LMFDB - Embedded newform 414.2.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [414,2,Mod(1,414)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(414, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("414.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 414 = 2 \cdot 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	414.a (trivial)

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:	yes
Analytic conductor:	$3.30580664368$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{7}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 7 $ x^2 - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.64575$ of defining polynomial
Character	$\chi$	$=$	414.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +1.64575 q^{5} +2.00000 q^{7} -1.00000 q^{8} -1.64575 q^{10} -1.64575 q^{11} +5.29150 q^{13} -2.00000 q^{14} +1.00000 q^{16} -3.29150 q^{17} +0.354249 q^{19} +1.64575 q^{20} +1.64575 q^{22} +1.00000 q^{23} -2.29150 q^{25} -5.29150 q^{26} +2.00000 q^{28} +9.29150 q^{29} -1.29150 q^{31} -1.00000 q^{32} +3.29150 q^{34} +3.29150 q^{35} +6.93725 q^{37} -0.354249 q^{38} -1.64575 q^{40} +6.00000 q^{41} +0.354249 q^{43} -1.64575 q^{44} -1.00000 q^{46} +6.00000 q^{47} -3.00000 q^{49} +2.29150 q^{50} +5.29150 q^{52} -1.64575 q^{53} -2.70850 q^{55} -2.00000 q^{56} -9.29150 q^{58} +0.354249 q^{61} +1.29150 q^{62} +1.00000 q^{64} +8.70850 q^{65} -14.9373 q^{67} -3.29150 q^{68} -3.29150 q^{70} -6.00000 q^{71} -7.29150 q^{73} -6.93725 q^{74} +0.354249 q^{76} -3.29150 q^{77} +8.58301 q^{79} +1.64575 q^{80} -6.00000 q^{82} -13.6458 q^{83} -5.41699 q^{85} -0.354249 q^{86} +1.64575 q^{88} +10.5830 q^{91} +1.00000 q^{92} -6.00000 q^{94} +0.583005 q^{95} -1.29150 q^{97} +3.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 4 q^{7} - 2 q^{8} + 2 q^{10} + 2 q^{11} - 4 q^{14} + 2 q^{16} + 4 q^{17} + 6 q^{19} - 2 q^{20} - 2 q^{22} + 2 q^{23} + 6 q^{25} + 4 q^{28} + 8 q^{29} + 8 q^{31} - 2 q^{32}+ \cdots + 6 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + 4 * q^7 - 2 * q^8 + 2 * q^10 + 2 * q^11 - 4 * q^14 + 2 * q^16 + 4 * q^17 + 6 * q^19 - 2 * q^20 - 2 * q^22 + 2 * q^23 + 6 * q^25 + 4 * q^28 + 8 * q^29 + 8 * q^31 - 2 * q^32 - 4 * q^34 - 4 * q^35 - 2 * q^37 - 6 * q^38 + 2 * q^40 + 12 * q^41 + 6 * q^43 + 2 * q^44 - 2 * q^46 + 12 * q^47 - 6 * q^49 - 6 * q^50 + 2 * q^53 - 16 * q^55 - 4 * q^56 - 8 * q^58 + 6 * q^61 - 8 * q^62 + 2 * q^64 + 28 * q^65 - 14 * q^67 + 4 * q^68 + 4 * q^70 - 12 * q^71 - 4 * q^73 + 2 * q^74 + 6 * q^76 + 4 * q^77 - 4 * q^79 - 2 * q^80 - 12 * q^82 - 22 * q^83 - 32 * q^85 - 6 * q^86 - 2 * q^88 + 2 * q^92 - 12 * q^94 - 20 * q^95 + 8 * q^97 + 6 * q^98

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
$5$	1.64575	0.736002	0.368001	−	0.929825i	$-0.380042\pi$
$5$	1.64575	0.736002	0.368001	+	0.929825i	$0.380042\pi$
$6$	0	0
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−1.64575	−0.520432
$11$	−1.64575	−0.496213	−0.248106	−	0.968733i	$-0.579808\pi$
$11$	−1.64575	−0.496213	−0.248106	+	0.968733i	$0.579808\pi$
$12$	0	0
$13$	5.29150	1.46760	0.733799	−	0.679366i	$-0.237745\pi$
$13$	5.29150	1.46760	0.733799	+	0.679366i	$0.237745\pi$
$14$	−2.00000	−0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.29150	−0.798307	−0.399153	−	0.916884i	$-0.630696\pi$
$17$	−3.29150	−0.798307	−0.399153	+	0.916884i	$0.630696\pi$
$18$	0	0
$19$	0.354249	0.0812702	0.0406351	−	0.999174i	$-0.487062\pi$
$19$	0.354249	0.0812702	0.0406351	+	0.999174i	$0.487062\pi$
$20$	1.64575	0.368001
$21$	0	0
$22$	1.64575	0.350875
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−2.29150	−0.458301
$26$	−5.29150	−1.03775
$27$	0	0
$28$	2.00000	0.377964
$29$	9.29150	1.72539	0.862694	−	0.505726i	$-0.168775\pi$
$29$	9.29150	1.72539	0.862694	+	0.505726i	$0.168775\pi$
$30$	0	0
$31$	−1.29150	−0.231961	−0.115980	−	0.993252i	$-0.537001\pi$
$31$	−1.29150	−0.231961	−0.115980	+	0.993252i	$0.537001\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	3.29150	0.564488
$35$	3.29150	0.556365
$36$	0	0
$37$	6.93725	1.14048	0.570239	−	0.821479i	$-0.306851\pi$
$37$	6.93725	1.14048	0.570239	+	0.821479i	$0.306851\pi$
$38$	−0.354249	−0.0574667
$39$	0	0
$40$	−1.64575	−0.260216
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	0.354249	0.0540224	0.0270112	−	0.999635i	$-0.491401\pi$
$43$	0.354249	0.0540224	0.0270112	+	0.999635i	$0.491401\pi$
$44$	−1.64575	−0.248106
$45$	0	0
$46$	−1.00000	−0.147442
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	2.29150	0.324067
$51$	0	0
$52$	5.29150	0.733799
$53$	−1.64575	−0.226061	−0.113031	−	0.993592i	$-0.536056\pi$
$53$	−1.64575	−0.226061	−0.113031	+	0.993592i	$0.536056\pi$
$54$	0	0
$55$	−2.70850	−0.365214
$56$	−2.00000	−0.267261
$57$	0	0
$58$	−9.29150	−1.22003
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	0.354249	0.0453569	0.0226784	−	0.999743i	$-0.492781\pi$
$61$	0.354249	0.0453569	0.0226784	+	0.999743i	$0.492781\pi$
$62$	1.29150	0.164021
$63$	0	0
$64$	1.00000	0.125000
$65$	8.70850	1.08016
$66$	0	0
$67$	−14.9373	−1.82488	−0.912438	−	0.409215i	$-0.865803\pi$
$67$	−14.9373	−1.82488	−0.912438	+	0.409215i	$0.865803\pi$
$68$	−3.29150	−0.399153
$69$	0	0
$70$	−3.29150	−0.393410
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	0	0
$73$	−7.29150	−0.853406	−0.426703	−	0.904392i	$-0.640325\pi$
$73$	−7.29150	−0.853406	−0.426703	+	0.904392i	$0.640325\pi$
$74$	−6.93725	−0.806439
$75$	0	0
$76$	0.354249	0.0406351
$77$	−3.29150	−0.375102
$78$	0	0
$79$	8.58301	0.965664	0.482832	−	0.875713i	$-0.339608\pi$
$79$	8.58301	0.965664	0.482832	+	0.875713i	$0.339608\pi$
$80$	1.64575	0.184001
$81$	0	0
$82$	−6.00000	−0.662589
$83$	−13.6458	−1.49782	−0.748908	−	0.662674i	$-0.769421\pi$
$83$	−13.6458	−1.49782	−0.748908	+	0.662674i	$0.769421\pi$
$84$	0	0
$85$	−5.41699	−0.587556
$86$	−0.354249	−0.0381996
$87$	0	0
$88$	1.64575	0.175438
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	10.5830	1.10940
$92$	1.00000	0.104257
$93$	0	0
$94$	−6.00000	−0.618853
$95$	0.583005	0.0598151
$96$	0	0
$97$	−1.29150	−0.131132	−0.0655661	−	0.997848i	$-0.520885\pi$
$97$	−1.29150	−0.131132	−0.0655661	+	0.997848i	$0.520885\pi$
$98$	3.00000	0.303046
$99$	0	0
$100$	−2.29150	−0.229150
$101$	9.29150	0.924539	0.462270	−	0.886739i	$-0.347035\pi$
$101$	9.29150	0.924539	0.462270	+	0.886739i	$0.347035\pi$
$102$	0	0
$103$	−6.70850	−0.661008	−0.330504	−	0.943805i	$-0.607219\pi$
$103$	−6.70850	−0.661008	−0.330504	+	0.943805i	$0.607219\pi$
$104$	−5.29150	−0.518875
$105$	0	0
$106$	1.64575	0.159849
$107$	−13.6458	−1.31918	−0.659592	−	0.751624i	$-0.729271\pi$
$107$	−13.6458	−1.31918	−0.659592	+	0.751624i	$0.729271\pi$
$108$	0	0
$109$	−18.2288	−1.74600	−0.872999	−	0.487722i	$-0.837828\pi$
$109$	−18.2288	−1.74600	−0.872999	+	0.487722i	$0.837828\pi$
$110$	2.70850	0.258245
$111$	0	0
$112$	2.00000	0.188982
$113$	6.58301	0.619277	0.309639	−	0.950854i	$-0.399792\pi$
$113$	6.58301	0.619277	0.309639	+	0.950854i	$0.399792\pi$
$114$	0	0
$115$	1.64575	0.153467
$116$	9.29150	0.862694
$117$	0	0
$118$	0	0
$119$	−6.58301	−0.603463
$120$	0	0
$121$	−8.29150	−0.753773
$122$	−0.354249	−0.0320722
$123$	0	0
$124$	−1.29150	−0.115980
$125$	−12.0000	−1.07331
$126$	0	0
$127$	−10.5830	−0.939090	−0.469545	−	0.882909i	$-0.655582\pi$
$127$	−10.5830	−0.939090	−0.469545	+	0.882909i	$0.655582\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−8.70850	−0.763786
$131$	−21.8745	−1.91118	−0.955592	−	0.294692i	$-0.904783\pi$
$131$	−21.8745	−1.91118	−0.955592	+	0.294692i	$0.904783\pi$
$132$	0	0
$133$	0.708497	0.0614345
$134$	14.9373	1.29038
$135$	0	0
$136$	3.29150	0.282244
$137$	21.8745	1.86887	0.934433	−	0.356140i	$-0.115907\pi$
$137$	21.8745	1.86887	0.934433	+	0.356140i	$0.115907\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	3.29150	0.278183
$141$	0	0
$142$	6.00000	0.503509
$143$	−8.70850	−0.728241
$144$	0	0
$145$	15.2915	1.26989
$146$	7.29150	0.603449
$147$	0	0
$148$	6.93725	0.570239
$149$	10.3542	0.848253	0.424127	−	0.905603i	$-0.360581\pi$
$149$	10.3542	0.848253	0.424127	+	0.905603i	$0.360581\pi$
$150$	0	0
$151$	17.2915	1.40716	0.703581	−	0.710615i	$-0.251583\pi$
$151$	17.2915	1.40716	0.703581	+	0.710615i	$0.251583\pi$
$152$	−0.354249	−0.0287334
$153$	0	0
$154$	3.29150	0.265237
$155$	−2.12549	−0.170724
$156$	0	0
$157$	3.64575	0.290963	0.145481	−	0.989361i	$-0.453527\pi$
$157$	3.64575	0.290963	0.145481	+	0.989361i	$0.453527\pi$
$158$	−8.58301	−0.682827
$159$	0	0
$160$	−1.64575	−0.130108
$161$	2.00000	0.157622
$162$	0	0
$163$	−0.708497	−0.0554938	−0.0277469	−	0.999615i	$-0.508833\pi$
$163$	−0.708497	−0.0554938	−0.0277469	+	0.999615i	$0.508833\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	13.6458	1.05912
$167$	−19.1660	−1.48311	−0.741555	−	0.670892i	$-0.765911\pi$
$167$	−19.1660	−1.48311	−0.741555	+	0.670892i	$0.765911\pi$
$168$	0	0
$169$	15.0000	1.15385
$170$	5.41699	0.415465
$171$	0	0
$172$	0.354249	0.0270112
$173$	−15.8745	−1.20692	−0.603458	−	0.797395i	$-0.706211\pi$
$173$	−15.8745	−1.20692	−0.603458	+	0.797395i	$0.706211\pi$
$174$	0	0
$175$	−4.58301	−0.346443
$176$	−1.64575	−0.124053
$177$	0	0
$178$	0	0
$179$	15.2915	1.14294	0.571470	−	0.820623i	$-0.306373\pi$
$179$	15.2915	1.14294	0.571470	+	0.820623i	$0.306373\pi$
$180$	0	0
$181$	15.6458	1.16294	0.581470	−	0.813568i	$-0.302478\pi$
$181$	15.6458	1.16294	0.581470	+	0.813568i	$0.302478\pi$
$182$	−10.5830	−0.784465
$183$	0	0
$184$	−1.00000	−0.0737210
$185$	11.4170	0.839394
$186$	0	0
$187$	5.41699	0.396130
$188$	6.00000	0.437595
$189$	0	0
$190$	−0.583005	−0.0422956
$191$	3.29150	0.238165	0.119082	−	0.992884i	$-0.462005\pi$
$191$	3.29150	0.238165	0.119082	+	0.992884i	$0.462005\pi$
$192$	0	0
$193$	−22.0000	−1.58359	−0.791797	−	0.610784i	$-0.790854\pi$
$193$	−22.0000	−1.58359	−0.791797	+	0.610784i	$0.790854\pi$
$194$	1.29150	0.0927245
$195$	0	0
$196$	−3.00000	−0.214286
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	0	0
$199$	14.0000	0.992434	0.496217	−	0.868199i	$-0.334722\pi$
$199$	14.0000	0.992434	0.496217	+	0.868199i	$0.334722\pi$
$200$	2.29150	0.162034
$201$	0	0
$202$	−9.29150	−0.653748
$203$	18.5830	1.30427
$204$	0	0
$205$	9.87451	0.689666
$206$	6.70850	0.467403
$207$	0	0
$208$	5.29150	0.366900
$209$	−0.583005	−0.0403273
$210$	0	0
$211$	11.2915	0.777339	0.388670	−	0.921377i	$-0.372935\pi$
$211$	11.2915	0.777339	0.388670	+	0.921377i	$0.372935\pi$
$212$	−1.64575	−0.113031
$213$	0	0
$214$	13.6458	0.932804
$215$	0.583005	0.0397606
$216$	0	0
$217$	−2.58301	−0.175346
$218$	18.2288	1.23461
$219$	0	0
$220$	−2.70850	−0.182607
$221$	−17.4170	−1.17159
$222$	0	0
$223$	−19.8745	−1.33090	−0.665448	−	0.746444i	$-0.731759\pi$
$223$	−19.8745	−1.33090	−0.665448	+	0.746444i	$0.731759\pi$
$224$	−2.00000	−0.133631
$225$	0	0
$226$	−6.58301	−0.437895
$227$	−7.06275	−0.468771	−0.234385	−	0.972144i	$-0.575308\pi$
$227$	−7.06275	−0.468771	−0.234385	+	0.972144i	$0.575308\pi$
$228$	0	0
$229$	9.06275	0.598883	0.299442	−	0.954115i	$-0.403200\pi$
$229$	9.06275	0.598883	0.299442	+	0.954115i	$0.403200\pi$
$230$	−1.64575	−0.108518
$231$	0	0
$232$	−9.29150	−0.610017
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	9.87451	0.644142
$236$	0	0
$237$	0	0
$238$	6.58301	0.426713
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	0	0
$241$	5.29150	0.340856	0.170428	−	0.985370i	$-0.445485\pi$
$241$	5.29150	0.340856	0.170428	+	0.985370i	$0.445485\pi$
$242$	8.29150	0.532998
$243$	0	0
$244$	0.354249	0.0226784
$245$	−4.93725	−0.315430
$246$	0	0
$247$	1.87451	0.119272
$248$	1.29150	0.0820105
$249$	0	0
$250$	12.0000	0.758947
$251$	−25.6458	−1.61875	−0.809373	−	0.587295i	$-0.800193\pi$
$251$	−25.6458	−1.61875	−0.809373	+	0.587295i	$0.800193\pi$
$252$	0	0
$253$	−1.64575	−0.103467
$254$	10.5830	0.664037
$255$	0	0
$256$	1.00000	0.0625000
$257$	24.5830	1.53345	0.766723	−	0.641978i	$-0.221886\pi$
$257$	24.5830	1.53345	0.766723	+	0.641978i	$0.221886\pi$
$258$	0	0
$259$	13.8745	0.862120
$260$	8.70850	0.540078
$261$	0	0
$262$	21.8745	1.35141
$263$	−30.5830	−1.88583	−0.942914	−	0.333035i	$-0.891927\pi$
$263$	−30.5830	−1.88583	−0.942914	+	0.333035i	$0.891927\pi$
$264$	0	0
$265$	−2.70850	−0.166382
$266$	−0.708497	−0.0434408
$267$	0	0
$268$	−14.9373	−0.912438
$269$	−11.4170	−0.696106	−0.348053	−	0.937475i	$-0.613157\pi$
$269$	−11.4170	−0.696106	−0.348053	+	0.937475i	$0.613157\pi$
$270$	0	0
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	−3.29150	−0.199577
$273$	0	0
$274$	−21.8745	−1.32149
$275$	3.77124	0.227415
$276$	0	0
$277$	5.29150	0.317936	0.158968	−	0.987284i	$-0.449183\pi$
$277$	5.29150	0.317936	0.158968	+	0.987284i	$0.449183\pi$
$278$	4.00000	0.239904
$279$	0	0
$280$	−3.29150	−0.196705
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	−21.5203	−1.27925	−0.639623	−	0.768689i	$-0.720910\pi$
$283$	−21.5203	−1.27925	−0.639623	+	0.768689i	$0.720910\pi$
$284$	−6.00000	−0.356034
$285$	0	0
$286$	8.70850	0.514944
$287$	12.0000	0.708338
$288$	0	0
$289$	−6.16601	−0.362706
$290$	−15.2915	−0.897948
$291$	0	0
$292$	−7.29150	−0.426703
$293$	−32.2288	−1.88282	−0.941412	−	0.337259i	$-0.890500\pi$
$293$	−32.2288	−1.88282	−0.941412	+	0.337259i	$0.890500\pi$
$294$	0	0
$295$	0	0
$296$	−6.93725	−0.403220
$297$	0	0
$298$	−10.3542	−0.599806
$299$	5.29150	0.306015
$300$	0	0
$301$	0.708497	0.0408371
$302$	−17.2915	−0.995014
$303$	0	0
$304$	0.354249	0.0203176
$305$	0.583005	0.0333828
$306$	0	0
$307$	29.8745	1.70503	0.852514	−	0.522704i	$-0.175077\pi$
$307$	29.8745	1.70503	0.852514	+	0.522704i	$0.175077\pi$
$308$	−3.29150	−0.187551
$309$	0	0
$310$	2.12549	0.120720
$311$	13.1660	0.746576	0.373288	−	0.927716i	$-0.378230\pi$
$311$	13.1660	0.746576	0.373288	+	0.927716i	$0.378230\pi$
$312$	0	0
$313$	5.29150	0.299093	0.149547	−	0.988755i	$-0.452219\pi$
$313$	5.29150	0.299093	0.149547	+	0.988755i	$0.452219\pi$
$314$	−3.64575	−0.205742
$315$	0	0
$316$	8.58301	0.482832
$317$	21.2915	1.19585	0.597925	−	0.801552i	$-0.295992\pi$
$317$	21.2915	1.19585	0.597925	+	0.801552i	$0.295992\pi$
$318$	0	0
$319$	−15.2915	−0.856160
$320$	1.64575	0.0920003
$321$	0	0
$322$	−2.00000	−0.111456
$323$	−1.16601	−0.0648786
$324$	0	0
$325$	−12.1255	−0.672601
$326$	0.708497	0.0392400
$327$	0	0
$328$	−6.00000	−0.331295
$329$	12.0000	0.661581
$330$	0	0
$331$	−25.8745	−1.42219	−0.711096	−	0.703095i	$-0.751801\pi$
$331$	−25.8745	−1.42219	−0.711096	+	0.703095i	$0.751801\pi$
$332$	−13.6458	−0.748908
$333$	0	0
$334$	19.1660	1.04872
$335$	−24.5830	−1.34311
$336$	0	0
$337$	15.1660	0.826145	0.413073	−	0.910698i	$-0.364456\pi$
$337$	15.1660	0.826145	0.413073	+	0.910698i	$0.364456\pi$
$338$	−15.0000	−0.815892
$339$	0	0
$340$	−5.41699	−0.293778
$341$	2.12549	0.115102
$342$	0	0
$343$	−20.0000	−1.07990
$344$	−0.354249	−0.0190998
$345$	0	0
$346$	15.8745	0.853419
$347$	15.2915	0.820891	0.410445	−	0.911885i	$-0.365373\pi$
$347$	15.2915	0.820891	0.410445	+	0.911885i	$0.365373\pi$
$348$	0	0
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	4.58301	0.244972
$351$	0	0
$352$	1.64575	0.0877188
$353$	−6.58301	−0.350378	−0.175189	−	0.984535i	$-0.556054\pi$
$353$	−6.58301	−0.350378	−0.175189	+	0.984535i	$0.556054\pi$
$354$	0	0
$355$	−9.87451	−0.524084
$356$	0	0
$357$	0	0
$358$	−15.2915	−0.808181
$359$	−21.8745	−1.15449	−0.577246	−	0.816570i	$-0.695873\pi$
$359$	−21.8745	−1.15449	−0.577246	+	0.816570i	$0.695873\pi$
$360$	0	0
$361$	−18.8745	−0.993395
$362$	−15.6458	−0.822322
$363$	0	0
$364$	10.5830	0.554700
$365$	−12.0000	−0.628109
$366$	0	0
$367$	−11.1660	−0.582861	−0.291431	−	0.956592i	$-0.594131\pi$
$367$	−11.1660	−0.582861	−0.291431	+	0.956592i	$0.594131\pi$
$368$	1.00000	0.0521286
$369$	0	0
$370$	−11.4170	−0.593541
$371$	−3.29150	−0.170886
$372$	0	0
$373$	−26.9373	−1.39476	−0.697379	−	0.716702i	$-0.745651\pi$
$373$	−26.9373	−1.39476	−0.697379	+	0.716702i	$0.745651\pi$
$374$	−5.41699	−0.280106
$375$	0	0
$376$	−6.00000	−0.309426
$377$	49.1660	2.53218
$378$	0	0
$379$	−33.5203	−1.72182	−0.860910	−	0.508757i	$-0.830105\pi$
$379$	−33.5203	−1.72182	−0.860910	+	0.508757i	$0.830105\pi$
$380$	0.583005	0.0299075
$381$	0	0
$382$	−3.29150	−0.168408
$383$	31.7490	1.62230	0.811149	−	0.584839i	$-0.198842\pi$
$383$	31.7490	1.62230	0.811149	+	0.584839i	$0.198842\pi$
$384$	0	0
$385$	−5.41699	−0.276076
$386$	22.0000	1.11977
$387$	0	0
$388$	−1.29150	−0.0655661
$389$	−7.06275	−0.358095	−0.179048	−	0.983840i	$-0.557302\pi$
$389$	−7.06275	−0.358095	−0.179048	+	0.983840i	$0.557302\pi$
$390$	0	0
$391$	−3.29150	−0.166458
$392$	3.00000	0.151523
$393$	0	0
$394$	6.00000	0.302276
$395$	14.1255	0.710731
$396$	0	0
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	−14.0000	−0.701757
$399$	0	0
$400$	−2.29150	−0.114575
$401$	9.87451	0.493109	0.246555	−	0.969129i	$-0.420701\pi$
$401$	9.87451	0.493109	0.246555	+	0.969129i	$0.420701\pi$
$402$	0	0
$403$	−6.83399	−0.340425
$404$	9.29150	0.462270
$405$	0	0
$406$	−18.5830	−0.922259
$407$	−11.4170	−0.565919
$408$	0	0
$409$	15.1660	0.749911	0.374955	−	0.927043i	$-0.377658\pi$
$409$	15.1660	0.749911	0.374955	+	0.927043i	$0.377658\pi$
$410$	−9.87451	−0.487667
$411$	0	0
$412$	−6.70850	−0.330504
$413$	0	0
$414$	0	0
$415$	−22.4575	−1.10240
$416$	−5.29150	−0.259437
$417$	0	0
$418$	0.583005	0.0285157
$419$	16.9373	0.827439	0.413720	−	0.910404i	$-0.364229\pi$
$419$	16.9373	0.827439	0.413720	+	0.910404i	$0.364229\pi$
$420$	0	0
$421$	22.2288	1.08336	0.541682	−	0.840584i	$-0.317788\pi$
$421$	22.2288	1.08336	0.541682	+	0.840584i	$0.317788\pi$
$422$	−11.2915	−0.549662
$423$	0	0
$424$	1.64575	0.0799247
$425$	7.54249	0.365864
$426$	0	0
$427$	0.708497	0.0342866
$428$	−13.6458	−0.659592
$429$	0	0
$430$	−0.583005	−0.0281150
$431$	−33.8745	−1.63168	−0.815839	−	0.578279i	$-0.803724\pi$
$431$	−33.8745	−1.63168	−0.815839	+	0.578279i	$0.803724\pi$
$432$	0	0
$433$	−18.7085	−0.899073	−0.449537	−	0.893262i	$-0.648411\pi$
$433$	−18.7085	−0.899073	−0.449537	+	0.893262i	$0.648411\pi$
$434$	2.58301	0.123988
$435$	0	0
$436$	−18.2288	−0.872999
$437$	0.354249	0.0169460
$438$	0	0
$439$	27.7490	1.32439	0.662194	−	0.749332i	$-0.269625\pi$
$439$	27.7490	1.32439	0.662194	+	0.749332i	$0.269625\pi$
$440$	2.70850	0.129123
$441$	0	0
$442$	17.4170	0.828442
$443$	25.1660	1.19567	0.597837	−	0.801618i	$-0.296027\pi$
$443$	25.1660	1.19567	0.597837	+	0.801618i	$0.296027\pi$
$444$	0	0
$445$	0	0
$446$	19.8745	0.941085
$447$	0	0
$448$	2.00000	0.0944911
$449$	11.4170	0.538801	0.269401	−	0.963028i	$-0.413174\pi$
$449$	11.4170	0.538801	0.269401	+	0.963028i	$0.413174\pi$
$450$	0	0
$451$	−9.87451	−0.464972
$452$	6.58301	0.309639
$453$	0	0
$454$	7.06275	0.331471
$455$	17.4170	0.816521
$456$	0	0
$457$	−0.125492	−0.00587027	−0.00293514	−	0.999996i	$-0.500934\pi$
$457$	−0.125492	−0.00587027	−0.00293514	+	0.999996i	$0.500934\pi$
$458$	−9.06275	−0.423474
$459$	0	0
$460$	1.64575	0.0767336
$461$	25.7490	1.19925	0.599626	−	0.800281i	$-0.295316\pi$
$461$	25.7490	1.19925	0.599626	+	0.800281i	$0.295316\pi$
$462$	0	0
$463$	21.1660	0.983668	0.491834	−	0.870689i	$-0.336327\pi$
$463$	21.1660	0.983668	0.491834	+	0.870689i	$0.336327\pi$
$464$	9.29150	0.431347
$465$	0	0
$466$	0	0
$467$	13.6458	0.631450	0.315725	−	0.948851i	$-0.397752\pi$
$467$	13.6458	0.631450	0.315725	+	0.948851i	$0.397752\pi$
$468$	0	0
$469$	−29.8745	−1.37948
$470$	−9.87451	−0.455477
$471$	0	0
$472$	0	0
$473$	−0.583005	−0.0268066
$474$	0	0
$475$	−0.811762	−0.0372462
$476$	−6.58301	−0.301732
$477$	0	0
$478$	−24.0000	−1.09773
$479$	42.5830	1.94567	0.972834	−	0.231506i	$-0.0743651\pi$
$479$	42.5830	1.94567	0.972834	+	0.231506i	$0.0743651\pi$
$480$	0	0
$481$	36.7085	1.67376
$482$	−5.29150	−0.241021
$483$	0	0
$484$	−8.29150	−0.376886
$485$	−2.12549	−0.0965136
$486$	0	0
$487$	21.1660	0.959123	0.479562	−	0.877508i	$-0.340796\pi$
$487$	21.1660	0.959123	0.479562	+	0.877508i	$0.340796\pi$
$488$	−0.354249	−0.0160361
$489$	0	0
$490$	4.93725	0.223042
$491$	−21.8745	−0.987183	−0.493591	−	0.869694i	$-0.664316\pi$
$491$	−21.8745	−0.987183	−0.493591	+	0.869694i	$0.664316\pi$
$492$	0	0
$493$	−30.5830	−1.37739
$494$	−1.87451	−0.0843381
$495$	0	0
$496$	−1.29150	−0.0579902
$497$	−12.0000	−0.538274
$498$	0	0
$499$	31.0405	1.38956	0.694782	−	0.719220i	$-0.255501\pi$
$499$	31.0405	1.38956	0.694782	+	0.719220i	$0.255501\pi$
$500$	−12.0000	−0.536656
$501$	0	0
$502$	25.6458	1.14463
$503$	−2.12549	−0.0947710	−0.0473855	−	0.998877i	$-0.515089\pi$
$503$	−2.12549	−0.0947710	−0.0473855	+	0.998877i	$0.515089\pi$
$504$	0	0
$505$	15.2915	0.680463
$506$	1.64575	0.0731626
$507$	0	0
$508$	−10.5830	−0.469545
$509$	−27.8745	−1.23552	−0.617758	−	0.786368i	$-0.711959\pi$
$509$	−27.8745	−1.23552	−0.617758	+	0.786368i	$0.711959\pi$
$510$	0	0
$511$	−14.5830	−0.645114
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−24.5830	−1.08431
$515$	−11.0405	−0.486503
$516$	0	0
$517$	−9.87451	−0.434280
$518$	−13.8745	−0.609611
$519$	0	0
$520$	−8.70850	−0.381893
$521$	−28.4575	−1.24675	−0.623373	−	0.781925i	$-0.714238\pi$
$521$	−28.4575	−1.24675	−0.623373	+	0.781925i	$0.714238\pi$
$522$	0	0
$523$	−5.06275	−0.221378	−0.110689	−	0.993855i	$-0.535306\pi$
$523$	−5.06275	−0.221378	−0.110689	+	0.993855i	$0.535306\pi$
$524$	−21.8745	−0.955592
$525$	0	0
$526$	30.5830	1.33348
$527$	4.25098	0.185176
$528$	0	0
$529$	1.00000	0.0434783
$530$	2.70850	0.117650
$531$	0	0
$532$	0.708497	0.0307173
$533$	31.7490	1.37520
$534$	0	0
$535$	−22.4575	−0.970923
$536$	14.9373	0.645191
$537$	0	0
$538$	11.4170	0.492222
$539$	4.93725	0.212663
$540$	0	0
$541$	20.5830	0.884933	0.442466	−	0.896785i	$-0.354104\pi$
$541$	20.5830	0.884933	0.442466	+	0.896785i	$0.354104\pi$
$542$	−20.0000	−0.859074
$543$	0	0
$544$	3.29150	0.141122
$545$	−30.0000	−1.28506
$546$	0	0
$547$	16.7085	0.714404	0.357202	−	0.934027i	$-0.383731\pi$
$547$	16.7085	0.714404	0.357202	+	0.934027i	$0.383731\pi$
$548$	21.8745	0.934433
$549$	0	0
$550$	−3.77124	−0.160806
$551$	3.29150	0.140223
$552$	0	0
$553$	17.1660	0.729973
$554$	−5.29150	−0.224814
$555$	0	0
$556$	−4.00000	−0.169638
$557$	−42.1033	−1.78397	−0.891986	−	0.452062i	$-0.850688\pi$
$557$	−42.1033	−1.78397	−0.891986	+	0.452062i	$0.850688\pi$
$558$	0	0
$559$	1.87451	0.0792832
$560$	3.29150	0.139091
$561$	0	0
$562$	0	0
$563$	45.3948	1.91316	0.956581	−	0.291468i	$-0.0941436\pi$
$563$	45.3948	1.91316	0.956581	+	0.291468i	$0.0941436\pi$
$564$	0	0
$565$	10.8340	0.455789
$566$	21.5203	0.904564
$567$	0	0
$568$	6.00000	0.251754
$569$	−8.70850	−0.365079	−0.182540	−	0.983199i	$-0.558432\pi$
$569$	−8.70850	−0.365079	−0.182540	+	0.983199i	$0.558432\pi$
$570$	0	0
$571$	18.9373	0.792499	0.396250	−	0.918143i	$-0.370311\pi$
$571$	18.9373	0.792499	0.396250	+	0.918143i	$0.370311\pi$
$572$	−8.70850	−0.364121
$573$	0	0
$574$	−12.0000	−0.500870
$575$	−2.29150	−0.0955623
$576$	0	0
$577$	28.7085	1.19515	0.597575	−	0.801813i	$-0.296131\pi$
$577$	28.7085	1.19515	0.597575	+	0.801813i	$0.296131\pi$
$578$	6.16601	0.256472
$579$	0	0
$580$	15.2915	0.634945
$581$	−27.2915	−1.13224
$582$	0	0
$583$	2.70850	0.112174
$584$	7.29150	0.301725
$585$	0	0
$586$	32.2288	1.33136
$587$	−20.7085	−0.854731	−0.427366	−	0.904079i	$-0.640558\pi$
$587$	−20.7085	−0.854731	−0.427366	+	0.904079i	$0.640558\pi$
$588$	0	0
$589$	−0.457513	−0.0188515
$590$	0	0
$591$	0	0
$592$	6.93725	0.285119
$593$	30.0000	1.23195	0.615976	−	0.787765i	$-0.288762\pi$
$593$	30.0000	1.23195	0.615976	+	0.787765i	$0.288762\pi$
$594$	0	0
$595$	−10.8340	−0.444150
$596$	10.3542	0.424127
$597$	0	0
$598$	−5.29150	−0.216386
$599$	30.5830	1.24959	0.624794	−	0.780790i	$-0.285183\pi$
$599$	30.5830	1.24959	0.624794	+	0.780790i	$0.285183\pi$
$600$	0	0
$601$	−32.4575	−1.32397	−0.661985	−	0.749517i	$-0.730286\pi$
$601$	−32.4575	−1.32397	−0.661985	+	0.749517i	$0.730286\pi$
$602$	−0.708497	−0.0288762
$603$	0	0
$604$	17.2915	0.703581
$605$	−13.6458	−0.554779
$606$	0	0
$607$	35.8745	1.45610	0.728051	−	0.685523i	$-0.240427\pi$
$607$	35.8745	1.45610	0.728051	+	0.685523i	$0.240427\pi$
$608$	−0.354249	−0.0143667
$609$	0	0
$610$	−0.583005	−0.0236052
$611$	31.7490	1.28443
$612$	0	0
$613$	−2.93725	−0.118635	−0.0593173	−	0.998239i	$-0.518892\pi$
$613$	−2.93725	−0.118635	−0.0593173	+	0.998239i	$0.518892\pi$
$614$	−29.8745	−1.20564
$615$	0	0
$616$	3.29150	0.132618
$617$	12.0000	0.483102	0.241551	−	0.970388i	$-0.422344\pi$
$617$	12.0000	0.483102	0.241551	+	0.970388i	$0.422344\pi$
$618$	0	0
$619$	12.3542	0.496559	0.248280	−	0.968688i	$-0.420135\pi$
$619$	12.3542	0.496559	0.248280	+	0.968688i	$0.420135\pi$
$620$	−2.12549	−0.0853618
$621$	0	0
$622$	−13.1660	−0.527909
$623$	0	0
$624$	0	0
$625$	−8.29150	−0.331660
$626$	−5.29150	−0.211491
$627$	0	0
$628$	3.64575	0.145481
$629$	−22.8340	−0.910451
$630$	0	0
$631$	23.8745	0.950429	0.475215	−	0.879870i	$-0.342370\pi$
$631$	23.8745	0.950429	0.475215	+	0.879870i	$0.342370\pi$
$632$	−8.58301	−0.341414
$633$	0	0
$634$	−21.2915	−0.845594
$635$	−17.4170	−0.691172
$636$	0	0
$637$	−15.8745	−0.628971
$638$	15.2915	0.605396
$639$	0	0
$640$	−1.64575	−0.0650540
$641$	−28.4575	−1.12400	−0.562002	−	0.827136i	$-0.689969\pi$
$641$	−28.4575	−1.12400	−0.562002	+	0.827136i	$0.689969\pi$
$642$	0	0
$643$	24.3542	0.960438	0.480219	−	0.877149i	$-0.340557\pi$
$643$	24.3542	0.960438	0.480219	+	0.877149i	$0.340557\pi$
$644$	2.00000	0.0788110
$645$	0	0
$646$	1.16601	0.0458761
$647$	13.1660	0.517609	0.258805	−	0.965930i	$-0.416671\pi$
$647$	13.1660	0.517609	0.258805	+	0.965930i	$0.416671\pi$
$648$	0	0
$649$	0	0
$650$	12.1255	0.475601
$651$	0	0
$652$	−0.708497	−0.0277469
$653$	−15.8745	−0.621217	−0.310609	−	0.950538i	$-0.600533\pi$
$653$	−15.8745	−0.621217	−0.310609	+	0.950538i	$0.600533\pi$
$654$	0	0
$655$	−36.0000	−1.40664
$656$	6.00000	0.234261
$657$	0	0
$658$	−12.0000	−0.467809
$659$	7.06275	0.275126	0.137563	−	0.990493i	$-0.456073\pi$
$659$	7.06275	0.275126	0.137563	+	0.990493i	$0.456073\pi$
$660$	0	0
$661$	27.6458	1.07530	0.537648	−	0.843170i	$-0.319313\pi$
$661$	27.6458	1.07530	0.537648	+	0.843170i	$0.319313\pi$
$662$	25.8745	1.00564
$663$	0	0
$664$	13.6458	0.529558
$665$	1.16601	0.0452159
$666$	0	0
$667$	9.29150	0.359768
$668$	−19.1660	−0.741555
$669$	0	0
$670$	24.5830	0.949724
$671$	−0.583005	−0.0225067
$672$	0	0
$673$	−12.7085	−0.489877	−0.244938	−	0.969539i	$-0.578768\pi$
$673$	−12.7085	−0.489877	−0.244938	+	0.969539i	$0.578768\pi$
$674$	−15.1660	−0.584173
$675$	0	0
$676$	15.0000	0.576923
$677$	−7.06275	−0.271443	−0.135722	−	0.990747i	$-0.543335\pi$
$677$	−7.06275	−0.271443	−0.135722	+	0.990747i	$0.543335\pi$
$678$	0	0
$679$	−2.58301	−0.0991266
$680$	5.41699	0.207732
$681$	0	0
$682$	−2.12549	−0.0813893
$683$	−33.8745	−1.29617	−0.648086	−	0.761567i	$-0.724430\pi$
$683$	−33.8745	−1.29617	−0.648086	+	0.761567i	$0.724430\pi$
$684$	0	0
$685$	36.0000	1.37549
$686$	20.0000	0.763604
$687$	0	0
$688$	0.354249	0.0135056
$689$	−8.70850	−0.331767
$690$	0	0
$691$	17.8745	0.679978	0.339989	−	0.940429i	$-0.389577\pi$
$691$	17.8745	0.679978	0.339989	+	0.940429i	$0.389577\pi$
$692$	−15.8745	−0.603458
$693$	0	0
$694$	−15.2915	−0.580458
$695$	−6.58301	−0.249708
$696$	0	0
$697$	−19.7490	−0.748047
$698$	−2.00000	−0.0757011
$699$	0	0
$700$	−4.58301	−0.173221
$701$	−19.0627	−0.719990	−0.359995	−	0.932954i	$-0.617222\pi$
$701$	−19.0627	−0.719990	−0.359995	+	0.932954i	$0.617222\pi$
$702$	0	0
$703$	2.45751	0.0926869
$704$	−1.64575	−0.0620266
$705$	0	0
$706$	6.58301	0.247755
$707$	18.5830	0.698886
$708$	0	0
$709$	−38.9373	−1.46232	−0.731160	−	0.682206i	$-0.761021\pi$
$709$	−38.9373	−1.46232	−0.731160	+	0.682206i	$0.761021\pi$
$710$	9.87451	0.370584
$711$	0	0
$712$	0	0
$713$	−1.29150	−0.0483672
$714$	0	0
$715$	−14.3320	−0.535987
$716$	15.2915	0.571470
$717$	0	0
$718$	21.8745	0.816349
$719$	−19.7490	−0.736514	−0.368257	−	0.929724i	$-0.620045\pi$
$719$	−19.7490	−0.736514	−0.368257	+	0.929724i	$0.620045\pi$
$720$	0	0
$721$	−13.4170	−0.499675
$722$	18.8745	0.702436
$723$	0	0
$724$	15.6458	0.581470
$725$	−21.2915	−0.790747
$726$	0	0
$727$	40.3320	1.49583	0.747916	−	0.663793i	$-0.231055\pi$
$727$	40.3320	1.49583	0.747916	+	0.663793i	$0.231055\pi$
$728$	−10.5830	−0.392232
$729$	0	0
$730$	12.0000	0.444140
$731$	−1.16601	−0.0431265
$732$	0	0
$733$	−31.3948	−1.15959	−0.579796	−	0.814762i	$-0.696868\pi$
$733$	−31.3948	−1.15959	−0.579796	+	0.814762i	$0.696868\pi$
$734$	11.1660	0.412145
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	24.5830	0.905527
$738$	0	0
$739$	−10.5830	−0.389302	−0.194651	−	0.980873i	$-0.562357\pi$
$739$	−10.5830	−0.389302	−0.194651	+	0.980873i	$0.562357\pi$
$740$	11.4170	0.419697
$741$	0	0
$742$	3.29150	0.120835
$743$	−19.7490	−0.724521	−0.362261	−	0.932077i	$-0.617995\pi$
$743$	−19.7490	−0.724521	−0.362261	+	0.932077i	$0.617995\pi$
$744$	0	0
$745$	17.0405	0.624316
$746$	26.9373	0.986243
$747$	0	0
$748$	5.41699	0.198065
$749$	−27.2915	−0.997210
$750$	0	0
$751$	42.4575	1.54930	0.774648	−	0.632392i	$-0.217927\pi$
$751$	42.4575	1.54930	0.774648	+	0.632392i	$0.217927\pi$
$752$	6.00000	0.218797
$753$	0	0
$754$	−49.1660	−1.79052
$755$	28.4575	1.03567
$756$	0	0
$757$	5.77124	0.209759	0.104880	−	0.994485i	$-0.466554\pi$
$757$	5.77124	0.209759	0.104880	+	0.994485i	$0.466554\pi$
$758$	33.5203	1.21751
$759$	0	0
$760$	−0.583005	−0.0211478
$761$	−12.0000	−0.435000	−0.217500	−	0.976060i	$-0.569790\pi$
$761$	−12.0000	−0.435000	−0.217500	+	0.976060i	$0.569790\pi$
$762$	0	0
$763$	−36.4575	−1.31985
$764$	3.29150	0.119082
$765$	0	0
$766$	−31.7490	−1.14714
$767$	0	0
$768$	0	0
$769$	25.0405	0.902984	0.451492	−	0.892275i	$-0.350892\pi$
$769$	25.0405	0.902984	0.451492	+	0.892275i	$0.350892\pi$
$770$	5.41699	0.195215
$771$	0	0
$772$	−22.0000	−0.791797
$773$	−1.64575	−0.0591936	−0.0295968	−	0.999562i	$-0.509422\pi$
$773$	−1.64575	−0.0591936	−0.0295968	+	0.999562i	$0.509422\pi$
$774$	0	0
$775$	2.95948	0.106308
$776$	1.29150	0.0463622
$777$	0	0
$778$	7.06275	0.253212
$779$	2.12549	0.0761537
$780$	0	0
$781$	9.87451	0.353338
$782$	3.29150	0.117704
$783$	0	0
$784$	−3.00000	−0.107143
$785$	6.00000	0.214149
$786$	0	0
$787$	−44.3542	−1.58106	−0.790529	−	0.612424i	$-0.790194\pi$
$787$	−44.3542	−1.58106	−0.790529	+	0.612424i	$0.790194\pi$
$788$	−6.00000	−0.213741
$789$	0	0
$790$	−14.1255	−0.502562
$791$	13.1660	0.468129
$792$	0	0
$793$	1.87451	0.0665657
$794$	−2.00000	−0.0709773
$795$	0	0
$796$	14.0000	0.496217
$797$	−35.5203	−1.25819	−0.629096	−	0.777328i	$-0.716575\pi$
$797$	−35.5203	−1.25819	−0.629096	+	0.777328i	$0.716575\pi$
$798$	0	0
$799$	−19.7490	−0.698670
$800$	2.29150	0.0810169
$801$	0	0
$802$	−9.87451	−0.348681
$803$	12.0000	0.423471
$804$	0	0
$805$	3.29150	0.116010
$806$	6.83399	0.240717
$807$	0	0
$808$	−9.29150	−0.326874
$809$	−12.0000	−0.421898	−0.210949	−	0.977497i	$-0.567655\pi$
$809$	−12.0000	−0.421898	−0.210949	+	0.977497i	$0.567655\pi$
$810$	0	0
$811$	29.8745	1.04904	0.524518	−	0.851399i	$-0.324246\pi$
$811$	29.8745	1.04904	0.524518	+	0.851399i	$0.324246\pi$
$812$	18.5830	0.652136
$813$	0	0
$814$	11.4170	0.400165
$815$	−1.16601	−0.0408436
$816$	0	0
$817$	0.125492	0.00439041
$818$	−15.1660	−0.530267
$819$	0	0
$820$	9.87451	0.344833
$821$	−8.12549	−0.283582	−0.141791	−	0.989897i	$-0.545286\pi$
$821$	−8.12549	−0.283582	−0.141791	+	0.989897i	$0.545286\pi$
$822$	0	0
$823$	−13.2915	−0.463313	−0.231656	−	0.972798i	$-0.574414\pi$
$823$	−13.2915	−0.463313	−0.231656	+	0.972798i	$0.574414\pi$
$824$	6.70850	0.233702
$825$	0	0
$826$	0	0
$827$	−14.8118	−0.515055	−0.257528	−	0.966271i	$-0.582908\pi$
$827$	−14.8118	−0.515055	−0.257528	+	0.966271i	$0.582908\pi$
$828$	0	0
$829$	−15.4170	−0.535454	−0.267727	−	0.963495i	$-0.586273\pi$
$829$	−15.4170	−0.535454	−0.267727	+	0.963495i	$0.586273\pi$
$830$	22.4575	0.779512
$831$	0	0
$832$	5.29150	0.183450
$833$	9.87451	0.342131
$834$	0	0
$835$	−31.5425	−1.09157
$836$	−0.583005	−0.0201637
$837$	0	0
$838$	−16.9373	−0.585088
$839$	43.7490	1.51038	0.755192	−	0.655504i	$-0.227544\pi$
$839$	43.7490	1.51038	0.755192	+	0.655504i	$0.227544\pi$
$840$	0	0
$841$	57.3320	1.97697
$842$	−22.2288	−0.766054
$843$	0	0
$844$	11.2915	0.388670
$845$	24.6863	0.849233
$846$	0	0
$847$	−16.5830	−0.569799
$848$	−1.64575	−0.0565153
$849$	0	0
$850$	−7.54249	−0.258705
$851$	6.93725	0.237806
$852$	0	0
$853$	27.1660	0.930146	0.465073	−	0.885272i	$-0.346028\pi$
$853$	27.1660	0.930146	0.465073	+	0.885272i	$0.346028\pi$
$854$	−0.708497	−0.0242443
$855$	0	0
$856$	13.6458	0.466402
$857$	36.0000	1.22974	0.614868	−	0.788630i	$-0.289209\pi$
$857$	36.0000	1.22974	0.614868	+	0.788630i	$0.289209\pi$
$858$	0	0
$859$	13.4170	0.457782	0.228891	−	0.973452i	$-0.426490\pi$
$859$	13.4170	0.457782	0.228891	+	0.973452i	$0.426490\pi$
$860$	0.583005	0.0198803
$861$	0	0
$862$	33.8745	1.15377
$863$	10.8340	0.368793	0.184397	−	0.982852i	$-0.440967\pi$
$863$	10.8340	0.368793	0.184397	+	0.982852i	$0.440967\pi$
$864$	0	0
$865$	−26.1255	−0.888293
$866$	18.7085	0.635741
$867$	0	0
$868$	−2.58301	−0.0876729
$869$	−14.1255	−0.479175
$870$	0	0
$871$	−79.0405	−2.67819
$872$	18.2288	0.617304
$873$	0	0
$874$	−0.354249	−0.0119826
$875$	−24.0000	−0.811348
$876$	0	0
$877$	−13.2915	−0.448822	−0.224411	−	0.974495i	$-0.572046\pi$
$877$	−13.2915	−0.448822	−0.224411	+	0.974495i	$0.572046\pi$
$878$	−27.7490	−0.936484
$879$	0	0
$880$	−2.70850	−0.0913034
$881$	31.7490	1.06965	0.534826	−	0.844962i	$-0.320377\pi$
$881$	31.7490	1.06965	0.534826	+	0.844962i	$0.320377\pi$
$882$	0	0
$883$	−7.29150	−0.245379	−0.122689	−	0.992445i	$-0.539152\pi$
$883$	−7.29150	−0.245379	−0.122689	+	0.992445i	$0.539152\pi$
$884$	−17.4170	−0.585797
$885$	0	0
$886$	−25.1660	−0.845469
$887$	25.7490	0.864567	0.432284	−	0.901738i	$-0.357708\pi$
$887$	25.7490	0.864567	0.432284	+	0.901738i	$0.357708\pi$
$888$	0	0
$889$	−21.1660	−0.709885
$890$	0	0
$891$	0	0
$892$	−19.8745	−0.665448
$893$	2.12549	0.0711269
$894$	0	0
$895$	25.1660	0.841207
$896$	−2.00000	−0.0668153
$897$	0	0
$898$	−11.4170	−0.380990
$899$	−12.0000	−0.400222
$900$	0	0
$901$	5.41699	0.180466
$902$	9.87451	0.328785
$903$	0	0
$904$	−6.58301	−0.218947
$905$	25.7490	0.855926
$906$	0	0
$907$	−45.5203	−1.51148	−0.755738	−	0.654874i	$-0.772722\pi$
$907$	−45.5203	−1.51148	−0.755738	+	0.654874i	$0.772722\pi$
$908$	−7.06275	−0.234385
$909$	0	0
$910$	−17.4170	−0.577368
$911$	3.29150	0.109052	0.0545262	−	0.998512i	$-0.482635\pi$
$911$	3.29150	0.109052	0.0545262	+	0.998512i	$0.482635\pi$
$912$	0	0
$913$	22.4575	0.743235
$914$	0.125492	0.00415091
$915$	0	0
$916$	9.06275	0.299442
$917$	−43.7490	−1.44472
$918$	0	0
$919$	−34.0000	−1.12156	−0.560778	−	0.827966i	$-0.689498\pi$
$919$	−34.0000	−1.12156	−0.560778	+	0.827966i	$0.689498\pi$
$920$	−1.64575	−0.0542588
$921$	0	0
$922$	−25.7490	−0.847999
$923$	−31.7490	−1.04503
$924$	0	0
$925$	−15.8967	−0.522681
$926$	−21.1660	−0.695558
$927$	0	0
$928$	−9.29150	−0.305009
$929$	−6.00000	−0.196854	−0.0984268	−	0.995144i	$-0.531381\pi$
$929$	−6.00000	−0.196854	−0.0984268	+	0.995144i	$0.531381\pi$
$930$	0	0
$931$	−1.06275	−0.0348301
$932$	0	0
$933$	0	0
$934$	−13.6458	−0.446503
$935$	8.91503	0.291553
$936$	0	0
$937$	−28.5830	−0.933766	−0.466883	−	0.884319i	$-0.654623\pi$
$937$	−28.5830	−0.933766	−0.466883	+	0.884319i	$0.654623\pi$
$938$	29.8745	0.975437
$939$	0	0
$940$	9.87451	0.322071
$941$	−26.8118	−0.874038	−0.437019	−	0.899452i	$-0.643966\pi$
$941$	−26.8118	−0.874038	−0.437019	+	0.899452i	$0.643966\pi$
$942$	0	0
$943$	6.00000	0.195387
$944$	0	0
$945$	0	0
$946$	0.583005	0.0189551
$947$	2.12549	0.0690692	0.0345346	−	0.999404i	$-0.489005\pi$
$947$	2.12549	0.0690692	0.0345346	+	0.999404i	$0.489005\pi$
$948$	0	0
$949$	−38.5830	−1.25246
$950$	0.811762	0.0263370
$951$	0	0
$952$	6.58301	0.213356
$953$	−23.0405	−0.746356	−0.373178	−	0.927760i	$-0.621732\pi$
$953$	−23.0405	−0.746356	−0.373178	+	0.927760i	$0.621732\pi$
$954$	0	0
$955$	5.41699	0.175290
$956$	24.0000	0.776215
$957$	0	0
$958$	−42.5830	−1.37579
$959$	43.7490	1.41273
$960$	0	0
$961$	−29.3320	−0.946194
$962$	−36.7085	−1.18353
$963$	0	0
$964$	5.29150	0.170428
$965$	−36.2065	−1.16553
$966$	0	0
$967$	23.8745	0.767752	0.383876	−	0.923385i	$-0.374589\pi$
$967$	23.8745	0.767752	0.383876	+	0.923385i	$0.374589\pi$
$968$	8.29150	0.266499
$969$	0	0
$970$	2.12549	0.0682454
$971$	−20.2288	−0.649172	−0.324586	−	0.945856i	$-0.605225\pi$
$971$	−20.2288	−0.649172	−0.324586	+	0.945856i	$0.605225\pi$
$972$	0	0
$973$	−8.00000	−0.256468
$974$	−21.1660	−0.678203
$975$	0	0
$976$	0.354249	0.0113392
$977$	−21.8745	−0.699828	−0.349914	−	0.936782i	$-0.613789\pi$
$977$	−21.8745	−0.699828	−0.349914	+	0.936782i	$0.613789\pi$
$978$	0	0
$979$	0	0
$980$	−4.93725	−0.157715
$981$	0	0
$982$	21.8745	0.698044
$983$	−21.8745	−0.697688	−0.348844	−	0.937181i	$-0.613426\pi$
$983$	−21.8745	−0.697688	−0.348844	+	0.937181i	$0.613426\pi$
$984$	0	0
$985$	−9.87451	−0.314628
$986$	30.5830	0.973961
$987$	0	0
$988$	1.87451	0.0596360
$989$	0.354249	0.0112645
$990$	0	0
$991$	−14.4575	−0.459258	−0.229629	−	0.973278i	$-0.573751\pi$
$991$	−14.4575	−0.459258	−0.229629	+	0.973278i	$0.573751\pi$
$992$	1.29150	0.0410052
$993$	0	0
$994$	12.0000	0.380617
$995$	23.0405	0.730434
$996$	0	0
$997$	−40.5830	−1.28528	−0.642638	−	0.766170i	$-0.722160\pi$
$997$	−40.5830	−1.28528	−0.642638	+	0.766170i	$0.722160\pi$
$998$	−31.0405	−0.982570
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	414.2.a.e.1.2	✓	2
3.2	odd	2		414.2.a.g.1.1	yes	2
4.3	odd	2		3312.2.a.v.1.2		2
12.11	even	2		3312.2.a.z.1.1		2
23.22	odd	2		9522.2.a.bb.1.1		2
69.68	even	2		9522.2.a.bc.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
414.2.a.e.1.2	✓	2	1.1	even	1	trivial
414.2.a.g.1.1	yes	2	3.2	odd	2
3312.2.a.v.1.2		2	4.3	odd	2
3312.2.a.z.1.1		2	12.11	even	2
9522.2.a.bb.1.1		2	23.22	odd	2
9522.2.a.bc.1.2		2	69.68	even	2

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:	\( N \)	\(=\)	\( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	414.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.64575 q^{5} +2.00000 q^{7} -1.00000 q^{8} -1.64575 q^{10} -1.64575 q^{11} +5.29150 q^{13} -2.00000 q^{14} +1.00000 q^{16} -3.29150 q^{17} +0.354249 q^{19} +1.64575 q^{20} +1.64575 q^{22} +1.00000 q^{23} -2.29150 q^{25} -5.29150 q^{26} +2.00000 q^{28} +9.29150 q^{29} -1.29150 q^{31} -1.00000 q^{32} +3.29150 q^{34} +3.29150 q^{35} +6.93725 q^{37} -0.354249 q^{38} -1.64575 q^{40} +6.00000 q^{41} +0.354249 q^{43} -1.64575 q^{44} -1.00000 q^{46} +6.00000 q^{47} -3.00000 q^{49} +2.29150 q^{50} +5.29150 q^{52} -1.64575 q^{53} -2.70850 q^{55} -2.00000 q^{56} -9.29150 q^{58} +0.354249 q^{61} +1.29150 q^{62} +1.00000 q^{64} +8.70850 q^{65} -14.9373 q^{67} -3.29150 q^{68} -3.29150 q^{70} -6.00000 q^{71} -7.29150 q^{73} -6.93725 q^{74} +0.354249 q^{76} -3.29150 q^{77} +8.58301 q^{79} +1.64575 q^{80} -6.00000 q^{82} -13.6458 q^{83} -5.41699 q^{85} -0.354249 q^{86} +1.64575 q^{88} +10.5830 q^{91} +1.00000 q^{92} -6.00000 q^{94} +0.583005 q^{95} -1.29150 q^{97} +3.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 4 q^{7} - 2 q^{8} + 2 q^{10} + 2 q^{11} - 4 q^{14} + 2 q^{16} + 4 q^{17} + 6 q^{19} - 2 q^{20} - 2 q^{22} + 2 q^{23} + 6 q^{25} + 4 q^{28} + 8 q^{29} + 8 q^{31} - 2 q^{32}+ \cdots + 6 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + 4 * q^7 - 2 * q^8 + 2 * q^10 + 2 * q^11 - 4 * q^14 + 2 * q^16 + 4 * q^17 + 6 * q^19 - 2 * q^20 - 2 * q^22 + 2 * q^23 + 6 * q^25 + 4 * q^28 + 8 * q^29 + 8 * q^31 - 2 * q^32 - 4 * q^34 - 4 * q^35 - 2 * q^37 - 6 * q^38 + 2 * q^40 + 12 * q^41 + 6 * q^43 + 2 * q^44 - 2 * q^46 + 12 * q^47 - 6 * q^49 - 6 * q^50 + 2 * q^53 - 16 * q^55 - 4 * q^56 - 8 * q^58 + 6 * q^61 - 8 * q^62 + 2 * q^64 + 28 * q^65 - 14 * q^67 + 4 * q^68 + 4 * q^70 - 12 * q^71 - 4 * q^73 + 2 * q^74 + 6 * q^76 + 4 * q^77 - 4 * q^79 - 2 * q^80 - 12 * q^82 - 22 * q^83 - 32 * q^85 - 6 * q^86 - 2 * q^88 + 2 * q^92 - 12 * q^94 - 20 * q^95 + 8 * q^97 + 6 * q^98

Introduction

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