LMFDB - Embedded newform 1840.2.i.c.1471.8

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			1471.8
Root			$-0.650065 - 0.125946i$ of defining polynomial
Character	$\chi$	$=$	1840.1471
Dual form			1840.2.i.c.1471.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.431920i q^{3} +1.00000i q^{5} +3.33035 q^{7} +2.81344 q^{9} +O(q^{10})$	$q-0.431920i q^{3} +1.00000i q^{5} +3.33035 q^{7} +2.81344 q^{9} +1.49387 q^{11} +5.60770 q^{13} +0.431920 q^{15} -0.587466i q^{17} -0.133749 q^{19} -1.43845i q^{21} +(-3.96039 + 2.70469i) q^{23} -1.00000 q^{25} -2.51095i q^{27} -1.54132 q^{29} -1.79205i q^{31} -0.645235i q^{33} +3.33035i q^{35} +5.62689i q^{37} -2.42208i q^{39} -1.87121 q^{41} -5.67688 q^{43} +2.81344i q^{45} -6.73883i q^{47} +4.09125 q^{49} -0.253738 q^{51} -0.671148i q^{53} +1.49387i q^{55} +0.0577688i q^{57} -3.46410i q^{59} +13.7510i q^{61} +9.36976 q^{63} +5.60770i q^{65} -3.10774 q^{67} +(1.16821 + 1.71057i) q^{69} -12.2043i q^{71} +15.3760 q^{73} +0.431920i q^{75} +4.97513 q^{77} +8.78461 q^{79} +7.35581 q^{81} -11.3849 q^{83} +0.587466 q^{85} +0.665728i q^{87} +1.49622i q^{89} +18.6756 q^{91} -0.774021 q^{93} -0.133749i q^{95} -1.91632i q^{97} +4.20293 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$		−	0.431920i		−	0.249369i	−0.992196	−	0.124685i	$-0.960208\pi$
$3$		−	0.431920i		−	0.249369i	0.992196	−	0.124685i	$-0.0397919\pi$
$4$		0			0
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$	3.33035			1.25876			0.629378	−	0.777100i	$-0.283310\pi$
$7$	3.33035			1.25876			0.629378	+	0.777100i	$0.283310\pi$
$8$		0			0
$9$	2.81344			0.937815
$10$		0			0
$11$	1.49387			0.450420			0.225210	−	0.974310i	$-0.427693\pi$
$11$	1.49387			0.450420			0.225210	+	0.974310i	$0.427693\pi$
$12$		0			0
$13$	5.60770			1.55530			0.777648	−	0.628700i	$-0.216413\pi$
$13$	5.60770			1.55530			0.777648	+	0.628700i	$0.216413\pi$
$14$		0			0
$15$	0.431920			0.111521
$16$		0			0
$17$		−	0.587466i		−	0.142481i	−0.997459	−	0.0712407i	$-0.977304\pi$
$17$		−	0.587466i		−	0.142481i	0.997459	−	0.0712407i	$-0.0226958\pi$
$18$		0			0
$19$	−0.133749			−0.0306841			−0.0153420	−	0.999882i	$-0.504884\pi$
$19$	−0.133749			−0.0306841			−0.0153420	+	0.999882i	$0.504884\pi$
$20$		0			0
$21$		−	1.43845i		−	0.313895i
$22$		0			0
$23$	−3.96039	+	2.70469i	−0.825798	+	0.563967i
$23$	−3.96039	+	2.70469i	−0.825798	+	0.563967i
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$		−	2.51095i		−	0.483232i
$28$		0			0
$29$	−1.54132			−0.286216			−0.143108	−	0.989707i	$-0.545710\pi$
$29$	−1.54132			−0.286216			−0.143108	+	0.989707i	$0.545710\pi$
$30$		0			0
$31$		−	1.79205i		−	0.321861i	−0.986966	−	0.160930i	$-0.948551\pi$
$31$		−	1.79205i		−	0.321861i	0.986966	−	0.160930i	$-0.0514495\pi$
$32$		0			0
$33$		−	0.645235i		−	0.112321i
$34$		0			0
$35$			3.33035i			0.562932i
$36$		0			0
$37$			5.62689i			0.925055i	0.886605	+	0.462527i	$0.153057\pi$
$37$			5.62689i			0.925055i	−0.886605	+	0.462527i	$0.846943\pi$
$38$		0			0
$39$		−	2.42208i		−	0.387843i
$40$		0			0
$41$	−1.87121			−0.292234			−0.146117	−	0.989267i	$-0.546678\pi$
$41$	−1.87121			−0.292234			−0.146117	+	0.989267i	$0.546678\pi$
$42$		0			0
$43$	−5.67688			−0.865715			−0.432858	−	0.901462i	$-0.642495\pi$
$43$	−5.67688			−0.865715			−0.432858	+	0.901462i	$0.642495\pi$
$44$		0			0
$45$			2.81344i			0.419404i
$46$		0			0
$47$		−	6.73883i		−	0.982959i	−0.870889	−	0.491480i	$-0.836456\pi$
$47$		−	6.73883i		−	0.982959i	0.870889	−	0.491480i	$-0.163544\pi$
$48$		0			0
$49$	4.09125			0.584464
$50$		0			0
$51$	−0.253738			−0.0355305
$52$		0			0
$53$		−	0.671148i		−	0.0921892i	−0.998937	−	0.0460946i	$-0.985322\pi$
$53$		−	0.671148i		−	0.0921892i	0.998937	−	0.0460946i	$-0.0146776\pi$
$54$		0			0
$55$			1.49387i			0.201434i
$56$		0			0
$57$			0.0577688i			0.00765167i
$58$		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
$61$			13.7510i			1.76064i	0.474381	+	0.880320i	$0.342672\pi$
$61$			13.7510i			1.76064i	−0.474381	+	0.880320i	$0.657328\pi$
$62$		0			0
$63$	9.36976			1.18048
$64$		0			0
$65$			5.60770i			0.695549i
$66$		0			0
$67$	−3.10774			−0.379671			−0.189835	−	0.981816i	$-0.560795\pi$
$67$	−3.10774			−0.379671			−0.189835	+	0.981816i	$0.560795\pi$
$68$		0			0
$69$	1.16821	+	1.71057i	0.140636	+	0.205929i
$70$		0			0
$71$		−	12.2043i		−	1.44838i	−0.689600	−	0.724191i	$-0.742213\pi$
$71$		−	12.2043i		−	1.44838i	0.689600	−	0.724191i	$-0.257787\pi$
$72$		0			0
$73$	15.3760			1.79963			0.899815	−	0.436273i	$-0.143702\pi$
$73$	15.3760			1.79963			0.899815	+	0.436273i	$0.143702\pi$
$74$		0			0
$75$			0.431920i			0.0498739i
$76$		0			0
$77$	4.97513			0.566968
$78$		0			0
$79$	8.78461			0.988346			0.494173	−	0.869364i	$-0.335471\pi$
$79$	8.78461			0.988346			0.494173	+	0.869364i	$0.335471\pi$
$80$		0			0
$81$	7.35581			0.817312
$82$		0			0
$83$	−11.3849			−1.24965			−0.624826	−	0.780764i	$-0.714830\pi$
$83$	−11.3849			−1.24965			−0.624826	+	0.780764i	$0.714830\pi$
$84$		0			0
$85$	0.587466			0.0637196
$86$		0			0
$87$			0.665728i			0.0713735i
$88$		0			0
$89$			1.49622i			0.158599i	0.996851	+	0.0792993i	$0.0252683\pi$
$89$			1.49622i			0.158599i	−0.996851	+	0.0792993i	$0.974732\pi$
$90$		0			0
$91$	18.6756			1.95774
$92$		0			0
$93$	−0.774021			−0.0802622
$94$		0			0
$95$		−	0.133749i		−	0.0137223i
$96$		0			0
$97$		−	1.91632i		−	0.194573i	−0.995256	−	0.0972863i	$-0.968984\pi$
$97$		−	1.91632i		−	0.194573i	0.995256	−	0.0972863i	$-0.0310163\pi$
$98$		0			0
$99$	4.20293			0.422411
$100$		0			0
$101$	−0.825068			−0.0820974			−0.0410487	−	0.999157i	$-0.513070\pi$
$101$	−0.825068			−0.0820974			−0.0410487	+	0.999157i	$0.513070\pi$
$102$		0			0
$103$	9.62350			0.948232			0.474116	−	0.880462i	$-0.342768\pi$
$103$	9.62350			0.948232			0.474116	+	0.880462i	$0.342768\pi$
$104$		0			0
$105$	1.43845			0.140378
$106$		0			0
$107$	−9.38969			−0.907736			−0.453868	−	0.891069i	$-0.649956\pi$
$107$	−9.38969			−0.907736			−0.453868	+	0.891069i	$0.649956\pi$
$108$		0			0
$109$			5.47030i			0.523960i	0.965073	+	0.261980i	$0.0843754\pi$
$109$			5.47030i			0.523960i	−0.965073	+	0.261980i	$0.915625\pi$
$110$		0			0
$111$	2.43037			0.230680
$112$		0			0
$113$			12.7116i			1.19581i	0.801568	+	0.597904i	$0.204000\pi$
$113$			12.7116i			1.19581i	−0.801568	+	0.597904i	$0.796000\pi$
$114$		0			0
$115$	−2.70469	−	3.96039i	−0.252214	−	0.369308i
$116$		0			0
$117$	15.7769			1.45858
$118$		0			0
$119$		−	1.95647i		−	0.179349i
$120$		0			0
$121$	−8.76834			−0.797122
$122$		0			0
$123$			0.808215i			0.0728743i
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$		−	17.1124i		−	1.51848i	−0.650813	−	0.759238i	$-0.725572\pi$
$127$		−	17.1124i		−	1.51848i	0.650813	−	0.759238i	$-0.274428\pi$
$128$		0			0
$129$			2.45196i			0.215883i
$130$		0			0
$131$			17.8475i			1.55934i	0.626190	+	0.779670i	$0.284613\pi$
$131$			17.8475i			1.55934i	−0.626190	+	0.779670i	$0.715387\pi$
$132$		0			0
$133$	−0.445431			−0.0386237
$134$		0			0
$135$	2.51095			0.216108
$136$		0			0
$137$		−	18.7759i		−	1.60413i	−0.597234	−	0.802067i	$-0.703734\pi$
$137$		−	18.7759i		−	1.60413i	0.597234	−	0.802067i	$-0.296266\pi$
$138$		0			0
$139$			5.01115i			0.425040i	0.977157	+	0.212520i	$0.0681671\pi$
$139$			5.01115i			0.425040i	−0.977157	+	0.212520i	$0.931833\pi$
$140$		0			0
$141$	−2.91064			−0.245120
$142$		0			0
$143$	8.37719			0.700536
$144$		0			0
$145$		−	1.54132i		−	0.128000i
$146$		0			0
$147$		−	1.76709i		−	0.145747i
$148$		0			0
$149$		−	10.9837i		−	0.899823i	−0.893073	−	0.449911i	$-0.851456\pi$
$149$		−	10.9837i		−	0.899823i	0.893073	−	0.449911i	$-0.148544\pi$
$150$		0			0
$151$		−	5.27608i		−	0.429361i	−0.976684	−	0.214681i	$-0.931129\pi$
$151$		−	5.27608i		−	0.429361i	0.976684	−	0.214681i	$-0.0688710\pi$
$152$		0			0
$153$		−	1.65280i		−	0.133621i
$154$		0			0
$155$	1.79205			0.143941
$156$		0			0
$157$		−	10.7116i		−	0.854880i	−0.904044	−	0.427440i	$-0.859416\pi$
$157$		−	10.7116i		−	0.854880i	0.904044	−	0.427440i	$-0.140584\pi$
$158$		0			0
$159$	−0.289882			−0.0229892
$160$		0			0
$161$	−13.1895	+	9.00757i	−1.03948	+	0.709896i
$162$		0			0
$163$		−	1.19570i		−	0.0936546i	−0.998903	−	0.0468273i	$-0.985089\pi$
$163$		−	1.19570i		−	0.0936546i	0.998903	−	0.0468273i	$-0.0149111\pi$
$164$		0			0
$165$	0.645235			0.0502314
$166$		0			0
$167$			3.28637i			0.254307i	0.991883	+	0.127153i	$0.0405841\pi$
$167$			3.28637i			0.254307i	−0.991883	+	0.127153i	$0.959416\pi$
$168$		0			0
$169$	18.4463			1.41894
$170$		0			0
$171$	−0.376295			−0.0287760
$172$		0			0
$173$	3.86449			0.293812			0.146906	−	0.989150i	$-0.453069\pi$
$173$	3.86449			0.293812			0.146906	+	0.989150i	$0.453069\pi$
$174$		0			0
$175$	−3.33035			−0.251751
$176$		0			0
$177$	−1.49622			−0.112462
$178$		0			0
$179$			4.52606i			0.338293i	0.985591	+	0.169147i	$0.0541011\pi$
$179$			4.52606i			0.338293i	−0.985591	+	0.169147i	$0.945899\pi$
$180$		0			0
$181$			20.4028i			1.51653i	0.651948	+	0.758264i	$0.273952\pi$
$181$			20.4028i			1.51653i	−0.651948	+	0.758264i	$0.726048\pi$
$182$		0			0
$183$	5.93935			0.439050
$184$		0			0
$185$	−5.62689			−0.413697
$186$		0			0
$187$		−	0.877600i		−	0.0641765i
$188$		0			0
$189$		−	8.36233i		−	0.608270i
$190$		0			0
$191$	7.56441			0.547341			0.273671	−	0.961823i	$-0.411762\pi$
$191$	7.56441			0.547341			0.273671	+	0.961823i	$0.411762\pi$
$192$		0			0
$193$	−13.1682			−0.947869			−0.473934	−	0.880560i	$-0.657167\pi$
$193$	−13.1682			−0.947869			−0.473934	+	0.880560i	$0.657167\pi$
$194$		0			0
$195$	2.42208			0.173449
$196$		0			0
$197$	22.4155			1.59704			0.798518	−	0.601971i	$-0.205618\pi$
$197$	22.4155			1.59704			0.798518	+	0.601971i	$0.205618\pi$
$198$		0			0
$199$	−19.3144			−1.36916			−0.684580	−	0.728938i	$-0.740014\pi$
$199$	−19.3144			−1.36916			−0.684580	+	0.728938i	$0.740014\pi$
$200$		0			0
$201$			1.34230i			0.0946782i
$202$		0			0
$203$	−5.13314			−0.360276
$204$		0			0
$205$		−	1.87121i		−	0.130691i
$206$		0			0
$207$	−11.1423	+	7.60949i	−0.774445	+	0.528896i
$208$		0			0
$209$	−0.199804			−0.0138207
$210$		0			0
$211$			7.44442i			0.512495i	0.966611	+	0.256247i	$0.0824862\pi$
$211$			7.44442i			0.512495i	−0.966611	+	0.256247i	$0.917514\pi$
$212$		0			0
$213$	−5.27128			−0.361182
$214$		0			0
$215$		−	5.67688i		−	0.387160i
$216$		0			0
$217$		−	5.96814i		−	0.405144i
$218$		0			0
$219$		−	6.64122i		−	0.448772i
$220$		0			0
$221$		−	3.29433i		−	0.221601i
$222$		0			0
$223$		−	4.45667i		−	0.298441i	−0.988804	−	0.149220i	$-0.952324\pi$
$223$		−	4.45667i		−	0.298441i	0.988804	−	0.149220i	$-0.0476764\pi$
$224$		0			0
$225$	−2.81344			−0.187563
$226$		0			0
$227$	−18.6195			−1.23582			−0.617911	−	0.786248i	$-0.712021\pi$
$227$	−18.6195			−1.23582			−0.617911	+	0.786248i	$0.712021\pi$
$228$		0			0
$229$		−	14.8271i		−	0.979806i	−0.871777	−	0.489903i	$-0.837032\pi$
$229$		−	14.8271i		−	0.979806i	0.871777	−	0.489903i	$-0.162968\pi$
$230$		0			0
$231$		−	2.14886i		−	0.141385i
$232$		0			0
$233$	−11.5818			−0.758748			−0.379374	−	0.925243i	$-0.623861\pi$
$233$	−11.5818			−0.758748			−0.379374	+	0.925243i	$0.623861\pi$
$234$		0			0
$235$	6.73883			0.439593
$236$		0			0
$237$		−	3.79425i		−	0.246463i
$238$		0			0
$239$		−	22.4516i		−	1.45228i	−0.687549	−	0.726138i	$-0.741313\pi$
$239$		−	22.4516i		−	1.45228i	0.687549	−	0.726138i	$-0.258687\pi$
$240$		0			0
$241$			15.7424i			1.01406i	0.861929	+	0.507029i	$0.169256\pi$
$241$			15.7424i			1.01406i	−0.861929	+	0.507029i	$0.830744\pi$
$242$		0			0
$243$		−	10.7100i		−	0.687044i
$244$		0			0
$245$			4.09125i			0.261380i
$246$		0			0
$247$	−0.750023			−0.0477228
$248$		0			0
$249$			4.91736i			0.311625i
$250$		0			0
$251$	0.641226			0.0404738			0.0202369	−	0.999795i	$-0.493558\pi$
$251$	0.641226			0.0404738			0.0202369	+	0.999795i	$0.493558\pi$
$252$		0			0
$253$	−5.91632	+	4.04046i	−0.371956	+	0.254022i
$254$		0			0
$255$		−	0.253738i		−	0.0158897i
$256$		0			0
$257$	1.60503			0.100119			0.0500596	−	0.998746i	$-0.484059\pi$
$257$	1.60503			0.100119			0.0500596	+	0.998746i	$0.484059\pi$
$258$		0			0
$259$			18.7395i			1.16442i
$260$		0			0
$261$	−4.33642			−0.268418
$262$		0			0
$263$	2.02013			0.124567			0.0622833	−	0.998059i	$-0.480162\pi$
$263$	2.02013			0.124567			0.0622833	+	0.998059i	$0.480162\pi$
$264$		0			0
$265$	0.671148			0.0412283
$266$		0			0
$267$	0.646246			0.0395496
$268$		0			0
$269$	−17.0434			−1.03915			−0.519577	−	0.854424i	$-0.673910\pi$
$269$	−17.0434			−1.03915			−0.519577	+	0.854424i	$0.673910\pi$
$270$		0			0
$271$			21.3260i			1.29546i	0.761868	+	0.647732i	$0.224282\pi$
$271$			21.3260i			1.29546i	−0.761868	+	0.647732i	$0.775718\pi$
$272$		0			0
$273$		−	8.06638i		−	0.488199i
$274$		0			0
$275$	−1.49387			−0.0900840
$276$		0			0
$277$	−1.79889			−0.108085			−0.0540425	−	0.998539i	$-0.517211\pi$
$277$	−1.79889			−0.108085			−0.0540425	+	0.998539i	$0.517211\pi$
$278$		0			0
$279$		−	5.04182i		−	0.301846i
$280$		0			0
$281$			17.8865i			1.06702i	0.845793	+	0.533511i	$0.179128\pi$
$281$			17.8865i			1.06702i	−0.845793	+	0.533511i	$0.820872\pi$
$282$		0			0
$283$	−0.693948			−0.0412509			−0.0206255	−	0.999787i	$-0.506566\pi$
$283$	−0.693948			−0.0412509			−0.0206255	+	0.999787i	$0.506566\pi$
$284$		0			0
$285$	−0.0577688			−0.00342193
$286$		0			0
$287$	−6.23180			−0.367852
$288$		0			0
$289$	16.6549			0.979699
$290$		0			0
$291$	−0.827697			−0.0485204
$292$		0			0
$293$		−	21.5923i		−	1.26143i	−0.776013	−	0.630717i	$-0.782761\pi$
$293$		−	21.5923i		−	1.26143i	0.776013	−	0.630717i	$-0.217239\pi$
$294$		0			0
$295$	3.46410			0.201688
$296$		0			0
$297$		−	3.75104i		−	0.217657i
$298$		0			0
$299$	−22.2086	+	15.1671i	−1.28436	+	0.877135i
$300$		0			0
$301$	−18.9060			−1.08972
$302$		0			0
$303$			0.356364i			0.0204726i
$304$		0			0
$305$	−13.7510			−0.787382
$306$		0			0
$307$		−	15.6567i		−	0.893577i	−0.894640	−	0.446789i	$-0.852568\pi$
$307$		−	15.6567i		−	0.893577i	0.894640	−	0.446789i	$-0.147432\pi$
$308$		0			0
$309$		−	4.15659i		−	0.236460i
$310$		0			0
$311$			21.3602i			1.21122i	0.795760	+	0.605612i	$0.207072\pi$
$311$			21.3602i			1.21122i	−0.795760	+	0.605612i	$0.792928\pi$
$312$		0			0
$313$		−	2.46274i		−	0.139202i	−0.997575	−	0.0696010i	$-0.977827\pi$
$313$		−	2.46274i		−	0.139202i	0.997575	−	0.0696010i	$-0.0221726\pi$
$314$		0			0
$315$			9.36976i			0.527926i
$316$		0			0
$317$	−14.2337			−0.799447			−0.399723	−	0.916636i	$-0.630894\pi$
$317$	−14.2337			−0.799447			−0.399723	+	0.916636i	$0.630894\pi$
$318$		0			0
$319$	−2.30254			−0.128917
$320$		0			0
$321$			4.05560i			0.226362i
$322$		0			0
$323$			0.0785729i			0.00437191i
$324$		0			0
$325$	−5.60770			−0.311059
$326$		0			0
$327$	2.36274			0.130660
$328$		0			0
$329$		−	22.4427i		−	1.23730i
$330$		0			0
$331$			33.1540i			1.82231i	0.412064	+	0.911155i	$0.364808\pi$
$331$			33.1540i			1.82231i	−0.412064	+	0.911155i	$0.635192\pi$
$332$		0			0
$333$			15.8309i			0.867530i
$334$		0			0
$335$		−	3.10774i		−	0.169794i
$336$		0			0
$337$		−	5.99130i		−	0.326367i	−0.986596	−	0.163184i	$-0.947824\pi$
$337$		−	5.99130i		−	0.326367i	0.986596	−	0.163184i	$-0.0521763\pi$
$338$		0			0
$339$	5.49040			0.298198
$340$		0			0
$341$		−	2.67709i		−	0.144973i
$342$		0			0
$343$	−9.68716			−0.523058
$344$		0			0
$345$	−1.71057	+	1.16821i	−0.0920941	+	0.0628943i
$346$		0			0
$347$			16.6892i			0.895922i	0.894053	+	0.447961i	$0.147850\pi$
$347$			16.6892i			0.895922i	−0.894053	+	0.447961i	$0.852150\pi$
$348$		0			0
$349$	20.2416			1.08351			0.541753	−	0.840538i	$-0.317761\pi$
$349$	20.2416			1.08351			0.541753	+	0.840538i	$0.317761\pi$
$350$		0			0
$351$		−	14.0806i		−	0.751568i
$352$		0			0
$353$	−8.58935			−0.457165			−0.228583	−	0.973525i	$-0.573409\pi$
$353$	−8.58935			−0.457165			−0.228583	+	0.973525i	$0.573409\pi$
$354$		0			0
$355$	12.2043			0.647736
$356$		0			0
$357$	−0.845039			−0.0447242
$358$		0			0
$359$	0.686108			0.0362114			0.0181057	−	0.999836i	$-0.494236\pi$
$359$	0.686108			0.0362114			0.0181057	+	0.999836i	$0.494236\pi$
$360$		0			0
$361$	−18.9821			−0.999058
$362$		0			0
$363$			3.78722i			0.198778i
$364$		0			0
$365$			15.3760i			0.804819i
$366$		0			0
$367$	10.9974			0.574059			0.287029	−	0.957922i	$-0.407332\pi$
$367$	10.9974			0.574059			0.287029	+	0.957922i	$0.407332\pi$
$368$		0			0
$369$	−5.26456			−0.274062
$370$		0			0
$371$		−	2.23516i		−	0.116044i
$372$		0			0
$373$		−	23.7710i		−	1.23082i	−0.788209	−	0.615408i	$-0.788991\pi$
$373$		−	23.7710i		−	1.23082i	0.788209	−	0.615408i	$-0.211009\pi$
$374$		0			0
$375$	−0.431920			−0.0223043
$376$		0			0
$377$	−8.64326			−0.445151
$378$		0			0
$379$	−20.1935			−1.03727			−0.518636	−	0.854995i	$-0.673560\pi$
$379$	−20.1935			−1.03727			−0.518636	+	0.854995i	$0.673560\pi$
$380$		0			0
$381$	−7.39117			−0.378661
$382$		0			0
$383$	26.4037			1.34917			0.674584	−	0.738198i	$-0.264323\pi$
$383$	26.4037			1.34917			0.674584	+	0.738198i	$0.264323\pi$
$384$		0			0
$385$			4.97513i			0.253556i
$386$		0			0
$387$	−15.9716			−0.811881
$388$		0			0
$389$		−	34.5010i		−	1.74927i	−0.484780	−	0.874636i	$-0.661101\pi$
$389$		−	34.5010i		−	1.74927i	0.484780	−	0.874636i	$-0.338899\pi$
$390$		0			0
$391$	1.58891	+	2.32659i	0.0803547	+	0.117661i
$392$		0			0
$393$	7.70868			0.388852
$394$		0			0
$395$			8.78461i			0.442002i
$396$		0			0
$397$	3.74054			0.187732			0.0938662	−	0.995585i	$-0.470077\pi$
$397$	3.74054			0.187732			0.0938662	+	0.995585i	$0.470077\pi$
$398$		0			0
$399$			0.192391i			0.00963158i
$400$		0			0
$401$		−	0.488648i		−	0.0244019i	−0.999926	−	0.0122010i	$-0.996116\pi$
$401$		−	0.488648i		−	0.0244019i	0.999926	−	0.0122010i	$-0.00388378\pi$
$402$		0			0
$403$		−	10.0493i		−	0.500589i
$404$		0			0
$405$			7.35581i			0.365513i
$406$		0			0
$407$			8.40586i			0.416663i
$408$		0			0
$409$	−17.7751			−0.878920			−0.439460	−	0.898262i	$-0.644830\pi$
$409$	−17.7751			−0.878920			−0.439460	+	0.898262i	$0.644830\pi$
$410$		0			0
$411$	−8.10970			−0.400022
$412$		0			0
$413$		−	11.5367i		−	0.567683i
$414$		0			0
$415$		−	11.3849i		−	0.558861i
$416$		0			0
$417$	2.16442			0.105992
$418$		0			0
$419$	27.1866			1.32815			0.664075	−	0.747666i	$-0.268826\pi$
$419$	27.1866			1.32815			0.664075	+	0.747666i	$0.268826\pi$
$420$		0			0
$421$			22.3412i			1.08884i	0.838812	+	0.544421i	$0.183251\pi$
$421$			22.3412i			1.08884i	−0.838812	+	0.544421i	$0.816749\pi$
$422$		0			0
$423$		−	18.9593i		−	0.921834i
$424$		0			0
$425$			0.587466i			0.0284963i
$426$		0			0
$427$			45.7958i			2.21621i
$428$		0			0
$429$		−	3.61828i		−	0.174692i
$430$		0			0
$431$	6.57184			0.316554			0.158277	−	0.987395i	$-0.449406\pi$
$431$	6.57184			0.316554			0.158277	+	0.987395i	$0.449406\pi$
$432$		0			0
$433$		−	19.3283i		−	0.928857i	−0.885611	−	0.464429i	$-0.846260\pi$
$433$		−	19.3283i		−	0.928857i	0.885611	−	0.464429i	$-0.153740\pi$
$434$		0			0
$435$	−0.665728			−0.0319192
$436$		0			0
$437$	0.529697	−	0.361749i	0.0253388	−	0.0173048i
$438$		0			0
$439$		−	23.1896i		−	1.10678i	−0.832923	−	0.553389i	$-0.813334\pi$
$439$		−	23.1896i		−	1.10678i	0.832923	−	0.553389i	$-0.186666\pi$
$440$		0			0
$441$	11.5105			0.548119
$442$		0			0
$443$			23.0342i			1.09439i	0.837005	+	0.547195i	$0.184304\pi$
$443$			23.0342i			1.09439i	−0.837005	+	0.547195i	$0.815696\pi$
$444$		0			0
$445$	−1.49622			−0.0709274
$446$		0			0
$447$	−4.74410			−0.224388
$448$		0			0
$449$	−8.77591			−0.414161			−0.207080	−	0.978324i	$-0.566396\pi$
$449$	−8.77591			−0.414161			−0.207080	+	0.978324i	$0.566396\pi$
$450$		0			0
$451$	−2.79536			−0.131628
$452$		0			0
$453$	−2.27885			−0.107070
$454$		0			0
$455$			18.6756i			0.875526i
$456$		0			0
$457$		−	19.6441i		−	0.918913i	−0.888200	−	0.459456i	$-0.848044\pi$
$457$		−	19.6441i		−	0.918913i	0.888200	−	0.459456i	$-0.151956\pi$
$458$		0			0
$459$	−1.47509			−0.0688515
$460$		0			0
$461$	−27.3933			−1.27583			−0.637916	−	0.770106i	$-0.720203\pi$
$461$	−27.3933			−1.27583			−0.637916	+	0.770106i	$0.720203\pi$
$462$		0			0
$463$			3.66422i			0.170291i	0.996369	+	0.0851453i	$0.0271355\pi$
$463$			3.66422i			0.170291i	−0.996369	+	0.0851453i	$0.972865\pi$
$464$		0			0
$465$		−	0.774021i		−	0.0358944i
$466$		0			0
$467$	−20.9046			−0.967349			−0.483675	−	0.875248i	$-0.660698\pi$
$467$	−20.9046			−0.967349			−0.483675	+	0.875248i	$0.660698\pi$
$468$		0			0
$469$	−10.3499			−0.477912
$470$		0			0
$471$	−4.62656			−0.213181
$472$		0			0
$473$	−8.48054			−0.389935
$474$		0			0
$475$	0.133749			0.00613682
$476$		0			0
$477$		−	1.88824i		−	0.0864564i
$478$		0			0
$479$	−5.37915			−0.245780			−0.122890	−	0.992420i	$-0.539216\pi$
$479$	−5.37915			−0.245780			−0.122890	+	0.992420i	$0.539216\pi$
$480$		0			0
$481$			31.5539i			1.43873i
$482$		0			0
$483$	3.89055	+	5.69681i	0.177026	+	0.259214i
$484$		0			0
$485$	1.91632			0.0870155
$486$		0			0
$487$		−	35.9018i		−	1.62687i	−0.581659	−	0.813433i	$-0.697596\pi$
$487$		−	35.9018i		−	1.62687i	0.581659	−	0.813433i	$-0.302404\pi$
$488$		0			0
$489$	−0.516448			−0.0233546
$490$		0			0
$491$			31.1465i			1.40562i	0.711377	+	0.702811i	$0.248072\pi$
$491$			31.1465i			1.40562i	−0.711377	+	0.702811i	$0.751928\pi$
$492$		0			0
$493$			0.905473i			0.0407805i
$494$		0			0
$495$			4.20293i			0.188908i
$496$		0			0
$497$		−	40.6446i		−	1.82316i
$498$		0			0
$499$			17.3624i			0.777247i	0.921397	+	0.388623i	$0.127049\pi$
$499$			17.3624i			0.777247i	−0.921397	+	0.388623i	$0.872951\pi$
$500$		0			0
$501$	1.41945			0.0634163
$502$		0			0
$503$	−36.6614			−1.63465			−0.817325	−	0.576176i	$-0.804544\pi$
$503$	−36.6614			−1.63465			−0.817325	+	0.576176i	$0.804544\pi$
$504$		0			0
$505$		−	0.825068i		−	0.0367151i
$506$		0			0
$507$		−	7.96732i		−	0.353841i
$508$		0			0
$509$	−16.3836			−0.726191			−0.363095	−	0.931752i	$-0.618280\pi$
$509$	−16.3836			−0.726191			−0.363095	+	0.931752i	$0.618280\pi$
$510$		0			0
$511$	51.2076			2.26529
$512$		0			0
$513$			0.335836i			0.0148275i
$514$		0			0
$515$			9.62350i			0.424062i
$516$		0			0
$517$		−	10.0670i		−	0.442744i
$518$		0			0
$519$		−	1.66915i		−	0.0732677i
$520$		0			0
$521$		−	32.2786i		−	1.41415i	−0.707139	−	0.707075i	$-0.750014\pi$
$521$		−	32.2786i		−	1.41415i	0.707139	−	0.707075i	$-0.249986\pi$
$522$		0			0
$523$	−33.5308			−1.46620			−0.733099	−	0.680122i	$-0.761927\pi$
$523$	−33.5308			−1.46620			−0.733099	+	0.680122i	$0.761927\pi$
$524$		0			0
$525$			1.43845i			0.0627790i
$526$		0			0
$527$	−1.05277			−0.0458592
$528$		0			0
$529$	8.36932	−	21.4232i	0.363883	−	0.931445i
$530$		0			0
$531$		−	9.74606i		−	0.422943i
$532$		0			0
$533$	−10.4932			−0.454511
$534$		0			0
$535$		−	9.38969i		−	0.405952i
$536$		0			0
$537$	1.95490			0.0843600
$538$		0			0
$539$	6.11181			0.263254
$540$		0			0
$541$	2.21455			0.0952109			0.0476055	−	0.998866i	$-0.484841\pi$
$541$	2.21455			0.0952109			0.0476055	+	0.998866i	$0.484841\pi$
$542$		0			0
$543$	8.81238			0.378176
$544$		0			0
$545$	−5.47030			−0.234322
$546$		0			0
$547$		−	26.4406i		−	1.13052i	−0.824914	−	0.565259i	$-0.808776\pi$
$547$		−	26.4406i		−	1.13052i	0.824914	−	0.565259i	$-0.191224\pi$
$548$		0			0
$549$			38.6878i			1.65115i
$550$		0			0
$551$	0.206150			0.00878228
$552$		0			0
$553$	29.2559			1.24409
$554$		0			0
$555$			2.43037i			0.103163i
$556$		0			0
$557$		−	36.3444i		−	1.53996i	−0.638068	−	0.769980i	$-0.720266\pi$
$557$		−	36.3444i		−	1.53996i	0.638068	−	0.769980i	$-0.279734\pi$
$558$		0			0
$559$	−31.8342			−1.34644
$560$		0			0
$561$	−0.379053			−0.0160036
$562$		0			0
$563$	−27.6226			−1.16416			−0.582078	−	0.813133i	$-0.697760\pi$
$563$	−27.6226			−1.16416			−0.582078	+	0.813133i	$0.697760\pi$
$564$		0			0
$565$	−12.7116			−0.534782
$566$		0			0
$567$	24.4974			1.02880
$568$		0			0
$569$			32.5696i			1.36539i	0.730705	+	0.682694i	$0.239192\pi$
$569$			32.5696i			1.36539i	−0.730705	+	0.682694i	$0.760808\pi$
$570$		0			0
$571$	−25.8227			−1.08065			−0.540323	−	0.841457i	$-0.681698\pi$
$571$	−25.8227			−1.08065			−0.540323	+	0.841457i	$0.681698\pi$
$572$		0			0
$573$		−	3.26722i		−	0.136490i
$574$		0			0
$575$	3.96039	−	2.70469i	0.165160	−	0.112793i
$576$		0			0
$577$	7.04131			0.293134			0.146567	−	0.989201i	$-0.453178\pi$
$577$	7.04131			0.293134			0.146567	+	0.989201i	$0.453178\pi$
$578$		0			0
$579$			5.68762i			0.236369i
$580$		0			0
$581$	−37.9156			−1.57301
$582$		0			0
$583$		−	1.00261i		−	0.0415239i
$584$		0			0
$585$			15.7769i			0.652297i
$586$		0			0
$587$		−	15.7385i		−	0.649596i	−0.945783	−	0.324798i	$-0.894704\pi$
$587$		−	15.7385i		−	0.649596i	0.945783	−	0.324798i	$-0.105296\pi$
$588$		0			0
$589$			0.239684i			0.00987601i
$590$		0			0
$591$		−	9.68169i		−	0.398252i
$592$		0			0
$593$	−18.3132			−0.752032			−0.376016	−	0.926613i	$-0.622706\pi$
$593$	−18.3132			−0.752032			−0.376016	+	0.926613i	$0.622706\pi$
$594$		0			0
$595$	1.95647			0.0802074
$596$		0			0
$597$			8.34228i			0.341427i
$598$		0			0
$599$			24.2851i			0.992263i	0.868248	+	0.496131i	$0.165247\pi$
$599$			24.2851i			0.992263i	−0.868248	+	0.496131i	$0.834753\pi$
$600$		0			0
$601$	−24.3344			−0.992623			−0.496311	−	0.868145i	$-0.665313\pi$
$601$	−24.3344			−0.992623			−0.496311	+	0.868145i	$0.665313\pi$
$602$		0			0
$603$	−8.74345			−0.356061
$604$		0			0
$605$		−	8.76834i		−	0.356484i
$606$		0			0
$607$		−	43.5580i		−	1.76796i	−0.467522	−	0.883982i	$-0.654853\pi$
$607$		−	43.5580i		−	1.76796i	0.467522	−	0.883982i	$-0.345147\pi$
$608$		0			0
$609$			2.21711i			0.0898418i
$610$		0			0
$611$		−	37.7893i		−	1.52879i
$612$		0			0
$613$			33.7850i			1.36456i	0.731090	+	0.682281i	$0.239012\pi$
$613$			33.7850i			1.36456i	−0.731090	+	0.682281i	$0.760988\pi$
$614$		0			0
$615$	−0.808215			−0.0325904
$616$		0			0
$617$			4.58747i			0.184684i	0.995727	+	0.0923422i	$0.0294354\pi$
$617$			4.58747i			0.184684i	−0.995727	+	0.0923422i	$0.970565\pi$
$618$		0			0
$619$	−20.4498			−0.821949			−0.410974	−	0.911647i	$-0.634811\pi$
$619$	−20.4498			−0.821949			−0.410974	+	0.911647i	$0.634811\pi$
$620$		0			0
$621$	6.79133	+	9.94431i	0.272526	+	0.399052i
$622$		0			0
$623$			4.98293i			0.199637i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$			0.0862994i			0.00344646i
$628$		0			0
$629$	3.30561			0.131803
$630$		0			0
$631$	15.8905			0.632593			0.316296	−	0.948660i	$-0.397561\pi$
$631$	15.8905			0.632593			0.316296	+	0.948660i	$0.397561\pi$
$632$		0			0
$633$	3.21540			0.127801
$634$		0			0
$635$	17.1124			0.679083
$636$		0			0
$637$	22.9425			0.909015
$638$		0			0
$639$		−	34.3361i		−	1.35831i
$640$		0			0
$641$		−	17.2424i		−	0.681035i	−0.940238	−	0.340517i	$-0.889398\pi$
$641$		−	17.2424i		−	0.681035i	0.940238	−	0.340517i	$-0.110602\pi$
$642$		0			0
$643$	10.1248			0.399283			0.199642	−	0.979869i	$-0.436022\pi$
$643$	10.1248			0.399283			0.199642	+	0.979869i	$0.436022\pi$
$644$		0			0
$645$	−2.45196			−0.0965458
$646$		0			0
$647$			41.9438i			1.64898i	0.565877	+	0.824490i	$0.308538\pi$
$647$			41.9438i			1.64898i	−0.565877	+	0.824490i	$0.691462\pi$
$648$		0			0
$649$		−	5.17493i		−	0.203134i
$650$		0			0
$651$	−2.57776			−0.101030
$652$		0			0
$653$	11.3615			0.444609			0.222305	−	0.974977i	$-0.428642\pi$
$653$	11.3615			0.444609			0.222305	+	0.974977i	$0.428642\pi$
$654$		0			0
$655$	−17.8475			−0.697358
$656$		0			0
$657$	43.2596			1.68772
$658$		0			0
$659$	44.9335			1.75036			0.875181	−	0.483796i	$-0.160742\pi$
$659$	44.9335			1.75036			0.875181	+	0.483796i	$0.160742\pi$
$660$		0			0
$661$			18.4201i			0.716459i	0.933634	+	0.358229i	$0.116619\pi$
$661$			18.4201i			0.716459i	−0.933634	+	0.358229i	$0.883381\pi$
$662$		0			0
$663$	−1.42289			−0.0552604
$664$		0			0
$665$		−	0.445431i		−	0.0172731i
$666$		0			0
$667$	6.10422	−	4.16879i	0.236357	−	0.161416i
$668$		0			0
$669$	−1.92493			−0.0744220
$670$		0			0
$671$			20.5423i			0.793027i
$672$		0			0
$673$	28.7183			1.10701			0.553505	−	0.832846i	$-0.313290\pi$
$673$	28.7183			1.10701			0.553505	+	0.832846i	$0.313290\pi$
$674$		0			0
$675$			2.51095i			0.0966463i
$676$		0			0
$677$		−	32.8807i		−	1.26371i	−0.775088	−	0.631853i	$-0.782294\pi$
$677$		−	32.8807i		−	1.26371i	0.775088	−	0.631853i	$-0.217706\pi$
$678$		0			0
$679$		−	6.38202i		−	0.244919i
$680$		0			0
$681$			8.04216i			0.308176i
$682$		0			0
$683$			18.9659i			0.725712i	0.931845	+	0.362856i	$0.118198\pi$
$683$			18.9659i			0.725712i	−0.931845	+	0.362856i	$0.881802\pi$
$684$		0			0
$685$	18.7759			0.717391
$686$		0			0
$687$	−6.40415			−0.244333
$688$		0			0
$689$		−	3.76359i		−	0.143381i
$690$		0			0
$691$		−	35.5474i		−	1.35229i	−0.736770	−	0.676144i	$-0.763650\pi$
$691$		−	35.5474i		−	1.35229i	0.736770	−	0.676144i	$-0.236350\pi$
$692$		0			0
$693$	13.9972			0.531711
$694$		0			0
$695$	−5.01115			−0.190084
$696$		0			0
$697$			1.09927i			0.0416380i
$698$		0			0
$699$			5.00241i			0.189208i
$700$		0			0
$701$			27.8627i			1.05236i	0.850373	+	0.526180i	$0.176376\pi$
$701$			27.8627i			1.05236i	−0.850373	+	0.526180i	$0.823624\pi$
$702$		0			0
$703$		−	0.752590i		−	0.0283845i
$704$		0			0
$705$		−	2.91064i		−	0.109621i
$706$		0			0
$707$	−2.74777			−0.103340
$708$		0			0
$709$			47.5723i			1.78662i	0.449445	+	0.893308i	$0.351622\pi$
$709$			47.5723i			1.78662i	−0.449445	+	0.893308i	$0.648378\pi$
$710$		0			0
$711$	24.7150			0.926886
$712$		0			0
$713$	4.84693	+	7.09719i	0.181519	+	0.265792i
$714$		0			0
$715$			8.37719i			0.313289i
$716$		0			0
$717$	−9.69732			−0.362153
$718$		0			0
$719$		−	0.801481i		−	0.0298902i	−0.999888	−	0.0149451i	$-0.995243\pi$
$719$		−	0.801481i		−	0.0298902i	0.999888	−	0.0149451i	$-0.00475735\pi$
$720$		0			0
$721$	32.0497			1.19359
$722$		0			0
$723$	6.79947			0.252875
$724$		0			0
$725$	1.54132			0.0572432
$726$		0			0
$727$	−10.3810			−0.385010			−0.192505	−	0.981296i	$-0.561661\pi$
$727$	−10.3810			−0.385010			−0.192505	+	0.981296i	$0.561661\pi$
$728$		0			0
$729$	17.4416			0.645984
$730$		0			0
$731$			3.33497i			0.123348i
$732$		0			0
$733$		−	0.713693i		−	0.0263609i	−0.999913	−	0.0131804i	$-0.995804\pi$
$733$		−	0.713693i		−	0.0263609i	0.999913	−	0.0131804i	$-0.00419558\pi$
$734$		0			0
$735$	1.76709			0.0651802
$736$		0			0
$737$	−4.64257			−0.171011
$738$		0			0
$739$		−	16.2261i		−	0.596888i	−0.954427	−	0.298444i	$-0.903532\pi$
$739$		−	16.2261i		−	0.596888i	0.954427	−	0.298444i	$-0.0964676\pi$
$740$		0			0
$741$			0.323950i			0.0119006i
$742$		0			0
$743$	18.2682			0.670195			0.335098	−	0.942183i	$-0.391231\pi$
$743$	18.2682			0.670195			0.335098	+	0.942183i	$0.391231\pi$
$744$		0			0
$745$	10.9837			0.402413
$746$		0			0
$747$	−32.0307			−1.17194
$748$		0			0
$749$	−31.2710			−1.14262
$750$		0			0
$751$	48.7265			1.77805			0.889027	−	0.457855i	$-0.151382\pi$
$751$	48.7265			1.77805			0.889027	+	0.457855i	$0.151382\pi$
$752$		0			0
$753$		−	0.276958i		−	0.0100929i
$754$		0			0
$755$	5.27608			0.192016
$756$		0			0
$757$		−	3.13655i		−	0.114000i	−0.998374	−	0.0569999i	$-0.981847\pi$
$757$		−	3.13655i		−	0.114000i	0.998374	−	0.0569999i	$-0.0181535\pi$
$758$		0			0
$759$	1.74516	+	2.55538i	0.0633453	+	0.0927544i
$760$		0			0
$761$	−37.0807			−1.34417			−0.672087	−	0.740473i	$-0.734602\pi$
$761$	−37.0807			−1.34417			−0.672087	+	0.740473i	$0.734602\pi$
$762$		0			0
$763$			18.2180i			0.659537i
$764$		0			0
$765$	1.65280			0.0597572
$766$		0			0
$767$		−	19.4256i		−	0.701419i
$768$		0			0
$769$		−	21.7326i		−	0.783698i	−0.920029	−	0.391849i	$-0.871835\pi$
$769$		−	21.7326i		−	0.783698i	0.920029	−	0.391849i	$-0.128165\pi$
$770$		0			0
$771$		−	0.693246i		−	0.0249666i
$772$		0			0
$773$		−	33.6176i		−	1.20914i	−0.796551	−	0.604571i	$-0.793345\pi$
$773$		−	33.6176i		−	1.20914i	0.796551	−	0.604571i	$-0.206655\pi$
$774$		0			0
$775$			1.79205i			0.0643722i
$776$		0			0
$777$	8.09398			0.290370
$778$		0			0
$779$	0.250273			0.00896695
$780$		0			0
$781$		−	18.2317i		−	0.652380i
$782$		0			0
$783$			3.87017i			0.138309i
$784$		0			0
$785$	10.7116			0.382314
$786$		0			0
$787$	41.4616			1.47795			0.738973	−	0.673735i	$-0.235311\pi$
$787$	41.4616			1.47795			0.738973	+	0.673735i	$0.235311\pi$
$788$		0			0
$789$		−	0.872536i		−	0.0310631i
$790$		0			0
$791$			42.3342i			1.50523i
$792$		0			0
$793$			77.1117i			2.73832i
$794$		0			0
$795$		−	0.289882i		−	0.0102811i
$796$		0			0
$797$		−	13.3574i		−	0.473144i	−0.971614	−	0.236572i	$-0.923976\pi$
$797$		−	13.3574i		−	0.473144i	0.971614	−	0.236572i	$-0.0760240\pi$
$798$		0			0
$799$	−3.95883			−0.140053
$800$		0			0
$801$			4.20952i			0.148736i
$802$		0			0
$803$	22.9699			0.810589
$804$		0			0
$805$	−9.00757	−	13.1895i	−0.317475	−	0.464868i
$806$		0			0
$807$			7.36139i			0.259133i
$808$		0			0
$809$	−15.7531			−0.553850			−0.276925	−	0.960892i	$-0.589315\pi$
$809$	−15.7531			−0.553850			−0.276925	+	0.960892i	$0.589315\pi$
$810$		0			0
$811$		−	40.3282i		−	1.41612i	−0.706154	−	0.708058i	$-0.749572\pi$
$811$		−	40.3282i		−	1.41612i	0.706154	−	0.708058i	$-0.250428\pi$
$812$		0			0
$813$	9.21115			0.323049
$814$		0			0
$815$	1.19570			0.0418836
$816$		0			0
$817$	0.759275			0.0265637
$818$		0			0
$819$	52.5428			1.83599
$820$		0			0
$821$	22.5615			0.787401			0.393700	−	0.919239i	$-0.371195\pi$
$821$	22.5615			0.787401			0.393700	+	0.919239i	$0.371195\pi$
$822$		0			0
$823$			39.4460i			1.37500i	0.726184	+	0.687501i	$0.241292\pi$
$823$			39.4460i			1.37500i	−0.726184	+	0.687501i	$0.758708\pi$
$824$		0			0
$825$			0.645235i			0.0224642i
$826$		0			0
$827$	−1.06266			−0.0369522			−0.0184761	−	0.999829i	$-0.505881\pi$
$827$	−1.06266			−0.0369522			−0.0184761	+	0.999829i	$0.505881\pi$
$828$		0			0
$829$	45.5193			1.58095			0.790475	−	0.612494i	$-0.209834\pi$
$829$	45.5193			1.58095			0.790475	+	0.612494i	$0.209834\pi$
$830$		0			0
$831$			0.776979i			0.0269531i
$832$		0			0
$833$		−	2.40347i		−	0.0832753i
$834$		0			0
$835$	−3.28637			−0.113729
$836$		0			0
$837$	−4.49973			−0.155533
$838$		0			0
$839$	−48.8776			−1.68744			−0.843721	−	0.536783i	$-0.819640\pi$
$839$	−48.8776			−1.68744			−0.843721	+	0.536783i	$0.819640\pi$
$840$		0			0
$841$	−26.6243			−0.918080
$842$		0			0
$843$	7.72556			0.266083
$844$		0			0
$845$			18.4463i			0.634571i
$846$		0			0
$847$	−29.2017			−1.00338
$848$		0			0
$849$			0.299730i			0.0102867i
$850$		0			0
$851$	−15.2190	−	22.2847i	−0.521700	−	0.763908i
$852$		0			0
$853$	−9.71443			−0.332616			−0.166308	−	0.986074i	$-0.553185\pi$
$853$	−9.71443			−0.332616			−0.166308	+	0.986074i	$0.553185\pi$
$854$		0			0
$855$		−	0.376295i		−	0.0128690i
$856$		0			0
$857$	12.3706			0.422570			0.211285	−	0.977424i	$-0.432235\pi$
$857$	12.3706			0.422570			0.211285	+	0.977424i	$0.432235\pi$
$858$		0			0
$859$			12.8422i			0.438169i	0.975706	+	0.219084i	$0.0703070\pi$
$859$			12.8422i			0.438169i	−0.975706	+	0.219084i	$0.929693\pi$
$860$		0			0
$861$			2.69164i			0.0917309i
$862$		0			0
$863$			12.0107i			0.408850i	0.978882	+	0.204425i	$0.0655325\pi$
$863$			12.0107i			0.408850i	−0.978882	+	0.204425i	$0.934467\pi$
$864$		0			0
$865$			3.86449i			0.131397i
$866$		0			0
$867$		−	7.19358i		−	0.244307i
$868$		0			0
$869$	13.1231			0.445171
$870$		0			0
$871$	−17.4273			−0.590500
$872$		0			0
$873$		−	5.39146i		−	0.182473i
$874$		0			0
$875$		−	3.33035i		−	0.112586i
$876$		0			0
$877$	24.6582			0.832648			0.416324	−	0.909216i	$-0.363318\pi$
$877$	24.6582			0.832648			0.416324	+	0.909216i	$0.363318\pi$
$878$		0			0
$879$	−9.32615			−0.314563
$880$		0			0
$881$			34.2078i			1.15249i	0.817277	+	0.576246i	$0.195483\pi$
$881$			34.2078i			1.15249i	−0.817277	+	0.576246i	$0.804517\pi$
$882$		0			0
$883$		−	24.0738i		−	0.810148i	−0.914284	−	0.405074i	$-0.867246\pi$
$883$		−	24.0738i		−	0.810148i	0.914284	−	0.405074i	$-0.132754\pi$
$884$		0			0
$885$		−	1.49622i		−	0.0502948i
$886$		0			0
$887$			51.5148i			1.72970i	0.502032	+	0.864849i	$0.332586\pi$
$887$			51.5148i			1.72970i	−0.502032	+	0.864849i	$0.667414\pi$
$888$		0			0
$889$		−	56.9902i		−	1.91139i
$890$		0			0
$891$	10.9886			0.368134
$892$		0			0
$893$			0.901310i			0.0301612i
$894$		0			0
$895$	−4.52606			−0.151289
$896$		0			0
$897$	6.55097	+	9.59237i	0.218731	+	0.320280i
$898$		0			0
$899$			2.76212i			0.0921218i
$900$		0			0
$901$	−0.394276			−0.0131352
$902$		0			0
$903$			8.16589i			0.271744i
$904$		0			0
$905$	−20.4028			−0.678212
$906$		0			0
$907$	10.3699			0.344328			0.172164	−	0.985068i	$-0.444924\pi$
$907$	10.3699			0.344328			0.172164	+	0.985068i	$0.444924\pi$
$908$		0			0
$909$	−2.32128			−0.0769921
$910$		0			0
$911$	42.8582			1.41996			0.709978	−	0.704224i	$-0.248705\pi$
$911$	42.8582			1.41996			0.709978	+	0.704224i	$0.248705\pi$
$912$		0			0
$913$	−17.0076			−0.562868
$914$		0			0
$915$			5.93935i			0.196349i
$916$		0			0
$917$			59.4384i			1.96283i
$918$		0			0
$919$	4.69304			0.154809			0.0774046	−	0.997000i	$-0.475337\pi$
$919$	4.69304			0.154809			0.0774046	+	0.997000i	$0.475337\pi$
$920$		0			0
$921$	−6.76247			−0.222831
$922$		0			0
$923$		−	68.4379i		−	2.25266i
$924$		0			0
$925$		−	5.62689i		−	0.185011i
$926$		0			0
$927$	27.0752			0.889266
$928$		0			0
$929$	−4.62774			−0.151831			−0.0759155	−	0.997114i	$-0.524188\pi$
$929$	−4.62774			−0.151831			−0.0759155	+	0.997114i	$0.524188\pi$
$930$		0			0
$931$	−0.547200			−0.0179337
$932$		0			0
$933$	9.22589			0.302042
$934$		0			0
$935$	0.877600			0.0287006
$936$		0			0
$937$			11.3202i			0.369813i	0.982756	+	0.184907i	$0.0591983\pi$
$937$			11.3202i			0.369813i	−0.982756	+	0.184907i	$0.940802\pi$
$938$		0			0
$939$	−1.06371			−0.0347127
$940$		0			0
$941$		−	3.77584i		−	0.123089i	−0.998104	−	0.0615444i	$-0.980397\pi$
$941$		−	3.77584i		−	0.123089i	0.998104	−	0.0615444i	$-0.0196026\pi$
$942$		0			0
$943$	7.41073	−	5.06105i	0.241327	−	0.164810i
$944$		0			0
$945$	8.36233			0.272027
$946$		0			0
$947$		−	15.8170i		−	0.513985i	−0.966413	−	0.256992i	$-0.917268\pi$
$947$		−	15.8170i		−	0.513985i	0.966413	−	0.256992i	$-0.0827315\pi$
$948$		0			0
$949$	86.2242			2.79896
$950$		0			0
$951$			6.14784i			0.199357i
$952$		0			0
$953$		−	19.5665i		−	0.633820i	−0.948456	−	0.316910i	$-0.897355\pi$
$953$		−	19.5665i		−	0.633820i	0.948456	−	0.316910i	$-0.102645\pi$
$954$		0			0
$955$			7.56441i			0.244779i
$956$		0			0
$957$			0.994513i			0.0321481i
$958$		0			0
$959$		−	62.5304i		−	2.01921i
$960$		0			0
$961$	27.7886			0.896406
$962$		0			0
$963$	−26.4174			−0.851288
$964$		0			0
$965$		−	13.1682i		−	0.423900i
$966$		0			0
$967$			11.8330i			0.380524i	0.981733	+	0.190262i	$0.0609337\pi$
$967$			11.8330i			0.380524i	−0.981733	+	0.190262i	$0.939066\pi$
$968$		0			0
$969$	0.0339372			0.00109022
$970$		0			0
$971$	−11.0700			−0.355254			−0.177627	−	0.984098i	$-0.556842\pi$
$971$	−11.0700			−0.355254			−0.177627	+	0.984098i	$0.556842\pi$
$972$		0			0
$973$			16.6889i			0.535021i
$974$		0			0
$975$			2.42208i			0.0775686i
$976$		0			0
$977$			41.7683i			1.33629i	0.744032	+	0.668144i	$0.232911\pi$
$977$			41.7683i			1.33629i	−0.744032	+	0.668144i	$0.767089\pi$
$978$		0			0
$979$			2.23516i			0.0714360i
$980$		0			0
$981$			15.3904i			0.491378i
$982$		0			0
$983$	−43.8359			−1.39815			−0.699074	−	0.715050i	$-0.746404\pi$
$983$	−43.8359			−1.39815			−0.699074	+	0.715050i	$0.746404\pi$
$984$		0			0
$985$			22.4155i			0.714216i
$986$		0			0
$987$	−9.69345			−0.308546
$988$		0			0
$989$	22.4826	−	15.3542i	0.714906	−	0.488235i
$990$		0			0
$991$		−	23.9970i		−	0.762291i	−0.924515	−	0.381145i	$-0.875530\pi$
$991$		−	23.9970i		−	0.762291i	0.924515	−	0.381145i	$-0.124470\pi$
$992$		0			0
$993$	14.3199			0.454428
$994$		0			0
$995$		−	19.3144i		−	0.612307i
$996$		0			0
$997$	−16.0961			−0.509767			−0.254884	−	0.966972i	$-0.582037\pi$
$997$	−16.0961			−0.509767			−0.254884	+	0.966972i	$0.582037\pi$
$998$		0			0
$999$	14.1288			0.447016

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.2.i.c.1471.8	yes	16
4.3	odd	2	inner	1840.2.i.c.1471.10	yes	16
23.22	odd	2	inner	1840.2.i.c.1471.7	✓	16
92.91	even	2	inner	1840.2.i.c.1471.9	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1840.2.i.c.1471.7	✓	16	23.22	odd	2	inner
1840.2.i.c.1471.8	yes	16	1.1	even	1	trivial
1840.2.i.c.1471.9	yes	16	92.91	even	2	inner
1840.2.i.c.1471.10	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)

\(f(q)\)	\(=\)	\(q-0.431920i q^{3} +1.00000i q^{5} +3.33035 q^{7} +2.81344 q^{9} +O(q^{10})\)	\(q-0.431920i q^{3} +1.00000i q^{5} +3.33035 q^{7} +2.81344 q^{9} +1.49387 q^{11} +5.60770 q^{13} +0.431920 q^{15} -0.587466i q^{17} -0.133749 q^{19} -1.43845i q^{21} +(-3.96039 + 2.70469i) q^{23} -1.00000 q^{25} -2.51095i q^{27} -1.54132 q^{29} -1.79205i q^{31} -0.645235i q^{33} +3.33035i q^{35} +5.62689i q^{37} -2.42208i q^{39} -1.87121 q^{41} -5.67688 q^{43} +2.81344i q^{45} -6.73883i q^{47} +4.09125 q^{49} -0.253738 q^{51} -0.671148i q^{53} +1.49387i q^{55} +0.0577688i q^{57} -3.46410i q^{59} +13.7510i q^{61} +9.36976 q^{63} +5.60770i q^{65} -3.10774 q^{67} +(1.16821 + 1.71057i) q^{69} -12.2043i q^{71} +15.3760 q^{73} +0.431920i q^{75} +4.97513 q^{77} +8.78461 q^{79} +7.35581 q^{81} -11.3849 q^{83} +0.587466 q^{85} +0.665728i q^{87} +1.49622i q^{89} +18.6756 q^{91} -0.774021 q^{93} -0.133749i q^{95} -1.91632i q^{97} +4.20293 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 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

Related objects

Downloads

Learn more