LMFDB - Embedded newform 315.2.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [315,2,Mod(1,315)]

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, 0]))

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

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

chi := DirichletCharacter("315.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 315 = 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	315.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:	$2.51528766367$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	315.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23607 q^{2} +3.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} -2.23607 q^{8} +O(q^{10})$	\(q-2.23607 q^{2} +3.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} -2.23607 q^{8} -2.23607 q^{10} +2.47214 q^{11} -4.47214 q^{13} -2.23607 q^{14} -1.00000 q^{16} +2.00000 q^{17} +6.47214 q^{19} +3.00000 q^{20} -5.52786 q^{22} -4.00000 q^{23} +1.00000 q^{25} +10.0000 q^{26} +3.00000 q^{28} +2.00000 q^{29} +10.4721 q^{31} +6.70820 q^{32} -4.47214 q^{34} +1.00000 q^{35} +10.9443 q^{37} -14.4721 q^{38} -2.23607 q^{40} +2.00000 q^{41} -8.94427 q^{43} +7.41641 q^{44} +8.94427 q^{46} +4.94427 q^{47} +1.00000 q^{49} -2.23607 q^{50} -13.4164 q^{52} +12.4721 q^{53} +2.47214 q^{55} -2.23607 q^{56} -4.47214 q^{58} -8.94427 q^{59} -2.00000 q^{61} -23.4164 q^{62} -13.0000 q^{64} -4.47214 q^{65} -4.00000 q^{67} +6.00000 q^{68} -2.23607 q^{70} -14.4721 q^{71} -3.52786 q^{73} -24.4721 q^{74} +19.4164 q^{76} +2.47214 q^{77} -4.94427 q^{79} -1.00000 q^{80} -4.47214 q^{82} -0.944272 q^{83} +2.00000 q^{85} +20.0000 q^{86} -5.52786 q^{88} +2.00000 q^{89} -4.47214 q^{91} -12.0000 q^{92} -11.0557 q^{94} +6.47214 q^{95} -0.472136 q^{97} -2.23607 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{4} + 2 q^{5} + 2 q^{7}+O(q^{10}) $ 2 * q + 6 * q^4 + 2 * q^5 + 2 * q^7	$ 2 q + 6 q^{4} + 2 q^{5} + 2 q^{7} - 4 q^{11} - 2 q^{16} + 4 q^{17} + 4 q^{19} + 6 q^{20} - 20 q^{22} - 8 q^{23} + 2 q^{25} + 20 q^{26} + 6 q^{28} + 4 q^{29} + 12 q^{31} + 2 q^{35} + 4 q^{37} - 20 q^{38} + 4 q^{41} - 12 q^{44} - 8 q^{47} + 2 q^{49} + 16 q^{53} - 4 q^{55} - 4 q^{61} - 20 q^{62} - 26 q^{64} - 8 q^{67} + 12 q^{68} - 20 q^{71} - 16 q^{73} - 40 q^{74} + 12 q^{76} - 4 q^{77} + 8 q^{79} - 2 q^{80} + 16 q^{83} + 4 q^{85} + 40 q^{86} - 20 q^{88} + 4 q^{89} - 24 q^{92} - 40 q^{94} + 4 q^{95} + 8 q^{97}+O(q^{100}) $ 2 * q + 6 * q^4 + 2 * q^5 + 2 * q^7 - 4 * q^11 - 2 * q^16 + 4 * q^17 + 4 * q^19 + 6 * q^20 - 20 * q^22 - 8 * q^23 + 2 * q^25 + 20 * q^26 + 6 * q^28 + 4 * q^29 + 12 * q^31 + 2 * q^35 + 4 * q^37 - 20 * q^38 + 4 * q^41 - 12 * q^44 - 8 * q^47 + 2 * q^49 + 16 * q^53 - 4 * q^55 - 4 * q^61 - 20 * q^62 - 26 * q^64 - 8 * q^67 + 12 * q^68 - 20 * q^71 - 16 * q^73 - 40 * q^74 + 12 * q^76 - 4 * q^77 + 8 * q^79 - 2 * q^80 + 16 * q^83 + 4 * q^85 + 40 * q^86 - 20 * q^88 + 4 * q^89 - 24 * q^92 - 40 * q^94 + 4 * q^95 + 8 * q^97

Display 100 coefficients 10 coefficients

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 $

$2$	−2.23607	−1.58114	−0.790569	−	0.612372i	$-0.790215\pi$
$2$	−2.23607	−1.58114	−0.790569	+	0.612372i	$0.790215\pi$
$3$	0	0
$3$	0	0
$4$	3.00000	1.50000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−2.23607	−0.790569
$9$	0	0
$10$	−2.23607	−0.707107
$11$	2.47214	0.745377	0.372689	−	0.927957i	$-0.378436\pi$
$11$	2.47214	0.745377	0.372689	+	0.927957i	$0.378436\pi$
$12$	0	0
$13$	−4.47214	−1.24035	−0.620174	−	0.784465i	$-0.712938\pi$
$13$	−4.47214	−1.24035	−0.620174	+	0.784465i	$0.712938\pi$
$14$	−2.23607	−0.597614
$15$	0	0
$16$	−1.00000	−0.250000
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	6.47214	1.48481	0.742405	−	0.669951i	$-0.233685\pi$
$19$	6.47214	1.48481	0.742405	+	0.669951i	$0.233685\pi$
$20$	3.00000	0.670820
$21$	0	0
$22$	−5.52786	−1.17854
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	10.0000	1.96116
$27$	0	0
$28$	3.00000	0.566947
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	10.4721	1.88085	0.940426	−	0.340000i	$-0.110427\pi$
$31$	10.4721	1.88085	0.940426	+	0.340000i	$0.110427\pi$
$32$	6.70820	1.18585
$33$	0	0
$34$	−4.47214	−0.766965
$35$	1.00000	0.169031
$36$	0	0
$37$	10.9443	1.79923	0.899614	−	0.436687i	$-0.143848\pi$
$37$	10.9443	1.79923	0.899614	+	0.436687i	$0.143848\pi$
$38$	−14.4721	−2.34769
$39$	0	0
$40$	−2.23607	−0.353553
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	−8.94427	−1.36399	−0.681994	−	0.731357i	$-0.738887\pi$
$43$	−8.94427	−1.36399	−0.681994	+	0.731357i	$0.738887\pi$
$44$	7.41641	1.11807
$45$	0	0
$46$	8.94427	1.31876
$47$	4.94427	0.721196	0.360598	−	0.932721i	$-0.382573\pi$
$47$	4.94427	0.721196	0.360598	+	0.932721i	$0.382573\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−2.23607	−0.316228
$51$	0	0
$52$	−13.4164	−1.86052
$53$	12.4721	1.71318	0.856590	−	0.515998i	$-0.172579\pi$
$53$	12.4721	1.71318	0.856590	+	0.515998i	$0.172579\pi$
$54$	0	0
$55$	2.47214	0.333343
$56$	−2.23607	−0.298807
$57$	0	0
$58$	−4.47214	−0.587220
$59$	−8.94427	−1.16445	−0.582223	−	0.813029i	$-0.697817\pi$
$59$	−8.94427	−1.16445	−0.582223	+	0.813029i	$0.697817\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	−23.4164	−2.97389
$63$	0	0
$64$	−13.0000	−1.62500
$65$	−4.47214	−0.554700
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	6.00000	0.727607
$69$	0	0
$70$	−2.23607	−0.267261
$71$	−14.4721	−1.71753	−0.858763	−	0.512373i	$-0.828767\pi$
$71$	−14.4721	−1.71753	−0.858763	+	0.512373i	$0.828767\pi$
$72$	0	0
$73$	−3.52786	−0.412905	−0.206453	−	0.978457i	$-0.566192\pi$
$73$	−3.52786	−0.412905	−0.206453	+	0.978457i	$0.566192\pi$
$74$	−24.4721	−2.84483
$75$	0	0
$76$	19.4164	2.22721
$77$	2.47214	0.281726
$78$	0	0
$79$	−4.94427	−0.556274	−0.278137	−	0.960541i	$-0.589717\pi$
$79$	−4.94427	−0.556274	−0.278137	+	0.960541i	$0.589717\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	−4.47214	−0.493865
$83$	−0.944272	−0.103647	−0.0518237	−	0.998656i	$-0.516503\pi$
$83$	−0.944272	−0.103647	−0.0518237	+	0.998656i	$0.516503\pi$
$84$	0	0
$85$	2.00000	0.216930
$86$	20.0000	2.15666
$87$	0	0
$88$	−5.52786	−0.589272
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	0	0
$91$	−4.47214	−0.468807
$92$	−12.0000	−1.25109
$93$	0	0
$94$	−11.0557	−1.14031
$95$	6.47214	0.664027
$96$	0	0
$97$	−0.472136	−0.0479381	−0.0239691	−	0.999713i	$-0.507630\pi$
$97$	−0.472136	−0.0479381	−0.0239691	+	0.999713i	$0.507630\pi$
$98$	−2.23607	−0.225877
$99$	0	0
$100$	3.00000	0.300000
$101$	14.0000	1.39305	0.696526	−	0.717532i	$-0.254728\pi$
$101$	14.0000	1.39305	0.696526	+	0.717532i	$0.254728\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	10.0000	0.980581
$105$	0	0
$106$	−27.8885	−2.70877
$107$	−4.94427	−0.477981	−0.238990	−	0.971022i	$-0.576816\pi$
$107$	−4.94427	−0.477981	−0.238990	+	0.971022i	$0.576816\pi$
$108$	0	0
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	−5.52786	−0.527061
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	8.47214	0.796992	0.398496	−	0.917170i	$-0.369532\pi$
$113$	8.47214	0.796992	0.398496	+	0.917170i	$0.369532\pi$
$114$	0	0
$115$	−4.00000	−0.373002
$116$	6.00000	0.557086
$117$	0	0
$118$	20.0000	1.84115
$119$	2.00000	0.183340
$120$	0	0
$121$	−4.88854	−0.444413
$122$	4.47214	0.404888
$123$	0	0
$124$	31.4164	2.82128
$125$	1.00000	0.0894427
$126$	0	0
$127$	12.9443	1.14862	0.574309	−	0.818638i	$-0.305271\pi$
$127$	12.9443	1.14862	0.574309	+	0.818638i	$0.305271\pi$
$128$	15.6525	1.38350
$129$	0	0
$130$	10.0000	0.877058
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	0	0
$133$	6.47214	0.561205
$134$	8.94427	0.772667
$135$	0	0
$136$	−4.47214	−0.383482
$137$	−12.4721	−1.06557	−0.532783	−	0.846252i	$-0.678854\pi$
$137$	−12.4721	−1.06557	−0.532783	+	0.846252i	$0.678854\pi$
$138$	0	0
$139$	−19.4164	−1.64688	−0.823439	−	0.567405i	$-0.807948\pi$
$139$	−19.4164	−1.64688	−0.823439	+	0.567405i	$0.807948\pi$
$140$	3.00000	0.253546
$141$	0	0
$142$	32.3607	2.71565
$143$	−11.0557	−0.924526
$144$	0	0
$145$	2.00000	0.166091
$146$	7.88854	0.652861
$147$	0	0
$148$	32.8328	2.69884
$149$	−2.94427	−0.241204	−0.120602	−	0.992701i	$-0.538483\pi$
$149$	−2.94427	−0.241204	−0.120602	+	0.992701i	$0.538483\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	−14.4721	−1.17385
$153$	0	0
$154$	−5.52786	−0.445448
$155$	10.4721	0.841142
$156$	0	0
$157$	8.47214	0.676150	0.338075	−	0.941119i	$-0.390224\pi$
$157$	8.47214	0.676150	0.338075	+	0.941119i	$0.390224\pi$
$158$	11.0557	0.879547
$159$	0	0
$160$	6.70820	0.530330
$161$	−4.00000	−0.315244
$162$	0	0
$163$	0.944272	0.0739611	0.0369805	−	0.999316i	$-0.488226\pi$
$163$	0.944272	0.0739611	0.0369805	+	0.999316i	$0.488226\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	2.11146	0.163881
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	0	0
$169$	7.00000	0.538462
$170$	−4.47214	−0.342997
$171$	0	0
$172$	−26.8328	−2.04598
$173$	−14.9443	−1.13619	−0.568096	−	0.822962i	$-0.692320\pi$
$173$	−14.9443	−1.13619	−0.568096	+	0.822962i	$0.692320\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	−2.47214	−0.186344
$177$	0	0
$178$	−4.47214	−0.335201
$179$	2.47214	0.184776	0.0923881	−	0.995723i	$-0.470550\pi$
$179$	2.47214	0.184776	0.0923881	+	0.995723i	$0.470550\pi$
$180$	0	0
$181$	18.9443	1.40812	0.704058	−	0.710142i	$-0.251369\pi$
$181$	18.9443	1.40812	0.704058	+	0.710142i	$0.251369\pi$
$182$	10.0000	0.741249
$183$	0	0
$184$	8.94427	0.659380
$185$	10.9443	0.804639
$186$	0	0
$187$	4.94427	0.361561
$188$	14.8328	1.08179
$189$	0	0
$190$	−14.4721	−1.04992
$191$	−27.4164	−1.98378	−0.991891	−	0.127093i	$-0.959435\pi$
$191$	−27.4164	−1.98378	−0.991891	+	0.127093i	$0.959435\pi$
$192$	0	0
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	1.05573	0.0757969
$195$	0	0
$196$	3.00000	0.214286
$197$	−24.4721	−1.74357	−0.871784	−	0.489891i	$-0.837037\pi$
$197$	−24.4721	−1.74357	−0.871784	+	0.489891i	$0.837037\pi$
$198$	0	0
$199$	0.583592	0.0413697	0.0206849	−	0.999786i	$-0.493415\pi$
$199$	0.583592	0.0413697	0.0206849	+	0.999786i	$0.493415\pi$
$200$	−2.23607	−0.158114
$201$	0	0
$202$	−31.3050	−2.20261
$203$	2.00000	0.140372
$204$	0	0
$205$	2.00000	0.139686
$206$	0	0
$207$	0	0
$208$	4.47214	0.310087
$209$	16.0000	1.10674
$210$	0	0
$211$	0.944272	0.0650064	0.0325032	−	0.999472i	$-0.489652\pi$
$211$	0.944272	0.0650064	0.0325032	+	0.999472i	$0.489652\pi$
$212$	37.4164	2.56977
$213$	0	0
$214$	11.0557	0.755754
$215$	−8.94427	−0.609994
$216$	0	0
$217$	10.4721	0.710895
$218$	4.47214	0.302891
$219$	0	0
$220$	7.41641	0.500014
$221$	−8.94427	−0.601657
$222$	0	0
$223$	4.94427	0.331093	0.165546	−	0.986202i	$-0.447061\pi$
$223$	4.94427	0.331093	0.165546	+	0.986202i	$0.447061\pi$
$224$	6.70820	0.448211
$225$	0	0
$226$	−18.9443	−1.26015
$227$	16.9443	1.12463	0.562315	−	0.826923i	$-0.309911\pi$
$227$	16.9443	1.12463	0.562315	+	0.826923i	$0.309911\pi$
$228$	0	0
$229$	−11.8885	−0.785617	−0.392809	−	0.919620i	$-0.628496\pi$
$229$	−11.8885	−0.785617	−0.392809	+	0.919620i	$0.628496\pi$
$230$	8.94427	0.589768
$231$	0	0
$232$	−4.47214	−0.293610
$233$	−17.4164	−1.14099	−0.570493	−	0.821302i	$-0.693248\pi$
$233$	−17.4164	−1.14099	−0.570493	+	0.821302i	$0.693248\pi$
$234$	0	0
$235$	4.94427	0.322529
$236$	−26.8328	−1.74667
$237$	0	0
$238$	−4.47214	−0.289886
$239$	1.52786	0.0988293	0.0494147	−	0.998778i	$-0.484264\pi$
$239$	1.52786	0.0988293	0.0494147	+	0.998778i	$0.484264\pi$
$240$	0	0
$241$	−1.05573	−0.0680054	−0.0340027	−	0.999422i	$-0.510825\pi$
$241$	−1.05573	−0.0680054	−0.0340027	+	0.999422i	$0.510825\pi$
$242$	10.9311	0.702679
$243$	0	0
$244$	−6.00000	−0.384111
$245$	1.00000	0.0638877
$246$	0	0
$247$	−28.9443	−1.84168
$248$	−23.4164	−1.48694
$249$	0	0
$250$	−2.23607	−0.141421
$251$	0.944272	0.0596019	0.0298010	−	0.999556i	$-0.490513\pi$
$251$	0.944272	0.0596019	0.0298010	+	0.999556i	$0.490513\pi$
$252$	0	0
$253$	−9.88854	−0.621687
$254$	−28.9443	−1.81613
$255$	0	0
$256$	−9.00000	−0.562500
$257$	−1.05573	−0.0658545	−0.0329273	−	0.999458i	$-0.510483\pi$
$257$	−1.05573	−0.0658545	−0.0329273	+	0.999458i	$0.510483\pi$
$258$	0	0
$259$	10.9443	0.680044
$260$	−13.4164	−0.832050
$261$	0	0
$262$	8.94427	0.552579
$263$	−24.9443	−1.53813	−0.769065	−	0.639171i	$-0.779278\pi$
$263$	−24.9443	−1.53813	−0.769065	+	0.639171i	$0.779278\pi$
$264$	0	0
$265$	12.4721	0.766157
$266$	−14.4721	−0.887344
$267$	0	0
$268$	−12.0000	−0.733017
$269$	23.8885	1.45651	0.728255	−	0.685306i	$-0.240332\pi$
$269$	23.8885	1.45651	0.728255	+	0.685306i	$0.240332\pi$
$270$	0	0
$271$	−10.4721	−0.636137	−0.318068	−	0.948068i	$-0.603034\pi$
$271$	−10.4721	−0.636137	−0.318068	+	0.948068i	$0.603034\pi$
$272$	−2.00000	−0.121268
$273$	0	0
$274$	27.8885	1.68481
$275$	2.47214	0.149075
$276$	0	0
$277$	1.05573	0.0634326	0.0317163	−	0.999497i	$-0.489903\pi$
$277$	1.05573	0.0634326	0.0317163	+	0.999497i	$0.489903\pi$
$278$	43.4164	2.60394
$279$	0	0
$280$	−2.23607	−0.133631
$281$	−6.94427	−0.414261	−0.207130	−	0.978313i	$-0.566412\pi$
$281$	−6.94427	−0.414261	−0.207130	+	0.978313i	$0.566412\pi$
$282$	0	0
$283$	12.0000	0.713326	0.356663	−	0.934233i	$-0.383914\pi$
$283$	12.0000	0.713326	0.356663	+	0.934233i	$0.383914\pi$
$284$	−43.4164	−2.57629
$285$	0	0
$286$	24.7214	1.46180
$287$	2.00000	0.118056
$288$	0	0
$289$	−13.0000	−0.764706
$290$	−4.47214	−0.262613
$291$	0	0
$292$	−10.5836	−0.619358
$293$	−22.9443	−1.34042	−0.670209	−	0.742172i	$-0.733796\pi$
$293$	−22.9443	−1.34042	−0.670209	+	0.742172i	$0.733796\pi$
$294$	0	0
$295$	−8.94427	−0.520756
$296$	−24.4721	−1.42241
$297$	0	0
$298$	6.58359	0.381377
$299$	17.8885	1.03452
$300$	0	0
$301$	−8.94427	−0.515539
$302$	35.7771	2.05874
$303$	0	0
$304$	−6.47214	−0.371202
$305$	−2.00000	−0.114520
$306$	0	0
$307$	−32.9443	−1.88023	−0.940114	−	0.340859i	$-0.889282\pi$
$307$	−32.9443	−1.88023	−0.940114	+	0.340859i	$0.889282\pi$
$308$	7.41641	0.422589
$309$	0	0
$310$	−23.4164	−1.32996
$311$	9.88854	0.560728	0.280364	−	0.959894i	$-0.409545\pi$
$311$	9.88854	0.560728	0.280364	+	0.959894i	$0.409545\pi$
$312$	0	0
$313$	9.41641	0.532247	0.266123	−	0.963939i	$-0.414257\pi$
$313$	9.41641	0.532247	0.266123	+	0.963939i	$0.414257\pi$
$314$	−18.9443	−1.06909
$315$	0	0
$316$	−14.8328	−0.834411
$317$	30.3607	1.70523	0.852613	−	0.522543i	$-0.175017\pi$
$317$	30.3607	1.70523	0.852613	+	0.522543i	$0.175017\pi$
$318$	0	0
$319$	4.94427	0.276826
$320$	−13.0000	−0.726722
$321$	0	0
$322$	8.94427	0.498445
$323$	12.9443	0.720239
$324$	0	0
$325$	−4.47214	−0.248069
$326$	−2.11146	−0.116943
$327$	0	0
$328$	−4.47214	−0.246932
$329$	4.94427	0.272587
$330$	0	0
$331$	−16.9443	−0.931341	−0.465671	−	0.884958i	$-0.654187\pi$
$331$	−16.9443	−0.931341	−0.465671	+	0.884958i	$0.654187\pi$
$332$	−2.83282	−0.155471
$333$	0	0
$334$	−17.8885	−0.978818
$335$	−4.00000	−0.218543
$336$	0	0
$337$	11.8885	0.647610	0.323805	−	0.946124i	$-0.395038\pi$
$337$	11.8885	0.647610	0.323805	+	0.946124i	$0.395038\pi$
$338$	−15.6525	−0.851382
$339$	0	0
$340$	6.00000	0.325396
$341$	25.8885	1.40194
$342$	0	0
$343$	1.00000	0.0539949
$344$	20.0000	1.07833
$345$	0	0
$346$	33.4164	1.79648
$347$	8.00000	0.429463	0.214731	−	0.976673i	$-0.431112\pi$
$347$	8.00000	0.429463	0.214731	+	0.976673i	$0.431112\pi$
$348$	0	0
$349$	23.8885	1.27872	0.639362	−	0.768906i	$-0.279198\pi$
$349$	23.8885	1.27872	0.639362	+	0.768906i	$0.279198\pi$
$350$	−2.23607	−0.119523
$351$	0	0
$352$	16.5836	0.883908
$353$	27.8885	1.48436	0.742179	−	0.670202i	$-0.233793\pi$
$353$	27.8885	1.48436	0.742179	+	0.670202i	$0.233793\pi$
$354$	0	0
$355$	−14.4721	−0.768101
$356$	6.00000	0.317999
$357$	0	0
$358$	−5.52786	−0.292157
$359$	−9.52786	−0.502861	−0.251431	−	0.967875i	$-0.580901\pi$
$359$	−9.52786	−0.502861	−0.251431	+	0.967875i	$0.580901\pi$
$360$	0	0
$361$	22.8885	1.20466
$362$	−42.3607	−2.22643
$363$	0	0
$364$	−13.4164	−0.703211
$365$	−3.52786	−0.184657
$366$	0	0
$367$	20.9443	1.09328	0.546641	−	0.837367i	$-0.315906\pi$
$367$	20.9443	1.09328	0.546641	+	0.837367i	$0.315906\pi$
$368$	4.00000	0.208514
$369$	0	0
$370$	−24.4721	−1.27225
$371$	12.4721	0.647521
$372$	0	0
$373$	6.00000	0.310668	0.155334	−	0.987862i	$-0.450355\pi$
$373$	6.00000	0.310668	0.155334	+	0.987862i	$0.450355\pi$
$374$	−11.0557	−0.571678
$375$	0	0
$376$	−11.0557	−0.570156
$377$	−8.94427	−0.460653
$378$	0	0
$379$	−2.11146	−0.108458	−0.0542291	−	0.998529i	$-0.517270\pi$
$379$	−2.11146	−0.108458	−0.0542291	+	0.998529i	$0.517270\pi$
$380$	19.4164	0.996041
$381$	0	0
$382$	61.3050	3.13663
$383$	8.00000	0.408781	0.204390	−	0.978889i	$-0.434479\pi$
$383$	8.00000	0.408781	0.204390	+	0.978889i	$0.434479\pi$
$384$	0	0
$385$	2.47214	0.125992
$386$	31.3050	1.59338
$387$	0	0
$388$	−1.41641	−0.0719072
$389$	−10.9443	−0.554897	−0.277448	−	0.960741i	$-0.589489\pi$
$389$	−10.9443	−0.554897	−0.277448	+	0.960741i	$0.589489\pi$
$390$	0	0
$391$	−8.00000	−0.404577
$392$	−2.23607	−0.112938
$393$	0	0
$394$	54.7214	2.75682
$395$	−4.94427	−0.248773
$396$	0	0
$397$	13.4164	0.673350	0.336675	−	0.941621i	$-0.390698\pi$
$397$	13.4164	0.673350	0.336675	+	0.941621i	$0.390698\pi$
$398$	−1.30495	−0.0654113
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	−10.0000	−0.499376	−0.249688	−	0.968326i	$-0.580328\pi$
$401$	−10.0000	−0.499376	−0.249688	+	0.968326i	$0.580328\pi$
$402$	0	0
$403$	−46.8328	−2.33291
$404$	42.0000	2.08958
$405$	0	0
$406$	−4.47214	−0.221948
$407$	27.0557	1.34110
$408$	0	0
$409$	−23.8885	−1.18121	−0.590606	−	0.806960i	$-0.701111\pi$
$409$	−23.8885	−1.18121	−0.590606	+	0.806960i	$0.701111\pi$
$410$	−4.47214	−0.220863
$411$	0	0
$412$	0	0
$413$	−8.94427	−0.440119
$414$	0	0
$415$	−0.944272	−0.0463525
$416$	−30.0000	−1.47087
$417$	0	0
$418$	−35.7771	−1.74991
$419$	−5.88854	−0.287674	−0.143837	−	0.989601i	$-0.545944\pi$
$419$	−5.88854	−0.287674	−0.143837	+	0.989601i	$0.545944\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	−2.11146	−0.102784
$423$	0	0
$424$	−27.8885	−1.35439
$425$	2.00000	0.0970143
$426$	0	0
$427$	−2.00000	−0.0967868
$428$	−14.8328	−0.716971
$429$	0	0
$430$	20.0000	0.964486
$431$	9.52786	0.458941	0.229471	−	0.973316i	$-0.426301\pi$
$431$	9.52786	0.458941	0.229471	+	0.973316i	$0.426301\pi$
$432$	0	0
$433$	7.52786	0.361766	0.180883	−	0.983505i	$-0.442104\pi$
$433$	7.52786	0.361766	0.180883	+	0.983505i	$0.442104\pi$
$434$	−23.4164	−1.12402
$435$	0	0
$436$	−6.00000	−0.287348
$437$	−25.8885	−1.23842
$438$	0	0
$439$	10.4721	0.499808	0.249904	−	0.968271i	$-0.419601\pi$
$439$	10.4721	0.499808	0.249904	+	0.968271i	$0.419601\pi$
$440$	−5.52786	−0.263531
$441$	0	0
$442$	20.0000	0.951303
$443$	8.00000	0.380091	0.190046	−	0.981775i	$-0.439136\pi$
$443$	8.00000	0.380091	0.190046	+	0.981775i	$0.439136\pi$
$444$	0	0
$445$	2.00000	0.0948091
$446$	−11.0557	−0.523504
$447$	0	0
$448$	−13.0000	−0.614192
$449$	14.0000	0.660701	0.330350	−	0.943858i	$-0.392833\pi$
$449$	14.0000	0.660701	0.330350	+	0.943858i	$0.392833\pi$
$450$	0	0
$451$	4.94427	0.232817
$452$	25.4164	1.19549
$453$	0	0
$454$	−37.8885	−1.77820
$455$	−4.47214	−0.209657
$456$	0	0
$457$	−10.9443	−0.511951	−0.255976	−	0.966683i	$-0.582397\pi$
$457$	−10.9443	−0.511951	−0.255976	+	0.966683i	$0.582397\pi$
$458$	26.5836	1.24217
$459$	0	0
$460$	−12.0000	−0.559503
$461$	31.8885	1.48520	0.742599	−	0.669737i	$-0.233593\pi$
$461$	31.8885	1.48520	0.742599	+	0.669737i	$0.233593\pi$
$462$	0	0
$463$	3.05573	0.142012	0.0710059	−	0.997476i	$-0.477379\pi$
$463$	3.05573	0.142012	0.0710059	+	0.997476i	$0.477379\pi$
$464$	−2.00000	−0.0928477
$465$	0	0
$466$	38.9443	1.80406
$467$	−8.94427	−0.413892	−0.206946	−	0.978352i	$-0.566352\pi$
$467$	−8.94427	−0.413892	−0.206946	+	0.978352i	$0.566352\pi$
$468$	0	0
$469$	−4.00000	−0.184703
$470$	−11.0557	−0.509963
$471$	0	0
$472$	20.0000	0.920575
$473$	−22.1115	−1.01669
$474$	0	0
$475$	6.47214	0.296962
$476$	6.00000	0.275010
$477$	0	0
$478$	−3.41641	−0.156263
$479$	−17.8885	−0.817348	−0.408674	−	0.912680i	$-0.634009\pi$
$479$	−17.8885	−0.817348	−0.408674	+	0.912680i	$0.634009\pi$
$480$	0	0
$481$	−48.9443	−2.23167
$482$	2.36068	0.107526
$483$	0	0
$484$	−14.6656	−0.666620
$485$	−0.472136	−0.0214386
$486$	0	0
$487$	−3.05573	−0.138468	−0.0692341	−	0.997600i	$-0.522056\pi$
$487$	−3.05573	−0.138468	−0.0692341	+	0.997600i	$0.522056\pi$
$488$	4.47214	0.202444
$489$	0	0
$490$	−2.23607	−0.101015
$491$	−41.3050	−1.86407	−0.932033	−	0.362373i	$-0.881967\pi$
$491$	−41.3050	−1.86407	−0.932033	+	0.362373i	$0.881967\pi$
$492$	0	0
$493$	4.00000	0.180151
$494$	64.7214	2.91195
$495$	0	0
$496$	−10.4721	−0.470213
$497$	−14.4721	−0.649164
$498$	0	0
$499$	21.8885	0.979866	0.489933	−	0.871760i	$-0.337021\pi$
$499$	21.8885	0.979866	0.489933	+	0.871760i	$0.337021\pi$
$500$	3.00000	0.134164
$501$	0	0
$502$	−2.11146	−0.0942389
$503$	−32.0000	−1.42681	−0.713405	−	0.700752i	$-0.752848\pi$
$503$	−32.0000	−1.42681	−0.713405	+	0.700752i	$0.752848\pi$
$504$	0	0
$505$	14.0000	0.622992
$506$	22.1115	0.982974
$507$	0	0
$508$	38.8328	1.72293
$509$	−11.8885	−0.526950	−0.263475	−	0.964666i	$-0.584869\pi$
$509$	−11.8885	−0.526950	−0.263475	+	0.964666i	$0.584869\pi$
$510$	0	0
$511$	−3.52786	−0.156064
$512$	−11.1803	−0.494106
$513$	0	0
$514$	2.36068	0.104125
$515$	0	0
$516$	0	0
$517$	12.2229	0.537563
$518$	−24.4721	−1.07524
$519$	0	0
$520$	10.0000	0.438529
$521$	−15.8885	−0.696090	−0.348045	−	0.937478i	$-0.613154\pi$
$521$	−15.8885	−0.696090	−0.348045	+	0.937478i	$0.613154\pi$
$522$	0	0
$523$	8.94427	0.391106	0.195553	−	0.980693i	$-0.437350\pi$
$523$	8.94427	0.391106	0.195553	+	0.980693i	$0.437350\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	55.7771	2.43200
$527$	20.9443	0.912347
$528$	0	0
$529$	−7.00000	−0.304348
$530$	−27.8885	−1.21140
$531$	0	0
$532$	19.4164	0.841808
$533$	−8.94427	−0.387419
$534$	0	0
$535$	−4.94427	−0.213760
$536$	8.94427	0.386334
$537$	0	0
$538$	−53.4164	−2.30294
$539$	2.47214	0.106482
$540$	0	0
$541$	23.8885	1.02705	0.513524	−	0.858075i	$-0.328340\pi$
$541$	23.8885	1.02705	0.513524	+	0.858075i	$0.328340\pi$
$542$	23.4164	1.00582
$543$	0	0
$544$	13.4164	0.575224
$545$	−2.00000	−0.0856706
$546$	0	0
$547$	−29.8885	−1.27794	−0.638971	−	0.769231i	$-0.720640\pi$
$547$	−29.8885	−1.27794	−0.638971	+	0.769231i	$0.720640\pi$
$548$	−37.4164	−1.59835
$549$	0	0
$550$	−5.52786	−0.235709
$551$	12.9443	0.551445
$552$	0	0
$553$	−4.94427	−0.210252
$554$	−2.36068	−0.100296
$555$	0	0
$556$	−58.2492	−2.47032
$557$	−11.5279	−0.488451	−0.244226	−	0.969718i	$-0.578534\pi$
$557$	−11.5279	−0.488451	−0.244226	+	0.969718i	$0.578534\pi$
$558$	0	0
$559$	40.0000	1.69182
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	15.5279	0.655003
$563$	21.8885	0.922492	0.461246	−	0.887272i	$-0.347403\pi$
$563$	21.8885	0.922492	0.461246	+	0.887272i	$0.347403\pi$
$564$	0	0
$565$	8.47214	0.356425
$566$	−26.8328	−1.12787
$567$	0	0
$568$	32.3607	1.35782
$569$	4.11146	0.172361	0.0861806	−	0.996280i	$-0.472534\pi$
$569$	4.11146	0.172361	0.0861806	+	0.996280i	$0.472534\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	−33.1672	−1.38679
$573$	0	0
$574$	−4.47214	−0.186663
$575$	−4.00000	−0.166812
$576$	0	0
$577$	−34.3607	−1.43045	−0.715227	−	0.698892i	$-0.753677\pi$
$577$	−34.3607	−1.43045	−0.715227	+	0.698892i	$0.753677\pi$
$578$	29.0689	1.20911
$579$	0	0
$580$	6.00000	0.249136
$581$	−0.944272	−0.0391750
$582$	0	0
$583$	30.8328	1.27696
$584$	7.88854	0.326430
$585$	0	0
$586$	51.3050	2.11939
$587$	4.00000	0.165098	0.0825488	−	0.996587i	$-0.473694\pi$
$587$	4.00000	0.165098	0.0825488	+	0.996587i	$0.473694\pi$
$588$	0	0
$589$	67.7771	2.79271
$590$	20.0000	0.823387
$591$	0	0
$592$	−10.9443	−0.449807
$593$	11.8885	0.488204	0.244102	−	0.969750i	$-0.421507\pi$
$593$	11.8885	0.488204	0.244102	+	0.969750i	$0.421507\pi$
$594$	0	0
$595$	2.00000	0.0819920
$596$	−8.83282	−0.361806
$597$	0	0
$598$	−40.0000	−1.63572
$599$	32.3607	1.32222	0.661111	−	0.750288i	$-0.270085\pi$
$599$	32.3607	1.32222	0.661111	+	0.750288i	$0.270085\pi$
$600$	0	0
$601$	21.0557	0.858881	0.429441	−	0.903095i	$-0.358711\pi$
$601$	21.0557	0.858881	0.429441	+	0.903095i	$0.358711\pi$
$602$	20.0000	0.815139
$603$	0	0
$604$	−48.0000	−1.95309
$605$	−4.88854	−0.198748
$606$	0	0
$607$	−14.8328	−0.602045	−0.301023	−	0.953617i	$-0.597328\pi$
$607$	−14.8328	−0.602045	−0.301023	+	0.953617i	$0.597328\pi$
$608$	43.4164	1.76077
$609$	0	0
$610$	4.47214	0.181071
$611$	−22.1115	−0.894534
$612$	0	0
$613$	10.9443	0.442035	0.221017	−	0.975270i	$-0.429062\pi$
$613$	10.9443	0.442035	0.221017	+	0.975270i	$0.429062\pi$
$614$	73.6656	2.97290
$615$	0	0
$616$	−5.52786	−0.222724
$617$	−7.52786	−0.303060	−0.151530	−	0.988453i	$-0.548420\pi$
$617$	−7.52786	−0.303060	−0.151530	+	0.988453i	$0.548420\pi$
$618$	0	0
$619$	12.5836	0.505777	0.252889	−	0.967495i	$-0.418619\pi$
$619$	12.5836	0.505777	0.252889	+	0.967495i	$0.418619\pi$
$620$	31.4164	1.26171
$621$	0	0
$622$	−22.1115	−0.886589
$623$	2.00000	0.0801283
$624$	0	0
$625$	1.00000	0.0400000
$626$	−21.0557	−0.841556
$627$	0	0
$628$	25.4164	1.01423
$629$	21.8885	0.872753
$630$	0	0
$631$	−22.8328	−0.908960	−0.454480	−	0.890757i	$-0.650175\pi$
$631$	−22.8328	−0.908960	−0.454480	+	0.890757i	$0.650175\pi$
$632$	11.0557	0.439773
$633$	0	0
$634$	−67.8885	−2.69620
$635$	12.9443	0.513678
$636$	0	0
$637$	−4.47214	−0.177192
$638$	−11.0557	−0.437700
$639$	0	0
$640$	15.6525	0.618718
$641$	36.8328	1.45481	0.727404	−	0.686209i	$-0.240726\pi$
$641$	36.8328	1.45481	0.727404	+	0.686209i	$0.240726\pi$
$642$	0	0
$643$	−32.9443	−1.29920	−0.649598	−	0.760278i	$-0.725063\pi$
$643$	−32.9443	−1.29920	−0.649598	+	0.760278i	$0.725063\pi$
$644$	−12.0000	−0.472866
$645$	0	0
$646$	−28.9443	−1.13880
$647$	−33.8885	−1.33230	−0.666148	−	0.745820i	$-0.732058\pi$
$647$	−33.8885	−1.33230	−0.666148	+	0.745820i	$0.732058\pi$
$648$	0	0
$649$	−22.1115	−0.867951
$650$	10.0000	0.392232
$651$	0	0
$652$	2.83282	0.110942
$653$	49.4164	1.93381	0.966907	−	0.255130i	$-0.0821183\pi$
$653$	49.4164	1.93381	0.966907	+	0.255130i	$0.0821183\pi$
$654$	0	0
$655$	−4.00000	−0.156293
$656$	−2.00000	−0.0780869
$657$	0	0
$658$	−11.0557	−0.430997
$659$	41.3050	1.60901	0.804506	−	0.593944i	$-0.202430\pi$
$659$	41.3050	1.60901	0.804506	+	0.593944i	$0.202430\pi$
$660$	0	0
$661$	−0.111456	−0.00433514	−0.00216757	−	0.999998i	$-0.500690\pi$
$661$	−0.111456	−0.00433514	−0.00216757	+	0.999998i	$0.500690\pi$
$662$	37.8885	1.47258
$663$	0	0
$664$	2.11146	0.0819404
$665$	6.47214	0.250979
$666$	0	0
$667$	−8.00000	−0.309761
$668$	24.0000	0.928588
$669$	0	0
$670$	8.94427	0.345547
$671$	−4.94427	−0.190872
$672$	0	0
$673$	−44.8328	−1.72818	−0.864089	−	0.503339i	$-0.832105\pi$
$673$	−44.8328	−1.72818	−0.864089	+	0.503339i	$0.832105\pi$
$674$	−26.5836	−1.02396
$675$	0	0
$676$	21.0000	0.807692
$677$	−38.9443	−1.49675	−0.748375	−	0.663276i	$-0.769166\pi$
$677$	−38.9443	−1.49675	−0.748375	+	0.663276i	$0.769166\pi$
$678$	0	0
$679$	−0.472136	−0.0181189
$680$	−4.47214	−0.171499
$681$	0	0
$682$	−57.8885	−2.21667
$683$	−33.8885	−1.29671	−0.648355	−	0.761339i	$-0.724543\pi$
$683$	−33.8885	−1.29671	−0.648355	+	0.761339i	$0.724543\pi$
$684$	0	0
$685$	−12.4721	−0.476536
$686$	−2.23607	−0.0853735
$687$	0	0
$688$	8.94427	0.340997
$689$	−55.7771	−2.12494
$690$	0	0
$691$	−0.360680	−0.0137209	−0.00686045	−	0.999976i	$-0.502184\pi$
$691$	−0.360680	−0.0137209	−0.00686045	+	0.999976i	$0.502184\pi$
$692$	−44.8328	−1.70429
$693$	0	0
$694$	−17.8885	−0.679040
$695$	−19.4164	−0.736506
$696$	0	0
$697$	4.00000	0.151511
$698$	−53.4164	−2.02184
$699$	0	0
$700$	3.00000	0.113389
$701$	34.0000	1.28416	0.642081	−	0.766637i	$-0.278071\pi$
$701$	34.0000	1.28416	0.642081	+	0.766637i	$0.278071\pi$
$702$	0	0
$703$	70.8328	2.67151
$704$	−32.1378	−1.21124
$705$	0	0
$706$	−62.3607	−2.34698
$707$	14.0000	0.526524
$708$	0	0
$709$	−45.7771	−1.71919	−0.859597	−	0.510972i	$-0.829286\pi$
$709$	−45.7771	−1.71919	−0.859597	+	0.510972i	$0.829286\pi$
$710$	32.3607	1.21447
$711$	0	0
$712$	−4.47214	−0.167600
$713$	−41.8885	−1.56874
$714$	0	0
$715$	−11.0557	−0.413461
$716$	7.41641	0.277164
$717$	0	0
$718$	21.3050	0.795094
$719$	−46.8328	−1.74657	−0.873285	−	0.487210i	$-0.838015\pi$
$719$	−46.8328	−1.74657	−0.873285	+	0.487210i	$0.838015\pi$
$720$	0	0
$721$	0	0
$722$	−51.1803	−1.90474
$723$	0	0
$724$	56.8328	2.11217
$725$	2.00000	0.0742781
$726$	0	0
$727$	14.8328	0.550119	0.275059	−	0.961427i	$-0.411302\pi$
$727$	14.8328	0.550119	0.275059	+	0.961427i	$0.411302\pi$
$728$	10.0000	0.370625
$729$	0	0
$730$	7.88854	0.291968
$731$	−17.8885	−0.661632
$732$	0	0
$733$	37.4164	1.38201	0.691003	−	0.722852i	$-0.257169\pi$
$733$	37.4164	1.38201	0.691003	+	0.722852i	$0.257169\pi$
$734$	−46.8328	−1.72863
$735$	0	0
$736$	−26.8328	−0.989071
$737$	−9.88854	−0.364249
$738$	0	0
$739$	29.8885	1.09947	0.549734	−	0.835340i	$-0.314729\pi$
$739$	29.8885	1.09947	0.549734	+	0.835340i	$0.314729\pi$
$740$	32.8328	1.20696
$741$	0	0
$742$	−27.8885	−1.02382
$743$	18.8328	0.690909	0.345455	−	0.938436i	$-0.387725\pi$
$743$	18.8328	0.690909	0.345455	+	0.938436i	$0.387725\pi$
$744$	0	0
$745$	−2.94427	−0.107870
$746$	−13.4164	−0.491210
$747$	0	0
$748$	14.8328	0.542341
$749$	−4.94427	−0.180660
$750$	0	0
$751$	−3.05573	−0.111505	−0.0557526	−	0.998445i	$-0.517756\pi$
$751$	−3.05573	−0.111505	−0.0557526	+	0.998445i	$0.517756\pi$
$752$	−4.94427	−0.180299
$753$	0	0
$754$	20.0000	0.728357
$755$	−16.0000	−0.582300
$756$	0	0
$757$	−3.88854	−0.141332	−0.0706658	−	0.997500i	$-0.522512\pi$
$757$	−3.88854	−0.141332	−0.0706658	+	0.997500i	$0.522512\pi$
$758$	4.72136	0.171488
$759$	0	0
$760$	−14.4721	−0.524960
$761$	−7.88854	−0.285959	−0.142980	−	0.989726i	$-0.545668\pi$
$761$	−7.88854	−0.285959	−0.142980	+	0.989726i	$0.545668\pi$
$762$	0	0
$763$	−2.00000	−0.0724049
$764$	−82.2492	−2.97567
$765$	0	0
$766$	−17.8885	−0.646339
$767$	40.0000	1.44432
$768$	0	0
$769$	0.832816	0.0300321	0.0150161	−	0.999887i	$-0.495220\pi$
$769$	0.832816	0.0300321	0.0150161	+	0.999887i	$0.495220\pi$
$770$	−5.52786	−0.199210
$771$	0	0
$772$	−42.0000	−1.51161
$773$	25.0557	0.901192	0.450596	−	0.892728i	$-0.351212\pi$
$773$	25.0557	0.901192	0.450596	+	0.892728i	$0.351212\pi$
$774$	0	0
$775$	10.4721	0.376170
$776$	1.05573	0.0378984
$777$	0	0
$778$	24.4721	0.877369
$779$	12.9443	0.463777
$780$	0	0
$781$	−35.7771	−1.28020
$782$	17.8885	0.639693
$783$	0	0
$784$	−1.00000	−0.0357143
$785$	8.47214	0.302383
$786$	0	0
$787$	48.9443	1.74467	0.872337	−	0.488904i	$-0.162603\pi$
$787$	48.9443	1.74467	0.872337	+	0.488904i	$0.162603\pi$
$788$	−73.4164	−2.61535
$789$	0	0
$790$	11.0557	0.393345
$791$	8.47214	0.301234
$792$	0	0
$793$	8.94427	0.317620
$794$	−30.0000	−1.06466
$795$	0	0
$796$	1.75078	0.0620546
$797$	1.05573	0.0373958	0.0186979	−	0.999825i	$-0.494048\pi$
$797$	1.05573	0.0373958	0.0186979	+	0.999825i	$0.494048\pi$
$798$	0	0
$799$	9.88854	0.349832
$800$	6.70820	0.237171
$801$	0	0
$802$	22.3607	0.789583
$803$	−8.72136	−0.307770
$804$	0	0
$805$	−4.00000	−0.140981
$806$	104.721	3.68865
$807$	0	0
$808$	−31.3050	−1.10130
$809$	−21.0557	−0.740280	−0.370140	−	0.928976i	$-0.620690\pi$
$809$	−21.0557	−0.740280	−0.370140	+	0.928976i	$0.620690\pi$
$810$	0	0
$811$	28.5836	1.00371	0.501853	−	0.864953i	$-0.332652\pi$
$811$	28.5836	1.00371	0.501853	+	0.864953i	$0.332652\pi$
$812$	6.00000	0.210559
$813$	0	0
$814$	−60.4984	−2.12047
$815$	0.944272	0.0330764
$816$	0	0
$817$	−57.8885	−2.02526
$818$	53.4164	1.86766
$819$	0	0
$820$	6.00000	0.209529
$821$	37.7771	1.31843	0.659215	−	0.751955i	$-0.270889\pi$
$821$	37.7771	1.31843	0.659215	+	0.751955i	$0.270889\pi$
$822$	0	0
$823$	27.0557	0.943103	0.471552	−	0.881838i	$-0.343694\pi$
$823$	27.0557	0.943103	0.471552	+	0.881838i	$0.343694\pi$
$824$	0	0
$825$	0	0
$826$	20.0000	0.695889
$827$	4.94427	0.171929	0.0859646	−	0.996298i	$-0.472603\pi$
$827$	4.94427	0.171929	0.0859646	+	0.996298i	$0.472603\pi$
$828$	0	0
$829$	−30.9443	−1.07474	−0.537369	−	0.843347i	$-0.680582\pi$
$829$	−30.9443	−1.07474	−0.537369	+	0.843347i	$0.680582\pi$
$830$	2.11146	0.0732897
$831$	0	0
$832$	58.1378	2.01556
$833$	2.00000	0.0692959
$834$	0	0
$835$	8.00000	0.276851
$836$	48.0000	1.66011
$837$	0	0
$838$	13.1672	0.454853
$839$	−1.16718	−0.0402957	−0.0201478	−	0.999797i	$-0.506414\pi$
$839$	−1.16718	−0.0402957	−0.0201478	+	0.999797i	$0.506414\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	−49.1935	−1.69532
$843$	0	0
$844$	2.83282	0.0975095
$845$	7.00000	0.240807
$846$	0	0
$847$	−4.88854	−0.167972
$848$	−12.4721	−0.428295
$849$	0	0
$850$	−4.47214	−0.153393
$851$	−43.7771	−1.50066
$852$	0	0
$853$	31.3050	1.07186	0.535931	−	0.844262i	$-0.319961\pi$
$853$	31.3050	1.07186	0.535931	+	0.844262i	$0.319961\pi$
$854$	4.47214	0.153033
$855$	0	0
$856$	11.0557	0.377877
$857$	16.8328	0.574998	0.287499	−	0.957781i	$-0.407176\pi$
$857$	16.8328	0.574998	0.287499	+	0.957781i	$0.407176\pi$
$858$	0	0
$859$	41.5279	1.41691	0.708456	−	0.705755i	$-0.249392\pi$
$859$	41.5279	1.41691	0.708456	+	0.705755i	$0.249392\pi$
$860$	−26.8328	−0.914991
$861$	0	0
$862$	−21.3050	−0.725650
$863$	13.8885	0.472772	0.236386	−	0.971659i	$-0.424037\pi$
$863$	13.8885	0.472772	0.236386	+	0.971659i	$0.424037\pi$
$864$	0	0
$865$	−14.9443	−0.508120
$866$	−16.8328	−0.572002
$867$	0	0
$868$	31.4164	1.06634
$869$	−12.2229	−0.414634
$870$	0	0
$871$	17.8885	0.606130
$872$	4.47214	0.151446
$873$	0	0
$874$	57.8885	1.95811
$875$	1.00000	0.0338062
$876$	0	0
$877$	−3.16718	−0.106948	−0.0534741	−	0.998569i	$-0.517029\pi$
$877$	−3.16718	−0.106948	−0.0534741	+	0.998569i	$0.517029\pi$
$878$	−23.4164	−0.790265
$879$	0	0
$880$	−2.47214	−0.0833357
$881$	−7.88854	−0.265772	−0.132886	−	0.991131i	$-0.542424\pi$
$881$	−7.88854	−0.265772	−0.132886	+	0.991131i	$0.542424\pi$
$882$	0	0
$883$	−2.11146	−0.0710562	−0.0355281	−	0.999369i	$-0.511311\pi$
$883$	−2.11146	−0.0710562	−0.0355281	+	0.999369i	$0.511311\pi$
$884$	−26.8328	−0.902485
$885$	0	0
$886$	−17.8885	−0.600977
$887$	−22.8328	−0.766651	−0.383325	−	0.923613i	$-0.625221\pi$
$887$	−22.8328	−0.766651	−0.383325	+	0.923613i	$0.625221\pi$
$888$	0	0
$889$	12.9443	0.434137
$890$	−4.47214	−0.149906
$891$	0	0
$892$	14.8328	0.496639
$893$	32.0000	1.07084
$894$	0	0
$895$	2.47214	0.0826344
$896$	15.6525	0.522913
$897$	0	0
$898$	−31.3050	−1.04466
$899$	20.9443	0.698531
$900$	0	0
$901$	24.9443	0.831014
$902$	−11.0557	−0.368115
$903$	0	0
$904$	−18.9443	−0.630077
$905$	18.9443	0.629729
$906$	0	0
$907$	18.1115	0.601381	0.300691	−	0.953722i	$-0.402783\pi$
$907$	18.1115	0.601381	0.300691	+	0.953722i	$0.402783\pi$
$908$	50.8328	1.68695
$909$	0	0
$910$	10.0000	0.331497
$911$	34.2492	1.13473	0.567364	−	0.823467i	$-0.307963\pi$
$911$	34.2492	1.13473	0.567364	+	0.823467i	$0.307963\pi$
$912$	0	0
$913$	−2.33437	−0.0772563
$914$	24.4721	0.809466
$915$	0	0
$916$	−35.6656	−1.17843
$917$	−4.00000	−0.132092
$918$	0	0
$919$	−52.9443	−1.74647	−0.873235	−	0.487299i	$-0.837982\pi$
$919$	−52.9443	−1.74647	−0.873235	+	0.487299i	$0.837982\pi$
$920$	8.94427	0.294884
$921$	0	0
$922$	−71.3050	−2.34830
$923$	64.7214	2.13033
$924$	0	0
$925$	10.9443	0.359845
$926$	−6.83282	−0.224540
$927$	0	0
$928$	13.4164	0.440415
$929$	51.8885	1.70241	0.851204	−	0.524835i	$-0.175873\pi$
$929$	51.8885	1.70241	0.851204	+	0.524835i	$0.175873\pi$
$930$	0	0
$931$	6.47214	0.212116
$932$	−52.2492	−1.71148
$933$	0	0
$934$	20.0000	0.654420
$935$	4.94427	0.161695
$936$	0	0
$937$	−43.5279	−1.42199	−0.710997	−	0.703195i	$-0.751756\pi$
$937$	−43.5279	−1.42199	−0.710997	+	0.703195i	$0.751756\pi$
$938$	8.94427	0.292041
$939$	0	0
$940$	14.8328	0.483793
$941$	30.0000	0.977972	0.488986	−	0.872292i	$-0.337367\pi$
$941$	30.0000	0.977972	0.488986	+	0.872292i	$0.337367\pi$
$942$	0	0
$943$	−8.00000	−0.260516
$944$	8.94427	0.291111
$945$	0	0
$946$	49.4427	1.60752
$947$	−17.8885	−0.581300	−0.290650	−	0.956830i	$-0.593871\pi$
$947$	−17.8885	−0.581300	−0.290650	+	0.956830i	$0.593871\pi$
$948$	0	0
$949$	15.7771	0.512146
$950$	−14.4721	−0.469538
$951$	0	0
$952$	−4.47214	−0.144943
$953$	6.58359	0.213263	0.106632	−	0.994299i	$-0.465993\pi$
$953$	6.58359	0.213263	0.106632	+	0.994299i	$0.465993\pi$
$954$	0	0
$955$	−27.4164	−0.887174
$956$	4.58359	0.148244
$957$	0	0
$958$	40.0000	1.29234
$959$	−12.4721	−0.402746
$960$	0	0
$961$	78.6656	2.53760
$962$	109.443	3.52857
$963$	0	0
$964$	−3.16718	−0.102008
$965$	−14.0000	−0.450676
$966$	0	0
$967$	−9.88854	−0.317994	−0.158997	−	0.987279i	$-0.550826\pi$
$967$	−9.88854	−0.317994	−0.158997	+	0.987279i	$0.550826\pi$
$968$	10.9311	0.351339
$969$	0	0
$970$	1.05573	0.0338974
$971$	−23.0557	−0.739894	−0.369947	−	0.929053i	$-0.620624\pi$
$971$	−23.0557	−0.739894	−0.369947	+	0.929053i	$0.620624\pi$
$972$	0	0
$973$	−19.4164	−0.622461
$974$	6.83282	0.218938
$975$	0	0
$976$	2.00000	0.0640184
$977$	−57.4164	−1.83691	−0.918457	−	0.395521i	$-0.870564\pi$
$977$	−57.4164	−1.83691	−0.918457	+	0.395521i	$0.870564\pi$
$978$	0	0
$979$	4.94427	0.158020
$980$	3.00000	0.0958315
$981$	0	0
$982$	92.3607	2.94735
$983$	−30.8328	−0.983414	−0.491707	−	0.870761i	$-0.663627\pi$
$983$	−30.8328	−0.983414	−0.491707	+	0.870761i	$0.663627\pi$
$984$	0	0
$985$	−24.4721	−0.779747
$986$	−8.94427	−0.284844
$987$	0	0
$988$	−86.8328	−2.76252
$989$	35.7771	1.13765
$990$	0	0
$991$	−12.9443	−0.411188	−0.205594	−	0.978637i	$-0.565913\pi$
$991$	−12.9443	−0.411188	−0.205594	+	0.978637i	$0.565913\pi$
$992$	70.2492	2.23042
$993$	0	0
$994$	32.3607	1.02642
$995$	0.583592	0.0185011
$996$	0	0
$997$	21.4164	0.678264	0.339132	−	0.940739i	$-0.389867\pi$
$997$	21.4164	0.678264	0.339132	+	0.940739i	$0.389867\pi$
$998$	−48.9443	−1.54930
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.2.a.d.1.1		2
3.2	odd	2		105.2.a.b.1.2	✓	2
4.3	odd	2		5040.2.a.bw.1.1		2
5.2	odd	4		1575.2.d.d.1324.2		4
5.3	odd	4		1575.2.d.d.1324.3		4
5.4	even	2		1575.2.a.r.1.2		2
7.6	odd	2		2205.2.a.w.1.1		2
12.11	even	2		1680.2.a.v.1.2		2
15.2	even	4		525.2.d.c.274.4		4
15.8	even	4		525.2.d.c.274.1		4
15.14	odd	2		525.2.a.g.1.1		2
21.2	odd	6		735.2.i.k.361.1		4
21.5	even	6		735.2.i.i.361.1		4
21.11	odd	6		735.2.i.k.226.1		4
21.17	even	6		735.2.i.i.226.1		4
21.20	even	2		735.2.a.k.1.2		2
24.5	odd	2		6720.2.a.cx.1.2		2
24.11	even	2		6720.2.a.cs.1.1		2
60.59	even	2		8400.2.a.cx.1.2		2
105.104	even	2		3675.2.a.y.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.a.b.1.2	✓	2	3.2	odd	2
315.2.a.d.1.1		2	1.1	even	1	trivial
525.2.a.g.1.1		2	15.14	odd	2
525.2.d.c.274.1		4	15.8	even	4
525.2.d.c.274.4		4	15.2	even	4
735.2.a.k.1.2		2	21.20	even	2
735.2.i.i.226.1		4	21.17	even	6
735.2.i.i.361.1		4	21.5	even	6
735.2.i.k.226.1		4	21.11	odd	6
735.2.i.k.361.1		4	21.2	odd	6
1575.2.a.r.1.2		2	5.4	even	2
1575.2.d.d.1324.2		4	5.2	odd	4
1575.2.d.d.1324.3		4	5.3	odd	4
1680.2.a.v.1.2		2	12.11	even	2
2205.2.a.w.1.1		2	7.6	odd	2
3675.2.a.y.1.1		2	105.104	even	2
5040.2.a.bw.1.1		2	4.3	odd	2
6720.2.a.cs.1.1		2	24.11	even	2
6720.2.a.cx.1.2		2	24.5	odd	2
8400.2.a.cx.1.2		2	60.59	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:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	315.a (trivial)

\(f(q)\)	\(=\)	\(q-2.23607 q^{2} +3.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} -2.23607 q^{8} +O(q^{10})\)	\(q-2.23607 q^{2} +3.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} -2.23607 q^{8} -2.23607 q^{10} +2.47214 q^{11} -4.47214 q^{13} -2.23607 q^{14} -1.00000 q^{16} +2.00000 q^{17} +6.47214 q^{19} +3.00000 q^{20} -5.52786 q^{22} -4.00000 q^{23} +1.00000 q^{25} +10.0000 q^{26} +3.00000 q^{28} +2.00000 q^{29} +10.4721 q^{31} +6.70820 q^{32} -4.47214 q^{34} +1.00000 q^{35} +10.9443 q^{37} -14.4721 q^{38} -2.23607 q^{40} +2.00000 q^{41} -8.94427 q^{43} +7.41641 q^{44} +8.94427 q^{46} +4.94427 q^{47} +1.00000 q^{49} -2.23607 q^{50} -13.4164 q^{52} +12.4721 q^{53} +2.47214 q^{55} -2.23607 q^{56} -4.47214 q^{58} -8.94427 q^{59} -2.00000 q^{61} -23.4164 q^{62} -13.0000 q^{64} -4.47214 q^{65} -4.00000 q^{67} +6.00000 q^{68} -2.23607 q^{70} -14.4721 q^{71} -3.52786 q^{73} -24.4721 q^{74} +19.4164 q^{76} +2.47214 q^{77} -4.94427 q^{79} -1.00000 q^{80} -4.47214 q^{82} -0.944272 q^{83} +2.00000 q^{85} +20.0000 q^{86} -5.52786 q^{88} +2.00000 q^{89} -4.47214 q^{91} -12.0000 q^{92} -11.0557 q^{94} +6.47214 q^{95} -0.472136 q^{97} -2.23607 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{4} + 2 q^{5} + 2 q^{7}+O(q^{10}) \) 2 * q + 6 * q^4 + 2 * q^5 + 2 * q^7	\( 2 q + 6 q^{4} + 2 q^{5} + 2 q^{7} - 4 q^{11} - 2 q^{16} + 4 q^{17} + 4 q^{19} + 6 q^{20} - 20 q^{22} - 8 q^{23} + 2 q^{25} + 20 q^{26} + 6 q^{28} + 4 q^{29} + 12 q^{31} + 2 q^{35} + 4 q^{37} - 20 q^{38} + 4 q^{41} - 12 q^{44} - 8 q^{47} + 2 q^{49} + 16 q^{53} - 4 q^{55} - 4 q^{61} - 20 q^{62} - 26 q^{64} - 8 q^{67} + 12 q^{68} - 20 q^{71} - 16 q^{73} - 40 q^{74} + 12 q^{76} - 4 q^{77} + 8 q^{79} - 2 q^{80} + 16 q^{83} + 4 q^{85} + 40 q^{86} - 20 q^{88} + 4 q^{89} - 24 q^{92} - 40 q^{94} + 4 q^{95} + 8 q^{97}+O(q^{100}) \) 2 * q + 6 * q^4 + 2 * q^5 + 2 * q^7 - 4 * q^11 - 2 * q^16 + 4 * q^17 + 4 * q^19 + 6 * q^20 - 20 * q^22 - 8 * q^23 + 2 * q^25 + 20 * q^26 + 6 * q^28 + 4 * q^29 + 12 * q^31 + 2 * q^35 + 4 * q^37 - 20 * q^38 + 4 * q^41 - 12 * q^44 - 8 * q^47 + 2 * q^49 + 16 * q^53 - 4 * q^55 - 4 * q^61 - 20 * q^62 - 26 * q^64 - 8 * q^67 + 12 * q^68 - 20 * q^71 - 16 * q^73 - 40 * q^74 + 12 * q^76 - 4 * q^77 + 8 * q^79 - 2 * q^80 + 16 * q^83 + 4 * q^85 + 40 * q^86 - 20 * q^88 + 4 * q^89 - 24 * q^92 - 40 * q^94 + 4 * q^95 + 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more