LMFDB - Embedded newform 1840.3.k.b.321.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,3,Mod(321,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1840.321");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1840.k (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$50.1363686423$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2 x^{9} - 10 x^{8} + 34 x^{7} + 346 x^{6} - 968 x^{5} + 165 x^{4} + 6972 x^{3} + 19344 x^{2} + \cdots + 225444 $ x^10 - 2x^9 - 10x^8 + 34x^7 + 346x^6 - 968x^5 + 165x^4 + 6972x^3 + 19344x^2 - 89080*x + 225444
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 5 $
Twist minimal:	no (minimal twist has level 115)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			321.2
Root			$2.24810 + 2.23607i$ of defining polynomial
Character	$\chi$	$=$	1840.321
Dual form			1840.3.k.b.321.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.98928 q^{3} +2.23607i q^{5} +6.68423i q^{7} +6.91435 q^{9} +O(q^{10})$	$q-3.98928 q^{3} +2.23607i q^{5} +6.68423i q^{7} +6.91435 q^{9} -10.4810i q^{11} -4.43198 q^{13} -8.92030i q^{15} +1.02893i q^{17} +24.8394i q^{19} -26.6653i q^{21} +(-0.739966 - 22.9881i) q^{23} -5.00000 q^{25} +8.32025 q^{27} +20.9027 q^{29} -35.4525 q^{31} +41.8117i q^{33} -14.9464 q^{35} -50.9422i q^{37} +17.6804 q^{39} -53.5201 q^{41} -64.2047i q^{43} +15.4609i q^{45} +2.03047 q^{47} +4.32106 q^{49} -4.10469i q^{51} -13.7344i q^{53} +23.4362 q^{55} -99.0912i q^{57} -7.09638 q^{59} +68.5368i q^{61} +46.2171i q^{63} -9.91021i q^{65} +80.9722i q^{67} +(2.95193 + 91.7059i) q^{69} -117.944 q^{71} +38.6374 q^{73} +19.9464 q^{75} +70.0575 q^{77} -11.9863i q^{79} -95.4209 q^{81} -10.8042i q^{83} -2.30076 q^{85} -83.3869 q^{87} -17.7703i q^{89} -29.6244i q^{91} +141.430 q^{93} -55.5425 q^{95} -5.84199i q^{97} -72.4693i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 2 q^{3} - 16 q^{9}+O(q^{10}) $ 10 * q + 2 * q^3 - 16 * q^9	$ 10 q + 2 q^{3} - 16 q^{9} - 2 q^{13} - 44 q^{23} - 50 q^{25} - 40 q^{27} - 46 q^{29} - 16 q^{31} + 60 q^{35} - 72 q^{39} - 84 q^{41} - 112 q^{47} + 50 q^{49} + 10 q^{55} + 262 q^{59} + 124 q^{69} - 236 q^{71} + 168 q^{73} - 10 q^{75} + 300 q^{77} - 258 q^{81} - 540 q^{87} + 100 q^{93} + 90 q^{95}+O(q^{100}) $ 10 * q + 2 * q^3 - 16 * q^9 - 2 * q^13 - 44 * q^23 - 50 * q^25 - 40 * q^27 - 46 * q^29 - 16 * q^31 + 60 * q^35 - 72 * q^39 - 84 * q^41 - 112 * q^47 + 50 * q^49 + 10 * q^55 + 262 * q^59 + 124 * q^69 - 236 * q^71 + 168 * q^73 - 10 * q^75 + 300 * q^77 - 258 * q^81 - 540 * q^87 + 100 * q^93 + 90 * q^95

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−3.98928			−1.32976			−0.664880	−	0.746950i	$-0.731517\pi$
$3$	−3.98928			−1.32976			−0.664880	+	0.746950i	$0.731517\pi$
$4$		0			0
$5$			2.23607i			0.447214i
$5$			2.23607i			0.447214i
$6$		0			0
$7$			6.68423i			0.954890i	0.878662	+	0.477445i	$0.158437\pi$
$7$			6.68423i			0.954890i	−0.878662	+	0.477445i	$0.841563\pi$
$8$		0			0
$9$	6.91435			0.768261
$10$		0			0
$11$		−	10.4810i		−	0.952819i	−0.879224	−	0.476409i	$-0.841938\pi$
$11$		−	10.4810i		−	0.952819i	0.879224	−	0.476409i	$-0.158062\pi$
$12$		0			0
$13$	−4.43198			−0.340922			−0.170461	−	0.985364i	$-0.554526\pi$
$13$	−4.43198			−0.340922			−0.170461	+	0.985364i	$0.554526\pi$
$14$		0			0
$15$		−	8.92030i		−	0.594687i
$16$		0			0
$17$			1.02893i			0.0605252i	0.999542	+	0.0302626i	$0.00963436\pi$
$17$			1.02893i			0.0605252i	−0.999542	+	0.0302626i	$0.990366\pi$
$18$		0			0
$19$			24.8394i			1.30734i	0.756782	+	0.653668i	$0.226771\pi$
$19$			24.8394i			1.30734i	−0.756782	+	0.653668i	$0.773229\pi$
$20$		0			0
$21$		−	26.6653i		−	1.26977i
$22$		0			0
$23$	−0.739966	−	22.9881i	−0.0321724	−	0.999482i
$23$	−0.739966	−	22.9881i	−0.0321724	−	0.999482i
$24$		0			0
$25$	−5.00000			−0.200000
$26$		0			0
$27$	8.32025			0.308158
$28$		0			0
$29$	20.9027			0.720784			0.360392	−	0.932801i	$-0.382643\pi$
$29$	20.9027			0.720784			0.360392	+	0.932801i	$0.382643\pi$
$30$		0			0
$31$	−35.4525			−1.14363			−0.571815	−	0.820383i	$-0.693760\pi$
$31$	−35.4525			−1.14363			−0.571815	+	0.820383i	$0.693760\pi$
$32$		0			0
$33$			41.8117i			1.26702i
$34$		0			0
$35$	−14.9464			−0.427040
$36$		0			0
$37$		−	50.9422i		−	1.37682i	−0.725323	−	0.688409i	$-0.758310\pi$
$37$		−	50.9422i		−	1.37682i	0.725323	−	0.688409i	$-0.241690\pi$
$38$		0			0
$39$	17.6804			0.453344
$40$		0			0
$41$	−53.5201			−1.30537			−0.652684	−	0.757630i	$-0.726357\pi$
$41$	−53.5201			−1.30537			−0.652684	+	0.757630i	$0.726357\pi$
$42$		0			0
$43$		−	64.2047i		−	1.49313i	−0.665311	−	0.746566i	$-0.731701\pi$
$43$		−	64.2047i		−	1.49313i	0.665311	−	0.746566i	$-0.268299\pi$
$44$		0			0
$45$			15.4609i			0.343577i
$46$		0			0
$47$	2.03047			0.0432015			0.0216008	−	0.999767i	$-0.493124\pi$
$47$	2.03047			0.0432015			0.0216008	+	0.999767i	$0.493124\pi$
$48$		0			0
$49$	4.32106			0.0881848
$50$		0			0
$51$		−	4.10469i		−	0.0804840i
$52$		0			0
$53$		−	13.7344i		−	0.259140i	−0.991570	−	0.129570i	$-0.958640\pi$
$53$		−	13.7344i		−	0.259140i	0.991570	−	0.129570i	$-0.0413597\pi$
$54$		0			0
$55$	23.4362			0.426114
$56$		0			0
$57$		−	99.0912i		−	1.73844i
$58$		0			0
$59$	−7.09638			−0.120278			−0.0601388	−	0.998190i	$-0.519154\pi$
$59$	−7.09638			−0.120278			−0.0601388	+	0.998190i	$0.519154\pi$
$60$		0			0
$61$			68.5368i			1.12355i	0.827289	+	0.561777i	$0.189882\pi$
$61$			68.5368i			1.12355i	−0.827289	+	0.561777i	$0.810118\pi$
$62$		0			0
$63$			46.2171i			0.733605i
$64$		0			0
$65$		−	9.91021i		−	0.152465i
$66$		0			0
$67$			80.9722i			1.20854i	0.796780	+	0.604270i	$0.206535\pi$
$67$			80.9722i			1.20854i	−0.796780	+	0.604270i	$0.793465\pi$
$68$		0			0
$69$	2.95193	+	91.7059i	0.0427816	+	1.32907i
$70$		0			0
$71$	−117.944			−1.66118			−0.830588	−	0.556887i	$-0.811996\pi$
$71$	−117.944			−1.66118			−0.830588	+	0.556887i	$0.811996\pi$
$72$		0			0
$73$	38.6374			0.529280			0.264640	−	0.964347i	$-0.414747\pi$
$73$	38.6374			0.529280			0.264640	+	0.964347i	$0.414747\pi$
$74$		0			0
$75$	19.9464			0.265952
$76$		0			0
$77$	70.0575			0.909837
$78$		0			0
$79$		−	11.9863i		−	0.151725i	−0.997118	−	0.0758624i	$-0.975829\pi$
$79$		−	11.9863i		−	0.151725i	0.997118	−	0.0758624i	$-0.0241710\pi$
$80$		0			0
$81$	−95.4209			−1.17804
$82$		0			0
$83$		−	10.8042i		−	0.130171i	−0.997880	−	0.0650854i	$-0.979268\pi$
$83$		−	10.8042i		−	0.130171i	0.997880	−	0.0650854i	$-0.0207320\pi$
$84$		0			0
$85$	−2.30076			−0.0270677
$86$		0			0
$87$	−83.3869			−0.958470
$88$		0			0
$89$		−	17.7703i		−	0.199667i	−0.995004	−	0.0998333i	$-0.968169\pi$
$89$		−	17.7703i		−	0.199667i	0.995004	−	0.0998333i	$-0.0318310\pi$
$90$		0			0
$91$		−	29.6244i		−	0.325543i
$92$		0			0
$93$	141.430			1.52075
$94$		0			0
$95$	−55.5425			−0.584658
$96$		0			0
$97$		−	5.84199i		−	0.0602267i	−0.999546	−	0.0301134i	$-0.990413\pi$
$97$		−	5.84199i		−	0.0602267i	0.999546	−	0.0301134i	$-0.00958683\pi$
$98$		0			0
$99$		−	72.4693i		−	0.732013i
$100$		0			0
$101$	110.564			1.09470			0.547348	−	0.836905i	$-0.315638\pi$
$101$	110.564			1.09470			0.547348	+	0.836905i	$0.315638\pi$
$102$		0			0
$103$			171.773i			1.66770i	0.551989	+	0.833851i	$0.313869\pi$
$103$			171.773i			1.66770i	−0.551989	+	0.833851i	$0.686131\pi$
$104$		0			0
$105$	59.6253			0.567860
$106$		0			0
$107$		−	179.226i		−	1.67501i	−0.546433	−	0.837503i	$-0.684015\pi$
$107$		−	179.226i		−	1.67501i	0.546433	−	0.837503i	$-0.315985\pi$
$108$		0			0
$109$			174.662i			1.60240i	0.598394	+	0.801202i	$0.295806\pi$
$109$			174.662i			1.60240i	−0.598394	+	0.801202i	$0.704194\pi$
$110$		0			0
$111$			203.223i			1.83084i
$112$		0			0
$113$		−	93.5781i		−	0.828125i	−0.910248	−	0.414063i	$-0.864109\pi$
$113$		−	93.5781i		−	0.828125i	0.910248	−	0.414063i	$-0.135891\pi$
$114$		0			0
$115$	51.4029	−	1.65461i	0.446982	−	0.0143879i
$116$		0			0
$117$	−30.6442			−0.261917
$118$		0			0
$119$	−6.87760			−0.0577950
$120$		0			0
$121$	11.1485			0.0921363
$122$		0			0
$123$	213.506			1.73583
$124$		0			0
$125$		−	11.1803i		−	0.0894427i
$126$		0			0
$127$	163.291			1.28575			0.642877	−	0.765969i	$-0.277740\pi$
$127$	163.291			1.28575			0.642877	+	0.765969i	$0.277740\pi$
$128$		0			0
$129$			256.130i			1.98551i
$130$		0			0
$131$	183.879			1.40365			0.701827	−	0.712348i	$-0.252368\pi$
$131$	183.879			1.40365			0.701827	+	0.712348i	$0.252368\pi$
$132$		0			0
$133$	−166.032			−1.24836
$134$		0			0
$135$			18.6047i			0.137812i
$136$		0			0
$137$		−	63.7907i		−	0.465625i	−0.972522	−	0.232813i	$-0.925207\pi$
$137$		−	63.7907i		−	0.465625i	0.972522	−	0.232813i	$-0.0747929\pi$
$138$		0			0
$139$	190.801			1.37267			0.686333	−	0.727287i	$-0.259219\pi$
$139$	190.801			1.37267			0.686333	+	0.727287i	$0.259219\pi$
$140$		0			0
$141$	−8.10012			−0.0574476
$142$		0			0
$143$			46.4516i			0.324836i
$144$		0			0
$145$			46.7399i			0.322344i
$146$		0			0
$147$	−17.2379			−0.117265
$148$		0			0
$149$		−	142.039i		−	0.953285i	−0.879097	−	0.476643i	$-0.841854\pi$
$149$		−	142.039i		−	0.953285i	0.879097	−	0.476643i	$-0.158146\pi$
$150$		0			0
$151$	176.984			1.17208			0.586041	−	0.810281i	$-0.300686\pi$
$151$	176.984			1.17208			0.586041	+	0.810281i	$0.300686\pi$
$152$		0			0
$153$			7.11437i			0.0464992i
$154$		0			0
$155$		−	79.2742i		−	0.511447i
$156$		0			0
$157$		−	149.409i		−	0.951649i	−0.879540	−	0.475825i	$-0.842150\pi$
$157$		−	149.409i		−	0.951649i	0.879540	−	0.475825i	$-0.157850\pi$
$158$		0			0
$159$			54.7904i			0.344594i
$160$		0			0
$161$	153.658	−	4.94610i	0.954396	−	0.0307211i
$162$		0			0
$163$	265.489			1.62877			0.814384	−	0.580327i	$-0.197075\pi$
$163$	265.489			1.62877			0.814384	+	0.580327i	$0.197075\pi$
$164$		0			0
$165$	−93.4937			−0.566629
$166$		0			0
$167$	218.554			1.30871			0.654353	−	0.756189i	$-0.272941\pi$
$167$	218.554			1.30871			0.654353	+	0.756189i	$0.272941\pi$
$168$		0			0
$169$	−149.358			−0.883772
$170$		0			0
$171$			171.748i			1.00437i
$172$		0			0
$173$	236.667			1.36802			0.684009	−	0.729473i	$-0.260235\pi$
$173$	236.667			1.36802			0.684009	+	0.729473i	$0.260235\pi$
$174$		0			0
$175$		−	33.4212i		−	0.190978i
$176$		0			0
$177$	28.3094			0.159940
$178$		0			0
$179$	−92.9645			−0.519355			−0.259677	−	0.965695i	$-0.583616\pi$
$179$	−92.9645			−0.519355			−0.259677	+	0.965695i	$0.583616\pi$
$180$		0			0
$181$		−	43.4494i		−	0.240052i	−0.992771	−	0.120026i	$-0.961702\pi$
$181$		−	43.4494i		−	0.240052i	0.992771	−	0.120026i	$-0.0382978\pi$
$182$		0			0
$183$		−	273.412i		−	1.49406i
$184$		0			0
$185$	113.910			0.615732
$186$		0			0
$187$	10.7842			0.0576696
$188$		0			0
$189$			55.6145i			0.294257i
$190$		0			0
$191$		−	200.585i		−	1.05018i	−0.851046	−	0.525092i	$-0.824031\pi$
$191$		−	200.585i		−	1.05018i	0.851046	−	0.525092i	$-0.175969\pi$
$192$		0			0
$193$	−14.3503			−0.0743539			−0.0371770	−	0.999309i	$-0.511837\pi$
$193$	−14.3503			−0.0743539			−0.0371770	+	0.999309i	$0.511837\pi$
$194$		0			0
$195$			39.5346i			0.202741i
$196$		0			0
$197$	160.019			0.812277			0.406139	−	0.913812i	$-0.366875\pi$
$197$	160.019			0.812277			0.406139	+	0.913812i	$0.366875\pi$
$198$		0			0
$199$		−	160.292i		−	0.805487i	−0.915313	−	0.402743i	$-0.868057\pi$
$199$		−	160.292i		−	0.805487i	0.915313	−	0.402743i	$-0.131943\pi$
$200$		0			0
$201$		−	323.021i		−	1.60707i
$202$		0			0
$203$			139.719i			0.688270i
$204$		0			0
$205$		−	119.675i		−	0.583778i
$206$		0			0
$207$	−5.11638	−	158.948i	−0.0247168	−	0.767863i
$208$		0			0
$209$	260.342			1.24565
$210$		0			0
$211$	297.476			1.40984			0.704920	−	0.709287i	$-0.250983\pi$
$211$	297.476			1.40984			0.704920	+	0.709287i	$0.250983\pi$
$212$		0			0
$213$	470.510			2.20896
$214$		0			0
$215$	143.566			0.667749
$216$		0			0
$217$		−	236.973i		−	1.09204i
$218$		0			0
$219$	−154.135			−0.703815
$220$		0			0
$221$		−	4.56019i		−	0.0206344i
$222$		0			0
$223$	−333.278			−1.49452			−0.747260	−	0.664532i	$-0.768631\pi$
$223$	−333.278			−1.49452			−0.747260	+	0.664532i	$0.768631\pi$
$224$		0			0
$225$	−34.5717			−0.153652
$226$		0			0
$227$		−	193.754i		−	0.853542i	−0.904360	−	0.426771i	$-0.859651\pi$
$227$		−	193.754i		−	0.853542i	0.904360	−	0.426771i	$-0.140349\pi$
$228$		0			0
$229$			412.487i			1.80125i	0.434596	+	0.900626i	$0.356891\pi$
$229$			412.487i			1.80125i	−0.434596	+	0.900626i	$0.643109\pi$
$230$		0			0
$231$	−279.479			−1.20986
$232$		0			0
$233$	−153.707			−0.659685			−0.329843	−	0.944036i	$-0.606996\pi$
$233$	−153.707			−0.659685			−0.329843	+	0.944036i	$0.606996\pi$
$234$		0			0
$235$			4.54027i			0.0193203i
$236$		0			0
$237$			47.8165i			0.201758i
$238$		0			0
$239$	153.943			0.644114			0.322057	−	0.946720i	$-0.395626\pi$
$239$	153.943			0.644114			0.322057	+	0.946720i	$0.395626\pi$
$240$		0			0
$241$			129.479i			0.537258i	0.963244	+	0.268629i	$0.0865705\pi$
$241$			129.479i			0.537258i	−0.963244	+	0.268629i	$0.913429\pi$
$242$		0			0
$243$	305.778			1.25835
$244$		0			0
$245$			9.66218i			0.0394375i
$246$		0			0
$247$		−	110.088i		−	0.445699i
$248$		0			0
$249$			43.1009i			0.173096i
$250$		0			0
$251$			431.971i			1.72100i	0.509451	+	0.860500i	$0.329849\pi$
$251$			431.971i			1.72100i	−0.509451	+	0.860500i	$0.670151\pi$
$252$		0			0
$253$	−240.938	+	7.75559i	−0.952326	+	0.0306545i
$254$		0			0
$255$	9.17836			0.0359936
$256$		0			0
$257$	48.3917			0.188295			0.0941473	−	0.995558i	$-0.469988\pi$
$257$	48.3917			0.188295			0.0941473	+	0.995558i	$0.469988\pi$
$258$		0			0
$259$	340.510			1.31471
$260$		0			0
$261$	144.529			0.553750
$262$		0			0
$263$			447.412i			1.70119i	0.525824	+	0.850594i	$0.323757\pi$
$263$			447.412i			1.70119i	−0.525824	+	0.850594i	$0.676243\pi$
$264$		0			0
$265$	30.7111			0.115891
$266$		0			0
$267$			70.8908i			0.265509i
$268$		0			0
$269$	−25.8356			−0.0960433			−0.0480216	−	0.998846i	$-0.515292\pi$
$269$	−25.8356			−0.0960433			−0.0480216	+	0.998846i	$0.515292\pi$
$270$		0			0
$271$	−95.4214			−0.352109			−0.176054	−	0.984380i	$-0.556333\pi$
$271$	−95.4214			−0.352109			−0.176054	+	0.984380i	$0.556333\pi$
$272$		0			0
$273$			118.180i			0.432893i
$274$		0			0
$275$			52.4050i			0.190564i
$276$		0			0
$277$	−99.6472			−0.359737			−0.179869	−	0.983691i	$-0.557567\pi$
$277$	−99.6472			−0.359737			−0.179869	+	0.983691i	$0.557567\pi$
$278$		0			0
$279$	−245.131			−0.878605
$280$		0			0
$281$			438.884i			1.56186i	0.624616	+	0.780932i	$0.285255\pi$
$281$			438.884i			1.56186i	−0.624616	+	0.780932i	$0.714745\pi$
$282$		0			0
$283$			248.165i			0.876907i	0.898754	+	0.438454i	$0.144474\pi$
$283$			248.165i			0.876907i	−0.898754	+	0.438454i	$0.855526\pi$
$284$		0			0
$285$	221.575			0.777455
$286$		0			0
$287$		−	357.741i		−	1.24648i
$288$		0			0
$289$	287.941			0.996337
$290$		0			0
$291$			23.3053i			0.0800871i
$292$		0			0
$293$			117.436i			0.400805i	0.979714	+	0.200402i	$0.0642249\pi$
$293$			117.436i			0.400805i	−0.979714	+	0.200402i	$0.935775\pi$
$294$		0			0
$295$		−	15.8680i		−	0.0537898i
$296$		0			0
$297$		−	87.2046i		−	0.293618i
$298$		0			0
$299$	3.27951	+	101.883i	0.0109683	+	0.340745i
$300$		0			0
$301$	429.159			1.42578
$302$		0			0
$303$	−441.071			−1.45568
$304$		0			0
$305$	−153.253			−0.502468
$306$		0			0
$307$	−479.192			−1.56089			−0.780444	−	0.625226i	$-0.785007\pi$
$307$	−479.192			−1.56089			−0.780444	+	0.625226i	$0.785007\pi$
$308$		0			0
$309$		−	685.252i		−	2.21764i
$310$		0			0
$311$	40.0444			0.128760			0.0643801	−	0.997925i	$-0.479493\pi$
$311$	40.0444			0.128760			0.0643801	+	0.997925i	$0.479493\pi$
$312$		0			0
$313$		−	441.860i		−	1.41169i	−0.708365	−	0.705846i	$-0.750567\pi$
$313$		−	441.860i		−	1.41169i	0.708365	−	0.705846i	$-0.249433\pi$
$314$		0			0
$315$	−103.345			−0.328078
$316$		0			0
$317$	−5.52692			−0.0174351			−0.00871754	−	0.999962i	$-0.502775\pi$
$317$	−5.52692			−0.0174351			−0.00871754	+	0.999962i	$0.502775\pi$
$318$		0			0
$319$		−	219.082i		−	0.686777i
$320$		0			0
$321$			714.981i			2.22736i
$322$		0			0
$323$	−25.5580			−0.0791268
$324$		0			0
$325$	22.1599			0.0681843
$326$		0			0
$327$		−	696.776i		−	2.13081i
$328$		0			0
$329$			13.5721i			0.0412527i
$330$		0			0
$331$	−82.5365			−0.249355			−0.124678	−	0.992197i	$-0.539790\pi$
$331$	−82.5365			−0.249355			−0.124678	+	0.992197i	$0.539790\pi$
$332$		0			0
$333$		−	352.232i		−	1.05775i
$334$		0			0
$335$	−181.059			−0.540476
$336$		0			0
$337$		−	285.729i		−	0.847860i	−0.905695	−	0.423930i	$-0.860650\pi$
$337$		−	285.729i		−	0.847860i	0.905695	−	0.423930i	$-0.139350\pi$
$338$		0			0
$339$			373.309i			1.10121i
$340$		0			0
$341$			371.578i			1.08967i
$342$		0			0
$343$			356.410i			1.03910i
$344$		0			0
$345$	−205.061	+	6.60072i	−0.594379	+	0.0191325i
$346$		0			0
$347$	452.930			1.30527			0.652637	−	0.757671i	$-0.273663\pi$
$347$	452.930			1.30527			0.652637	+	0.757671i	$0.273663\pi$
$348$		0			0
$349$	408.430			1.17029			0.585143	−	0.810930i	$-0.301038\pi$
$349$	408.430			1.17029			0.585143	+	0.810930i	$0.301038\pi$
$350$		0			0
$351$	−36.8752			−0.105058
$352$		0			0
$353$	311.821			0.883346			0.441673	−	0.897176i	$-0.354385\pi$
$353$	311.821			0.883346			0.441673	+	0.897176i	$0.354385\pi$
$354$		0			0
$355$		−	263.730i		−	0.742901i
$356$		0			0
$357$	27.4367			0.0768534
$358$		0			0
$359$			469.380i			1.30746i	0.756726	+	0.653732i	$0.226798\pi$
$359$			469.380i			1.30746i	−0.756726	+	0.653732i	$0.773202\pi$
$360$		0			0
$361$	−255.995			−0.709127
$362$		0			0
$363$	−44.4744			−0.122519
$364$		0			0
$365$			86.3959i			0.236701i
$366$		0			0
$367$			116.830i			0.318339i	0.987251	+	0.159169i	$0.0508816\pi$
$367$			116.830i			0.318339i	−0.987251	+	0.159169i	$0.949118\pi$
$368$		0			0
$369$	−370.056			−1.00286
$370$		0			0
$371$	91.8040			0.247450
$372$		0			0
$373$			281.784i			0.755454i	0.925917	+	0.377727i	$0.123294\pi$
$373$			281.784i			0.755454i	−0.925917	+	0.377727i	$0.876706\pi$
$374$		0			0
$375$			44.6015i			0.118937i
$376$		0			0
$377$	−92.6405			−0.245731
$378$		0			0
$379$		−	458.059i		−	1.20860i	−0.796757	−	0.604299i	$-0.793453\pi$
$379$		−	458.059i		−	1.20860i	0.796757	−	0.604299i	$-0.206547\pi$
$380$		0			0
$381$	−651.413			−1.70974
$382$		0			0
$383$			166.686i			0.435212i	0.976037	+	0.217606i	$0.0698248\pi$
$383$			166.686i			0.435212i	−0.976037	+	0.217606i	$0.930175\pi$
$384$		0			0
$385$			156.653i			0.406892i
$386$		0			0
$387$		−	443.933i		−	1.14711i
$388$		0			0
$389$			247.739i			0.636861i	0.947946	+	0.318430i	$0.103156\pi$
$389$			247.739i			0.636861i	−0.947946	+	0.318430i	$0.896844\pi$
$390$		0			0
$391$	23.6531	−	0.761372i	0.0604939	−	0.00194724i
$392$		0			0
$393$	−733.543			−1.86652
$394$		0			0
$395$	26.8021			0.0678534
$396$		0			0
$397$	158.384			0.398951			0.199476	−	0.979903i	$-0.436076\pi$
$397$	158.384			0.398951			0.199476	+	0.979903i	$0.436076\pi$
$398$		0			0
$399$	662.349			1.66002
$400$		0			0
$401$		−	349.566i		−	0.871737i	−0.900010	−	0.435868i	$-0.856441\pi$
$401$		−	349.566i		−	0.871737i	0.900010	−	0.435868i	$-0.143559\pi$
$402$		0			0
$403$	157.125			0.389888
$404$		0			0
$405$		−	213.368i		−	0.526834i
$406$		0			0
$407$	−533.926			−1.31186
$408$		0			0
$409$	114.004			0.278739			0.139370	−	0.990240i	$-0.455492\pi$
$409$	114.004			0.278739			0.139370	+	0.990240i	$0.455492\pi$
$410$		0			0
$411$			254.479i			0.619170i
$412$		0			0
$413$		−	47.4338i		−	0.114852i
$414$		0			0
$415$	24.1589			0.0582142
$416$		0			0
$417$	−761.157			−1.82532
$418$		0			0
$419$			92.8600i			0.221623i	0.993841	+	0.110811i	$0.0353450\pi$
$419$			92.8600i			0.221623i	−0.993841	+	0.110811i	$0.964655\pi$
$420$		0			0
$421$			40.7090i			0.0966960i	0.998831	+	0.0483480i	$0.0153957\pi$
$421$			40.7090i			0.0966960i	−0.998831	+	0.0483480i	$0.984604\pi$
$422$		0			0
$423$	14.0394			0.0331900
$424$		0			0
$425$		−	5.14465i		−	0.0121050i
$426$		0			0
$427$	−458.116			−1.07287
$428$		0			0
$429$		−	185.308i		−	0.431954i
$430$		0			0
$431$		−	719.900i		−	1.67030i	−0.550021	−	0.835151i	$-0.685380\pi$
$431$		−	719.900i		−	1.67030i	0.550021	−	0.835151i	$-0.314620\pi$
$432$		0			0
$433$			272.237i			0.628723i	0.949303	+	0.314361i	$0.101790\pi$
$433$			272.237i			0.628723i	−0.949303	+	0.314361i	$0.898210\pi$
$434$		0			0
$435$		−	186.459i		−	0.428641i
$436$		0			0
$437$	571.010	−	18.3803i	1.30666	−	0.0420602i
$438$		0			0
$439$	−746.299			−1.70000			−0.849999	−	0.526784i	$-0.823398\pi$
$439$	−746.299			−1.70000			−0.849999	+	0.526784i	$0.823398\pi$
$440$		0			0
$441$	29.8773			0.0677489
$442$		0			0
$443$	−502.782			−1.13495			−0.567474	−	0.823391i	$-0.692079\pi$
$443$	−502.782			−1.13495			−0.567474	+	0.823391i	$0.692079\pi$
$444$		0			0
$445$	39.7357			0.0892936
$446$		0			0
$447$			566.635i			1.26764i
$448$		0			0
$449$	−144.711			−0.322296			−0.161148	−	0.986930i	$-0.551520\pi$
$449$	−144.711			−0.322296			−0.161148	+	0.986930i	$0.551520\pi$
$450$		0			0
$451$			560.944i			1.24378i
$452$		0			0
$453$	−706.040			−1.55859
$454$		0			0
$455$	66.2421			0.145587
$456$		0			0
$457$		−	541.305i		−	1.18447i	−0.805764	−	0.592237i	$-0.798245\pi$
$457$		−	541.305i		−	1.18447i	0.805764	−	0.592237i	$-0.201755\pi$
$458$		0			0
$459$			8.56095i			0.0186513i
$460$		0			0
$461$	533.587			1.15746			0.578728	−	0.815521i	$-0.303549\pi$
$461$	533.587			1.15746			0.578728	+	0.815521i	$0.303549\pi$
$462$		0			0
$463$	−75.4471			−0.162953			−0.0814763	−	0.996675i	$-0.525964\pi$
$463$	−75.4471			−0.162953			−0.0814763	+	0.996675i	$0.525964\pi$
$464$		0			0
$465$			316.247i			0.680101i
$466$		0			0
$467$		−	248.347i		−	0.531792i	−0.964002	−	0.265896i	$-0.914332\pi$
$467$		−	248.347i		−	0.531792i	0.964002	−	0.265896i	$-0.0856678\pi$
$468$		0			0
$469$	−541.237			−1.15402
$470$		0			0
$471$			596.034i			1.26546i
$472$		0			0
$473$	−672.930			−1.42268
$474$		0			0
$475$		−	124.197i		−	0.261467i
$476$		0			0
$477$		−	94.9645i		−	0.199087i
$478$		0			0
$479$		−	173.159i		−	0.361501i	−0.983529	−	0.180751i	$-0.942147\pi$
$479$		−	173.159i		−	0.361501i	0.983529	−	0.180751i	$-0.0578527\pi$
$480$		0			0
$481$			225.775i			0.469387i
$482$		0			0
$483$	−612.984	+	19.7314i	−1.26912	+	0.0408517i
$484$		0			0
$485$	13.0631			0.0269342
$486$		0			0
$487$	611.054			1.25473			0.627365	−	0.778725i	$-0.284133\pi$
$487$	611.054			1.25473			0.627365	+	0.778725i	$0.284133\pi$
$488$		0			0
$489$	−1059.11			−2.16587
$490$		0			0
$491$	−0.789626			−0.00160820			−0.000804100	−	1.00000i	$-0.500256\pi$
$491$	−0.789626			−0.00160820			−0.000804100		1.00000i	$0.500256\pi$
$492$		0			0
$493$			21.5074i			0.0436256i
$494$		0			0
$495$	162.046			0.327366
$496$		0			0
$497$		−	788.362i		−	1.58624i
$498$		0			0
$499$	−467.175			−0.936223			−0.468112	−	0.883669i	$-0.655065\pi$
$499$	−467.175			−0.936223			−0.468112	+	0.883669i	$0.655065\pi$
$500$		0			0
$501$	−871.872			−1.74026
$502$		0			0
$503$		−	474.467i		−	0.943275i	−0.881792	−	0.471638i	$-0.843663\pi$
$503$		−	474.467i		−	0.943275i	0.881792	−	0.471638i	$-0.156337\pi$
$504$		0			0
$505$			247.229i			0.489563i
$506$		0			0
$507$	595.829			1.17521
$508$		0			0
$509$	434.172			0.852989			0.426495	−	0.904490i	$-0.359748\pi$
$509$	434.172			0.852989			0.426495	+	0.904490i	$0.359748\pi$
$510$		0			0
$511$			258.261i			0.505404i
$512$		0			0
$513$			206.670i			0.402865i
$514$		0			0
$515$	−384.097			−0.745819
$516$		0			0
$517$		−	21.2814i		−	0.0411632i
$518$		0			0
$519$	−944.131			−1.81914
$520$		0			0
$521$		−	53.4153i		−	0.102525i	−0.998685	−	0.0512623i	$-0.983676\pi$
$521$		−	53.4153i		−	0.102525i	0.998685	−	0.0512623i	$-0.0163244\pi$
$522$		0			0
$523$		−	993.809i		−	1.90021i	−0.311934	−	0.950104i	$-0.600977\pi$
$523$		−	993.809i		−	1.90021i	0.311934	−	0.950104i	$-0.399023\pi$
$524$		0			0
$525$			133.326i			0.253955i
$526$		0			0
$527$		−	36.4781i		−	0.0692184i
$528$		0			0
$529$	−527.905	+	34.0208i	−0.997930	+	0.0643115i
$530$		0			0
$531$	−49.0668			−0.0924045
$532$		0			0
$533$	237.200			0.445028
$534$		0			0
$535$	400.761			0.749085
$536$		0			0
$537$	370.861			0.690617
$538$		0			0
$539$		−	45.2890i		−	0.0840242i
$540$		0			0
$541$	813.317			1.50336			0.751679	−	0.659529i	$-0.229244\pi$
$541$	813.317			1.50336			0.751679	+	0.659529i	$0.229244\pi$
$542$		0			0
$543$			173.332i			0.319211i
$544$		0			0
$545$	−390.556			−0.716617
$546$		0			0
$547$	42.0335			0.0768437			0.0384219	−	0.999262i	$-0.487767\pi$
$547$	42.0335			0.0768437			0.0384219	+	0.999262i	$0.487767\pi$
$548$		0			0
$549$			473.887i			0.863182i
$550$		0			0
$551$			519.211i			0.942307i
$552$		0			0
$553$	80.1190			0.144881
$554$		0			0
$555$	−454.420			−0.818775
$556$		0			0
$557$		−	1086.25i		−	1.95018i	−0.221806	−	0.975091i	$-0.571195\pi$
$557$		−	1086.25i		−	1.95018i	0.221806	−	0.975091i	$-0.428805\pi$
$558$		0			0
$559$			284.554i			0.509041i
$560$		0			0
$561$	−43.0212			−0.0766867
$562$		0			0
$563$		−	932.386i		−	1.65610i	−0.560652	−	0.828051i	$-0.689450\pi$
$563$		−	932.386i		−	1.65610i	0.560652	−	0.828051i	$-0.310550\pi$
$564$		0			0
$565$	209.247			0.370349
$566$		0			0
$567$		−	637.816i		−	1.12490i
$568$		0			0
$569$		−	233.192i		−	0.409827i	−0.978780	−	0.204914i	$-0.934309\pi$
$569$		−	233.192i		−	0.409827i	0.978780	−	0.204914i	$-0.0656913\pi$
$570$		0			0
$571$			326.457i			0.571729i	0.958270	+	0.285864i	$0.0922807\pi$
$571$			326.457i			0.571729i	−0.958270	+	0.285864i	$0.907719\pi$
$572$		0			0
$573$			800.189i			1.39649i
$574$		0			0
$575$	3.69983	+	114.940i	0.00643449	+	0.199896i
$576$		0			0
$577$	−907.490			−1.57277			−0.786387	−	0.617735i	$-0.788051\pi$
$577$	−907.490			−1.57277			−0.786387	+	0.617735i	$0.788051\pi$
$578$		0			0
$579$	57.2474			0.0988728
$580$		0			0
$581$	72.2177			0.124299
$582$		0			0
$583$	−143.951			−0.246913
$584$		0			0
$585$		−	68.5226i		−	0.117133i
$586$		0			0
$587$	79.5946			0.135596			0.0677978	−	0.997699i	$-0.478403\pi$
$587$	79.5946			0.135596			0.0677978	+	0.997699i	$0.478403\pi$
$588$		0			0
$589$		−	880.618i		−	1.49511i
$590$		0			0
$591$	−638.359			−1.08013
$592$		0			0
$593$	1106.32			1.86564			0.932819	−	0.360345i	$-0.117341\pi$
$593$	1106.32			1.86564			0.932819	+	0.360345i	$0.117341\pi$
$594$		0			0
$595$		−	15.3788i		−	0.0258467i
$596$		0			0
$597$			639.449i			1.07110i
$598$		0			0
$599$	987.575			1.64871			0.824353	−	0.566076i	$-0.191539\pi$
$599$	987.575			1.64871			0.824353	+	0.566076i	$0.191539\pi$
$600$		0			0
$601$	−539.148			−0.897085			−0.448543	−	0.893761i	$-0.648057\pi$
$601$	−539.148			−0.897085			−0.448543	+	0.893761i	$0.648057\pi$
$602$		0			0
$603$			559.870i			0.928474i
$604$		0			0
$605$			24.9288i			0.0412046i
$606$		0			0
$607$	472.163			0.777864			0.388932	−	0.921267i	$-0.372844\pi$
$607$	472.163			0.777864			0.388932	+	0.921267i	$0.372844\pi$
$608$		0			0
$609$		−	557.377i		−	0.915233i
$610$		0			0
$611$	−8.99901			−0.0147283
$612$		0			0
$613$		−	8.85817i		−	0.0144505i	−0.999974	−	0.00722526i	$-0.997700\pi$
$613$		−	8.85817i		−	0.0144505i	0.999974	−	0.00722526i	$-0.00229989\pi$
$614$		0			0
$615$			477.415i			0.776285i
$616$		0			0
$617$			1079.90i			1.75024i	0.483905	+	0.875121i	$0.339218\pi$
$617$			1079.90i			1.75024i	−0.483905	+	0.875121i	$0.660782\pi$
$618$		0			0
$619$			88.9802i			0.143748i	0.997414	+	0.0718742i	$0.0228980\pi$
$619$			88.9802i			0.143748i	−0.997414	+	0.0718742i	$0.977102\pi$
$620$		0			0
$621$	−6.15670	−	191.267i	−0.00991418	−	0.307998i
$622$		0			0
$623$	118.781			0.190660
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$	−1038.58			−1.65642
$628$		0			0
$629$	52.4160			0.0833322
$630$		0			0
$631$		−	181.198i		−	0.287161i	−0.989639	−	0.143580i	$-0.954138\pi$
$631$		−	181.198i		−	0.287161i	0.989639	−	0.143580i	$-0.0458616\pi$
$632$		0			0
$633$	−1186.72			−1.87475
$634$		0			0
$635$			365.129i			0.575007i
$636$		0			0
$637$	−19.1508			−0.0300641
$638$		0			0
$639$	−815.502			−1.27622
$640$		0			0
$641$			668.352i			1.04267i	0.853352	+	0.521336i	$0.174566\pi$
$641$			668.352i			1.04267i	−0.853352	+	0.521336i	$0.825434\pi$
$642$		0			0
$643$		−	607.807i		−	0.945268i	−0.881259	−	0.472634i	$-0.843303\pi$
$643$		−	607.807i		−	0.945268i	0.881259	−	0.472634i	$-0.156697\pi$
$644$		0			0
$645$	−572.725			−0.887946
$646$		0			0
$647$	−336.091			−0.519461			−0.259730	−	0.965681i	$-0.583634\pi$
$647$	−336.091			−0.519461			−0.259730	+	0.965681i	$0.583634\pi$
$648$		0			0
$649$			74.3772i			0.114603i
$650$		0			0
$651$			945.350i			1.45215i
$652$		0			0
$653$	470.281			0.720185			0.360093	−	0.932917i	$-0.382745\pi$
$653$	470.281			0.720185			0.360093	+	0.932917i	$0.382745\pi$
$654$		0			0
$655$			411.165i			0.627733i
$656$		0			0
$657$	267.152			0.406625
$658$		0			0
$659$			545.138i			0.827220i	0.910454	+	0.413610i	$0.135732\pi$
$659$			545.138i			0.827220i	−0.910454	+	0.413610i	$0.864268\pi$
$660$		0			0
$661$		−	957.624i		−	1.44875i	−0.689406	−	0.724375i	$-0.742128\pi$
$661$		−	957.624i		−	1.44875i	0.689406	−	0.724375i	$-0.257872\pi$
$662$		0			0
$663$			18.1919i			0.0274387i
$664$		0			0
$665$		−	371.259i		−	0.558284i
$666$		0			0
$667$	−15.4673	−	480.514i	−0.0231894	−	0.720411i
$668$		0			0
$669$	1329.54			1.98735
$670$		0			0
$671$	718.334			1.07054
$672$		0			0
$673$	194.792			0.289439			0.144719	−	0.989473i	$-0.453772\pi$
$673$	194.792			0.289439			0.144719	+	0.989473i	$0.453772\pi$
$674$		0			0
$675$	−41.6013			−0.0616315
$676$		0			0
$677$			464.694i			0.686402i	0.939262	+	0.343201i	$0.111511\pi$
$677$			464.694i			0.686402i	−0.939262	+	0.343201i	$0.888489\pi$
$678$		0			0
$679$	39.0492			0.0575099
$680$		0			0
$681$			772.939i			1.13501i
$682$		0			0
$683$	1081.91			1.58405			0.792027	−	0.610486i	$-0.209026\pi$
$683$	1081.91			1.58405			0.792027	+	0.610486i	$0.209026\pi$
$684$		0			0
$685$	142.640			0.208234
$686$		0			0
$687$		−	1645.52i		−	2.39523i
$688$		0			0
$689$			60.8707i			0.0883464i
$690$		0			0
$691$	−892.056			−1.29096			−0.645482	−	0.763776i	$-0.723343\pi$
$691$	−892.056			−1.29096			−0.645482	+	0.763776i	$0.723343\pi$
$692$		0			0
$693$	484.402			0.698992
$694$		0			0
$695$			426.643i			0.613875i
$696$		0			0
$697$		−	55.0684i		−	0.0790077i
$698$		0			0
$699$	613.179			0.877223
$700$		0			0
$701$			408.305i			0.582461i	0.956653	+	0.291231i	$0.0940647\pi$
$701$			408.305i			0.582461i	−0.956653	+	0.291231i	$0.905935\pi$
$702$		0			0
$703$	1265.37			1.79996
$704$		0			0
$705$		−	18.1124i		−	0.0256914i
$706$		0			0
$707$			739.037i			1.04531i
$708$		0			0
$709$			221.424i			0.312305i	0.987733	+	0.156152i	$0.0499091\pi$
$709$			221.424i			0.312305i	−0.987733	+	0.156152i	$0.950091\pi$
$710$		0			0
$711$		−	82.8772i		−	0.116564i
$712$		0			0
$713$	26.2336	+	814.985i	0.0367933	+	1.14304i
$714$		0			0
$715$	−103.869			−0.145271
$716$		0			0
$717$	−614.122			−0.856517
$718$		0			0
$719$	−548.707			−0.763153			−0.381577	−	0.924337i	$-0.624619\pi$
$719$	−548.707			−0.763153			−0.381577	+	0.924337i	$0.624619\pi$
$720$		0			0
$721$	−1148.17			−1.59247
$722$		0			0
$723$		−	516.528i		−	0.714424i
$724$		0			0
$725$	−104.514			−0.144157
$726$		0			0
$727$			591.127i			0.813105i	0.913628	+	0.406552i	$0.133269\pi$
$727$			591.127i			0.813105i	−0.913628	+	0.406552i	$0.866731\pi$
$728$		0			0
$729$	−361.047			−0.495263
$730$		0			0
$731$	66.0621			0.0903722
$732$		0			0
$733$		−	561.314i		−	0.765776i	−0.923795	−	0.382888i	$-0.874929\pi$
$733$		−	561.314i		−	0.765776i	0.923795	−	0.382888i	$-0.125071\pi$
$734$		0			0
$735$		−	38.5451i		−	0.0524423i
$736$		0			0
$737$	848.670			1.15152
$738$		0			0
$739$	880.148			1.19100			0.595500	−	0.803356i	$-0.296954\pi$
$739$	880.148			1.19100			0.595500	+	0.803356i	$0.296954\pi$
$740$		0			0
$741$			439.170i			0.592672i
$742$		0			0
$743$			738.685i			0.994193i	0.867695	+	0.497097i	$0.165601\pi$
$743$			738.685i			0.994193i	−0.867695	+	0.497097i	$0.834399\pi$
$744$		0			0
$745$	317.610			0.426322
$746$		0			0
$747$		−	74.7039i		−	0.100005i
$748$		0			0
$749$	1197.99			1.59945
$750$		0			0
$751$		−	962.584i		−	1.28174i	−0.767651	−	0.640868i	$-0.778574\pi$
$751$		−	962.584i		−	1.28174i	0.767651	−	0.640868i	$-0.221426\pi$
$752$		0			0
$753$		−	1723.25i		−	2.28852i
$754$		0			0
$755$			395.749i			0.524171i
$756$		0			0
$757$		−	553.711i		−	0.731454i	−0.930722	−	0.365727i	$-0.880820\pi$
$757$		−	553.711i		−	0.731454i	0.930722	−	0.365727i	$-0.119180\pi$
$758$		0			0
$759$	961.170	−	30.9392i	1.26636	−	0.0407631i
$760$		0			0
$761$	−429.314			−0.564145			−0.282072	−	0.959393i	$-0.591022\pi$
$761$	−429.314			−0.564145			−0.282072	+	0.959393i	$0.591022\pi$
$762$		0			0
$763$	−1167.48			−1.53012
$764$		0			0
$765$	−15.9082			−0.0207951
$766$		0			0
$767$	31.4510			0.0410052
$768$		0			0
$769$			254.304i			0.330695i	0.986235	+	0.165348i	$0.0528746\pi$
$769$			254.304i			0.330695i	−0.986235	+	0.165348i	$0.947125\pi$
$770$		0			0
$771$	−193.048			−0.250387
$772$		0			0
$773$			247.004i			0.319540i	0.987154	+	0.159770i	$0.0510752\pi$
$773$			247.004i			0.319540i	−0.987154	+	0.159770i	$0.948925\pi$
$774$		0			0
$775$	177.263			0.228726
$776$		0			0
$777$	−1358.39			−1.74825
$778$		0			0
$779$		−	1329.41i		−	1.70655i
$780$		0			0
$781$			1236.17i			1.58280i
$782$		0			0
$783$	173.916			0.222115
$784$		0			0
$785$	334.089			0.425590
$786$		0			0
$787$			37.5937i			0.0477683i	0.999715	+	0.0238842i	$0.00760329\pi$
$787$			37.5937i			0.0477683i	−0.999715	+	0.0238842i	$0.992397\pi$
$788$		0			0
$789$		−	1784.85i		−	2.26217i
$790$		0			0
$791$	625.498			0.790768
$792$		0			0
$793$		−	303.754i		−	0.383044i
$794$		0			0
$795$	−122.515			−0.154107
$796$		0			0
$797$		−	1022.90i		−	1.28343i	−0.766942	−	0.641716i	$-0.778223\pi$
$797$		−	1022.90i		−	1.28343i	0.766942	−	0.641716i	$-0.221777\pi$
$798$		0			0
$799$			2.08921i			0.00261478i
$800$		0			0
$801$		−	122.870i		−	0.153396i
$802$		0			0
$803$		−	404.959i		−	0.504308i
$804$		0			0
$805$	11.0598	+	343.589i	0.0137389	+	0.426819i
$806$		0			0
$807$	103.066			0.127714
$808$		0			0
$809$	−206.973			−0.255839			−0.127919	−	0.991785i	$-0.540830\pi$
$809$	−206.973			−0.255839			−0.127919	+	0.991785i	$0.540830\pi$
$810$		0			0
$811$	−605.472			−0.746575			−0.373288	−	0.927716i	$-0.621770\pi$
$811$	−605.472			−0.746575			−0.373288	+	0.927716i	$0.621770\pi$
$812$		0			0
$813$	380.663			0.468220
$814$		0			0
$815$			593.652i			0.728407i
$816$		0			0
$817$	1594.80			1.95203
$818$		0			0
$819$		−	204.833i		−	0.250102i
$820$		0			0
$821$	884.992			1.07794			0.538972	−	0.842324i	$-0.318813\pi$
$821$	884.992			1.07794			0.538972	+	0.842324i	$0.318813\pi$
$822$		0			0
$823$	−381.663			−0.463746			−0.231873	−	0.972746i	$-0.574485\pi$
$823$	−381.663			−0.463746			−0.231873	+	0.972746i	$0.574485\pi$
$824$		0			0
$825$		−	209.058i		−	0.253404i
$826$		0			0
$827$			1216.83i			1.47137i	0.677322	+	0.735687i	$0.263141\pi$
$827$			1216.83i			1.47137i	−0.677322	+	0.735687i	$0.736859\pi$
$828$		0			0
$829$	−801.202			−0.966468			−0.483234	−	0.875491i	$-0.660538\pi$
$829$	−801.202			−0.966468			−0.483234	+	0.875491i	$0.660538\pi$
$830$		0			0
$831$	397.521			0.478364
$832$		0			0
$833$			4.44606i			0.00533741i
$834$		0			0
$835$			488.701i			0.585271i
$836$		0			0
$837$	−294.974			−0.352418
$838$		0			0
$839$		−	1624.13i		−	1.93579i	−0.251355	−	0.967895i	$-0.580876\pi$
$839$		−	1624.13i		−	1.93579i	0.251355	−	0.967895i	$-0.419124\pi$
$840$		0			0
$841$	−404.076			−0.480470
$842$		0			0
$843$		−	1750.83i		−	2.07690i
$844$		0			0
$845$		−	333.974i		−	0.395235i
$846$		0			0
$847$			74.5191i			0.0879800i
$848$		0			0
$849$		−	989.999i		−	1.16608i
$850$		0			0
$851$	−1171.07	+	37.6955i	−1.37610	+	0.0442956i
$852$		0			0
$853$	−459.407			−0.538579			−0.269289	−	0.963059i	$-0.586789\pi$
$853$	−459.407			−0.538579			−0.269289	+	0.963059i	$0.586789\pi$
$854$		0			0
$855$	−384.040			−0.449170
$856$		0			0
$857$	−1071.34			−1.25010			−0.625052	−	0.780583i	$-0.714922\pi$
$857$	−1071.34			−1.25010			−0.625052	+	0.780583i	$0.714922\pi$
$858$		0			0
$859$	−602.078			−0.700906			−0.350453	−	0.936580i	$-0.613972\pi$
$859$	−602.078			−0.700906			−0.350453	+	0.936580i	$0.613972\pi$
$860$		0			0
$861$			1427.13i			1.65752i
$862$		0			0
$863$	1300.54			1.50700			0.753502	−	0.657446i	$-0.228363\pi$
$863$	1300.54			1.50700			0.753502	+	0.657446i	$0.228363\pi$
$864$		0			0
$865$			529.204i			0.611796i
$866$		0			0
$867$	−1148.68			−1.32489
$868$		0			0
$869$	−125.628			−0.144566
$870$		0			0
$871$		−	358.867i		−	0.412017i
$872$		0			0
$873$		−	40.3936i		−	0.0462698i
$874$		0			0
$875$	74.7320			0.0854080
$876$		0			0
$877$	1457.85			1.66231			0.831155	−	0.556042i	$-0.187680\pi$
$877$	1457.85			1.66231			0.831155	+	0.556042i	$0.187680\pi$
$878$		0			0
$879$		−	468.484i		−	0.532974i
$880$		0			0
$881$		−	533.589i		−	0.605662i	−0.953044	−	0.302831i	$-0.902068\pi$
$881$		−	533.589i		−	0.605662i	0.953044	−	0.302831i	$-0.0979318\pi$
$882$		0			0
$883$	−1364.02			−1.54475			−0.772377	−	0.635165i	$-0.780932\pi$
$883$	−1364.02			−1.54475			−0.772377	+	0.635165i	$0.780932\pi$
$884$		0			0
$885$			63.3018i			0.0715275i
$886$		0			0
$887$	409.111			0.461230			0.230615	−	0.973045i	$-0.425926\pi$
$887$	409.111			0.461230			0.230615	+	0.973045i	$0.425926\pi$
$888$		0			0
$889$			1091.47i			1.22775i
$890$		0			0
$891$			1000.11i			1.12246i
$892$		0			0
$893$			50.4357i			0.0564789i
$894$		0			0
$895$		−	207.875i		−	0.232263i
$896$		0			0
$897$	−13.0829	−	406.439i	−0.0145852	−	0.453109i
$898$		0			0
$899$	−741.054			−0.824310
$900$		0			0
$901$	14.1317			0.0156845
$902$		0			0
$903$	−1712.03			−1.89594
$904$		0			0
$905$	97.1558			0.107355
$906$		0			0
$907$		−	231.536i		−	0.255277i	−0.991821	−	0.127638i	$-0.959260\pi$
$907$		−	231.536i		−	0.255277i	0.991821	−	0.127638i	$-0.0407397\pi$
$908$		0			0
$909$	764.479			0.841011
$910$		0			0
$911$			1061.17i			1.16484i	0.812888	+	0.582420i	$0.197894\pi$
$911$			1061.17i			1.16484i	−0.812888	+	0.582420i	$0.802106\pi$
$912$		0			0
$913$	−113.239			−0.124029
$914$		0			0
$915$	611.368			0.668162
$916$		0			0
$917$			1229.09i			1.34033i
$918$		0			0
$919$		−	7.16111i		−	0.00779229i	−0.999992	−	0.00389614i	$-0.998760\pi$
$919$		−	7.16111i		−	0.00779229i	0.999992	−	0.00389614i	$-0.00124018\pi$
$920$		0			0
$921$	1911.63			2.07561
$922$		0			0
$923$	522.723			0.566331
$924$		0			0
$925$			254.711i			0.275364i
$926$		0			0
$927$			1187.70i			1.28123i
$928$		0			0
$929$	−101.980			−0.109773			−0.0548867	−	0.998493i	$-0.517480\pi$
$929$	−101.980			−0.109773			−0.0548867	+	0.998493i	$0.517480\pi$
$930$		0			0
$931$			107.332i			0.115287i
$932$		0			0
$933$	−159.748			−0.171220
$934$		0			0
$935$			24.1142i			0.0257906i
$936$		0			0
$937$		−	953.155i		−	1.01724i	−0.860991	−	0.508620i	$-0.830156\pi$
$937$		−	953.155i		−	1.01724i	0.860991	−	0.508620i	$-0.169844\pi$
$938$		0			0
$939$			1762.70i			1.87721i
$940$		0			0
$941$			350.749i			0.372740i	0.982480	+	0.186370i	$0.0596724\pi$
$941$			350.749i			0.372740i	−0.982480	+	0.186370i	$0.940328\pi$
$942$		0			0
$943$	39.6030	+	1230.32i	0.0419968	+	1.30469i
$944$		0			0
$945$	−124.358			−0.131596
$946$		0			0
$947$	237.687			0.250989			0.125495	−	0.992094i	$-0.459948\pi$
$947$	237.687			0.250989			0.125495	+	0.992094i	$0.459948\pi$
$948$		0			0
$949$	−171.240			−0.180443
$950$		0			0
$951$	22.0484			0.0231845
$952$		0			0
$953$			139.892i			0.146791i	0.997303	+	0.0733957i	$0.0233836\pi$
$953$			139.892i			0.146791i	−0.997303	+	0.0733957i	$0.976616\pi$
$954$		0			0
$955$	448.522			0.469656
$956$		0			0
$957$			873.978i			0.913248i
$958$		0			0
$959$	426.392			0.444621
$960$		0			0
$961$	295.880			0.307888
$962$		0			0
$963$		−	1239.23i		−	1.28684i
$964$		0			0
$965$		−	32.0883i		−	0.0332521i
$966$		0			0
$967$	158.892			0.164314			0.0821571	−	0.996619i	$-0.473819\pi$
$967$	158.892			0.164314			0.0821571	+	0.996619i	$0.473819\pi$
$968$		0			0
$969$	101.958			0.105220
$970$		0			0
$971$		−	1021.55i		−	1.05206i	−0.850467	−	0.526028i	$-0.823681\pi$
$971$		−	1021.55i		−	1.05206i	0.850467	−	0.526028i	$-0.176319\pi$
$972$		0			0
$973$			1275.36i			1.31075i
$974$		0			0
$975$	−88.4020			−0.0906687
$976$		0			0
$977$		−	4.11462i		−	0.00421148i	−0.999998	−	0.00210574i	$-0.999330\pi$
$977$		−	4.11462i		−	0.00421148i	0.999998	−	0.00210574i	$-0.000670279\pi$
$978$		0			0
$979$	−186.251			−0.190246
$980$		0			0
$981$			1207.67i			1.23106i
$982$		0			0
$983$			1804.00i			1.83520i	0.397510	+	0.917598i	$0.369874\pi$
$983$			1804.00i			1.83520i	−0.397510	+	0.917598i	$0.630126\pi$
$984$		0			0
$985$			357.812i			0.363261i
$986$		0			0
$987$		−	54.1431i		−	0.0548562i
$988$		0			0
$989$	−1475.94	+	47.5093i	−1.49236	+	0.0480377i
$990$		0			0
$991$	485.829			0.490241			0.245120	−	0.969493i	$-0.421173\pi$
$991$	485.829			0.490241			0.245120	+	0.969493i	$0.421173\pi$
$992$		0			0
$993$	329.261			0.331582
$994$		0			0
$995$	358.423			0.360225
$996$		0			0
$997$	502.354			0.503866			0.251933	−	0.967745i	$-0.418934\pi$
$997$	502.354			0.503866			0.251933	+	0.967745i	$0.418934\pi$
$998$		0			0
$999$		−	423.852i		−	0.424277i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.3.k.b.321.2		10
4.3	odd	2		115.3.d.b.91.8	yes	10
12.11	even	2		1035.3.g.b.91.3		10
20.3	even	4		575.3.c.d.574.7		20
20.7	even	4		575.3.c.d.574.14		20
20.19	odd	2		575.3.d.g.551.4		10
23.22	odd	2	inner	1840.3.k.b.321.1		10
92.91	even	2		115.3.d.b.91.7	✓	10
276.275	odd	2		1035.3.g.b.91.4		10
460.183	odd	4		575.3.c.d.574.8		20
460.367	odd	4		575.3.c.d.574.13		20
460.459	even	2		575.3.d.g.551.3		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.3.d.b.91.7	✓	10	92.91	even	2
115.3.d.b.91.8	yes	10	4.3	odd	2
575.3.c.d.574.7		20	20.3	even	4
575.3.c.d.574.8		20	460.183	odd	4
575.3.c.d.574.13		20	460.367	odd	4
575.3.c.d.574.14		20	20.7	even	4
575.3.d.g.551.3		10	460.459	even	2
575.3.d.g.551.4		10	20.19	odd	2
1035.3.g.b.91.3		10	12.11	even	2
1035.3.g.b.91.4		10	276.275	odd	2
1840.3.k.b.321.1		10	23.22	odd	2	inner
1840.3.k.b.321.2		10	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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1840.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.98928 q^{3} +2.23607i q^{5} +6.68423i q^{7} +6.91435 q^{9} +O(q^{10})\)	\(q-3.98928 q^{3} +2.23607i q^{5} +6.68423i q^{7} +6.91435 q^{9} -10.4810i q^{11} -4.43198 q^{13} -8.92030i q^{15} +1.02893i q^{17} +24.8394i q^{19} -26.6653i q^{21} +(-0.739966 - 22.9881i) q^{23} -5.00000 q^{25} +8.32025 q^{27} +20.9027 q^{29} -35.4525 q^{31} +41.8117i q^{33} -14.9464 q^{35} -50.9422i q^{37} +17.6804 q^{39} -53.5201 q^{41} -64.2047i q^{43} +15.4609i q^{45} +2.03047 q^{47} +4.32106 q^{49} -4.10469i q^{51} -13.7344i q^{53} +23.4362 q^{55} -99.0912i q^{57} -7.09638 q^{59} +68.5368i q^{61} +46.2171i q^{63} -9.91021i q^{65} +80.9722i q^{67} +(2.95193 + 91.7059i) q^{69} -117.944 q^{71} +38.6374 q^{73} +19.9464 q^{75} +70.0575 q^{77} -11.9863i q^{79} -95.4209 q^{81} -10.8042i q^{83} -2.30076 q^{85} -83.3869 q^{87} -17.7703i q^{89} -29.6244i q^{91} +141.430 q^{93} -55.5425 q^{95} -5.84199i q^{97} -72.4693i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 2 q^{3} - 16 q^{9}+O(q^{10}) \) 10 * q + 2 * q^3 - 16 * q^9	\( 10 q + 2 q^{3} - 16 q^{9} - 2 q^{13} - 44 q^{23} - 50 q^{25} - 40 q^{27} - 46 q^{29} - 16 q^{31} + 60 q^{35} - 72 q^{39} - 84 q^{41} - 112 q^{47} + 50 q^{49} + 10 q^{55} + 262 q^{59} + 124 q^{69} - 236 q^{71} + 168 q^{73} - 10 q^{75} + 300 q^{77} - 258 q^{81} - 540 q^{87} + 100 q^{93} + 90 q^{95}+O(q^{100}) \) 10 * q + 2 * q^3 - 16 * q^9 - 2 * q^13 - 44 * q^23 - 50 * q^25 - 40 * q^27 - 46 * q^29 - 16 * q^31 + 60 * q^35 - 72 * q^39 - 84 * q^41 - 112 * q^47 + 50 * q^49 + 10 * q^55 + 262 * q^59 + 124 * q^69 - 236 * q^71 + 168 * q^73 - 10 * q^75 + 300 * q^77 - 258 * q^81 - 540 * q^87 + 100 * q^93 + 90 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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