LMFDB - Embedded newform 1840.2.i.a.1471.2

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} - 32x^{14} + 400x^{12} - 2398x^{10} + 7128x^{8} - 9200x^{6} + 4705x^{4} + 2696x^{2} + 784 $ x^16 - 32x^14 + 400x^12 - 2398x^10 + 7128x^8 - 9200x^6 + 4705x^4 + 2696*x^2 + 784
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1471.2
Root			$2.25010 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1840.1471
Dual form			1840.2.i.a.1471.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.11612i q^{3} +1.00000i q^{5} -1.38407 q^{7} -6.71022 q^{9} +O(q^{10})$	$q-3.11612i q^{3} +1.00000i q^{5} -1.38407 q^{7} -6.71022 q^{9} +5.13521 q^{11} +5.71022 q^{13} +3.11612 q^{15} +4.71022i q^{17} +8.44538 q^{19} +4.31294i q^{21} +(0.287039 + 4.78723i) q^{23} -1.00000 q^{25} +11.5615i q^{27} -4.10750 q^{29} +6.74538i q^{31} -16.0020i q^{33} -1.38407i q^{35} +1.81577i q^{37} -17.7937i q^{39} +4.20739 q^{41} +2.76814 q^{43} -6.71022i q^{45} -1.44501i q^{47} -5.08434 q^{49} +14.6776 q^{51} +9.60467i q^{53} +5.13521i q^{55} -26.3168i q^{57} -13.9895i q^{59} -5.90012i q^{61} +9.28743 q^{63} +5.71022i q^{65} +13.4154 q^{67} +(14.9176 - 0.894449i) q^{69} -3.94519i q^{71} -3.12871 q^{73} +3.11612i q^{75} -7.10750 q^{77} +9.45259 q^{79} +15.8964 q^{81} -13.4154 q^{83} -4.71022 q^{85} +12.7995i q^{87} -13.4204i q^{89} -7.90336 q^{91} +21.0194 q^{93} +8.44538i q^{95} -4.26857i q^{97} -34.4584 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 32 q^{9}+O(q^{10}) $ 16 * q - 32 * q^9	$ 16 q - 32 q^{9} + 16 q^{13} - 16 q^{25} + 72 q^{29} - 4 q^{41} - 32 q^{49} + 92 q^{69} - 20 q^{73} + 24 q^{77} + 60 q^{93}+O(q^{100}) $ 16 * q - 32 * q^9 + 16 * q^13 - 16 * q^25 + 72 * q^29 - 4 * q^41 - 32 * q^49 + 92 * q^69 - 20 * q^73 + 24 * q^77 + 60 * 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
$3$		−	3.11612i		−	1.79909i	−0.436824	−	0.899547i	$-0.643897\pi$
$3$		−	3.11612i		−	1.79909i	0.436824	−	0.899547i	$-0.356103\pi$
$4$		0			0
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$	−1.38407			−0.523130			−0.261565	−	0.965186i	$-0.584239\pi$
$7$	−1.38407			−0.523130			−0.261565	+	0.965186i	$0.584239\pi$
$8$		0			0
$9$	−6.71022			−2.23674
$10$		0			0
$11$	5.13521			1.54832			0.774162	−	0.632987i	$-0.218171\pi$
$11$	5.13521			1.54832			0.774162	+	0.632987i	$0.218171\pi$
$12$		0			0
$13$	5.71022			1.58373			0.791865	−	0.610696i	$-0.209110\pi$
$13$	5.71022			1.58373			0.791865	+	0.610696i	$0.209110\pi$
$14$		0			0
$15$	3.11612			0.804579
$16$		0			0
$17$			4.71022i			1.14240i	0.820812	+	0.571198i	$0.193521\pi$
$17$			4.71022i			1.14240i	−0.820812	+	0.571198i	$0.806479\pi$
$18$		0			0
$19$	8.44538			1.93750			0.968752	−	0.248033i	$-0.0797840\pi$
$19$	8.44538			1.93750			0.968752	+	0.248033i	$0.0797840\pi$
$20$		0			0
$21$			4.31294i			0.941160i
$22$		0			0
$23$	0.287039	+	4.78723i	0.0598518	+	0.998207i
$23$	0.287039	+	4.78723i	0.0598518	+	0.998207i
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$			11.5615i			2.22501i
$28$		0			0
$29$	−4.10750			−0.762744			−0.381372	−	0.924422i	$-0.624548\pi$
$29$	−4.10750			−0.762744			−0.381372	+	0.924422i	$0.624548\pi$
$30$		0			0
$31$			6.74538i			1.21151i	0.795652	+	0.605753i	$0.207128\pi$
$31$			6.74538i			1.21151i	−0.795652	+	0.605753i	$0.792872\pi$
$32$		0			0
$33$		−	16.0020i		−	2.78558i
$34$		0			0
$35$		−	1.38407i		−	0.233951i
$36$		0			0
$37$			1.81577i			0.298511i	0.988799	+	0.149256i	$0.0476877\pi$
$37$			1.81577i			0.298511i	−0.988799	+	0.149256i	$0.952312\pi$
$38$		0			0
$39$		−	17.7937i		−	2.84928i
$40$		0			0
$41$	4.20739			0.657084			0.328542	−	0.944489i	$-0.393443\pi$
$41$	4.20739			0.657084			0.328542	+	0.944489i	$0.393443\pi$
$42$		0			0
$43$	2.76814			0.422138			0.211069	−	0.977471i	$-0.432306\pi$
$43$	2.76814			0.422138			0.211069	+	0.977471i	$0.432306\pi$
$44$		0			0
$45$		−	6.71022i		−	1.00030i
$46$		0			0
$47$		−	1.44501i		−	0.210777i	−0.994431	−	0.105388i	$-0.966391\pi$
$47$		−	1.44501i		−	0.210777i	0.994431	−	0.105388i	$-0.0336085\pi$
$48$		0			0
$49$	−5.08434			−0.726335
$50$		0			0
$51$	14.6776			2.05528
$52$		0			0
$53$			9.60467i			1.31930i	0.751572	+	0.659651i	$0.229296\pi$
$53$			9.60467i			1.31930i	−0.751572	+	0.659651i	$0.770704\pi$
$54$		0			0
$55$			5.13521i			0.692432i
$56$		0			0
$57$		−	26.3168i		−	3.48575i
$58$		0			0
$59$		−	13.9895i		−	1.82128i	−0.413201	−	0.910640i	$-0.635589\pi$
$59$		−	13.9895i		−	1.82128i	0.413201	−	0.910640i	$-0.364411\pi$
$60$		0			0
$61$		−	5.90012i		−	0.755432i	−0.925921	−	0.377716i	$-0.876709\pi$
$61$		−	5.90012i		−	0.755432i	0.925921	−	0.377716i	$-0.123291\pi$
$62$		0			0
$63$	9.28743			1.17011
$64$		0			0
$65$			5.71022i			0.708266i
$66$		0			0
$67$	13.4154			1.63896			0.819478	−	0.573111i	$-0.194263\pi$
$67$	13.4154			1.63896			0.819478	+	0.573111i	$0.194263\pi$
$68$		0			0
$69$	14.9176	−	0.894449i	1.79587	−	0.107679i
$70$		0			0
$71$		−	3.94519i		−	0.468208i	−0.972212	−	0.234104i	$-0.924784\pi$
$71$		−	3.94519i		−	0.468208i	0.972212	−	0.234104i	$-0.0752155\pi$
$72$		0			0
$73$	−3.12871			−0.366188			−0.183094	−	0.983095i	$-0.558611\pi$
$73$	−3.12871			−0.366188			−0.183094	+	0.983095i	$0.558611\pi$
$74$		0			0
$75$			3.11612i			0.359819i
$76$		0			0
$77$	−7.10750			−0.809975
$78$		0			0
$79$	9.45259			1.06350			0.531750	−	0.846902i	$-0.321535\pi$
$79$	9.45259			1.06350			0.531750	+	0.846902i	$0.321535\pi$
$80$		0			0
$81$	15.8964			1.76627
$82$		0			0
$83$	−13.4154			−1.47254			−0.736268	−	0.676690i	$-0.763414\pi$
$83$	−13.4154			−1.47254			−0.736268	+	0.676690i	$0.763414\pi$
$84$		0			0
$85$	−4.71022			−0.510895
$86$		0			0
$87$			12.7995i			1.37225i
$88$		0			0
$89$		−	13.4204i		−	1.42256i	−0.702907	−	0.711282i	$-0.748115\pi$
$89$		−	13.4204i		−	1.42256i	0.702907	−	0.711282i	$-0.251885\pi$
$90$		0			0
$91$	−7.90336			−0.828497
$92$		0			0
$93$	21.0194			2.17961
$94$		0			0
$95$			8.44538i			0.866478i
$96$		0			0
$97$		−	4.26857i		−	0.433408i	−0.976237	−	0.216704i	$-0.930469\pi$
$97$		−	4.26857i		−	0.433408i	0.976237	−	0.216704i	$-0.0695306\pi$
$98$		0			0
$99$	−34.4584			−3.46320
$100$		0			0
$101$	1.18990			0.118399			0.0591995	−	0.998246i	$-0.481145\pi$
$101$	1.18990			0.118399			0.0591995	+	0.998246i	$0.481145\pi$
$102$		0			0
$103$	−11.7555			−1.15831			−0.579154	−	0.815218i	$-0.696617\pi$
$103$	−11.7555			−1.15831			−0.579154	+	0.815218i	$0.696617\pi$
$104$		0			0
$105$	−4.31294			−0.420900
$106$		0			0
$107$	−13.2936			−1.28514			−0.642568	−	0.766228i	$-0.722131\pi$
$107$	−13.2936			−1.28514			−0.642568	+	0.766228i	$0.722131\pi$
$108$		0			0
$109$		−	4.58151i		−	0.438829i	−0.975632	−	0.219415i	$-0.929585\pi$
$109$		−	4.58151i		−	0.438829i	0.975632	−	0.219415i	$-0.0704147\pi$
$110$		0			0
$111$	5.65817			0.537049
$112$		0			0
$113$			10.6103i			0.998137i	0.866563	+	0.499068i	$0.166324\pi$
$113$			10.6103i			0.998137i	−0.866563	+	0.499068i	$0.833676\pi$
$114$		0			0
$115$	−4.78723	+	0.287039i	−0.446412	+	0.0267665i
$116$		0			0
$117$	−38.3168			−3.54239
$118$		0			0
$119$		−	6.51928i		−	0.597622i
$120$		0			0
$121$	15.3704			1.39731
$122$		0			0
$123$		−	13.1107i		−	1.18216i
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$			2.71505i			0.240922i	0.992718	+	0.120461i	$0.0384372\pi$
$127$			2.71505i			0.240922i	−0.992718	+	0.120461i	$0.961563\pi$
$128$		0			0
$129$		−	8.62588i		−	0.759466i
$130$		0			0
$131$		−	4.78723i		−	0.418263i	−0.977888	−	0.209131i	$-0.932936\pi$
$131$		−	4.78723i		−	0.418263i	0.977888	−	0.209131i	$-0.0670636\pi$
$132$		0			0
$133$	−11.6890			−1.01357
$134$		0			0
$135$	−11.5615			−0.995056
$136$		0			0
$137$			7.58718i			0.648216i	0.946020	+	0.324108i	$0.105064\pi$
$137$			7.58718i			0.648216i	−0.946020	+	0.324108i	$0.894936\pi$
$138$		0			0
$139$			1.88598i			0.159967i	0.996796	+	0.0799835i	$0.0254868\pi$
$139$			1.88598i			0.159967i	−0.996796	+	0.0799835i	$0.974513\pi$
$140$		0			0
$141$	−4.50283			−0.379207
$142$		0			0
$143$	29.3232			2.45213
$144$		0			0
$145$		−	4.10750i		−	0.341110i
$146$		0			0
$147$			15.8434i			1.30675i
$148$		0			0
$149$			5.83326i			0.477880i	0.971034	+	0.238940i	$0.0767999\pi$
$149$			5.83326i			0.477880i	−0.971034	+	0.238940i	$0.923200\pi$
$150$		0			0
$151$			19.3109i			1.57150i	0.618544	+	0.785751i	$0.287723\pi$
$151$			19.3109i			1.57150i	−0.618544	+	0.785751i	$0.712277\pi$
$152$		0			0
$153$		−	31.6066i		−	2.55524i
$154$		0			0
$155$	−6.74538			−0.541802
$156$		0			0
$157$			16.6103i			1.32565i	0.748775	+	0.662825i	$0.230643\pi$
$157$			16.6103i			1.32565i	−0.748775	+	0.662825i	$0.769357\pi$
$158$		0			0
$159$	29.9293			2.37355
$160$		0			0
$161$	−0.397283	−	6.62588i	−0.0313103	−	0.522192i
$162$		0			0
$163$		−	19.5867i		−	1.53415i	−0.641556	−	0.767076i	$-0.721711\pi$
$163$		−	19.5867i		−	1.53415i	0.641556	−	0.767076i	$-0.278289\pi$
$164$		0			0
$165$	16.0020			1.24575
$166$		0			0
$167$		−	9.51036i		−	0.735934i	−0.929839	−	0.367967i	$-0.880054\pi$
$167$		−	9.51036i		−	0.735934i	0.929839	−	0.367967i	$-0.119946\pi$
$168$		0			0
$169$	19.6066			1.50820
$170$		0			0
$171$	−56.6704			−4.33369
$172$		0			0
$173$	13.3206			1.01274			0.506372	−	0.862315i	$-0.330986\pi$
$173$	13.3206			1.01274			0.506372	+	0.862315i	$0.330986\pi$
$174$		0			0
$175$	1.38407			0.104626
$176$		0			0
$177$	−43.5930			−3.27665
$178$		0			0
$179$			0.749054i			0.0559869i	0.999608	+	0.0279935i	$0.00891176\pi$
$179$			0.749054i			0.0559869i	−0.999608	+	0.0279935i	$0.991088\pi$
$180$		0			0
$181$		−	20.7965i		−	1.54579i	−0.634532	−	0.772897i	$-0.718807\pi$
$181$		−	20.7965i		−	1.54579i	0.634532	−	0.772897i	$-0.281193\pi$
$182$		0			0
$183$	−18.3855			−1.35909
$184$		0			0
$185$	−1.81577			−0.133498
$186$		0			0
$187$			24.1880i			1.76880i
$188$		0			0
$189$		−	16.0020i		−	1.16397i
$190$		0			0
$191$	15.1108			1.09338			0.546688	−	0.837336i	$-0.315888\pi$
$191$	15.1108			1.09338			0.546688	+	0.837336i	$0.315888\pi$
$192$		0			0
$193$	−11.9233			−0.858256			−0.429128	−	0.903244i	$-0.641179\pi$
$193$	−11.9233			−0.858256			−0.429128	+	0.903244i	$0.641179\pi$
$194$		0			0
$195$	17.7937			1.27424
$196$		0			0
$197$	−9.59705			−0.683762			−0.341881	−	0.939743i	$-0.611064\pi$
$197$	−9.59705			−0.683762			−0.341881	+	0.939743i	$0.611064\pi$
$198$		0			0
$199$	2.64626			0.187589			0.0937944	−	0.995592i	$-0.470100\pi$
$199$	2.64626			0.187589			0.0937944	+	0.995592i	$0.470100\pi$
$200$		0			0
$201$		−	41.8041i		−	2.94864i
$202$		0			0
$203$	5.68508			0.399014
$204$		0			0
$205$			4.20739i			0.293857i
$206$		0			0
$207$	−1.92610	−	32.1234i	−0.133873	−	2.23273i
$208$		0			0
$209$	43.3688			2.99988
$210$		0			0
$211$		−	4.29317i		−	0.295554i	−0.989021	−	0.147777i	$-0.952788\pi$
$211$		−	4.29317i		−	0.295554i	0.989021	−	0.147777i	$-0.0472117\pi$
$212$		0			0
$213$	−12.2937			−0.842349
$214$		0			0
$215$			2.76814i			0.188786i
$216$		0			0
$217$		−	9.33610i		−	0.633776i
$218$		0			0
$219$			9.74944i			0.658806i
$220$		0			0
$221$			26.8964i			1.80925i
$222$		0			0
$223$		−	2.89002i		−	0.193530i	−0.995307	−	0.0967651i	$-0.969150\pi$
$223$		−	2.89002i		−	0.193530i	0.995307	−	0.0967651i	$-0.0308496\pi$
$224$		0			0
$225$	6.71022			0.447348
$226$		0			0
$227$	13.7345			0.911593			0.455796	−	0.890084i	$-0.349355\pi$
$227$	13.7345			0.911593			0.455796	+	0.890084i	$0.349355\pi$
$228$		0			0
$229$			18.2150i			1.20368i	0.798616	+	0.601841i	$0.205566\pi$
$229$			18.2150i			1.20368i	−0.798616	+	0.601841i	$0.794434\pi$
$230$		0			0
$231$			22.1479i			1.45722i
$232$		0			0
$233$	−22.0806			−1.44655			−0.723275	−	0.690560i	$-0.757364\pi$
$233$	−22.0806			−1.44655			−0.723275	+	0.690560i	$0.757364\pi$
$234$		0			0
$235$	1.44501			0.0942622
$236$		0			0
$237$		−	29.4554i		−	1.91334i
$238$		0			0
$239$		−	8.69231i		−	0.562259i	−0.959670	−	0.281129i	$-0.909291\pi$
$239$		−	8.69231i		−	0.562259i	0.959670	−	0.281129i	$-0.0907090\pi$
$240$		0			0
$241$			12.7832i			0.823440i	0.911310	+	0.411720i	$0.135072\pi$
$241$			12.7832i			0.823440i	−0.911310	+	0.411720i	$0.864928\pi$
$242$		0			0
$243$		−	14.8506i		−	0.952668i
$244$		0			0
$245$		−	5.08434i		−	0.324827i
$246$		0			0
$247$	48.2250			3.06848
$248$		0			0
$249$			41.8041i			2.64923i
$250$		0			0
$251$	18.7736			1.18498			0.592489	−	0.805579i	$-0.298145\pi$
$251$	18.7736			1.18498			0.592489	+	0.805579i	$0.298145\pi$
$252$		0			0
$253$	1.47401	+	24.5835i	0.0926701	+	1.54555i
$254$		0			0
$255$			14.6776i			0.919149i
$256$		0			0
$257$	17.9233			1.11802			0.559012	−	0.829160i	$-0.311181\pi$
$257$	17.9233			1.11802			0.559012	+	0.829160i	$0.311181\pi$
$258$		0			0
$259$		−	2.51316i		−	0.156160i
$260$		0			0
$261$	27.5623			1.70606
$262$		0			0
$263$	−10.8046			−0.666241			−0.333120	−	0.942884i	$-0.608102\pi$
$263$	−10.8046			−0.666241			−0.333120	+	0.942884i	$0.608102\pi$
$264$		0			0
$265$	−9.60467			−0.590010
$266$		0			0
$267$	−41.8197			−2.55933
$268$		0			0
$269$	−17.8964			−1.09116			−0.545581	−	0.838058i	$-0.683691\pi$
$269$	−17.8964			−1.09116			−0.545581	+	0.838058i	$0.683691\pi$
$270$		0			0
$271$		−	23.4808i		−	1.42636i	−0.700982	−	0.713179i	$-0.747255\pi$
$271$		−	23.4808i		−	1.42636i	0.700982	−	0.713179i	$-0.252745\pi$
$272$		0			0
$273$			24.6278i			1.49054i
$274$		0			0
$275$	−5.13521			−0.309665
$276$		0			0
$277$	−1.28607			−0.0772722			−0.0386361	−	0.999253i	$-0.512301\pi$
$277$	−1.28607			−0.0772722			−0.0386361	+	0.999253i	$0.512301\pi$
$278$		0			0
$279$		−	45.2630i		−	2.70983i
$280$		0			0
$281$			14.6298i			0.872740i	0.899767	+	0.436370i	$0.143736\pi$
$281$			14.6298i			0.872740i	−0.899767	+	0.436370i	$0.856264\pi$
$282$		0			0
$283$	4.80314			0.285517			0.142758	−	0.989758i	$-0.454403\pi$
$283$	4.80314			0.285517			0.142758	+	0.989758i	$0.454403\pi$
$284$		0			0
$285$	26.3168			1.55888
$286$		0			0
$287$	−5.82333			−0.343740
$288$		0			0
$289$	−5.18618			−0.305070
$290$		0			0
$291$	−13.3014			−0.779742
$292$		0			0
$293$			9.80444i			0.572781i	0.958113	+	0.286391i	$0.0924555\pi$
$293$			9.80444i			0.572781i	−0.958113	+	0.286391i	$0.907544\pi$
$294$		0			0
$295$	13.9895			0.814501
$296$		0			0
$297$			59.3708i			3.44504i
$298$		0			0
$299$	1.63906	+	27.3362i	0.0947891	+	1.58089i
$300$		0			0
$301$	−3.83131			−0.220833
$302$		0			0
$303$		−	3.70786i		−	0.213011i
$304$		0			0
$305$	5.90012			0.337840
$306$		0			0
$307$		−	12.8813i		−	0.735172i	−0.929990	−	0.367586i	$-0.880184\pi$
$307$		−	12.8813i		−	0.735172i	0.929990	−	0.367586i	$-0.119816\pi$
$308$		0			0
$309$			36.6317i			2.08391i
$310$		0			0
$311$			0.813161i			0.0461102i	0.999734	+	0.0230551i	$0.00733931\pi$
$311$			0.813161i			0.0461102i	−0.999734	+	0.0230551i	$0.992661\pi$
$312$		0			0
$313$		−	19.0695i		−	1.07787i	−0.842347	−	0.538935i	$-0.818827\pi$
$313$		−	19.0695i		−	1.07787i	0.842347	−	0.538935i	$-0.181173\pi$
$314$		0			0
$315$			9.28743i			0.523287i
$316$		0			0
$317$	−8.62783			−0.484587			−0.242294	−	0.970203i	$-0.577900\pi$
$317$	−8.62783			−0.484587			−0.242294	+	0.970203i	$0.577900\pi$
$318$		0			0
$319$	−21.0929			−1.18098
$320$		0			0
$321$			41.4243i			2.31208i
$322$		0			0
$323$			39.7796i			2.21340i
$324$		0			0
$325$	−5.71022			−0.316746
$326$		0			0
$327$	−14.2766			−0.789495
$328$		0			0
$329$			2.00000i			0.110264i
$330$		0			0
$331$		−	15.8226i		−	0.869690i	−0.900505	−	0.434845i	$-0.856803\pi$
$331$		−	15.8226i		−	0.869690i	0.900505	−	0.434845i	$-0.143197\pi$
$332$		0			0
$333$		−	12.1842i		−	0.667692i
$334$		0			0
$335$			13.4154i			0.732963i
$336$		0			0
$337$			7.25371i			0.395135i	0.980289	+	0.197567i	$0.0633041\pi$
$337$			7.25371i			0.395135i	−0.980289	+	0.197567i	$0.936696\pi$
$338$		0			0
$339$	33.0631			1.79574
$340$		0			0
$341$			34.6390i			1.87581i
$342$		0			0
$343$	16.7256			0.903098
$344$		0			0
$345$	0.894449	+	14.9176i	0.0481555	+	0.803137i
$346$		0			0
$347$		−	7.65960i		−	0.411189i	−0.978637	−	0.205594i	$-0.934087\pi$
$347$		−	7.65960i		−	0.411189i	0.978637	−	0.205594i	$-0.0659128\pi$
$348$		0			0
$349$	−26.5223			−1.41971			−0.709853	−	0.704350i	$-0.751238\pi$
$349$	−26.5223			−1.41971			−0.709853	+	0.704350i	$0.751238\pi$
$350$		0			0
$351$			66.0187i			3.52382i
$352$		0			0
$353$	−23.7008			−1.26147			−0.630734	−	0.775999i	$-0.717246\pi$
$353$	−23.7008			−1.26147			−0.630734	+	0.775999i	$0.717246\pi$
$354$		0			0
$355$	3.94519			0.209389
$356$		0			0
$357$	−20.3149			−1.07518
$358$		0			0
$359$	−21.0768			−1.11239			−0.556195	−	0.831052i	$-0.687739\pi$
$359$	−21.0768			−1.11239			−0.556195	+	0.831052i	$0.687739\pi$
$360$		0			0
$361$	52.3245			2.75392
$362$		0			0
$363$		−	47.8961i		−	2.51389i
$364$		0			0
$365$		−	3.12871i		−	0.163764i
$366$		0			0
$367$	1.79123			0.0935016			0.0467508	−	0.998907i	$-0.485113\pi$
$367$	1.79123			0.0935016			0.0467508	+	0.998907i	$0.485113\pi$
$368$		0			0
$369$	−28.2325			−1.46973
$370$		0			0
$371$		−	13.2936i		−	0.690167i
$372$		0			0
$373$		−	1.80023i		−	0.0932125i	−0.998913	−	0.0466063i	$-0.985159\pi$
$373$		−	1.80023i		−	0.0932125i	0.998913	−	0.0466063i	$-0.0148406\pi$
$374$		0			0
$375$	−3.11612			−0.160916
$376$		0			0
$377$	−23.4548			−1.20798
$378$		0			0
$379$	22.6838			1.16519			0.582594	−	0.812763i	$-0.302038\pi$
$379$	22.6838			1.16519			0.582594	+	0.812763i	$0.302038\pi$
$380$		0			0
$381$	8.46042			0.433441
$382$		0			0
$383$	29.1003			1.48695			0.743477	−	0.668761i	$-0.233175\pi$
$383$	29.1003			1.48695			0.743477	+	0.668761i	$0.233175\pi$
$384$		0			0
$385$		−	7.10750i		−	0.362232i
$386$		0			0
$387$	−18.5749			−0.944213
$388$		0			0
$389$		−	14.3612i		−	0.728142i	−0.931371	−	0.364071i	$-0.881387\pi$
$389$		−	14.3612i		−	0.728142i	0.931371	−	0.364071i	$-0.118613\pi$
$390$		0			0
$391$	−22.5489	+	1.35202i	−1.14035	+	0.0683745i
$392$		0			0
$393$	−14.9176			−0.752494
$394$		0			0
$395$			9.45259i			0.475611i
$396$		0			0
$397$	37.9047			1.90238			0.951191	−	0.308603i	$-0.0998613\pi$
$397$	37.9047			1.90238			0.951191	+	0.308603i	$0.0998613\pi$
$398$		0			0
$399$			36.4244i			1.82350i
$400$		0			0
$401$		−	14.0577i		−	0.702006i	−0.936374	−	0.351003i	$-0.885841\pi$
$401$		−	14.0577i		−	0.702006i	0.936374	−	0.351003i	$-0.114159\pi$
$402$		0			0
$403$			38.5176i			1.91870i
$404$		0			0
$405$			15.8964i			0.789899i
$406$		0			0
$407$			9.32437i			0.462192i
$408$		0			0
$409$	14.1519			0.699765			0.349882	−	0.936794i	$-0.386222\pi$
$409$	14.1519			0.699765			0.349882	+	0.936794i	$0.386222\pi$
$410$		0			0
$411$	23.6426			1.16620
$412$		0			0
$413$			19.3625i			0.952766i
$414$		0			0
$415$		−	13.4154i		−	0.658538i
$416$		0			0
$417$	5.87696			0.287796
$418$		0			0
$419$	−0.307866			−0.0150402			−0.00752012	−	0.999972i	$-0.502394\pi$
$419$	−0.307866			−0.0150402			−0.00752012	+	0.999972i	$0.502394\pi$
$420$		0			0
$421$		−	3.99805i		−	0.194853i	−0.995243	−	0.0974264i	$-0.968939\pi$
$421$		−	3.99805i		−	0.194853i	0.995243	−	0.0974264i	$-0.0310611\pi$
$422$		0			0
$423$			9.69635i			0.471453i
$424$		0			0
$425$		−	4.71022i		−	0.228479i
$426$		0			0
$427$			8.16619i			0.395189i
$428$		0			0
$429$		−	91.3747i		−	4.41161i
$430$		0			0
$431$	−7.76849			−0.374195			−0.187098	−	0.982341i	$-0.559908\pi$
$431$	−7.76849			−0.374195			−0.187098	+	0.982341i	$0.559908\pi$
$432$		0			0
$433$			8.86580i			0.426063i	0.977045	+	0.213032i	$0.0683337\pi$
$433$			8.86580i			0.426063i	−0.977045	+	0.213032i	$0.931666\pi$
$434$		0			0
$435$	−12.7995			−0.613688
$436$		0			0
$437$	2.42416	+	40.4300i	0.115963	+	1.93403i
$438$		0			0
$439$		−	22.9290i		−	1.09434i	−0.837022	−	0.547170i	$-0.815705\pi$
$439$		−	22.9290i		−	1.09434i	0.837022	−	0.547170i	$-0.184295\pi$
$440$		0			0
$441$	34.1171			1.62462
$442$		0			0
$443$		−	1.94225i		−	0.0922790i	−0.998935	−	0.0461395i	$-0.985308\pi$
$443$		−	1.94225i		−	0.0922790i	0.998935	−	0.0461395i	$-0.0146919\pi$
$444$		0			0
$445$	13.4204			0.636190
$446$		0			0
$447$	18.1772			0.859750
$448$		0			0
$449$	−36.2788			−1.71210			−0.856052	−	0.516890i	$-0.827090\pi$
$449$	−36.2788			−1.71210			−0.856052	+	0.516890i	$0.827090\pi$
$450$		0			0
$451$	21.6058			1.01738
$452$		0			0
$453$	60.1752			2.82728
$454$		0			0
$455$		−	7.90336i		−	0.370515i
$456$		0			0
$457$		−	18.1881i		−	0.850805i	−0.905004	−	0.425403i	$-0.860133\pi$
$457$		−	18.1881i		−	0.850805i	0.905004	−	0.425403i	$-0.139867\pi$
$458$		0			0
$459$	−54.4572			−2.54185
$460$		0			0
$461$	−24.9176			−1.16053			−0.580264	−	0.814428i	$-0.697051\pi$
$461$	−24.9176			−1.16053			−0.580264	+	0.814428i	$0.697051\pi$
$462$		0			0
$463$		−	1.76432i		−	0.0819949i	−0.999159	−	0.0409975i	$-0.986946\pi$
$463$		−	1.76432i		−	0.0819949i	0.999159	−	0.0409975i	$-0.0130536\pi$
$464$		0			0
$465$			21.0194i			0.974753i
$466$		0			0
$467$	−12.4533			−0.576268			−0.288134	−	0.957590i	$-0.593035\pi$
$467$	−12.4533			−0.576268			−0.288134	+	0.957590i	$0.593035\pi$
$468$		0			0
$469$	−18.5679			−0.857387
$470$		0			0
$471$	51.7599			2.38497
$472$		0			0
$473$	14.2150			0.653607
$474$		0			0
$475$	−8.44538			−0.387501
$476$		0			0
$477$		−	64.4495i		−	2.95094i
$478$		0			0
$479$	−27.2674			−1.24588			−0.622939	−	0.782270i	$-0.714062\pi$
$479$	−27.2674			−1.24588			−0.622939	+	0.782270i	$0.714062\pi$
$480$		0			0
$481$			10.3685i			0.472761i
$482$		0			0
$483$	−20.6470	+	1.23798i	−0.939473	+	0.0563301i
$484$		0			0
$485$	4.26857			0.193826
$486$		0			0
$487$		−	37.2566i		−	1.68826i	−0.536139	−	0.844130i	$-0.680118\pi$
$487$		−	37.2566i		−	1.68826i	0.536139	−	0.844130i	$-0.319882\pi$
$488$		0			0
$489$	−61.0347			−2.76008
$490$		0			0
$491$		−	10.3504i		−	0.467109i	−0.972344	−	0.233554i	$-0.924964\pi$
$491$		−	10.3504i		−	0.467109i	0.972344	−	0.233554i	$-0.0750357\pi$
$492$		0			0
$493$		−	19.3473i		−	0.871356i
$494$		0			0
$495$		−	34.4584i		−	1.54879i
$496$		0			0
$497$			5.46042i			0.244933i
$498$		0			0
$499$		−	23.0430i		−	1.03155i	−0.856725	−	0.515773i	$-0.827505\pi$
$499$		−	23.0430i		−	1.03155i	0.856725	−	0.515773i	$-0.172495\pi$
$500$		0			0
$501$	−29.6354			−1.32401
$502$		0			0
$503$	3.12594			0.139379			0.0696893	−	0.997569i	$-0.477799\pi$
$503$	3.12594			0.139379			0.0696893	+	0.997569i	$0.477799\pi$
$504$		0			0
$505$			1.18990i			0.0529497i
$506$		0			0
$507$		−	61.0966i		−	2.71340i
$508$		0			0
$509$	−5.11738			−0.226824			−0.113412	−	0.993548i	$-0.536178\pi$
$509$	−5.11738			−0.226824			−0.113412	+	0.993548i	$0.536178\pi$
$510$		0			0
$511$	4.33036			0.191564
$512$		0			0
$513$			97.6413i			4.31097i
$514$		0			0
$515$		−	11.7555i		−	0.518011i
$516$		0			0
$517$		−	7.42044i		−	0.326351i
$518$		0			0
$519$		−	41.5085i		−	1.82202i
$520$		0			0
$521$		−	7.43178i		−	0.325592i	−0.986660	−	0.162796i	$-0.947949\pi$
$521$		−	7.43178i		−	0.325592i	0.986660	−	0.162796i	$-0.0520512\pi$
$522$		0			0
$523$	−7.89037			−0.345022			−0.172511	−	0.985008i	$-0.555188\pi$
$523$	−7.89037			−0.345022			−0.172511	+	0.985008i	$0.555188\pi$
$524$		0			0
$525$		−	4.31294i		−	0.188232i
$526$		0			0
$527$	−31.7723			−1.38402
$528$		0			0
$529$	−22.8352	+	2.74825i	−0.992836	+	0.119489i
$530$		0			0
$531$			93.8727i			4.07373i
$532$		0			0
$533$	24.0251			1.04064
$534$		0			0
$535$		−	13.2936i		−	0.574730i
$536$		0			0
$537$	2.33414			0.100726
$538$		0			0
$539$	−26.1092			−1.12460
$540$		0			0
$541$	30.7528			1.32217			0.661084	−	0.750312i	$-0.270097\pi$
$541$	30.7528			1.32217			0.661084	+	0.750312i	$0.270097\pi$
$542$		0			0
$543$	−64.8045			−2.78103
$544$		0			0
$545$	4.58151			0.196250
$546$		0			0
$547$			13.8904i			0.593912i	0.954891	+	0.296956i	$0.0959715\pi$
$547$			13.8904i			0.593912i	−0.954891	+	0.296956i	$0.904029\pi$
$548$		0			0
$549$			39.5911i			1.68971i
$550$		0			0
$551$	−34.6894			−1.47782
$552$		0			0
$553$	−13.0831			−0.556348
$554$		0			0
$555$			5.65817i			0.240176i
$556$		0			0
$557$		−	8.97489i		−	0.380278i	−0.981757	−	0.190139i	$-0.939106\pi$
$557$		−	8.97489i		−	0.380278i	0.981757	−	0.190139i	$-0.0608939\pi$
$558$		0			0
$559$	15.8067			0.668553
$560$		0			0
$561$	75.3727			3.18224
$562$		0			0
$563$	18.3356			0.772752			0.386376	−	0.922341i	$-0.373727\pi$
$563$	18.3356			0.772752			0.386376	+	0.922341i	$0.373727\pi$
$564$		0			0
$565$	−10.6103			−0.446380
$566$		0			0
$567$	−22.0018			−0.923987
$568$		0			0
$569$			28.8448i			1.20924i	0.796515	+	0.604618i	$0.206674\pi$
$569$			28.8448i			1.20924i	−0.796515	+	0.604618i	$0.793326\pi$
$570$		0			0
$571$	−28.1139			−1.17653			−0.588265	−	0.808668i	$-0.700189\pi$
$571$	−28.1139			−1.17653			−0.588265	+	0.808668i	$0.700189\pi$
$572$		0			0
$573$		−	47.0870i		−	1.96709i
$574$		0			0
$575$	−0.287039	−	4.78723i	−0.0119704	−	0.199641i
$576$		0			0
$577$	−15.4661			−0.643862			−0.321931	−	0.946763i	$-0.604332\pi$
$577$	−15.4661			−0.643862			−0.321931	+	0.946763i	$0.604332\pi$
$578$		0			0
$579$			37.1544i			1.54408i
$580$		0			0
$581$	18.5679			0.770327
$582$		0			0
$583$			49.3220i			2.04271i
$584$		0			0
$585$		−	38.3168i		−	1.58421i
$586$		0			0
$587$		−	17.8769i		−	0.737860i	−0.929457	−	0.368930i	$-0.879724\pi$
$587$		−	17.8769i		−	0.737860i	0.929457	−	0.368930i	$-0.120276\pi$
$588$		0			0
$589$			56.9673i			2.34730i
$590$		0			0
$591$			29.9056i			1.23015i
$592$		0			0
$593$	−15.2557			−0.626475			−0.313237	−	0.949675i	$-0.601414\pi$
$593$	−15.2557			−0.626475			−0.313237	+	0.949675i	$0.601414\pi$
$594$		0			0
$595$	6.51928			0.267265
$596$		0			0
$597$		−	8.24609i		−	0.337490i
$598$		0			0
$599$			25.7916i			1.05382i	0.849922	+	0.526908i	$0.176649\pi$
$599$			25.7916i			1.05382i	−0.849922	+	0.526908i	$0.823351\pi$
$600$		0			0
$601$	2.09422			0.0854249			0.0427124	−	0.999087i	$-0.486400\pi$
$601$	2.09422			0.0854249			0.0427124	+	0.999087i	$0.486400\pi$
$602$		0			0
$603$	−90.0205			−3.66592
$604$		0			0
$605$			15.3704i			0.624896i
$606$		0			0
$607$		−	25.0509i		−	1.01678i	−0.861126	−	0.508392i	$-0.830240\pi$
$607$		−	25.0509i		−	1.01678i	0.861126	−	0.508392i	$-0.169760\pi$
$608$		0			0
$609$		−	17.7154i		−	0.717865i
$610$		0			0
$611$		−	8.25134i		−	0.333813i
$612$		0			0
$613$		−	29.3741i		−	1.18641i	−0.805052	−	0.593205i	$-0.797862\pi$
$613$		−	29.3741i		−	1.18641i	0.805052	−	0.593205i	$-0.202138\pi$
$614$		0			0
$615$	13.1107			0.528676
$616$		0			0
$617$			25.8308i			1.03991i	0.854194	+	0.519955i	$0.174051\pi$
$617$			25.8308i			1.03991i	−0.854194	+	0.519955i	$0.825949\pi$
$618$		0			0
$619$	−1.51718			−0.0609805			−0.0304903	−	0.999535i	$-0.509707\pi$
$619$	−1.51718			−0.0609805			−0.0304903	+	0.999535i	$0.509707\pi$
$620$		0			0
$621$	−55.3476	+	3.31861i	−2.22102	+	0.133171i
$622$		0			0
$623$			18.5749i			0.744186i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		−	135.143i		−	5.39708i
$628$		0			0
$629$	−8.55269			−0.341018
$630$		0			0
$631$	12.9588			0.515881			0.257940	−	0.966161i	$-0.416956\pi$
$631$	12.9588			0.515881			0.257940	+	0.966161i	$0.416956\pi$
$632$		0			0
$633$	−13.3780			−0.531729
$634$		0			0
$635$	−2.71505			−0.107743
$636$		0			0
$637$	−29.0327			−1.15032
$638$		0			0
$639$			26.4731i			1.04726i
$640$		0			0
$641$			30.2500i			1.19480i	0.801942	+	0.597401i	$0.203800\pi$
$641$			30.2500i			1.19480i	−0.801942	+	0.597401i	$0.796200\pi$
$642$		0			0
$643$	−19.2977			−0.761028			−0.380514	−	0.924775i	$-0.624253\pi$
$643$	−19.2977			−0.761028			−0.380514	+	0.924775i	$0.624253\pi$
$644$		0			0
$645$	8.62588			0.339644
$646$		0			0
$647$			33.1803i			1.30445i	0.758024	+	0.652226i	$0.226165\pi$
$647$			33.1803i			1.30445i	−0.758024	+	0.652226i	$0.773835\pi$
$648$		0			0
$649$		−	71.8391i		−	2.81993i
$650$		0			0
$651$	−29.0924			−1.14022
$652$		0			0
$653$	−6.28978			−0.246138			−0.123069	−	0.992398i	$-0.539274\pi$
$653$	−6.28978			−0.246138			−0.123069	+	0.992398i	$0.539274\pi$
$654$		0			0
$655$	4.78723			0.187053
$656$		0			0
$657$	20.9943			0.819067
$658$		0			0
$659$	−17.3430			−0.675586			−0.337793	−	0.941220i	$-0.609680\pi$
$659$	−17.3430			−0.675586			−0.337793	+	0.941220i	$0.609680\pi$
$660$		0			0
$661$			12.4036i			0.482445i	0.970470	+	0.241222i	$0.0775483\pi$
$661$			12.4036i			0.482445i	−0.970470	+	0.241222i	$0.922452\pi$
$662$		0			0
$663$	83.8125			3.25501
$664$		0			0
$665$		−	11.6890i		−	0.453281i
$666$		0			0
$667$	−1.17901	−	19.6636i	−0.0456516	−	0.761377i
$668$		0			0
$669$	−9.00567			−0.348179
$670$		0			0
$671$		−	30.2984i		−	1.16965i
$672$		0			0
$673$	−0.656283			−0.0252978			−0.0126489	−	0.999920i	$-0.504026\pi$
$673$	−0.656283			−0.0252978			−0.0126489	+	0.999920i	$0.504026\pi$
$674$		0			0
$675$		−	11.5615i		−	0.445002i
$676$		0			0
$677$			14.8214i			0.569634i	0.958582	+	0.284817i	$0.0919329\pi$
$677$			14.8214i			0.569634i	−0.958582	+	0.284817i	$0.908067\pi$
$678$		0			0
$679$			5.90801i			0.226729i
$680$		0			0
$681$		−	42.7985i		−	1.64004i
$682$		0			0
$683$			4.47598i			0.171269i	0.996327	+	0.0856344i	$0.0272917\pi$
$683$			4.47598i			0.171269i	−0.996327	+	0.0856344i	$0.972708\pi$
$684$		0			0
$685$	−7.58718			−0.289891
$686$		0			0
$687$	56.7602			2.16554
$688$		0			0
$689$			54.8448i			2.08942i
$690$		0			0
$691$			3.46410i			0.131781i	0.997827	+	0.0658903i	$0.0209887\pi$
$691$			3.46410i			0.131781i	−0.997827	+	0.0658903i	$0.979011\pi$
$692$		0			0
$693$	47.6929			1.81170
$694$		0			0
$695$	−1.88598			−0.0715394
$696$		0			0
$697$			19.8177i			0.750650i
$698$		0			0
$699$			68.8060i			2.60248i
$700$		0			0
$701$			9.24980i			0.349360i	0.984625	+	0.174680i	$0.0558891\pi$
$701$			9.24980i			0.349360i	−0.984625	+	0.174680i	$0.944111\pi$
$702$		0			0
$703$			15.3349i			0.578366i
$704$		0			0
$705$		−	4.50283i		−	0.169587i
$706$		0			0
$707$	−1.64690			−0.0619381
$708$		0			0
$709$		−	34.6073i		−	1.29970i	−0.760061	−	0.649852i	$-0.774831\pi$
$709$		−	34.6073i		−	1.29970i	0.760061	−	0.649852i	$-0.225169\pi$
$710$		0			0
$711$	−63.4290			−2.37877
$712$		0			0
$713$	−32.2917	+	1.93619i	−1.20933	+	0.0725109i
$714$		0			0
$715$			29.3232i			1.09663i
$716$		0			0
$717$	−27.0863			−1.01156
$718$		0			0
$719$			31.5572i			1.17688i	0.808540	+	0.588442i	$0.200258\pi$
$719$			31.5572i			1.17688i	−0.808540	+	0.588442i	$0.799742\pi$
$720$		0			0
$721$	16.2705			0.605946
$722$		0			0
$723$	39.8341			1.48145
$724$		0			0
$725$	4.10750			0.152549
$726$		0			0
$727$	−3.15799			−0.117123			−0.0585617	−	0.998284i	$-0.518651\pi$
$727$	−3.15799			−0.117123			−0.0585617	+	0.998284i	$0.518651\pi$
$728$		0			0
$729$	1.41282			0.0523267
$730$		0			0
$731$			13.0386i			0.482249i
$732$		0			0
$733$		−	39.8349i		−	1.47134i	−0.677342	−	0.735668i	$-0.736868\pi$
$733$		−	39.8349i		−	1.47134i	0.677342	−	0.735668i	$-0.263132\pi$
$734$		0			0
$735$	−15.8434			−0.584394
$736$		0			0
$737$	68.8911			2.53764
$738$		0			0
$739$			20.7912i			0.764815i	0.923994	+	0.382408i	$0.124905\pi$
$739$			20.7912i			0.764815i	−0.923994	+	0.382408i	$0.875095\pi$
$740$		0			0
$741$		−	150.275i		−	5.52049i
$742$		0			0
$743$	−41.6592			−1.52833			−0.764163	−	0.645023i	$-0.776848\pi$
$743$	−41.6592			−1.52833			−0.764163	+	0.645023i	$0.776848\pi$
$744$		0			0
$745$	−5.83326			−0.213714
$746$		0			0
$747$	90.0205			3.29368
$748$		0			0
$749$	18.3992			0.672293
$750$		0			0
$751$	28.1430			1.02695			0.513476	−	0.858104i	$-0.328357\pi$
$751$	28.1430			1.02695			0.513476	+	0.858104i	$0.328357\pi$
$752$		0			0
$753$		−	58.5008i		−	2.13189i
$754$		0			0
$755$	−19.3109			−0.702797
$756$		0			0
$757$			2.01554i			0.0732560i	0.999329	+	0.0366280i	$0.0116617\pi$
$757$			2.01554i			0.0732560i	−0.999329	+	0.0366280i	$0.988338\pi$
$758$		0			0
$759$	76.6051	−	4.59319i	2.78059	−	0.166722i
$760$		0			0
$761$	26.4316			0.958145			0.479072	−	0.877775i	$-0.340973\pi$
$761$	26.4316			0.958145			0.479072	+	0.877775i	$0.340973\pi$
$762$		0			0
$763$			6.34114i			0.229565i
$764$		0			0
$765$	31.6066			1.14274
$766$		0			0
$767$		−	79.8832i		−	2.88442i
$768$		0			0
$769$		−	37.5778i		−	1.35509i	−0.735481	−	0.677545i	$-0.763044\pi$
$769$		−	37.5778i		−	1.35509i	0.735481	−	0.677545i	$-0.236956\pi$
$770$		0			0
$771$		−	55.8511i		−	2.01143i
$772$		0			0
$773$			39.0096i			1.40308i	0.712632	+	0.701538i	$0.247503\pi$
$773$			39.0096i			1.40308i	−0.712632	+	0.701538i	$0.752497\pi$
$774$		0			0
$775$		−	6.74538i		−	0.242301i
$776$		0			0
$777$	−7.83131			−0.280947
$778$		0			0
$779$	35.5330			1.27310
$780$		0			0
$781$		−	20.2594i		−	0.724937i
$782$		0			0
$783$		−	47.4889i		−	1.69712i
$784$		0			0
$785$	−16.6103			−0.592848
$786$		0			0
$787$	−36.2081			−1.29068			−0.645340	−	0.763895i	$-0.723284\pi$
$787$	−36.2081			−1.29068			−0.645340	+	0.763895i	$0.723284\pi$
$788$		0			0
$789$			33.6685i			1.19863i
$790$		0			0
$791$		−	14.6855i		−	0.522155i
$792$		0			0
$793$		−	33.6910i		−	1.19640i
$794$		0			0
$795$			29.9293i			1.06148i
$796$		0			0
$797$		−	14.3643i		−	0.508808i	−0.967098	−	0.254404i	$-0.918121\pi$
$797$		−	14.3643i		−	0.508808i	0.967098	−	0.254404i	$-0.0818793\pi$
$798$		0			0
$799$	6.80632			0.240790
$800$		0			0
$801$			90.0541i			3.18191i
$802$		0			0
$803$	−16.0666			−0.566978
$804$		0			0
$805$	6.62588	−	0.397283i	0.233531	−	0.0140024i
$806$		0			0
$807$			55.7674i			1.96311i
$808$		0			0
$809$	9.70455			0.341194			0.170597	−	0.985341i	$-0.445430\pi$
$809$	9.70455			0.341194			0.170597	+	0.985341i	$0.445430\pi$
$810$		0			0
$811$			32.5534i			1.14310i	0.820566	+	0.571552i	$0.193658\pi$
$811$			32.5534i			1.14310i	−0.820566	+	0.571552i	$0.806342\pi$
$812$		0			0
$813$	−73.1691			−2.56615
$814$		0			0
$815$	19.5867			0.686094
$816$		0			0
$817$	23.3780			0.817894
$818$		0			0
$819$	53.0333			1.85313
$820$		0			0
$821$	48.5524			1.69449			0.847245	−	0.531202i	$-0.178260\pi$
$821$	48.5524			1.69449			0.847245	+	0.531202i	$0.178260\pi$
$822$		0			0
$823$			36.8269i			1.28370i	0.766828	+	0.641852i	$0.221834\pi$
$823$			36.8269i			1.28370i	−0.766828	+	0.641852i	$0.778166\pi$
$824$		0			0
$825$			16.0020i			0.557117i
$826$		0			0
$827$	29.1227			1.01270			0.506348	−	0.862329i	$-0.330995\pi$
$827$	29.1227			1.01270			0.506348	+	0.862329i	$0.330995\pi$
$828$		0			0
$829$	−28.2305			−0.980487			−0.490244	−	0.871585i	$-0.663092\pi$
$829$	−28.2305			−0.980487			−0.490244	+	0.871585i	$0.663092\pi$
$830$		0			0
$831$			4.00754i			0.139020i
$832$		0			0
$833$		−	23.9484i		−	0.829762i
$834$		0			0
$835$	9.51036			0.329120
$836$		0			0
$837$	−77.9868			−2.69562
$838$		0			0
$839$	24.8228			0.856978			0.428489	−	0.903547i	$-0.359046\pi$
$839$	24.8228			0.856978			0.428489	+	0.903547i	$0.359046\pi$
$840$		0			0
$841$	−12.1284			−0.418221
$842$		0			0
$843$	45.5882			1.57014
$844$		0			0
$845$			19.6066i			0.674488i
$846$		0			0
$847$	−21.2738			−0.730975
$848$		0			0
$849$		−	14.9672i		−	0.513672i
$850$		0			0
$851$	−8.69252	+	0.521198i	−0.297976	+	0.0178664i
$852$		0			0
$853$	43.8333			1.50082			0.750411	−	0.660971i	$-0.229856\pi$
$853$	43.8333			1.50082			0.750411	+	0.660971i	$0.229856\pi$
$854$		0			0
$855$		−	56.6704i		−	1.93809i
$856$		0			0
$857$	−32.8328			−1.12155			−0.560773	−	0.827969i	$-0.689496\pi$
$857$	−32.8328			−1.12155			−0.560773	+	0.827969i	$0.689496\pi$
$858$		0			0
$859$			24.2489i			0.827363i	0.910422	+	0.413681i	$0.135757\pi$
$859$			24.2489i			0.827363i	−0.910422	+	0.413681i	$0.864243\pi$
$860$		0			0
$861$			18.1462i			0.618421i
$862$		0			0
$863$		−	16.3698i		−	0.557234i	−0.960402	−	0.278617i	$-0.910124\pi$
$863$		−	16.3698i		−	0.557234i	0.960402	−	0.278617i	$-0.0898760\pi$
$864$		0			0
$865$			13.3206i			0.452913i
$866$		0			0
$867$			16.1608i			0.548849i
$868$		0			0
$869$	48.5411			1.64664
$870$		0			0
$871$	76.6051			2.59566
$872$		0			0
$873$			28.6431i			0.969421i
$874$		0			0
$875$			1.38407i			0.0467902i
$876$		0			0
$877$	−1.32056			−0.0445921			−0.0222960	−	0.999751i	$-0.507098\pi$
$877$	−1.32056			−0.0445921			−0.0222960	+	0.999751i	$0.507098\pi$
$878$		0			0
$879$	30.5518			1.03049
$880$		0			0
$881$		−	0.188433i		−	0.00634848i	−0.999995	−	0.00317424i	$-0.998990\pi$
$881$		−	0.188433i		−	0.00634848i	0.999995	−	0.00317424i	$-0.00101039\pi$
$882$		0			0
$883$			11.4152i			0.384152i	0.981380	+	0.192076i	$0.0615220\pi$
$883$			11.4152i			0.384152i	−0.981380	+	0.192076i	$0.938478\pi$
$884$		0			0
$885$		−	43.5930i		−	1.46536i
$886$		0			0
$887$			7.34694i			0.246686i	0.992364	+	0.123343i	$0.0393615\pi$
$887$			7.34694i			0.246686i	−0.992364	+	0.123343i	$0.960638\pi$
$888$		0			0
$889$		−	3.75782i		−	0.126033i
$890$		0			0
$891$	81.6314			2.73476
$892$		0			0
$893$		−	12.2037i		−	0.408380i
$894$		0			0
$895$	−0.749054			−0.0250381
$896$		0			0
$897$	85.1828	−	5.10750i	2.84417	−	0.170535i
$898$		0			0
$899$		−	27.7067i		−	0.924070i
$900$		0			0
$901$	−45.2401			−1.50717
$902$		0			0
$903$			11.9388i			0.397299i
$904$		0			0
$905$	20.7965			0.691300
$906$		0			0
$907$	−48.8008			−1.62040			−0.810202	−	0.586150i	$-0.800643\pi$
$907$	−48.8008			−1.62040			−0.810202	+	0.586150i	$0.800643\pi$
$908$		0			0
$909$	−7.98446			−0.264828
$910$		0			0
$911$	−29.6250			−0.981519			−0.490760	−	0.871295i	$-0.663281\pi$
$911$	−29.6250			−0.981519			−0.490760	+	0.871295i	$0.663281\pi$
$912$		0			0
$913$	−68.8911			−2.27996
$914$		0			0
$915$		−	18.3855i		−	0.607805i
$916$		0			0
$917$			6.62588i			0.218806i
$918$		0			0
$919$	−38.2691			−1.26238			−0.631190	−	0.775628i	$-0.717433\pi$
$919$	−38.2691			−1.26238			−0.631190	+	0.775628i	$0.717433\pi$
$920$		0			0
$921$	−40.1396			−1.32264
$922$		0			0
$923$		−	22.5279i		−	0.741514i
$924$		0			0
$925$		−	1.81577i		−	0.0597022i
$926$		0			0
$927$	78.8823			2.59084
$928$		0			0
$929$	12.5223			0.410843			0.205421	−	0.978674i	$-0.434144\pi$
$929$	12.5223			0.410843			0.205421	+	0.978674i	$0.434144\pi$
$930$		0			0
$931$	−42.9392			−1.40728
$932$		0			0
$933$	2.53391			0.0829565
$934$		0			0
$935$	−24.1880			−0.791032
$936$		0			0
$937$		−	18.4928i		−	0.604133i	−0.953287	−	0.302066i	$-0.902324\pi$
$937$		−	18.4928i		−	0.604133i	0.953287	−	0.302066i	$-0.0976764\pi$
$938$		0			0
$939$	−59.4228			−1.93919
$940$		0			0
$941$			16.2725i			0.530468i	0.964184	+	0.265234i	$0.0854492\pi$
$941$			16.2725i			0.530468i	−0.964184	+	0.265234i	$0.914551\pi$
$942$		0			0
$943$	1.20769	+	20.1417i	0.0393276	+	0.655906i
$944$		0			0
$945$	16.0020			0.520544
$946$		0			0
$947$		−	33.0330i		−	1.07343i	−0.843764	−	0.536715i	$-0.819665\pi$
$947$		−	33.0330i		−	1.07343i	0.843764	−	0.536715i	$-0.180335\pi$
$948$		0			0
$949$	−17.8656			−0.579943
$950$		0			0
$951$			26.8854i			0.871818i
$952$		0			0
$953$			19.1558i			0.620516i	0.950652	+	0.310258i	$0.100416\pi$
$953$			19.1558i			0.620516i	−0.950652	+	0.310258i	$0.899584\pi$
$954$		0			0
$955$			15.1108i			0.488973i
$956$		0			0
$957$			65.7281i			2.12469i
$958$		0			0
$959$		−	10.5012i		−	0.339101i
$960$		0			0
$961$	−14.5002			−0.467749
$962$		0			0
$963$	89.2027			2.87452
$964$		0			0
$965$		−	11.9233i		−	0.383824i
$966$		0			0
$967$		−	6.05093i		−	0.194585i	−0.995256	−	0.0972925i	$-0.968982\pi$
$967$		−	6.05093i		−	0.194585i	0.995256	−	0.0972925i	$-0.0310182\pi$
$968$		0			0
$969$	123.958			3.98211
$970$		0			0
$971$	−26.6158			−0.854141			−0.427071	−	0.904218i	$-0.640454\pi$
$971$	−26.6158			−0.854141			−0.427071	+	0.904218i	$0.640454\pi$
$972$		0			0
$973$		−	2.61034i		−	0.0836836i
$974$		0			0
$975$			17.7937i			0.569856i
$976$		0			0
$977$		−	9.69535i		−	0.310182i	−0.987900	−	0.155091i	$-0.950433\pi$
$977$		−	9.69535i		−	0.310182i	0.987900	−	0.155091i	$-0.0495671\pi$
$978$		0			0
$979$		−	68.9168i		−	2.20259i
$980$		0			0
$981$			30.7430i			0.981547i
$982$		0			0
$983$	−39.4218			−1.25736			−0.628680	−	0.777664i	$-0.716405\pi$
$983$	−39.4218			−1.25736			−0.628680	+	0.777664i	$0.716405\pi$
$984$		0			0
$985$		−	9.59705i		−	0.305787i
$986$		0			0
$987$	6.23225			0.198375
$988$		0			0
$989$	0.794566	+	13.2518i	0.0252657	+	0.421381i
$990$		0			0
$991$		−	46.3661i		−	1.47287i	−0.676509	−	0.736434i	$-0.736508\pi$
$991$		−	46.3661i		−	1.47287i	0.676509	−	0.736434i	$-0.263492\pi$
$992$		0			0
$993$	−49.3052			−1.56465
$994$		0			0
$995$			2.64626i			0.0838922i
$996$		0			0
$997$	28.4724			0.901731			0.450865	−	0.892592i	$-0.351115\pi$
$997$	28.4724			0.901731			0.450865	+	0.892592i	$0.351115\pi$
$998$		0			0
$999$	−20.9931			−0.664191

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.2.i.a.1471.2	yes	16
4.3	odd	2	inner	1840.2.i.a.1471.16	yes	16
23.22	odd	2	inner	1840.2.i.a.1471.1	✓	16
92.91	even	2	inner	1840.2.i.a.1471.15	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1840.2.i.a.1471.1	✓	16	23.22	odd	2	inner
1840.2.i.a.1471.2	yes	16	1.1	even	1	trivial
1840.2.i.a.1471.15	yes	16	92.91	even	2	inner
1840.2.i.a.1471.16	yes	16	4.3	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1840.i (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.11612i q^{3} +1.00000i q^{5} -1.38407 q^{7} -6.71022 q^{9} +O(q^{10})\)	\(q-3.11612i q^{3} +1.00000i q^{5} -1.38407 q^{7} -6.71022 q^{9} +5.13521 q^{11} +5.71022 q^{13} +3.11612 q^{15} +4.71022i q^{17} +8.44538 q^{19} +4.31294i q^{21} +(0.287039 + 4.78723i) q^{23} -1.00000 q^{25} +11.5615i q^{27} -4.10750 q^{29} +6.74538i q^{31} -16.0020i q^{33} -1.38407i q^{35} +1.81577i q^{37} -17.7937i q^{39} +4.20739 q^{41} +2.76814 q^{43} -6.71022i q^{45} -1.44501i q^{47} -5.08434 q^{49} +14.6776 q^{51} +9.60467i q^{53} +5.13521i q^{55} -26.3168i q^{57} -13.9895i q^{59} -5.90012i q^{61} +9.28743 q^{63} +5.71022i q^{65} +13.4154 q^{67} +(14.9176 - 0.894449i) q^{69} -3.94519i q^{71} -3.12871 q^{73} +3.11612i q^{75} -7.10750 q^{77} +9.45259 q^{79} +15.8964 q^{81} -13.4154 q^{83} -4.71022 q^{85} +12.7995i q^{87} -13.4204i q^{89} -7.90336 q^{91} +21.0194 q^{93} +8.44538i q^{95} -4.26857i q^{97} -34.4584 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 32 q^{9}+O(q^{10}) \) 16 * q - 32 * q^9	\( 16 q - 32 q^{9} + 16 q^{13} - 16 q^{25} + 72 q^{29} - 4 q^{41} - 32 q^{49} + 92 q^{69} - 20 q^{73} + 24 q^{77} + 60 q^{93}+O(q^{100}) \) 16 * q - 32 * q^9 + 16 * q^13 - 16 * q^25 + 72 * q^29 - 4 * q^41 - 32 * q^49 + 92 * q^69 - 20 * q^73 + 24 * q^77 + 60 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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