LMFDB - Embedded newform 1088.4.a.bi.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1088,4,Mod(1,1088)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1088, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1088.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1088 = 2^{6} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1088.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:	$64.1940780862$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 34x^{6} + 116x^{5} + 383x^{4} - 964x^{3} - 1476x^{2} + 1978x + 1028 $ x^8 - 4x^7 - 34x^6 + 116x^5 + 383x^4 - 964x^3 - 1476x^2 + 1978*x + 1028
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	no (minimal twist has level 544)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.42708$ of defining polynomial
Character	$\chi$	$=$	1088.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.16565 q^{3} +16.7810 q^{5} +31.7490 q^{7} -25.6413 q^{9} +O(q^{10})$	$q-1.16565 q^{3} +16.7810 q^{5} +31.7490 q^{7} -25.6413 q^{9} -10.5557 q^{11} +53.0327 q^{13} -19.5608 q^{15} +17.0000 q^{17} -26.4898 q^{19} -37.0082 q^{21} +200.849 q^{23} +156.603 q^{25} +61.3613 q^{27} +4.00185 q^{29} -243.235 q^{31} +12.3043 q^{33} +532.781 q^{35} +158.425 q^{37} -61.8176 q^{39} +360.289 q^{41} -476.569 q^{43} -430.287 q^{45} +418.157 q^{47} +664.999 q^{49} -19.8161 q^{51} +110.022 q^{53} -177.136 q^{55} +30.8779 q^{57} -455.621 q^{59} -31.1780 q^{61} -814.084 q^{63} +889.943 q^{65} -333.081 q^{67} -234.120 q^{69} +113.476 q^{71} +124.858 q^{73} -182.544 q^{75} -335.134 q^{77} -38.8745 q^{79} +620.788 q^{81} -1190.94 q^{83} +285.278 q^{85} -4.66476 q^{87} -978.287 q^{89} +1683.73 q^{91} +283.527 q^{93} -444.527 q^{95} +391.132 q^{97} +270.662 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{5} + 136 q^{9}+O(q^{10}) $ 8 * q - 16 * q^5 + 136 * q^9	$ 8 q - 16 q^{5} + 136 q^{9} + 56 q^{13} + 136 q^{17} - 312 q^{21} + 600 q^{25} + 128 q^{29} - 304 q^{33} + 112 q^{37} + 1328 q^{41} - 2592 q^{45} + 1624 q^{49} - 1216 q^{53} + 2560 q^{57} - 752 q^{61} + 1824 q^{65} - 1448 q^{69} + 3952 q^{73} + 1480 q^{77} + 6824 q^{81} - 272 q^{85} + 2304 q^{89} + 2680 q^{93} + 4560 q^{97}+O(q^{100}) $ 8 * q - 16 * q^5 + 136 * q^9 + 56 * q^13 + 136 * q^17 - 312 * q^21 + 600 * q^25 + 128 * q^29 - 304 * q^33 + 112 * q^37 + 1328 * q^41 - 2592 * q^45 + 1624 * q^49 - 1216 * q^53 + 2560 * q^57 - 752 * q^61 + 1824 * q^65 - 1448 * q^69 + 3952 * q^73 + 1480 * q^77 + 6824 * q^81 - 272 * q^85 + 2304 * q^89 + 2680 * q^93 + 4560 * 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 $

$2$	0	0
$2$	0	0
$3$	−1.16565	−0.224330	−0.112165	−	0.993690i	$-0.535778\pi$
$3$	−1.16565	−0.224330	−0.112165	+	0.993690i	$0.535778\pi$
$4$	0	0
$5$	16.7810	1.50094	0.750471	−	0.660904i	$-0.229827\pi$
$5$	16.7810	1.50094	0.750471	+	0.660904i	$0.229827\pi$
$6$	0	0
$7$	31.7490	1.71428	0.857142	−	0.515080i	$-0.172238\pi$
$7$	31.7490	1.71428	0.857142	+	0.515080i	$0.172238\pi$
$8$	0	0
$9$	−25.6413	−0.949676
$10$	0	0
$11$	−10.5557	−0.289334	−0.144667	−	0.989480i	$-0.546211\pi$
$11$	−10.5557	−0.289334	−0.144667	+	0.989480i	$0.546211\pi$
$12$	0	0
$13$	53.0327	1.13143	0.565716	−	0.824600i	$-0.308600\pi$
$13$	53.0327	1.13143	0.565716	+	0.824600i	$0.308600\pi$
$14$	0	0
$15$	−19.5608	−0.336705
$16$	0	0
$17$	17.0000	0.242536
$17$	17.0000	0.242536
$18$	0	0
$19$	−26.4898	−0.319852	−0.159926	−	0.987129i	$-0.551126\pi$
$19$	−26.4898	−0.319852	−0.159926	+	0.987129i	$0.551126\pi$
$20$	0	0
$21$	−37.0082	−0.384565
$22$	0	0
$23$	200.849	1.82086	0.910432	−	0.413659i	$-0.135749\pi$
$23$	200.849	1.82086	0.910432	+	0.413659i	$0.135749\pi$
$24$	0	0
$25$	156.603	1.25282
$26$	0	0
$27$	61.3613	0.437370
$28$	0	0
$29$	4.00185	0.0256250	0.0128125	−	0.999918i	$-0.495922\pi$
$29$	4.00185	0.0256250	0.0128125	+	0.999918i	$0.495922\pi$
$30$	0	0
$31$	−243.235	−1.40924	−0.704619	−	0.709586i	$-0.748882\pi$
$31$	−243.235	−1.40924	−0.704619	+	0.709586i	$0.748882\pi$
$32$	0	0
$33$	12.3043	0.0649061
$34$	0	0
$35$	532.781	2.57304
$36$	0	0
$37$	158.425	0.703916	0.351958	−	0.936016i	$-0.385516\pi$
$37$	158.425	0.703916	0.351958	+	0.936016i	$0.385516\pi$
$38$	0	0
$39$	−61.8176	−0.253814
$40$	0	0
$41$	360.289	1.37238	0.686192	−	0.727421i	$-0.259281\pi$
$41$	360.289	1.37238	0.686192	+	0.727421i	$0.259281\pi$
$42$	0	0
$43$	−476.569	−1.69014	−0.845072	−	0.534653i	$-0.820442\pi$
$43$	−476.569	−1.69014	−0.845072	+	0.534653i	$0.820442\pi$
$44$	0	0
$45$	−430.287	−1.42541
$46$	0	0
$47$	418.157	1.29776	0.648878	−	0.760893i	$-0.275239\pi$
$47$	418.157	1.29776	0.648878	+	0.760893i	$0.275239\pi$
$48$	0	0
$49$	664.999	1.93877
$50$	0	0
$51$	−19.8161	−0.0544079
$52$	0	0
$53$	110.022	0.285146	0.142573	−	0.989784i	$-0.454462\pi$
$53$	110.022	0.285146	0.142573	+	0.989784i	$0.454462\pi$
$54$	0	0
$55$	−177.136	−0.434273
$56$	0	0
$57$	30.8779	0.0717523
$58$	0	0
$59$	−455.621	−1.00537	−0.502685	−	0.864470i	$-0.667654\pi$
$59$	−455.621	−1.00537	−0.502685	+	0.864470i	$0.667654\pi$
$60$	0	0
$61$	−31.1780	−0.0654416	−0.0327208	−	0.999465i	$-0.510417\pi$
$61$	−31.1780	−0.0654416	−0.0327208	+	0.999465i	$0.510417\pi$
$62$	0	0
$63$	−814.084	−1.62802
$64$	0	0
$65$	889.943	1.69821
$66$	0	0
$67$	−333.081	−0.607348	−0.303674	−	0.952776i	$-0.598213\pi$
$67$	−333.081	−0.607348	−0.303674	+	0.952776i	$0.598213\pi$
$68$	0	0
$69$	−234.120	−0.408474
$70$	0	0
$71$	113.476	0.189677	0.0948386	−	0.995493i	$-0.469766\pi$
$71$	113.476	0.189677	0.0948386	+	0.995493i	$0.469766\pi$
$72$	0	0
$73$	124.858	0.200186	0.100093	−	0.994978i	$-0.468086\pi$
$73$	124.858	0.200186	0.100093	+	0.994978i	$0.468086\pi$
$74$	0	0
$75$	−182.544	−0.281045
$76$	0	0
$77$	−335.134	−0.496001
$78$	0	0
$79$	−38.8745	−0.0553636	−0.0276818	−	0.999617i	$-0.508813\pi$
$79$	−38.8745	−0.0553636	−0.0276818	+	0.999617i	$0.508813\pi$
$80$	0	0
$81$	620.788	0.851561
$82$	0	0
$83$	−1190.94	−1.57498	−0.787489	−	0.616329i	$-0.788619\pi$
$83$	−1190.94	−1.57498	−0.787489	+	0.616329i	$0.788619\pi$
$84$	0	0
$85$	285.278	0.364032
$86$	0	0
$87$	−4.66476	−0.00574844
$88$	0	0
$89$	−978.287	−1.16515	−0.582574	−	0.812777i	$-0.697954\pi$
$89$	−978.287	−1.16515	−0.582574	+	0.812777i	$0.697954\pi$
$90$	0	0
$91$	1683.73	1.93960
$92$	0	0
$93$	283.527	0.316134
$94$	0	0
$95$	−444.527	−0.480079
$96$	0	0
$97$	391.132	0.409417	0.204708	−	0.978823i	$-0.434375\pi$
$97$	391.132	0.409417	0.204708	+	0.978823i	$0.434375\pi$
$98$	0	0
$99$	270.662	0.274774
$100$	0	0
$101$	−1405.47	−1.38465	−0.692323	−	0.721588i	$-0.743412\pi$
$101$	−1405.47	−1.38465	−0.692323	+	0.721588i	$0.743412\pi$
$102$	0	0
$103$	414.514	0.396536	0.198268	−	0.980148i	$-0.436468\pi$
$103$	414.514	0.396536	0.198268	+	0.980148i	$0.436468\pi$
$104$	0	0
$105$	−621.036	−0.577209
$106$	0	0
$107$	−1029.42	−0.930071	−0.465035	−	0.885292i	$-0.653958\pi$
$107$	−1029.42	−0.930071	−0.465035	+	0.885292i	$0.653958\pi$
$108$	0	0
$109$	846.834	0.744147	0.372073	−	0.928203i	$-0.378647\pi$
$109$	846.834	0.744147	0.372073	+	0.928203i	$0.378647\pi$
$110$	0	0
$111$	−184.668	−0.157909
$112$	0	0
$113$	868.186	0.722762	0.361381	−	0.932418i	$-0.382305\pi$
$113$	868.186	0.722762	0.361381	+	0.932418i	$0.382305\pi$
$114$	0	0
$115$	3370.45	2.73301
$116$	0	0
$117$	−1359.83	−1.07449
$118$	0	0
$119$	539.733	0.415775
$120$	0	0
$121$	−1219.58	−0.916286
$122$	0	0
$123$	−419.971	−0.307866
$124$	0	0
$125$	530.331	0.379474
$126$	0	0
$127$	−404.863	−0.282880	−0.141440	−	0.989947i	$-0.545173\pi$
$127$	−404.863	−0.282880	−0.141440	+	0.989947i	$0.545173\pi$
$128$	0	0
$129$	555.514	0.379149
$130$	0	0
$131$	361.572	0.241151	0.120575	−	0.992704i	$-0.461526\pi$
$131$	361.572	0.241151	0.120575	+	0.992704i	$0.461526\pi$
$132$	0	0
$133$	−841.026	−0.548317
$134$	0	0
$135$	1029.71	0.656467
$136$	0	0
$137$	3072.27	1.91593	0.957963	−	0.286892i	$-0.0926220\pi$
$137$	3072.27	1.91593	0.957963	+	0.286892i	$0.0926220\pi$
$138$	0	0
$139$	634.213	0.387002	0.193501	−	0.981100i	$-0.438016\pi$
$139$	634.213	0.387002	0.193501	+	0.981100i	$0.438016\pi$
$140$	0	0
$141$	−487.425	−0.291125
$142$	0	0
$143$	−559.799	−0.327362
$144$	0	0
$145$	67.1551	0.0384616
$146$	0	0
$147$	−775.156	−0.434924
$148$	0	0
$149$	2211.32	1.21583	0.607915	−	0.794002i	$-0.292006\pi$
$149$	2211.32	1.21583	0.607915	+	0.794002i	$0.292006\pi$
$150$	0	0
$151$	2760.95	1.48796	0.743982	−	0.668200i	$-0.232935\pi$
$151$	2760.95	1.48796	0.743982	+	0.668200i	$0.232935\pi$
$152$	0	0
$153$	−435.901	−0.230330
$154$	0	0
$155$	−4081.74	−2.11518
$156$	0	0
$157$	3327.61	1.69154	0.845771	−	0.533546i	$-0.179141\pi$
$157$	3327.61	1.69154	0.845771	+	0.533546i	$0.179141\pi$
$158$	0	0
$159$	−128.248	−0.0639667
$160$	0	0
$161$	6376.75	3.12148
$162$	0	0
$163$	454.281	0.218295	0.109147	−	0.994026i	$-0.465188\pi$
$163$	454.281	0.218295	0.109147	+	0.994026i	$0.465188\pi$
$164$	0	0
$165$	206.479	0.0974203
$166$	0	0
$167$	−2146.91	−0.994809	−0.497405	−	0.867519i	$-0.665714\pi$
$167$	−2146.91	−0.994809	−0.497405	+	0.867519i	$0.665714\pi$
$168$	0	0
$169$	615.467	0.280140
$170$	0	0
$171$	679.233	0.303756
$172$	0	0
$173$	1442.68	0.634018	0.317009	−	0.948423i	$-0.397321\pi$
$173$	1442.68	0.634018	0.317009	+	0.948423i	$0.397321\pi$
$174$	0	0
$175$	4971.99	2.14770
$176$	0	0
$177$	531.095	0.225534
$178$	0	0
$179$	4377.52	1.82788	0.913941	−	0.405846i	$-0.133023\pi$
$179$	4377.52	1.82788	0.913941	+	0.405846i	$0.133023\pi$
$180$	0	0
$181$	4095.15	1.68171	0.840857	−	0.541257i	$-0.182051\pi$
$181$	4095.15	1.68171	0.840857	+	0.541257i	$0.182051\pi$
$182$	0	0
$183$	36.3427	0.0146805
$184$	0	0
$185$	2658.53	1.05654
$186$	0	0
$187$	−179.447	−0.0701738
$188$	0	0
$189$	1948.16	0.749777
$190$	0	0
$191$	4643.64	1.75917	0.879587	−	0.475737i	$-0.157819\pi$
$191$	4643.64	1.75917	0.879587	+	0.475737i	$0.157819\pi$
$192$	0	0
$193$	1341.79	0.500435	0.250217	−	0.968190i	$-0.419498\pi$
$193$	1341.79	0.500435	0.250217	+	0.968190i	$0.419498\pi$
$194$	0	0
$195$	−1037.36	−0.380960
$196$	0	0
$197$	−2202.95	−0.796719	−0.398359	−	0.917229i	$-0.630420\pi$
$197$	−2202.95	−0.796719	−0.398359	+	0.917229i	$0.630420\pi$
$198$	0	0
$199$	−2073.15	−0.738502	−0.369251	−	0.929330i	$-0.620386\pi$
$199$	−2073.15	−0.738502	−0.369251	+	0.929330i	$0.620386\pi$
$200$	0	0
$201$	388.256	0.136246
$202$	0	0
$203$	127.055	0.0439285
$204$	0	0
$205$	6046.02	2.05987
$206$	0	0
$207$	−5150.02	−1.72923
$208$	0	0
$209$	279.620	0.0925440
$210$	0	0
$211$	−5538.65	−1.80709	−0.903545	−	0.428493i	$-0.859045\pi$
$211$	−5538.65	−1.80709	−0.903545	+	0.428493i	$0.859045\pi$
$212$	0	0
$213$	−132.273	−0.0425502
$214$	0	0
$215$	−7997.33	−2.53681
$216$	0	0
$217$	−7722.48	−2.41583
$218$	0	0
$219$	−145.541	−0.0449076
$220$	0	0
$221$	901.556	0.274413
$222$	0	0
$223$	−845.390	−0.253863	−0.126932	−	0.991911i	$-0.540513\pi$
$223$	−845.390	−0.253863	−0.126932	+	0.991911i	$0.540513\pi$
$224$	0	0
$225$	−4015.50	−1.18978
$226$	0	0
$227$	−3854.89	−1.12713	−0.563565	−	0.826072i	$-0.690570\pi$
$227$	−3854.89	−1.12713	−0.563565	+	0.826072i	$0.690570\pi$
$228$	0	0
$229$	3996.61	1.15329	0.576646	−	0.816994i	$-0.304361\pi$
$229$	3996.61	1.15329	0.576646	+	0.816994i	$0.304361\pi$
$230$	0	0
$231$	390.649	0.111268
$232$	0	0
$233$	−2408.59	−0.677220	−0.338610	−	0.940927i	$-0.609957\pi$
$233$	−2408.59	−0.677220	−0.338610	+	0.940927i	$0.609957\pi$
$234$	0	0
$235$	7017.11	1.94785
$236$	0	0
$237$	45.3141	0.0124197
$238$	0	0
$239$	−5823.34	−1.57607	−0.788034	−	0.615631i	$-0.788901\pi$
$239$	−5823.34	−1.57607	−0.788034	+	0.615631i	$0.788901\pi$
$240$	0	0
$241$	−3082.74	−0.823971	−0.411985	−	0.911190i	$-0.635165\pi$
$241$	−3082.74	−0.823971	−0.411985	+	0.911190i	$0.635165\pi$
$242$	0	0
$243$	−2380.38	−0.628400
$244$	0	0
$245$	11159.4	2.90998
$246$	0	0
$247$	−1404.83	−0.361891
$248$	0	0
$249$	1388.23	0.353314
$250$	0	0
$251$	−36.0373	−0.00906237	−0.00453118	−	0.999990i	$-0.501442\pi$
$251$	−36.0373	−0.00906237	−0.00453118	+	0.999990i	$0.501442\pi$
$252$	0	0
$253$	−2120.11	−0.526838
$254$	0	0
$255$	−332.534	−0.0816631
$256$	0	0
$257$	−2437.56	−0.591637	−0.295819	−	0.955244i	$-0.595592\pi$
$257$	−2437.56	−0.591637	−0.295819	+	0.955244i	$0.595592\pi$
$258$	0	0
$259$	5029.83	1.20671
$260$	0	0
$261$	−102.612	−0.0243354
$262$	0	0
$263$	5240.19	1.22861	0.614305	−	0.789069i	$-0.289437\pi$
$263$	5240.19	1.22861	0.614305	+	0.789069i	$0.289437\pi$
$264$	0	0
$265$	1846.29	0.427987
$266$	0	0
$267$	1140.34	0.261377
$268$	0	0
$269$	614.470	0.139275	0.0696374	−	0.997572i	$-0.477816\pi$
$269$	614.470	0.139275	0.0696374	+	0.997572i	$0.477816\pi$
$270$	0	0
$271$	3886.26	0.871121	0.435560	−	0.900160i	$-0.356550\pi$
$271$	3886.26	0.871121	0.435560	+	0.900160i	$0.356550\pi$
$272$	0	0
$273$	−1962.65	−0.435109
$274$	0	0
$275$	−1653.06	−0.362484
$276$	0	0
$277$	−7086.52	−1.53714	−0.768570	−	0.639766i	$-0.779031\pi$
$277$	−7086.52	−1.53714	−0.768570	+	0.639766i	$0.779031\pi$
$278$	0	0
$279$	6236.86	1.33832
$280$	0	0
$281$	−2265.56	−0.480967	−0.240483	−	0.970653i	$-0.577306\pi$
$281$	−2265.56	−0.480967	−0.240483	+	0.970653i	$0.577306\pi$
$282$	0	0
$283$	−5024.39	−1.05537	−0.527684	−	0.849441i	$-0.676939\pi$
$283$	−5024.39	−1.05537	−0.527684	+	0.849441i	$0.676939\pi$
$284$	0	0
$285$	518.163	0.107696
$286$	0	0
$287$	11438.8	2.35266
$288$	0	0
$289$	289.000	0.0588235
$290$	0	0
$291$	−455.923	−0.0918443
$292$	0	0
$293$	3341.65	0.666285	0.333142	−	0.942876i	$-0.391891\pi$
$293$	3341.65	0.666285	0.333142	+	0.942876i	$0.391891\pi$
$294$	0	0
$295$	−7645.80	−1.50900
$296$	0	0
$297$	−647.714	−0.126546
$298$	0	0
$299$	10651.6	2.06019
$300$	0	0
$301$	−15130.6	−2.89739
$302$	0	0
$303$	1638.28	0.310617
$304$	0	0
$305$	−523.200	−0.0982240
$306$	0	0
$307$	−3513.66	−0.653209	−0.326604	−	0.945161i	$-0.605904\pi$
$307$	−3513.66	−0.653209	−0.326604	+	0.945161i	$0.605904\pi$
$308$	0	0
$309$	−483.178	−0.0889548
$310$	0	0
$311$	−5859.37	−1.06834	−0.534171	−	0.845376i	$-0.679376\pi$
$311$	−5859.37	−1.06834	−0.534171	+	0.845376i	$0.679376\pi$
$312$	0	0
$313$	−7167.54	−1.29436	−0.647178	−	0.762339i	$-0.724051\pi$
$313$	−7167.54	−1.29436	−0.647178	+	0.762339i	$0.724051\pi$
$314$	0	0
$315$	−13661.2	−2.44356
$316$	0	0
$317$	−8474.00	−1.50141	−0.750705	−	0.660637i	$-0.770286\pi$
$317$	−8474.00	−1.50141	−0.750705	+	0.660637i	$0.770286\pi$
$318$	0	0
$319$	−42.2424	−0.00741418
$320$	0	0
$321$	1199.94	0.208642
$322$	0	0
$323$	−450.327	−0.0775755
$324$	0	0
$325$	8305.08	1.41749
$326$	0	0
$327$	−987.112	−0.166934
$328$	0	0
$329$	13276.1	2.22472
$330$	0	0
$331$	8036.70	1.33455	0.667277	−	0.744810i	$-0.267460\pi$
$331$	8036.70	1.33455	0.667277	+	0.744810i	$0.267460\pi$
$332$	0	0
$333$	−4062.22	−0.668493
$334$	0	0
$335$	−5589.44	−0.911593
$336$	0	0
$337$	8387.76	1.35582	0.677908	−	0.735146i	$-0.262887\pi$
$337$	8387.76	1.35582	0.677908	+	0.735146i	$0.262887\pi$
$338$	0	0
$339$	−1012.00	−0.162137
$340$	0	0
$341$	2567.53	0.407740
$342$	0	0
$343$	10223.1	1.60932
$344$	0	0
$345$	−3928.77	−0.613095
$346$	0	0
$347$	6393.65	0.989133	0.494567	−	0.869140i	$-0.335327\pi$
$347$	6393.65	0.989133	0.494567	+	0.869140i	$0.335327\pi$
$348$	0	0
$349$	2803.61	0.430010	0.215005	−	0.976613i	$-0.431023\pi$
$349$	2803.61	0.430010	0.215005	+	0.976613i	$0.431023\pi$
$350$	0	0
$351$	3254.16	0.494855
$352$	0	0
$353$	−5410.85	−0.815837	−0.407918	−	0.913018i	$-0.633745\pi$
$353$	−5410.85	−0.815837	−0.407918	+	0.913018i	$0.633745\pi$
$354$	0	0
$355$	1904.24	0.284694
$356$	0	0
$357$	−629.140	−0.0932707
$358$	0	0
$359$	−9090.49	−1.33643	−0.668214	−	0.743969i	$-0.732941\pi$
$359$	−9090.49	−1.33643	−0.668214	+	0.743969i	$0.732941\pi$
$360$	0	0
$361$	−6157.29	−0.897695
$362$	0	0
$363$	1421.60	0.205550
$364$	0	0
$365$	2095.25	0.300467
$366$	0	0
$367$	13209.5	1.87883	0.939415	−	0.342782i	$-0.111369\pi$
$367$	13209.5	1.87883	0.939415	+	0.342782i	$0.111369\pi$
$368$	0	0
$369$	−9238.27	−1.30332
$370$	0	0
$371$	3493.10	0.488822
$372$	0	0
$373$	1415.84	0.196540	0.0982700	−	0.995160i	$-0.468669\pi$
$373$	1415.84	0.196540	0.0982700	+	0.995160i	$0.468669\pi$
$374$	0	0
$375$	−618.181	−0.0851272
$376$	0	0
$377$	212.229	0.0289929
$378$	0	0
$379$	−3342.97	−0.453079	−0.226539	−	0.974002i	$-0.572741\pi$
$379$	−3342.97	−0.453079	−0.226539	+	0.974002i	$0.572741\pi$
$380$	0	0
$381$	471.929	0.0634584
$382$	0	0
$383$	−9486.89	−1.26569	−0.632843	−	0.774280i	$-0.718112\pi$
$383$	−9486.89	−1.26569	−0.632843	+	0.774280i	$0.718112\pi$
$384$	0	0
$385$	−5623.89	−0.744468
$386$	0	0
$387$	12219.8	1.60509
$388$	0	0
$389$	−5561.74	−0.724914	−0.362457	−	0.932001i	$-0.618062\pi$
$389$	−5561.74	−0.724914	−0.362457	+	0.932001i	$0.618062\pi$
$390$	0	0
$391$	3414.43	0.441624
$392$	0	0
$393$	−421.467	−0.0540972
$394$	0	0
$395$	−652.354	−0.0830974
$396$	0	0
$397$	−7956.47	−1.00585	−0.502927	−	0.864329i	$-0.667743\pi$
$397$	−7956.47	−1.00585	−0.502927	+	0.864329i	$0.667743\pi$
$398$	0	0
$399$	980.343	0.123004
$400$	0	0
$401$	5835.73	0.726739	0.363370	−	0.931645i	$-0.381626\pi$
$401$	5835.73	0.726739	0.363370	+	0.931645i	$0.381626\pi$
$402$	0	0
$403$	−12899.4	−1.59446
$404$	0	0
$405$	10417.5	1.27814
$406$	0	0
$407$	−1672.29	−0.203667
$408$	0	0
$409$	4161.62	0.503126	0.251563	−	0.967841i	$-0.419055\pi$
$409$	4161.62	0.503126	0.251563	+	0.967841i	$0.419055\pi$
$410$	0	0
$411$	−3581.20	−0.429799
$412$	0	0
$413$	−14465.5	−1.72349
$414$	0	0
$415$	−19985.3	−2.36395
$416$	0	0
$417$	−739.271	−0.0868159
$418$	0	0
$419$	526.799	0.0614220	0.0307110	−	0.999528i	$-0.490223\pi$
$419$	526.799	0.0614220	0.0307110	+	0.999528i	$0.490223\pi$
$420$	0	0
$421$	6302.16	0.729569	0.364785	−	0.931092i	$-0.381143\pi$
$421$	6302.16	0.729569	0.364785	+	0.931092i	$0.381143\pi$
$422$	0	0
$423$	−10722.1	−1.23245
$424$	0	0
$425$	2662.25	0.303854
$426$	0	0
$427$	−989.871	−0.112186
$428$	0	0
$429$	652.530	0.0734369
$430$	0	0
$431$	5624.67	0.628610	0.314305	−	0.949322i	$-0.398229\pi$
$431$	5624.67	0.628610	0.314305	+	0.949322i	$0.398229\pi$
$432$	0	0
$433$	14364.6	1.59427	0.797133	−	0.603803i	$-0.206349\pi$
$433$	14364.6	1.59427	0.797133	+	0.603803i	$0.206349\pi$
$434$	0	0
$435$	−78.2794	−0.00862807
$436$	0	0
$437$	−5320.46	−0.582407
$438$	0	0
$439$	4655.07	0.506092	0.253046	−	0.967454i	$-0.418568\pi$
$439$	4655.07	0.506092	0.253046	+	0.967454i	$0.418568\pi$
$440$	0	0
$441$	−17051.4	−1.84121
$442$	0	0
$443$	−12335.4	−1.32296	−0.661479	−	0.749963i	$-0.730071\pi$
$443$	−12335.4	−1.32296	−0.661479	+	0.749963i	$0.730071\pi$
$444$	0	0
$445$	−16416.7	−1.74882
$446$	0	0
$447$	−2577.63	−0.272746
$448$	0	0
$449$	−4298.43	−0.451793	−0.225897	−	0.974151i	$-0.572531\pi$
$449$	−4298.43	−0.451793	−0.225897	+	0.974151i	$0.572531\pi$
$450$	0	0
$451$	−3803.12	−0.397077
$452$	0	0
$453$	−3218.30	−0.333794
$454$	0	0
$455$	28254.8	2.91122
$456$	0	0
$457$	1036.67	0.106113	0.0530564	−	0.998592i	$-0.483104\pi$
$457$	1036.67	0.106113	0.0530564	+	0.998592i	$0.483104\pi$
$458$	0	0
$459$	1043.14	0.106078
$460$	0	0
$461$	−7019.41	−0.709168	−0.354584	−	0.935024i	$-0.615378\pi$
$461$	−7019.41	−0.709168	−0.354584	+	0.935024i	$0.615378\pi$
$462$	0	0
$463$	−4016.72	−0.403181	−0.201590	−	0.979470i	$-0.564611\pi$
$463$	−4016.72	−0.403181	−0.201590	+	0.979470i	$0.564611\pi$
$464$	0	0
$465$	4757.88	0.474498
$466$	0	0
$467$	−684.278	−0.0678043	−0.0339022	−	0.999425i	$-0.510793\pi$
$467$	−684.278	−0.0678043	−0.0339022	+	0.999425i	$0.510793\pi$
$468$	0	0
$469$	−10575.0	−1.04117
$470$	0	0
$471$	−3878.83	−0.379463
$472$	0	0
$473$	5030.54	0.489016
$474$	0	0
$475$	−4148.39	−0.400718
$476$	0	0
$477$	−2821.11	−0.270796
$478$	0	0
$479$	−4168.81	−0.397657	−0.198829	−	0.980034i	$-0.563714\pi$
$479$	−4168.81	−0.397657	−0.198829	+	0.980034i	$0.563714\pi$
$480$	0	0
$481$	8401.70	0.796434
$482$	0	0
$483$	−7433.06	−0.700240
$484$	0	0
$485$	6563.60	0.614511
$486$	0	0
$487$	1831.22	0.170391	0.0851957	−	0.996364i	$-0.472848\pi$
$487$	1831.22	0.170391	0.0851957	+	0.996364i	$0.472848\pi$
$488$	0	0
$489$	−529.533	−0.0489700
$490$	0	0
$491$	9449.95	0.868575	0.434287	−	0.900774i	$-0.357000\pi$
$491$	9449.95	0.868575	0.434287	+	0.900774i	$0.357000\pi$
$492$	0	0
$493$	68.0314	0.00621497
$494$	0	0
$495$	4541.99	0.412419
$496$	0	0
$497$	3602.74	0.325161
$498$	0	0
$499$	−6524.39	−0.585315	−0.292657	−	0.956217i	$-0.594539\pi$
$499$	−6524.39	−0.585315	−0.292657	+	0.956217i	$0.594539\pi$
$500$	0	0
$501$	2502.55	0.223165
$502$	0	0
$503$	−385.856	−0.0342037	−0.0171019	−	0.999854i	$-0.505444\pi$
$503$	−385.856	−0.0342037	−0.0171019	+	0.999854i	$0.505444\pi$
$504$	0	0
$505$	−23585.2	−2.07827
$506$	0	0
$507$	−717.420	−0.0628436
$508$	0	0
$509$	−16377.7	−1.42619	−0.713094	−	0.701068i	$-0.752707\pi$
$509$	−16377.7	−1.42619	−0.713094	+	0.701068i	$0.752707\pi$
$510$	0	0
$511$	3964.12	0.343175
$512$	0	0
$513$	−1625.45	−0.139894
$514$	0	0
$515$	6955.97	0.595178
$516$	0	0
$517$	−4413.96	−0.375485
$518$	0	0
$519$	−1681.66	−0.142229
$520$	0	0
$521$	20078.7	1.68841	0.844205	−	0.536020i	$-0.180073\pi$
$521$	20078.7	1.68841	0.844205	+	0.536020i	$0.180073\pi$
$522$	0	0
$523$	−2073.49	−0.173360	−0.0866801	−	0.996236i	$-0.527626\pi$
$523$	−2073.49	−0.173360	−0.0866801	+	0.996236i	$0.527626\pi$
$524$	0	0
$525$	−5795.60	−0.481792
$526$	0	0
$527$	−4135.00	−0.341790
$528$	0	0
$529$	28173.3	2.31555
$530$	0	0
$531$	11682.7	0.954776
$532$	0	0
$533$	19107.1	1.55276
$534$	0	0
$535$	−17274.7	−1.39598
$536$	0	0
$537$	−5102.66	−0.410048
$538$	0	0
$539$	−7019.55	−0.560952
$540$	0	0
$541$	−12600.1	−1.00133	−0.500665	−	0.865641i	$-0.666911\pi$
$541$	−12600.1	−1.00133	−0.500665	+	0.865641i	$0.666911\pi$
$542$	0	0
$543$	−4773.52	−0.377258
$544$	0	0
$545$	14210.7	1.11692
$546$	0	0
$547$	25467.7	1.99071	0.995357	−	0.0962488i	$-0.0306844\pi$
$547$	25467.7	1.99071	0.995357	+	0.0962488i	$0.0306844\pi$
$548$	0	0
$549$	799.444	0.0621483
$550$	0	0
$551$	−106.008	−0.00819620
$552$	0	0
$553$	−1234.23	−0.0949089
$554$	0	0
$555$	−3098.92	−0.237012
$556$	0	0
$557$	−17825.0	−1.35596	−0.677980	−	0.735081i	$-0.737144\pi$
$557$	−17825.0	−1.35596	−0.677980	+	0.735081i	$0.737144\pi$
$558$	0	0
$559$	−25273.8	−1.91228
$560$	0	0
$561$	209.173	0.0157421
$562$	0	0
$563$	1662.34	0.124439	0.0622196	−	0.998062i	$-0.480182\pi$
$563$	1662.34	0.124439	0.0622196	+	0.998062i	$0.480182\pi$
$564$	0	0
$565$	14569.1	1.08482
$566$	0	0
$567$	19709.4	1.45982
$568$	0	0
$569$	−6076.82	−0.447721	−0.223861	−	0.974621i	$-0.571866\pi$
$569$	−6076.82	−0.447721	−0.223861	+	0.974621i	$0.571866\pi$
$570$	0	0
$571$	17112.8	1.25420	0.627099	−	0.778939i	$-0.284242\pi$
$571$	17112.8	1.25420	0.627099	+	0.778939i	$0.284242\pi$
$572$	0	0
$573$	−5412.87	−0.394635
$574$	0	0
$575$	31453.5	2.28122
$576$	0	0
$577$	86.2098	0.00622004	0.00311002	−	0.999995i	$-0.499010\pi$
$577$	86.2098	0.00622004	0.00311002	+	0.999995i	$0.499010\pi$
$578$	0	0
$579$	−1564.05	−0.112262
$580$	0	0
$581$	−37811.3	−2.69996
$582$	0	0
$583$	−1161.37	−0.0825024
$584$	0	0
$585$	−22819.3	−1.61275
$586$	0	0
$587$	−5264.07	−0.370139	−0.185070	−	0.982725i	$-0.559251\pi$
$587$	−5264.07	−0.370139	−0.185070	+	0.982725i	$0.559251\pi$
$588$	0	0
$589$	6443.27	0.450747
$590$	0	0
$591$	2567.87	0.178728
$592$	0	0
$593$	−8460.14	−0.585863	−0.292931	−	0.956133i	$-0.594631\pi$
$593$	−8460.14	−0.585863	−0.292931	+	0.956133i	$0.594631\pi$
$594$	0	0
$595$	9057.28	0.624054
$596$	0	0
$597$	2416.57	0.165668
$598$	0	0
$599$	21638.6	1.47601	0.738004	−	0.674796i	$-0.235769\pi$
$599$	21638.6	1.47601	0.738004	+	0.674796i	$0.235769\pi$
$600$	0	0
$601$	−26366.0	−1.78950	−0.894750	−	0.446567i	$-0.852647\pi$
$601$	−26366.0	−1.78950	−0.894750	+	0.446567i	$0.852647\pi$
$602$	0	0
$603$	8540.61	0.576784
$604$	0	0
$605$	−20465.8	−1.37529
$606$	0	0
$607$	20621.8	1.37893	0.689467	−	0.724317i	$-0.257845\pi$
$607$	20621.8	1.37893	0.689467	+	0.724317i	$0.257845\pi$
$608$	0	0
$609$	−148.101	−0.00985447
$610$	0	0
$611$	22176.0	1.46832
$612$	0	0
$613$	−26254.2	−1.72985	−0.864926	−	0.501900i	$-0.832635\pi$
$613$	−26254.2	−1.72985	−0.864926	+	0.501900i	$0.832635\pi$
$614$	0	0
$615$	−7047.55	−0.462089
$616$	0	0
$617$	1009.02	0.0658371	0.0329185	−	0.999458i	$-0.489520\pi$
$617$	1009.02	0.0658371	0.0329185	+	0.999458i	$0.489520\pi$
$618$	0	0
$619$	−13650.9	−0.886392	−0.443196	−	0.896425i	$-0.646155\pi$
$619$	−13650.9	−0.886392	−0.443196	+	0.896425i	$0.646155\pi$
$620$	0	0
$621$	12324.3	0.796391
$622$	0	0
$623$	−31059.6	−1.99740
$624$	0	0
$625$	−10675.9	−0.683256
$626$	0	0
$627$	−325.939	−0.0207604
$628$	0	0
$629$	2693.22	0.170725
$630$	0	0
$631$	−23738.3	−1.49763	−0.748816	−	0.662778i	$-0.769378\pi$
$631$	−23738.3	−1.49763	−0.748816	+	0.662778i	$0.769378\pi$
$632$	0	0
$633$	6456.13	0.405384
$634$	0	0
$635$	−6794.02	−0.424586
$636$	0	0
$637$	35266.7	2.19359
$638$	0	0
$639$	−2909.66	−0.180132
$640$	0	0
$641$	−18313.9	−1.12848	−0.564240	−	0.825611i	$-0.690831\pi$
$641$	−18313.9	−1.12848	−0.564240	+	0.825611i	$0.690831\pi$
$642$	0	0
$643$	−14535.0	−0.891451	−0.445725	−	0.895170i	$-0.647054\pi$
$643$	−14535.0	−0.891451	−0.445725	+	0.895170i	$0.647054\pi$
$644$	0	0
$645$	9322.09	0.569081
$646$	0	0
$647$	23237.7	1.41201	0.706004	−	0.708207i	$-0.250496\pi$
$647$	23237.7	1.41201	0.706004	+	0.708207i	$0.250496\pi$
$648$	0	0
$649$	4809.42	0.290888
$650$	0	0
$651$	9001.71	0.541943
$652$	0	0
$653$	−24776.7	−1.48482	−0.742410	−	0.669946i	$-0.766317\pi$
$653$	−24776.7	−1.48482	−0.742410	+	0.669946i	$0.766317\pi$
$654$	0	0
$655$	6067.56	0.361953
$656$	0	0
$657$	−3201.52	−0.190112
$658$	0	0
$659$	−11063.6	−0.653987	−0.326994	−	0.945027i	$-0.606036\pi$
$659$	−11063.6	−0.653987	−0.326994	+	0.945027i	$0.606036\pi$
$660$	0	0
$661$	−11086.1	−0.652341	−0.326171	−	0.945311i	$-0.605758\pi$
$661$	−11086.1	−0.652341	−0.326171	+	0.945311i	$0.605758\pi$
$662$	0	0
$663$	−1050.90	−0.0615589
$664$	0	0
$665$	−14113.3	−0.822992
$666$	0	0
$667$	803.767	0.0466596
$668$	0	0
$669$	985.429	0.0569490
$670$	0	0
$671$	329.107	0.0189345
$672$	0	0
$673$	−30924.7	−1.77126	−0.885632	−	0.464388i	$-0.846274\pi$
$673$	−30924.7	−1.77126	−0.885632	+	0.464388i	$0.846274\pi$
$674$	0	0
$675$	9609.37	0.547948
$676$	0	0
$677$	16434.3	0.932970	0.466485	−	0.884529i	$-0.345520\pi$
$677$	16434.3	0.932970	0.466485	+	0.884529i	$0.345520\pi$
$678$	0	0
$679$	12418.1	0.701857
$680$	0	0
$681$	4493.46	0.252848
$682$	0	0
$683$	−25129.7	−1.40785	−0.703926	−	0.710273i	$-0.748571\pi$
$683$	−25129.7	−1.40785	−0.703926	+	0.710273i	$0.748571\pi$
$684$	0	0
$685$	51555.9	2.87569
$686$	0	0
$687$	−4658.66	−0.258717
$688$	0	0
$689$	5834.79	0.322624
$690$	0	0
$691$	−20139.5	−1.10874	−0.554371	−	0.832269i	$-0.687041\pi$
$691$	−20139.5	−1.10874	−0.554371	+	0.832269i	$0.687041\pi$
$692$	0	0
$693$	8593.25	0.471040
$694$	0	0
$695$	10642.7	0.580867
$696$	0	0
$697$	6124.92	0.332852
$698$	0	0
$699$	2807.58	0.151920
$700$	0	0
$701$	−14457.6	−0.778965	−0.389482	−	0.921034i	$-0.627346\pi$
$701$	−14457.6	−0.778965	−0.389482	+	0.921034i	$0.627346\pi$
$702$	0	0
$703$	−4196.65	−0.225149
$704$	0	0
$705$	−8179.50	−0.436961
$706$	0	0
$707$	−44622.2	−2.37368
$708$	0	0
$709$	11418.8	0.604855	0.302427	−	0.953172i	$-0.402203\pi$
$709$	11418.8	0.604855	0.302427	+	0.953172i	$0.402203\pi$
$710$	0	0
$711$	996.791	0.0525775
$712$	0	0
$713$	−48853.5	−2.56603
$714$	0	0
$715$	−9394.00	−0.491351
$716$	0	0
$717$	6787.98	0.353559
$718$	0	0
$719$	−7890.05	−0.409248	−0.204624	−	0.978841i	$-0.565597\pi$
$719$	−7890.05	−0.409248	−0.204624	+	0.978841i	$0.565597\pi$
$720$	0	0
$721$	13160.4	0.679776
$722$	0	0
$723$	3593.40	0.184841
$724$	0	0
$725$	626.701	0.0321036
$726$	0	0
$727$	−5368.41	−0.273870	−0.136935	−	0.990580i	$-0.543725\pi$
$727$	−5368.41	−0.273870	−0.136935	+	0.990580i	$0.543725\pi$
$728$	0	0
$729$	−13986.6	−0.710592
$730$	0	0
$731$	−8101.68	−0.409920
$732$	0	0
$733$	34845.6	1.75587	0.877935	−	0.478780i	$-0.158921\pi$
$733$	34845.6	1.75587	0.877935	+	0.478780i	$0.158921\pi$
$734$	0	0
$735$	−13007.9	−0.652795
$736$	0	0
$737$	3515.91	0.175726
$738$	0	0
$739$	−14670.5	−0.730264	−0.365132	−	0.930956i	$-0.618976\pi$
$739$	−14670.5	−0.730264	−0.365132	+	0.930956i	$0.618976\pi$
$740$	0	0
$741$	1637.54	0.0811829
$742$	0	0
$743$	−12358.7	−0.610226	−0.305113	−	0.952316i	$-0.598694\pi$
$743$	−12358.7	−0.610226	−0.305113	+	0.952316i	$0.598694\pi$
$744$	0	0
$745$	37108.3	1.82489
$746$	0	0
$747$	30537.3	1.49572
$748$	0	0
$749$	−32683.0	−1.59441
$750$	0	0
$751$	27659.8	1.34397	0.671984	−	0.740566i	$-0.265443\pi$
$751$	27659.8	1.34397	0.671984	+	0.740566i	$0.265443\pi$
$752$	0	0
$753$	42.0069	0.00203296
$754$	0	0
$755$	46331.5	2.23335
$756$	0	0
$757$	−1662.45	−0.0798188	−0.0399094	−	0.999203i	$-0.512707\pi$
$757$	−1662.45	−0.0798188	−0.0399094	+	0.999203i	$0.512707\pi$
$758$	0	0
$759$	2471.30	0.118185
$760$	0	0
$761$	−13399.0	−0.638255	−0.319128	−	0.947712i	$-0.603390\pi$
$761$	−13399.0	−0.638255	−0.319128	+	0.947712i	$0.603390\pi$
$762$	0	0
$763$	26886.1	1.27568
$764$	0	0
$765$	−7314.87	−0.345712
$766$	0	0
$767$	−24162.8	−1.13751
$768$	0	0
$769$	11257.8	0.527917	0.263958	−	0.964534i	$-0.414972\pi$
$769$	11257.8	0.527917	0.263958	+	0.964534i	$0.414972\pi$
$770$	0	0
$771$	2841.34	0.132722
$772$	0	0
$773$	13596.8	0.632658	0.316329	−	0.948650i	$-0.397550\pi$
$773$	13596.8	0.632658	0.316329	+	0.948650i	$0.397550\pi$
$774$	0	0
$775$	−38091.4	−1.76553
$776$	0	0
$777$	−5863.03	−0.270701
$778$	0	0
$779$	−9544.01	−0.438960
$780$	0	0
$781$	−1197.82	−0.0548800
$782$	0	0
$783$	245.559	0.0112076
$784$	0	0
$785$	55840.7	2.53890
$786$	0	0
$787$	8294.83	0.375704	0.187852	−	0.982197i	$-0.439848\pi$
$787$	8294.83	0.375704	0.187852	+	0.982197i	$0.439848\pi$
$788$	0	0
$789$	−6108.24	−0.275613
$790$	0	0
$791$	27564.0	1.23902
$792$	0	0
$793$	−1653.46	−0.0740428
$794$	0	0
$795$	−2152.13	−0.0960102
$796$	0	0
$797$	−12296.9	−0.546522	−0.273261	−	0.961940i	$-0.588102\pi$
$797$	−12296.9	−0.546522	−0.273261	+	0.961940i	$0.588102\pi$
$798$	0	0
$799$	7108.67	0.314752
$800$	0	0
$801$	25084.5	1.10651
$802$	0	0
$803$	−1317.97	−0.0579205
$804$	0	0
$805$	107008.	4.68516
$806$	0	0
$807$	−716.258	−0.0312434
$808$	0	0
$809$	−9058.67	−0.393679	−0.196839	−	0.980436i	$-0.563068\pi$
$809$	−9058.67	−0.393679	−0.196839	+	0.980436i	$0.563068\pi$
$810$	0	0
$811$	−12695.8	−0.549706	−0.274853	−	0.961486i	$-0.588629\pi$
$811$	−12695.8	−0.549706	−0.274853	+	0.961486i	$0.588629\pi$
$812$	0	0
$813$	−4530.03	−0.195418
$814$	0	0
$815$	7623.31	0.327648
$816$	0	0
$817$	12624.3	0.540596
$818$	0	0
$819$	−43173.1	−1.84199
$820$	0	0
$821$	−22827.6	−0.970387	−0.485193	−	0.874407i	$-0.661251\pi$
$821$	−22827.6	−0.970387	−0.485193	+	0.874407i	$0.661251\pi$
$822$	0	0
$823$	−27839.6	−1.17913	−0.589567	−	0.807719i	$-0.700702\pi$
$823$	−27839.6	−1.17913	−0.589567	+	0.807719i	$0.700702\pi$
$824$	0	0
$825$	1926.89	0.0813160
$826$	0	0
$827$	7309.12	0.307331	0.153666	−	0.988123i	$-0.450892\pi$
$827$	7309.12	0.307331	0.153666	+	0.988123i	$0.450892\pi$
$828$	0	0
$829$	36503.0	1.52931	0.764656	−	0.644438i	$-0.222909\pi$
$829$	36503.0	1.52931	0.764656	+	0.644438i	$0.222909\pi$
$830$	0	0
$831$	8260.41	0.344826
$832$	0	0
$833$	11305.0	0.470221
$834$	0	0
$835$	−36027.4	−1.49315
$836$	0	0
$837$	−14925.2	−0.616358
$838$	0	0
$839$	−11483.7	−0.472542	−0.236271	−	0.971687i	$-0.575925\pi$
$839$	−11483.7	−0.472542	−0.236271	+	0.971687i	$0.575925\pi$
$840$	0	0
$841$	−24373.0	−0.999343
$842$	0	0
$843$	2640.85	0.107895
$844$	0	0
$845$	10328.2	0.420473
$846$	0	0
$847$	−38720.3	−1.57077
$848$	0	0
$849$	5856.68	0.236750
$850$	0	0
$851$	31819.5	1.28174
$852$	0	0
$853$	14099.2	0.565939	0.282970	−	0.959129i	$-0.408680\pi$
$853$	14099.2	0.565939	0.282970	+	0.959129i	$0.408680\pi$
$854$	0	0
$855$	11398.2	0.455920
$856$	0	0
$857$	−15633.7	−0.623148	−0.311574	−	0.950222i	$-0.600856\pi$
$857$	−15633.7	−0.623148	−0.311574	+	0.950222i	$0.600856\pi$
$858$	0	0
$859$	25320.8	1.00574	0.502872	−	0.864361i	$-0.332277\pi$
$859$	25320.8	1.00574	0.502872	+	0.864361i	$0.332277\pi$
$860$	0	0
$861$	−13333.7	−0.527770
$862$	0	0
$863$	−31444.7	−1.24031	−0.620156	−	0.784479i	$-0.712931\pi$
$863$	−31444.7	−1.24031	−0.620156	+	0.784479i	$0.712931\pi$
$864$	0	0
$865$	24209.7	0.951623
$866$	0	0
$867$	−336.873	−0.0131959
$868$	0	0
$869$	410.349	0.0160186
$870$	0	0
$871$	−17664.2	−0.687173
$872$	0	0
$873$	−10029.1	−0.388814
$874$	0	0
$875$	16837.5	0.650526
$876$	0	0
$877$	32359.1	1.24594	0.622969	−	0.782246i	$-0.285926\pi$
$877$	32359.1	1.24594	0.622969	+	0.782246i	$0.285926\pi$
$878$	0	0
$879$	−3895.20	−0.149467
$880$	0	0
$881$	17110.9	0.654348	0.327174	−	0.944964i	$-0.393904\pi$
$881$	17110.9	0.654348	0.327174	+	0.944964i	$0.393904\pi$
$882$	0	0
$883$	−24437.9	−0.931369	−0.465685	−	0.884951i	$-0.654192\pi$
$883$	−24437.9	−0.931369	−0.465685	+	0.884951i	$0.654192\pi$
$884$	0	0
$885$	8912.33	0.338514
$886$	0	0
$887$	−27679.7	−1.04779	−0.523897	−	0.851781i	$-0.675522\pi$
$887$	−27679.7	−1.04779	−0.523897	+	0.851781i	$0.675522\pi$
$888$	0	0
$889$	−12854.0	−0.484937
$890$	0	0
$891$	−6552.87	−0.246386
$892$	0	0
$893$	−11076.9	−0.415090
$894$	0	0
$895$	73459.3	2.74354
$896$	0	0
$897$	−12416.0	−0.462160
$898$	0	0
$899$	−973.391	−0.0361117
$900$	0	0
$901$	1870.38	0.0691581
$902$	0	0
$903$	17637.0	0.649970
$904$	0	0
$905$	68720.9	2.52415
$906$	0	0
$907$	−23724.5	−0.868531	−0.434266	−	0.900785i	$-0.642992\pi$
$907$	−23724.5	−0.868531	−0.434266	+	0.900785i	$0.642992\pi$
$908$	0	0
$909$	36037.9	1.31496
$910$	0	0
$911$	12782.2	0.464866	0.232433	−	0.972612i	$-0.425331\pi$
$911$	12782.2	0.464866	0.232433	+	0.972612i	$0.425331\pi$
$912$	0	0
$913$	12571.3	0.455695
$914$	0	0
$915$	609.868	0.0220345
$916$	0	0
$917$	11479.6	0.413401
$918$	0	0
$919$	14177.2	0.508881	0.254441	−	0.967088i	$-0.418109\pi$
$919$	14177.2	0.508881	0.254441	+	0.967088i	$0.418109\pi$
$920$	0	0
$921$	4095.70	0.146534
$922$	0	0
$923$	6017.92	0.214607
$924$	0	0
$925$	24809.8	0.881883
$926$	0	0
$927$	−10628.7	−0.376581
$928$	0	0
$929$	−27706.5	−0.978492	−0.489246	−	0.872146i	$-0.662728\pi$
$929$	−27706.5	−0.978492	−0.489246	+	0.872146i	$0.662728\pi$
$930$	0	0
$931$	−17615.7	−0.620120
$932$	0	0
$933$	6829.98	0.239661
$934$	0	0
$935$	−3011.31	−0.105327
$936$	0	0
$937$	−6043.38	−0.210703	−0.105351	−	0.994435i	$-0.533597\pi$
$937$	−6043.38	−0.210703	−0.105351	+	0.994435i	$0.533597\pi$
$938$	0	0
$939$	8354.85	0.290362
$940$	0	0
$941$	48734.8	1.68832	0.844160	−	0.536092i	$-0.180100\pi$
$941$	48734.8	1.68832	0.844160	+	0.536092i	$0.180100\pi$
$942$	0	0
$943$	72363.7	2.49892
$944$	0	0
$945$	32692.1	1.12537
$946$	0	0
$947$	16251.6	0.557661	0.278831	−	0.960340i	$-0.410053\pi$
$947$	16251.6	0.557661	0.278831	+	0.960340i	$0.410053\pi$
$948$	0	0
$949$	6621.57	0.226497
$950$	0	0
$951$	9877.72	0.336811
$952$	0	0
$953$	−3428.14	−0.116525	−0.0582625	−	0.998301i	$-0.518556\pi$
$953$	−3428.14	−0.116525	−0.0582625	+	0.998301i	$0.518556\pi$
$954$	0	0
$955$	77925.1	2.64042
$956$	0	0
$957$	49.2399	0.00166322
$958$	0	0
$959$	97541.5	3.28444
$960$	0	0
$961$	29372.4	0.985950
$962$	0	0
$963$	26395.6	0.883266
$964$	0	0
$965$	22516.6	0.751123
$966$	0	0
$967$	8400.92	0.279375	0.139687	−	0.990196i	$-0.455390\pi$
$967$	8400.92	0.279375	0.139687	+	0.990196i	$0.455390\pi$
$968$	0	0
$969$	524.924	0.0174025
$970$	0	0
$971$	43523.7	1.43846	0.719229	−	0.694773i	$-0.244495\pi$
$971$	43523.7	1.43846	0.719229	+	0.694773i	$0.244495\pi$
$972$	0	0
$973$	20135.6	0.663431
$974$	0	0
$975$	−9680.82	−0.317984
$976$	0	0
$977$	−15850.0	−0.519024	−0.259512	−	0.965740i	$-0.583562\pi$
$977$	−15850.0	−0.519024	−0.259512	+	0.965740i	$0.583562\pi$
$978$	0	0
$979$	10326.5	0.337117
$980$	0	0
$981$	−21713.9	−0.706698
$982$	0	0
$983$	−34943.1	−1.13379	−0.566894	−	0.823791i	$-0.691855\pi$
$983$	−34943.1	−1.13379	−0.566894	+	0.823791i	$0.691855\pi$
$984$	0	0
$985$	−36967.8	−1.19583
$986$	0	0
$987$	−15475.3	−0.499071
$988$	0	0
$989$	−95718.4	−3.07752
$990$	0	0
$991$	−33028.6	−1.05872	−0.529358	−	0.848399i	$-0.677567\pi$
$991$	−33028.6	−1.05872	−0.529358	+	0.848399i	$0.677567\pi$
$992$	0	0
$993$	−9367.99	−0.299380
$994$	0	0
$995$	−34789.6	−1.10845
$996$	0	0
$997$	22849.2	0.725819	0.362910	−	0.931824i	$-0.381783\pi$
$997$	22849.2	0.725819	0.362910	+	0.931824i	$0.381783\pi$
$998$	0	0
$999$	9721.17	0.307872

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1088.4.a.bi.1.4		8
4.3	odd	2	inner	1088.4.a.bi.1.5		8
8.3	odd	2		544.4.a.m.1.4	✓	8
8.5	even	2		544.4.a.m.1.5	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
544.4.a.m.1.4	✓	8	8.3	odd	2
544.4.a.m.1.5	yes	8	8.5	even	2
1088.4.a.bi.1.4		8	1.1	even	1	trivial
1088.4.a.bi.1.5		8	4.3	odd	2	inner

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:	\( N \)	\(=\)	\( 1088 = 2^{6} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1088.a (trivial)

\(f(q)\)	\(=\)	\(q-1.16565 q^{3} +16.7810 q^{5} +31.7490 q^{7} -25.6413 q^{9} +O(q^{10})\)	\(q-1.16565 q^{3} +16.7810 q^{5} +31.7490 q^{7} -25.6413 q^{9} -10.5557 q^{11} +53.0327 q^{13} -19.5608 q^{15} +17.0000 q^{17} -26.4898 q^{19} -37.0082 q^{21} +200.849 q^{23} +156.603 q^{25} +61.3613 q^{27} +4.00185 q^{29} -243.235 q^{31} +12.3043 q^{33} +532.781 q^{35} +158.425 q^{37} -61.8176 q^{39} +360.289 q^{41} -476.569 q^{43} -430.287 q^{45} +418.157 q^{47} +664.999 q^{49} -19.8161 q^{51} +110.022 q^{53} -177.136 q^{55} +30.8779 q^{57} -455.621 q^{59} -31.1780 q^{61} -814.084 q^{63} +889.943 q^{65} -333.081 q^{67} -234.120 q^{69} +113.476 q^{71} +124.858 q^{73} -182.544 q^{75} -335.134 q^{77} -38.8745 q^{79} +620.788 q^{81} -1190.94 q^{83} +285.278 q^{85} -4.66476 q^{87} -978.287 q^{89} +1683.73 q^{91} +283.527 q^{93} -444.527 q^{95} +391.132 q^{97} +270.662 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{5} + 136 q^{9}+O(q^{10}) \) 8 * q - 16 * q^5 + 136 * q^9	\( 8 q - 16 q^{5} + 136 q^{9} + 56 q^{13} + 136 q^{17} - 312 q^{21} + 600 q^{25} + 128 q^{29} - 304 q^{33} + 112 q^{37} + 1328 q^{41} - 2592 q^{45} + 1624 q^{49} - 1216 q^{53} + 2560 q^{57} - 752 q^{61} + 1824 q^{65} - 1448 q^{69} + 3952 q^{73} + 1480 q^{77} + 6824 q^{81} - 272 q^{85} + 2304 q^{89} + 2680 q^{93} + 4560 q^{97}+O(q^{100}) \) 8 * q - 16 * q^5 + 136 * q^9 + 56 * q^13 + 136 * q^17 - 312 * q^21 + 600 * q^25 + 128 * q^29 - 304 * q^33 + 112 * q^37 + 1328 * q^41 - 2592 * q^45 + 1624 * q^49 - 1216 * q^53 + 2560 * q^57 - 752 * q^61 + 1824 * q^65 - 1448 * q^69 + 3952 * q^73 + 1480 * q^77 + 6824 * q^81 - 272 * q^85 + 2304 * q^89 + 2680 * q^93 + 4560 * q^97

Introduction

L-functions

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