LMFDB - Embedded newform 8624.2.a.bf.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8624,2,Mod(1,8624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8624, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8624.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8624 = 2^{4} \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8624.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:	$68.8629867032$
Analytic rank:	$1$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 154)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	8624.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.23607 q^{3} -3.23607 q^{5} +7.47214 q^{9} +O(q^{10})$	$q-3.23607 q^{3} -3.23607 q^{5} +7.47214 q^{9} -1.00000 q^{11} -1.23607 q^{13} +10.4721 q^{15} +6.47214 q^{17} -2.76393 q^{19} -4.00000 q^{23} +5.47214 q^{25} -14.4721 q^{27} -4.47214 q^{29} +2.00000 q^{31} +3.23607 q^{33} -10.9443 q^{37} +4.00000 q^{39} -6.47214 q^{41} +1.52786 q^{43} -24.1803 q^{45} -2.00000 q^{47} -20.9443 q^{51} -0.472136 q^{53} +3.23607 q^{55} +8.94427 q^{57} +7.23607 q^{59} +5.23607 q^{61} +4.00000 q^{65} +15.4164 q^{67} +12.9443 q^{69} +2.47214 q^{71} +4.94427 q^{73} -17.7082 q^{75} +24.4164 q^{81} +10.1803 q^{83} -20.9443 q^{85} +14.4721 q^{87} -10.0000 q^{89} -6.47214 q^{93} +8.94427 q^{95} -3.52786 q^{97} -7.47214 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{5} + 6 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 2 * q^5 + 6 * q^9	$ 2 q - 2 q^{3} - 2 q^{5} + 6 q^{9} - 2 q^{11} + 2 q^{13} + 12 q^{15} + 4 q^{17} - 10 q^{19} - 8 q^{23} + 2 q^{25} - 20 q^{27} + 4 q^{31} + 2 q^{33} - 4 q^{37} + 8 q^{39} - 4 q^{41} + 12 q^{43} - 26 q^{45} - 4 q^{47} - 24 q^{51} + 8 q^{53} + 2 q^{55} + 10 q^{59} + 6 q^{61} + 8 q^{65} + 4 q^{67} + 8 q^{69} - 4 q^{71} - 8 q^{73} - 22 q^{75} + 22 q^{81} - 2 q^{83} - 24 q^{85} + 20 q^{87} - 20 q^{89} - 4 q^{93} - 16 q^{97} - 6 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^5 + 6 * q^9 - 2 * q^11 + 2 * q^13 + 12 * q^15 + 4 * q^17 - 10 * q^19 - 8 * q^23 + 2 * q^25 - 20 * q^27 + 4 * q^31 + 2 * q^33 - 4 * q^37 + 8 * q^39 - 4 * q^41 + 12 * q^43 - 26 * q^45 - 4 * q^47 - 24 * q^51 + 8 * q^53 + 2 * q^55 + 10 * q^59 + 6 * q^61 + 8 * q^65 + 4 * q^67 + 8 * q^69 - 4 * q^71 - 8 * q^73 - 22 * q^75 + 22 * q^81 - 2 * q^83 - 24 * q^85 + 20 * q^87 - 20 * q^89 - 4 * q^93 - 16 * q^97 - 6 * q^99

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
$3$	−3.23607	−1.86834	−0.934172	−	0.356822i	$-0.883860\pi$
$3$	−3.23607	−1.86834	−0.934172	+	0.356822i	$0.883860\pi$
$4$	0	0
$5$	−3.23607	−1.44721	−0.723607	−	0.690212i	$-0.757517\pi$
$5$	−3.23607	−1.44721	−0.723607	+	0.690212i	$0.757517\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	7.47214	2.49071
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−1.23607	−0.342824	−0.171412	−	0.985199i	$-0.554833\pi$
$13$	−1.23607	−0.342824	−0.171412	+	0.985199i	$0.554833\pi$
$14$	0	0
$15$	10.4721	2.70389
$16$	0	0
$17$	6.47214	1.56972	0.784862	−	0.619671i	$-0.212734\pi$
$17$	6.47214	1.56972	0.784862	+	0.619671i	$0.212734\pi$
$18$	0	0
$19$	−2.76393	−0.634089	−0.317045	−	0.948411i	$-0.602691\pi$
$19$	−2.76393	−0.634089	−0.317045	+	0.948411i	$0.602691\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−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
$25$	5.47214	1.09443
$26$	0	0
$27$	−14.4721	−2.78516
$28$	0	0
$29$	−4.47214	−0.830455	−0.415227	−	0.909718i	$-0.636298\pi$
$29$	−4.47214	−0.830455	−0.415227	+	0.909718i	$0.636298\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	0	0
$33$	3.23607	0.563327
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−10.9443	−1.79923	−0.899614	−	0.436687i	$-0.856152\pi$
$37$	−10.9443	−1.79923	−0.899614	+	0.436687i	$0.856152\pi$
$38$	0	0
$39$	4.00000	0.640513
$40$	0	0
$41$	−6.47214	−1.01078	−0.505389	−	0.862892i	$-0.668651\pi$
$41$	−6.47214	−1.01078	−0.505389	+	0.862892i	$0.668651\pi$
$42$	0	0
$43$	1.52786	0.232997	0.116499	−	0.993191i	$-0.462833\pi$
$43$	1.52786	0.232997	0.116499	+	0.993191i	$0.462833\pi$
$44$	0	0
$45$	−24.1803	−3.60459
$46$	0	0
$47$	−2.00000	−0.291730	−0.145865	−	0.989305i	$-0.546597\pi$
$47$	−2.00000	−0.291730	−0.145865	+	0.989305i	$0.546597\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−20.9443	−2.93278
$52$	0	0
$53$	−0.472136	−0.0648529	−0.0324264	−	0.999474i	$-0.510323\pi$
$53$	−0.472136	−0.0648529	−0.0324264	+	0.999474i	$0.510323\pi$
$54$	0	0
$55$	3.23607	0.436351
$56$	0	0
$57$	8.94427	1.18470
$58$	0	0
$59$	7.23607	0.942056	0.471028	−	0.882118i	$-0.343883\pi$
$59$	7.23607	0.942056	0.471028	+	0.882118i	$0.343883\pi$
$60$	0	0
$61$	5.23607	0.670410	0.335205	−	0.942145i	$-0.391194\pi$
$61$	5.23607	0.670410	0.335205	+	0.942145i	$0.391194\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	4.00000	0.496139
$66$	0	0
$67$	15.4164	1.88341	0.941707	−	0.336434i	$-0.109221\pi$
$67$	15.4164	1.88341	0.941707	+	0.336434i	$0.109221\pi$
$68$	0	0
$69$	12.9443	1.55831
$70$	0	0
$71$	2.47214	0.293389	0.146694	−	0.989182i	$-0.453137\pi$
$71$	2.47214	0.293389	0.146694	+	0.989182i	$0.453137\pi$
$72$	0	0
$73$	4.94427	0.578683	0.289342	−	0.957226i	$-0.406564\pi$
$73$	4.94427	0.578683	0.289342	+	0.957226i	$0.406564\pi$
$74$	0	0
$75$	−17.7082	−2.04477
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	24.4164	2.71293
$82$	0	0
$83$	10.1803	1.11744	0.558719	−	0.829357i	$-0.311293\pi$
$83$	10.1803	1.11744	0.558719	+	0.829357i	$0.311293\pi$
$84$	0	0
$85$	−20.9443	−2.27173
$86$	0	0
$87$	14.4721	1.55158
$88$	0	0
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−6.47214	−0.671129
$94$	0	0
$95$	8.94427	0.917663
$96$	0	0
$97$	−3.52786	−0.358200	−0.179100	−	0.983831i	$-0.557319\pi$
$97$	−3.52786	−0.358200	−0.179100	+	0.983831i	$0.557319\pi$
$98$	0	0
$99$	−7.47214	−0.750978
$100$	0	0
$101$	14.1803	1.41100	0.705498	−	0.708712i	$-0.250723\pi$
$101$	14.1803	1.41100	0.705498	+	0.708712i	$0.250723\pi$
$102$	0	0
$103$	2.94427	0.290108	0.145054	−	0.989424i	$-0.453664\pi$
$103$	2.94427	0.290108	0.145054	+	0.989424i	$0.453664\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.47214	0.625685	0.312842	−	0.949805i	$-0.398719\pi$
$107$	6.47214	0.625685	0.312842	+	0.949805i	$0.398719\pi$
$108$	0	0
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	35.4164	3.36158
$112$	0	0
$113$	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
$115$	12.9443	1.20706
$116$	0	0
$117$	−9.23607	−0.853875
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	20.9443	1.88848
$124$	0	0
$125$	−1.52786	−0.136656
$126$	0	0
$127$	12.0000	1.06483	0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000	1.06483	0.532414	+	0.846484i	$0.321285\pi$
$128$	0	0
$129$	−4.94427	−0.435319
$130$	0	0
$131$	9.23607	0.806959	0.403480	−	0.914989i	$-0.367801\pi$
$131$	9.23607	0.806959	0.403480	+	0.914989i	$0.367801\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	46.8328	4.03073
$136$	0	0
$137$	15.8885	1.35745	0.678725	−	0.734393i	$-0.262533\pi$
$137$	15.8885	1.35745	0.678725	+	0.734393i	$0.262533\pi$
$138$	0	0
$139$	−8.29180	−0.703301	−0.351650	−	0.936131i	$-0.614379\pi$
$139$	−8.29180	−0.703301	−0.351650	+	0.936131i	$0.614379\pi$
$140$	0	0
$141$	6.47214	0.545052
$142$	0	0
$143$	1.23607	0.103365
$144$	0	0
$145$	14.4721	1.20185
$146$	0	0
$147$	0	0
$148$	0	0
$149$	22.3607	1.83186	0.915929	−	0.401340i	$-0.131455\pi$
$149$	22.3607	1.83186	0.915929	+	0.401340i	$0.131455\pi$
$150$	0	0
$151$	−12.0000	−0.976546	−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000	−0.976546	−0.488273	+	0.872691i	$0.662373\pi$
$152$	0	0
$153$	48.3607	3.90973
$154$	0	0
$155$	−6.47214	−0.519854
$156$	0	0
$157$	−18.6525	−1.48863	−0.744315	−	0.667829i	$-0.767224\pi$
$157$	−18.6525	−1.48863	−0.744315	+	0.667829i	$0.767224\pi$
$158$	0	0
$159$	1.52786	0.121168
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−7.41641	−0.580898	−0.290449	−	0.956890i	$-0.593805\pi$
$163$	−7.41641	−0.580898	−0.290449	+	0.956890i	$0.593805\pi$
$164$	0	0
$165$	−10.4721	−0.815255
$166$	0	0
$167$	−15.4164	−1.19296	−0.596479	−	0.802629i	$-0.703434\pi$
$167$	−15.4164	−1.19296	−0.596479	+	0.802629i	$0.703434\pi$
$168$	0	0
$169$	−11.4721	−0.882472
$170$	0	0
$171$	−20.6525	−1.57933
$172$	0	0
$173$	−1.23607	−0.0939765	−0.0469883	−	0.998895i	$-0.514962\pi$
$173$	−1.23607	−0.0939765	−0.0469883	+	0.998895i	$0.514962\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−23.4164	−1.76008
$178$	0	0
$179$	−8.94427	−0.668526	−0.334263	−	0.942480i	$-0.608487\pi$
$179$	−8.94427	−0.668526	−0.334263	+	0.942480i	$0.608487\pi$
$180$	0	0
$181$	−4.76393	−0.354100	−0.177050	−	0.984202i	$-0.556655\pi$
$181$	−4.76393	−0.354100	−0.177050	+	0.984202i	$0.556655\pi$
$182$	0	0
$183$	−16.9443	−1.25256
$184$	0	0
$185$	35.4164	2.60387
$186$	0	0
$187$	−6.47214	−0.473289
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−6.47214	−0.468307	−0.234154	−	0.972200i	$-0.575232\pi$
$191$	−6.47214	−0.468307	−0.234154	+	0.972200i	$0.575232\pi$
$192$	0	0
$193$	2.94427	0.211933	0.105967	−	0.994370i	$-0.466206\pi$
$193$	2.94427	0.211933	0.105967	+	0.994370i	$0.466206\pi$
$194$	0	0
$195$	−12.9443	−0.926959
$196$	0	0
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	0	0
$199$	1.05573	0.0748386	0.0374193	−	0.999300i	$-0.488086\pi$
$199$	1.05573	0.0748386	0.0374193	+	0.999300i	$0.488086\pi$
$200$	0	0
$201$	−49.8885	−3.51887
$202$	0	0
$203$	0	0
$204$	0	0
$205$	20.9443	1.46281
$206$	0	0
$207$	−29.8885	−2.07740
$208$	0	0
$209$	2.76393	0.191185
$210$	0	0
$211$	22.4721	1.54705	0.773523	−	0.633768i	$-0.218493\pi$
$211$	22.4721	1.54705	0.773523	+	0.633768i	$0.218493\pi$
$212$	0	0
$213$	−8.00000	−0.548151
$214$	0	0
$215$	−4.94427	−0.337197
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−16.0000	−1.08118
$220$	0	0
$221$	−8.00000	−0.538138
$222$	0	0
$223$	8.47214	0.567336	0.283668	−	0.958923i	$-0.408449\pi$
$223$	8.47214	0.567336	0.283668	+	0.958923i	$0.408449\pi$
$224$	0	0
$225$	40.8885	2.72590
$226$	0	0
$227$	−14.7639	−0.979917	−0.489958	−	0.871746i	$-0.662988\pi$
$227$	−14.7639	−0.979917	−0.489958	+	0.871746i	$0.662988\pi$
$228$	0	0
$229$	−12.7639	−0.843464	−0.421732	−	0.906720i	$-0.638578\pi$
$229$	−12.7639	−0.843464	−0.421732	+	0.906720i	$0.638578\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2.94427	0.192886	0.0964428	−	0.995339i	$-0.469254\pi$
$233$	2.94427	0.192886	0.0964428	+	0.995339i	$0.469254\pi$
$234$	0	0
$235$	6.47214	0.422196
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−20.0000	−1.29369	−0.646846	−	0.762620i	$-0.723912\pi$
$239$	−20.0000	−1.29369	−0.646846	+	0.762620i	$0.723912\pi$
$240$	0	0
$241$	11.4164	0.735395	0.367698	−	0.929945i	$-0.380146\pi$
$241$	11.4164	0.735395	0.367698	+	0.929945i	$0.380146\pi$
$242$	0	0
$243$	−35.5967	−2.28353
$244$	0	0
$245$	0	0
$246$	0	0
$247$	3.41641	0.217381
$248$	0	0
$249$	−32.9443	−2.08776
$250$	0	0
$251$	24.7639	1.56309	0.781543	−	0.623852i	$-0.214433\pi$
$251$	24.7639	1.56309	0.781543	+	0.623852i	$0.214433\pi$
$252$	0	0
$253$	4.00000	0.251478
$254$	0	0
$255$	67.7771	4.24437
$256$	0	0
$257$	10.9443	0.682685	0.341342	−	0.939939i	$-0.389118\pi$
$257$	10.9443	0.682685	0.341342	+	0.939939i	$0.389118\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−33.4164	−2.06842
$262$	0	0
$263$	−12.9443	−0.798178	−0.399089	−	0.916912i	$-0.630674\pi$
$263$	−12.9443	−0.798178	−0.399089	+	0.916912i	$0.630674\pi$
$264$	0	0
$265$	1.52786	0.0938559
$266$	0	0
$267$	32.3607	1.98044
$268$	0	0
$269$	27.2361	1.66061	0.830306	−	0.557307i	$-0.188166\pi$
$269$	27.2361	1.66061	0.830306	+	0.557307i	$0.188166\pi$
$270$	0	0
$271$	−16.9443	−1.02929	−0.514646	−	0.857403i	$-0.672076\pi$
$271$	−16.9443	−1.02929	−0.514646	+	0.857403i	$0.672076\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−5.47214	−0.329982
$276$	0	0
$277$	12.4721	0.749378	0.374689	−	0.927151i	$-0.377749\pi$
$277$	12.4721	0.749378	0.374689	+	0.927151i	$0.377749\pi$
$278$	0	0
$279$	14.9443	0.894690
$280$	0	0
$281$	−24.8328	−1.48140	−0.740701	−	0.671835i	$-0.765506\pi$
$281$	−24.8328	−1.48140	−0.740701	+	0.671835i	$0.765506\pi$
$282$	0	0
$283$	−16.6525	−0.989887	−0.494943	−	0.868925i	$-0.664811\pi$
$283$	−16.6525	−0.989887	−0.494943	+	0.868925i	$0.664811\pi$
$284$	0	0
$285$	−28.9443	−1.71451
$286$	0	0
$287$	0	0
$288$	0	0
$289$	24.8885	1.46403
$290$	0	0
$291$	11.4164	0.669242
$292$	0	0
$293$	−4.65248	−0.271801	−0.135900	−	0.990723i	$-0.543393\pi$
$293$	−4.65248	−0.271801	−0.135900	+	0.990723i	$0.543393\pi$
$294$	0	0
$295$	−23.4164	−1.36336
$296$	0	0
$297$	14.4721	0.839759
$298$	0	0
$299$	4.94427	0.285935
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−45.8885	−2.63623
$304$	0	0
$305$	−16.9443	−0.970226
$306$	0	0
$307$	32.0689	1.83027	0.915134	−	0.403150i	$-0.132085\pi$
$307$	32.0689	1.83027	0.915134	+	0.403150i	$0.132085\pi$
$308$	0	0
$309$	−9.52786	−0.542021
$310$	0	0
$311$	5.41641	0.307136	0.153568	−	0.988138i	$-0.450924\pi$
$311$	5.41641	0.307136	0.153568	+	0.988138i	$0.450924\pi$
$312$	0	0
$313$	−28.4721	−1.60934	−0.804670	−	0.593722i	$-0.797658\pi$
$313$	−28.4721	−1.60934	−0.804670	+	0.593722i	$0.797658\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−13.0557	−0.733283	−0.366641	−	0.930362i	$-0.619492\pi$
$317$	−13.0557	−0.733283	−0.366641	+	0.930362i	$0.619492\pi$
$318$	0	0
$319$	4.47214	0.250392
$320$	0	0
$321$	−20.9443	−1.16900
$322$	0	0
$323$	−17.8885	−0.995345
$324$	0	0
$325$	−6.76393	−0.375195
$326$	0	0
$327$	32.3607	1.78955
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−0.944272	−0.0519019	−0.0259509	−	0.999663i	$-0.508261\pi$
$331$	−0.944272	−0.0519019	−0.0259509	+	0.999663i	$0.508261\pi$
$332$	0	0
$333$	−81.7771	−4.48136
$334$	0	0
$335$	−49.8885	−2.72570
$336$	0	0
$337$	18.0000	0.980522	0.490261	−	0.871576i	$-0.336901\pi$
$337$	18.0000	0.980522	0.490261	+	0.871576i	$0.336901\pi$
$338$	0	0
$339$	−27.4164	−1.48905
$340$	0	0
$341$	−2.00000	−0.108306
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−41.8885	−2.25520
$346$	0	0
$347$	6.47214	0.347442	0.173721	−	0.984795i	$-0.444421\pi$
$347$	6.47214	0.347442	0.173721	+	0.984795i	$0.444421\pi$
$348$	0	0
$349$	−8.29180	−0.443850	−0.221925	−	0.975064i	$-0.571234\pi$
$349$	−8.29180	−0.443850	−0.221925	+	0.975064i	$0.571234\pi$
$350$	0	0
$351$	17.8885	0.954820
$352$	0	0
$353$	34.9443	1.85990	0.929948	−	0.367691i	$-0.119852\pi$
$353$	34.9443	1.85990	0.929948	+	0.367691i	$0.119852\pi$
$354$	0	0
$355$	−8.00000	−0.424596
$356$	0	0
$357$	0	0
$358$	0	0
$359$	26.8328	1.41618	0.708091	−	0.706121i	$-0.249557\pi$
$359$	26.8328	1.41618	0.708091	+	0.706121i	$0.249557\pi$
$360$	0	0
$361$	−11.3607	−0.597931
$362$	0	0
$363$	−3.23607	−0.169850
$364$	0	0
$365$	−16.0000	−0.837478
$366$	0	0
$367$	21.4164	1.11793	0.558964	−	0.829192i	$-0.311199\pi$
$367$	21.4164	1.11793	0.558964	+	0.829192i	$0.311199\pi$
$368$	0	0
$369$	−48.3607	−2.51756
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−6.00000	−0.310668	−0.155334	−	0.987862i	$-0.549645\pi$
$373$	−6.00000	−0.310668	−0.155334	+	0.987862i	$0.549645\pi$
$374$	0	0
$375$	4.94427	0.255321
$376$	0	0
$377$	5.52786	0.284699
$378$	0	0
$379$	−5.52786	−0.283947	−0.141974	−	0.989870i	$-0.545345\pi$
$379$	−5.52786	−0.283947	−0.141974	+	0.989870i	$0.545345\pi$
$380$	0	0
$381$	−38.8328	−1.98947
$382$	0	0
$383$	11.8885	0.607476	0.303738	−	0.952756i	$-0.401765\pi$
$383$	11.8885	0.607476	0.303738	+	0.952756i	$0.401765\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	11.4164	0.580329
$388$	0	0
$389$	6.58359	0.333801	0.166901	−	0.985974i	$-0.446624\pi$
$389$	6.58359	0.333801	0.166901	+	0.985974i	$0.446624\pi$
$390$	0	0
$391$	−25.8885	−1.30924
$392$	0	0
$393$	−29.8885	−1.50768
$394$	0	0
$395$	0	0
$396$	0	0
$397$	10.2918	0.516530	0.258265	−	0.966074i	$-0.416849\pi$
$397$	10.2918	0.516530	0.258265	+	0.966074i	$0.416849\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−30.3607	−1.51614	−0.758070	−	0.652173i	$-0.773857\pi$
$401$	−30.3607	−1.51614	−0.758070	+	0.652173i	$0.773857\pi$
$402$	0	0
$403$	−2.47214	−0.123146
$404$	0	0
$405$	−79.0132	−3.92620
$406$	0	0
$407$	10.9443	0.542487
$408$	0	0
$409$	23.4164	1.15787	0.578933	−	0.815375i	$-0.303469\pi$
$409$	23.4164	1.15787	0.578933	+	0.815375i	$0.303469\pi$
$410$	0	0
$411$	−51.4164	−2.53618
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−32.9443	−1.61717
$416$	0	0
$417$	26.8328	1.31401
$418$	0	0
$419$	12.7639	0.623559	0.311779	−	0.950155i	$-0.399075\pi$
$419$	12.7639	0.623559	0.311779	+	0.950155i	$0.399075\pi$
$420$	0	0
$421$	7.52786	0.366886	0.183443	−	0.983030i	$-0.441276\pi$
$421$	7.52786	0.366886	0.183443	+	0.983030i	$0.441276\pi$
$422$	0	0
$423$	−14.9443	−0.726615
$424$	0	0
$425$	35.4164	1.71795
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−4.00000	−0.193122
$430$	0	0
$431$	−40.9443	−1.97222	−0.986108	−	0.166105i	$-0.946881\pi$
$431$	−40.9443	−1.97222	−0.986108	+	0.166105i	$0.946881\pi$
$432$	0	0
$433$	−19.5279	−0.938449	−0.469225	−	0.883079i	$-0.655467\pi$
$433$	−19.5279	−0.938449	−0.469225	+	0.883079i	$0.655467\pi$
$434$	0	0
$435$	−46.8328	−2.24546
$436$	0	0
$437$	11.0557	0.528867
$438$	0	0
$439$	−8.94427	−0.426887	−0.213443	−	0.976955i	$-0.568468\pi$
$439$	−8.94427	−0.426887	−0.213443	+	0.976955i	$0.568468\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	7.05573	0.335228	0.167614	−	0.985853i	$-0.446394\pi$
$443$	7.05573	0.335228	0.167614	+	0.985853i	$0.446394\pi$
$444$	0	0
$445$	32.3607	1.53404
$446$	0	0
$447$	−72.3607	−3.42254
$448$	0	0
$449$	1.05573	0.0498229	0.0249114	−	0.999690i	$-0.492070\pi$
$449$	1.05573	0.0498229	0.0249114	+	0.999690i	$0.492070\pi$
$450$	0	0
$451$	6.47214	0.304761
$452$	0	0
$453$	38.8328	1.82452
$454$	0	0
$455$	0	0
$456$	0	0
$457$	9.05573	0.423609	0.211805	−	0.977312i	$-0.432066\pi$
$457$	9.05573	0.423609	0.211805	+	0.977312i	$0.432066\pi$
$458$	0	0
$459$	−93.6656	−4.37194
$460$	0	0
$461$	−29.2361	−1.36166	−0.680830	−	0.732442i	$-0.738381\pi$
$461$	−29.2361	−1.36166	−0.680830	+	0.732442i	$0.738381\pi$
$462$	0	0
$463$	21.5279	1.00048	0.500242	−	0.865885i	$-0.333244\pi$
$463$	21.5279	1.00048	0.500242	+	0.865885i	$0.333244\pi$
$464$	0	0
$465$	20.9443	0.971267
$466$	0	0
$467$	13.1246	0.607335	0.303667	−	0.952778i	$-0.401789\pi$
$467$	13.1246	0.607335	0.303667	+	0.952778i	$0.401789\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	60.3607	2.78127
$472$	0	0
$473$	−1.52786	−0.0702513
$474$	0	0
$475$	−15.1246	−0.693965
$476$	0	0
$477$	−3.52786	−0.161530
$478$	0	0
$479$	−32.3607	−1.47860	−0.739299	−	0.673378i	$-0.764843\pi$
$479$	−32.3607	−1.47860	−0.739299	+	0.673378i	$0.764843\pi$
$480$	0	0
$481$	13.5279	0.616818
$482$	0	0
$483$	0	0
$484$	0	0
$485$	11.4164	0.518392
$486$	0	0
$487$	0.944272	0.0427890	0.0213945	−	0.999771i	$-0.493189\pi$
$487$	0.944272	0.0427890	0.0213945	+	0.999771i	$0.493189\pi$
$488$	0	0
$489$	24.0000	1.08532
$490$	0	0
$491$	−0.944272	−0.0426144	−0.0213072	−	0.999773i	$-0.506783\pi$
$491$	−0.944272	−0.0426144	−0.0213072	+	0.999773i	$0.506783\pi$
$492$	0	0
$493$	−28.9443	−1.30358
$494$	0	0
$495$	24.1803	1.08683
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−12.3607	−0.553340	−0.276670	−	0.960965i	$-0.589231\pi$
$499$	−12.3607	−0.553340	−0.276670	+	0.960965i	$0.589231\pi$
$500$	0	0
$501$	49.8885	2.22886
$502$	0	0
$503$	4.00000	0.178351	0.0891756	−	0.996016i	$-0.471577\pi$
$503$	4.00000	0.178351	0.0891756	+	0.996016i	$0.471577\pi$
$504$	0	0
$505$	−45.8885	−2.04201
$506$	0	0
$507$	37.1246	1.64876
$508$	0	0
$509$	34.0689	1.51008	0.755038	−	0.655681i	$-0.227618\pi$
$509$	34.0689	1.51008	0.755038	+	0.655681i	$0.227618\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	40.0000	1.76604
$514$	0	0
$515$	−9.52786	−0.419848
$516$	0	0
$517$	2.00000	0.0879599
$518$	0	0
$519$	4.00000	0.175581
$520$	0	0
$521$	−34.3607	−1.50537	−0.752684	−	0.658382i	$-0.771241\pi$
$521$	−34.3607	−1.50537	−0.752684	+	0.658382i	$0.771241\pi$
$522$	0	0
$523$	−27.7082	−1.21160	−0.605798	−	0.795619i	$-0.707146\pi$
$523$	−27.7082	−1.21160	−0.605798	+	0.795619i	$0.707146\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	12.9443	0.563861
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	54.0689	2.34639
$532$	0	0
$533$	8.00000	0.346518
$534$	0	0
$535$	−20.9443	−0.905500
$536$	0	0
$537$	28.9443	1.24904
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−9.05573	−0.389336	−0.194668	−	0.980869i	$-0.562363\pi$
$541$	−9.05573	−0.389336	−0.194668	+	0.980869i	$0.562363\pi$
$542$	0	0
$543$	15.4164	0.661581
$544$	0	0
$545$	32.3607	1.38618
$546$	0	0
$547$	−16.9443	−0.724485	−0.362242	−	0.932084i	$-0.617989\pi$
$547$	−16.9443	−0.724485	−0.362242	+	0.932084i	$0.617989\pi$
$548$	0	0
$549$	39.1246	1.66980
$550$	0	0
$551$	12.3607	0.526583
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−114.610	−4.86492
$556$	0	0
$557$	−28.8328	−1.22169	−0.610843	−	0.791752i	$-0.709169\pi$
$557$	−28.8328	−1.22169	−0.610843	+	0.791752i	$0.709169\pi$
$558$	0	0
$559$	−1.88854	−0.0798769
$560$	0	0
$561$	20.9443	0.884268
$562$	0	0
$563$	26.7639	1.12797	0.563983	−	0.825787i	$-0.309268\pi$
$563$	26.7639	1.12797	0.563983	+	0.825787i	$0.309268\pi$
$564$	0	0
$565$	−27.4164	−1.15342
$566$	0	0
$567$	0	0
$568$	0	0
$569$	16.8328	0.705668	0.352834	−	0.935686i	$-0.385218\pi$
$569$	16.8328	0.705668	0.352834	+	0.935686i	$0.385218\pi$
$570$	0	0
$571$	45.8885	1.92038	0.960188	−	0.279355i	$-0.0901206\pi$
$571$	45.8885	1.92038	0.960188	+	0.279355i	$0.0901206\pi$
$572$	0	0
$573$	20.9443	0.874960
$574$	0	0
$575$	−21.8885	−0.912815
$576$	0	0
$577$	−9.05573	−0.376995	−0.188497	−	0.982074i	$-0.560362\pi$
$577$	−9.05573	−0.376995	−0.188497	+	0.982074i	$0.560362\pi$
$578$	0	0
$579$	−9.52786	−0.395965
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0.472136	0.0195539
$584$	0	0
$585$	29.8885	1.23574
$586$	0	0
$587$	−28.1803	−1.16313	−0.581564	−	0.813501i	$-0.697559\pi$
$587$	−28.1803	−1.16313	−0.581564	+	0.813501i	$0.697559\pi$
$588$	0	0
$589$	−5.52786	−0.227772
$590$	0	0
$591$	−58.2492	−2.39605
$592$	0	0
$593$	−24.0000	−0.985562	−0.492781	−	0.870153i	$-0.664020\pi$
$593$	−24.0000	−0.985562	−0.492781	+	0.870153i	$0.664020\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−3.41641	−0.139824
$598$	0	0
$599$	−12.3607	−0.505044	−0.252522	−	0.967591i	$-0.581260\pi$
$599$	−12.3607	−0.505044	−0.252522	+	0.967591i	$0.581260\pi$
$600$	0	0
$601$	−18.8328	−0.768207	−0.384103	−	0.923290i	$-0.625489\pi$
$601$	−18.8328	−0.768207	−0.384103	+	0.923290i	$0.625489\pi$
$602$	0	0
$603$	115.193	4.69104
$604$	0	0
$605$	−3.23607	−0.131565
$606$	0	0
$607$	−32.0000	−1.29884	−0.649420	−	0.760430i	$-0.724988\pi$
$607$	−32.0000	−1.29884	−0.649420	+	0.760430i	$0.724988\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2.47214	0.100012
$612$	0	0
$613$	19.5279	0.788723	0.394362	−	0.918955i	$-0.370966\pi$
$613$	19.5279	0.788723	0.394362	+	0.918955i	$0.370966\pi$
$614$	0	0
$615$	−67.7771	−2.73304
$616$	0	0
$617$	−5.41641	−0.218056	−0.109028	−	0.994039i	$-0.534774\pi$
$617$	−5.41641	−0.218056	−0.109028	+	0.994039i	$0.534774\pi$
$618$	0	0
$619$	−48.5410	−1.95103	−0.975514	−	0.219937i	$-0.929415\pi$
$619$	−48.5410	−1.95103	−0.975514	+	0.219937i	$0.929415\pi$
$620$	0	0
$621$	57.8885	2.32299
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−22.4164	−0.896656
$626$	0	0
$627$	−8.94427	−0.357200
$628$	0	0
$629$	−70.8328	−2.82429
$630$	0	0
$631$	4.58359	0.182470	0.0912350	−	0.995829i	$-0.470919\pi$
$631$	4.58359	0.182470	0.0912350	+	0.995829i	$0.470919\pi$
$632$	0	0
$633$	−72.7214	−2.89041
$634$	0	0
$635$	−38.8328	−1.54103
$636$	0	0
$637$	0	0
$638$	0	0
$639$	18.4721	0.730746
$640$	0	0
$641$	36.4721	1.44056	0.720281	−	0.693682i	$-0.244013\pi$
$641$	36.4721	1.44056	0.720281	+	0.693682i	$0.244013\pi$
$642$	0	0
$643$	−23.2361	−0.916341	−0.458171	−	0.888864i	$-0.651495\pi$
$643$	−23.2361	−0.916341	−0.458171	+	0.888864i	$0.651495\pi$
$644$	0	0
$645$	16.0000	0.629999
$646$	0	0
$647$	24.8328	0.976279	0.488139	−	0.872766i	$-0.337676\pi$
$647$	24.8328	0.976279	0.488139	+	0.872766i	$0.337676\pi$
$648$	0	0
$649$	−7.23607	−0.284041
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.63932	0.0641516	0.0320758	−	0.999485i	$-0.489788\pi$
$653$	1.63932	0.0641516	0.0320758	+	0.999485i	$0.489788\pi$
$654$	0	0
$655$	−29.8885	−1.16784
$656$	0	0
$657$	36.9443	1.44133
$658$	0	0
$659$	−43.4164	−1.69126	−0.845632	−	0.533767i	$-0.820776\pi$
$659$	−43.4164	−1.69126	−0.845632	+	0.533767i	$0.820776\pi$
$660$	0	0
$661$	−37.1246	−1.44398	−0.721990	−	0.691903i	$-0.756772\pi$
$661$	−37.1246	−1.44398	−0.721990	+	0.691903i	$0.756772\pi$
$662$	0	0
$663$	25.8885	1.00543
$664$	0	0
$665$	0	0
$666$	0	0
$667$	17.8885	0.692647
$668$	0	0
$669$	−27.4164	−1.05998
$670$	0	0
$671$	−5.23607	−0.202136
$672$	0	0
$673$	31.8885	1.22921	0.614607	−	0.788834i	$-0.289315\pi$
$673$	31.8885	1.22921	0.614607	+	0.788834i	$0.289315\pi$
$674$	0	0
$675$	−79.1935	−3.04816
$676$	0	0
$677$	−32.0689	−1.23251	−0.616254	−	0.787548i	$-0.711350\pi$
$677$	−32.0689	−1.23251	−0.616254	+	0.787548i	$0.711350\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	47.7771	1.83082
$682$	0	0
$683$	−15.0557	−0.576091	−0.288046	−	0.957617i	$-0.593006\pi$
$683$	−15.0557	−0.576091	−0.288046	+	0.957617i	$0.593006\pi$
$684$	0	0
$685$	−51.4164	−1.96452
$686$	0	0
$687$	41.3050	1.57588
$688$	0	0
$689$	0.583592	0.0222331
$690$	0	0
$691$	−18.6525	−0.709574	−0.354787	−	0.934947i	$-0.615447\pi$
$691$	−18.6525	−0.709574	−0.354787	+	0.934947i	$0.615447\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	26.8328	1.01783
$696$	0	0
$697$	−41.8885	−1.58664
$698$	0	0
$699$	−9.52786	−0.360377
$700$	0	0
$701$	46.7214	1.76464	0.882321	−	0.470649i	$-0.155980\pi$
$701$	46.7214	1.76464	0.882321	+	0.470649i	$0.155980\pi$
$702$	0	0
$703$	30.2492	1.14087
$704$	0	0
$705$	−20.9443	−0.788807
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−4.47214	−0.167955	−0.0839773	−	0.996468i	$-0.526762\pi$
$709$	−4.47214	−0.167955	−0.0839773	+	0.996468i	$0.526762\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−8.00000	−0.299602
$714$	0	0
$715$	−4.00000	−0.149592
$716$	0	0
$717$	64.7214	2.41706
$718$	0	0
$719$	−36.8328	−1.37363	−0.686816	−	0.726831i	$-0.740992\pi$
$719$	−36.8328	−1.37363	−0.686816	+	0.726831i	$0.740992\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−36.9443	−1.37397
$724$	0	0
$725$	−24.4721	−0.908872
$726$	0	0
$727$	18.0000	0.667583	0.333792	−	0.942647i	$-0.391672\pi$
$727$	18.0000	0.667583	0.333792	+	0.942647i	$0.391672\pi$
$728$	0	0
$729$	41.9443	1.55349
$730$	0	0
$731$	9.88854	0.365741
$732$	0	0
$733$	−8.87539	−0.327820	−0.163910	−	0.986475i	$-0.552411\pi$
$733$	−8.87539	−0.327820	−0.163910	+	0.986475i	$0.552411\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−15.4164	−0.567871
$738$	0	0
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	0	0
$741$	−11.0557	−0.406142
$742$	0	0
$743$	13.8885	0.509521	0.254761	−	0.967004i	$-0.418003\pi$
$743$	13.8885	0.509521	0.254761	+	0.967004i	$0.418003\pi$
$744$	0	0
$745$	−72.3607	−2.65109
$746$	0	0
$747$	76.0689	2.78321
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−0.944272	−0.0344570	−0.0172285	−	0.999852i	$-0.505484\pi$
$751$	−0.944272	−0.0344570	−0.0172285	+	0.999852i	$0.505484\pi$
$752$	0	0
$753$	−80.1378	−2.92038
$754$	0	0
$755$	38.8328	1.41327
$756$	0	0
$757$	39.3050	1.42856	0.714281	−	0.699859i	$-0.246754\pi$
$757$	39.3050	1.42856	0.714281	+	0.699859i	$0.246754\pi$
$758$	0	0
$759$	−12.9443	−0.469847
$760$	0	0
$761$	−15.4164	−0.558844	−0.279422	−	0.960168i	$-0.590143\pi$
$761$	−15.4164	−0.558844	−0.279422	+	0.960168i	$0.590143\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−156.498	−5.65821
$766$	0	0
$767$	−8.94427	−0.322959
$768$	0	0
$769$	16.5836	0.598020	0.299010	−	0.954250i	$-0.403344\pi$
$769$	16.5836	0.598020	0.299010	+	0.954250i	$0.403344\pi$
$770$	0	0
$771$	−35.4164	−1.27549
$772$	0	0
$773$	−2.29180	−0.0824302	−0.0412151	−	0.999150i	$-0.513123\pi$
$773$	−2.29180	−0.0824302	−0.0412151	+	0.999150i	$0.513123\pi$
$774$	0	0
$775$	10.9443	0.393130
$776$	0	0
$777$	0	0
$778$	0	0
$779$	17.8885	0.640924
$780$	0	0
$781$	−2.47214	−0.0884600
$782$	0	0
$783$	64.7214	2.31295
$784$	0	0
$785$	60.3607	2.15437
$786$	0	0
$787$	−5.81966	−0.207448	−0.103724	−	0.994606i	$-0.533076\pi$
$787$	−5.81966	−0.207448	−0.103724	+	0.994606i	$0.533076\pi$
$788$	0	0
$789$	41.8885	1.49127
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−6.47214	−0.229832
$794$	0	0
$795$	−4.94427	−0.175355
$796$	0	0
$797$	−7.59675	−0.269091	−0.134545	−	0.990907i	$-0.542957\pi$
$797$	−7.59675	−0.269091	−0.134545	+	0.990907i	$0.542957\pi$
$798$	0	0
$799$	−12.9443	−0.457935
$800$	0	0
$801$	−74.7214	−2.64015
$802$	0	0
$803$	−4.94427	−0.174480
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−88.1378	−3.10260
$808$	0	0
$809$	−38.9443	−1.36921	−0.684604	−	0.728915i	$-0.740025\pi$
$809$	−38.9443	−1.36921	−0.684604	+	0.728915i	$0.740025\pi$
$810$	0	0
$811$	9.23607	0.324322	0.162161	−	0.986764i	$-0.448154\pi$
$811$	9.23607	0.324322	0.162161	+	0.986764i	$0.448154\pi$
$812$	0	0
$813$	54.8328	1.92307
$814$	0	0
$815$	24.0000	0.840683
$816$	0	0
$817$	−4.22291	−0.147741
$818$	0	0
$819$	0	0
$820$	0	0
$821$	25.4164	0.887039	0.443519	−	0.896265i	$-0.353730\pi$
$821$	25.4164	0.887039	0.443519	+	0.896265i	$0.353730\pi$
$822$	0	0
$823$	−34.2492	−1.19385	−0.596926	−	0.802296i	$-0.703612\pi$
$823$	−34.2492	−1.19385	−0.596926	+	0.802296i	$0.703612\pi$
$824$	0	0
$825$	17.7082	0.616521
$826$	0	0
$827$	0.944272	0.0328356	0.0164178	−	0.999865i	$-0.494774\pi$
$827$	0.944272	0.0328356	0.0164178	+	0.999865i	$0.494774\pi$
$828$	0	0
$829$	1.70820	0.0593284	0.0296642	−	0.999560i	$-0.490556\pi$
$829$	1.70820	0.0593284	0.0296642	+	0.999560i	$0.490556\pi$
$830$	0	0
$831$	−40.3607	−1.40010
$832$	0	0
$833$	0	0
$834$	0	0
$835$	49.8885	1.72646
$836$	0	0
$837$	−28.9443	−1.00046
$838$	0	0
$839$	−36.8328	−1.27161	−0.635805	−	0.771850i	$-0.719332\pi$
$839$	−36.8328	−1.27161	−0.635805	+	0.771850i	$0.719332\pi$
$840$	0	0
$841$	−9.00000	−0.310345
$842$	0	0
$843$	80.3607	2.76777
$844$	0	0
$845$	37.1246	1.27713
$846$	0	0
$847$	0	0
$848$	0	0
$849$	53.8885	1.84945
$850$	0	0
$851$	43.7771	1.50066
$852$	0	0
$853$	34.5410	1.18266	0.591331	−	0.806429i	$-0.298603\pi$
$853$	34.5410	1.18266	0.591331	+	0.806429i	$0.298603\pi$
$854$	0	0
$855$	66.8328	2.28563
$856$	0	0
$857$	37.5279	1.28193	0.640964	−	0.767571i	$-0.278535\pi$
$857$	37.5279	1.28193	0.640964	+	0.767571i	$0.278535\pi$
$858$	0	0
$859$	−25.1246	−0.857241	−0.428620	−	0.903485i	$-0.641000\pi$
$859$	−25.1246	−0.857241	−0.428620	+	0.903485i	$0.641000\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−27.4164	−0.933265	−0.466633	−	0.884451i	$-0.654533\pi$
$863$	−27.4164	−0.933265	−0.466633	+	0.884451i	$0.654533\pi$
$864$	0	0
$865$	4.00000	0.136004
$866$	0	0
$867$	−80.5410	−2.73532
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−19.0557	−0.645679
$872$	0	0
$873$	−26.3607	−0.892174
$874$	0	0
$875$	0	0
$876$	0	0
$877$	26.9443	0.909843	0.454922	−	0.890531i	$-0.349667\pi$
$877$	26.9443	0.909843	0.454922	+	0.890531i	$0.349667\pi$
$878$	0	0
$879$	15.0557	0.507817
$880$	0	0
$881$	24.8328	0.836639	0.418319	−	0.908300i	$-0.362619\pi$
$881$	24.8328	0.836639	0.418319	+	0.908300i	$0.362619\pi$
$882$	0	0
$883$	−50.8328	−1.71066	−0.855330	−	0.518083i	$-0.826646\pi$
$883$	−50.8328	−1.71066	−0.855330	+	0.518083i	$0.826646\pi$
$884$	0	0
$885$	75.7771	2.54722
$886$	0	0
$887$	0.360680	0.0121104	0.00605522	−	0.999982i	$-0.498073\pi$
$887$	0.360680	0.0121104	0.00605522	+	0.999982i	$0.498073\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−24.4164	−0.817980
$892$	0	0
$893$	5.52786	0.184983
$894$	0	0
$895$	28.9443	0.967500
$896$	0	0
$897$	−16.0000	−0.534224
$898$	0	0
$899$	−8.94427	−0.298308
$900$	0	0
$901$	−3.05573	−0.101801
$902$	0	0
$903$	0	0
$904$	0	0
$905$	15.4164	0.512459
$906$	0	0
$907$	−20.3607	−0.676065	−0.338033	−	0.941134i	$-0.609761\pi$
$907$	−20.3607	−0.676065	−0.338033	+	0.941134i	$0.609761\pi$
$908$	0	0
$909$	105.957	3.51439
$910$	0	0
$911$	28.0000	0.927681	0.463841	−	0.885919i	$-0.346471\pi$
$911$	28.0000	0.927681	0.463841	+	0.885919i	$0.346471\pi$
$912$	0	0
$913$	−10.1803	−0.336920
$914$	0	0
$915$	54.8328	1.81272
$916$	0	0
$917$	0	0
$918$	0	0
$919$	57.8885	1.90957	0.954783	−	0.297302i	$-0.0960869\pi$
$919$	57.8885	1.90957	0.954783	+	0.297302i	$0.0960869\pi$
$920$	0	0
$921$	−103.777	−3.41957
$922$	0	0
$923$	−3.05573	−0.100581
$924$	0	0
$925$	−59.8885	−1.96912
$926$	0	0
$927$	22.0000	0.722575
$928$	0	0
$929$	40.2492	1.32053	0.660267	−	0.751031i	$-0.270443\pi$
$929$	40.2492	1.32053	0.660267	+	0.751031i	$0.270443\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−17.5279	−0.573837
$934$	0	0
$935$	20.9443	0.684951
$936$	0	0
$937$	20.9443	0.684220	0.342110	−	0.939660i	$-0.388859\pi$
$937$	20.9443	0.684220	0.342110	+	0.939660i	$0.388859\pi$
$938$	0	0
$939$	92.1378	3.00680
$940$	0	0
$941$	34.1803	1.11425	0.557124	−	0.830430i	$-0.311905\pi$
$941$	34.1803	1.11425	0.557124	+	0.830430i	$0.311905\pi$
$942$	0	0
$943$	25.8885	0.843047
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0.944272	0.0306847	0.0153424	−	0.999882i	$-0.495116\pi$
$947$	0.944272	0.0306847	0.0153424	+	0.999882i	$0.495116\pi$
$948$	0	0
$949$	−6.11146	−0.198386
$950$	0	0
$951$	42.2492	1.37002
$952$	0	0
$953$	5.05573	0.163771	0.0818855	−	0.996642i	$-0.473906\pi$
$953$	5.05573	0.163771	0.0818855	+	0.996642i	$0.473906\pi$
$954$	0	0
$955$	20.9443	0.677741
$956$	0	0
$957$	−14.4721	−0.467818
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	48.3607	1.55840
$964$	0	0
$965$	−9.52786	−0.306713
$966$	0	0
$967$	−10.1115	−0.325163	−0.162581	−	0.986695i	$-0.551982\pi$
$967$	−10.1115	−0.325163	−0.162581	+	0.986695i	$0.551982\pi$
$968$	0	0
$969$	57.8885	1.85965
$970$	0	0
$971$	−16.5410	−0.530827	−0.265413	−	0.964135i	$-0.585508\pi$
$971$	−16.5410	−0.530827	−0.265413	+	0.964135i	$0.585508\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	21.8885	0.700994
$976$	0	0
$977$	24.8328	0.794472	0.397236	−	0.917716i	$-0.369969\pi$
$977$	24.8328	0.794472	0.397236	+	0.917716i	$0.369969\pi$
$978$	0	0
$979$	10.0000	0.319601
$980$	0	0
$981$	−74.7214	−2.38567
$982$	0	0
$983$	14.0000	0.446531	0.223265	−	0.974758i	$-0.428328\pi$
$983$	14.0000	0.446531	0.223265	+	0.974758i	$0.428328\pi$
$984$	0	0
$985$	−58.2492	−1.85597
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−6.11146	−0.194333
$990$	0	0
$991$	−44.3607	−1.40916	−0.704582	−	0.709623i	$-0.748865\pi$
$991$	−44.3607	−1.40916	−0.704582	+	0.709623i	$0.748865\pi$
$992$	0	0
$993$	3.05573	0.0969706
$994$	0	0
$995$	−3.41641	−0.108307
$996$	0	0
$997$	−1.81966	−0.0576292	−0.0288146	−	0.999585i	$-0.509173\pi$
$997$	−1.81966	−0.0576292	−0.0288146	+	0.999585i	$0.509173\pi$
$998$	0	0
$999$	158.387	5.01114

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8624.2.a.bf.1.1		2
4.3	odd	2		1078.2.a.w.1.2		2
7.6	odd	2		1232.2.a.p.1.2		2
12.11	even	2		9702.2.a.cu.1.2		2
28.3	even	6		1078.2.e.q.177.2		4
28.11	odd	6		1078.2.e.n.177.1		4
28.19	even	6		1078.2.e.q.67.2		4
28.23	odd	6		1078.2.e.n.67.1		4
28.27	even	2		154.2.a.d.1.1	✓	2
56.13	odd	2		4928.2.a.bk.1.1		2
56.27	even	2		4928.2.a.bt.1.2		2
84.83	odd	2		1386.2.a.m.1.1		2
140.27	odd	4		3850.2.c.q.1849.4		4
140.83	odd	4		3850.2.c.q.1849.1		4
140.139	even	2		3850.2.a.bj.1.2		2
308.307	odd	2		1694.2.a.l.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
154.2.a.d.1.1	✓	2	28.27	even	2
1078.2.a.w.1.2		2	4.3	odd	2
1078.2.e.n.67.1		4	28.23	odd	6
1078.2.e.n.177.1		4	28.11	odd	6
1078.2.e.q.67.2		4	28.19	even	6
1078.2.e.q.177.2		4	28.3	even	6
1232.2.a.p.1.2		2	7.6	odd	2
1386.2.a.m.1.1		2	84.83	odd	2
1694.2.a.l.1.1		2	308.307	odd	2
3850.2.a.bj.1.2		2	140.139	even	2
3850.2.c.q.1849.1		4	140.83	odd	4
3850.2.c.q.1849.4		4	140.27	odd	4
4928.2.a.bk.1.1		2	56.13	odd	2
4928.2.a.bt.1.2		2	56.27	even	2
8624.2.a.bf.1.1		2	1.1	even	1	trivial
9702.2.a.cu.1.2		2	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8624 = 2^{4} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8624.a (trivial)

\(f(q)\)	\(=\)	\(q-3.23607 q^{3} -3.23607 q^{5} +7.47214 q^{9} +O(q^{10})\)	\(q-3.23607 q^{3} -3.23607 q^{5} +7.47214 q^{9} -1.00000 q^{11} -1.23607 q^{13} +10.4721 q^{15} +6.47214 q^{17} -2.76393 q^{19} -4.00000 q^{23} +5.47214 q^{25} -14.4721 q^{27} -4.47214 q^{29} +2.00000 q^{31} +3.23607 q^{33} -10.9443 q^{37} +4.00000 q^{39} -6.47214 q^{41} +1.52786 q^{43} -24.1803 q^{45} -2.00000 q^{47} -20.9443 q^{51} -0.472136 q^{53} +3.23607 q^{55} +8.94427 q^{57} +7.23607 q^{59} +5.23607 q^{61} +4.00000 q^{65} +15.4164 q^{67} +12.9443 q^{69} +2.47214 q^{71} +4.94427 q^{73} -17.7082 q^{75} +24.4164 q^{81} +10.1803 q^{83} -20.9443 q^{85} +14.4721 q^{87} -10.0000 q^{89} -6.47214 q^{93} +8.94427 q^{95} -3.52786 q^{97} -7.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5} + 6 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 2 * q^5 + 6 * q^9	\( 2 q - 2 q^{3} - 2 q^{5} + 6 q^{9} - 2 q^{11} + 2 q^{13} + 12 q^{15} + 4 q^{17} - 10 q^{19} - 8 q^{23} + 2 q^{25} - 20 q^{27} + 4 q^{31} + 2 q^{33} - 4 q^{37} + 8 q^{39} - 4 q^{41} + 12 q^{43} - 26 q^{45} - 4 q^{47} - 24 q^{51} + 8 q^{53} + 2 q^{55} + 10 q^{59} + 6 q^{61} + 8 q^{65} + 4 q^{67} + 8 q^{69} - 4 q^{71} - 8 q^{73} - 22 q^{75} + 22 q^{81} - 2 q^{83} - 24 q^{85} + 20 q^{87} - 20 q^{89} - 4 q^{93} - 16 q^{97} - 6 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^5 + 6 * q^9 - 2 * q^11 + 2 * q^13 + 12 * q^15 + 4 * q^17 - 10 * q^19 - 8 * q^23 + 2 * q^25 - 20 * q^27 + 4 * q^31 + 2 * q^33 - 4 * q^37 + 8 * q^39 - 4 * q^41 + 12 * q^43 - 26 * q^45 - 4 * q^47 - 24 * q^51 + 8 * q^53 + 2 * q^55 + 10 * q^59 + 6 * q^61 + 8 * q^65 + 4 * q^67 + 8 * q^69 - 4 * q^71 - 8 * q^73 - 22 * q^75 + 22 * q^81 - 2 * q^83 - 24 * q^85 + 20 * q^87 - 20 * q^89 - 4 * q^93 - 16 * q^97 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more