LMFDB - Embedded newform 315.3.h.b.181.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [315,3,Mod(181,315)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(315, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("315.181");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 315 = 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	315.h (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:	$8.58312832735$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-5}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 5 $ x^2 + 5
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 35)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			181.2
Root			$2.23607i$ of defining polynomial
Character	$\chi$	$=$	315.181
Dual form			315.3.h.b.181.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -3.00000 q^{4} +2.23607i q^{5} +7.00000 q^{7} -7.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} -3.00000 q^{4} +2.23607i q^{5} +7.00000 q^{7} -7.00000 q^{8} +2.23607i q^{10} -2.00000 q^{11} +13.4164i q^{13} +7.00000 q^{14} +5.00000 q^{16} +26.8328i q^{17} +13.4164i q^{19} -6.70820i q^{20} -2.00000 q^{22} -26.0000 q^{23} -5.00000 q^{25} +13.4164i q^{26} -21.0000 q^{28} +22.0000 q^{29} +53.6656i q^{31} +33.0000 q^{32} +26.8328i q^{34} +15.6525i q^{35} +14.0000 q^{37} +13.4164i q^{38} -15.6525i q^{40} -26.8328i q^{41} -34.0000 q^{43} +6.00000 q^{44} -26.0000 q^{46} -26.8328i q^{47} +49.0000 q^{49} -5.00000 q^{50} -40.2492i q^{52} +34.0000 q^{53} -4.47214i q^{55} -49.0000 q^{56} +22.0000 q^{58} -40.2492i q^{59} -93.9149i q^{61} +53.6656i q^{62} +13.0000 q^{64} -30.0000 q^{65} +14.0000 q^{67} -80.4984i q^{68} +15.6525i q^{70} -62.0000 q^{71} +53.6656i q^{73} +14.0000 q^{74} -40.2492i q^{76} -14.0000 q^{77} +38.0000 q^{79} +11.1803i q^{80} -26.8328i q^{82} +40.2492i q^{83} -60.0000 q^{85} -34.0000 q^{86} +14.0000 q^{88} -26.8328i q^{89} +93.9149i q^{91} +78.0000 q^{92} -26.8328i q^{94} -30.0000 q^{95} +26.8328i q^{97} +49.0000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 6 q^{4} + 14 q^{7} - 14 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 - 6 * q^4 + 14 * q^7 - 14 * q^8	$ 2 q + 2 q^{2} - 6 q^{4} + 14 q^{7} - 14 q^{8} - 4 q^{11} + 14 q^{14} + 10 q^{16} - 4 q^{22} - 52 q^{23} - 10 q^{25} - 42 q^{28} + 44 q^{29} + 66 q^{32} + 28 q^{37} - 68 q^{43} + 12 q^{44} - 52 q^{46} + 98 q^{49} - 10 q^{50} + 68 q^{53} - 98 q^{56} + 44 q^{58} + 26 q^{64} - 60 q^{65} + 28 q^{67} - 124 q^{71} + 28 q^{74} - 28 q^{77} + 76 q^{79} - 120 q^{85} - 68 q^{86} + 28 q^{88} + 156 q^{92} - 60 q^{95} + 98 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 - 6 * q^4 + 14 * q^7 - 14 * q^8 - 4 * q^11 + 14 * q^14 + 10 * q^16 - 4 * q^22 - 52 * q^23 - 10 * q^25 - 42 * q^28 + 44 * q^29 + 66 * q^32 + 28 * q^37 - 68 * q^43 + 12 * q^44 - 52 * q^46 + 98 * q^49 - 10 * q^50 + 68 * q^53 - 98 * q^56 + 44 * q^58 + 26 * q^64 - 60 * q^65 + 28 * q^67 - 124 * q^71 + 28 * q^74 - 28 * q^77 + 76 * q^79 - 120 * q^85 - 68 * q^86 + 28 * q^88 + 156 * q^92 - 60 * q^95 + 98 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/315\mathbb{Z}\right)^\times$.

$n$	$127$	$136$	$281$
$\chi(n)$	$1$	$-1$	$1$

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	0.500000	0.250000	−	0.968246i	$-0.419569\pi$
$2$	1.00000	0.500000	0.250000	+	0.968246i	$0.419569\pi$
$3$	0	0
$3$	0	0
$4$	−3.00000	−0.750000
$5$	2.23607i	0.447214i
$5$	2.23607i	0.447214i
$6$	0	0
$7$	7.00000	1.00000
$7$	7.00000	1.00000
$8$	−7.00000	−0.875000
$9$	0	0
$10$	2.23607i	0.223607i
$11$	−2.00000	−0.181818	−0.0909091	−	0.995859i	$-0.528977\pi$
$11$	−2.00000	−0.181818	−0.0909091	+	0.995859i	$0.528977\pi$
$12$	0	0
$13$	13.4164i	1.03203i	0.856579	+	0.516016i	$0.172585\pi$
$13$	13.4164i	1.03203i	−0.856579	+	0.516016i	$0.827415\pi$
$14$	7.00000	0.500000
$15$	0	0
$16$	5.00000	0.312500
$17$	26.8328i	1.57840i	0.614136	+	0.789200i	$0.289505\pi$
$17$	26.8328i	1.57840i	−0.614136	+	0.789200i	$0.710495\pi$
$18$	0	0
$19$	13.4164i	0.706127i	0.935599	+	0.353063i	$0.114860\pi$
$19$	13.4164i	0.706127i	−0.935599	+	0.353063i	$0.885140\pi$
$20$	− 6.70820i	− 0.335410i
$21$	0	0
$22$	−2.00000	−0.0909091
$23$	−26.0000	−1.13043	−0.565217	−	0.824942i	$-0.691208\pi$
$23$	−26.0000	−1.13043	−0.565217	+	0.824942i	$0.691208\pi$
$24$	0	0
$25$	−5.00000	−0.200000
$26$	13.4164i	0.516016i
$27$	0	0
$28$	−21.0000	−0.750000
$29$	22.0000	0.758621	0.379310	−	0.925270i	$-0.376161\pi$
$29$	22.0000	0.758621	0.379310	+	0.925270i	$0.376161\pi$
$30$	0	0
$31$	53.6656i	1.73115i	0.500780	+	0.865575i	$0.333047\pi$
$31$	53.6656i	1.73115i	−0.500780	+	0.865575i	$0.666953\pi$
$32$	33.0000	1.03125
$33$	0	0
$34$	26.8328i	0.789200i
$35$	15.6525i	0.447214i
$36$	0	0
$37$	14.0000	0.378378	0.189189	−	0.981941i	$-0.439414\pi$
$37$	14.0000	0.378378	0.189189	+	0.981941i	$0.439414\pi$
$38$	13.4164i	0.353063i
$39$	0	0
$40$	− 15.6525i	− 0.391312i
$41$	− 26.8328i	− 0.654459i	−0.944945	−	0.327229i	$-0.893885\pi$
$41$	− 26.8328i	− 0.654459i	0.944945	−	0.327229i	$-0.106115\pi$
$42$	0	0
$43$	−34.0000	−0.790698	−0.395349	−	0.918531i	$-0.629376\pi$
$43$	−34.0000	−0.790698	−0.395349	+	0.918531i	$0.629376\pi$
$44$	6.00000	0.136364
$45$	0	0
$46$	−26.0000	−0.565217
$47$	− 26.8328i	− 0.570911i	−0.958392	−	0.285455i	$-0.907855\pi$
$47$	− 26.8328i	− 0.570911i	0.958392	−	0.285455i	$-0.0921449\pi$
$48$	0	0
$49$	49.0000	1.00000
$50$	−5.00000	−0.100000
$51$	0	0
$52$	− 40.2492i	− 0.774024i
$53$	34.0000	0.641509	0.320755	−	0.947162i	$-0.396063\pi$
$53$	34.0000	0.641509	0.320755	+	0.947162i	$0.396063\pi$
$54$	0	0
$55$	− 4.47214i	− 0.0813116i
$56$	−49.0000	−0.875000
$57$	0	0
$58$	22.0000	0.379310
$59$	− 40.2492i	− 0.682190i	−0.940029	−	0.341095i	$-0.889202\pi$
$59$	− 40.2492i	− 0.682190i	0.940029	−	0.341095i	$-0.110798\pi$
$60$	0	0
$61$	− 93.9149i	− 1.53959i	−0.638293	−	0.769794i	$-0.720359\pi$
$61$	− 93.9149i	− 1.53959i	0.638293	−	0.769794i	$-0.279641\pi$
$62$	53.6656i	0.865575i
$63$	0	0
$64$	13.0000	0.203125
$65$	−30.0000	−0.461538
$66$	0	0
$67$	14.0000	0.208955	0.104478	−	0.994527i	$-0.466683\pi$
$67$	14.0000	0.208955	0.104478	+	0.994527i	$0.466683\pi$
$68$	− 80.4984i	− 1.18380i
$69$	0	0
$70$	15.6525i	0.223607i
$71$	−62.0000	−0.873239	−0.436620	−	0.899646i	$-0.643824\pi$
$71$	−62.0000	−0.873239	−0.436620	+	0.899646i	$0.643824\pi$
$72$	0	0
$73$	53.6656i	0.735146i	0.929995	+	0.367573i	$0.119811\pi$
$73$	53.6656i	0.735146i	−0.929995	+	0.367573i	$0.880189\pi$
$74$	14.0000	0.189189
$75$	0	0
$76$	− 40.2492i	− 0.529595i
$77$	−14.0000	−0.181818
$78$	0	0
$79$	38.0000	0.481013	0.240506	−	0.970648i	$-0.422687\pi$
$79$	38.0000	0.481013	0.240506	+	0.970648i	$0.422687\pi$
$80$	11.1803i	0.139754i
$81$	0	0
$82$	− 26.8328i	− 0.327229i
$83$	40.2492i	0.484930i	0.970160	+	0.242465i	$0.0779560\pi$
$83$	40.2492i	0.484930i	−0.970160	+	0.242465i	$0.922044\pi$
$84$	0	0
$85$	−60.0000	−0.705882
$86$	−34.0000	−0.395349
$87$	0	0
$88$	14.0000	0.159091
$89$	− 26.8328i	− 0.301492i	−0.988573	−	0.150746i	$-0.951832\pi$
$89$	− 26.8328i	− 0.301492i	0.988573	−	0.150746i	$-0.0481676\pi$
$90$	0	0
$91$	93.9149i	1.03203i
$92$	78.0000	0.847826
$93$	0	0
$94$	− 26.8328i	− 0.285455i
$95$	−30.0000	−0.315789
$96$	0	0
$97$	26.8328i	0.276627i	0.990388	+	0.138313i	$0.0441681\pi$
$97$	26.8328i	0.276627i	−0.990388	+	0.138313i	$0.955832\pi$
$98$	49.0000	0.500000
$99$	0	0
$100$	15.0000	0.150000
$101$	67.0820i	0.664179i	0.943248	+	0.332089i	$0.107754\pi$
$101$	67.0820i	0.664179i	−0.943248	+	0.332089i	$0.892246\pi$
$102$	0	0
$103$	− 160.997i	− 1.56308i	−0.623857	−	0.781538i	$-0.714435\pi$
$103$	− 160.997i	− 1.56308i	0.623857	−	0.781538i	$-0.285565\pi$
$104$	− 93.9149i	− 0.903027i
$105$	0	0
$106$	34.0000	0.320755
$107$	106.000	0.990654	0.495327	−	0.868707i	$-0.335048\pi$
$107$	106.000	0.990654	0.495327	+	0.868707i	$0.335048\pi$
$108$	0	0
$109$	−142.000	−1.30275	−0.651376	−	0.758755i	$-0.725808\pi$
$109$	−142.000	−1.30275	−0.651376	+	0.758755i	$0.725808\pi$
$110$	− 4.47214i	− 0.0406558i
$111$	0	0
$112$	35.0000	0.312500
$113$	34.0000	0.300885	0.150442	−	0.988619i	$-0.451930\pi$
$113$	34.0000	0.300885	0.150442	+	0.988619i	$0.451930\pi$
$114$	0	0
$115$	− 58.1378i	− 0.505546i
$116$	−66.0000	−0.568966
$117$	0	0
$118$	− 40.2492i	− 0.341095i
$119$	187.830i	1.57840i
$120$	0	0
$121$	−117.000	−0.966942
$122$	− 93.9149i	− 0.769794i
$123$	0	0
$124$	− 160.997i	− 1.29836i
$125$	− 11.1803i	− 0.0894427i
$126$	0	0
$127$	194.000	1.52756	0.763780	−	0.645477i	$-0.223341\pi$
$127$	194.000	1.52756	0.763780	+	0.645477i	$0.223341\pi$
$128$	−119.000	−0.929688
$129$	0	0
$130$	−30.0000	−0.230769
$131$	− 120.748i	− 0.921738i	−0.887468	−	0.460869i	$-0.847538\pi$
$131$	− 120.748i	− 0.921738i	0.887468	−	0.460869i	$-0.152462\pi$
$132$	0	0
$133$	93.9149i	0.706127i
$134$	14.0000	0.104478
$135$	0	0
$136$	− 187.830i	− 1.38110i
$137$	166.000	1.21168	0.605839	−	0.795587i	$-0.292837\pi$
$137$	166.000	1.21168	0.605839	+	0.795587i	$0.292837\pi$
$138$	0	0
$139$	93.9149i	0.675646i	0.941210	+	0.337823i	$0.109691\pi$
$139$	93.9149i	0.675646i	−0.941210	+	0.337823i	$0.890309\pi$
$140$	− 46.9574i	− 0.335410i
$141$	0	0
$142$	−62.0000	−0.436620
$143$	− 26.8328i	− 0.187642i
$144$	0	0
$145$	49.1935i	0.339265i
$146$	53.6656i	0.367573i
$147$	0	0
$148$	−42.0000	−0.283784
$149$	142.000	0.953020	0.476510	−	0.879169i	$-0.341902\pi$
$149$	142.000	0.953020	0.476510	+	0.879169i	$0.341902\pi$
$150$	0	0
$151$	2.00000	0.0132450	0.00662252	−	0.999978i	$-0.497892\pi$
$151$	2.00000	0.0132450	0.00662252	+	0.999978i	$0.497892\pi$
$152$	− 93.9149i	− 0.617861i
$153$	0	0
$154$	−14.0000	−0.0909091
$155$	−120.000	−0.774194
$156$	0	0
$157$	67.0820i	0.427274i	0.976913	+	0.213637i	$0.0685310\pi$
$157$	67.0820i	0.427274i	−0.976913	+	0.213637i	$0.931469\pi$
$158$	38.0000	0.240506
$159$	0	0
$160$	73.7902i	0.461189i
$161$	−182.000	−1.13043
$162$	0	0
$163$	−34.0000	−0.208589	−0.104294	−	0.994546i	$-0.533258\pi$
$163$	−34.0000	−0.208589	−0.104294	+	0.994546i	$0.533258\pi$
$164$	80.4984i	0.490844i
$165$	0	0
$166$	40.2492i	0.242465i
$167$	− 107.331i	− 0.642702i	−0.946960	−	0.321351i	$-0.895863\pi$
$167$	− 107.331i	− 0.642702i	0.946960	−	0.321351i	$-0.104137\pi$
$168$	0	0
$169$	−11.0000	−0.0650888
$170$	−60.0000	−0.352941
$171$	0	0
$172$	102.000	0.593023
$173$	147.580i	0.853066i	0.904472	+	0.426533i	$0.140265\pi$
$173$	147.580i	0.853066i	−0.904472	+	0.426533i	$0.859735\pi$
$174$	0	0
$175$	−35.0000	−0.200000
$176$	−10.0000	−0.0568182
$177$	0	0
$178$	− 26.8328i	− 0.150746i
$179$	−218.000	−1.21788	−0.608939	−	0.793217i	$-0.708404\pi$
$179$	−218.000	−1.21788	−0.608939	+	0.793217i	$0.708404\pi$
$180$	0	0
$181$	254.912i	1.40835i	0.710025	+	0.704176i	$0.248683\pi$
$181$	254.912i	1.40835i	−0.710025	+	0.704176i	$0.751317\pi$
$182$	93.9149i	0.516016i
$183$	0	0
$184$	182.000	0.989130
$185$	31.3050i	0.169216i
$186$	0	0
$187$	− 53.6656i	− 0.286982i
$188$	80.4984i	0.428183i
$189$	0	0
$190$	−30.0000	−0.157895
$191$	58.0000	0.303665	0.151832	−	0.988406i	$-0.451483\pi$
$191$	58.0000	0.303665	0.151832	+	0.988406i	$0.451483\pi$
$192$	0	0
$193$	206.000	1.06736	0.533679	−	0.845687i	$-0.320809\pi$
$193$	206.000	1.06736	0.533679	+	0.845687i	$0.320809\pi$
$194$	26.8328i	0.138313i
$195$	0	0
$196$	−147.000	−0.750000
$197$	226.000	1.14721	0.573604	−	0.819133i	$-0.305545\pi$
$197$	226.000	1.14721	0.573604	+	0.819133i	$0.305545\pi$
$198$	0	0
$199$	− 134.164i	− 0.674191i	−0.941470	−	0.337096i	$-0.890555\pi$
$199$	− 134.164i	− 0.674191i	0.941470	−	0.337096i	$-0.109445\pi$
$200$	35.0000	0.175000
$201$	0	0
$202$	67.0820i	0.332089i
$203$	154.000	0.758621
$204$	0	0
$205$	60.0000	0.292683
$206$	− 160.997i	− 0.781538i
$207$	0	0
$208$	67.0820i	0.322510i
$209$	− 26.8328i	− 0.128387i
$210$	0	0
$211$	−118.000	−0.559242	−0.279621	−	0.960111i	$-0.590209\pi$
$211$	−118.000	−0.559242	−0.279621	+	0.960111i	$0.590209\pi$
$212$	−102.000	−0.481132
$213$	0	0
$214$	106.000	0.495327
$215$	− 76.0263i	− 0.353611i
$216$	0	0
$217$	375.659i	1.73115i
$218$	−142.000	−0.651376
$219$	0	0
$220$	13.4164i	0.0609837i
$221$	−360.000	−1.62896
$222$	0	0
$223$	− 80.4984i	− 0.360980i	−0.983577	−	0.180490i	$-0.942232\pi$
$223$	− 80.4984i	− 0.360980i	0.983577	−	0.180490i	$-0.0577683\pi$
$224$	231.000	1.03125
$225$	0	0
$226$	34.0000	0.150442
$227$	254.912i	1.12296i	0.827491	+	0.561480i	$0.189768\pi$
$227$	254.912i	1.12296i	−0.827491	+	0.561480i	$0.810232\pi$
$228$	0	0
$229$	− 13.4164i	− 0.0585869i	−0.999571	−	0.0292935i	$-0.990674\pi$
$229$	− 13.4164i	− 0.0585869i	0.999571	−	0.0292935i	$-0.00932573\pi$
$230$	− 58.1378i	− 0.252773i
$231$	0	0
$232$	−154.000	−0.663793
$233$	214.000	0.918455	0.459227	−	0.888319i	$-0.348126\pi$
$233$	214.000	0.918455	0.459227	+	0.888319i	$0.348126\pi$
$234$	0	0
$235$	60.0000	0.255319
$236$	120.748i	0.511643i
$237$	0	0
$238$	187.830i	0.789200i
$239$	−98.0000	−0.410042	−0.205021	−	0.978758i	$-0.565726\pi$
$239$	−98.0000	−0.410042	−0.205021	+	0.978758i	$0.565726\pi$
$240$	0	0
$241$	160.997i	0.668037i	0.942567	+	0.334018i	$0.108405\pi$
$241$	160.997i	0.668037i	−0.942567	+	0.334018i	$0.891595\pi$
$242$	−117.000	−0.483471
$243$	0	0
$244$	281.745i	1.15469i
$245$	109.567i	0.447214i
$246$	0	0
$247$	−180.000	−0.728745
$248$	− 375.659i	− 1.51476i
$249$	0	0
$250$	− 11.1803i	− 0.0447214i
$251$	335.410i	1.33630i	0.744029	+	0.668148i	$0.232913\pi$
$251$	335.410i	1.33630i	−0.744029	+	0.668148i	$0.767087\pi$
$252$	0	0
$253$	52.0000	0.205534
$254$	194.000	0.763780
$255$	0	0
$256$	−171.000	−0.667969
$257$	− 134.164i	− 0.522039i	−0.965333	−	0.261020i	$-0.915941\pi$
$257$	− 134.164i	− 0.522039i	0.965333	−	0.261020i	$-0.0840587\pi$
$258$	0	0
$259$	98.0000	0.378378
$260$	90.0000	0.346154
$261$	0	0
$262$	− 120.748i	− 0.460869i
$263$	34.0000	0.129278	0.0646388	−	0.997909i	$-0.479410\pi$
$263$	34.0000	0.129278	0.0646388	+	0.997909i	$0.479410\pi$
$264$	0	0
$265$	76.0263i	0.286892i
$266$	93.9149i	0.353063i
$267$	0	0
$268$	−42.0000	−0.156716
$269$	254.912i	0.947627i	0.880625	+	0.473814i	$0.157123\pi$
$269$	254.912i	0.947627i	−0.880625	+	0.473814i	$0.842877\pi$
$270$	0	0
$271$	− 321.994i	− 1.18817i	−0.804403	−	0.594084i	$-0.797514\pi$
$271$	− 321.994i	− 1.18817i	0.804403	−	0.594084i	$-0.202486\pi$
$272$	134.164i	0.493250i
$273$	0	0
$274$	166.000	0.605839
$275$	10.0000	0.0363636
$276$	0	0
$277$	14.0000	0.0505415	0.0252708	−	0.999681i	$-0.491955\pi$
$277$	14.0000	0.0505415	0.0252708	+	0.999681i	$0.491955\pi$
$278$	93.9149i	0.337823i
$279$	0	0
$280$	− 109.567i	− 0.391312i
$281$	−2.00000	−0.00711744	−0.00355872	−	0.999994i	$-0.501133\pi$
$281$	−2.00000	−0.00711744	−0.00355872	+	0.999994i	$0.501133\pi$
$282$	0	0
$283$	93.9149i	0.331855i	0.986138	+	0.165927i	$0.0530617\pi$
$283$	93.9149i	0.331855i	−0.986138	+	0.165927i	$0.946938\pi$
$284$	186.000	0.654930
$285$	0	0
$286$	− 26.8328i	− 0.0938210i
$287$	− 187.830i	− 0.654459i
$288$	0	0
$289$	−431.000	−1.49135
$290$	49.1935i	0.169633i
$291$	0	0
$292$	− 160.997i	− 0.551359i
$293$	335.410i	1.14474i	0.819994	+	0.572372i	$0.193977\pi$
$293$	335.410i	1.14474i	−0.819994	+	0.572372i	$0.806023\pi$
$294$	0	0
$295$	90.0000	0.305085
$296$	−98.0000	−0.331081
$297$	0	0
$298$	142.000	0.476510
$299$	− 348.827i	− 1.16664i
$300$	0	0
$301$	−238.000	−0.790698
$302$	2.00000	0.00662252
$303$	0	0
$304$	67.0820i	0.220665i
$305$	210.000	0.688525
$306$	0	0
$307$	− 201.246i	− 0.655525i	−0.944760	−	0.327762i	$-0.893705\pi$
$307$	− 201.246i	− 0.655525i	0.944760	−	0.327762i	$-0.106295\pi$
$308$	42.0000	0.136364
$309$	0	0
$310$	−120.000	−0.387097
$311$	− 509.823i	− 1.63930i	−0.572862	−	0.819652i	$-0.694167\pi$
$311$	− 509.823i	− 1.63930i	0.572862	−	0.819652i	$-0.305833\pi$
$312$	0	0
$313$	− 321.994i	− 1.02873i	−0.857570	−	0.514367i	$-0.828027\pi$
$313$	− 321.994i	− 1.02873i	0.857570	−	0.514367i	$-0.171973\pi$
$314$	67.0820i	0.213637i
$315$	0	0
$316$	−114.000	−0.360759
$317$	−374.000	−1.17981	−0.589905	−	0.807472i	$-0.700835\pi$
$317$	−374.000	−1.17981	−0.589905	+	0.807472i	$0.700835\pi$
$318$	0	0
$319$	−44.0000	−0.137931
$320$	29.0689i	0.0908403i
$321$	0	0
$322$	−182.000	−0.565217
$323$	−360.000	−1.11455
$324$	0	0
$325$	− 67.0820i	− 0.206406i
$326$	−34.0000	−0.104294
$327$	0	0
$328$	187.830i	0.572652i
$329$	− 187.830i	− 0.570911i
$330$	0	0
$331$	482.000	1.45619	0.728097	−	0.685474i	$-0.240405\pi$
$331$	482.000	1.45619	0.728097	+	0.685474i	$0.240405\pi$
$332$	− 120.748i	− 0.363698i
$333$	0	0
$334$	− 107.331i	− 0.321351i
$335$	31.3050i	0.0934476i
$336$	0	0
$337$	494.000	1.46588	0.732938	−	0.680296i	$-0.238149\pi$
$337$	494.000	1.46588	0.732938	+	0.680296i	$0.238149\pi$
$338$	−11.0000	−0.0325444
$339$	0	0
$340$	180.000	0.529412
$341$	− 107.331i	− 0.314754i
$342$	0	0
$343$	343.000	1.00000
$344$	238.000	0.691860
$345$	0	0
$346$	147.580i	0.426533i
$347$	346.000	0.997118	0.498559	−	0.866856i	$-0.333863\pi$
$347$	346.000	0.997118	0.498559	+	0.866856i	$0.333863\pi$
$348$	0	0
$349$	335.410i	0.961061i	0.876978	+	0.480530i	$0.159556\pi$
$349$	335.410i	0.961061i	−0.876978	+	0.480530i	$0.840444\pi$
$350$	−35.0000	−0.100000
$351$	0	0
$352$	−66.0000	−0.187500
$353$	26.8328i	0.0760136i	0.999277	+	0.0380068i	$0.0121009\pi$
$353$	26.8328i	0.0760136i	−0.999277	+	0.0380068i	$0.987899\pi$
$354$	0	0
$355$	− 138.636i	− 0.390525i
$356$	80.4984i	0.226119i
$357$	0	0
$358$	−218.000	−0.608939
$359$	−338.000	−0.941504	−0.470752	−	0.882266i	$-0.656017\pi$
$359$	−338.000	−0.941504	−0.470752	+	0.882266i	$0.656017\pi$
$360$	0	0
$361$	181.000	0.501385
$362$	254.912i	0.704176i
$363$	0	0
$364$	− 281.745i	− 0.774024i
$365$	−120.000	−0.328767
$366$	0	0
$367$	− 295.161i	− 0.804253i	−0.915584	−	0.402127i	$-0.868271\pi$
$367$	− 295.161i	− 0.804253i	0.915584	−	0.402127i	$-0.131729\pi$
$368$	−130.000	−0.353261
$369$	0	0
$370$	31.3050i	0.0846080i
$371$	238.000	0.641509
$372$	0	0
$373$	86.0000	0.230563	0.115282	−	0.993333i	$-0.463223\pi$
$373$	86.0000	0.230563	0.115282	+	0.993333i	$0.463223\pi$
$374$	− 53.6656i	− 0.143491i
$375$	0	0
$376$	187.830i	0.499547i
$377$	295.161i	0.782920i
$378$	0	0
$379$	−262.000	−0.691293	−0.345646	−	0.938365i	$-0.612340\pi$
$379$	−262.000	−0.691293	−0.345646	+	0.938365i	$0.612340\pi$
$380$	90.0000	0.236842
$381$	0	0
$382$	58.0000	0.151832
$383$	− 563.489i	− 1.47125i	−0.677388	−	0.735625i	$-0.736888\pi$
$383$	− 563.489i	− 1.47125i	0.677388	−	0.735625i	$-0.263112\pi$
$384$	0	0
$385$	− 31.3050i	− 0.0813116i
$386$	206.000	0.533679
$387$	0	0
$388$	− 80.4984i	− 0.207470i
$389$	−698.000	−1.79434	−0.897172	−	0.441681i	$-0.854382\pi$
$389$	−698.000	−1.79434	−0.897172	+	0.441681i	$0.854382\pi$
$390$	0	0
$391$	− 697.653i	− 1.78428i
$392$	−343.000	−0.875000
$393$	0	0
$394$	226.000	0.573604
$395$	84.9706i	0.215115i
$396$	0	0
$397$	− 308.577i	− 0.777273i	−0.921391	−	0.388636i	$-0.872946\pi$
$397$	− 308.577i	− 0.777273i	0.921391	−	0.388636i	$-0.127054\pi$
$398$	− 134.164i	− 0.337096i
$399$	0	0
$400$	−25.0000	−0.0625000
$401$	538.000	1.34165	0.670823	−	0.741618i	$-0.265941\pi$
$401$	538.000	1.34165	0.670823	+	0.741618i	$0.265941\pi$
$402$	0	0
$403$	−720.000	−1.78660
$404$	− 201.246i	− 0.498134i
$405$	0	0
$406$	154.000	0.379310
$407$	−28.0000	−0.0687961
$408$	0	0
$409$	− 295.161i	− 0.721665i	−0.932631	−	0.360832i	$-0.882493\pi$
$409$	− 295.161i	− 0.721665i	0.932631	−	0.360832i	$-0.117507\pi$
$410$	60.0000	0.146341
$411$	0	0
$412$	482.991i	1.17231i
$413$	− 281.745i	− 0.682190i
$414$	0	0
$415$	−90.0000	−0.216867
$416$	442.741i	1.06428i
$417$	0	0
$418$	− 26.8328i	− 0.0641933i
$419$	− 818.401i	− 1.95322i	−0.215009	−	0.976612i	$-0.568978\pi$
$419$	− 818.401i	− 1.95322i	0.215009	−	0.976612i	$-0.431022\pi$
$420$	0	0
$421$	−118.000	−0.280285	−0.140143	−	0.990131i	$-0.544756\pi$
$421$	−118.000	−0.280285	−0.140143	+	0.990131i	$0.544756\pi$
$422$	−118.000	−0.279621
$423$	0	0
$424$	−238.000	−0.561321
$425$	− 134.164i	− 0.315680i
$426$	0	0
$427$	− 657.404i	− 1.53959i
$428$	−318.000	−0.742991
$429$	0	0
$430$	− 76.0263i	− 0.176805i
$431$	718.000	1.66589	0.832947	−	0.553353i	$-0.186652\pi$
$431$	718.000	1.66589	0.832947	+	0.553353i	$0.186652\pi$
$432$	0	0
$433$	− 509.823i	− 1.17742i	−0.808344	−	0.588711i	$-0.799636\pi$
$433$	− 509.823i	− 1.17742i	0.808344	−	0.588711i	$-0.200364\pi$
$434$	375.659i	0.865575i
$435$	0	0
$436$	426.000	0.977064
$437$	− 348.827i	− 0.798230i
$438$	0	0
$439$	26.8328i	0.0611226i	0.999533	+	0.0305613i	$0.00972948\pi$
$439$	26.8328i	0.0611226i	−0.999533	+	0.0305613i	$0.990271\pi$
$440$	31.3050i	0.0711476i
$441$	0	0
$442$	−360.000	−0.814480
$443$	634.000	1.43115	0.715576	−	0.698535i	$-0.246164\pi$
$443$	634.000	1.43115	0.715576	+	0.698535i	$0.246164\pi$
$444$	0	0
$445$	60.0000	0.134831
$446$	− 80.4984i	− 0.180490i
$447$	0	0
$448$	91.0000	0.203125
$449$	−338.000	−0.752784	−0.376392	−	0.926461i	$-0.622835\pi$
$449$	−338.000	−0.752784	−0.376392	+	0.926461i	$0.622835\pi$
$450$	0	0
$451$	53.6656i	0.118993i
$452$	−102.000	−0.225664
$453$	0	0
$454$	254.912i	0.561480i
$455$	−210.000	−0.461538
$456$	0	0
$457$	−466.000	−1.01969	−0.509847	−	0.860265i	$-0.670298\pi$
$457$	−466.000	−1.01969	−0.509847	+	0.860265i	$0.670298\pi$
$458$	− 13.4164i	− 0.0292935i
$459$	0	0
$460$	174.413i	0.379159i
$461$	− 442.741i	− 0.960394i	−0.877161	−	0.480197i	$-0.840565\pi$
$461$	− 442.741i	− 0.960394i	0.877161	−	0.480197i	$-0.159435\pi$
$462$	0	0
$463$	206.000	0.444924	0.222462	−	0.974941i	$-0.428591\pi$
$463$	206.000	0.444924	0.222462	+	0.974941i	$0.428591\pi$
$464$	110.000	0.237069
$465$	0	0
$466$	214.000	0.459227
$467$	362.243i	0.775681i	0.921727	+	0.387840i	$0.126779\pi$
$467$	362.243i	0.775681i	−0.921727	+	0.387840i	$0.873221\pi$
$468$	0	0
$469$	98.0000	0.208955
$470$	60.0000	0.127660
$471$	0	0
$472$	281.745i	0.596916i
$473$	68.0000	0.143763
$474$	0	0
$475$	− 67.0820i	− 0.141225i
$476$	− 563.489i	− 1.18380i
$477$	0	0
$478$	−98.0000	−0.205021
$479$	214.663i	0.448147i	0.974572	+	0.224074i	$0.0719356\pi$
$479$	214.663i	0.448147i	−0.974572	+	0.224074i	$0.928064\pi$
$480$	0	0
$481$	187.830i	0.390498i
$482$	160.997i	0.334018i
$483$	0	0
$484$	351.000	0.725207
$485$	−60.0000	−0.123711
$486$	0	0
$487$	−166.000	−0.340862	−0.170431	−	0.985370i	$-0.554516\pi$
$487$	−166.000	−0.340862	−0.170431	+	0.985370i	$0.554516\pi$
$488$	657.404i	1.34714i
$489$	0	0
$490$	109.567i	0.223607i
$491$	838.000	1.70672	0.853360	−	0.521321i	$-0.174561\pi$
$491$	838.000	1.70672	0.853360	+	0.521321i	$0.174561\pi$
$492$	0	0
$493$	590.322i	1.19741i
$494$	−180.000	−0.364372
$495$	0	0
$496$	268.328i	0.540984i
$497$	−434.000	−0.873239
$498$	0	0
$499$	−262.000	−0.525050	−0.262525	−	0.964925i	$-0.584555\pi$
$499$	−262.000	−0.525050	−0.262525	+	0.964925i	$0.584555\pi$
$500$	33.5410i	0.0670820i
$501$	0	0
$502$	335.410i	0.668148i
$503$	429.325i	0.853529i	0.904363	+	0.426764i	$0.140347\pi$
$503$	429.325i	0.853529i	−0.904363	+	0.426764i	$0.859653\pi$
$504$	0	0
$505$	−150.000	−0.297030
$506$	52.0000	0.102767
$507$	0	0
$508$	−582.000	−1.14567
$509$	898.899i	1.76601i	0.469363	+	0.883005i	$0.344484\pi$
$509$	898.899i	1.76601i	−0.469363	+	0.883005i	$0.655516\pi$
$510$	0	0
$511$	375.659i	0.735146i
$512$	305.000	0.595703
$513$	0	0
$514$	− 134.164i	− 0.261020i
$515$	360.000	0.699029
$516$	0	0
$517$	53.6656i	0.103802i
$518$	98.0000	0.189189
$519$	0	0
$520$	210.000	0.403846
$521$	724.486i	1.39057i	0.718735	+	0.695284i	$0.244721\pi$
$521$	724.486i	1.39057i	−0.718735	+	0.695284i	$0.755279\pi$
$522$	0	0
$523$	523.240i	1.00046i	0.865893	+	0.500229i	$0.166751\pi$
$523$	523.240i	1.00046i	−0.865893	+	0.500229i	$0.833249\pi$
$524$	362.243i	0.691303i
$525$	0	0
$526$	34.0000	0.0646388
$527$	−1440.00	−2.73245
$528$	0	0
$529$	147.000	0.277883
$530$	76.0263i	0.143446i
$531$	0	0
$532$	− 281.745i	− 0.529595i
$533$	360.000	0.675422
$534$	0	0
$535$	237.023i	0.443034i
$536$	−98.0000	−0.182836
$537$	0	0
$538$	254.912i	0.473814i
$539$	−98.0000	−0.181818
$540$	0	0
$541$	842.000	1.55638	0.778189	−	0.628031i	$-0.216139\pi$
$541$	842.000	1.55638	0.778189	+	0.628031i	$0.216139\pi$
$542$	− 321.994i	− 0.594084i
$543$	0	0
$544$	885.483i	1.62773i
$545$	− 317.522i	− 0.582609i
$546$	0	0
$547$	134.000	0.244973	0.122486	−	0.992470i	$-0.460913\pi$
$547$	134.000	0.244973	0.122486	+	0.992470i	$0.460913\pi$
$548$	−498.000	−0.908759
$549$	0	0
$550$	10.0000	0.0181818
$551$	295.161i	0.535682i
$552$	0	0
$553$	266.000	0.481013
$554$	14.0000	0.0252708
$555$	0	0
$556$	− 281.745i	− 0.506735i
$557$	706.000	1.26750	0.633752	−	0.773536i	$-0.281514\pi$
$557$	706.000	1.26750	0.633752	+	0.773536i	$0.281514\pi$
$558$	0	0
$559$	− 456.158i	− 0.816025i
$560$	78.2624i	0.139754i
$561$	0	0
$562$	−2.00000	−0.00355872
$563$	− 13.4164i	− 0.0238302i	−0.999929	−	0.0119151i	$-0.996207\pi$
$563$	− 13.4164i	− 0.0238302i	0.999929	−	0.0119151i	$-0.00379279\pi$
$564$	0	0
$565$	76.0263i	0.134560i
$566$	93.9149i	0.165927i
$567$	0	0
$568$	434.000	0.764085
$569$	82.0000	0.144112	0.0720562	−	0.997401i	$-0.477044\pi$
$569$	82.0000	0.144112	0.0720562	+	0.997401i	$0.477044\pi$
$570$	0	0
$571$	−118.000	−0.206655	−0.103327	−	0.994647i	$-0.532949\pi$
$571$	−118.000	−0.206655	−0.103327	+	0.994647i	$0.532949\pi$
$572$	80.4984i	0.140732i
$573$	0	0
$574$	− 187.830i	− 0.327229i
$575$	130.000	0.226087
$576$	0	0
$577$	885.483i	1.53463i	0.641269	+	0.767316i	$0.278408\pi$
$577$	885.483i	1.53463i	−0.641269	+	0.767316i	$0.721592\pi$
$578$	−431.000	−0.745675
$579$	0	0
$580$	− 147.580i	− 0.254449i
$581$	281.745i	0.484930i
$582$	0	0
$583$	−68.0000	−0.116638
$584$	− 375.659i	− 0.643252i
$585$	0	0
$586$	335.410i	0.572372i
$587$	− 791.568i	− 1.34850i	−0.738504	−	0.674249i	$-0.764468\pi$
$587$	− 791.568i	− 1.34850i	0.738504	−	0.674249i	$-0.235532\pi$
$588$	0	0
$589$	−720.000	−1.22241
$590$	90.0000	0.152542
$591$	0	0
$592$	70.0000	0.118243
$593$	− 134.164i	− 0.226246i	−0.993581	−	0.113123i	$-0.963915\pi$
$593$	− 134.164i	− 0.226246i	0.993581	−	0.113123i	$-0.0360855\pi$
$594$	0	0
$595$	−420.000	−0.705882
$596$	−426.000	−0.714765
$597$	0	0
$598$	− 348.827i	− 0.583322i
$599$	−398.000	−0.664441	−0.332220	−	0.943202i	$-0.607798\pi$
$599$	−398.000	−0.664441	−0.332220	+	0.943202i	$0.607798\pi$
$600$	0	0
$601$	− 134.164i	− 0.223235i	−0.993751	−	0.111617i	$-0.964397\pi$
$601$	− 134.164i	− 0.223235i	0.993751	−	0.111617i	$-0.0356031\pi$
$602$	−238.000	−0.395349
$603$	0	0
$604$	−6.00000	−0.00993377
$605$	− 261.620i	− 0.432430i
$606$	0	0
$607$	939.149i	1.54720i	0.633676	+	0.773598i	$0.281545\pi$
$607$	939.149i	1.54720i	−0.633676	+	0.773598i	$0.718455\pi$
$608$	442.741i	0.728193i
$609$	0	0
$610$	210.000	0.344262
$611$	360.000	0.589198
$612$	0	0
$613$	206.000	0.336052	0.168026	−	0.985783i	$-0.446261\pi$
$613$	206.000	0.336052	0.168026	+	0.985783i	$0.446261\pi$
$614$	− 201.246i	− 0.327762i
$615$	0	0
$616$	98.0000	0.159091
$617$	−494.000	−0.800648	−0.400324	−	0.916374i	$-0.631102\pi$
$617$	−494.000	−0.800648	−0.400324	+	0.916374i	$0.631102\pi$
$618$	0	0
$619$	− 120.748i	− 0.195069i	−0.995232	−	0.0975345i	$-0.968904\pi$
$619$	− 120.748i	− 0.195069i	0.995232	−	0.0975345i	$-0.0310956\pi$
$620$	360.000	0.580645
$621$	0	0
$622$	− 509.823i	− 0.819652i
$623$	− 187.830i	− 0.301492i
$624$	0	0
$625$	25.0000	0.0400000
$626$	− 321.994i	− 0.514367i
$627$	0	0
$628$	− 201.246i	− 0.320456i
$629$	375.659i	0.597233i
$630$	0	0
$631$	542.000	0.858954	0.429477	−	0.903078i	$-0.358698\pi$
$631$	542.000	0.858954	0.429477	+	0.903078i	$0.358698\pi$
$632$	−266.000	−0.420886
$633$	0	0
$634$	−374.000	−0.589905
$635$	433.797i	0.683145i
$636$	0	0
$637$	657.404i	1.03203i
$638$	−44.0000	−0.0689655
$639$	0	0
$640$	− 266.092i	− 0.415769i
$641$	298.000	0.464899	0.232449	−	0.972609i	$-0.425326\pi$
$641$	298.000	0.464899	0.232449	+	0.972609i	$0.425326\pi$
$642$	0	0
$643$	− 1006.23i	− 1.56490i	−0.622714	−	0.782450i	$-0.713970\pi$
$643$	− 1006.23i	− 1.56490i	0.622714	−	0.782450i	$-0.286030\pi$
$644$	546.000	0.847826
$645$	0	0
$646$	−360.000	−0.557276
$647$	643.988i	0.995344i	0.867365	+	0.497672i	$0.165812\pi$
$647$	643.988i	0.995344i	−0.867365	+	0.497672i	$0.834188\pi$
$648$	0	0
$649$	80.4984i	0.124035i
$650$	− 67.0820i	− 0.103203i
$651$	0	0
$652$	102.000	0.156442
$653$	154.000	0.235835	0.117917	−	0.993023i	$-0.462378\pi$
$653$	154.000	0.235835	0.117917	+	0.993023i	$0.462378\pi$
$654$	0	0
$655$	270.000	0.412214
$656$	− 134.164i	− 0.204518i
$657$	0	0
$658$	− 187.830i	− 0.285455i
$659$	−338.000	−0.512898	−0.256449	−	0.966558i	$-0.582553\pi$
$659$	−338.000	−0.512898	−0.256449	+	0.966558i	$0.582553\pi$
$660$	0	0
$661$	576.906i	0.872777i	0.899758	+	0.436388i	$0.143743\pi$
$661$	576.906i	0.872777i	−0.899758	+	0.436388i	$0.856257\pi$
$662$	482.000	0.728097
$663$	0	0
$664$	− 281.745i	− 0.424314i
$665$	−210.000	−0.315789
$666$	0	0
$667$	−572.000	−0.857571
$668$	321.994i	0.482027i
$669$	0	0
$670$	31.3050i	0.0467238i
$671$	187.830i	0.279925i
$672$	0	0
$673$	−814.000	−1.20951	−0.604755	−	0.796412i	$-0.706729\pi$
$673$	−814.000	−1.20951	−0.604755	+	0.796412i	$0.706729\pi$
$674$	494.000	0.732938
$675$	0	0
$676$	33.0000	0.0488166
$677$	− 684.237i	− 1.01069i	−0.862918	−	0.505345i	$-0.831365\pi$
$677$	− 684.237i	− 1.01069i	0.862918	−	0.505345i	$-0.168635\pi$
$678$	0	0
$679$	187.830i	0.276627i
$680$	420.000	0.617647
$681$	0	0
$682$	− 107.331i	− 0.157377i
$683$	−926.000	−1.35578	−0.677892	−	0.735162i	$-0.737106\pi$
$683$	−926.000	−1.35578	−0.677892	+	0.735162i	$0.737106\pi$
$684$	0	0
$685$	371.187i	0.541879i
$686$	343.000	0.500000
$687$	0	0
$688$	−170.000	−0.247093
$689$	456.158i	0.662058i
$690$	0	0
$691$	− 576.906i	− 0.834885i	−0.908703	−	0.417443i	$-0.862927\pi$
$691$	− 576.906i	− 0.834885i	0.908703	−	0.417443i	$-0.137073\pi$
$692$	− 442.741i	− 0.639800i
$693$	0	0
$694$	346.000	0.498559
$695$	−210.000	−0.302158
$696$	0	0
$697$	720.000	1.03300
$698$	335.410i	0.480530i
$699$	0	0
$700$	105.000	0.150000
$701$	−362.000	−0.516405	−0.258203	−	0.966091i	$-0.583130\pi$
$701$	−362.000	−0.516405	−0.258203	+	0.966091i	$0.583130\pi$
$702$	0	0
$703$	187.830i	0.267183i
$704$	−26.0000	−0.0369318
$705$	0	0
$706$	26.8328i	0.0380068i
$707$	469.574i	0.664179i
$708$	0	0
$709$	1058.00	1.49224	0.746121	−	0.665810i	$-0.231914\pi$
$709$	1058.00	1.49224	0.746121	+	0.665810i	$0.231914\pi$
$710$	− 138.636i	− 0.195262i
$711$	0	0
$712$	187.830i	0.263806i
$713$	− 1395.31i	− 1.95695i
$714$	0	0
$715$	60.0000	0.0839161
$716$	654.000	0.913408
$717$	0	0
$718$	−338.000	−0.470752
$719$	− 482.991i	− 0.671753i	−0.941906	−	0.335877i	$-0.890967\pi$
$719$	− 482.991i	− 0.671753i	0.941906	−	0.335877i	$-0.109033\pi$
$720$	0	0
$721$	− 1126.98i	− 1.56308i
$722$	181.000	0.250693
$723$	0	0
$724$	− 764.735i	− 1.05626i
$725$	−110.000	−0.151724
$726$	0	0
$727$	− 1126.98i	− 1.55018i	−0.631853	−	0.775088i	$-0.717705\pi$
$727$	− 1126.98i	− 1.55018i	0.631853	−	0.775088i	$-0.282295\pi$
$728$	− 657.404i	− 0.903027i
$729$	0	0
$730$	−120.000	−0.164384
$731$	− 912.316i	− 1.24804i
$732$	0	0
$733$	1301.39i	1.77543i	0.460392	+	0.887716i	$0.347709\pi$
$733$	1301.39i	1.77543i	−0.460392	+	0.887716i	$0.652291\pi$
$734$	− 295.161i	− 0.402127i
$735$	0	0
$736$	−858.000	−1.16576
$737$	−28.0000	−0.0379919
$738$	0	0
$739$	−982.000	−1.32882	−0.664411	−	0.747367i	$-0.731318\pi$
$739$	−982.000	−1.32882	−0.664411	+	0.747367i	$0.731318\pi$
$740$	− 93.9149i	− 0.126912i
$741$	0	0
$742$	238.000	0.320755
$743$	694.000	0.934051	0.467026	−	0.884244i	$-0.345326\pi$
$743$	694.000	0.934051	0.467026	+	0.884244i	$0.345326\pi$
$744$	0	0
$745$	317.522i	0.426204i
$746$	86.0000	0.115282
$747$	0	0
$748$	160.997i	0.215236i
$749$	742.000	0.990654
$750$	0	0
$751$	242.000	0.322237	0.161119	−	0.986935i	$-0.448490\pi$
$751$	242.000	0.322237	0.161119	+	0.986935i	$0.448490\pi$
$752$	− 134.164i	− 0.178410i
$753$	0	0
$754$	295.161i	0.391460i
$755$	4.47214i	0.00592336i
$756$	0	0
$757$	−106.000	−0.140026	−0.0700132	−	0.997546i	$-0.522304\pi$
$757$	−106.000	−0.140026	−0.0700132	+	0.997546i	$0.522304\pi$
$758$	−262.000	−0.345646
$759$	0	0
$760$	210.000	0.276316
$761$	1100.15i	1.44566i	0.691027	+	0.722829i	$0.257158\pi$
$761$	1100.15i	1.44566i	−0.691027	+	0.722829i	$0.742842\pi$
$762$	0	0
$763$	−994.000	−1.30275
$764$	−174.000	−0.227749
$765$	0	0
$766$	− 563.489i	− 0.735625i
$767$	540.000	0.704042
$768$	0	0
$769$	− 1126.98i	− 1.46551i	−0.680492	−	0.732756i	$-0.738234\pi$
$769$	− 1126.98i	− 1.46551i	0.680492	−	0.732756i	$-0.261766\pi$
$770$	− 31.3050i	− 0.0406558i
$771$	0	0
$772$	−618.000	−0.800518
$773$	818.401i	1.05873i	0.848393	+	0.529367i	$0.177570\pi$
$773$	818.401i	1.05873i	−0.848393	+	0.529367i	$0.822430\pi$
$774$	0	0
$775$	− 268.328i	− 0.346230i
$776$	− 187.830i	− 0.242049i
$777$	0	0
$778$	−698.000	−0.897172
$779$	360.000	0.462131
$780$	0	0
$781$	124.000	0.158771
$782$	− 697.653i	− 0.892140i
$783$	0	0
$784$	245.000	0.312500
$785$	−150.000	−0.191083
$786$	0	0
$787$	− 684.237i	− 0.869424i	−0.900569	−	0.434712i	$-0.856850\pi$
$787$	− 684.237i	− 0.869424i	0.900569	−	0.434712i	$-0.143150\pi$
$788$	−678.000	−0.860406
$789$	0	0
$790$	84.9706i	0.107558i
$791$	238.000	0.300885
$792$	0	0
$793$	1260.00	1.58890
$794$	− 308.577i	− 0.388636i
$795$	0	0
$796$	402.492i	0.505644i
$797$	308.577i	0.387174i	0.981083	+	0.193587i	$0.0620121\pi$
$797$	308.577i	0.387174i	−0.981083	+	0.193587i	$0.937988\pi$
$798$	0	0
$799$	720.000	0.901126
$800$	−165.000	−0.206250
$801$	0	0
$802$	538.000	0.670823
$803$	− 107.331i	− 0.133663i
$804$	0	0
$805$	− 406.964i	− 0.505546i
$806$	−720.000	−0.893300
$807$	0	0
$808$	− 469.574i	− 0.581156i
$809$	−1358.00	−1.67862	−0.839308	−	0.543657i	$-0.817039\pi$
$809$	−1358.00	−1.67862	−0.839308	+	0.543657i	$0.817039\pi$
$810$	0	0
$811$	308.577i	0.380490i	0.981737	+	0.190245i	$0.0609282\pi$
$811$	308.577i	0.380490i	−0.981737	+	0.190245i	$0.939072\pi$
$812$	−462.000	−0.568966
$813$	0	0
$814$	−28.0000	−0.0343980
$815$	− 76.0263i	− 0.0932838i
$816$	0	0
$817$	− 456.158i	− 0.558333i
$818$	− 295.161i	− 0.360832i
$819$	0	0
$820$	−180.000	−0.219512
$821$	−482.000	−0.587089	−0.293544	−	0.955945i	$-0.594835\pi$
$821$	−482.000	−0.587089	−0.293544	+	0.955945i	$0.594835\pi$
$822$	0	0
$823$	926.000	1.12515	0.562576	−	0.826746i	$-0.309810\pi$
$823$	926.000	1.12515	0.562576	+	0.826746i	$0.309810\pi$
$824$	1126.98i	1.36769i
$825$	0	0
$826$	− 281.745i	− 0.341095i
$827$	226.000	0.273277	0.136638	−	0.990621i	$-0.456370\pi$
$827$	226.000	0.273277	0.136638	+	0.990621i	$0.456370\pi$
$828$	0	0
$829$	1462.39i	1.76404i	0.471213	+	0.882020i	$0.343816\pi$
$829$	1462.39i	1.76404i	−0.471213	+	0.882020i	$0.656184\pi$
$830$	−90.0000	−0.108434
$831$	0	0
$832$	174.413i	0.209631i
$833$	1314.81i	1.57840i
$834$	0	0
$835$	240.000	0.287425
$836$	80.4984i	0.0962900i
$837$	0	0
$838$	− 818.401i	− 0.976612i
$839$	− 831.817i	− 0.991439i	−0.868483	−	0.495719i	$-0.834904\pi$
$839$	− 831.817i	− 0.991439i	0.868483	−	0.495719i	$-0.165096\pi$
$840$	0	0
$841$	−357.000	−0.424495
$842$	−118.000	−0.140143
$843$	0	0
$844$	354.000	0.419431
$845$	− 24.5967i	− 0.0291086i
$846$	0	0
$847$	−819.000	−0.966942
$848$	170.000	0.200472
$849$	0	0
$850$	− 134.164i	− 0.157840i
$851$	−364.000	−0.427732
$852$	0	0
$853$	40.2492i	0.0471855i	0.999722	+	0.0235927i	$0.00751050\pi$
$853$	40.2492i	0.0471855i	−0.999722	+	0.0235927i	$0.992489\pi$
$854$	− 657.404i	− 0.769794i
$855$	0	0
$856$	−742.000	−0.866822
$857$	268.328i	0.313102i	0.987670	+	0.156551i	$0.0500375\pi$
$857$	268.328i	0.313102i	−0.987670	+	0.156551i	$0.949962\pi$
$858$	0	0
$859$	308.577i	0.359229i	0.983737	+	0.179614i	$0.0574850\pi$
$859$	308.577i	0.359229i	−0.983737	+	0.179614i	$0.942515\pi$
$860$	228.079i	0.265208i
$861$	0	0
$862$	718.000	0.832947
$863$	514.000	0.595597	0.297798	−	0.954629i	$-0.403748\pi$
$863$	514.000	0.595597	0.297798	+	0.954629i	$0.403748\pi$
$864$	0	0
$865$	−330.000	−0.381503
$866$	− 509.823i	− 0.588711i
$867$	0	0
$868$	− 1126.98i	− 1.29836i
$869$	−76.0000	−0.0874568
$870$	0	0
$871$	187.830i	0.215648i
$872$	994.000	1.13991
$873$	0	0
$874$	− 348.827i	− 0.399115i
$875$	− 78.2624i	− 0.0894427i
$876$	0	0
$877$	−1306.00	−1.48917	−0.744584	−	0.667529i	$-0.767352\pi$
$877$	−1306.00	−1.48917	−0.744584	+	0.667529i	$0.767352\pi$
$878$	26.8328i	0.0305613i
$879$	0	0
$880$	− 22.3607i	− 0.0254099i
$881$	− 1126.98i	− 1.27920i	−0.768706	−	0.639602i	$-0.779099\pi$
$881$	− 1126.98i	− 1.27920i	0.768706	−	0.639602i	$-0.220901\pi$
$882$	0	0
$883$	1526.00	1.72820	0.864100	−	0.503321i	$-0.167889\pi$
$883$	1526.00	1.72820	0.864100	+	0.503321i	$0.167889\pi$
$884$	1080.00	1.22172
$885$	0	0
$886$	634.000	0.715576
$887$	1556.30i	1.75457i	0.479970	+	0.877285i	$0.340648\pi$
$887$	1556.30i	1.75457i	−0.479970	+	0.877285i	$0.659352\pi$
$888$	0	0
$889$	1358.00	1.52756
$890$	60.0000	0.0674157
$891$	0	0
$892$	241.495i	0.270735i
$893$	360.000	0.403135
$894$	0	0
$895$	− 487.463i	− 0.544651i
$896$	−833.000	−0.929688
$897$	0	0
$898$	−338.000	−0.376392
$899$	1180.64i	1.31329i
$900$	0	0
$901$	912.316i	1.01256i
$902$	53.6656i	0.0594963i
$903$	0	0
$904$	−238.000	−0.263274
$905$	−570.000	−0.629834
$906$	0	0
$907$	734.000	0.809261	0.404631	−	0.914480i	$-0.367400\pi$
$907$	734.000	0.809261	0.404631	+	0.914480i	$0.367400\pi$
$908$	− 764.735i	− 0.842219i
$909$	0	0
$910$	−210.000	−0.230769
$911$	−1202.00	−1.31943	−0.659715	−	0.751516i	$-0.729323\pi$
$911$	−1202.00	−1.31943	−0.659715	+	0.751516i	$0.729323\pi$
$912$	0	0
$913$	− 80.4984i	− 0.0881692i
$914$	−466.000	−0.509847
$915$	0	0
$916$	40.2492i	0.0439402i
$917$	− 845.234i	− 0.921738i
$918$	0	0
$919$	−1282.00	−1.39499	−0.697497	−	0.716587i	$-0.745703\pi$
$919$	−1282.00	−1.39499	−0.697497	+	0.716587i	$0.745703\pi$
$920$	406.964i	0.442353i
$921$	0	0
$922$	− 442.741i	− 0.480197i
$923$	− 831.817i	− 0.901210i
$924$	0	0
$925$	−70.0000	−0.0756757
$926$	206.000	0.222462
$927$	0	0
$928$	726.000	0.782328
$929$	− 1126.98i	− 1.21311i	−0.795042	−	0.606554i	$-0.792551\pi$
$929$	− 1126.98i	− 1.21311i	0.795042	−	0.606554i	$-0.207449\pi$
$930$	0	0
$931$	657.404i	0.706127i
$932$	−642.000	−0.688841
$933$	0	0
$934$	362.243i	0.387840i
$935$	120.000	0.128342
$936$	0	0
$937$	− 214.663i	− 0.229096i	−0.993418	−	0.114548i	$-0.963458\pi$
$937$	− 214.663i	− 0.229096i	0.993418	−	0.114548i	$-0.0365419\pi$
$938$	98.0000	0.104478
$939$	0	0
$940$	−180.000	−0.191489
$941$	845.234i	0.898229i	0.893474	+	0.449115i	$0.148261\pi$
$941$	845.234i	0.898229i	−0.893474	+	0.449115i	$0.851739\pi$
$942$	0	0
$943$	697.653i	0.739823i
$944$	− 201.246i	− 0.213184i
$945$	0	0
$946$	68.0000	0.0718816
$947$	−734.000	−0.775079	−0.387540	−	0.921853i	$-0.626675\pi$
$947$	−734.000	−0.775079	−0.387540	+	0.921853i	$0.626675\pi$
$948$	0	0
$949$	−720.000	−0.758693
$950$	− 67.0820i	− 0.0706127i
$951$	0	0
$952$	− 1314.81i	− 1.38110i
$953$	934.000	0.980063	0.490031	−	0.871705i	$-0.336985\pi$
$953$	934.000	0.980063	0.490031	+	0.871705i	$0.336985\pi$
$954$	0	0
$955$	129.692i	0.135803i
$956$	294.000	0.307531
$957$	0	0
$958$	214.663i	0.224074i
$959$	1162.00	1.21168
$960$	0	0
$961$	−1919.00	−1.99688
$962$	187.830i	0.195249i
$963$	0	0
$964$	− 482.991i	− 0.501028i
$965$	460.630i	0.477337i
$966$	0	0
$967$	314.000	0.324716	0.162358	−	0.986732i	$-0.448090\pi$
$967$	314.000	0.324716	0.162358	+	0.986732i	$0.448090\pi$
$968$	819.000	0.846074
$969$	0	0
$970$	−60.0000	−0.0618557
$971$	− 147.580i	− 0.151988i	−0.997108	−	0.0759941i	$-0.975787\pi$
$971$	− 147.580i	− 0.151988i	0.997108	−	0.0759941i	$-0.0242130\pi$
$972$	0	0
$973$	657.404i	0.675646i
$974$	−166.000	−0.170431
$975$	0	0
$976$	− 469.574i	− 0.481121i
$977$	1486.00	1.52098	0.760491	−	0.649348i	$-0.224958\pi$
$977$	1486.00	1.52098	0.760491	+	0.649348i	$0.224958\pi$
$978$	0	0
$979$	53.6656i	0.0548168i
$980$	− 328.702i	− 0.335410i
$981$	0	0
$982$	838.000	0.853360
$983$	− 965.981i	− 0.982687i	−0.870966	−	0.491344i	$-0.836506\pi$
$983$	− 965.981i	− 0.982687i	0.870966	−	0.491344i	$-0.163494\pi$
$984$	0	0
$985$	505.351i	0.513047i
$986$	590.322i	0.598704i
$987$	0	0
$988$	540.000	0.546559
$989$	884.000	0.893832
$990$	0	0
$991$	−58.0000	−0.0585267	−0.0292634	−	0.999572i	$-0.509316\pi$
$991$	−58.0000	−0.0585267	−0.0292634	+	0.999572i	$0.509316\pi$
$992$	1770.97i	1.78525i
$993$	0	0
$994$	−434.000	−0.436620
$995$	300.000	0.301508
$996$	0	0
$997$	630.571i	0.632469i	0.948681	+	0.316234i	$0.102419\pi$
$997$	630.571i	0.632469i	−0.948681	+	0.316234i	$0.897581\pi$
$998$	−262.000	−0.262525
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.3.h.b.181.2		2
3.2	odd	2		35.3.d.a.6.1	✓	2
7.6	odd	2	inner	315.3.h.b.181.1		2
12.11	even	2		560.3.f.a.321.2		2
15.2	even	4		175.3.c.d.174.1		4
15.8	even	4		175.3.c.d.174.4		4
15.14	odd	2		175.3.d.f.76.2		2
21.2	odd	6		245.3.h.b.31.2		4
21.5	even	6		245.3.h.b.31.1		4
21.11	odd	6		245.3.h.b.166.1		4
21.17	even	6		245.3.h.b.166.2		4
21.20	even	2		35.3.d.a.6.2	yes	2
84.83	odd	2		560.3.f.a.321.1		2
105.62	odd	4		175.3.c.d.174.2		4
105.83	odd	4		175.3.c.d.174.3		4
105.104	even	2		175.3.d.f.76.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.3.d.a.6.1	✓	2	3.2	odd	2
35.3.d.a.6.2	yes	2	21.20	even	2
175.3.c.d.174.1		4	15.2	even	4
175.3.c.d.174.2		4	105.62	odd	4
175.3.c.d.174.3		4	105.83	odd	4
175.3.c.d.174.4		4	15.8	even	4
175.3.d.f.76.1		2	105.104	even	2
175.3.d.f.76.2		2	15.14	odd	2
245.3.h.b.31.1		4	21.5	even	6
245.3.h.b.31.2		4	21.2	odd	6
245.3.h.b.166.1		4	21.11	odd	6
245.3.h.b.166.2		4	21.17	even	6
315.3.h.b.181.1		2	7.6	odd	2	inner
315.3.h.b.181.2		2	1.1	even	1	trivial
560.3.f.a.321.1		2	84.83	odd	2
560.3.f.a.321.2		2	12.11	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:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	315.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -3.00000 q^{4} +2.23607i q^{5} +7.00000 q^{7} -7.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} -3.00000 q^{4} +2.23607i q^{5} +7.00000 q^{7} -7.00000 q^{8} +2.23607i q^{10} -2.00000 q^{11} +13.4164i q^{13} +7.00000 q^{14} +5.00000 q^{16} +26.8328i q^{17} +13.4164i q^{19} -6.70820i q^{20} -2.00000 q^{22} -26.0000 q^{23} -5.00000 q^{25} +13.4164i q^{26} -21.0000 q^{28} +22.0000 q^{29} +53.6656i q^{31} +33.0000 q^{32} +26.8328i q^{34} +15.6525i q^{35} +14.0000 q^{37} +13.4164i q^{38} -15.6525i q^{40} -26.8328i q^{41} -34.0000 q^{43} +6.00000 q^{44} -26.0000 q^{46} -26.8328i q^{47} +49.0000 q^{49} -5.00000 q^{50} -40.2492i q^{52} +34.0000 q^{53} -4.47214i q^{55} -49.0000 q^{56} +22.0000 q^{58} -40.2492i q^{59} -93.9149i q^{61} +53.6656i q^{62} +13.0000 q^{64} -30.0000 q^{65} +14.0000 q^{67} -80.4984i q^{68} +15.6525i q^{70} -62.0000 q^{71} +53.6656i q^{73} +14.0000 q^{74} -40.2492i q^{76} -14.0000 q^{77} +38.0000 q^{79} +11.1803i q^{80} -26.8328i q^{82} +40.2492i q^{83} -60.0000 q^{85} -34.0000 q^{86} +14.0000 q^{88} -26.8328i q^{89} +93.9149i q^{91} +78.0000 q^{92} -26.8328i q^{94} -30.0000 q^{95} +26.8328i q^{97} +49.0000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 6 q^{4} + 14 q^{7} - 14 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 - 6 * q^4 + 14 * q^7 - 14 * q^8	\( 2 q + 2 q^{2} - 6 q^{4} + 14 q^{7} - 14 q^{8} - 4 q^{11} + 14 q^{14} + 10 q^{16} - 4 q^{22} - 52 q^{23} - 10 q^{25} - 42 q^{28} + 44 q^{29} + 66 q^{32} + 28 q^{37} - 68 q^{43} + 12 q^{44} - 52 q^{46} + 98 q^{49} - 10 q^{50} + 68 q^{53} - 98 q^{56} + 44 q^{58} + 26 q^{64} - 60 q^{65} + 28 q^{67} - 124 q^{71} + 28 q^{74} - 28 q^{77} + 76 q^{79} - 120 q^{85} - 68 q^{86} + 28 q^{88} + 156 q^{92} - 60 q^{95} + 98 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 - 6 * q^4 + 14 * q^7 - 14 * q^8 - 4 * q^11 + 14 * q^14 + 10 * q^16 - 4 * q^22 - 52 * q^23 - 10 * q^25 - 42 * q^28 + 44 * q^29 + 66 * q^32 + 28 * q^37 - 68 * q^43 + 12 * q^44 - 52 * q^46 + 98 * q^49 - 10 * q^50 + 68 * q^53 - 98 * q^56 + 44 * q^58 + 26 * q^64 - 60 * q^65 + 28 * q^67 - 124 * q^71 + 28 * q^74 - 28 * q^77 + 76 * q^79 - 120 * q^85 - 68 * q^86 + 28 * q^88 + 156 * q^92 - 60 * q^95 + 98 * q^98

Introduction

L-functions

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