LMFDB - Embedded newform 2394.2.b.j.1709.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2394,2,Mod(1709,2394)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2394, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("2394.1709");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2394.b (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:	$19.1161862439$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 10x^{6} + 30x^{4} + 24x^{2} + 1 $ x^8 + 10x^6 + 30x^4 + 24*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{5}\cdot 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1709.6
Root			$0.209899i$ of defining polynomial
Character	$\chi$	$=$	2394.1709
Dual form			2394.2.b.j.1709.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +2.76612i q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +2.76612i q^{5} -1.00000 q^{7} +1.00000 q^{8} +2.76612i q^{10} +4.92827i q^{11} -2.93944i q^{13} -1.00000 q^{14} +1.00000 q^{16} +4.92827i q^{17} +(1.87745 - 3.93385i) q^{19} +2.76612i q^{20} +4.92827i q^{22} -0.685627i q^{23} -2.65142 q^{25} -2.93944i q^{26} -1.00000 q^{28} -1.65142 q^{29} +2.16215i q^{31} +1.00000 q^{32} +4.92827i q^{34} -2.76612i q^{35} +9.77488i q^{37} +(1.87745 - 3.93385i) q^{38} +2.76612i q^{40} -2.37594 q^{41} -8.15699 q^{43} +4.92827i q^{44} -0.685627i q^{46} +5.87887i q^{47} +1.00000 q^{49} -2.65142 q^{50} -2.93944i q^{52} -3.47519 q^{53} -13.6322 q^{55} -1.00000 q^{56} -1.65142 q^{58} -4.26046 q^{61} +2.16215i q^{62} +1.00000 q^{64} +8.13083 q^{65} -6.56450i q^{67} +4.92827i q^{68} -2.76612i q^{70} -11.1266 q^{71} +2.85415 q^{73} +9.77488i q^{74} +(1.87745 - 3.93385i) q^{76} -4.92827i q^{77} +11.0781i q^{79} +2.76612i q^{80} -2.37594 q^{82} +7.07681i q^{83} -13.6322 q^{85} -8.15699 q^{86} +4.92827i q^{88} +13.5026 q^{89} +2.93944i q^{91} -0.685627i q^{92} +5.87887i q^{94} +(10.8815 + 5.19325i) q^{95} -3.11275i q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} + 8 q^{4} - 8 q^{7} + 8 q^{8}+O(q^{10}) $ 8 * q + 8 * q^2 + 8 * q^4 - 8 * q^7 + 8 * q^8	$ 8 q + 8 q^{2} + 8 q^{4} - 8 q^{7} + 8 q^{8} - 8 q^{14} + 8 q^{16} + 12 q^{19} - 24 q^{25} - 8 q^{28} - 16 q^{29} + 8 q^{32} + 12 q^{38} + 24 q^{41} - 32 q^{43} + 8 q^{49} - 24 q^{50} + 48 q^{53} - 8 q^{56} - 16 q^{58} + 8 q^{61} + 8 q^{64} + 16 q^{65} - 16 q^{71} - 16 q^{73} + 12 q^{76} + 24 q^{82} - 32 q^{86} - 8 q^{89} + 8 q^{95} + 8 q^{98}+O(q^{100}) $ 8 * q + 8 * q^2 + 8 * q^4 - 8 * q^7 + 8 * q^8 - 8 * q^14 + 8 * q^16 + 12 * q^19 - 24 * q^25 - 8 * q^28 - 16 * q^29 + 8 * q^32 + 12 * q^38 + 24 * q^41 - 32 * q^43 + 8 * q^49 - 24 * q^50 + 48 * q^53 - 8 * q^56 - 16 * q^58 + 8 * q^61 + 8 * q^64 + 16 * q^65 - 16 * q^71 - 16 * q^73 + 12 * q^76 + 24 * q^82 - 32 * q^86 - 8 * q^89 + 8 * q^95 + 8 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2394\mathbb{Z}\right)^\times$.

$n$	$533$	$1009$	$1711$
$\chi(n)$	$-1$	$-1$	$1$

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 $

$2$	1.00000			0.707107
$2$	1.00000			0.707107
$3$		0			0
$3$		0			0
$4$	1.00000			0.500000
$5$			2.76612i			1.23705i	0.785766	+	0.618523i	$0.212269\pi$
$5$			2.76612i			1.23705i	−0.785766	+	0.618523i	$0.787731\pi$
$6$		0			0
$7$	−1.00000			−0.377964
$7$	−1.00000			−0.377964
$8$	1.00000			0.353553
$9$		0			0
$10$			2.76612i			0.874724i
$11$			4.92827i			1.48593i	0.669331	+	0.742964i	$0.266581\pi$
$11$			4.92827i			1.48593i	−0.669331	+	0.742964i	$0.733419\pi$
$12$		0			0
$13$		−	2.93944i		−	0.815253i	−0.913149	−	0.407626i	$-0.866357\pi$
$13$		−	2.93944i		−	0.815253i	0.913149	−	0.407626i	$-0.133643\pi$
$14$	−1.00000			−0.267261
$15$		0			0
$16$	1.00000			0.250000
$17$			4.92827i			1.19528i	0.801764	+	0.597640i	$0.203895\pi$
$17$			4.92827i			1.19528i	−0.801764	+	0.597640i	$0.796105\pi$
$18$		0			0
$19$	1.87745	−	3.93385i	0.430716	−	0.902488i
$19$	1.87745	−	3.93385i	0.430716	−	0.902488i
$20$			2.76612i			0.618523i
$21$		0			0
$22$			4.92827i			1.05071i
$23$		−	0.685627i		−	0.142963i	−0.997442	−	0.0714816i	$-0.977227\pi$
$23$		−	0.685627i		−	0.142963i	0.997442	−	0.0714816i	$-0.0227727\pi$
$24$		0			0
$25$	−2.65142			−0.530284
$26$		−	2.93944i		−	0.576471i
$27$		0			0
$28$	−1.00000			−0.188982
$29$	−1.65142			−0.306661			−0.153331	−	0.988175i	$-0.549000\pi$
$29$	−1.65142			−0.306661			−0.153331	+	0.988175i	$0.549000\pi$
$30$		0			0
$31$			2.16215i			0.388333i	0.980969	+	0.194167i	$0.0622002\pi$
$31$			2.16215i			0.388333i	−0.980969	+	0.194167i	$0.937800\pi$
$32$	1.00000			0.176777
$33$		0			0
$34$			4.92827i			0.845191i
$35$		−	2.76612i		−	0.467560i
$36$		0			0
$37$			9.77488i			1.60698i	0.595318	+	0.803490i	$0.297026\pi$
$37$			9.77488i			1.60698i	−0.595318	+	0.803490i	$0.702974\pi$
$38$	1.87745	−	3.93385i	0.304562	−	0.638155i
$39$		0			0
$40$			2.76612i			0.437362i
$41$	−2.37594			−0.371059			−0.185530	−	0.982639i	$-0.559400\pi$
$41$	−2.37594			−0.371059			−0.185530	+	0.982639i	$0.559400\pi$
$42$		0			0
$43$	−8.15699			−1.24393			−0.621965	−	0.783045i	$-0.713665\pi$
$43$	−8.15699			−1.24393			−0.621965	+	0.783045i	$0.713665\pi$
$44$			4.92827i			0.742964i
$45$		0			0
$46$		−	0.685627i		−	0.101090i
$47$			5.87887i			0.857522i	0.903418	+	0.428761i	$0.141050\pi$
$47$			5.87887i			0.857522i	−0.903418	+	0.428761i	$0.858950\pi$
$48$		0			0
$49$	1.00000			0.142857
$50$	−2.65142			−0.374968
$51$		0			0
$52$		−	2.93944i		−	0.407626i
$53$	−3.47519			−0.477354			−0.238677	−	0.971099i	$-0.576714\pi$
$53$	−3.47519			−0.477354			−0.238677	+	0.971099i	$0.576714\pi$
$54$		0			0
$55$	−13.6322			−1.83816
$56$	−1.00000			−0.133631
$57$		0			0
$58$	−1.65142			−0.216842
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$	−4.26046			−0.545496			−0.272748	−	0.962085i	$-0.587933\pi$
$61$	−4.26046			−0.545496			−0.272748	+	0.962085i	$0.587933\pi$
$62$			2.16215i			0.274593i
$63$		0			0
$64$	1.00000			0.125000
$65$	8.13083			1.00851
$66$		0			0
$67$		−	6.56450i		−	0.801981i	−0.916082	−	0.400991i	$-0.868666\pi$
$67$		−	6.56450i		−	0.801981i	0.916082	−	0.400991i	$-0.131334\pi$
$68$			4.92827i			0.597640i
$69$		0			0
$70$		−	2.76612i		−	0.330615i
$71$	−11.1266			−1.32049			−0.660243	−	0.751052i	$-0.729547\pi$
$71$	−11.1266			−1.32049			−0.660243	+	0.751052i	$0.729547\pi$
$72$		0			0
$73$	2.85415			0.334053			0.167026	−	0.985952i	$-0.446584\pi$
$73$	2.85415			0.334053			0.167026	+	0.985952i	$0.446584\pi$
$74$			9.77488i			1.13631i
$75$		0			0
$76$	1.87745	−	3.93385i	0.215358	−	0.451244i
$77$		−	4.92827i		−	0.561628i
$78$		0			0
$79$			11.0781i			1.24638i	0.782070	+	0.623191i	$0.214164\pi$
$79$			11.0781i			1.24638i	−0.782070	+	0.623191i	$0.785836\pi$
$80$			2.76612i			0.309262i
$81$		0			0
$82$	−2.37594			−0.262379
$83$			7.07681i			0.776781i	0.921495	+	0.388390i	$0.126969\pi$
$83$			7.07681i			0.776781i	−0.921495	+	0.388390i	$0.873031\pi$
$84$		0			0
$85$	−13.6322			−1.47862
$86$	−8.15699			−0.879591
$87$		0			0
$88$			4.92827i			0.525355i
$89$	13.5026			1.43127			0.715634	−	0.698476i	$-0.246138\pi$
$89$	13.5026			1.43127			0.715634	+	0.698476i	$0.246138\pi$
$90$		0			0
$91$			2.93944i			0.308137i
$92$		−	0.685627i		−	0.0714816i
$93$		0			0
$94$			5.87887i			0.606359i
$95$	10.8815	+	5.19325i	1.11642	+	0.532816i
$96$		0			0
$97$		−	3.11275i		−	0.316052i	−0.987435	−	0.158026i	$-0.949487\pi$
$97$		−	3.11275i		−	0.316052i	0.987435	−	0.158026i	$-0.0505130\pi$
$98$	1.00000			0.101015
$99$		0			0
$100$	−2.65142			−0.265142
$101$		−	8.64499i		−	0.860209i	−0.902779	−	0.430104i	$-0.858477\pi$
$101$		−	8.64499i		−	0.860209i	0.902779	−	0.430104i	$-0.141523\pi$
$102$		0			0
$103$		−	4.49761i		−	0.443163i	−0.975142	−	0.221581i	$-0.928878\pi$
$103$		−	4.49761i		−	0.443163i	0.975142	−	0.221581i	$-0.0711218\pi$
$104$		−	2.93944i		−	0.288235i
$105$		0			0
$106$	−3.47519			−0.337540
$107$	9.66678			0.934523			0.467261	−	0.884119i	$-0.345241\pi$
$107$	9.66678			0.934523			0.467261	+	0.884119i	$0.345241\pi$
$108$		0			0
$109$			14.6895i			1.40700i	0.710694	+	0.703502i	$0.248381\pi$
$109$			14.6895i			1.40700i	−0.710694	+	0.703502i	$0.751619\pi$
$110$	−13.6322			−1.29978
$111$		0			0
$112$	−1.00000			−0.0944911
$113$	−11.0111			−1.03584			−0.517920	−	0.855429i	$-0.673294\pi$
$113$	−11.0111			−1.03584			−0.517920	+	0.855429i	$0.673294\pi$
$114$		0			0
$115$	1.89653			0.176852
$116$	−1.65142			−0.153331
$117$		0			0
$118$		0			0
$119$		−	4.92827i		−	0.451774i
$120$		0			0
$121$	−13.2878			−1.20798
$122$	−4.26046			−0.385724
$123$		0			0
$124$			2.16215i			0.194167i
$125$			6.49645i			0.581060i
$126$		0			0
$127$			1.56818i			0.139154i	0.997577	+	0.0695768i	$0.0221649\pi$
$127$			1.56818i			0.139154i	−0.997577	+	0.0695768i	$0.977835\pi$
$128$	1.00000			0.0883883
$129$		0			0
$130$	8.13083			0.713121
$131$		−	15.8330i		−	1.38334i	−0.722214	−	0.691669i	$-0.756876\pi$
$131$		−	15.8330i		−	1.38334i	0.722214	−	0.691669i	$-0.243124\pi$
$132$		0			0
$133$	−1.87745	+	3.93385i	−0.162795	+	0.341108i
$134$		−	6.56450i		−	0.567086i
$135$		0			0
$136$			4.92827i			0.422595i
$137$			5.61390i			0.479627i	0.970819	+	0.239814i	$0.0770864\pi$
$137$			5.61390i			0.479627i	−0.970819	+	0.239814i	$0.922914\pi$
$138$		0			0
$139$	22.0770			1.87255			0.936273	−	0.351273i	$-0.114251\pi$
$139$	22.0770			1.87255			0.936273	+	0.351273i	$0.114251\pi$
$140$		−	2.76612i		−	0.233780i
$141$		0			0
$142$	−11.1266			−0.933725
$143$	14.4863			1.21141
$144$		0			0
$145$		−	4.56803i		−	0.379354i
$146$	2.85415			0.236211
$147$		0			0
$148$			9.77488i			0.803490i
$149$			1.37125i			0.112338i	0.998421	+	0.0561688i	$0.0178885\pi$
$149$			1.37125i			0.112338i	−0.998421	+	0.0561688i	$0.982112\pi$
$150$		0			0
$151$		−	22.8358i		−	1.85835i	−0.369636	−	0.929177i	$-0.620518\pi$
$151$		−	22.8358i		−	1.85835i	0.369636	−	0.929177i	$-0.379482\pi$
$152$	1.87745	−	3.93385i	0.152281	−	0.319078i
$153$		0			0
$154$		−	4.92827i		−	0.397131i
$155$	−5.98076			−0.480386
$156$		0			0
$157$	5.53595			0.441817			0.220908	−	0.975295i	$-0.429098\pi$
$157$	5.53595			0.441817			0.220908	+	0.975295i	$0.429098\pi$
$158$			11.0781i			0.881325i
$159$		0			0
$160$			2.76612i			0.218681i
$161$			0.685627i			0.0540350i
$162$		0			0
$163$	−21.9200			−1.71691			−0.858454	−	0.512891i	$-0.828574\pi$
$163$	−21.9200			−1.71691			−0.858454	+	0.512891i	$0.828574\pi$
$164$	−2.37594			−0.185530
$165$		0			0
$166$			7.07681i			0.549267i
$167$	17.7892			1.37657			0.688284	−	0.725442i	$-0.258364\pi$
$167$	17.7892			1.37657			0.688284	+	0.725442i	$0.258364\pi$
$168$		0			0
$169$	4.35972			0.335363
$170$	−13.6322			−1.04554
$171$		0			0
$172$	−8.15699			−0.621965
$173$	9.66678			0.734952			0.367476	−	0.930033i	$-0.380222\pi$
$173$	9.66678			0.734952			0.367476	+	0.930033i	$0.380222\pi$
$174$		0			0
$175$	2.65142			0.200429
$176$			4.92827i			0.371482i
$177$		0			0
$178$	13.5026			1.01206
$179$	14.4295			1.07851			0.539254	−	0.842143i	$-0.318706\pi$
$179$	14.4295			1.07851			0.539254	+	0.842143i	$0.318706\pi$
$180$		0			0
$181$			11.5880i			0.861331i	0.902512	+	0.430666i	$0.141721\pi$
$181$			11.5880i			0.861331i	−0.902512	+	0.430666i	$0.858279\pi$
$182$			2.93944i			0.217886i
$183$		0			0
$184$		−	0.685627i		−	0.0505451i
$185$	−27.0385			−1.98791
$186$		0			0
$187$	−24.2878			−1.77610
$188$			5.87887i			0.428761i
$189$		0			0
$190$	10.8815	+	5.19325i	0.789428	+	0.376758i
$191$			3.36772i			0.243680i	0.992550	+	0.121840i	$0.0388794\pi$
$191$			3.36772i			0.243680i	−0.992550	+	0.121840i	$0.961121\pi$
$192$		0			0
$193$			16.0064i			1.15216i	0.817392	+	0.576081i	$0.195419\pi$
$193$			16.0064i			1.15216i	−0.817392	+	0.576081i	$0.804581\pi$
$194$		−	3.11275i		−	0.223483i
$195$		0			0
$196$	1.00000			0.0714286
$197$		−	5.87887i		−	0.418852i	−0.977824	−	0.209426i	$-0.932840\pi$
$197$		−	5.87887i		−	0.418852i	0.977824	−	0.209426i	$-0.0671596\pi$
$198$		0			0
$199$	7.43368			0.526959			0.263480	−	0.964665i	$-0.415130\pi$
$199$	7.43368			0.526959			0.263480	+	0.964665i	$0.415130\pi$
$200$	−2.65142			−0.187484
$201$		0			0
$202$		−	8.64499i		−	0.608260i
$203$	1.65142			0.115907
$204$		0			0
$205$		−	6.57214i		−	0.459018i
$206$		−	4.49761i		−	0.313363i
$207$		0			0
$208$		−	2.93944i		−	0.203813i
$209$	19.3871	+	9.25256i	1.34103	+	0.640013i
$210$		0			0
$211$		−	7.18208i		−	0.494435i	−0.968960	−	0.247217i	$-0.920484\pi$
$211$		−	7.18208i		−	0.494435i	0.968960	−	0.247217i	$-0.0795161\pi$
$212$	−3.47519			−0.238677
$213$		0			0
$214$	9.66678			0.660807
$215$		−	22.5632i		−	1.53880i
$216$		0			0
$217$		−	2.16215i		−	0.146776i
$218$			14.6895i			0.994902i
$219$		0			0
$220$	−13.6322			−0.919082
$221$	14.4863			0.974456
$222$		0			0
$223$		−	25.7653i		−	1.72537i	−0.505742	−	0.862685i	$-0.668781\pi$
$223$		−	25.7653i		−	1.72537i	0.505742	−	0.862685i	$-0.331219\pi$
$224$	−1.00000			−0.0668153
$225$		0			0
$226$	−11.0111			−0.732450
$227$	16.9350			1.12402			0.562009	−	0.827131i	$-0.310029\pi$
$227$	16.9350			1.12402			0.562009	+	0.827131i	$0.310029\pi$
$228$		0			0
$229$	10.2878			0.679839			0.339919	−	0.940455i	$-0.389600\pi$
$229$	10.2878			0.679839			0.339919	+	0.940455i	$0.389600\pi$
$230$	1.89653			0.125053
$231$		0			0
$232$	−1.65142			−0.108421
$233$			3.97169i			0.260194i	0.991501	+	0.130097i	$0.0415289\pi$
$233$			3.97169i			0.260194i	−0.991501	+	0.130097i	$0.958471\pi$
$234$		0			0
$235$	−16.2617			−1.06079
$236$		0			0
$237$		0			0
$238$		−	4.92827i		−	0.319452i
$239$			16.8753i			1.09157i	0.837924	+	0.545786i	$0.183769\pi$
$239$			16.8753i			1.09157i	−0.837924	+	0.545786i	$0.816231\pi$
$240$		0			0
$241$		−	9.88015i		−	0.636436i	−0.948018	−	0.318218i	$-0.896916\pi$
$241$		−	9.88015i		−	0.636436i	0.948018	−	0.318218i	$-0.103084\pi$
$242$	−13.2878			−0.854174
$243$		0			0
$244$	−4.26046			−0.272748
$245$			2.76612i			0.176721i
$246$		0			0
$247$	−11.5633	−	5.51864i	−0.735756	−	0.351142i
$248$			2.16215i			0.137297i
$249$		0			0
$250$			6.49645i			0.410872i
$251$			10.6202i			0.670342i	0.942157	+	0.335171i	$0.108794\pi$
$251$			10.6202i			0.670342i	−0.942157	+	0.335171i	$0.891206\pi$
$252$		0			0
$253$	3.37895			0.212433
$254$			1.56818i			0.0983965i
$255$		0			0
$256$	1.00000			0.0625000
$257$	−5.18857			−0.323654			−0.161827	−	0.986819i	$-0.551739\pi$
$257$	−5.18857			−0.323654			−0.161827	+	0.986819i	$0.551739\pi$
$258$		0			0
$259$		−	9.77488i		−	0.607382i
$260$	8.13083			0.504253
$261$		0			0
$262$		−	15.8330i		−	0.978168i
$263$		−	1.13990i		−	0.0702890i	−0.999382	−	0.0351445i	$-0.988811\pi$
$263$		−	1.13990i		−	0.0702890i	0.999382	−	0.0351445i	$-0.0111891\pi$
$264$		0			0
$265$		−	9.61280i		−	0.590510i
$266$	−1.87745	+	3.93385i	−0.115114	+	0.241200i
$267$		0			0
$268$		−	6.56450i		−	0.400991i
$269$	−8.79339			−0.536143			−0.268071	−	0.963399i	$-0.586386\pi$
$269$	−8.79339			−0.536143			−0.268071	+	0.963399i	$0.586386\pi$
$270$		0			0
$271$	−9.88452			−0.600442			−0.300221	−	0.953870i	$-0.597060\pi$
$271$	−9.88452			−0.600442			−0.300221	+	0.953870i	$0.597060\pi$
$272$			4.92827i			0.298820i
$273$		0			0
$274$			5.61390i			0.339148i
$275$		−	13.0669i		−	0.787965i
$276$		0			0
$277$	26.3687			1.58434			0.792171	−	0.610299i	$-0.208951\pi$
$277$	26.3687			1.58434			0.792171	+	0.610299i	$0.208951\pi$
$278$	22.0770			1.32409
$279$		0			0
$280$		−	2.76612i		−	0.165307i
$281$	12.9504			0.772555			0.386278	−	0.922383i	$-0.373761\pi$
$281$	12.9504			0.772555			0.386278	+	0.922383i	$0.373761\pi$
$282$		0			0
$283$	6.88151			0.409063			0.204532	−	0.978860i	$-0.434433\pi$
$283$	6.88151			0.409063			0.204532	+	0.978860i	$0.434433\pi$
$284$	−11.1266			−0.660243
$285$		0			0
$286$	14.4863			0.856595
$287$	2.37594			0.140247
$288$		0			0
$289$	−7.28782			−0.428696
$290$		−	4.56803i		−	0.268244i
$291$		0			0
$292$	2.85415			0.167026
$293$	−2.67792			−0.156446			−0.0782228	−	0.996936i	$-0.524925\pi$
$293$	−2.67792			−0.156446			−0.0782228	+	0.996936i	$0.524925\pi$
$294$		0			0
$295$		0			0
$296$			9.77488i			0.568153i
$297$		0			0
$298$			1.37125i			0.0794346i
$299$	−2.01536			−0.116551
$300$		0			0
$301$	8.15699			0.470161
$302$		−	22.8358i		−	1.31405i
$303$		0			0
$304$	1.87745	−	3.93385i	0.107679	−	0.225622i
$305$		−	11.7850i		−	0.674804i
$306$		0			0
$307$			29.0597i			1.65852i	0.558861	+	0.829261i	$0.311239\pi$
$307$			29.0597i			1.65852i	−0.558861	+	0.829261i	$0.688761\pi$
$308$		−	4.92827i		−	0.280814i
$309$		0			0
$310$	−5.98076			−0.339684
$311$		−	13.1562i		−	0.746020i	−0.927827	−	0.373010i	$-0.878326\pi$
$311$		−	13.1562i		−	0.746020i	0.927827	−	0.373010i	$-0.121674\pi$
$312$		0			0
$313$	−1.50979			−0.0853383			−0.0426692	−	0.999089i	$-0.513586\pi$
$313$	−1.50979			−0.0853383			−0.0426692	+	0.999089i	$0.513586\pi$
$314$	5.53595			0.312411
$315$		0			0
$316$			11.0781i			0.623191i
$317$	27.1573			1.52531			0.762654	−	0.646807i	$-0.223896\pi$
$317$	27.1573			1.52531			0.762654	+	0.646807i	$0.223896\pi$
$318$		0			0
$319$		−	8.13865i		−	0.455677i
$320$			2.76612i			0.154631i
$321$		0			0
$322$			0.685627i			0.0382085i
$323$	19.3871	+	9.25256i	1.07873	+	0.514826i
$324$		0			0
$325$			7.79369i			0.432316i
$326$	−21.9200			−1.21404
$327$		0			0
$328$	−2.37594			−0.131189
$329$		−	5.87887i		−	0.324113i
$330$		0			0
$331$		−	5.35655i		−	0.294423i	−0.989105	−	0.147211i	$-0.952970\pi$
$331$		−	5.35655i		−	0.294423i	0.989105	−	0.147211i	$-0.0470298\pi$
$332$			7.07681i			0.388390i
$333$		0			0
$334$	17.7892			0.973380
$335$	18.1582			0.992088
$336$		0			0
$337$			17.8875i			0.974397i	0.873291	+	0.487198i	$0.161981\pi$
$337$			17.8875i			0.974397i	−0.873291	+	0.487198i	$0.838019\pi$
$338$	4.35972			0.237137
$339$		0			0
$340$	−13.6322			−0.739309
$341$	−10.6556			−0.577035
$342$		0			0
$343$	−1.00000			−0.0539949
$344$	−8.15699			−0.439795
$345$		0			0
$346$	9.66678			0.519689
$347$		−	18.5992i		−	0.998455i	−0.866471	−	0.499228i	$-0.833617\pi$
$347$		−	18.5992i		−	0.998455i	0.866471	−	0.499228i	$-0.166383\pi$
$348$		0			0
$349$	−12.7507			−0.682528			−0.341264	−	0.939968i	$-0.610855\pi$
$349$	−12.7507			−0.682528			−0.341264	+	0.939968i	$0.610855\pi$
$350$	2.65142			0.141724
$351$		0			0
$352$			4.92827i			0.262678i
$353$			17.5745i			0.935398i	0.883888	+	0.467699i	$0.154917\pi$
$353$			17.5745i			0.935398i	−0.883888	+	0.467699i	$0.845083\pi$
$354$		0			0
$355$		−	30.7776i		−	1.63350i
$356$	13.5026			0.715634
$357$		0			0
$358$	14.4295			0.762621
$359$			17.6290i			0.930422i	0.885200	+	0.465211i	$0.154022\pi$
$359$			17.6290i			0.930422i	−0.885200	+	0.465211i	$0.845978\pi$
$360$		0			0
$361$	−11.9504	−	14.7712i	−0.628968	−	0.777432i
$362$			11.5880i			0.609053i
$363$		0			0
$364$			2.93944i			0.154068i
$365$			7.89491i			0.413239i
$366$		0			0
$367$	32.1925			1.68043			0.840217	−	0.542251i	$-0.182428\pi$
$367$	32.1925			1.68043			0.840217	+	0.542251i	$0.182428\pi$
$368$		−	0.685627i		−	0.0357408i
$369$		0			0
$370$	−27.0385			−1.40566
$371$	3.47519			0.180423
$372$		0			0
$373$			23.4458i			1.21398i	0.794711	+	0.606988i	$0.207622\pi$
$373$			23.4458i			1.21398i	−0.794711	+	0.606988i	$0.792378\pi$
$374$	−24.2878			−1.25589
$375$		0			0
$376$			5.87887i			0.303180i
$377$			4.85425i			0.250007i
$378$		0			0
$379$			1.27600i			0.0655435i	0.999463	+	0.0327717i	$0.0104334\pi$
$379$			1.27600i			0.0655435i	−0.999463	+	0.0327717i	$0.989567\pi$
$380$	10.8815	+	5.19325i	0.558210	+	0.266408i
$381$		0			0
$382$			3.36772i			0.172308i
$383$	9.66343			0.493778			0.246889	−	0.969044i	$-0.420592\pi$
$383$	9.66343			0.493778			0.246889	+	0.969044i	$0.420592\pi$
$384$		0			0
$385$	13.6322			0.694760
$386$			16.0064i			0.814702i
$387$		0			0
$388$		−	3.11275i		−	0.158026i
$389$		−	27.3098i		−	1.38466i	−0.721579	−	0.692332i	$-0.756584\pi$
$389$		−	27.3098i		−	1.38466i	0.721579	−	0.692332i	$-0.243416\pi$
$390$		0			0
$391$	3.37895			0.170881
$392$	1.00000			0.0505076
$393$		0			0
$394$		−	5.87887i		−	0.296173i
$395$	−30.6433			−1.54183
$396$		0			0
$397$	22.6018			1.13435			0.567176	−	0.823597i	$-0.308036\pi$
$397$	22.6018			1.13435			0.567176	+	0.823597i	$0.308036\pi$
$398$	7.43368			0.372617
$399$		0			0
$400$	−2.65142			−0.132571
$401$	28.1771			1.40710			0.703549	−	0.710647i	$-0.251598\pi$
$401$	28.1771			1.40710			0.703549	+	0.710647i	$0.251598\pi$
$402$		0			0
$403$	6.35549			0.316590
$404$		−	8.64499i		−	0.430104i
$405$		0			0
$406$	1.65142			0.0819587
$407$	−48.1732			−2.38786
$408$		0			0
$409$		−	23.9581i		−	1.18465i	−0.805699	−	0.592325i	$-0.798210\pi$
$409$		−	23.9581i		−	1.18465i	0.805699	−	0.592325i	$-0.201790\pi$
$410$		−	6.57214i		−	0.324575i
$411$		0			0
$412$		−	4.49761i		−	0.221581i
$413$		0			0
$414$		0			0
$415$	−19.5753			−0.960914
$416$		−	2.93944i		−	0.144118i
$417$		0			0
$418$	19.3871	+	9.25256i	0.948253	+	0.452558i
$419$		−	29.5796i		−	1.44506i	−0.691340	−	0.722529i	$-0.742979\pi$
$419$		−	29.5796i		−	1.44506i	0.691340	−	0.722529i	$-0.257021\pi$
$420$		0			0
$421$			15.2467i			0.743079i	0.928417	+	0.371540i	$0.121170\pi$
$421$			15.2467i			0.743079i	−0.928417	+	0.371540i	$0.878830\pi$
$422$		−	7.18208i		−	0.349618i
$423$		0			0
$424$	−3.47519			−0.168770
$425$		−	13.0669i		−	0.633839i
$426$		0			0
$427$	4.26046			0.206178
$428$	9.66678			0.467261
$429$		0			0
$430$		−	22.5632i		−	1.08809i
$431$	−23.7215			−1.14262			−0.571312	−	0.820733i	$-0.693565\pi$
$431$	−23.7215			−1.14262			−0.571312	+	0.820733i	$0.693565\pi$
$432$		0			0
$433$		−	13.8306i		−	0.664656i	−0.943164	−	0.332328i	$-0.892166\pi$
$433$		−	13.8306i		−	0.664656i	0.943164	−	0.332328i	$-0.107834\pi$
$434$		−	2.16215i		−	0.103786i
$435$		0			0
$436$			14.6895i			0.703502i
$437$	−2.69716	−	1.28723i	−0.129022	−	0.0615765i
$438$		0			0
$439$		−	29.5039i		−	1.40815i	−0.710128	−	0.704073i	$-0.751363\pi$
$439$		−	29.5039i		−	1.40815i	0.710128	−	0.704073i	$-0.248637\pi$
$440$	−13.6322			−0.649889
$441$		0			0
$442$	14.4863			0.689044
$443$		−	0.257341i		−	0.0122267i	−0.999981	−	0.00611333i	$-0.998054\pi$
$443$		−	0.257341i		−	0.0122267i	0.999981	−	0.00611333i	$-0.00194594\pi$
$444$		0			0
$445$			37.3497i			1.77055i
$446$		−	25.7653i		−	1.22002i
$447$		0			0
$448$	−1.00000			−0.0472456
$449$	−22.3687			−1.05564			−0.527822	−	0.849355i	$-0.676991\pi$
$449$	−22.3687			−1.05564			−0.527822	+	0.849355i	$0.676991\pi$
$450$		0			0
$451$		−	11.7093i		−	0.551368i
$452$	−11.0111			−0.517920
$453$		0			0
$454$	16.9350			0.794800
$455$	−8.13083			−0.381179
$456$		0			0
$457$	−33.2990			−1.55766			−0.778830	−	0.627235i	$-0.784186\pi$
$457$	−33.2990			−1.55766			−0.778830	+	0.627235i	$0.784186\pi$
$458$	10.2878			0.480718
$459$		0			0
$460$	1.89653			0.0884260
$461$		−	1.90481i		−	0.0887157i	−0.999016	−	0.0443578i	$-0.985876\pi$
$461$		−	1.90481i		−	0.0887157i	0.999016	−	0.0443578i	$-0.0141242\pi$
$462$		0			0
$463$	−5.22888			−0.243007			−0.121503	−	0.992591i	$-0.538772\pi$
$463$	−5.22888			−0.243007			−0.121503	+	0.992591i	$0.538772\pi$
$464$	−1.65142			−0.0766653
$465$		0			0
$466$			3.97169i			0.183985i
$467$		−	31.1342i		−	1.44072i	−0.693601	−	0.720359i	$-0.743977\pi$
$467$		−	31.1342i		−	1.44072i	0.693601	−	0.720359i	$-0.256023\pi$
$468$		0			0
$469$			6.56450i			0.303120i
$470$	−16.2617			−0.750095
$471$		0			0
$472$		0			0
$473$		−	40.1998i		−	1.84839i
$474$		0			0
$475$	−4.97790	+	10.4303i	−0.228402	+	0.478575i
$476$		−	4.92827i		−	0.225887i
$477$		0			0
$478$			16.8753i			0.771858i
$479$		−	7.25013i		−	0.331267i	−0.986187	−	0.165633i	$-0.947033\pi$
$479$		−	7.25013i		−	0.331267i	0.986187	−	0.165633i	$-0.0529668\pi$
$480$		0			0
$481$	28.7326			1.31010
$482$		−	9.88015i		−	0.450028i
$483$		0			0
$484$	−13.2878			−0.603992
$485$	8.61025			0.390971
$486$		0			0
$487$		−	16.2637i		−	0.736978i	−0.929632	−	0.368489i	$-0.879875\pi$
$487$		−	16.2637i		−	0.736978i	0.929632	−	0.368489i	$-0.120125\pi$
$488$	−4.26046			−0.192862
$489$		0			0
$490$			2.76612i			0.124961i
$491$		−	31.7282i		−	1.43187i	−0.698166	−	0.715936i	$-0.746000\pi$
$491$		−	31.7282i		−	1.43187i	0.698166	−	0.715936i	$-0.254000\pi$
$492$		0			0
$493$		−	8.13865i		−	0.366546i
$494$	−11.5633	−	5.51864i	−0.520258	−	0.248295i
$495$		0			0
$496$			2.16215i			0.0970833i
$497$	11.1266			0.499097
$498$		0			0
$499$	2.29171			0.102591			0.0512954	−	0.998684i	$-0.483665\pi$
$499$	2.29171			0.102591			0.0512954	+	0.998684i	$0.483665\pi$
$500$			6.49645i			0.290530i
$501$		0			0
$502$			10.6202i			0.474004i
$503$			21.0239i			0.937410i	0.883355	+	0.468705i	$0.155279\pi$
$503$			21.0239i			0.937410i	−0.883355	+	0.468705i	$0.844721\pi$
$504$		0			0
$505$	23.9131			1.06412
$506$	3.37895			0.150213
$507$		0			0
$508$			1.56818i			0.0695768i
$509$	−40.6364			−1.80118			−0.900588	−	0.434673i	$-0.856864\pi$
$509$	−40.6364			−1.80118			−0.900588	+	0.434673i	$0.856864\pi$
$510$		0			0
$511$	−2.85415			−0.126260
$512$	1.00000			0.0441942
$513$		0			0
$514$	−5.18857			−0.228858
$515$	12.4409			0.548213
$516$		0			0
$517$	−28.9727			−1.27422
$518$		−	9.77488i		−	0.429484i
$519$		0			0
$520$	8.13083			0.356561
$521$	−40.8857			−1.79124			−0.895619	−	0.444823i	$-0.853267\pi$
$521$	−40.8857			−1.79124			−0.895619	+	0.444823i	$0.853267\pi$
$522$		0			0
$523$		−	19.1427i		−	0.837053i	−0.908205	−	0.418526i	$-0.862547\pi$
$523$		−	19.1427i		−	0.837053i	0.908205	−	0.418526i	$-0.137453\pi$
$524$		−	15.8330i		−	0.691669i
$525$		0			0
$526$		−	1.13990i		−	0.0497018i
$527$	−10.6556			−0.464167
$528$		0			0
$529$	22.5299			0.979562
$530$		−	9.61280i		−	0.417553i
$531$		0			0
$532$	−1.87745	+	3.93385i	−0.0813977	+	0.170554i
$533$			6.98392i			0.302507i
$534$		0			0
$535$			26.7395i			1.15605i
$536$		−	6.56450i		−	0.283543i
$537$		0			0
$538$	−8.79339			−0.379110
$539$			4.92827i			0.212276i
$540$		0			0
$541$	45.2190			1.94412			0.972058	−	0.234742i	$-0.0754246\pi$
$541$	45.2190			1.94412			0.972058	+	0.234742i	$0.0754246\pi$
$542$	−9.88452			−0.424577
$543$		0			0
$544$			4.92827i			0.211298i
$545$	−40.6330			−1.74053
$546$		0			0
$547$		−	19.0274i		−	0.813555i	−0.913527	−	0.406777i	$-0.866652\pi$
$547$		−	19.0274i		−	0.813555i	0.913527	−	0.406777i	$-0.133348\pi$
$548$			5.61390i			0.239814i
$549$		0			0
$550$		−	13.0669i		−	0.557175i
$551$	−3.10046	+	6.49645i	−0.132084	+	0.276758i
$552$		0			0
$553$		−	11.0781i		−	0.471088i
$554$	26.3687			1.12030
$555$		0			0
$556$	22.0770			0.936273
$557$			26.1219i			1.10682i	0.832909	+	0.553410i	$0.186674\pi$
$557$			26.1219i			1.10682i	−0.832909	+	0.553410i	$0.813326\pi$
$558$		0			0
$559$			23.9770i			1.01412i
$560$		−	2.76612i		−	0.116890i
$561$		0			0
$562$	12.9504			0.546279
$563$	23.9615			1.00986			0.504929	−	0.863161i	$-0.331519\pi$
$563$	23.9615			1.00986			0.504929	+	0.863161i	$0.331519\pi$
$564$		0			0
$565$		−	30.4581i		−	1.28138i
$566$	6.88151			0.289251
$567$		0			0
$568$	−11.1266			−0.466862
$569$	−28.0316			−1.17515			−0.587573	−	0.809171i	$-0.699916\pi$
$569$	−28.0316			−1.17515			−0.587573	+	0.809171i	$0.699916\pi$
$570$		0			0
$571$	−8.54017			−0.357395			−0.178697	−	0.983904i	$-0.557188\pi$
$571$	−8.54017			−0.357395			−0.178697	+	0.983904i	$0.557188\pi$
$572$	14.4863			0.605704
$573$		0			0
$574$	2.37594			0.0991698
$575$			1.81789i			0.0758111i
$576$		0			0
$577$	38.2424			1.59205			0.796027	−	0.605262i	$-0.206932\pi$
$577$	38.2424			1.59205			0.796027	+	0.605262i	$0.206932\pi$
$578$	−7.28782			−0.303134
$579$		0			0
$580$		−	4.56803i		−	0.189677i
$581$		−	7.07681i		−	0.293596i
$582$		0			0
$583$		−	17.1267i		−	0.709314i
$584$	2.85415			0.118105
$585$		0			0
$586$	−2.67792			−0.110624
$587$		−	20.0697i		−	0.828366i	−0.910194	−	0.414183i	$-0.864067\pi$
$587$		−	20.0697i		−	0.828366i	0.910194	−	0.414183i	$-0.135933\pi$
$588$		0			0
$589$	8.50557	+	4.05932i	0.350466	+	0.167261i
$590$		0			0
$591$		0			0
$592$			9.77488i			0.401745i
$593$			32.8000i			1.34693i	0.739217	+	0.673467i	$0.235196\pi$
$593$			32.8000i			1.34693i	−0.739217	+	0.673467i	$0.764804\pi$
$594$		0			0
$595$	13.6322			0.558865
$596$			1.37125i			0.0561688i
$597$		0			0
$598$	−2.01536			−0.0824141
$599$	5.73230			0.234215			0.117108	−	0.993119i	$-0.462638\pi$
$599$	5.73230			0.234215			0.117108	+	0.993119i	$0.462638\pi$
$600$		0			0
$601$		−	19.3428i		−	0.789010i	−0.918894	−	0.394505i	$-0.870916\pi$
$601$		−	19.3428i		−	0.789010i	0.918894	−	0.394505i	$-0.129084\pi$
$602$	8.15699			0.332454
$603$		0			0
$604$		−	22.8358i		−	0.929177i
$605$		−	36.7557i		−	1.49433i
$606$		0			0
$607$		−	19.9940i		−	0.811532i	−0.913977	−	0.405766i	$-0.867005\pi$
$607$		−	19.9940i		−	0.811532i	0.913977	−	0.405766i	$-0.132995\pi$
$608$	1.87745	−	3.93385i	0.0761405	−	0.159539i
$609$		0			0
$610$		−	11.7850i		−	0.477159i
$611$	17.2806			0.699097
$612$		0			0
$613$	6.43790			0.260024			0.130012	−	0.991512i	$-0.458498\pi$
$613$	6.43790			0.260024			0.130012	+	0.991512i	$0.458498\pi$
$614$			29.0597i			1.17275i
$615$		0			0
$616$		−	4.92827i		−	0.198566i
$617$			32.9557i			1.32675i	0.748289	+	0.663373i	$0.230876\pi$
$617$			32.9557i			1.32675i	−0.748289	+	0.663373i	$0.769124\pi$
$618$		0			0
$619$	12.8995			0.518476			0.259238	−	0.965813i	$-0.416529\pi$
$619$	12.8995			0.518476			0.259238	+	0.965813i	$0.416529\pi$
$620$	−5.98076			−0.240193
$621$		0			0
$622$		−	13.1562i		−	0.527516i
$623$	−13.5026			−0.540968
$624$		0			0
$625$	−31.2271			−1.24908
$626$	−1.50979			−0.0603433
$627$		0			0
$628$	5.53595			0.220908
$629$	−48.1732			−1.92079
$630$		0			0
$631$	20.0307			0.797410			0.398705	−	0.917079i	$-0.369460\pi$
$631$	20.0307			0.797410			0.398705	+	0.917079i	$0.369460\pi$
$632$			11.0781i			0.440663i
$633$		0			0
$634$	27.1573			1.07856
$635$	−4.33778			−0.172140
$636$		0			0
$637$		−	2.93944i		−	0.116465i
$638$		−	8.13865i		−	0.322212i
$639$		0			0
$640$			2.76612i			0.109341i
$641$	−2.95641			−0.116771			−0.0583857	−	0.998294i	$-0.518595\pi$
$641$	−2.95641			−0.116771			−0.0583857	+	0.998294i	$0.518595\pi$
$642$		0			0
$643$	−3.45292			−0.136170			−0.0680849	−	0.997680i	$-0.521689\pi$
$643$	−3.45292			−0.136170			−0.0680849	+	0.997680i	$0.521689\pi$
$644$			0.685627i			0.0270175i
$645$		0			0
$646$	19.3871	+	9.25256i	0.762774	+	0.364037i
$647$			19.8964i			0.782208i	0.920346	+	0.391104i	$0.127907\pi$
$647$			19.8964i			0.782208i	−0.920346	+	0.391104i	$0.872093\pi$
$648$		0			0
$649$		0			0
$650$			7.79369i			0.305693i
$651$		0			0
$652$	−21.9200			−0.858454
$653$			6.90350i			0.270155i	0.990835	+	0.135077i	$0.0431283\pi$
$653$			6.90350i			0.270155i	−0.990835	+	0.135077i	$0.956872\pi$
$654$		0			0
$655$	43.7961			1.71125
$656$	−2.37594			−0.0927649
$657$		0			0
$658$		−	5.87887i		−	0.229182i
$659$	25.4214			0.990275			0.495138	−	0.868815i	$-0.335118\pi$
$659$	25.4214			0.990275			0.495138	+	0.868815i	$0.335118\pi$
$660$		0			0
$661$			14.4866i			0.563465i	0.959493	+	0.281733i	$0.0909091\pi$
$661$			14.4866i			0.563465i	−0.959493	+	0.281733i	$0.909091\pi$
$662$		−	5.35655i		−	0.208188i
$663$		0			0
$664$			7.07681i			0.274634i
$665$	−10.8815	−	5.19325i	−0.421967	−	0.201385i
$666$		0			0
$667$			1.13226i			0.0438413i
$668$	17.7892			0.688284
$669$		0			0
$670$	18.1582			0.701512
$671$		−	20.9967i		−	0.810569i
$672$		0			0
$673$			36.1465i			1.39334i	0.717390	+	0.696672i	$0.245337\pi$
$673$			36.1465i			1.39334i	−0.717390	+	0.696672i	$0.754663\pi$
$674$			17.8875i			0.689003i
$675$		0			0
$676$	4.35972			0.167681
$677$	42.5564			1.63558			0.817788	−	0.575520i	$-0.195200\pi$
$677$	42.5564			1.63558			0.817788	+	0.575520i	$0.195200\pi$
$678$		0			0
$679$			3.11275i			0.119456i
$680$	−13.6322			−0.522770
$681$		0			0
$682$	−10.6556			−0.408026
$683$	14.0355			0.537053			0.268526	−	0.963272i	$-0.413463\pi$
$683$	14.0355			0.537053			0.268526	+	0.963272i	$0.413463\pi$
$684$		0			0
$685$	−15.5287			−0.593322
$686$	−1.00000			−0.0381802
$687$		0			0
$688$	−8.15699			−0.310982
$689$			10.2151i			0.389164i
$690$		0			0
$691$	−8.90410			−0.338728			−0.169364	−	0.985554i	$-0.554171\pi$
$691$	−8.90410			−0.338728			−0.169364	+	0.985554i	$0.554171\pi$
$692$	9.66678			0.367476
$693$		0			0
$694$		−	18.5992i		−	0.706014i
$695$			61.0676i			2.31643i
$696$		0			0
$697$		−	11.7093i		−	0.443520i
$698$	−12.7507			−0.482620
$699$		0			0
$700$	2.65142			0.100214
$701$			47.9871i			1.81245i	0.422798	+	0.906224i	$0.361048\pi$
$701$			47.9871i			1.81245i	−0.422798	+	0.906224i	$0.638952\pi$
$702$		0			0
$703$	38.4529	+	18.3518i	1.45028	+	0.692152i
$704$			4.92827i			0.185741i
$705$		0			0
$706$			17.5745i			0.661426i
$707$			8.64499i			0.325128i
$708$		0			0
$709$	−24.3533			−0.914609			−0.457305	−	0.889310i	$-0.651185\pi$
$709$	−24.3533			−0.914609			−0.457305	+	0.889310i	$0.651185\pi$
$710$		−	30.7776i		−	1.15506i
$711$		0			0
$712$	13.5026			0.506030
$713$	1.48243			0.0555173
$714$		0			0
$715$			40.0709i			1.49857i
$716$	14.4295			0.539254
$717$		0			0
$718$			17.6290i			0.657908i
$719$			8.86512i			0.330613i	0.986242	+	0.165307i	$0.0528614\pi$
$719$			8.86512i			0.330613i	−0.986242	+	0.165307i	$0.947139\pi$
$720$		0			0
$721$			4.49761i			0.167500i
$722$	−11.9504	−	14.7712i	−0.444747	−	0.549727i
$723$		0			0
$724$			11.5880i			0.430666i
$725$	4.37862			0.162618
$726$		0			0
$727$	−19.1573			−0.710506			−0.355253	−	0.934770i	$-0.615605\pi$
$727$	−19.1573			−0.710506			−0.355253	+	0.934770i	$0.615605\pi$
$728$			2.93944i			0.108943i
$729$		0			0
$730$			7.89491i			0.292204i
$731$		−	40.1998i		−	1.48684i
$732$		0			0
$733$	40.2220			1.48563			0.742816	−	0.669495i	$-0.233490\pi$
$733$	40.2220			1.48563			0.742816	+	0.669495i	$0.233490\pi$
$734$	32.1925			1.18825
$735$		0			0
$736$		−	0.685627i		−	0.0252726i
$737$	32.3516			1.19169
$738$		0			0
$739$	47.6138			1.75150			0.875752	−	0.482762i	$-0.160366\pi$
$739$	47.6138			1.75150			0.875752	+	0.482762i	$0.160366\pi$
$740$	−27.0385			−0.993955
$741$		0			0
$742$	3.47519			0.127578
$743$	−39.8593			−1.46229			−0.731147	−	0.682220i	$-0.761015\pi$
$743$	−39.8593			−1.46229			−0.731147	+	0.682220i	$0.761015\pi$
$744$		0			0
$745$	−3.79305			−0.138967
$746$			23.4458i			0.858411i
$747$		0			0
$748$	−24.2878			−0.888051
$749$	−9.66678			−0.353216
$750$		0			0
$751$		−	47.3831i		−	1.72904i	−0.502603	−	0.864518i	$-0.667624\pi$
$751$		−	47.3831i		−	1.72904i	0.502603	−	0.864518i	$-0.332376\pi$
$752$			5.87887i			0.214380i
$753$		0			0
$754$			4.85425i			0.176781i
$755$	63.1667			2.29887
$756$		0			0
$757$	39.0966			1.42099			0.710495	−	0.703703i	$-0.248471\pi$
$757$	39.0966			1.42099			0.710495	+	0.703703i	$0.248471\pi$
$758$			1.27600i			0.0463462i
$759$		0			0
$760$	10.8815	+	5.19325i	0.394714	+	0.188379i
$761$		−	53.0230i		−	1.92208i	−0.276404	−	0.961041i	$-0.589143\pi$
$761$		−	53.0230i		−	1.92208i	0.276404	−	0.961041i	$-0.410857\pi$
$762$		0			0
$763$		−	14.6895i		−	0.531797i
$764$			3.36772i			0.121840i
$765$		0			0
$766$	9.66343			0.349154
$767$		0			0
$768$		0			0
$769$	2.90410			0.104725			0.0523623	−	0.998628i	$-0.483325\pi$
$769$	2.90410			0.104725			0.0523623	+	0.998628i	$0.483325\pi$
$770$	13.6322			0.491270
$771$		0			0
$772$			16.0064i			0.576081i
$773$	−38.0162			−1.36735			−0.683675	−	0.729787i	$-0.739619\pi$
$773$	−38.0162			−1.36735			−0.683675	+	0.729787i	$0.739619\pi$
$774$		0			0
$775$		−	5.73277i		−	0.205927i
$776$		−	3.11275i		−	0.111741i
$777$		0			0
$778$		−	27.3098i		−	0.979105i
$779$	−4.46070	+	9.34660i	−0.159821	+	0.334877i
$780$		0			0
$781$		−	54.8349i		−	1.96215i
$782$	3.37895			0.120831
$783$		0			0
$784$	1.00000			0.0357143
$785$			15.3131i			0.546548i
$786$		0			0
$787$			53.0094i			1.88958i	0.327674	+	0.944791i	$0.393735\pi$
$787$			53.0094i			1.88958i	−0.327674	+	0.944791i	$0.606265\pi$
$788$		−	5.87887i		−	0.209426i
$789$		0			0
$790$	−30.6433			−1.09024
$791$	11.0111			0.391511
$792$		0			0
$793$			12.5234i			0.444718i
$794$	22.6018			0.802108
$795$		0			0
$796$	7.43368			0.263480
$797$	−46.9727			−1.66386			−0.831928	−	0.554883i	$-0.812763\pi$
$797$	−46.9727			−1.66386			−0.831928	+	0.554883i	$0.812763\pi$
$798$		0			0
$799$	−28.9727			−1.02498
$800$	−2.65142			−0.0937419
$801$		0			0
$802$	28.1771			0.994969
$803$			14.0660i			0.496378i
$804$		0			0
$805$	−1.89653			−0.0668438
$806$	6.35549			0.223863
$807$		0			0
$808$		−	8.64499i		−	0.304130i
$809$		−	3.41453i		−	0.120049i	−0.998197	−	0.0600243i	$-0.980882\pi$
$809$		−	3.41453i		−	0.120049i	0.998197	−	0.0600243i	$-0.0191178\pi$
$810$		0			0
$811$			19.5025i			0.684827i	0.939549	+	0.342413i	$0.111244\pi$
$811$			19.5025i			0.684827i	−0.939549	+	0.342413i	$0.888756\pi$
$812$	1.65142			0.0579535
$813$		0			0
$814$	−48.1732			−1.68847
$815$		−	60.6334i		−	2.12389i
$816$		0			0
$817$	−15.3143	+	32.0884i	−0.535780	+	1.12263i
$818$		−	23.9581i		−	0.837675i
$819$		0			0
$820$		−	6.57214i		−	0.229509i
$821$			48.8484i			1.70482i	0.522873	+	0.852411i	$0.324860\pi$
$821$			48.8484i			1.70482i	−0.522873	+	0.852411i	$0.675140\pi$
$822$		0			0
$823$	−46.9774			−1.63753			−0.818765	−	0.574129i	$-0.805341\pi$
$823$	−46.9774			−1.63753			−0.818765	+	0.574129i	$0.805341\pi$
$824$		−	4.49761i		−	0.156682i
$825$		0			0
$826$		0			0
$827$	44.8896			1.56097			0.780483	−	0.625177i	$-0.214973\pi$
$827$	44.8896			1.56097			0.780483	+	0.625177i	$0.214973\pi$
$828$		0			0
$829$			25.7689i			0.894989i	0.894287	+	0.447495i	$0.147684\pi$
$829$			25.7689i			0.894989i	−0.894287	+	0.447495i	$0.852316\pi$
$830$	−19.5753			−0.679469
$831$		0			0
$832$		−	2.93944i		−	0.101907i
$833$			4.92827i			0.170754i
$834$		0			0
$835$			49.2070i			1.70288i
$836$	19.3871	+	9.25256i	0.670516	+	0.320007i
$837$		0			0
$838$		−	29.5796i		−	1.02181i
$839$	−35.2764			−1.21788			−0.608938	−	0.793218i	$-0.708404\pi$
$839$	−35.2764			−1.21788			−0.608938	+	0.793218i	$0.708404\pi$
$840$		0			0
$841$	−26.2728			−0.905959
$842$			15.2467i			0.525436i
$843$		0			0
$844$		−	7.18208i		−	0.247217i
$845$			12.0595i			0.414859i
$846$		0			0
$847$	13.2878			0.456575
$848$	−3.47519			−0.119339
$849$		0			0
$850$		−	13.0669i		−	0.448192i
$851$	6.70192			0.229739
$852$		0			0
$853$	−45.9046			−1.57175			−0.785873	−	0.618388i	$-0.787786\pi$
$853$	−45.9046			−1.57175			−0.785873	+	0.618388i	$0.787786\pi$
$854$	4.26046			0.145790
$855$		0			0
$856$	9.66678			0.330404
$857$	−34.4307			−1.17613			−0.588065	−	0.808814i	$-0.700110\pi$
$857$	−34.4307			−1.17613			−0.588065	+	0.808814i	$0.700110\pi$
$858$		0			0
$859$	49.0920			1.67500			0.837499	−	0.546439i	$-0.184017\pi$
$859$	49.0920			1.67500			0.837499	+	0.546439i	$0.184017\pi$
$860$		−	22.5632i		−	0.769399i
$861$		0			0
$862$	−23.7215			−0.807958
$863$	44.5324			1.51590			0.757950	−	0.652312i	$-0.226201\pi$
$863$	44.5324			1.51590			0.757950	+	0.652312i	$0.226201\pi$
$864$		0			0
$865$			26.7395i			0.909169i
$866$		−	13.8306i		−	0.469983i
$867$		0			0
$868$		−	2.16215i		−	0.0733881i
$869$	−54.5958			−1.85203
$870$		0			0
$871$	−19.2959			−0.653818
$872$			14.6895i			0.497451i
$873$		0			0
$874$	−2.69716	−	1.28723i	−0.0912327	−	0.0435412i
$875$		−	6.49645i		−	0.219620i
$876$		0			0
$877$			28.3876i			0.958583i	0.877656	+	0.479291i	$0.159106\pi$
$877$			28.3876i			0.958583i	−0.877656	+	0.479291i	$0.840894\pi$
$878$		−	29.5039i		−	0.995709i
$879$		0			0
$880$	−13.6322			−0.459541
$881$			4.25028i			0.143195i	0.997434	+	0.0715977i	$0.0228098\pi$
$881$			4.25028i			0.143195i	−0.997434	+	0.0715977i	$0.977190\pi$
$882$		0			0
$883$	−22.7742			−0.766411			−0.383206	−	0.923663i	$-0.625180\pi$
$883$	−22.7742			−0.766411			−0.383206	+	0.923663i	$0.625180\pi$
$884$	14.4863			0.487228
$885$		0			0
$886$		−	0.257341i		−	0.00864555i
$887$	2.41531			0.0810980			0.0405490	−	0.999178i	$-0.487089\pi$
$887$	2.41531			0.0810980			0.0405490	+	0.999178i	$0.487089\pi$
$888$		0			0
$889$		−	1.56818i		−	0.0525951i
$890$			37.3497i			1.25196i
$891$		0			0
$892$		−	25.7653i		−	0.862685i
$893$	23.1266	+	11.0373i	0.773903	+	0.369348i
$894$		0			0
$895$			39.9136i			1.33417i
$896$	−1.00000			−0.0334077
$897$		0			0
$898$	−22.3687			−0.746453
$899$		−	3.57062i		−	0.119087i
$900$		0			0
$901$		−	17.1267i		−	0.570572i
$902$		−	11.7093i		−	0.389876i
$903$		0			0
$904$	−11.0111			−0.366225
$905$	−32.0539			−1.06551
$906$		0			0
$907$		−	48.5971i		−	1.61364i	−0.590798	−	0.806819i	$-0.701187\pi$
$907$		−	48.5971i		−	1.61364i	0.590798	−	0.806819i	$-0.298813\pi$
$908$	16.9350			0.562009
$909$		0			0
$910$	−8.13083			−0.269535
$911$	−17.0111			−0.563604			−0.281802	−	0.959473i	$-0.590932\pi$
$911$	−17.0111			−0.563604			−0.281802	+	0.959473i	$0.590932\pi$
$912$		0			0
$913$	−34.8764			−1.15424
$914$	−33.2990			−1.10143
$915$		0			0
$916$	10.2878			0.339919
$917$			15.8330i			0.522853i
$918$		0			0
$919$	−32.5480			−1.07366			−0.536829	−	0.843691i	$-0.680378\pi$
$919$	−32.5480			−1.07366			−0.536829	+	0.843691i	$0.680378\pi$
$920$	1.89653			0.0625267
$921$		0			0
$922$		−	1.90481i		−	0.0627315i
$923$			32.7060i			1.07653i
$924$		0			0
$925$		−	25.9173i		−	0.852157i
$926$	−5.22888			−0.171832
$927$		0			0
$928$	−1.65142			−0.0542106
$929$		−	27.0725i		−	0.888220i	−0.895972	−	0.444110i	$-0.853520\pi$
$929$		−	27.0725i		−	0.888220i	0.895972	−	0.444110i	$-0.146480\pi$
$930$		0			0
$931$	1.87745	−	3.93385i	0.0615308	−	0.128927i
$932$			3.97169i			0.130097i
$933$		0			0
$934$		−	31.1342i		−	1.01874i
$935$		−	67.1830i		−	2.19712i
$936$		0			0
$937$	−17.6523			−0.576676			−0.288338	−	0.957529i	$-0.593103\pi$
$937$	−17.6523			−0.576676			−0.288338	+	0.957529i	$0.593103\pi$
$938$			6.56450i			0.214339i
$939$		0			0
$940$	−16.2617			−0.530397
$941$	−4.06075			−0.132377			−0.0661884	−	0.997807i	$-0.521084\pi$
$941$	−4.06075			−0.132377			−0.0661884	+	0.997807i	$0.521084\pi$
$942$		0			0
$943$			1.62901i			0.0530478i
$944$		0			0
$945$		0			0
$946$		−	40.1998i		−	1.30701i
$947$			3.87644i			0.125967i	0.998015	+	0.0629836i	$0.0200616\pi$
$947$			3.87644i			0.125967i	−0.998015	+	0.0629836i	$0.979938\pi$
$948$		0			0
$949$		−	8.38958i		−	0.272337i
$950$	−4.97790	+	10.4303i	−0.161505	+	0.338404i
$951$		0			0
$952$		−	4.92827i		−	0.159726i
$953$	54.0854			1.75200			0.876000	−	0.482312i	$-0.160203\pi$
$953$	54.0854			1.75200			0.876000	+	0.482312i	$0.160203\pi$
$954$		0			0
$955$	−9.31552			−0.301443
$956$			16.8753i			0.545786i
$957$		0			0
$958$		−	7.25013i		−	0.234241i
$959$		−	5.61390i		−	0.181282i
$960$		0			0
$961$	26.3251			0.849197
$962$	28.7326			0.926377
$963$		0			0
$964$		−	9.88015i		−	0.318218i
$965$	−44.2755			−1.42528
$966$		0			0
$967$	6.18771			0.198983			0.0994916	−	0.995038i	$-0.468278\pi$
$967$	6.18771			0.198983			0.0994916	+	0.995038i	$0.468278\pi$
$968$	−13.2878			−0.427087
$969$		0			0
$970$	8.61025			0.276458
$971$	12.7588			0.409449			0.204725	−	0.978820i	$-0.434370\pi$
$971$	12.7588			0.409449			0.204725	+	0.978820i	$0.434370\pi$
$972$		0			0
$973$	−22.0770			−0.707756
$974$		−	16.2637i		−	0.521122i
$975$		0			0
$976$	−4.26046			−0.136374
$977$	−44.6980			−1.43002			−0.715009	−	0.699116i	$-0.753577\pi$
$977$	−44.6980			−1.43002			−0.715009	+	0.699116i	$0.753577\pi$
$978$		0			0
$979$			66.5442i			2.12676i
$980$			2.76612i			0.0883605i
$981$		0			0
$982$		−	31.7282i		−	1.01249i
$983$	−5.35159			−0.170689			−0.0853447	−	0.996351i	$-0.527199\pi$
$983$	−5.35159			−0.170689			−0.0853447	+	0.996351i	$0.527199\pi$
$984$		0			0
$985$	16.2617			0.518140
$986$		−	8.13865i		−	0.259187i
$987$		0			0
$988$	−11.5633	−	5.51864i	−0.367878	−	0.175571i
$989$			5.59265i			0.177836i
$990$		0			0
$991$			15.9323i			0.506107i	0.967452	+	0.253054i	$0.0814349\pi$
$991$			15.9323i			0.506107i	−0.967452	+	0.253054i	$0.918565\pi$
$992$			2.16215i			0.0686483i
$993$		0			0
$994$	11.1266			0.352915
$995$			20.5624i			0.651873i
$996$		0			0
$997$	17.2767			0.547158			0.273579	−	0.961850i	$-0.411792\pi$
$997$	17.2767			0.547158			0.273579	+	0.961850i	$0.411792\pi$
$998$	2.29171			0.0725427
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2394.2.b.j.1709.6	yes	8
3.2	odd	2		2394.2.b.i.1709.3	✓	8
19.18	odd	2		2394.2.b.i.1709.6	yes	8
57.56	even	2	inner	2394.2.b.j.1709.3	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2394.2.b.i.1709.3	✓	8	3.2	odd	2
2394.2.b.i.1709.6	yes	8	19.18	odd	2
2394.2.b.j.1709.3	yes	8	57.56	even	2	inner
2394.2.b.j.1709.6	yes	8	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:	\( N \)	\(=\)	\( 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2394.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +2.76612i q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +2.76612i q^{5} -1.00000 q^{7} +1.00000 q^{8} +2.76612i q^{10} +4.92827i q^{11} -2.93944i q^{13} -1.00000 q^{14} +1.00000 q^{16} +4.92827i q^{17} +(1.87745 - 3.93385i) q^{19} +2.76612i q^{20} +4.92827i q^{22} -0.685627i q^{23} -2.65142 q^{25} -2.93944i q^{26} -1.00000 q^{28} -1.65142 q^{29} +2.16215i q^{31} +1.00000 q^{32} +4.92827i q^{34} -2.76612i q^{35} +9.77488i q^{37} +(1.87745 - 3.93385i) q^{38} +2.76612i q^{40} -2.37594 q^{41} -8.15699 q^{43} +4.92827i q^{44} -0.685627i q^{46} +5.87887i q^{47} +1.00000 q^{49} -2.65142 q^{50} -2.93944i q^{52} -3.47519 q^{53} -13.6322 q^{55} -1.00000 q^{56} -1.65142 q^{58} -4.26046 q^{61} +2.16215i q^{62} +1.00000 q^{64} +8.13083 q^{65} -6.56450i q^{67} +4.92827i q^{68} -2.76612i q^{70} -11.1266 q^{71} +2.85415 q^{73} +9.77488i q^{74} +(1.87745 - 3.93385i) q^{76} -4.92827i q^{77} +11.0781i q^{79} +2.76612i q^{80} -2.37594 q^{82} +7.07681i q^{83} -13.6322 q^{85} -8.15699 q^{86} +4.92827i q^{88} +13.5026 q^{89} +2.93944i q^{91} -0.685627i q^{92} +5.87887i q^{94} +(10.8815 + 5.19325i) q^{95} -3.11275i q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} + 8 q^{4} - 8 q^{7} + 8 q^{8}+O(q^{10}) \) 8 * q + 8 * q^2 + 8 * q^4 - 8 * q^7 + 8 * q^8	\( 8 q + 8 q^{2} + 8 q^{4} - 8 q^{7} + 8 q^{8} - 8 q^{14} + 8 q^{16} + 12 q^{19} - 24 q^{25} - 8 q^{28} - 16 q^{29} + 8 q^{32} + 12 q^{38} + 24 q^{41} - 32 q^{43} + 8 q^{49} - 24 q^{50} + 48 q^{53} - 8 q^{56} - 16 q^{58} + 8 q^{61} + 8 q^{64} + 16 q^{65} - 16 q^{71} - 16 q^{73} + 12 q^{76} + 24 q^{82} - 32 q^{86} - 8 q^{89} + 8 q^{95} + 8 q^{98}+O(q^{100}) \) 8 * q + 8 * q^2 + 8 * q^4 - 8 * q^7 + 8 * q^8 - 8 * q^14 + 8 * q^16 + 12 * q^19 - 24 * q^25 - 8 * q^28 - 16 * q^29 + 8 * q^32 + 12 * q^38 + 24 * q^41 - 32 * q^43 + 8 * q^49 - 24 * q^50 + 48 * q^53 - 8 * q^56 - 16 * q^58 + 8 * q^61 + 8 * q^64 + 16 * q^65 - 16 * q^71 - 16 * q^73 + 12 * q^76 + 24 * q^82 - 32 * q^86 - 8 * q^89 + 8 * q^95 + 8 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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