LMFDB - Embedded newform 5265.2.a.t.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5265,2,Mod(1,5265)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5265, 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("5265.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5265 = 3^{4} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5265.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:	$42.0412366642$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.53209$ of defining polynomial
Character	$\chi$	$=$	5265.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.53209 q^{2} +4.41147 q^{4} -1.00000 q^{5} +2.41147 q^{7} +6.10607 q^{8} +O(q^{10})$	\(q+2.53209 q^{2} +4.41147 q^{4} -1.00000 q^{5} +2.41147 q^{7} +6.10607 q^{8} -2.53209 q^{10} -1.65270 q^{11} +1.00000 q^{13} +6.10607 q^{14} +6.63816 q^{16} +4.29086 q^{17} -3.22668 q^{19} -4.41147 q^{20} -4.18479 q^{22} +0.305407 q^{23} +1.00000 q^{25} +2.53209 q^{26} +10.6382 q^{28} +3.71688 q^{29} +0.453363 q^{31} +4.59627 q^{32} +10.8648 q^{34} -2.41147 q^{35} +5.26857 q^{37} -8.17024 q^{38} -6.10607 q^{40} +12.3550 q^{41} +7.00774 q^{43} -7.29086 q^{44} +0.773318 q^{46} +4.59627 q^{47} -1.18479 q^{49} +2.53209 q^{50} +4.41147 q^{52} -1.59627 q^{53} +1.65270 q^{55} +14.7246 q^{56} +9.41147 q^{58} +8.83750 q^{59} -1.89899 q^{61} +1.14796 q^{62} -1.63816 q^{64} -1.00000 q^{65} -8.72193 q^{67} +18.9290 q^{68} -6.10607 q^{70} +6.87939 q^{71} -2.04189 q^{73} +13.3405 q^{74} -14.2344 q^{76} -3.98545 q^{77} -0.319955 q^{79} -6.63816 q^{80} +31.2841 q^{82} +3.89899 q^{83} -4.29086 q^{85} +17.7442 q^{86} -10.0915 q^{88} -13.0223 q^{89} +2.41147 q^{91} +1.34730 q^{92} +11.6382 q^{94} +3.22668 q^{95} -11.8152 q^{97} -3.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} - 3 q^{7} + 6 q^{8}+O(q^{10}) $ 3 * q + 3 * q^2 + 3 * q^4 - 3 * q^5 - 3 * q^7 + 6 * q^8	$ 3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} - 3 q^{7} + 6 q^{8} - 3 q^{10} - 6 q^{11} + 3 q^{13} + 6 q^{14} + 3 q^{16} - 3 q^{17} - 3 q^{19} - 3 q^{20} - 9 q^{22} + 3 q^{23} + 3 q^{25} + 3 q^{26} + 15 q^{28} + 3 q^{29} - 12 q^{31} + 9 q^{34} + 3 q^{35} + 6 q^{37} - 3 q^{38} - 6 q^{40} + 12 q^{41} - 3 q^{43} - 6 q^{44} + 9 q^{46} + 3 q^{50} + 3 q^{52} + 9 q^{53} + 6 q^{55} + 12 q^{56} + 18 q^{58} + 24 q^{59} - 3 q^{61} - 12 q^{62} + 12 q^{64} - 3 q^{65} - 3 q^{67} + 24 q^{68} - 6 q^{70} + 15 q^{71} - 3 q^{73} - 3 q^{74} - 12 q^{76} + 6 q^{77} - 21 q^{79} - 3 q^{80} + 36 q^{82} + 9 q^{83} + 3 q^{85} + 24 q^{86} - 33 q^{89} - 3 q^{91} + 3 q^{92} + 18 q^{94} + 3 q^{95} - 39 q^{97} - 9 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 - 3 * q^5 - 3 * q^7 + 6 * q^8 - 3 * q^10 - 6 * q^11 + 3 * q^13 + 6 * q^14 + 3 * q^16 - 3 * q^17 - 3 * q^19 - 3 * q^20 - 9 * q^22 + 3 * q^23 + 3 * q^25 + 3 * q^26 + 15 * q^28 + 3 * q^29 - 12 * q^31 + 9 * q^34 + 3 * q^35 + 6 * q^37 - 3 * q^38 - 6 * q^40 + 12 * q^41 - 3 * q^43 - 6 * q^44 + 9 * q^46 + 3 * q^50 + 3 * q^52 + 9 * q^53 + 6 * q^55 + 12 * q^56 + 18 * q^58 + 24 * q^59 - 3 * q^61 - 12 * q^62 + 12 * q^64 - 3 * q^65 - 3 * q^67 + 24 * q^68 - 6 * q^70 + 15 * q^71 - 3 * q^73 - 3 * q^74 - 12 * q^76 + 6 * q^77 - 21 * q^79 - 3 * q^80 + 36 * q^82 + 9 * q^83 + 3 * q^85 + 24 * q^86 - 33 * q^89 - 3 * q^91 + 3 * q^92 + 18 * q^94 + 3 * q^95 - 39 * q^97 - 9 * q^98

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 $

$2$	2.53209	1.79046	0.895229	−	0.445607i	$-0.147012\pi$
$2$	2.53209	1.79046	0.895229	+	0.445607i	$0.147012\pi$
$3$	0	0
$3$	0	0
$4$	4.41147	2.20574
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	2.41147	0.911452	0.455726	−	0.890120i	$-0.349380\pi$
$7$	2.41147	0.911452	0.455726	+	0.890120i	$0.349380\pi$
$8$	6.10607	2.15882
$9$	0	0
$10$	−2.53209	−0.800717
$11$	−1.65270	−0.498309	−0.249154	−	0.968464i	$-0.580153\pi$
$11$	−1.65270	−0.498309	−0.249154	+	0.968464i	$0.580153\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	6.10607	1.63192
$15$	0	0
$16$	6.63816	1.65954
$17$	4.29086	1.04069	0.520343	−	0.853957i	$-0.325804\pi$
$17$	4.29086	1.04069	0.520343	+	0.853957i	$0.325804\pi$
$18$	0	0
$19$	−3.22668	−0.740252	−0.370126	−	0.928982i	$-0.620685\pi$
$19$	−3.22668	−0.740252	−0.370126	+	0.928982i	$0.620685\pi$
$20$	−4.41147	−0.986436
$21$	0	0
$22$	−4.18479	−0.892201
$23$	0.305407	0.0636818	0.0318409	−	0.999493i	$-0.489863\pi$
$23$	0.305407	0.0636818	0.0318409	+	0.999493i	$0.489863\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	2.53209	0.496583
$27$	0	0
$28$	10.6382	2.01042
$29$	3.71688	0.690208	0.345104	−	0.938565i	$-0.387844\pi$
$29$	3.71688	0.690208	0.345104	+	0.938565i	$0.387844\pi$
$30$	0	0
$31$	0.453363	0.0814264	0.0407132	−	0.999171i	$-0.487037\pi$
$31$	0.453363	0.0814264	0.0407132	+	0.999171i	$0.487037\pi$
$32$	4.59627	0.812513
$33$	0	0
$34$	10.8648	1.86330
$35$	−2.41147	−0.407614
$36$	0	0
$37$	5.26857	0.866148	0.433074	−	0.901358i	$-0.357429\pi$
$37$	5.26857	0.866148	0.433074	+	0.901358i	$0.357429\pi$
$38$	−8.17024	−1.32539
$39$	0	0
$40$	−6.10607	−0.965454
$41$	12.3550	1.92953	0.964766	−	0.263108i	$-0.0847476\pi$
$41$	12.3550	1.92953	0.964766	+	0.263108i	$0.0847476\pi$
$42$	0	0
$43$	7.00774	1.06867	0.534335	−	0.845273i	$-0.320562\pi$
$43$	7.00774	1.06867	0.534335	+	0.845273i	$0.320562\pi$
$44$	−7.29086	−1.09914
$45$	0	0
$46$	0.773318	0.114020
$47$	4.59627	0.670434	0.335217	−	0.942141i	$-0.391190\pi$
$47$	4.59627	0.670434	0.335217	+	0.942141i	$0.391190\pi$
$48$	0	0
$49$	−1.18479	−0.169256
$50$	2.53209	0.358091
$51$	0	0
$52$	4.41147	0.611761
$53$	−1.59627	−0.219264	−0.109632	−	0.993972i	$-0.534967\pi$
$53$	−1.59627	−0.219264	−0.109632	+	0.993972i	$0.534967\pi$
$54$	0	0
$55$	1.65270	0.222851
$56$	14.7246	1.96766
$57$	0	0
$58$	9.41147	1.23579
$59$	8.83750	1.15054	0.575272	−	0.817962i	$-0.304896\pi$
$59$	8.83750	1.15054	0.575272	+	0.817962i	$0.304896\pi$
$60$	0	0
$61$	−1.89899	−0.243140	−0.121570	−	0.992583i	$-0.538793\pi$
$61$	−1.89899	−0.243140	−0.121570	+	0.992583i	$0.538793\pi$
$62$	1.14796	0.145791
$63$	0	0
$64$	−1.63816	−0.204769
$65$	−1.00000	−0.124035
$66$	0	0
$67$	−8.72193	−1.06555	−0.532777	−	0.846256i	$-0.678852\pi$
$67$	−8.72193	−1.06555	−0.532777	+	0.846256i	$0.678852\pi$
$68$	18.9290	2.29548
$69$	0	0
$70$	−6.10607	−0.729815
$71$	6.87939	0.816433	0.408216	−	0.912885i	$-0.366151\pi$
$71$	6.87939	0.816433	0.408216	+	0.912885i	$0.366151\pi$
$72$	0	0
$73$	−2.04189	−0.238985	−0.119493	−	0.992835i	$-0.538127\pi$
$73$	−2.04189	−0.238985	−0.119493	+	0.992835i	$0.538127\pi$
$74$	13.3405	1.55080
$75$	0	0
$76$	−14.2344	−1.63280
$77$	−3.98545	−0.454184
$78$	0	0
$79$	−0.319955	−0.0359978	−0.0179989	−	0.999838i	$-0.505730\pi$
$79$	−0.319955	−0.0359978	−0.0179989	+	0.999838i	$0.505730\pi$
$80$	−6.63816	−0.742168
$81$	0	0
$82$	31.2841	3.45475
$83$	3.89899	0.427969	0.213985	−	0.976837i	$-0.431356\pi$
$83$	3.89899	0.427969	0.213985	+	0.976837i	$0.431356\pi$
$84$	0	0
$85$	−4.29086	−0.465409
$86$	17.7442	1.91341
$87$	0	0
$88$	−10.0915	−1.07576
$89$	−13.0223	−1.38036	−0.690180	−	0.723638i	$-0.742469\pi$
$89$	−13.0223	−1.38036	−0.690180	+	0.723638i	$0.742469\pi$
$90$	0	0
$91$	2.41147	0.252791
$92$	1.34730	0.140465
$93$	0	0
$94$	11.6382	1.20038
$95$	3.22668	0.331051
$96$	0	0
$97$	−11.8152	−1.19965	−0.599826	−	0.800130i	$-0.704764\pi$
$97$	−11.8152	−1.19965	−0.599826	+	0.800130i	$0.704764\pi$
$98$	−3.00000	−0.303046
$99$	0	0
$100$	4.41147	0.441147
$101$	2.26352	0.225228	0.112614	−	0.993639i	$-0.464078\pi$
$101$	2.26352	0.225228	0.112614	+	0.993639i	$0.464078\pi$
$102$	0	0
$103$	1.31996	0.130059	0.0650295	−	0.997883i	$-0.479286\pi$
$103$	1.31996	0.130059	0.0650295	+	0.997883i	$0.479286\pi$
$104$	6.10607	0.598749
$105$	0	0
$106$	−4.04189	−0.392583
$107$	−15.5672	−1.50494	−0.752468	−	0.658629i	$-0.771137\pi$
$107$	−15.5672	−1.50494	−0.752468	+	0.658629i	$0.771137\pi$
$108$	0	0
$109$	−9.90673	−0.948892	−0.474446	−	0.880285i	$-0.657352\pi$
$109$	−9.90673	−0.948892	−0.474446	+	0.880285i	$0.657352\pi$
$110$	4.18479	0.399004
$111$	0	0
$112$	16.0077	1.51259
$113$	13.4834	1.26841	0.634205	−	0.773165i	$-0.281327\pi$
$113$	13.4834	1.26841	0.634205	+	0.773165i	$0.281327\pi$
$114$	0	0
$115$	−0.305407	−0.0284794
$116$	16.3969	1.52242
$117$	0	0
$118$	22.3773	2.06000
$119$	10.3473	0.948535
$120$	0	0
$121$	−8.26857	−0.751688
$122$	−4.80840	−0.435332
$123$	0	0
$124$	2.00000	0.179605
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−4.00000	−0.354943	−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000	−0.354943	−0.177471	+	0.984126i	$0.556792\pi$
$128$	−13.3405	−1.17914
$129$	0	0
$130$	−2.53209	−0.222079
$131$	10.3277	0.902335	0.451167	−	0.892439i	$-0.351008\pi$
$131$	10.3277	0.902335	0.451167	+	0.892439i	$0.351008\pi$
$132$	0	0
$133$	−7.78106	−0.674703
$134$	−22.0847	−1.90783
$135$	0	0
$136$	26.2003	2.24665
$137$	0.467911	0.0399763	0.0199882	−	0.999800i	$-0.493637\pi$
$137$	0.467911	0.0399763	0.0199882	+	0.999800i	$0.493637\pi$
$138$	0	0
$139$	−18.7374	−1.58929	−0.794643	−	0.607076i	$-0.792342\pi$
$139$	−18.7374	−1.58929	−0.794643	+	0.607076i	$0.792342\pi$
$140$	−10.6382	−0.899088
$141$	0	0
$142$	17.4192	1.46179
$143$	−1.65270	−0.138206
$144$	0	0
$145$	−3.71688	−0.308670
$146$	−5.17024	−0.427892
$147$	0	0
$148$	23.2422	1.91049
$149$	8.34998	0.684057	0.342029	−	0.939690i	$-0.388886\pi$
$149$	8.34998	0.684057	0.342029	+	0.939690i	$0.388886\pi$
$150$	0	0
$151$	16.4688	1.34022	0.670108	−	0.742264i	$-0.266248\pi$
$151$	16.4688	1.34022	0.670108	+	0.742264i	$0.266248\pi$
$152$	−19.7023	−1.59807
$153$	0	0
$154$	−10.0915	−0.813198
$155$	−0.453363	−0.0364150
$156$	0	0
$157$	−8.59627	−0.686057	−0.343028	−	0.939325i	$-0.611453\pi$
$157$	−8.59627	−0.686057	−0.343028	+	0.939325i	$0.611453\pi$
$158$	−0.810155	−0.0644525
$159$	0	0
$160$	−4.59627	−0.363367
$161$	0.736482	0.0580429
$162$	0	0
$163$	−17.0155	−1.33276	−0.666378	−	0.745614i	$-0.732156\pi$
$163$	−17.0155	−1.33276	−0.666378	+	0.745614i	$0.732156\pi$
$164$	54.5039	4.25604
$165$	0	0
$166$	9.87258	0.766261
$167$	5.89393	0.456086	0.228043	−	0.973651i	$-0.426767\pi$
$167$	5.89393	0.456086	0.228043	+	0.973651i	$0.426767\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−10.8648	−0.833295
$171$	0	0
$172$	30.9145	2.35721
$173$	−4.68273	−0.356022	−0.178011	−	0.984029i	$-0.556966\pi$
$173$	−4.68273	−0.356022	−0.178011	+	0.984029i	$0.556966\pi$
$174$	0	0
$175$	2.41147	0.182290
$176$	−10.9709	−0.826963
$177$	0	0
$178$	−32.9736	−2.47148
$179$	10.2044	0.762712	0.381356	−	0.924428i	$-0.375457\pi$
$179$	10.2044	0.762712	0.381356	+	0.924428i	$0.375457\pi$
$180$	0	0
$181$	−25.7297	−1.91247	−0.956236	−	0.292597i	$-0.905481\pi$
$181$	−25.7297	−1.91247	−0.956236	+	0.292597i	$0.905481\pi$
$182$	6.10607	0.452612
$183$	0	0
$184$	1.86484	0.137478
$185$	−5.26857	−0.387353
$186$	0	0
$187$	−7.09152	−0.518583
$188$	20.2763	1.47880
$189$	0	0
$190$	8.17024	0.592732
$191$	0.182104	0.0131766	0.00658830	−	0.999978i	$-0.497903\pi$
$191$	0.182104	0.0131766	0.00658830	+	0.999978i	$0.497903\pi$
$192$	0	0
$193$	16.4953	1.18735	0.593677	−	0.804703i	$-0.297676\pi$
$193$	16.4953	1.18735	0.593677	+	0.804703i	$0.297676\pi$
$194$	−29.9172	−2.14793
$195$	0	0
$196$	−5.22668	−0.373334
$197$	−23.8699	−1.70066	−0.850330	−	0.526250i	$-0.823597\pi$
$197$	−23.8699	−1.70066	−0.850330	+	0.526250i	$0.823597\pi$
$198$	0	0
$199$	−1.00000	−0.0708881	−0.0354441	−	0.999372i	$-0.511285\pi$
$199$	−1.00000	−0.0708881	−0.0354441	+	0.999372i	$0.511285\pi$
$200$	6.10607	0.431764
$201$	0	0
$202$	5.73143	0.403262
$203$	8.96316	0.629091
$204$	0	0
$205$	−12.3550	−0.862913
$206$	3.34224	0.232865
$207$	0	0
$208$	6.63816	0.460273
$209$	5.33275	0.368874
$210$	0	0
$211$	25.8135	1.77707	0.888536	−	0.458808i	$-0.151723\pi$
$211$	25.8135	1.77707	0.888536	+	0.458808i	$0.151723\pi$
$212$	−7.04189	−0.483639
$213$	0	0
$214$	−39.4175	−2.69452
$215$	−7.00774	−0.477924
$216$	0	0
$217$	1.09327	0.0742162
$218$	−25.0847	−1.69895
$219$	0	0
$220$	7.29086	0.491550
$221$	4.29086	0.288634
$222$	0	0
$223$	22.3259	1.49506	0.747528	−	0.664231i	$-0.231241\pi$
$223$	22.3259	1.49506	0.747528	+	0.664231i	$0.231241\pi$
$224$	11.0838	0.740566
$225$	0	0
$226$	34.1411	2.27104
$227$	−14.4388	−0.958338	−0.479169	−	0.877723i	$-0.659062\pi$
$227$	−14.4388	−0.958338	−0.479169	+	0.877723i	$0.659062\pi$
$228$	0	0
$229$	11.1257	0.735205	0.367602	−	0.929983i	$-0.380179\pi$
$229$	11.1257	0.735205	0.367602	+	0.929983i	$0.380179\pi$
$230$	−0.773318	−0.0509911
$231$	0	0
$232$	22.6955	1.49003
$233$	13.9905	0.916548	0.458274	−	0.888811i	$-0.348468\pi$
$233$	13.9905	0.916548	0.458274	+	0.888811i	$0.348468\pi$
$234$	0	0
$235$	−4.59627	−0.299827
$236$	38.9864	2.53780
$237$	0	0
$238$	26.2003	1.69831
$239$	−8.41921	−0.544594	−0.272297	−	0.962213i	$-0.587783\pi$
$239$	−8.41921	−0.544594	−0.272297	+	0.962213i	$0.587783\pi$
$240$	0	0
$241$	16.4688	1.06085	0.530426	−	0.847731i	$-0.322032\pi$
$241$	16.4688	1.06085	0.530426	+	0.847731i	$0.322032\pi$
$242$	−20.9368	−1.34587
$243$	0	0
$244$	−8.37733	−0.536303
$245$	1.18479	0.0756936
$246$	0	0
$247$	−3.22668	−0.205309
$248$	2.76827	0.175785
$249$	0	0
$250$	−2.53209	−0.160143
$251$	23.3327	1.47275	0.736375	−	0.676574i	$-0.236536\pi$
$251$	23.3327	1.47275	0.736375	+	0.676574i	$0.236536\pi$
$252$	0	0
$253$	−0.504748	−0.0317332
$254$	−10.1284	−0.635510
$255$	0	0
$256$	−30.5030	−1.90644
$257$	14.5321	0.906487	0.453243	−	0.891387i	$-0.350267\pi$
$257$	14.5321	0.906487	0.453243	+	0.891387i	$0.350267\pi$
$258$	0	0
$259$	12.7050	0.789452
$260$	−4.41147	−0.273588
$261$	0	0
$262$	26.1506	1.61559
$263$	−19.7520	−1.21796	−0.608979	−	0.793186i	$-0.708421\pi$
$263$	−19.7520	−1.21796	−0.608979	+	0.793186i	$0.708421\pi$
$264$	0	0
$265$	1.59627	0.0980579
$266$	−19.7023	−1.20803
$267$	0	0
$268$	−38.4766	−2.35033
$269$	−3.03684	−0.185159	−0.0925796	−	0.995705i	$-0.529511\pi$
$269$	−3.03684	−0.185159	−0.0925796	+	0.995705i	$0.529511\pi$
$270$	0	0
$271$	9.77156	0.593580	0.296790	−	0.954943i	$-0.404084\pi$
$271$	9.77156	0.593580	0.296790	+	0.954943i	$0.404084\pi$
$272$	28.4834	1.72706
$273$	0	0
$274$	1.18479	0.0715759
$275$	−1.65270	−0.0996618
$276$	0	0
$277$	−19.6536	−1.18087	−0.590436	−	0.807084i	$-0.701044\pi$
$277$	−19.6536	−1.18087	−0.590436	+	0.807084i	$0.701044\pi$
$278$	−47.4448	−2.84555
$279$	0	0
$280$	−14.7246	−0.879964
$281$	−18.0993	−1.07971	−0.539856	−	0.841758i	$-0.681521\pi$
$281$	−18.0993	−1.07971	−0.539856	+	0.841758i	$0.681521\pi$
$282$	0	0
$283$	12.2517	0.728286	0.364143	−	0.931343i	$-0.381362\pi$
$283$	12.2517	0.728286	0.364143	+	0.931343i	$0.381362\pi$
$284$	30.3482	1.80084
$285$	0	0
$286$	−4.18479	−0.247452
$287$	29.7939	1.75868
$288$	0	0
$289$	1.41147	0.0830279
$290$	−9.41147	−0.552661
$291$	0	0
$292$	−9.00774	−0.527138
$293$	−12.5912	−0.735587	−0.367793	−	0.929908i	$-0.619887\pi$
$293$	−12.5912	−0.735587	−0.367793	+	0.929908i	$0.619887\pi$
$294$	0	0
$295$	−8.83750	−0.514539
$296$	32.1702	1.86986
$297$	0	0
$298$	21.1429	1.22478
$299$	0.305407	0.0176622
$300$	0	0
$301$	16.8990	0.974041
$302$	41.7006	2.39960
$303$	0	0
$304$	−21.4192	−1.22848
$305$	1.89899	0.108736
$306$	0	0
$307$	17.5371	1.00090	0.500449	−	0.865766i	$-0.333168\pi$
$307$	17.5371	1.00090	0.500449	+	0.865766i	$0.333168\pi$
$308$	−17.5817	−1.00181
$309$	0	0
$310$	−1.14796	−0.0651995
$311$	−25.4270	−1.44183	−0.720915	−	0.693023i	$-0.756278\pi$
$311$	−25.4270	−1.44183	−0.720915	+	0.693023i	$0.756278\pi$
$312$	0	0
$313$	−20.1584	−1.13942	−0.569710	−	0.821846i	$-0.692944\pi$
$313$	−20.1584	−1.13942	−0.569710	+	0.821846i	$0.692944\pi$
$314$	−21.7665	−1.22836
$315$	0	0
$316$	−1.41147	−0.0794016
$317$	−19.9145	−1.11851	−0.559254	−	0.828996i	$-0.688912\pi$
$317$	−19.9145	−1.11851	−0.559254	+	0.828996i	$0.688912\pi$
$318$	0	0
$319$	−6.14290	−0.343937
$320$	1.63816	0.0915757
$321$	0	0
$322$	1.86484	0.103923
$323$	−13.8452	−0.770370
$324$	0	0
$325$	1.00000	0.0554700
$326$	−43.0847	−2.38624
$327$	0	0
$328$	75.4407	4.16551
$329$	11.0838	0.611068
$330$	0	0
$331$	−8.18479	−0.449877	−0.224938	−	0.974373i	$-0.572218\pi$
$331$	−8.18479	−0.449877	−0.224938	+	0.974373i	$0.572218\pi$
$332$	17.2003	0.943988
$333$	0	0
$334$	14.9240	0.816603
$335$	8.72193	0.476530
$336$	0	0
$337$	−12.3259	−0.671437	−0.335718	−	0.941962i	$-0.608979\pi$
$337$	−12.3259	−0.671437	−0.335718	+	0.941962i	$0.608979\pi$
$338$	2.53209	0.137727
$339$	0	0
$340$	−18.9290	−1.02657
$341$	−0.749275	−0.0405755
$342$	0	0
$343$	−19.7374	−1.06572
$344$	42.7897	2.30707
$345$	0	0
$346$	−11.8571	−0.637441
$347$	12.8033	0.687320	0.343660	−	0.939094i	$-0.388333\pi$
$347$	12.8033	0.687320	0.343660	+	0.939094i	$0.388333\pi$
$348$	0	0
$349$	−18.7811	−1.00533	−0.502664	−	0.864482i	$-0.667647\pi$
$349$	−18.7811	−1.00533	−0.502664	+	0.864482i	$0.667647\pi$
$350$	6.10607	0.326383
$351$	0	0
$352$	−7.59627	−0.404882
$353$	−29.5749	−1.57411	−0.787057	−	0.616880i	$-0.788396\pi$
$353$	−29.5749	−1.57411	−0.787057	+	0.616880i	$0.788396\pi$
$354$	0	0
$355$	−6.87939	−0.365120
$356$	−57.4475	−3.04471
$357$	0	0
$358$	25.8384	1.36560
$359$	−0.344608	−0.0181877	−0.00909386	−	0.999959i	$-0.502895\pi$
$359$	−0.344608	−0.0181877	−0.00909386	+	0.999959i	$0.502895\pi$
$360$	0	0
$361$	−8.58853	−0.452028
$362$	−65.1498	−3.42420
$363$	0	0
$364$	10.6382	0.557591
$365$	2.04189	0.106877
$366$	0	0
$367$	−16.5107	−0.861853	−0.430927	−	0.902387i	$-0.641813\pi$
$367$	−16.5107	−0.861853	−0.430927	+	0.902387i	$0.641813\pi$
$368$	2.02734	0.105682
$369$	0	0
$370$	−13.3405	−0.693539
$371$	−3.84936	−0.199849
$372$	0	0
$373$	2.72967	0.141337	0.0706686	−	0.997500i	$-0.477487\pi$
$373$	2.72967	0.141337	0.0706686	+	0.997500i	$0.477487\pi$
$374$	−17.9564	−0.928501
$375$	0	0
$376$	28.0651	1.44735
$377$	3.71688	0.191429
$378$	0	0
$379$	−16.0232	−0.823058	−0.411529	−	0.911397i	$-0.635005\pi$
$379$	−16.0232	−0.823058	−0.411529	+	0.911397i	$0.635005\pi$
$380$	14.2344	0.730210
$381$	0	0
$382$	0.461104	0.0235921
$383$	−23.3131	−1.19125	−0.595623	−	0.803264i	$-0.703095\pi$
$383$	−23.3131	−1.19125	−0.595623	+	0.803264i	$0.703095\pi$
$384$	0	0
$385$	3.98545	0.203117
$386$	41.7674	2.12591
$387$	0	0
$388$	−52.1225	−2.64612
$389$	−17.9564	−0.910423	−0.455212	−	0.890383i	$-0.650436\pi$
$389$	−17.9564	−0.910423	−0.455212	+	0.890383i	$0.650436\pi$
$390$	0	0
$391$	1.31046	0.0662728
$392$	−7.23442	−0.365394
$393$	0	0
$394$	−60.4407	−3.04496
$395$	0.319955	0.0160987
$396$	0	0
$397$	−15.9736	−0.801692	−0.400846	−	0.916146i	$-0.631284\pi$
$397$	−15.9736	−0.801692	−0.400846	+	0.916146i	$0.631284\pi$
$398$	−2.53209	−0.126922
$399$	0	0
$400$	6.63816	0.331908
$401$	23.0009	1.14861	0.574306	−	0.818641i	$-0.305272\pi$
$401$	23.0009	1.14861	0.574306	+	0.818641i	$0.305272\pi$
$402$	0	0
$403$	0.453363	0.0225836
$404$	9.98545	0.496795
$405$	0	0
$406$	22.6955	1.12636
$407$	−8.70739	−0.431609
$408$	0	0
$409$	−7.03239	−0.347729	−0.173865	−	0.984770i	$-0.555626\pi$
$409$	−7.03239	−0.347729	−0.173865	+	0.984770i	$0.555626\pi$
$410$	−31.2841	−1.54501
$411$	0	0
$412$	5.82295	0.286876
$413$	21.3114	1.04867
$414$	0	0
$415$	−3.89899	−0.191394
$416$	4.59627	0.225351
$417$	0	0
$418$	13.5030	0.660453
$419$	22.4730	1.09788	0.548938	−	0.835863i	$-0.315032\pi$
$419$	22.4730	1.09788	0.548938	+	0.835863i	$0.315032\pi$
$420$	0	0
$421$	21.2344	1.03490	0.517451	−	0.855713i	$-0.326881\pi$
$421$	21.2344	1.03490	0.517451	+	0.855713i	$0.326881\pi$
$422$	65.3620	3.18177
$423$	0	0
$424$	−9.74691	−0.473352
$425$	4.29086	0.208137
$426$	0	0
$427$	−4.57935	−0.221611
$428$	−68.6742	−3.31949
$429$	0	0
$430$	−17.7442	−0.855702
$431$	−23.2395	−1.11941	−0.559703	−	0.828693i	$-0.689085\pi$
$431$	−23.2395	−1.11941	−0.559703	+	0.828693i	$0.689085\pi$
$432$	0	0
$433$	32.6536	1.56923	0.784617	−	0.619981i	$-0.212860\pi$
$433$	32.6536	1.56923	0.784617	+	0.619981i	$0.212860\pi$
$434$	2.76827	0.132881
$435$	0	0
$436$	−43.7033	−2.09301
$437$	−0.985452	−0.0471406
$438$	0	0
$439$	4.49525	0.214547	0.107273	−	0.994230i	$-0.465788\pi$
$439$	4.49525	0.214547	0.107273	+	0.994230i	$0.465788\pi$
$440$	10.0915	0.481094
$441$	0	0
$442$	10.8648	0.516788
$443$	−27.6709	−1.31468	−0.657341	−	0.753593i	$-0.728319\pi$
$443$	−27.6709	−1.31468	−0.657341	+	0.753593i	$0.728319\pi$
$444$	0	0
$445$	13.0223	0.617316
$446$	56.5313	2.67683
$447$	0	0
$448$	−3.95037	−0.186637
$449$	−35.3628	−1.66887	−0.834436	−	0.551104i	$-0.814207\pi$
$449$	−35.3628	−1.66887	−0.834436	+	0.551104i	$0.814207\pi$
$450$	0	0
$451$	−20.4192	−0.961503
$452$	59.4816	2.79778
$453$	0	0
$454$	−36.5604	−1.71586
$455$	−2.41147	−0.113052
$456$	0	0
$457$	−22.3851	−1.04713	−0.523565	−	0.851986i	$-0.675398\pi$
$457$	−22.3851	−1.04713	−0.523565	+	0.851986i	$0.675398\pi$
$458$	28.1712	1.31635
$459$	0	0
$460$	−1.34730	−0.0628180
$461$	41.9864	1.95550	0.977750	−	0.209771i	$-0.0672720\pi$
$461$	41.9864	1.95550	0.977750	+	0.209771i	$0.0672720\pi$
$462$	0	0
$463$	13.7050	0.636926	0.318463	−	0.947935i	$-0.396833\pi$
$463$	13.7050	0.636926	0.318463	+	0.947935i	$0.396833\pi$
$464$	24.6732	1.14543
$465$	0	0
$466$	35.4252	1.64104
$467$	16.4534	0.761371	0.380685	−	0.924705i	$-0.375688\pi$
$467$	16.4534	0.761371	0.380685	+	0.924705i	$0.375688\pi$
$468$	0	0
$469$	−21.0327	−0.971201
$470$	−11.6382	−0.536828
$471$	0	0
$472$	53.9623	2.48382
$473$	−11.5817	−0.532528
$474$	0	0
$475$	−3.22668	−0.148050
$476$	45.6468	2.09222
$477$	0	0
$478$	−21.3182	−0.975072
$479$	−0.172933	−0.00790151	−0.00395075	−	0.999992i	$-0.501258\pi$
$479$	−0.172933	−0.00790151	−0.00395075	+	0.999992i	$0.501258\pi$
$480$	0	0
$481$	5.26857	0.240226
$482$	41.7006	1.89941
$483$	0	0
$484$	−36.4766	−1.65803
$485$	11.8152	0.536501
$486$	0	0
$487$	−12.3351	−0.558957	−0.279479	−	0.960152i	$-0.590162\pi$
$487$	−12.3351	−0.558957	−0.279479	+	0.960152i	$0.590162\pi$
$488$	−11.5953	−0.524896
$489$	0	0
$490$	3.00000	0.135526
$491$	15.3723	0.693741	0.346870	−	0.937913i	$-0.387244\pi$
$491$	15.3723	0.693741	0.346870	+	0.937913i	$0.387244\pi$
$492$	0	0
$493$	15.9486	0.718290
$494$	−8.17024	−0.367597
$495$	0	0
$496$	3.00950	0.135130
$497$	16.5895	0.744139
$498$	0	0
$499$	−26.4037	−1.18199	−0.590997	−	0.806674i	$-0.701265\pi$
$499$	−26.4037	−1.18199	−0.590997	+	0.806674i	$0.701265\pi$
$500$	−4.41147	−0.197287
$501$	0	0
$502$	59.0806	2.63690
$503$	34.7692	1.55028	0.775141	−	0.631788i	$-0.217679\pi$
$503$	34.7692	1.55028	0.775141	+	0.631788i	$0.217679\pi$
$504$	0	0
$505$	−2.26352	−0.100725
$506$	−1.27807	−0.0568170
$507$	0	0
$508$	−17.6459	−0.782910
$509$	25.7151	1.13980	0.569901	−	0.821713i	$-0.306981\pi$
$509$	25.7151	1.13980	0.569901	+	0.821713i	$0.306981\pi$
$510$	0	0
$511$	−4.92396	−0.217823
$512$	−50.5553	−2.23425
$513$	0	0
$514$	36.7965	1.62303
$515$	−1.31996	−0.0581642
$516$	0	0
$517$	−7.59627	−0.334083
$518$	32.1702	1.41348
$519$	0	0
$520$	−6.10607	−0.267769
$521$	20.6382	0.904174	0.452087	−	0.891974i	$-0.350680\pi$
$521$	20.6382	0.904174	0.452087	+	0.891974i	$0.350680\pi$
$522$	0	0
$523$	−4.50475	−0.196979	−0.0984894	−	0.995138i	$-0.531401\pi$
$523$	−4.50475	−0.196979	−0.0984894	+	0.995138i	$0.531401\pi$
$524$	45.5604	1.99031
$525$	0	0
$526$	−50.0137	−2.18070
$527$	1.94532	0.0847394
$528$	0	0
$529$	−22.9067	−0.995945
$530$	4.04189	0.175568
$531$	0	0
$532$	−34.3259	−1.48822
$533$	12.3550	0.535156
$534$	0	0
$535$	15.5672	0.673027
$536$	−53.2567	−2.30034
$537$	0	0
$538$	−7.68954	−0.331520
$539$	1.95811	0.0843418
$540$	0	0
$541$	14.4783	0.622472	0.311236	−	0.950333i	$-0.399257\pi$
$541$	14.4783	0.622472	0.311236	+	0.950333i	$0.399257\pi$
$542$	24.7425	1.06278
$543$	0	0
$544$	19.7219	0.845571
$545$	9.90673	0.424358
$546$	0	0
$547$	−3.27631	−0.140085	−0.0700425	−	0.997544i	$-0.522313\pi$
$547$	−3.27631	−0.140085	−0.0700425	+	0.997544i	$0.522313\pi$
$548$	2.06418	0.0881773
$549$	0	0
$550$	−4.18479	−0.178440
$551$	−11.9932	−0.510927
$552$	0	0
$553$	−0.771564	−0.0328102
$554$	−49.7648	−2.11430
$555$	0	0
$556$	−82.6596	−3.50555
$557$	−18.6432	−0.789938	−0.394969	−	0.918694i	$-0.629245\pi$
$557$	−18.6432	−0.789938	−0.394969	+	0.918694i	$0.629245\pi$
$558$	0	0
$559$	7.00774	0.296396
$560$	−16.0077	−0.676451
$561$	0	0
$562$	−45.8289	−1.93318
$563$	7.79561	0.328546	0.164273	−	0.986415i	$-0.447472\pi$
$563$	7.79561	0.328546	0.164273	+	0.986415i	$0.447472\pi$
$564$	0	0
$565$	−13.4834	−0.567251
$566$	31.0223	1.30396
$567$	0	0
$568$	42.0060	1.76253
$569$	−19.9445	−0.836117	−0.418058	−	0.908420i	$-0.637289\pi$
$569$	−19.9445	−0.836117	−0.418058	+	0.908420i	$0.637289\pi$
$570$	0	0
$571$	−35.7888	−1.49771	−0.748857	−	0.662731i	$-0.769397\pi$
$571$	−35.7888	−1.49771	−0.748857	+	0.662731i	$0.769397\pi$
$572$	−7.29086	−0.304846
$573$	0	0
$574$	75.4407	3.14883
$575$	0.305407	0.0127364
$576$	0	0
$577$	−5.10876	−0.212680	−0.106340	−	0.994330i	$-0.533913\pi$
$577$	−5.10876	−0.212680	−0.106340	+	0.994330i	$0.533913\pi$
$578$	3.57398	0.148658
$579$	0	0
$580$	−16.3969	−0.680845
$581$	9.40230	0.390073
$582$	0	0
$583$	2.63816	0.109261
$584$	−12.4679	−0.515926
$585$	0	0
$586$	−31.8821	−1.31704
$587$	37.8580	1.56257	0.781284	−	0.624176i	$-0.214565\pi$
$587$	37.8580	1.56257	0.781284	+	0.624176i	$0.214565\pi$
$588$	0	0
$589$	−1.46286	−0.0602760
$590$	−22.3773	−0.921260
$591$	0	0
$592$	34.9736	1.43741
$593$	−45.5431	−1.87023	−0.935116	−	0.354342i	$-0.884705\pi$
$593$	−45.5431	−1.87023	−0.935116	+	0.354342i	$0.884705\pi$
$594$	0	0
$595$	−10.3473	−0.424198
$596$	36.8357	1.50885
$597$	0	0
$598$	0.773318	0.0316233
$599$	0.647651	0.0264623	0.0132312	−	0.999912i	$-0.495788\pi$
$599$	0.647651	0.0264623	0.0132312	+	0.999912i	$0.495788\pi$
$600$	0	0
$601$	37.3928	1.52528	0.762642	−	0.646821i	$-0.223902\pi$
$601$	37.3928	1.52528	0.762642	+	0.646821i	$0.223902\pi$
$602$	42.7897	1.74398
$603$	0	0
$604$	72.6519	2.95616
$605$	8.26857	0.336165
$606$	0	0
$607$	6.11793	0.248319	0.124159	−	0.992262i	$-0.460377\pi$
$607$	6.11793	0.248319	0.124159	+	0.992262i	$0.460377\pi$
$608$	−14.8307	−0.601464
$609$	0	0
$610$	4.80840	0.194686
$611$	4.59627	0.185945
$612$	0	0
$613$	12.0479	0.486609	0.243305	−	0.969950i	$-0.421769\pi$
$613$	12.0479	0.486609	0.243305	+	0.969950i	$0.421769\pi$
$614$	44.4056	1.79206
$615$	0	0
$616$	−24.3354	−0.980503
$617$	−2.74186	−0.110383	−0.0551915	−	0.998476i	$-0.517577\pi$
$617$	−2.74186	−0.110383	−0.0551915	+	0.998476i	$0.517577\pi$
$618$	0	0
$619$	−12.2763	−0.493427	−0.246713	−	0.969089i	$-0.579351\pi$
$619$	−12.2763	−0.493427	−0.246713	+	0.969089i	$0.579351\pi$
$620$	−2.00000	−0.0803219
$621$	0	0
$622$	−64.3833	−2.58154
$623$	−31.4029	−1.25813
$624$	0	0
$625$	1.00000	0.0400000
$626$	−51.0428	−2.04008
$627$	0	0
$628$	−37.9222	−1.51326
$629$	22.6067	0.901388
$630$	0	0
$631$	−31.7033	−1.26209	−0.631044	−	0.775747i	$-0.717373\pi$
$631$	−31.7033	−1.26209	−0.631044	+	0.775747i	$0.717373\pi$
$632$	−1.95367	−0.0777127
$633$	0	0
$634$	−50.4252	−2.00264
$635$	4.00000	0.158735
$636$	0	0
$637$	−1.18479	−0.0469432
$638$	−15.5544	−0.615804
$639$	0	0
$640$	13.3405	0.527329
$641$	29.6509	1.17114	0.585571	−	0.810621i	$-0.300870\pi$
$641$	29.6509	1.17114	0.585571	+	0.810621i	$0.300870\pi$
$642$	0	0
$643$	−30.2327	−1.19226	−0.596130	−	0.802888i	$-0.703296\pi$
$643$	−30.2327	−1.19226	−0.596130	+	0.802888i	$0.703296\pi$
$644$	3.24897	0.128027
$645$	0	0
$646$	−35.0574	−1.37931
$647$	35.7246	1.40448	0.702240	−	0.711940i	$-0.252183\pi$
$647$	35.7246	1.40448	0.702240	+	0.711940i	$0.252183\pi$
$648$	0	0
$649$	−14.6058	−0.573326
$650$	2.53209	0.0993167
$651$	0	0
$652$	−75.0634	−2.93971
$653$	0.282185	0.0110427	0.00552137	−	0.999985i	$-0.498242\pi$
$653$	0.282185	0.0110427	0.00552137	+	0.999985i	$0.498242\pi$
$654$	0	0
$655$	−10.3277	−0.403536
$656$	82.0147	3.20213
$657$	0	0
$658$	28.0651	1.09409
$659$	−14.6081	−0.569052	−0.284526	−	0.958668i	$-0.591836\pi$
$659$	−14.6081	−0.569052	−0.284526	+	0.958668i	$0.591836\pi$
$660$	0	0
$661$	−45.9299	−1.78647	−0.893234	−	0.449592i	$-0.851569\pi$
$661$	−45.9299	−1.78647	−0.893234	+	0.449592i	$0.851569\pi$
$662$	−20.7246	−0.805485
$663$	0	0
$664$	23.8075	0.923909
$665$	7.78106	0.301737
$666$	0	0
$667$	1.13516	0.0439537
$668$	26.0009	1.00601
$669$	0	0
$670$	22.0847	0.853207
$671$	3.13846	0.121159
$672$	0	0
$673$	−21.3756	−0.823968	−0.411984	−	0.911191i	$-0.635164\pi$
$673$	−21.3756	−0.823968	−0.411984	+	0.911191i	$0.635164\pi$
$674$	−31.2104	−1.20218
$675$	0	0
$676$	4.41147	0.169672
$677$	11.4780	0.441136	0.220568	−	0.975372i	$-0.429209\pi$
$677$	11.4780	0.441136	0.220568	+	0.975372i	$0.429209\pi$
$678$	0	0
$679$	−28.4921	−1.09343
$680$	−26.2003	−1.00473
$681$	0	0
$682$	−1.89723	−0.0726487
$683$	−35.3688	−1.35335	−0.676674	−	0.736283i	$-0.736579\pi$
$683$	−35.3688	−1.35335	−0.676674	+	0.736283i	$0.736579\pi$
$684$	0	0
$685$	−0.467911	−0.0178780
$686$	−49.9769	−1.90813
$687$	0	0
$688$	46.5185	1.77350
$689$	−1.59627	−0.0608129
$690$	0	0
$691$	−40.2746	−1.53212	−0.766058	−	0.642771i	$-0.777785\pi$
$691$	−40.2746	−1.53212	−0.766058	+	0.642771i	$0.777785\pi$
$692$	−20.6578	−0.785290
$693$	0	0
$694$	32.4192	1.23062
$695$	18.7374	0.710751
$696$	0	0
$697$	53.0137	2.00804
$698$	−47.5553	−1.80000
$699$	0	0
$700$	10.6382	0.402084
$701$	−37.4826	−1.41570	−0.707849	−	0.706364i	$-0.750334\pi$
$701$	−37.4826	−1.41570	−0.707849	+	0.706364i	$0.750334\pi$
$702$	0	0
$703$	−17.0000	−0.641167
$704$	2.70739	0.102038
$705$	0	0
$706$	−74.8863	−2.81838
$707$	5.45842	0.205285
$708$	0	0
$709$	40.1566	1.50811	0.754057	−	0.656809i	$-0.228094\pi$
$709$	40.1566	1.50811	0.754057	+	0.656809i	$0.228094\pi$
$710$	−17.4192	−0.653731
$711$	0	0
$712$	−79.5150	−2.97995
$713$	0.138460	0.00518538
$714$	0	0
$715$	1.65270	0.0618076
$716$	45.0164	1.68234
$717$	0	0
$718$	−0.872578	−0.0325643
$719$	−17.4884	−0.652209	−0.326104	−	0.945334i	$-0.605736\pi$
$719$	−17.4884	−0.652209	−0.326104	+	0.945334i	$0.605736\pi$
$720$	0	0
$721$	3.18304	0.118543
$722$	−21.7469	−0.809336
$723$	0	0
$724$	−113.506	−4.21841
$725$	3.71688	0.138042
$726$	0	0
$727$	49.1070	1.82128	0.910639	−	0.413203i	$-0.135590\pi$
$727$	49.1070	1.82128	0.910639	+	0.413203i	$0.135590\pi$
$728$	14.7246	0.545731
$729$	0	0
$730$	5.17024	0.191359
$731$	30.0692	1.11215
$732$	0	0
$733$	−17.5790	−0.649296	−0.324648	−	0.945835i	$-0.605246\pi$
$733$	−17.5790	−0.649296	−0.324648	+	0.945835i	$0.605246\pi$
$734$	−41.8066	−1.54311
$735$	0	0
$736$	1.40373	0.0517423
$737$	14.4148	0.530975
$738$	0	0
$739$	−48.9067	−1.79906	−0.899532	−	0.436856i	$-0.856092\pi$
$739$	−48.9067	−1.79906	−0.899532	+	0.436856i	$0.856092\pi$
$740$	−23.2422	−0.854399
$741$	0	0
$742$	−9.74691	−0.357820
$743$	29.2523	1.07316	0.536581	−	0.843849i	$-0.319716\pi$
$743$	29.2523	1.07316	0.536581	+	0.843849i	$0.319716\pi$
$744$	0	0
$745$	−8.34998	−0.305920
$746$	6.91178	0.253058
$747$	0	0
$748$	−31.2841	−1.14386
$749$	−37.5398	−1.37168
$750$	0	0
$751$	−4.96585	−0.181207	−0.0906033	−	0.995887i	$-0.528880\pi$
$751$	−4.96585	−0.181207	−0.0906033	+	0.995887i	$0.528880\pi$
$752$	30.5107	1.11261
$753$	0	0
$754$	9.41147	0.342746
$755$	−16.4688	−0.599363
$756$	0	0
$757$	18.4766	0.671543	0.335771	−	0.941943i	$-0.391003\pi$
$757$	18.4766	0.671543	0.335771	+	0.941943i	$0.391003\pi$
$758$	−40.5722	−1.47365
$759$	0	0
$760$	19.7023	0.714679
$761$	10.9273	0.396113	0.198056	−	0.980191i	$-0.436537\pi$
$761$	10.9273	0.396113	0.198056	+	0.980191i	$0.436537\pi$
$762$	0	0
$763$	−23.8898	−0.864869
$764$	0.803348	0.0290641
$765$	0	0
$766$	−59.0310	−2.13288
$767$	8.83750	0.319103
$768$	0	0
$769$	40.1070	1.44630	0.723148	−	0.690693i	$-0.242695\pi$
$769$	40.1070	1.44630	0.723148	+	0.690693i	$0.242695\pi$
$770$	10.0915	0.363673
$771$	0	0
$772$	72.7684	2.61899
$773$	13.4397	0.483394	0.241697	−	0.970352i	$-0.422296\pi$
$773$	13.4397	0.483394	0.241697	+	0.970352i	$0.422296\pi$
$774$	0	0
$775$	0.453363	0.0162853
$776$	−72.1444	−2.58983
$777$	0	0
$778$	−45.4671	−1.63007
$779$	−39.8658	−1.42834
$780$	0	0
$781$	−11.3696	−0.406836
$782$	3.31820	0.118659
$783$	0	0
$784$	−7.86484	−0.280887
$785$	8.59627	0.306814
$786$	0	0
$787$	−20.0327	−0.714089	−0.357045	−	0.934087i	$-0.616216\pi$
$787$	−20.0327	−0.714089	−0.357045	+	0.934087i	$0.616216\pi$
$788$	−105.301	−3.75121
$789$	0	0
$790$	0.810155	0.0288240
$791$	32.5149	1.15610
$792$	0	0
$793$	−1.89899	−0.0674350
$794$	−40.4466	−1.43539
$795$	0	0
$796$	−4.41147	−0.156361
$797$	−3.38743	−0.119989	−0.0599945	−	0.998199i	$-0.519108\pi$
$797$	−3.38743	−0.119989	−0.0599945	+	0.998199i	$0.519108\pi$
$798$	0	0
$799$	19.7219	0.697712
$800$	4.59627	0.162503
$801$	0	0
$802$	58.2404	2.05654
$803$	3.37464	0.119088
$804$	0	0
$805$	−0.736482	−0.0259576
$806$	1.14796	0.0404350
$807$	0	0
$808$	13.8212	0.486228
$809$	40.0259	1.40724	0.703618	−	0.710578i	$-0.251567\pi$
$809$	40.0259	1.40724	0.703618	+	0.710578i	$0.251567\pi$
$810$	0	0
$811$	5.06687	0.177922	0.0889609	−	0.996035i	$-0.471645\pi$
$811$	5.06687	0.177922	0.0889609	+	0.996035i	$0.471645\pi$
$812$	39.5408	1.38761
$813$	0	0
$814$	−22.0479	−0.772778
$815$	17.0155	0.596026
$816$	0	0
$817$	−22.6117	−0.791085
$818$	−17.8066	−0.622595
$819$	0	0
$820$	−54.5039	−1.90336
$821$	−37.3233	−1.30259	−0.651295	−	0.758824i	$-0.725774\pi$
$821$	−37.3233	−1.30259	−0.651295	+	0.758824i	$0.725774\pi$
$822$	0	0
$823$	25.4688	0.887788	0.443894	−	0.896079i	$-0.353597\pi$
$823$	25.4688	0.887788	0.443894	+	0.896079i	$0.353597\pi$
$824$	8.05973	0.280774
$825$	0	0
$826$	53.9623	1.87759
$827$	−32.4020	−1.12673	−0.563364	−	0.826209i	$-0.690493\pi$
$827$	−32.4020	−1.12673	−0.563364	+	0.826209i	$0.690493\pi$
$828$	0	0
$829$	29.8786	1.03773	0.518863	−	0.854858i	$-0.326356\pi$
$829$	29.8786	1.03773	0.518863	+	0.854858i	$0.326356\pi$
$830$	−9.87258	−0.342682
$831$	0	0
$832$	−1.63816	−0.0567928
$833$	−5.08378	−0.176142
$834$	0	0
$835$	−5.89393	−0.203968
$836$	23.5253	0.813639
$837$	0	0
$838$	56.9035	1.96570
$839$	−20.1002	−0.693936	−0.346968	−	0.937877i	$-0.612789\pi$
$839$	−20.1002	−0.693936	−0.346968	+	0.937877i	$0.612789\pi$
$840$	0	0
$841$	−15.1848	−0.523614
$842$	53.7674	1.85295
$843$	0	0
$844$	113.875	3.91975
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	−19.9394	−0.685127
$848$	−10.5963	−0.363877
$849$	0	0
$850$	10.8648	0.372661
$851$	1.60906	0.0551579
$852$	0	0
$853$	8.59802	0.294391	0.147195	−	0.989107i	$-0.452975\pi$
$853$	8.59802	0.294391	0.147195	+	0.989107i	$0.452975\pi$
$854$	−11.5953	−0.396784
$855$	0	0
$856$	−95.0542	−3.24889
$857$	50.9796	1.74143	0.870715	−	0.491789i	$-0.163657\pi$
$857$	50.9796	1.74143	0.870715	+	0.491789i	$0.163657\pi$
$858$	0	0
$859$	4.40198	0.150194	0.0750968	−	0.997176i	$-0.476073\pi$
$859$	4.40198	0.150194	0.0750968	+	0.997176i	$0.476073\pi$
$860$	−30.9145	−1.05417
$861$	0	0
$862$	−58.8444	−2.00425
$863$	30.0152	1.02173	0.510864	−	0.859662i	$-0.329326\pi$
$863$	30.0152	1.02173	0.510864	+	0.859662i	$0.329326\pi$
$864$	0	0
$865$	4.68273	0.159218
$866$	82.6819	2.80965
$867$	0	0
$868$	4.82295	0.163702
$869$	0.528791	0.0179380
$870$	0	0
$871$	−8.72193	−0.295531
$872$	−60.4911	−2.04849
$873$	0	0
$874$	−2.49525	−0.0844032
$875$	−2.41147	−0.0815227
$876$	0	0
$877$	41.1753	1.39039	0.695195	−	0.718821i	$-0.255318\pi$
$877$	41.1753	1.39039	0.695195	+	0.718821i	$0.255318\pi$
$878$	11.3824	0.384137
$879$	0	0
$880$	10.9709	0.369829
$881$	−23.1726	−0.780705	−0.390353	−	0.920665i	$-0.627647\pi$
$881$	−23.1726	−0.780705	−0.390353	+	0.920665i	$0.627647\pi$
$882$	0	0
$883$	−4.69728	−0.158076	−0.0790380	−	0.996872i	$-0.525185\pi$
$883$	−4.69728	−0.158076	−0.0790380	+	0.996872i	$0.525185\pi$
$884$	18.9290	0.636652
$885$	0	0
$886$	−70.0651	−2.35388
$887$	53.8411	1.80781	0.903904	−	0.427736i	$-0.140689\pi$
$887$	53.8411	1.80781	0.903904	+	0.427736i	$0.140689\pi$
$888$	0	0
$889$	−9.64590	−0.323513
$890$	32.9736	1.10528
$891$	0	0
$892$	98.4903	3.29770
$893$	−14.8307	−0.496290
$894$	0	0
$895$	−10.2044	−0.341095
$896$	−32.1702	−1.07473
$897$	0	0
$898$	−89.5417	−2.98805
$899$	1.68510	0.0562011
$900$	0	0
$901$	−6.84936	−0.228185
$902$	−51.7033	−1.72153
$903$	0	0
$904$	82.3305	2.73827
$905$	25.7297	0.855283
$906$	0	0
$907$	49.6705	1.64928	0.824642	−	0.565655i	$-0.191377\pi$
$907$	49.6705	1.64928	0.824642	+	0.565655i	$0.191377\pi$
$908$	−63.6965	−2.11384
$909$	0	0
$910$	−6.10607	−0.202414
$911$	−52.5586	−1.74134	−0.870672	−	0.491864i	$-0.836316\pi$
$911$	−52.5586	−1.74134	−0.870672	+	0.491864i	$0.836316\pi$
$912$	0	0
$913$	−6.44387	−0.213261
$914$	−56.6810	−1.87484
$915$	0	0
$916$	49.0806	1.62167
$917$	24.9050	0.822435
$918$	0	0
$919$	−9.80428	−0.323413	−0.161707	−	0.986839i	$-0.551700\pi$
$919$	−9.80428	−0.323413	−0.161707	+	0.986839i	$0.551700\pi$
$920$	−1.86484	−0.0614819
$921$	0	0
$922$	106.313	3.50124
$923$	6.87939	0.226438
$924$	0	0
$925$	5.26857	0.173230
$926$	34.7023	1.14039
$927$	0	0
$928$	17.0838	0.560802
$929$	0.0472658	0.00155074	0.000775370	−	1.00000i	$-0.499753\pi$
$929$	0.0472658	0.00155074	0.000775370		1.00000i	$0.499753\pi$
$930$	0	0
$931$	3.82295	0.125292
$932$	61.7187	2.02166
$933$	0	0
$934$	41.6614	1.36320
$935$	7.09152	0.231917
$936$	0	0
$937$	−13.6364	−0.445482	−0.222741	−	0.974878i	$-0.571500\pi$
$937$	−13.6364	−0.445482	−0.222741	+	0.974878i	$0.571500\pi$
$938$	−53.2567	−1.73889
$939$	0	0
$940$	−20.2763	−0.661340
$941$	22.4329	0.731293	0.365647	−	0.930754i	$-0.380848\pi$
$941$	22.4329	0.731293	0.365647	+	0.930754i	$0.380848\pi$
$942$	0	0
$943$	3.77332	0.122876
$944$	58.6647	1.90937
$945$	0	0
$946$	−29.3259	−0.953469
$947$	22.1780	0.720688	0.360344	−	0.932820i	$-0.382659\pi$
$947$	22.1780	0.720688	0.360344	+	0.932820i	$0.382659\pi$
$948$	0	0
$949$	−2.04189	−0.0662825
$950$	−8.17024	−0.265078
$951$	0	0
$952$	63.1813	2.04772
$953$	−8.65177	−0.280258	−0.140129	−	0.990133i	$-0.544752\pi$
$953$	−8.65177	−0.280258	−0.140129	+	0.990133i	$0.544752\pi$
$954$	0	0
$955$	−0.182104	−0.00589275
$956$	−37.1411	−1.20123
$957$	0	0
$958$	−0.437882	−0.0141473
$959$	1.12836	0.0364365
$960$	0	0
$961$	−30.7945	−0.993370
$962$	13.3405	0.430115
$963$	0	0
$964$	72.6519	2.33996
$965$	−16.4953	−0.531001
$966$	0	0
$967$	16.3696	0.526410	0.263205	−	0.964740i	$-0.415220\pi$
$967$	16.3696	0.526410	0.263205	+	0.964740i	$0.415220\pi$
$968$	−50.4884	−1.62276
$969$	0	0
$970$	29.9172	0.960582
$971$	−1.44057	−0.0462301	−0.0231150	−	0.999733i	$-0.507358\pi$
$971$	−1.44057	−0.0462301	−0.0231150	+	0.999733i	$0.507358\pi$
$972$	0	0
$973$	−45.1848	−1.44856
$974$	−31.2336	−1.00079
$975$	0	0
$976$	−12.6058	−0.403501
$977$	5.58172	0.178575	0.0892875	−	0.996006i	$-0.471541\pi$
$977$	5.58172	0.178575	0.0892875	+	0.996006i	$0.471541\pi$
$978$	0	0
$979$	21.5220	0.687846
$980$	5.22668	0.166960
$981$	0	0
$982$	38.9240	1.24211
$983$	57.3911	1.83049	0.915245	−	0.402897i	$-0.131997\pi$
$983$	57.3911	1.83049	0.915245	+	0.402897i	$0.131997\pi$
$984$	0	0
$985$	23.8699	0.760558
$986$	40.3833	1.28607
$987$	0	0
$988$	−14.2344	−0.452857
$989$	2.14022	0.0680549
$990$	0	0
$991$	−1.14290	−0.0363055	−0.0181528	−	0.999835i	$-0.505779\pi$
$991$	−1.14290	−0.0363055	−0.0181528	+	0.999835i	$0.505779\pi$
$992$	2.08378	0.0661600
$993$	0	0
$994$	42.0060	1.33235
$995$	1.00000	0.0317021
$996$	0	0
$997$	−29.8120	−0.944156	−0.472078	−	0.881557i	$-0.656496\pi$
$997$	−29.8120	−0.944156	−0.472078	+	0.881557i	$0.656496\pi$
$998$	−66.8566	−2.11631
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5265.2.a.t.1.3	yes	3
3.2	odd	2		5265.2.a.s.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5265.2.a.s.1.1	✓	3	3.2	odd	2
5265.2.a.t.1.3	yes	3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 5265 = 3^{4} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5265.a (trivial)

\(f(q)\)	\(=\)	\(q+2.53209 q^{2} +4.41147 q^{4} -1.00000 q^{5} +2.41147 q^{7} +6.10607 q^{8} +O(q^{10})\)	\(q+2.53209 q^{2} +4.41147 q^{4} -1.00000 q^{5} +2.41147 q^{7} +6.10607 q^{8} -2.53209 q^{10} -1.65270 q^{11} +1.00000 q^{13} +6.10607 q^{14} +6.63816 q^{16} +4.29086 q^{17} -3.22668 q^{19} -4.41147 q^{20} -4.18479 q^{22} +0.305407 q^{23} +1.00000 q^{25} +2.53209 q^{26} +10.6382 q^{28} +3.71688 q^{29} +0.453363 q^{31} +4.59627 q^{32} +10.8648 q^{34} -2.41147 q^{35} +5.26857 q^{37} -8.17024 q^{38} -6.10607 q^{40} +12.3550 q^{41} +7.00774 q^{43} -7.29086 q^{44} +0.773318 q^{46} +4.59627 q^{47} -1.18479 q^{49} +2.53209 q^{50} +4.41147 q^{52} -1.59627 q^{53} +1.65270 q^{55} +14.7246 q^{56} +9.41147 q^{58} +8.83750 q^{59} -1.89899 q^{61} +1.14796 q^{62} -1.63816 q^{64} -1.00000 q^{65} -8.72193 q^{67} +18.9290 q^{68} -6.10607 q^{70} +6.87939 q^{71} -2.04189 q^{73} +13.3405 q^{74} -14.2344 q^{76} -3.98545 q^{77} -0.319955 q^{79} -6.63816 q^{80} +31.2841 q^{82} +3.89899 q^{83} -4.29086 q^{85} +17.7442 q^{86} -10.0915 q^{88} -13.0223 q^{89} +2.41147 q^{91} +1.34730 q^{92} +11.6382 q^{94} +3.22668 q^{95} -11.8152 q^{97} -3.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} - 3 q^{7} + 6 q^{8}+O(q^{10}) \) 3 * q + 3 * q^2 + 3 * q^4 - 3 * q^5 - 3 * q^7 + 6 * q^8	\( 3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} - 3 q^{7} + 6 q^{8} - 3 q^{10} - 6 q^{11} + 3 q^{13} + 6 q^{14} + 3 q^{16} - 3 q^{17} - 3 q^{19} - 3 q^{20} - 9 q^{22} + 3 q^{23} + 3 q^{25} + 3 q^{26} + 15 q^{28} + 3 q^{29} - 12 q^{31} + 9 q^{34} + 3 q^{35} + 6 q^{37} - 3 q^{38} - 6 q^{40} + 12 q^{41} - 3 q^{43} - 6 q^{44} + 9 q^{46} + 3 q^{50} + 3 q^{52} + 9 q^{53} + 6 q^{55} + 12 q^{56} + 18 q^{58} + 24 q^{59} - 3 q^{61} - 12 q^{62} + 12 q^{64} - 3 q^{65} - 3 q^{67} + 24 q^{68} - 6 q^{70} + 15 q^{71} - 3 q^{73} - 3 q^{74} - 12 q^{76} + 6 q^{77} - 21 q^{79} - 3 q^{80} + 36 q^{82} + 9 q^{83} + 3 q^{85} + 24 q^{86} - 33 q^{89} - 3 q^{91} + 3 q^{92} + 18 q^{94} + 3 q^{95} - 39 q^{97} - 9 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 - 3 * q^5 - 3 * q^7 + 6 * q^8 - 3 * q^10 - 6 * q^11 + 3 * q^13 + 6 * q^14 + 3 * q^16 - 3 * q^17 - 3 * q^19 - 3 * q^20 - 9 * q^22 + 3 * q^23 + 3 * q^25 + 3 * q^26 + 15 * q^28 + 3 * q^29 - 12 * q^31 + 9 * q^34 + 3 * q^35 + 6 * q^37 - 3 * q^38 - 6 * q^40 + 12 * q^41 - 3 * q^43 - 6 * q^44 + 9 * q^46 + 3 * q^50 + 3 * q^52 + 9 * q^53 + 6 * q^55 + 12 * q^56 + 18 * q^58 + 24 * q^59 - 3 * q^61 - 12 * q^62 + 12 * q^64 - 3 * q^65 - 3 * q^67 + 24 * q^68 - 6 * q^70 + 15 * q^71 - 3 * q^73 - 3 * q^74 - 12 * q^76 + 6 * q^77 - 21 * q^79 - 3 * q^80 + 36 * q^82 + 9 * q^83 + 3 * q^85 + 24 * q^86 - 33 * q^89 - 3 * q^91 + 3 * q^92 + 18 * q^94 + 3 * q^95 - 39 * q^97 - 9 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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