$
$104$		0			0
$105$	5.56155			0.542752
$106$		0			0
$107$	4.17047			0.403174			0.201587	−	0.979471i	$-0.435390\pi$
$107$	4.17047			0.403174			0.201587	+	0.979471i	$0.435390\pi$
$108$		0			0
$109$		−	8.67371i		−	0.830790i	−0.909641	−	0.415395i	$-0.863643\pi$
$109$		−	8.67371i		−	0.830790i	0.909641	−	0.415395i	$-0.136357\pi$
$110$		0			0
$111$	−3.71576			−0.352684
$112$		0			0
$113$		−	6.07845i		−	0.571813i	−0.958258	−	0.285906i	$-0.907705\pi$
$113$		−	6.07845i		−	0.571813i	0.958258	−	0.285906i	$-0.0922946\pi$
$114$		0			0
$115$	4.43674	−	1.82081i	0.413728	−	0.169791i
$116$		0			0
$117$	−1.44705			−0.133779
$118$		0			0
$119$			22.6027i			2.07199i
$120$		0			0
$121$	4.95842			0.450766
$122$		0			0
$123$			11.1594i			1.00621i
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$			8.15486i			0.723627i	0.932251	+	0.361813i	$0.117842\pi$
$127$			8.15486i			0.723627i	−0.932251	+	0.361813i	$0.882158\pi$
$128$		0			0
$129$		−	8.76854i		−	0.772027i
$130$		0			0
$131$		−	9.70988i		−	0.848356i	−0.905579	−	0.424178i	$-0.860563\pi$
$131$		−	9.70988i		−	0.848356i	0.905579	−	0.424178i	$-0.139437\pi$
$132$		0			0
$133$	37.0290			3.21082
$134$		0			0
$135$	−5.48909			−0.472425
$136$		0			0
$137$		−	16.4275i		−	1.40350i	−0.712425	−	0.701748i	$-0.752403\pi$
$137$		−	16.4275i		−	1.40350i	0.712425	−	0.701748i	$-0.247597\pi$
$138$		0			0
$139$			17.2911i			1.46661i	0.679897	+	0.733307i	$0.262024\pi$
$139$			17.2911i			1.46661i	−0.679897	+	0.733307i	$0.737976\pi$
$140$		0			0
$141$	15.0689			1.26903
$142$		0			0
$143$	−3.76608			−0.314936
$144$		0			0
$145$		−	8.30039i		−	0.689310i
$146$		0			0
$147$			17.0814i			1.40885i
$148$		0			0
$149$			15.8439i			1.29798i	0.760795	+	0.648992i	$0.224809\pi$
$149$			15.8439i			1.29798i	−0.760795	+	0.648992i	$0.775191\pi$
$150$		0			0
$151$		−	1.07180i		−	0.0872221i	−0.999049	−	0.0436111i	$-0.986114\pi$
$151$		−	1.07180i		−	0.0872221i	0.999049	−	0.0436111i	$-0.0138862\pi$
$152$		0			0
$153$		−	7.55060i		−	0.610430i
$154$		0			0
$155$	−5.27450			−0.423658
$156$		0			0
$157$			8.07845i			0.644731i	0.946615	+	0.322365i	$0.104478\pi$
$157$			8.07845i			0.644731i	−0.946615	+	0.322365i	$0.895522\pi$
$158$		0			0
$159$	4.41244			0.349929
$160$		0			0
$161$	8.36624	+	20.3859i	0.659352	+	1.60664i
$162$		0			0
$163$		−	13.2641i		−	1.03892i	−0.854493	−	0.519462i	$-0.826132\pi$
$163$		−	13.2641i		−	1.03892i	0.854493	−	0.519462i	$-0.173868\pi$
$164$		0			0
$165$	−4.83531			−0.376429
$166$		0			0
$167$			16.7245i			1.29418i	0.762414	+	0.647090i	$0.224014\pi$
$167$			16.7245i			1.29418i	−0.762414	+	0.647090i	$0.775986\pi$
$168$		0			0
$169$	−12.1112			−0.931633
$170$		0			0
$171$	−12.3698			−0.945943
$172$		0			0
$173$	−11.6877			−0.888602			−0.444301	−	0.895878i	$-0.646548\pi$
$173$	−11.6877			−0.888602			−0.444301	+	0.895878i	$0.646548\pi$
$174$		0			0
$175$	4.59480			0.347334
$176$		0			0
$177$	4.19296			0.315162
$178$		0			0
$179$			8.66930i			0.647974i	0.946062	+	0.323987i	$0.105023\pi$
$179$			8.66930i			0.647974i	−0.946062	+	0.323987i	$0.894977\pi$
$180$		0			0
$181$			15.4973i			1.15191i	0.817482	+	0.575954i	$0.195369\pi$
$181$			15.4973i			1.15191i	−0.817482	+	0.575954i	$0.804631\pi$
$182$		0			0
$183$	14.4374			1.06725
$184$		0			0
$185$	−3.06985			−0.225700
$186$		0			0
$187$		−	19.6512i		−	1.43704i
$188$		0			0
$189$		−	25.2212i		−	1.83457i
$190$		0			0
$191$	−5.84599			−0.423001			−0.211501	−	0.977378i	$-0.567835\pi$
$191$	−5.84599			−0.423001			−0.211501	+	0.977378i	$0.567835\pi$
$192$		0			0
$193$	−17.3702			−1.25034			−0.625169	−	0.780490i	$-0.714970\pi$
$193$	−17.3702			−1.25034			−0.625169	+	0.780490i	$0.714970\pi$
$194$		0			0
$195$	1.14110			0.0817161
$196$		0			0
$197$	−4.75051			−0.338460			−0.169230	−	0.985577i	$-0.554128\pi$
$197$	−4.75051			−0.338460			−0.169230	+	0.985577i	$0.554128\pi$
$198$		0			0
$199$	−23.8947			−1.69385			−0.846924	−	0.531715i	$-0.821548\pi$
$199$	−23.8947			−1.69385			−0.846924	+	0.531715i	$0.821548\pi$
$200$		0			0
$201$			7.29087i			0.514258i
$202$		0			0
$203$	38.1386			2.67680
$204$		0			0
$205$			9.21958i			0.643924i
$206$		0			0
$207$	−2.79481	−	6.81007i	−0.194252	−	0.473332i
$208$		0			0
$209$	−32.1937			−2.22688
$210$		0			0
$211$			8.16711i			0.562247i	0.959672	+	0.281124i	$0.0907071\pi$
$211$			8.16711i			0.562247i	−0.959672	+	0.281124i	$0.909293\pi$
$212$		0			0
$213$	9.68323			0.663484
$214$		0			0
$215$		−	7.24432i		−	0.494058i
$216$		0			0
$217$		−	24.2353i		−	1.64520i
$218$		0			0
$219$		−	5.93239i		−	0.400873i
$220$		0			0
$221$			4.63756i			0.311956i
$222$		0			0
$223$		−	0.177514i		−	0.0118872i	−0.999982	−	0.00594362i	$-0.998108\pi$
$223$		−	0.177514i		−	0.0118872i	0.999982	−	0.00594362i	$-0.00189192\pi$
$224$		0			0
$225$	−1.53493			−0.102328
$226$		0			0
$227$	0.0922387			0.00612210			0.00306105	−	0.999995i	$-0.499026\pi$
$227$	0.0922387			0.00612210			0.00306105	+	0.999995i	$0.499026\pi$
$228$		0			0
$229$			23.5875i			1.55870i	0.626586	+	0.779352i	$0.284452\pi$
$229$			23.5875i			1.55870i	−0.626586	+	0.779352i	$0.715548\pi$
$230$		0			0
$231$		−	22.2173i		−	1.46179i
$232$		0			0
$233$	3.42349			0.224281			0.112140	−	0.993692i	$-0.464229\pi$
$233$	3.42349			0.224281			0.112140	+	0.993692i	$0.464229\pi$
$234$		0			0
$235$	12.4495			0.812117
$236$		0			0
$237$			1.47767i			0.0959851i
$238$		0			0
$239$		−	20.5985i		−	1.33241i	−0.745769	−	0.666204i	$-0.767918\pi$
$239$		−	20.5985i		−	1.33241i	0.745769	−	0.666204i	$-0.232082\pi$
$240$		0			0
$241$		−	6.43917i		−	0.414783i	−0.978258	−	0.207392i	$-0.933503\pi$
$241$		−	6.43917i		−	0.414783i	0.978258	−	0.207392i	$-0.0664974\pi$
$242$		0			0
$243$			13.9990i			0.898035i
$244$		0			0
$245$			14.1122i			0.901592i
$246$		0			0
$247$	7.59750			0.483417
$248$		0			0
$249$		−	8.60078i		−	0.545052i
$250$		0			0
$251$	−3.84951			−0.242979			−0.121490	−	0.992593i	$-0.538767\pi$
$251$	−3.84951			−0.242979			−0.121490	+	0.992593i	$0.538767\pi$
$252$		0			0
$253$	−7.27376	−	17.7239i	−0.457298	−	1.11429i
$254$		0			0
$255$			5.95421i			0.372867i
$256$		0			0
$257$	−28.1701			−1.75720			−0.878602	−	0.477555i	$-0.841523\pi$
$257$	−28.1701			−1.75720			−0.878602	+	0.477555i	$0.841523\pi$
$258$		0			0
$259$		−	14.1053i		−	0.876463i
$260$		0			0
$261$	−12.7405			−0.788616
$262$		0			0
$263$	23.0831			1.42336			0.711682	−	0.702502i	$-0.247934\pi$
$263$	23.0831			1.42336			0.711682	+	0.702502i	$0.247934\pi$
$264$		0			0
$265$	3.64543			0.223937
$266$		0			0
$267$	5.07517			0.310595
$268$		0			0
$269$	27.5552			1.68007			0.840035	−	0.542532i	$-0.182534\pi$
$269$	27.5552			1.68007			0.840035	+	0.542532i	$0.182534\pi$
$270$		0			0
$271$		−	30.2924i		−	1.84013i	−0.391760	−	0.920067i	$-0.628134\pi$
$271$		−	30.2924i		−	1.84013i	0.391760	−	0.920067i	$-0.371866\pi$
$272$		0			0
$273$			5.24314i			0.317329i
$274$		0			0
$275$	−3.99480			−0.240895
$276$		0			0
$277$	−30.7396			−1.84696			−0.923481	−	0.383645i	$-0.874669\pi$
$277$	−30.7396			−1.84696			−0.923481	+	0.383645i	$0.874669\pi$
$278$		0			0
$279$			8.09597i			0.484693i
$280$		0			0
$281$			7.75994i			0.462919i	0.972845	+	0.231460i	$0.0743501\pi$
$281$			7.75994i			0.462919i	−0.972845	+	0.231460i	$0.925650\pi$
$282$		0			0
$283$	12.0215			0.714602			0.357301	−	0.933989i	$-0.383697\pi$
$283$	12.0215			0.714602			0.357301	+	0.933989i	$0.383697\pi$
$284$		0			0
$285$	9.75451			0.577807
$286$		0			0
$287$	−42.3621			−2.50056
$288$		0			0
$289$	−7.19848			−0.423440
$290$		0			0
$291$	3.96257			0.232290
$292$		0			0
$293$			4.86890i			0.284444i	0.989835	+	0.142222i	$0.0454247\pi$
$293$			4.86890i			0.284444i	−0.989835	+	0.142222i	$0.954575\pi$
$294$		0			0
$295$	3.46410			0.201688
$296$		0			0
$297$			21.9278i			1.27238i
$298$		0			0
$299$	1.71656	+	4.18272i	0.0992713	+	0.241893i
$300$		0			0
$301$	33.2862			1.91858
$302$		0			0
$303$			9.48761i			0.545049i
$304$		0			0
$305$	11.9278			0.682984
$306$		0			0
$307$		−	3.72505i		−	0.212600i	−0.994334	−	0.106300i	$-0.966100\pi$
$307$		−	3.72505i		−	0.212600i	0.994334	−	0.106300i	$-0.0339003\pi$
$308$		0			0
$309$		−	15.7436i		−	0.895620i
$310$		0			0
$311$			12.6659i			0.718220i	0.933295	+	0.359110i	$0.116920\pi$
$311$			12.6659i			0.718220i	−0.933295	+	0.359110i	$0.883080\pi$
$312$		0			0
$313$			23.0596i			1.30341i	0.758474	+	0.651703i	$0.225945\pi$
$313$			23.0596i			1.30341i	−0.758474	+	0.651703i	$0.774055\pi$
$314$		0			0
$315$		−	7.05267i		−	0.397373i
$316$		0			0
$317$	1.79066			0.100573			0.0502867	−	0.998735i	$-0.483986\pi$
$317$	1.79066			0.100573			0.0502867	+	0.998735i	$0.483986\pi$
$318$		0			0
$319$	−33.1584			−1.85651
$320$		0			0
$321$			5.04795i			0.281749i
$322$		0			0
$323$			39.6433i			2.20581i
$324$		0			0
$325$	0.942747			0.0522942
$326$		0			0
$327$	10.4987			0.580578
$328$		0			0
$329$			57.2030i			3.15370i
$330$		0			0
$331$		−	34.6622i		−	1.90521i	−0.304211	−	0.952605i	$-0.598393\pi$
$331$		−	34.6622i		−	1.90521i	0.304211	−	0.952605i	$-0.401607\pi$
$332$		0			0
$333$			4.71199i			0.258216i
$334$		0			0
$335$			6.02351i			0.329099i
$336$		0			0
$337$			9.45800i			0.515210i	0.966250	+	0.257605i	$0.0829333\pi$
$337$			9.45800i			0.515210i	−0.966250	+	0.257605i	$0.917067\pi$
$338$		0			0
$339$	7.35737			0.399598
$340$		0			0
$341$			21.0706i			1.14104i
$342$		0			0
$343$	−32.6789			−1.76449
$344$		0			0
$345$	2.20391	+	5.37024i	0.118655	+	0.289124i
$346$		0			0
$347$			6.20287i			0.332987i	0.986043	+	0.166494i	$0.0532445\pi$
$347$			6.20287i			0.332987i	−0.986043	+	0.166494i	$0.946755\pi$
$348$		0			0
$349$	−30.4634			−1.63067			−0.815335	−	0.578989i	$-0.803447\pi$
$349$	−30.4634			−1.63067			−0.815335	+	0.578989i	$0.803447\pi$
$350$		0			0
$351$		−	5.17482i		−	0.276211i
$352$		0			0
$353$	−4.96242			−0.264123			−0.132061	−	0.991242i	$-0.542160\pi$
$353$	−4.96242			−0.264123			−0.132061	+	0.991242i	$0.542160\pi$
$354$		0			0
$355$	8.00001			0.424596
$356$		0			0
$357$	−27.3584			−1.44796
$358$		0			0
$359$	10.8396			0.572091			0.286045	−	0.958216i	$-0.407659\pi$
$359$	10.8396			0.572091			0.286045	+	0.958216i	$0.407659\pi$
$360$		0			0
$361$	45.9458			2.41820
$362$		0			0
$363$			6.00168i			0.315007i
$364$		0			0
$365$		−	4.90117i		−	0.256539i
$366$		0			0
$367$	5.00103			0.261051			0.130526	−	0.991445i	$-0.458333\pi$
$367$	5.00103			0.261051			0.130526	+	0.991445i	$0.458333\pi$
$368$		0			0
$369$	14.1514			0.736691
$370$		0			0
$371$			16.7500i			0.869618i
$372$		0			0
$373$		−	33.2690i		−	1.72260i	−0.508096	−	0.861301i	$-0.669650\pi$
$373$		−	33.2690i		−	1.72260i	0.508096	−	0.861301i	$-0.330350\pi$
$374$		0			0
$375$	1.21040			0.0625049
$376$		0			0
$377$	7.82516			0.403016
$378$		0			0
$379$	15.8749			0.815441			0.407721	−	0.913107i	$-0.366324\pi$
$379$	15.8749			0.815441			0.407721	+	0.913107i	$0.366324\pi$
$380$		0			0
$381$	−9.87066			−0.505689
$382$		0			0
$383$	27.2762			1.39375			0.696874	−	0.717193i	$-0.254574\pi$
$383$	27.2762			1.39375			0.696874	+	0.717193i	$0.254574\pi$
$384$		0			0
$385$		−	18.3553i		−	0.935472i
$386$		0			0
$387$	−11.1195			−0.565235
$388$		0			0
$389$			1.98105i			0.100443i	0.998738	+	0.0502216i	$0.0159927\pi$
$389$			1.98105i			0.100443i	−0.998738	+	0.0502216i	$0.984007\pi$
$390$		0			0
$391$	−21.8252	+	8.95691i	−1.10375	+	0.452970i
$392$		0			0
$393$	11.7529			0.592854
$394$		0			0
$395$			1.22081i			0.0614257i
$396$		0			0
$397$	−10.4824			−0.526096			−0.263048	−	0.964783i	$-0.584728\pi$
$397$	−10.4824			−0.526096			−0.263048	+	0.964783i	$0.584728\pi$
$398$		0			0
$399$			44.8200i			2.24381i
$400$		0			0
$401$			16.5789i			0.827909i	0.910297	+	0.413955i	$0.135853\pi$
$401$			16.5789i			0.827909i	−0.910297	+	0.413955i	$0.864147\pi$
$402$		0			0
$403$		−	4.97252i		−	0.247699i
$404$		0			0
$405$		−	2.03923i		−	0.101330i
$406$		0			0
$407$			12.2634i			0.607876i
$408$		0			0
$409$	−8.50973			−0.420779			−0.210390	−	0.977618i	$-0.567473\pi$
$409$	−8.50973			−0.420779			−0.210390	+	0.977618i	$0.567473\pi$
$410$		0			0
$411$	19.8839			0.980799
$412$		0			0
$413$			15.9168i			0.783217i
$414$		0			0
$415$		−	7.10572i		−	0.348806i
$416$		0			0
$417$	−20.9292			−1.02491
$418$		0			0
$419$	8.62454			0.421337			0.210668	−	0.977558i	$-0.432436\pi$
$419$	8.62454			0.421337			0.210668	+	0.977558i	$0.432436\pi$
$420$		0			0
$421$			24.2188i			1.18035i	0.807275	+	0.590176i	$0.200941\pi$
$421$			24.2188i			1.18035i	−0.807275	+	0.590176i	$0.799059\pi$
$422$		0			0
$423$		−	19.1091i		−	0.929116i
$424$		0			0
$425$			4.91920i			0.238616i
$426$		0			0
$427$			54.8058i			2.65224i
$428$		0			0
$429$		−	4.55848i		−	0.220085i
$430$		0			0
$431$	−2.55940			−0.123282			−0.0616411	−	0.998098i	$-0.519633\pi$
$431$	−2.55940			−0.123282			−0.0616411	+	0.998098i	$0.519633\pi$
$432$		0			0
$433$			36.6219i			1.75994i	0.475033	+	0.879968i	$0.342436\pi$
$433$			36.6219i			1.75994i	−0.475033	+	0.879968i	$0.657564\pi$
$434$		0			0
$435$	10.0468			0.481708
$436$		0			0
$437$	14.6737	+	35.7552i	0.701939	+	1.71040i
$438$		0			0
$439$			12.7422i			0.608153i	0.952648	+	0.304076i	$0.0983478\pi$
$439$			12.7422i			0.608153i	−0.952648	+	0.304076i	$0.901652\pi$
$440$		0			0
$441$	21.6611			1.03148
$442$		0			0
$443$		−	20.0121i		−	0.950802i	−0.879769	−	0.475401i	$-0.842303\pi$
$443$		−	20.0121i		−	0.950802i	0.879769	−	0.475401i	$-0.157697\pi$
$444$		0			0
$445$	4.19296			0.198765
$446$		0			0
$447$	−19.1775			−0.907065
$448$		0			0
$449$	−6.42749			−0.303332			−0.151666	−	0.988432i	$-0.548464\pi$
$449$	−6.42749			−0.303332			−0.151666	+	0.988432i	$0.548464\pi$
$450$		0			0
$451$	36.8304			1.73428
$452$		0			0
$453$	1.29731			0.0609531
$454$		0			0
$455$			4.33173i			0.203075i
$456$		0			0
$457$		−	10.0926i		−	0.472112i	−0.971739	−	0.236056i	$-0.924145\pi$
$457$		−	10.0926i		−	0.472112i	0.971739	−	0.236056i	$-0.0758549\pi$
$458$		0			0
$459$	27.0019			1.26034
$460$		0			0
$461$	−0.121581			−0.00566260			−0.00283130	−	0.999996i	$-0.500901\pi$
$461$	−0.121581			−0.00566260			−0.00283130	+	0.999996i	$0.500901\pi$
$462$		0			0
$463$		−	30.3265i		−	1.40939i	−0.709509	−	0.704696i	$-0.751083\pi$
$463$		−	30.3265i		−	1.40939i	0.709509	−	0.704696i	$-0.248917\pi$
$464$		0			0
$465$		−	6.38427i		−	0.296063i
$466$		0			0
$467$	−6.77151			−0.313348			−0.156674	−	0.987650i	$-0.550077\pi$
$467$	−6.77151			−0.313348			−0.156674	+	0.987650i	$0.550077\pi$
$468$		0			0
$469$	−27.6768			−1.27799
$470$		0			0
$471$	−9.77818			−0.450555
$472$		0			0
$473$	−28.9396			−1.33064
$474$		0			0
$475$	8.05890			0.369768
$476$		0			0
$477$		−	5.59547i		−	0.256199i
$478$		0			0
$479$	−34.5178			−1.57716			−0.788579	−	0.614933i	$-0.789183\pi$
$479$	−34.5178			−1.57716			−0.788579	+	0.614933i	$0.789183\pi$
$480$		0			0
$481$		−	2.89409i		−	0.131959i
$482$		0			0
$483$	−24.6752	+	10.1265i	−1.12276	+	0.460773i
$484$		0			0
$485$	3.27376			0.148654
$486$		0			0
$487$		−	1.35356i		−	0.0613359i	−0.999530	−	0.0306679i	$-0.990237\pi$
$487$		−	1.35356i		−	0.0613359i	0.999530	−	0.0306679i	$-0.00976344\pi$
$488$		0			0
$489$	16.0549			0.726028
$490$		0			0
$491$		−	7.55883i		−	0.341125i	−0.985347	−	0.170563i	$-0.945441\pi$
$491$		−	7.55883i		−	0.341125i	0.985347	−	0.170563i	$-0.0545585\pi$
$492$		0			0
$493$			40.8312i			1.83895i
$494$		0			0
$495$			6.13172i			0.275600i
$496$		0			0
$497$			36.7584i			1.64884i
$498$		0			0
$499$		−	18.3317i		−	0.820640i	−0.911942	−	0.410320i	$-0.865417\pi$
$499$		−	18.3317i		−	0.820640i	0.911942	−	0.410320i	$-0.134583\pi$
$500$		0			0
$501$	−20.2434			−0.904406
$502$		0			0
$503$	6.39875			0.285306			0.142653	−	0.989773i	$-0.454437\pi$
$503$	6.39875			0.285306			0.142653	+	0.989773i	$0.454437\pi$
$504$		0			0
$505$			7.83839i			0.348804i
$506$		0			0
$507$		−	14.6595i		−	0.651050i
$508$		0			0
$509$	−7.48475			−0.331756			−0.165878	−	0.986146i	$-0.553046\pi$
$509$	−7.48475			−0.331756			−0.165878	+	0.986146i	$0.553046\pi$
$510$		0			0
$511$	22.5199			0.996220
$512$		0			0
$513$		−	44.2360i		−	1.95307i
$514$		0			0
$515$		−	13.0069i		−	0.573151i
$516$		0			0
$517$		−	49.7333i		−	2.18727i
$518$		0			0
$519$		−	14.1469i		−	0.620978i
$520$		0			0
$521$		−	31.5484i		−	1.38216i	−0.722779	−	0.691080i	$-0.757135\pi$
$521$		−	31.5484i		−	1.38216i	0.722779	−	0.691080i	$-0.242865\pi$
$522$		0			0
$523$	4.28825			0.187512			0.0937560	−	0.995595i	$-0.470113\pi$
$523$	4.28825			0.187512			0.0937560	+	0.995595i	$0.470113\pi$
$524$		0			0
$525$			5.56155i			0.242726i
$526$		0			0
$527$	25.9463			1.13024
$528$		0			0
$529$	−16.3693	+	16.1569i	−0.711709	+	0.702474i
$530$		0			0
$531$		−	5.31714i		−	0.230744i
$532$		0			0
$533$	−8.69173			−0.376481
$534$		0			0
$535$			4.17047i			0.180305i
$536$		0			0
$537$	−10.4933			−0.452821
$538$		0			0
$539$	56.3752			2.42825
$540$		0			0
$541$	−17.8033			−0.765422			−0.382711	−	0.923868i	$-0.625009\pi$
$541$	−17.8033			−0.765422			−0.382711	+	0.923868i	$0.625009\pi$
$542$		0			0
$543$	−18.7580			−0.804984
$544$		0			0
$545$	8.67371			0.371541
$546$		0			0
$547$			17.3007i			0.739724i	0.929087	+	0.369862i	$0.120595\pi$
$547$			17.3007i			0.739724i	−0.929087	+	0.369862i	$0.879405\pi$
$548$		0			0
$549$		−	18.3083i		−	0.781378i
$550$		0			0
$551$	66.8920			2.84969
$552$		0			0
$553$	−5.60938			−0.238535
$554$		0			0
$555$		−	3.71576i		−	0.157725i
$556$		0			0
$557$			30.6712i			1.29958i	0.760113	+	0.649791i	$0.225144\pi$
$557$			30.6712i			1.29958i	−0.760113	+	0.649791i	$0.774856\pi$
$558$		0			0
$559$	6.82956			0.288859
$560$		0			0
$561$	23.7859			1.00424
$562$		0			0
$563$	−29.6410			−1.24922			−0.624609	−	0.780938i	$-0.714742\pi$
$563$	−29.6410			−1.24922			−0.624609	+	0.780938i	$0.714742\pi$
$564$		0			0
$565$	6.07845			0.255722
$566$		0			0
$567$	9.36983			0.393496
$568$		0			0
$569$		−	28.0266i		−	1.17494i	−0.809247	−	0.587469i	$-0.800124\pi$
$569$		−	28.0266i		−	1.17494i	0.809247	−	0.587469i	$-0.199876\pi$
$570$		0			0
$571$	−8.46600			−0.354291			−0.177146	−	0.984185i	$-0.556686\pi$
$571$	−8.46600			−0.354291			−0.177146	+	0.984185i	$0.556686\pi$
$572$		0			0
$573$		−	7.07600i		−	0.295604i
$574$		0			0
$575$	1.82081	+	4.43674i	0.0759329	+	0.185025i
$576$		0			0
$577$	−7.80612			−0.324973			−0.162486	−	0.986711i	$-0.551951\pi$
$577$	−7.80612			−0.324973			−0.162486	+	0.986711i	$0.551951\pi$
$578$		0			0
$579$		−	21.0250i		−	0.873768i
$580$		0			0
$581$	32.6493			1.35452
$582$		0			0
$583$		−	14.5628i		−	0.603128i
$584$		0			0
$585$		−	1.44705i		−	0.0598280i
$586$		0			0
$587$			26.1628i			1.07985i	0.841712	+	0.539926i	$0.181548\pi$
$587$			26.1628i			1.07985i	−0.841712	+	0.539926i	$0.818452\pi$
$588$		0			0
$589$		−	42.5067i		−	1.75146i
$590$		0			0
$591$		−	5.75003i		−	0.236525i
$592$		0			0
$593$	−41.4871			−1.70367			−0.851836	−	0.523809i	$-0.824511\pi$
$593$	−41.4871			−1.70367			−0.851836	+	0.523809i	$0.824511\pi$
$594$		0			0
$595$	−22.6027			−0.926620
$596$		0			0
$597$		−	28.9221i		−	1.18370i
$598$		0			0
$599$		−	38.3324i		−	1.56622i	−0.621885	−	0.783109i	$-0.713633\pi$
$599$		−	38.3324i		−	1.56622i	0.621885	−	0.783109i	$-0.286367\pi$
$600$		0			0
$601$	−21.7506			−0.887227			−0.443613	−	0.896218i	$-0.646304\pi$
$601$	−21.7506			−0.887227			−0.443613	+	0.896218i	$0.646304\pi$
$602$		0			0
$603$	9.24563			0.376511
$604$		0			0
$605$			4.95842i			0.201588i
$606$		0			0
$607$			16.5309i			0.670967i	0.942046	+	0.335484i	$0.108900\pi$
$607$			16.5309i			0.670967i	−0.942046	+	0.335484i	$0.891100\pi$
$608$		0			0
$609$			46.1631i			1.87062i
$610$		0			0
$611$			11.7367i			0.474818i
$612$		0			0
$613$		−	39.9121i		−	1.61204i	−0.591891	−	0.806018i	$-0.701618\pi$
$613$		−	39.9121i		−	1.61204i	0.591891	−	0.806018i	$-0.298382\pi$
$614$		0			0
$615$	−11.1594			−0.449991
$616$		0			0
$617$			8.91920i			0.359073i	0.983751	+	0.179537i	$0.0574598\pi$
$617$			8.91920i			0.359073i	−0.983751	+	0.179537i	$0.942540\pi$
$618$		0			0
$619$	−10.5080			−0.422351			−0.211175	−	0.977448i	$-0.567729\pi$
$619$	−10.5080			−0.422351			−0.211175	+	0.977448i	$0.567729\pi$
$620$		0			0
$621$	24.3536	−	9.99457i	0.977278	−	0.401068i
$622$		0			0
$623$			19.2658i			0.771867i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		−	38.9673i		−	1.55620i
$628$		0			0
$629$	15.1012			0.602124
$630$		0			0
$631$	−5.11136			−0.203480			−0.101740	−	0.994811i	$-0.532441\pi$
$631$	−5.11136			−0.203480			−0.101740	+	0.994811i	$0.532441\pi$
$632$		0			0
$633$	−9.88549			−0.392913
$634$		0			0
$635$	−8.15486			−0.323616
$636$		0			0
$637$	−13.3042			−0.527131
$638$		0			0
$639$		−	12.2794i		−	0.485766i
$640$		0			0
$641$			26.5457i			1.04849i	0.851567	+	0.524246i	$0.175653\pi$
$641$			26.5457i			1.04849i	−0.851567	+	0.524246i	$0.824347\pi$
$642$		0			0
$643$	−5.72549			−0.225791			−0.112896	−	0.993607i	$-0.536013\pi$
$643$	−5.72549			−0.225791			−0.112896	+	0.993607i	$0.536013\pi$
$644$		0			0
$645$	8.76854			0.345261
$646$		0			0
$647$			31.2328i			1.22789i	0.789349	+	0.613944i	$0.210418\pi$
$647$			31.2328i			1.22789i	−0.789349	+	0.613944i	$0.789582\pi$
$648$		0			0
$649$		−	13.8384i		−	0.543204i
$650$		0			0
$651$	29.3344			1.14971
$652$		0			0
$653$	21.3035			0.833669			0.416834	−	0.908982i	$-0.363139\pi$
$653$	21.3035			0.833669			0.416834	+	0.908982i	$0.363139\pi$
$654$		0			0
$655$	9.70988			0.379396
$656$		0			0
$657$	−7.52293			−0.293497
$658$		0			0
$659$	−2.19421			−0.0854744			−0.0427372	−	0.999086i	$-0.513608\pi$
$659$	−2.19421			−0.0854744			−0.0427372	+	0.999086i	$0.513608\pi$
$660$		0			0
$661$			25.4667i			0.990541i	0.868739	+	0.495270i	$0.164931\pi$
$661$			25.4667i			0.990541i	−0.868739	+	0.495270i	$0.835069\pi$
$662$		0			0
$663$	−5.61331			−0.218003
$664$		0			0
$665$			37.0290i			1.43592i
$666$		0			0
$667$	15.1134	+	36.8267i	0.585194	+	1.42593i
$668$		0			0
$669$	0.214864			0.00830711
$670$		0			0
$671$		−	47.6491i		−	1.83947i
$672$		0			0
$673$	14.3897			0.554682			0.277341	−	0.960772i	$-0.410547\pi$
$673$	14.3897			0.554682			0.277341	+	0.960772i	$0.410547\pi$
$674$		0			0
$675$		−	5.48909i		−	0.211275i
$676$		0			0
$677$		−	25.2096i		−	0.968882i	−0.874824	−	0.484441i	$-0.839023\pi$
$677$		−	25.2096i		−	0.968882i	0.874824	−	0.484441i	$-0.160977\pi$
$678$		0			0
$679$			15.0423i			0.577269i
$680$		0			0
$681$			0.111646i			0.00427828i
$682$		0			0
$683$			17.9235i			0.685822i	0.939368	+	0.342911i	$0.111413\pi$
$683$			17.9235i			0.685822i	−0.939368	+	0.342911i	$0.888587\pi$
$684$		0			0
$685$	16.4275			0.627662
$686$		0			0
$687$	−28.5503			−1.08926
$688$		0			0
$689$			3.43672i			0.130929i
$690$		0			0
$691$			49.5507i			1.88500i	0.334212	+	0.942498i	$0.391530\pi$
$691$			49.5507i			1.88500i	−0.334212	+	0.942498i	$0.608470\pi$
$692$		0			0
$693$	−28.1740			−1.07024
$694$		0			0
$695$	−17.2911			−0.655890
$696$		0			0
$697$		−	45.3529i		−	1.71786i
$698$		0			0
$699$			4.14381i			0.156733i
$700$		0			0
$701$		−	20.4699i		−	0.773137i	−0.922261	−	0.386569i	$-0.873660\pi$
$701$		−	20.4699i		−	0.773137i	0.922261	−	0.386569i	$-0.126340\pi$
$702$		0			0
$703$		−	24.7396i		−	0.933072i
$704$		0			0
$705$			15.0689i			0.567529i
$706$		0			0
$707$	−36.0158			−1.35451
$708$		0			0
$709$		−	14.0656i		−	0.528247i	−0.964489	−	0.264123i	$-0.914917\pi$
$709$		−	14.0656i		−	0.528247i	0.964489	−	0.264123i	$-0.0850826\pi$
$710$		0			0
$711$	1.87385			0.0702750
$712$		0			0
$713$	23.4016	−	9.60385i	0.876396	−	0.359667i
$714$		0			0
$715$		−	3.76608i		−	0.140844i
$716$		0			0
$717$	24.9325			0.931122
$718$		0			0
$719$			43.4045i			1.61872i	0.587316	+	0.809358i	$0.300185\pi$
$719$			43.4045i			1.61872i	−0.587316	+	0.809358i	$0.699815\pi$
$720$		0			0
$721$	59.7640			2.22573
$722$		0			0
$723$	7.79399			0.289861
$724$		0			0
$725$	8.30039			0.308269
$726$		0			0
$727$	35.0921			1.30150			0.650748	−	0.759294i	$-0.274456\pi$
$727$	35.0921			1.30150			0.650748	+	0.759294i	$0.274456\pi$
$728$		0			0
$729$	−23.0621			−0.854151
$730$		0			0
$731$			35.6362i			1.31805i
$732$		0			0
$733$			47.8275i			1.76655i	0.468855	+	0.883275i	$0.344667\pi$
$733$			47.8275i			1.76655i	−0.468855	+	0.883275i	$0.655333\pi$
$734$		0			0
$735$	−17.0814			−0.630056
$736$		0			0
$737$	24.0627			0.886361
$738$		0			0
$739$		−	14.7962i		−	0.544286i	−0.962257	−	0.272143i	$-0.912268\pi$
$739$		−	14.7962i		−	0.544286i	0.962257	−	0.272143i	$-0.0877324\pi$
$740$		0			0
$741$			9.19603i			0.337825i
$742$		0			0
$743$	−0.655166			−0.0240357			−0.0120178	−	0.999928i	$-0.503825\pi$
$743$	−0.655166			−0.0240357			−0.0120178	+	0.999928i	$0.503825\pi$
$744$		0			0
$745$	−15.8439			−0.580476
$746$		0			0
$747$	−10.9067			−0.399057
$748$		0			0
$749$	−19.1625			−0.700181
$750$		0			0
$751$	−33.3865			−1.21829			−0.609145	−	0.793059i	$-0.708487\pi$
$751$	−33.3865			−1.21829			−0.609145	+	0.793059i	$0.708487\pi$
$752$		0			0
$753$		−	4.65946i		−	0.169800i
$754$		0			0
$755$	1.07180			0.0390069
$756$		0			0
$757$		−	3.81319i		−	0.138593i	−0.997596	−	0.0692964i	$-0.977925\pi$
$757$		−	3.81319i		−	0.138593i	0.997596	−	0.0692964i	$-0.0220754\pi$
$758$		0			0
$759$	21.4530	−	8.80418i	0.778696	−	0.319571i
$760$		0			0
$761$	−39.6109			−1.43590			−0.717948	−	0.696097i	$-0.754918\pi$
$761$	−39.6109			−1.43590			−0.717948	+	0.696097i	$0.754918\pi$
$762$		0			0
$763$			39.8539i			1.44281i
$764$		0			0
$765$	7.55060			0.272993
$766$		0			0
$767$			3.26577i			0.117920i
$768$		0			0
$769$		−	23.2438i		−	0.838192i	−0.907942	−	0.419096i	$-0.862347\pi$
$769$		−	23.2438i		−	0.838192i	0.907942	−	0.419096i	$-0.137653\pi$
$770$		0			0
$771$		−	34.0972i		−	1.22798i
$772$		0			0
$773$			37.3646i			1.34391i	0.740591	+	0.671956i	$0.234546\pi$
$773$			37.3646i			1.34391i	−0.740591	+	0.671956i	$0.765454\pi$
$774$		0			0
$775$		−	5.27450i		−	0.189466i
$776$		0			0
$777$	17.0731			0.612496
$778$		0			0
$779$	−74.2997			−2.66206
$780$		0			0
$781$		−	31.9584i		−	1.14356i
$782$		0			0
$783$		−	45.5615i		−	1.62824i
$784$		0			0
$785$	−8.07845			−0.288332
$786$		0			0
$787$	17.2027			0.613211			0.306605	−	0.951837i	$-0.400807\pi$
$787$	17.2027			0.613211			0.306605	+	0.951837i	$0.400807\pi$
$788$		0			0
$789$			27.9398i			0.994683i
$790$		0			0
$791$			27.9292i			0.993050i
$792$		0			0
$793$			11.2449i			0.399318i
$794$		0			0
$795$			4.41244i			0.156493i
$796$		0			0
$797$		−	42.0627i		−	1.48994i	−0.667100	−	0.744969i	$-0.732464\pi$
$797$		−	42.0627i		−	1.48994i	0.667100	−	0.744969i	$-0.267536\pi$
$798$		0			0
$799$	−61.2416			−2.16657
$800$		0			0
$801$		−	6.43588i		−	0.227401i
$802$		0			0
$803$	−19.5792			−0.690934
$804$		0			0
$805$	−20.3859	+	8.36624i	−0.718509	+	0.294871i
$806$		0			0
$807$			33.3529i			1.17408i
$808$		0			0
$809$	39.6769			1.39496			0.697482	−	0.716602i	$-0.254304\pi$
$809$	39.6769			1.39496			0.697482	+	0.716602i	$0.254304\pi$
$810$		0			0
$811$		−	44.9224i		−	1.57744i	−0.614754	−	0.788719i	$-0.710745\pi$
$811$		−	44.9224i		−	1.57744i	0.614754	−	0.788719i	$-0.289255\pi$
$812$		0			0
$813$	36.6660			1.28593
$814$		0			0
$815$	13.2641			0.464621
$816$		0			0
$817$	58.3812			2.04250
$818$		0			0
$819$	6.64888			0.232331
$820$		0			0
$821$	−0.508178			−0.0177355			−0.00886777	−	0.999961i	$-0.502823\pi$
$821$	−0.508178			−0.0177355			−0.00886777	+	0.999961i	$0.502823\pi$
$822$		0			0
$823$		−	14.9598i		−	0.521467i	−0.965411	−	0.260734i	$-0.916036\pi$
$823$		−	14.9598i		−	0.521467i	0.965411	−	0.260734i	$-0.0839644\pi$
$824$		0			0
$825$		−	4.83531i		−	0.168344i
$826$		0			0
$827$	40.9135			1.42270			0.711350	−	0.702838i	$-0.248084\pi$
$827$	40.9135			1.42270			0.711350	+	0.702838i	$0.248084\pi$
$828$		0			0
$829$	−12.8328			−0.445703			−0.222851	−	0.974852i	$-0.571536\pi$
$829$	−12.8328			−0.445703			−0.222851	+	0.974852i	$0.571536\pi$
$830$		0			0
$831$		−	37.2072i		−	1.29070i
$832$		0			0
$833$		−	69.4204i		−	2.40528i
$834$		0			0
$835$	−16.7245			−0.578774
$836$		0			0
$837$	−28.9522			−1.00073
$838$		0			0
$839$	54.7825			1.89130			0.945651	−	0.325182i	$-0.105426\pi$
$839$	54.7825			1.89130			0.945651	+	0.325182i	$0.105426\pi$
$840$		0			0
$841$	39.8965			1.37574
$842$		0			0
$843$	−9.39265			−0.323500
$844$		0			0
$845$		−	12.1112i		−	0.416639i
$846$		0			0
$847$	−22.7829			−0.782831
$848$		0			0
$849$			14.5508i			0.499383i
$850$		0			0
$851$	13.6201	−	5.58961i	0.466892	−	0.191609i
$852$		0			0
$853$	−8.82916			−0.302305			−0.151152	−	0.988510i	$-0.548298\pi$
$853$	−8.82916			−0.302305			−0.151152	+	0.988510i	$0.548298\pi$
$854$		0			0
$855$		−	12.3698i		−	0.423039i
$856$		0			0
$857$	−49.0362			−1.67504			−0.837522	−	0.546404i	$-0.815996\pi$
$857$	−49.0362			−1.67504			−0.837522	+	0.546404i	$0.815996\pi$
$858$		0			0
$859$		−	8.44543i		−	0.288154i	−0.989566	−	0.144077i	$-0.953979\pi$
$859$		−	8.44543i		−	0.288154i	0.989566	−	0.144077i	$-0.0460213\pi$
$860$		0			0
$861$		−	51.2752i		−	1.74745i
$862$		0			0
$863$		−	8.70550i		−	0.296339i	−0.988962	−	0.148169i	$-0.952662\pi$
$863$		−	8.70550i		−	0.296339i	0.988962	−	0.148169i	$-0.0473381\pi$
$864$		0			0
$865$		−	11.6877i		−	0.397395i
$866$		0			0
$867$		−	8.71306i		−	0.295911i
$868$		0			0
$869$	4.87689			0.165437
$870$		0			0
$871$	−5.67864			−0.192413
$872$		0			0
$873$		−	5.02498i		−	0.170070i
$874$		0			0
$875$			4.59480i			0.155332i
$876$		0			0
$877$	52.7390			1.78087			0.890434	−	0.455112i	$-0.150401\pi$
$877$	52.7390			1.78087			0.890434	+	0.455112i	$0.150401\pi$
$878$		0			0
$879$	−5.89332			−0.198777
$880$		0			0
$881$			9.72859i			0.327765i	0.986480	+	0.163882i	$0.0524017\pi$
$881$			9.72859i			0.327765i	−0.986480	+	0.163882i	$0.947598\pi$
$882$		0			0
$883$		−	5.72337i		−	0.192607i	−0.995352	−	0.0963033i	$-0.969298\pi$
$883$		−	5.72337i		−	0.192607i	0.995352	−	0.0963033i	$-0.0307019\pi$
$884$		0			0
$885$			4.19296i			0.140945i
$886$		0			0
$887$		−	37.7250i		−	1.26668i	−0.773874	−	0.633340i	$-0.781683\pi$
$887$		−	37.7250i		−	1.26668i	0.773874	−	0.633340i	$-0.218317\pi$
$888$		0			0
$889$		−	37.4699i		−	1.25670i
$890$		0			0
$891$	−8.14630			−0.272911
$892$		0			0
$893$			100.329i			3.35740i
$894$		0			0
$895$	−8.66930			−0.289783
$896$		0			0
$897$	−5.06278	+	2.07773i	−0.169041	+	0.0693734i
$898$		0			0
$899$		−	43.7804i		−	1.46016i
$900$		0			0
$901$	−17.9326			−0.597421
$902$		0			0
$903$			40.2896i			1.34076i
$904$		0			0
$905$	−15.4973			−0.515149
$906$		0			0
$907$	30.9225			1.02677			0.513383	−	0.858160i	$-0.328392\pi$
$907$	30.9225			1.02677			0.513383	+	0.858160i	$0.328392\pi$
$908$		0			0
$909$	12.0313			0.399055
$910$		0			0
$911$	6.30711			0.208964			0.104482	−	0.994527i	$-0.466682\pi$
$911$	6.30711			0.208964			0.104482	+	0.994527i	$0.466682\pi$
$912$		0			0
$913$	−28.3859			−0.939437
$914$		0			0
$915$			14.4374i			0.477287i
$916$		0			0
$917$			44.6149i			1.47331i
$918$		0			0
$919$	23.6782			0.781073			0.390536	−	0.920588i	$-0.372290\pi$
$919$	23.6782			0.781073			0.390536	+	0.920588i	$0.372290\pi$
$920$		0			0
$921$	4.50881			0.148570
$922$		0			0
$923$			7.54198i			0.248247i
$924$		0			0
$925$		−	3.06985i		−	0.100936i
$926$		0			0
$927$	−19.9646			−0.655723
$928$		0			0
$929$	−8.98762			−0.294874			−0.147437	−	0.989071i	$-0.547102\pi$
$929$	−8.98762			−0.294874			−0.147437	+	0.989071i	$0.547102\pi$
$930$		0			0
$931$	−113.728			−3.72730
$932$		0			0
$933$	−15.3309			−0.501911
$934$		0			0
$935$	19.6512			0.642663
$936$		0			0
$937$		−	7.10343i		−	0.232059i	−0.993246	−	0.116030i	$-0.962983\pi$
$937$		−	7.10343i		−	0.232059i	0.993246	−	0.116030i	$-0.0370167\pi$
$938$		0			0
$939$	−27.9114			−0.910855
$940$		0			0
$941$		−	25.6939i		−	0.837597i	−0.908079	−	0.418798i	$-0.862451\pi$
$941$		−	25.6939i		−	0.837597i	0.908079	−	0.418798i	$-0.137549\pi$
$942$		0			0
$943$	−16.7871	−	40.9049i	−0.546663	−	1.33205i
$944$		0			0
$945$	25.2212			0.820447
$946$		0			0
$947$		−	13.4805i		−	0.438058i	−0.975718	−	0.219029i	$-0.929711\pi$
$947$		−	13.4805i		−	0.438058i	0.975718	−	0.219029i	$-0.0702890\pi$
$948$		0			0
$949$	4.62056			0.149990
$950$		0			0
$951$			2.16742i			0.0702834i
$952$		0			0
$953$		−	3.59698i		−	0.116518i	−0.998302	−	0.0582588i	$-0.981445\pi$
$953$		−	3.59698i		−	0.116518i	0.998302	−	0.0582588i	$-0.0185549\pi$
$954$		0			0
$955$		−	5.84599i		−	0.189172i
$956$		0			0
$957$		−	40.1350i		−	1.29738i
$958$		0			0
$959$			75.4810i			2.43741i
$960$		0			0
$961$	3.17964			0.102569
$962$		0			0
$963$	6.40136			0.206281
$964$		0			0
$965$		−	17.3702i		−	0.559168i
$966$		0			0
$967$			4.55488i			0.146475i	0.997315	+	0.0732375i	$0.0233331\pi$
$967$			4.55488i			0.146475i	−0.997315	+	0.0732375i	$0.976667\pi$
$968$		0			0
$969$	−47.9843			−1.54148
$970$		0			0
$971$	0.288725			0.00926561			0.00463281	−	0.999989i	$-0.498525\pi$
$971$	0.288725			0.00926561			0.00463281	+	0.999989i	$0.498525\pi$
$972$		0			0
$973$		−	79.4492i		−	2.54702i
$974$		0			0
$975$			1.14110i			0.0365445i
$976$		0			0
$977$			28.0416i			0.897130i	0.893750	+	0.448565i	$0.148065\pi$
$977$			28.0416i			0.897130i	−0.893750	+	0.448565i	$0.851935\pi$
$978$		0			0
$979$		−	16.7500i		−	0.535333i
$980$		0			0
$981$		−	13.3135i		−	0.425067i
$982$		0			0
$983$	−9.06300			−0.289065			−0.144532	−	0.989500i	$-0.546168\pi$
$983$	−9.06300			−0.289065			−0.144532	+	0.989500i	$0.546168\pi$
$984$		0			0
$985$		−	4.75051i		−	0.151364i
$986$		0			0
$987$	−69.2387			−2.20389
$988$		0			0
$989$	13.1905	+	32.1411i	0.419434	+	1.02203i
$990$		0			0
$991$		−	38.0968i		−	1.21018i	−0.796155	−	0.605092i	$-0.793136\pi$
$991$		−	38.0968i		−	1.21018i	0.796155	−	0.605092i	$-0.206864\pi$
$992$		0			0
$993$	41.9553			1.33141
$994$		0			0
$995$		−	23.8947i		−	0.757511i
$996$		0			0
$997$	4.67594			0.148088			0.0740442	−	0.997255i	$-0.476409\pi$
$997$	4.67594			0.148088			0.0740442	+	0.997255i	$0.476409\pi$
$998$		0			0
$999$	−16.8507			−0.533132

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.2.i.c.1471.12	yes	16
4.3	odd	2	inner	1840.2.i.c.1471.6	yes	16
23.22	odd	2	inner	1840.2.i.c.1471.11	yes	16
92.91	even	2	inner	1840.2.i.c.1471.5	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1840.2.i.c.1471.5	✓	16	92.91	even	2	inner
1840.2.i.c.1471.6	yes	16	4.3	odd	2	inner
1840.2.i.c.1471.11	yes	16	23.22	odd	2	inner
1840.2.i.c.1471.12	yes	16	1.1	even	1	trivial

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.i (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.21040i q^{3} +1.00000i q^{5} -4.59480 q^{7} +1.53493 q^{9} +O(q^{10})\)	\(q+1.21040i q^{3} +1.00000i q^{5} -4.59480 q^{7} +1.53493 q^{9} +3.99480 q^{11} -0.942747 q^{13} -1.21040 q^{15} -4.91920i q^{17} -8.05890 q^{19} -5.56155i q^{21} +(-1.82081 - 4.43674i) q^{23} -1.00000 q^{25} +5.48909i q^{27} -8.30039 q^{29} +5.27450i q^{31} +4.83531i q^{33} -4.59480i q^{35} +3.06985i q^{37} -1.14110i q^{39} +9.21958 q^{41} -7.24432 q^{43} +1.53493i q^{45} -12.4495i q^{47} +14.1122 q^{49} +5.95421 q^{51} -3.64543i q^{53} +3.99480i q^{55} -9.75451i q^{57} -3.46410i q^{59} -11.9278i q^{61} -7.05267 q^{63} -0.942747i q^{65} +6.02351 q^{67} +(5.37024 - 2.20391i) q^{69} -8.00001i q^{71} -4.90117 q^{73} -1.21040i q^{75} -18.3553 q^{77} +1.22081 q^{79} -2.03923 q^{81} -7.10572 q^{83} +4.91920 q^{85} -10.0468i q^{87} -4.19296i q^{89} +4.33173 q^{91} -6.38427 q^{93} -8.05890i q^{95} -3.27376i q^{97} +6.13172 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 8 q^{13} - 16 q^{25} - 84 q^{29} + 24 q^{41} + 36 q^{49} - 12 q^{69} + 68 q^{73} - 48 q^{77} + 32 q^{81} + 4 q^{85} - 52 q^{93}+O(q^{100}) \) 16 * q - 8 * q^13 - 16 * q^25 - 84 * q^29 + 24 * q^41 + 36 * q^49 - 12 * q^69 + 68 * q^73 - 48 * q^77 + 32 * q^81 + 4 * q^85 - 52 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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