LMFDB - Embedded newform 1840.3.k.e.321.16

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:	$48$
Twist minimal:	no (minimal twist has level 920)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			321.16
Character	$\chi$	$=$	1840.321
Dual form			1840.3.k.e.321.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.49795 q^{3} +2.23607i q^{5} -1.88865i q^{7} -2.76024 q^{9} +O(q^{10})$	$q-2.49795 q^{3} +2.23607i q^{5} -1.88865i q^{7} -2.76024 q^{9} +16.3573i q^{11} +15.4705 q^{13} -5.58559i q^{15} -12.7499i q^{17} +34.9190i q^{19} +4.71776i q^{21} +(16.4761 - 16.0480i) q^{23} -5.00000 q^{25} +29.3765 q^{27} +8.12080 q^{29} -53.1280 q^{31} -40.8596i q^{33} +4.22315 q^{35} -30.4645i q^{37} -38.6445 q^{39} +63.4237 q^{41} -3.36178i q^{43} -6.17208i q^{45} -31.3541 q^{47} +45.4330 q^{49} +31.8485i q^{51} +31.9732i q^{53} -36.5759 q^{55} -87.2261i q^{57} +27.0792 q^{59} +31.2946i q^{61} +5.21312i q^{63} +34.5931i q^{65} +28.7646i q^{67} +(-41.1564 + 40.0872i) q^{69} -38.5352 q^{71} -45.1786 q^{73} +12.4898 q^{75} +30.8931 q^{77} +87.8817i q^{79} -48.5389 q^{81} -73.7016i q^{83} +28.5096 q^{85} -20.2854 q^{87} -12.1631i q^{89} -29.2183i q^{91} +132.711 q^{93} -78.0814 q^{95} +35.0691i q^{97} -45.1499i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q + 128 q^{9}+O(q^{10}) $ 48 * q + 128 * q^9	$ 48 q + 128 q^{9} - 8 q^{23} - 240 q^{25} + 72 q^{29} - 32 q^{31} + 40 q^{35} + 96 q^{39} - 104 q^{41} - 128 q^{47} - 344 q^{49} - 80 q^{55} - 248 q^{59} + 292 q^{69} - 208 q^{71} + 224 q^{73} - 288 q^{77} + 184 q^{81} + 48 q^{87} - 672 q^{93}+O(q^{100}) $ 48 * q + 128 * q^9 - 8 * q^23 - 240 * q^25 + 72 * q^29 - 32 * q^31 + 40 * q^35 + 96 * q^39 - 104 * q^41 - 128 * q^47 - 344 * q^49 - 80 * q^55 - 248 * q^59 + 292 * q^69 - 208 * q^71 + 224 * q^73 - 288 * q^77 + 184 * q^81 + 48 * q^87 - 672 * 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
$3$	−2.49795			−0.832650			−0.416325	−	0.909216i	$-0.636682\pi$
$3$	−2.49795			−0.832650			−0.416325	+	0.909216i	$0.636682\pi$
$4$		0			0
$5$			2.23607i			0.447214i
$5$			2.23607i			0.447214i
$6$		0			0
$7$		−	1.88865i		−	0.269807i	−0.990859	−	0.134904i	$-0.956928\pi$
$7$		−	1.88865i		−	0.269807i	0.990859	−	0.134904i	$-0.0430725\pi$
$8$		0			0
$9$	−2.76024			−0.306693
$10$		0			0
$11$			16.3573i			1.48702i	0.668723	+	0.743512i	$0.266841\pi$
$11$			16.3573i			1.48702i	−0.668723	+	0.743512i	$0.733159\pi$
$12$		0			0
$13$	15.4705			1.19004			0.595019	−	0.803712i	$-0.297145\pi$
$13$	15.4705			1.19004			0.595019	+	0.803712i	$0.297145\pi$
$14$		0			0
$15$		−	5.58559i		−	0.372373i
$16$		0			0
$17$		−	12.7499i		−	0.749992i	−0.927026	−	0.374996i	$-0.877644\pi$
$17$		−	12.7499i		−	0.749992i	0.927026	−	0.374996i	$-0.122356\pi$
$18$		0			0
$19$			34.9190i			1.83784i	0.394439	+	0.918922i	$0.370939\pi$
$19$			34.9190i			1.83784i	−0.394439	+	0.918922i	$0.629061\pi$
$20$		0			0
$21$			4.71776i			0.224655i
$22$		0			0
$23$	16.4761	−	16.0480i	0.716350	−	0.697741i
$23$	16.4761	−	16.0480i	0.716350	−	0.697741i
$24$		0			0
$25$	−5.00000			−0.200000
$26$		0			0
$27$	29.3765			1.08802
$28$		0			0
$29$	8.12080			0.280028			0.140014	−	0.990150i	$-0.455285\pi$
$29$	8.12080			0.280028			0.140014	+	0.990150i	$0.455285\pi$
$30$		0			0
$31$	−53.1280			−1.71381			−0.856903	−	0.515478i	$-0.827614\pi$
$31$	−53.1280			−1.71381			−0.856903	+	0.515478i	$0.827614\pi$
$32$		0			0
$33$		−	40.8596i		−	1.23817i
$34$		0			0
$35$	4.22315			0.120661
$36$		0			0
$37$		−	30.4645i		−	0.823365i	−0.911327	−	0.411682i	$-0.864941\pi$
$37$		−	30.4645i		−	0.823365i	0.911327	−	0.411682i	$-0.135059\pi$
$38$		0			0
$39$	−38.6445			−0.990886
$40$		0			0
$41$	63.4237			1.54692			0.773460	−	0.633845i	$-0.218524\pi$
$41$	63.4237			1.54692			0.773460	+	0.633845i	$0.218524\pi$
$42$		0			0
$43$		−	3.36178i		−	0.0781810i	−0.999236	−	0.0390905i	$-0.987554\pi$
$43$		−	3.36178i		−	0.0781810i	0.999236	−	0.0390905i	$-0.0124461\pi$
$44$		0			0
$45$		−	6.17208i		−	0.137157i
$46$		0			0
$47$	−31.3541			−0.667108			−0.333554	−	0.942731i	$-0.608248\pi$
$47$	−31.3541			−0.667108			−0.333554	+	0.942731i	$0.608248\pi$
$48$		0			0
$49$	45.4330			0.927204
$50$		0			0
$51$			31.8485i			0.624481i
$52$		0			0
$53$			31.9732i			0.603269i	0.953424	+	0.301634i	$0.0975322\pi$
$53$			31.9732i			0.603269i	−0.953424	+	0.301634i	$0.902468\pi$
$54$		0			0
$55$	−36.5759			−0.665017
$56$		0			0
$57$		−	87.2261i		−	1.53028i
$58$		0			0
$59$	27.0792			0.458969			0.229485	−	0.973312i	$-0.426296\pi$
$59$	27.0792			0.458969			0.229485	+	0.973312i	$0.426296\pi$
$60$		0			0
$61$			31.2946i			0.513026i	0.966541	+	0.256513i	$0.0825736\pi$
$61$			31.2946i			0.513026i	−0.966541	+	0.256513i	$0.917426\pi$
$62$		0			0
$63$			5.21312i			0.0827480i
$64$		0			0
$65$			34.5931i			0.532201i
$66$		0			0
$67$			28.7646i			0.429322i	0.976689	+	0.214661i	$0.0688646\pi$
$67$			28.7646i			0.429322i	−0.976689	+	0.214661i	$0.931135\pi$
$68$		0			0
$69$	−41.1564	+	40.0872i	−0.596470	+	0.580974i
$70$		0			0
$71$	−38.5352			−0.542749			−0.271374	−	0.962474i	$-0.587478\pi$
$71$	−38.5352			−0.542749			−0.271374	+	0.962474i	$0.587478\pi$
$72$		0			0
$73$	−45.1786			−0.618884			−0.309442	−	0.950918i	$-0.600142\pi$
$73$	−45.1786			−0.618884			−0.309442	+	0.950918i	$0.600142\pi$
$74$		0			0
$75$	12.4898			0.166530
$76$		0			0
$77$	30.8931			0.401210
$78$		0			0
$79$			87.8817i			1.11243i	0.831040	+	0.556213i	$0.187746\pi$
$79$			87.8817i			1.11243i	−0.831040	+	0.556213i	$0.812254\pi$
$80$		0			0
$81$	−48.5389			−0.599246
$82$		0			0
$83$		−	73.7016i		−	0.887971i	−0.896034	−	0.443986i	$-0.853564\pi$
$83$		−	73.7016i		−	0.887971i	0.896034	−	0.443986i	$-0.146436\pi$
$84$		0			0
$85$	28.5096			0.335407
$86$		0			0
$87$	−20.2854			−0.233165
$88$		0			0
$89$		−	12.1631i		−	0.136664i	−0.997663	−	0.0683321i	$-0.978232\pi$
$89$		−	12.1631i		−	0.136664i	0.997663	−	0.0683321i	$-0.0217677\pi$
$90$		0			0
$91$		−	29.2183i		−	0.321081i
$92$		0			0
$93$	132.711			1.42700
$94$		0			0
$95$	−78.0814			−0.821909
$96$		0			0
$97$			35.0691i			0.361537i	0.983526	+	0.180769i	$0.0578585\pi$
$97$			35.0691i			0.361537i	−0.983526	+	0.180769i	$0.942141\pi$
$98$		0			0
$99$		−	45.1499i		−	0.456060i
$100$		0			0
$101$	−156.349			−1.54801			−0.774006	−	0.633179i	$-0.781750\pi$
$101$	−156.349			−1.54801			−0.774006	+	0.633179i	$0.781750\pi$
$102$		0			0
$103$			23.5864i			0.228994i	0.993424	+	0.114497i	$0.0365257\pi$
$103$			23.5864i			0.228994i	−0.993424	+	0.114497i	$0.963474\pi$
$104$		0			0
$105$	−10.5492			−0.100469
$106$		0			0
$107$			201.387i			1.88212i	0.338243	+	0.941059i	$0.390168\pi$
$107$			201.387i			1.88212i	−0.338243	+	0.941059i	$0.609832\pi$
$108$		0			0
$109$		−	157.698i		−	1.44677i	−0.690443	−	0.723387i	$-0.742584\pi$
$109$		−	157.698i		−	1.44677i	0.690443	−	0.723387i	$-0.257416\pi$
$110$		0			0
$111$			76.0988i			0.685575i
$112$		0			0
$113$			48.0470i			0.425195i	0.977140	+	0.212598i	$0.0681923\pi$
$113$			48.0470i			0.425195i	−0.977140	+	0.212598i	$0.931808\pi$
$114$		0			0
$115$	35.8845	+	36.8416i	0.312039	+	0.320362i
$116$		0			0
$117$	−42.7023			−0.364977
$118$		0			0
$119$	−24.0800			−0.202353
$120$		0			0
$121$	−146.560			−1.21124
$122$		0			0
$123$	−158.429			−1.28804
$124$		0			0
$125$		−	11.1803i		−	0.0894427i
$126$		0			0
$127$	163.498			1.28738			0.643692	−	0.765285i	$-0.277402\pi$
$127$	163.498			1.28738			0.643692	+	0.765285i	$0.277402\pi$
$128$		0			0
$129$			8.39758i			0.0650975i
$130$		0			0
$131$	−91.3814			−0.697568			−0.348784	−	0.937203i	$-0.613405\pi$
$131$	−91.3814			−0.697568			−0.348784	+	0.937203i	$0.613405\pi$
$132$		0			0
$133$	65.9499			0.495864
$134$		0			0
$135$			65.6879i			0.486577i
$136$		0			0
$137$			223.446i			1.63100i	0.578760	+	0.815498i	$0.303537\pi$
$137$			223.446i			1.63100i	−0.578760	+	0.815498i	$0.696463\pi$
$138$		0			0
$139$	120.873			0.869591			0.434795	−	0.900529i	$-0.356821\pi$
$139$	120.873			0.869591			0.434795	+	0.900529i	$0.356821\pi$
$140$		0			0
$141$	78.3210			0.555468
$142$		0			0
$143$			253.055i			1.76961i
$144$		0			0
$145$			18.1587i			0.125232i
$146$		0			0
$147$	−113.489			−0.772037
$148$		0			0
$149$			29.4581i			0.197706i	0.995102	+	0.0988528i	$0.0315173\pi$
$149$			29.4581i			0.197706i	−0.995102	+	0.0988528i	$0.968483\pi$
$150$		0			0
$151$	−168.269			−1.11436			−0.557182	−	0.830391i	$-0.688117\pi$
$151$	−168.269			−1.11436			−0.557182	+	0.830391i	$0.688117\pi$
$152$		0			0
$153$			35.1927i			0.230017i
$154$		0			0
$155$		−	118.798i		−	0.766437i
$156$		0			0
$157$			76.7379i			0.488776i	0.969678	+	0.244388i	$0.0785871\pi$
$157$			76.7379i			0.488776i	−0.969678	+	0.244388i	$0.921413\pi$
$158$		0			0
$159$		−	79.8676i		−	0.502312i
$160$		0			0
$161$	−30.3091	−	31.1175i	−0.188255	−	0.193276i
$162$		0			0
$163$	−293.853			−1.80278			−0.901390	−	0.433009i	$-0.857452\pi$
$163$	−293.853			−1.80278			−0.901390	+	0.433009i	$0.857452\pi$
$164$		0			0
$165$	91.3649			0.553727
$166$		0			0
$167$	114.215			0.683925			0.341962	−	0.939714i	$-0.388908\pi$
$167$	114.215			0.683925			0.341962	+	0.939714i	$0.388908\pi$
$168$		0			0
$169$	70.3362			0.416191
$170$		0			0
$171$		−	96.3849i		−	0.563654i
$172$		0			0
$173$	−203.732			−1.17764			−0.588821	−	0.808263i	$-0.700408\pi$
$173$	−203.732			−1.17764			−0.588821	+	0.808263i	$0.700408\pi$
$174$		0			0
$175$			9.44325i			0.0539614i
$176$		0			0
$177$	−67.6425			−0.382161
$178$		0			0
$179$	−242.678			−1.35574			−0.677872	−	0.735180i	$-0.737097\pi$
$179$	−242.678			−1.35574			−0.677872	+	0.735180i	$0.737097\pi$
$180$		0			0
$181$			49.3045i			0.272401i	0.990681	+	0.136200i	$0.0434891\pi$
$181$			49.3045i			0.272401i	−0.990681	+	0.136200i	$0.956511\pi$
$182$		0			0
$183$		−	78.1723i		−	0.427171i
$184$		0			0
$185$	68.1207			0.368220
$186$		0			0
$187$	208.553			1.11526
$188$		0			0
$189$		−	55.4819i		−	0.293555i
$190$		0			0
$191$			68.9313i			0.360897i	0.983584	+	0.180448i	$0.0577549\pi$
$191$			68.9313i			0.360897i	−0.983584	+	0.180448i	$0.942245\pi$
$192$		0			0
$193$	−28.8509			−0.149487			−0.0747433	−	0.997203i	$-0.523814\pi$
$193$	−28.8509			−0.149487			−0.0747433	+	0.997203i	$0.523814\pi$
$194$		0			0
$195$		−	86.4118i		−	0.443138i
$196$		0			0
$197$	−71.9661			−0.365310			−0.182655	−	0.983177i	$-0.558469\pi$
$197$	−71.9661			−0.365310			−0.182655	+	0.983177i	$0.558469\pi$
$198$		0			0
$199$			171.280i			0.860705i	0.902661	+	0.430352i	$0.141611\pi$
$199$			171.280i			0.860705i	−0.902661	+	0.430352i	$0.858389\pi$
$200$		0			0
$201$		−	71.8525i		−	0.357475i
$202$		0			0
$203$		−	15.3373i		−	0.0755534i
$204$		0			0
$205$			141.820i			0.691804i
$206$		0			0
$207$	−45.4779	+	44.2964i	−0.219700	+	0.213992i
$208$		0			0
$209$	−571.180			−2.73292
$210$		0			0
$211$	259.108			1.22800			0.614001	−	0.789305i	$-0.289559\pi$
$211$	259.108			1.22800			0.614001	+	0.789305i	$0.289559\pi$
$212$		0			0
$213$	96.2589			0.451920
$214$		0			0
$215$	7.51718			0.0349636
$216$		0			0
$217$			100.340i			0.462397i
$218$		0			0
$219$	112.854			0.515314
$220$		0			0
$221$		−	197.247i		−	0.892519i
$222$		0			0
$223$	−320.896			−1.43899			−0.719497	−	0.694495i	$-0.755628\pi$
$223$	−320.896			−1.43899			−0.719497	+	0.694495i	$0.755628\pi$
$224$		0			0
$225$	13.8012			0.0613386
$226$		0			0
$227$			364.377i			1.60519i	0.596527	+	0.802593i	$0.296547\pi$
$227$			364.377i			1.60519i	−0.596527	+	0.802593i	$0.703453\pi$
$228$		0			0
$229$			34.0484i			0.148683i	0.997233	+	0.0743414i	$0.0236855\pi$
$229$			34.0484i			0.148683i	−0.997233	+	0.0743414i	$0.976315\pi$
$230$		0			0
$231$	−77.1696			−0.334067
$232$		0			0
$233$	−354.635			−1.52204			−0.761019	−	0.648729i	$-0.775301\pi$
$233$	−354.635			−1.52204			−0.761019	+	0.648729i	$0.775301\pi$
$234$		0			0
$235$		−	70.1099i		−	0.298340i
$236$		0			0
$237$		−	219.524i		−	0.926263i
$238$		0			0
$239$	−353.247			−1.47802			−0.739010	−	0.673695i	$-0.764706\pi$
$239$	−353.247			−1.47802			−0.739010	+	0.673695i	$0.764706\pi$
$240$		0			0
$241$			247.713i			1.02786i	0.857833	+	0.513928i	$0.171810\pi$
$241$			247.713i			1.02786i	−0.857833	+	0.513928i	$0.828190\pi$
$242$		0			0
$243$	−143.141			−0.589056
$244$		0			0
$245$			101.591i			0.414658i
$246$		0			0
$247$			540.215i			2.18711i
$248$		0			0
$249$			184.103i			0.739370i
$250$		0			0
$251$			67.1804i			0.267651i	0.991005	+	0.133825i	$0.0427261\pi$
$251$			67.1804i			0.267651i	−0.991005	+	0.133825i	$0.957274\pi$
$252$		0			0
$253$	262.502	+	269.503i	1.03756	+	1.06523i
$254$		0			0
$255$	−71.2155			−0.279276
$256$		0			0
$257$	−51.2268			−0.199326			−0.0996630	−	0.995021i	$-0.531776\pi$
$257$	−51.2268			−0.199326			−0.0996630	+	0.995021i	$0.531776\pi$
$258$		0			0
$259$	−57.5367			−0.222150
$260$		0			0
$261$	−22.4153			−0.0858825
$262$		0			0
$263$		−	361.137i		−	1.37314i	−0.727062	−	0.686572i	$-0.759115\pi$
$263$		−	361.137i		−	1.37314i	0.727062	−	0.686572i	$-0.240885\pi$
$264$		0			0
$265$	−71.4944			−0.269790
$266$		0			0
$267$			30.3828i			0.113793i
$268$		0			0
$269$	−412.540			−1.53361			−0.766804	−	0.641881i	$-0.778154\pi$
$269$	−412.540			−1.53361			−0.766804	+	0.641881i	$0.778154\pi$
$270$		0			0
$271$	11.4335			0.0421902			0.0210951	−	0.999777i	$-0.493285\pi$
$271$	11.4335			0.0421902			0.0210951	+	0.999777i	$0.493285\pi$
$272$		0			0
$273$			72.9860i			0.267348i
$274$		0			0
$275$		−	81.7863i		−	0.297405i
$276$		0			0
$277$	277.729			1.00263			0.501316	−	0.865264i	$-0.332849\pi$
$277$	277.729			1.00263			0.501316	+	0.865264i	$0.332849\pi$
$278$		0			0
$279$	146.646			0.525613
$280$		0			0
$281$			233.496i			0.830946i	0.909606	+	0.415473i	$0.136384\pi$
$281$			233.496i			0.830946i	−0.909606	+	0.415473i	$0.863616\pi$
$282$		0			0
$283$		−	330.296i		−	1.16712i	−0.812069	−	0.583561i	$-0.801659\pi$
$283$		−	330.296i		−	1.16712i	0.812069	−	0.583561i	$-0.198341\pi$
$284$		0			0
$285$	195.043			0.684363
$286$		0			0
$287$		−	119.785i		−	0.417370i
$288$		0			0
$289$	126.441			0.437512
$290$		0			0
$291$		−	87.6009i		−	0.301034i
$292$		0			0
$293$			449.177i			1.53303i	0.642228	+	0.766513i	$0.278010\pi$
$293$			449.177i			1.53303i	−0.642228	+	0.766513i	$0.721990\pi$
$294$		0			0
$295$			60.5509i			0.205257i
$296$		0			0
$297$			480.519i			1.61791i
$298$		0			0
$299$	254.893	−	248.271i	0.852484	−	0.830338i
$300$		0			0
$301$	−6.34923			−0.0210938
$302$		0			0
$303$	390.553			1.28895
$304$		0			0
$305$	−69.9768			−0.229432
$306$		0			0
$307$	480.586			1.56543			0.782713	−	0.622382i	$-0.213835\pi$
$307$	480.586			1.56543			0.782713	+	0.622382i	$0.213835\pi$
$308$		0			0
$309$		−	58.9178i		−	0.190672i
$310$		0			0
$311$	141.186			0.453975			0.226987	−	0.973898i	$-0.427112\pi$
$311$	141.186			0.453975			0.226987	+	0.973898i	$0.427112\pi$
$312$		0			0
$313$			320.893i			1.02522i	0.858622	+	0.512609i	$0.171321\pi$
$313$			320.893i			1.02522i	−0.858622	+	0.512609i	$0.828679\pi$
$314$		0			0
$315$	−11.6569			−0.0370060
$316$		0			0
$317$	−134.421			−0.424041			−0.212020	−	0.977265i	$-0.568004\pi$
$317$	−134.421			−0.424041			−0.212020	+	0.977265i	$0.568004\pi$
$318$		0			0
$319$			132.834i			0.416408i
$320$		0			0
$321$		−	503.054i		−	1.56715i
$322$		0			0
$323$	445.213			1.37837
$324$		0			0
$325$	−77.3525			−0.238008
$326$		0			0
$327$			393.923i			1.20466i
$328$		0			0
$329$			59.2169i			0.179991i
$330$		0			0
$331$	21.9398			0.0662832			0.0331416	−	0.999451i	$-0.489449\pi$
$331$	21.9398			0.0662832			0.0331416	+	0.999451i	$0.489449\pi$
$332$		0			0
$333$			84.0893i			0.252520i
$334$		0			0
$335$	−64.3195			−0.191999
$336$		0			0
$337$			338.102i			1.00327i	0.865079	+	0.501635i	$0.167268\pi$
$337$			338.102i			1.00327i	−0.865079	+	0.501635i	$0.832732\pi$
$338$		0			0
$339$		−	120.019i		−	0.354039i
$340$		0			0
$341$		−	869.028i		−	2.54847i
$342$		0			0
$343$		−	178.351i		−	0.519973i
$344$		0			0
$345$	−89.6377	−	92.0285i	−0.259819	−	0.266749i
$346$		0			0
$347$	136.115			0.392264			0.196132	−	0.980578i	$-0.437162\pi$
$347$	136.115			0.392264			0.196132	+	0.980578i	$0.437162\pi$
$348$		0			0
$349$	193.379			0.554094			0.277047	−	0.960856i	$-0.410644\pi$
$349$	193.379			0.554094			0.277047	+	0.960856i	$0.410644\pi$
$350$		0			0
$351$	454.469			1.29478
$352$		0			0
$353$	493.294			1.39743			0.698717	−	0.715398i	$-0.253755\pi$
$353$	493.294			1.39743			0.698717	+	0.715398i	$0.253755\pi$
$354$		0			0
$355$		−	86.1672i		−	0.242725i
$356$		0			0
$357$	60.1507			0.168489
$358$		0			0
$359$			40.2226i			0.112041i	0.998430	+	0.0560203i	$0.0178412\pi$
$359$			40.2226i			0.112041i	−0.998430	+	0.0560203i	$0.982159\pi$
$360$		0			0
$361$	−858.340			−2.37767
$362$		0			0
$363$	366.100			1.00854
$364$		0			0
$365$		−	101.022i		−	0.276773i
$366$		0			0
$367$		−	542.407i		−	1.47795i	−0.673734	−	0.738974i	$-0.735311\pi$
$367$		−	542.407i		−	1.47795i	0.673734	−	0.738974i	$-0.264689\pi$
$368$		0			0
$369$	−175.065			−0.474430
$370$		0			0
$371$	60.3863			0.162766
$372$		0			0
$373$		−	162.621i		−	0.435981i	−0.975951	−	0.217990i	$-0.930050\pi$
$373$		−	162.621i		−	0.435981i	0.975951	−	0.217990i	$-0.0699501\pi$
$374$		0			0
$375$			27.9279i			0.0744745i
$376$		0			0
$377$	125.633			0.333243
$378$		0			0
$379$			258.524i			0.682122i	0.940041	+	0.341061i	$0.110786\pi$
$379$			258.524i			0.682122i	−0.940041	+	0.341061i	$0.889214\pi$
$380$		0			0
$381$	−408.409			−1.07194
$382$		0			0
$383$		−	672.047i		−	1.75469i	−0.479858	−	0.877346i	$-0.659312\pi$
$383$		−	672.047i		−	1.75469i	0.479858	−	0.877346i	$-0.340688\pi$
$384$		0			0
$385$			69.0792i			0.179426i
$386$		0			0
$387$			9.27933i			0.0239776i
$388$		0			0
$389$			645.973i			1.66060i	0.557317	+	0.830300i	$0.311831\pi$
$389$			645.973i			1.66060i	−0.557317	+	0.830300i	$0.688169\pi$
$390$		0			0
$391$	−204.610	−	210.068i	−0.523300	−	0.537257i
$392$		0			0
$393$	228.266			0.580831
$394$		0			0
$395$	−196.509			−0.497492
$396$		0			0
$397$	103.345			0.260316			0.130158	−	0.991493i	$-0.458452\pi$
$397$	103.345			0.260316			0.130158	+	0.991493i	$0.458452\pi$
$398$		0			0
$399$	−164.740			−0.412881
$400$		0			0
$401$		−	267.087i		−	0.666053i	−0.942917	−	0.333027i	$-0.891930\pi$
$401$		−	267.087i		−	0.666053i	0.942917	−	0.333027i	$-0.108070\pi$
$402$		0			0
$403$	−821.916			−2.03949
$404$		0			0
$405$		−	108.536i		−	0.267991i
$406$		0			0
$407$	498.316			1.22436
$408$		0			0
$409$	189.493			0.463309			0.231654	−	0.972798i	$-0.425586\pi$
$409$	189.493			0.463309			0.231654	+	0.972798i	$0.425586\pi$
$410$		0			0
$411$		−	558.158i		−	1.35805i
$412$		0			0
$413$		−	51.1431i		−	0.123833i
$414$		0			0
$415$	164.802			0.397113
$416$		0			0
$417$	−301.935			−0.724065
$418$		0			0
$419$		−	407.155i		−	0.971730i	−0.874034	−	0.485865i	$-0.838505\pi$
$419$		−	407.155i		−	0.971730i	0.874034	−	0.485865i	$-0.161495\pi$
$420$		0			0
$421$		−	660.663i		−	1.56927i	−0.619958	−	0.784635i	$-0.712850\pi$
$421$		−	660.663i		−	1.56927i	0.619958	−	0.784635i	$-0.287150\pi$
$422$		0			0
$423$	86.5447			0.204598
$424$		0			0
$425$			63.7493i			0.149998i
$426$		0			0
$427$	59.1045			0.138418
$428$		0			0
$429$		−	632.119i		−	1.47347i
$430$		0			0
$431$		−	342.436i		−	0.794515i	−0.917707	−	0.397258i	$-0.869962\pi$
$431$		−	342.436i		−	0.794515i	0.917707	−	0.397258i	$-0.130038\pi$
$432$		0			0
$433$		−	656.046i		−	1.51512i	−0.652767	−	0.757559i	$-0.726392\pi$
$433$		−	656.046i		−	1.51512i	0.652767	−	0.757559i	$-0.273608\pi$
$434$		0			0
$435$		−	45.3594i		−	0.104275i
$436$		0			0
$437$	560.382	+	575.328i	1.28234	+	1.31654i
$438$		0			0
$439$	−262.359			−0.597630			−0.298815	−	0.954311i	$-0.596591\pi$
$439$	−262.359			−0.597630			−0.298815	+	0.954311i	$0.596591\pi$
$440$		0			0
$441$	−125.406			−0.284367
$442$		0			0
$443$	−703.571			−1.58820			−0.794098	−	0.607790i	$-0.792056\pi$
$443$	−703.571			−1.58820			−0.794098	+	0.607790i	$0.792056\pi$
$444$		0			0
$445$	27.1975			0.0611180
$446$		0			0
$447$		−	73.5850i		−	0.164620i
$448$		0			0
$449$	1.47225			0.00327895			0.00163948	−	0.999999i	$-0.499478\pi$
$449$	1.47225			0.00327895			0.00163948	+	0.999999i	$0.499478\pi$
$450$		0			0
$451$			1037.44i			2.30031i
$452$		0			0
$453$	420.327			0.927875
$454$		0			0
$455$	65.3342			0.143592
$456$		0			0
$457$		−	275.848i		−	0.603605i	−0.953370	−	0.301803i	$-0.902412\pi$
$457$		−	275.848i		−	0.603605i	0.953370	−	0.301803i	$-0.0975884\pi$
$458$		0			0
$459$		−	374.546i		−	0.816005i
$460$		0			0
$461$	−859.215			−1.86381			−0.931904	−	0.362705i	$-0.881853\pi$
$461$	−859.215			−1.86381			−0.931904	+	0.362705i	$0.881853\pi$
$462$		0			0
$463$	−254.851			−0.550433			−0.275217	−	0.961382i	$-0.588750\pi$
$463$	−254.851			−0.550433			−0.275217	+	0.961382i	$0.588750\pi$
$464$		0			0
$465$			296.751i			0.638174i
$466$		0			0
$467$		−	593.941i		−	1.27182i	−0.771763	−	0.635911i	$-0.780625\pi$
$467$		−	593.941i		−	1.27182i	0.771763	−	0.635911i	$-0.219375\pi$
$468$		0			0
$469$	54.3262			0.115834
$470$		0			0
$471$		−	191.687i		−	0.406980i
$472$		0			0
$473$	54.9896			0.116257
$474$		0			0
$475$		−	174.595i		−	0.367569i
$476$		0			0
$477$		−	88.2538i		−	0.185018i
$478$		0			0
$479$			115.295i			0.240698i	0.992732	+	0.120349i	$0.0384014\pi$
$479$			115.295i			0.240698i	−0.992732	+	0.120349i	$0.961599\pi$
$480$		0			0
$481$		−	471.301i		−	0.979835i
$482$		0			0
$483$	75.7107	+	77.7300i	0.156751	+	0.160932i
$484$		0			0
$485$	−78.4169			−0.161684
$486$		0			0
$487$	77.5434			0.159227			0.0796133	−	0.996826i	$-0.474631\pi$
$487$	77.5434			0.159227			0.0796133	+	0.996826i	$0.474631\pi$
$488$		0			0
$489$	734.031			1.50109
$490$		0			0
$491$	718.847			1.46405			0.732024	−	0.681279i	$-0.238576\pi$
$491$	718.847			1.46405			0.732024	+	0.681279i	$0.238576\pi$
$492$		0			0
$493$		−	103.539i		−	0.210018i
$494$		0			0
$495$	100.958			0.203956
$496$		0			0
$497$			72.7794i			0.146437i
$498$		0			0
$499$	521.195			1.04448			0.522240	−	0.852799i	$-0.325097\pi$
$499$	521.195			1.04448			0.522240	+	0.852799i	$0.325097\pi$
$500$		0			0
$501$	−285.305			−0.569470
$502$		0			0
$503$		−	38.0710i		−	0.0756879i	−0.999284	−	0.0378440i	$-0.987951\pi$
$503$		−	38.0710i		−	0.0756879i	0.999284	−	0.0378440i	$-0.0120490\pi$
$504$		0			0
$505$		−	349.607i		−	0.692292i
$506$		0			0
$507$	−175.696			−0.346541
$508$		0			0
$509$	659.053			1.29480			0.647400	−	0.762150i	$-0.275856\pi$
$509$	659.053			1.29480			0.647400	+	0.762150i	$0.275856\pi$
$510$		0			0
$511$			85.3265i			0.166979i
$512$		0			0
$513$			1025.80i			1.99961i
$514$		0			0
$515$	−52.7409			−0.102409
$516$		0			0
$517$		−	512.867i		−	0.992006i
$518$		0			0
$519$	508.913			0.980565
$520$		0			0
$521$			665.115i			1.27661i	0.769783	+	0.638306i	$0.220364\pi$
$521$			665.115i			1.27661i	−0.769783	+	0.638306i	$0.779636\pi$
$522$		0			0
$523$			573.320i			1.09621i	0.836408	+	0.548107i	$0.184651\pi$
$523$			573.320i			1.09621i	−0.836408	+	0.548107i	$0.815349\pi$
$524$		0			0
$525$		−	23.5888i		−	0.0449310i
$526$		0			0
$527$			677.374i			1.28534i
$528$		0			0
$529$	13.9212	−	528.817i	0.0263161	−	0.999654i
$530$		0			0
$531$	−74.7450			−0.140763
$532$		0			0
$533$	981.196			1.84089
$534$		0			0
$535$	−450.314			−0.841709
$536$		0			0
$537$	606.198			1.12886
$538$		0			0
$539$			743.160i			1.37877i
$540$		0			0
$541$	970.216			1.79338			0.896688	−	0.442663i	$-0.145966\pi$
$541$	970.216			1.79338			0.896688	+	0.442663i	$0.145966\pi$
$542$		0			0
$543$		−	123.160i		−	0.226815i
$544$		0			0
$545$	352.624			0.647017
$546$		0			0
$547$	−270.046			−0.493685			−0.246843	−	0.969056i	$-0.579393\pi$
$547$	−270.046			−0.493685			−0.246843	+	0.969056i	$0.579393\pi$
$548$		0			0
$549$		−	86.3805i		−	0.157341i
$550$		0			0
$551$			283.571i			0.514647i
$552$		0			0
$553$	165.978			0.300141
$554$		0			0
$555$	−170.162			−0.306598
$556$		0			0
$557$		−	668.284i		−	1.19979i	−0.800078	−	0.599896i	$-0.795209\pi$
$557$		−	668.284i		−	1.19979i	0.800078	−	0.599896i	$-0.204791\pi$
$558$		0			0
$559$		−	52.0085i		−	0.0930384i
$560$		0			0
$561$	−520.955			−0.928618
$562$		0			0
$563$		−	178.187i		−	0.316496i	−0.987399	−	0.158248i	$-0.949415\pi$
$563$		−	178.187i		−	0.316496i	0.987399	−	0.158248i	$-0.0505846\pi$
$564$		0			0
$565$	−107.436			−0.190153
$566$		0			0
$567$			91.6730i			0.161681i
$568$		0			0
$569$			947.752i			1.66564i	0.553540	+	0.832822i	$0.313276\pi$
$569$			947.752i			1.66564i	−0.553540	+	0.832822i	$0.686724\pi$
$570$		0			0
$571$			1112.11i			1.94766i	0.227279	+	0.973830i	$0.427017\pi$
$571$			1112.11i			1.94766i	−0.227279	+	0.973830i	$0.572983\pi$
$572$		0			0
$573$		−	172.187i		−	0.300501i
$574$		0			0
$575$	−82.3803	+	80.2402i	−0.143270	+	0.139548i
$576$		0			0
$577$	469.044			0.812901			0.406450	−	0.913673i	$-0.366766\pi$
$577$	469.044			0.812901			0.406450	+	0.913673i	$0.366766\pi$
$578$		0			0
$579$	72.0682			0.124470
$580$		0			0
$581$	−139.197			−0.239581
$582$		0			0
$583$	−522.995			−0.897075
$584$		0			0
$585$		−	95.4852i		−	0.163222i
$586$		0			0
$587$	232.570			0.396201			0.198101	−	0.980182i	$-0.436523\pi$
$587$	232.570			0.396201			0.198101	+	0.980182i	$0.436523\pi$
$588$		0			0
$589$		−	1855.18i		−	3.14971i
$590$		0			0
$591$	179.768			0.304176
$592$		0			0
$593$	11.8324			0.0199535			0.00997677	−	0.999950i	$-0.496824\pi$
$593$	11.8324			0.0199535			0.00997677	+	0.999950i	$0.496824\pi$
$594$		0			0
$595$		−	53.8446i		−	0.0904951i
$596$		0			0
$597$		−	427.850i		−	0.716666i
$598$		0			0
$599$	247.699			0.413521			0.206760	−	0.978392i	$-0.433708\pi$
$599$	247.699			0.413521			0.206760	+	0.978392i	$0.433708\pi$
$600$		0			0
$601$	−278.178			−0.462859			−0.231429	−	0.972852i	$-0.574340\pi$
$601$	−278.178			−0.462859			−0.231429	+	0.972852i	$0.574340\pi$
$602$		0			0
$603$		−	79.3970i		−	0.131670i
$604$		0			0
$605$		−	327.718i		−	0.541683i
$606$		0			0
$607$	48.9065			0.0805708			0.0402854	−	0.999188i	$-0.487173\pi$
$607$	48.9065			0.0805708			0.0402854	+	0.999188i	$0.487173\pi$
$608$		0			0
$609$			38.3119i			0.0629096i
$610$		0			0
$611$	−485.063			−0.793884
$612$		0			0
$613$		−	108.956i		−	0.177742i	−0.996043	−	0.0888709i	$-0.971674\pi$
$613$		−	108.956i		−	0.177742i	0.996043	−	0.0888709i	$-0.0283259\pi$
$614$		0			0
$615$		−	354.259i		−	0.576031i
$616$		0			0
$617$			250.668i			0.406269i	0.979151	+	0.203135i	$0.0651129\pi$
$617$			250.668i			0.406269i	−0.979151	+	0.203135i	$0.934887\pi$
$618$		0			0
$619$		−	1201.15i		−	1.94046i	−0.242183	−	0.970231i	$-0.577863\pi$
$619$		−	1201.15i		−	1.94046i	0.242183	−	0.970231i	$-0.422137\pi$
$620$		0			0
$621$	484.009	−	471.435i	0.779403	−	0.759155i
$622$		0			0
$623$	−22.9718			−0.0368729
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$	1426.78			2.27557
$628$		0			0
$629$	−388.418			−0.617517
$630$		0			0
$631$		−	425.651i		−	0.674566i	−0.941403	−	0.337283i	$-0.890492\pi$
$631$		−	425.651i		−	0.674566i	0.941403	−	0.337283i	$-0.109508\pi$
$632$		0			0
$633$	−647.240			−1.02250
$634$		0			0
$635$			365.592i			0.575735i
$636$		0			0
$637$	702.871			1.10341
$638$		0			0
$639$	106.366			0.166457
$640$		0			0
$641$		−	1013.74i		−	1.58149i	−0.612143	−	0.790747i	$-0.709692\pi$
$641$		−	1013.74i		−	1.58149i	0.612143	−	0.790747i	$-0.290308\pi$
$642$		0			0
$643$		−	216.835i		−	0.337224i	−0.985683	−	0.168612i	$-0.946072\pi$
$643$		−	216.835i		−	0.337224i	0.985683	−	0.168612i	$-0.0539285\pi$
$644$		0			0
$645$	−18.7775			−0.0291125
$646$		0			0
$647$	−1083.81			−1.67513			−0.837564	−	0.546340i	$-0.816021\pi$
$647$	−1083.81			−1.67513			−0.837564	+	0.546340i	$0.816021\pi$
$648$		0			0
$649$			442.941i			0.682498i
$650$		0			0
$651$		−	250.645i		−	0.385015i
$652$		0			0
$653$	866.299			1.32664			0.663322	−	0.748334i	$-0.269146\pi$
$653$	866.299			1.32664			0.663322	+	0.748334i	$0.269146\pi$
$654$		0			0
$655$		−	204.335i		−	0.311962i
$656$		0			0
$657$	124.704			0.189808
$658$		0			0
$659$			892.140i			1.35378i	0.736085	+	0.676890i	$0.236673\pi$
$659$			892.140i			1.35378i	−0.736085	+	0.676890i	$0.763327\pi$
$660$		0			0
$661$			24.7064i			0.0373774i	0.999825	+	0.0186887i	$0.00594914\pi$
$661$			24.7064i			0.0373774i	−0.999825	+	0.0186887i	$0.994051\pi$
$662$		0			0
$663$			492.713i			0.743156i
$664$		0			0
$665$			147.468i			0.221757i
$666$		0			0
$667$	133.799	−	130.323i	0.200598	−	0.195387i
$668$		0			0
$669$	801.582			1.19818
$670$		0			0
$671$	−511.893			−0.762881
$672$		0			0
$673$	−168.088			−0.249759			−0.124879	−	0.992172i	$-0.539854\pi$
$673$	−168.088			−0.249759			−0.124879	+	0.992172i	$0.539854\pi$
$674$		0			0
$675$	−146.883			−0.217604
$676$		0			0
$677$			502.335i			0.742002i	0.928632	+	0.371001i	$0.120985\pi$
$677$			502.335i			0.742002i	−0.928632	+	0.371001i	$0.879015\pi$
$678$		0			0
$679$	66.2333			0.0975453
$680$		0			0
$681$		−	910.197i		−	1.33656i
$682$		0			0
$683$	−495.091			−0.724878			−0.362439	−	0.932008i	$-0.618056\pi$
$683$	−495.091			−0.724878			−0.362439	+	0.932008i	$0.618056\pi$
$684$		0			0
$685$	−499.641			−0.729403
$686$		0			0
$687$		−	85.0511i		−	0.123801i
$688$		0			0
$689$			494.642i			0.717913i
$690$		0			0
$691$	−961.643			−1.39167			−0.695834	−	0.718203i	$-0.744965\pi$
$691$	−961.643			−1.39167			−0.695834	+	0.718203i	$0.744965\pi$
$692$		0			0
$693$	−85.2724			−0.123048
$694$		0			0
$695$			270.281i			0.388893i
$696$		0			0
$697$		−	808.644i		−	1.16018i
$698$		0			0
$699$	885.861			1.26733
$700$		0			0
$701$			870.699i			1.24208i	0.783778	+	0.621041i	$0.213290\pi$
$701$			870.699i			1.24208i	−0.783778	+	0.621041i	$0.786710\pi$
$702$		0			0
$703$	1063.79			1.51322
$704$		0			0
$705$			175.131i			0.248413i
$706$		0			0
$707$			295.289i			0.417665i
$708$		0			0
$709$		−	350.121i		−	0.493824i	−0.969038	−	0.246912i	$-0.920584\pi$
$709$		−	350.121i		−	0.493824i	0.969038	−	0.246912i	$-0.0794158\pi$
$710$		0			0
$711$		−	242.575i		−	0.341174i
$712$		0			0
$713$	−875.340	+	852.600i	−1.22769	+	1.19579i
$714$		0			0
$715$	−565.848			−0.791396
$716$		0			0
$717$	882.393			1.23067
$718$		0			0
$719$	710.999			0.988872			0.494436	−	0.869214i	$-0.335375\pi$
$719$	710.999			0.988872			0.494436	+	0.869214i	$0.335375\pi$
$720$		0			0
$721$	44.5465			0.0617843
$722$		0			0
$723$		−	618.776i		−	0.855845i
$724$		0			0
$725$	−40.6040			−0.0560055
$726$		0			0
$727$			207.713i			0.285713i	0.989743	+	0.142856i	$0.0456287\pi$
$727$			207.713i			0.285713i	−0.989743	+	0.142856i	$0.954371\pi$
$728$		0			0
$729$	794.409			1.08972
$730$		0			0
$731$	−42.8623			−0.0586352
$732$		0			0
$733$			442.741i			0.604012i	0.953306	+	0.302006i	$0.0976563\pi$
$733$			442.741i			0.604012i	−0.953306	+	0.302006i	$0.902344\pi$
$734$		0			0
$735$		−	253.770i		−	0.345265i
$736$		0			0
$737$	−470.509			−0.638412
$738$		0			0
$739$	−925.255			−1.25204			−0.626019	−	0.779808i	$-0.715317\pi$
$739$	−925.255			−1.25204			−0.626019	+	0.779808i	$0.715317\pi$
$740$		0			0
$741$		−	1349.43i		−	1.82109i
$742$		0			0
$743$			616.966i			0.830371i	0.909737	+	0.415186i	$0.136283\pi$
$743$			616.966i			0.830371i	−0.909737	+	0.415186i	$0.863717\pi$
$744$		0			0
$745$	−65.8704			−0.0884166
$746$		0			0
$747$			203.434i			0.272335i
$748$		0			0
$749$	380.349			0.507809
$750$		0			0
$751$		−	886.992i		−	1.18108i	−0.807008	−	0.590541i	$-0.798915\pi$
$751$		−	886.992i		−	1.18108i	0.807008	−	0.590541i	$-0.201085\pi$
$752$		0			0
$753$		−	167.813i		−	0.222860i
$754$		0			0
$755$		−	376.261i		−	0.498358i
$756$		0			0
$757$			790.797i			1.04465i	0.852748	+	0.522323i	$0.174934\pi$
$757$			790.797i			1.04465i	−0.852748	+	0.522323i	$0.825066\pi$
$758$		0			0
$759$	−655.717	−	673.206i	−0.863922	−	0.886964i
$760$		0			0
$761$	−1396.15			−1.83463			−0.917313	−	0.398167i	$-0.869646\pi$
$761$	−1396.15			−1.83463			−0.917313	+	0.398167i	$0.869646\pi$
$762$		0			0
$763$	−297.837			−0.390350
$764$		0			0
$765$	−78.6932			−0.102867
$766$		0			0
$767$	418.928			0.546191
$768$		0			0
$769$			679.294i			0.883348i	0.897176	+	0.441674i	$0.145615\pi$
$769$			679.294i			0.883348i	−0.897176	+	0.441674i	$0.854385\pi$
$770$		0			0
$771$	127.962			0.165969
$772$		0			0
$773$		−	796.530i		−	1.03044i	−0.857058	−	0.515220i	$-0.827710\pi$
$773$		−	796.530i		−	1.03044i	0.857058	−	0.515220i	$-0.172290\pi$
$774$		0			0
$775$	265.640			0.342761
$776$		0			0
$777$	143.724			0.184973
$778$		0			0
$779$			2214.70i			2.84300i
$780$		0			0
$781$		−	630.330i		−	0.807080i
$782$		0			0
$783$	238.561			0.304675
$784$		0			0
$785$	−171.591			−0.218587
$786$		0			0
$787$			1394.93i			1.77246i	0.463242	+	0.886232i	$0.346686\pi$
$787$			1394.93i			1.77246i	−0.463242	+	0.886232i	$0.653314\pi$
$788$		0			0
$789$			902.102i			1.14335i
$790$		0			0
$791$	90.7440			0.114721
$792$		0			0
$793$			484.142i			0.610520i
$794$		0			0
$795$	178.589			0.224641
$796$		0			0
$797$			677.231i			0.849726i	0.905258	+	0.424863i	$0.139678\pi$
$797$			677.231i			0.849726i	−0.905258	+	0.424863i	$0.860322\pi$
$798$		0			0
$799$			399.760i			0.500326i
$800$		0			0
$801$			33.5731i			0.0419139i
$802$		0			0
$803$		−	738.997i		−	0.920296i
$804$		0			0
$805$	69.5809	−	67.7732i	0.0864359	−	0.0841904i
$806$		0			0
$807$	1030.51			1.27696
$808$		0			0
$809$	−639.936			−0.791021			−0.395511	−	0.918461i	$-0.629432\pi$
$809$	−639.936			−0.791021			−0.395511	+	0.918461i	$0.629432\pi$
$810$		0			0
$811$	−343.399			−0.423427			−0.211713	−	0.977332i	$-0.567904\pi$
$811$	−343.399			−0.423427			−0.211713	+	0.977332i	$0.567904\pi$
$812$		0			0
$813$	−28.5604			−0.0351297
$814$		0			0
$815$		−	657.075i		−	0.806228i
$816$		0			0
$817$	117.390			0.143685
$818$		0			0
$819$			80.6496i			0.0984733i
$820$		0			0
$821$	−0.945559			−0.00115172			−0.000575858	−	1.00000i	$-0.500183\pi$
$821$	−0.945559			−0.00115172			−0.000575858		1.00000i	$0.500183\pi$
$822$		0			0
$823$	115.753			0.140647			0.0703236	−	0.997524i	$-0.477597\pi$
$823$	115.753			0.140647			0.0703236	+	0.997524i	$0.477597\pi$
$824$		0			0
$825$			204.298i			0.247634i
$826$		0			0
$827$		−	875.896i		−	1.05912i	−0.848271	−	0.529562i	$-0.822356\pi$
$827$		−	875.896i		−	1.05912i	0.848271	−	0.529562i	$-0.177644\pi$
$828$		0			0
$829$	1151.15			1.38860			0.694299	−	0.719686i	$-0.255714\pi$
$829$	1151.15			1.38860			0.694299	+	0.719686i	$0.255714\pi$
$830$		0			0
$831$	−693.754			−0.834843
$832$		0			0
$833$		−	579.265i		−	0.695396i
$834$		0			0
$835$			255.393i			0.305860i
$836$		0			0
$837$	−1560.71			−1.86465
$838$		0			0
$839$			1043.67i			1.24394i	0.783040	+	0.621972i	$0.213668\pi$
$839$			1043.67i			1.24394i	−0.783040	+	0.621972i	$0.786332\pi$
$840$		0			0
$841$	−775.053			−0.921585
$842$		0			0
$843$		−	583.261i		−	0.691887i
$844$		0			0
$845$			157.277i			0.186126i
$846$		0			0
$847$			276.801i			0.326801i
$848$		0			0
$849$			825.063i			0.971805i
$850$		0			0
$851$	−488.895	−	501.935i	−0.574495	−	0.589818i
$852$		0			0
$853$	1462.10			1.71407			0.857035	−	0.515259i	$-0.172304\pi$
$853$	1462.10			1.71407			0.857035	+	0.515259i	$0.172304\pi$
$854$		0			0
$855$	215.523			0.252074
$856$		0			0
$857$	529.936			0.618362			0.309181	−	0.951003i	$-0.399945\pi$
$857$	529.936			0.618362			0.309181	+	0.951003i	$0.399945\pi$
$858$		0			0
$859$	78.8721			0.0918185			0.0459092	−	0.998946i	$-0.485381\pi$
$859$	78.8721			0.0918185			0.0459092	+	0.998946i	$0.485381\pi$
$860$		0			0
$861$			299.218i			0.347523i
$862$		0			0
$863$	760.548			0.881284			0.440642	−	0.897683i	$-0.354751\pi$
$863$	760.548			0.881284			0.440642	+	0.897683i	$0.354751\pi$
$864$		0			0
$865$		−	455.559i		−	0.526658i
$866$		0			0
$867$	−315.843			−0.364295
$868$		0			0
$869$	−1437.50			−1.65421
$870$		0			0
$871$			445.002i			0.510909i
$872$		0			0
$873$		−	96.7991i		−	0.110881i
$874$		0			0
$875$	−21.1157			−0.0241323
$876$		0			0
$877$	352.299			0.401709			0.200855	−	0.979621i	$-0.435628\pi$
$877$	352.299			0.401709			0.200855	+	0.979621i	$0.435628\pi$
$878$		0			0
$879$		−	1122.02i		−	1.27648i
$880$		0			0
$881$		−	64.2357i		−	0.0729122i	−0.999335	−	0.0364561i	$-0.988393\pi$
$881$		−	64.2357i		−	0.0729122i	0.999335	−	0.0364561i	$-0.0116069\pi$
$882$		0			0
$883$	−51.8656			−0.0587379			−0.0293690	−	0.999569i	$-0.509350\pi$
$883$	−51.8656			−0.0587379			−0.0293690	+	0.999569i	$0.509350\pi$
$884$		0			0
$885$		−	151.253i		−	0.170908i
$886$		0			0
$887$	−86.4219			−0.0974317			−0.0487159	−	0.998813i	$-0.515513\pi$
$887$	−86.4219			−0.0974317			−0.0487159	+	0.998813i	$0.515513\pi$
$888$		0			0
$889$		−	308.790i		−	0.347345i
$890$		0			0
$891$		−	793.964i		−	0.891093i
$892$		0			0
$893$		−	1094.85i		−	1.22604i
$894$		0			0
$895$		−	542.645i		−	0.606307i
$896$		0			0
$897$	−636.710	+	620.169i	−0.709821	+	0.691381i
$898$		0			0
$899$	−431.442			−0.479913
$900$		0			0
$901$	407.655			0.452447
$902$		0			0
$903$	15.8601			0.0175638
$904$		0			0
$905$	−110.248			−0.121821
$906$		0			0
$907$		−	221.543i		−	0.244259i	−0.992514	−	0.122130i	$-0.961028\pi$
$907$		−	221.543i		−	0.244259i	0.992514	−	0.122130i	$-0.0389724\pi$
$908$		0			0
$909$	431.561			0.474765
$910$		0			0
$911$		−	1813.25i		−	1.99039i	−0.0979133	−	0.995195i	$-0.531217\pi$
$911$		−	1813.25i		−	1.99039i	0.0979133	−	0.995195i	$-0.468783\pi$
$912$		0			0
$913$	1205.56			1.32043
$914$		0			0
$915$	174.799			0.191037
$916$		0			0
$917$			172.588i			0.188209i
$918$		0			0
$919$			165.392i			0.179969i	0.995943	+	0.0899846i	$0.0286818\pi$
$919$			165.392i			0.179969i	−0.995943	+	0.0899846i	$0.971318\pi$
$920$		0			0
$921$	−1200.48			−1.30345
$922$		0			0
$923$	−596.158			−0.645891
$924$		0			0
$925$			152.322i			0.164673i
$926$		0			0
$927$		−	65.1042i		−	0.0702311i
$928$		0			0
$929$	−569.605			−0.613137			−0.306569	−	0.951849i	$-0.599181\pi$
$929$	−569.605			−0.613137			−0.306569	+	0.951849i	$0.599181\pi$
$930$		0			0
$931$			1586.48i			1.70406i
$932$		0			0
$933$	−352.676			−0.378002
$934$		0			0
$935$			466.338i			0.498758i
$936$		0			0
$937$		−	388.041i		−	0.414131i	−0.978327	−	0.207066i	$-0.933609\pi$
$937$		−	388.041i		−	0.414131i	0.978327	−	0.207066i	$-0.0663914\pi$
$938$		0			0
$939$		−	801.576i		−	0.853649i
$940$		0			0
$941$			295.417i			0.313939i	0.987603	+	0.156970i	$0.0501725\pi$
$941$			295.417i			0.313939i	−0.987603	+	0.156970i	$0.949828\pi$
$942$		0			0
$943$	1044.97	−	1017.83i	1.10814	−	1.07935i
$944$		0			0
$945$	124.061			0.131282
$946$		0			0
$947$	−115.729			−0.122206			−0.0611028	−	0.998131i	$-0.519462\pi$
$947$	−115.729			−0.122206			−0.0611028	+	0.998131i	$0.519462\pi$
$948$		0			0
$949$	−698.935			−0.736496
$950$		0			0
$951$	335.777			0.353078
$952$		0			0
$953$			1378.90i			1.44691i	0.690372	+	0.723455i	$0.257447\pi$
$953$			1378.90i			1.44691i	−0.690372	+	0.723455i	$0.742553\pi$
$954$		0			0
$955$	−154.135			−0.161398
$956$		0			0
$957$		−	331.813i		−	0.346722i
$958$		0			0
$959$	422.012			0.440054
$960$		0			0
$961$	1861.58			1.93713
$962$		0			0
$963$		−	555.875i		−	0.577233i
$964$		0			0
$965$		−	64.5126i		−	0.0668525i
$966$		0			0
$967$	583.791			0.603714			0.301857	−	0.953353i	$-0.402394\pi$
$967$	583.791			0.603714			0.301857	+	0.953353i	$0.402394\pi$
$968$		0			0
$969$	−1112.12			−1.14770
$970$		0			0
$971$		−	845.838i		−	0.871099i	−0.900165	−	0.435550i	$-0.856554\pi$
$971$		−	845.838i		−	0.871099i	0.900165	−	0.435550i	$-0.143446\pi$
$972$		0			0
$973$		−	228.287i		−	0.234622i
$974$		0			0
$975$	193.223			0.198177
$976$		0			0
$977$			801.000i			0.819857i	0.912118	+	0.409928i	$0.134446\pi$
$977$			801.000i			0.819857i	−0.912118	+	0.409928i	$0.865554\pi$
$978$		0			0
$979$	198.955			0.203223
$980$		0			0
$981$			435.285i			0.443716i
$982$		0			0
$983$			1548.06i			1.57484i	0.616420	+	0.787418i	$0.288582\pi$
$983$			1548.06i			1.57484i	−0.616420	+	0.787418i	$0.711418\pi$
$984$		0			0
$985$		−	160.921i		−	0.163372i
$986$		0			0
$987$		−	147.921i		−	0.149869i
$988$		0			0
$989$	−53.9500	−	55.3890i	−0.0545501	−	0.0560050i
$990$		0			0
$991$	−832.973			−0.840538			−0.420269	−	0.907400i	$-0.638064\pi$
$991$	−832.973			−0.840538			−0.420269	+	0.907400i	$0.638064\pi$
$992$		0			0
$993$	−54.8044			−0.0551908
$994$		0			0
$995$	−382.994			−0.384919
$996$		0			0
$997$	−809.404			−0.811839			−0.405920	−	0.913909i	$-0.633049\pi$
$997$	−809.404			−0.811839			−0.405920	+	0.913909i	$0.633049\pi$
$998$		0			0
$999$		−	894.940i		−	0.895836i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.3.k.e.321.16		48
4.3	odd	2		920.3.k.a.321.34	yes	48
23.22	odd	2	inner	1840.3.k.e.321.15		48
92.91	even	2		920.3.k.a.321.33	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.3.k.a.321.33	✓	48	92.91	even	2
920.3.k.a.321.34	yes	48	4.3	odd	2
1840.3.k.e.321.15		48	23.22	odd	2	inner
1840.3.k.e.321.16		48	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)

\(f(q)\)	\(=\)	\(q-2.49795 q^{3} +2.23607i q^{5} -1.88865i q^{7} -2.76024 q^{9} +O(q^{10})\)	\(q-2.49795 q^{3} +2.23607i q^{5} -1.88865i q^{7} -2.76024 q^{9} +16.3573i q^{11} +15.4705 q^{13} -5.58559i q^{15} -12.7499i q^{17} +34.9190i q^{19} +4.71776i q^{21} +(16.4761 - 16.0480i) q^{23} -5.00000 q^{25} +29.3765 q^{27} +8.12080 q^{29} -53.1280 q^{31} -40.8596i q^{33} +4.22315 q^{35} -30.4645i q^{37} -38.6445 q^{39} +63.4237 q^{41} -3.36178i q^{43} -6.17208i q^{45} -31.3541 q^{47} +45.4330 q^{49} +31.8485i q^{51} +31.9732i q^{53} -36.5759 q^{55} -87.2261i q^{57} +27.0792 q^{59} +31.2946i q^{61} +5.21312i q^{63} +34.5931i q^{65} +28.7646i q^{67} +(-41.1564 + 40.0872i) q^{69} -38.5352 q^{71} -45.1786 q^{73} +12.4898 q^{75} +30.8931 q^{77} +87.8817i q^{79} -48.5389 q^{81} -73.7016i q^{83} +28.5096 q^{85} -20.2854 q^{87} -12.1631i q^{89} -29.2183i q^{91} +132.711 q^{93} -78.0814 q^{95} +35.0691i q^{97} -45.1499i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 128 q^{9}+O(q^{10}) \) 48 * q + 128 * q^9	\( 48 q + 128 q^{9} - 8 q^{23} - 240 q^{25} + 72 q^{29} - 32 q^{31} + 40 q^{35} + 96 q^{39} - 104 q^{41} - 128 q^{47} - 344 q^{49} - 80 q^{55} - 248 q^{59} + 292 q^{69} - 208 q^{71} + 224 q^{73} - 288 q^{77} + 184 q^{81} + 48 q^{87} - 672 q^{93}+O(q^{100}) \) 48 * q + 128 * q^9 - 8 * q^23 - 240 * q^25 + 72 * q^29 - 32 * q^31 + 40 * q^35 + 96 * q^39 - 104 * q^41 - 128 * q^47 - 344 * q^49 - 80 * q^55 - 248 * q^59 + 292 * q^69 - 208 * q^71 + 224 * q^73 - 288 * q^77 + 184 * q^81 + 48 * q^87 - 672 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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