LMFDB - Embedded newform 7605.2.a.bq.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7605,2,Mod(1,7605)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7605, 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("7605.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7605 = 3^{2} \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7605.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:	$60.7262307372$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2535)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.80194$ of defining polynomial
Character	$\chi$	$=$	7605.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.80194 q^{2} +1.24698 q^{4} +1.00000 q^{5} +0.801938 q^{7} +1.35690 q^{8} +O(q^{10})$	$q-1.80194 q^{2} +1.24698 q^{4} +1.00000 q^{5} +0.801938 q^{7} +1.35690 q^{8} -1.80194 q^{10} -1.00000 q^{11} -1.44504 q^{14} -4.93900 q^{16} +0.396125 q^{17} -2.74094 q^{19} +1.24698 q^{20} +1.80194 q^{22} -1.55496 q^{23} +1.00000 q^{25} +1.00000 q^{28} +7.34481 q^{29} +2.38404 q^{31} +6.18598 q^{32} -0.713792 q^{34} +0.801938 q^{35} -5.96077 q^{37} +4.93900 q^{38} +1.35690 q^{40} -8.80194 q^{41} -6.80194 q^{43} -1.24698 q^{44} +2.80194 q^{46} +11.7899 q^{47} -6.35690 q^{49} -1.80194 q^{50} -0.868313 q^{53} -1.00000 q^{55} +1.08815 q^{56} -13.2349 q^{58} +8.11960 q^{59} -4.11529 q^{61} -4.29590 q^{62} -1.26875 q^{64} +9.93900 q^{67} +0.493959 q^{68} -1.44504 q^{70} +6.07069 q^{71} -4.29052 q^{73} +10.7409 q^{74} -3.41789 q^{76} -0.801938 q^{77} -14.1075 q^{79} -4.93900 q^{80} +15.8605 q^{82} -1.83877 q^{83} +0.396125 q^{85} +12.2567 q^{86} -1.35690 q^{88} +6.52111 q^{89} -1.93900 q^{92} -21.2446 q^{94} -2.74094 q^{95} -13.1521 q^{97} +11.4547 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} - q^{4} + 3 q^{5} - 2 q^{7}+O(q^{10}) $ 3 * q - q^2 - q^4 + 3 * q^5 - 2 * q^7	$ 3 q - q^{2} - q^{4} + 3 q^{5} - 2 q^{7} - q^{10} - 3 q^{11} - 4 q^{14} - 5 q^{16} + 10 q^{17} + 6 q^{19} - q^{20} + q^{22} - 5 q^{23} + 3 q^{25} + 3 q^{28} - q^{29} - 3 q^{31} + 4 q^{32} + 6 q^{34} - 2 q^{35} - 5 q^{37} + 5 q^{38} - 22 q^{41} - 16 q^{43} + q^{44} + 4 q^{46} + 12 q^{47} - 15 q^{49} - q^{50} - 5 q^{53} - 3 q^{55} + 7 q^{56} - 16 q^{58} + 3 q^{59} - 10 q^{61} + q^{62} + 4 q^{64} + 20 q^{67} - 8 q^{68} - 4 q^{70} + 6 q^{71} - 2 q^{73} + 18 q^{74} - 16 q^{76} + 2 q^{77} - 2 q^{79} - 5 q^{80} + 12 q^{82} + 27 q^{83} + 10 q^{85} + 10 q^{86} + 4 q^{89} + 4 q^{92} - 18 q^{94} + 6 q^{95} - 9 q^{97} + 12 q^{98}+O(q^{100}) $ 3 * q - q^2 - q^4 + 3 * q^5 - 2 * q^7 - q^10 - 3 * q^11 - 4 * q^14 - 5 * q^16 + 10 * q^17 + 6 * q^19 - q^20 + q^22 - 5 * q^23 + 3 * q^25 + 3 * q^28 - q^29 - 3 * q^31 + 4 * q^32 + 6 * q^34 - 2 * q^35 - 5 * q^37 + 5 * q^38 - 22 * q^41 - 16 * q^43 + q^44 + 4 * q^46 + 12 * q^47 - 15 * q^49 - q^50 - 5 * q^53 - 3 * q^55 + 7 * q^56 - 16 * q^58 + 3 * q^59 - 10 * q^61 + q^62 + 4 * q^64 + 20 * q^67 - 8 * q^68 - 4 * q^70 + 6 * q^71 - 2 * q^73 + 18 * q^74 - 16 * q^76 + 2 * q^77 - 2 * q^79 - 5 * q^80 + 12 * q^82 + 27 * q^83 + 10 * q^85 + 10 * q^86 + 4 * q^89 + 4 * q^92 - 18 * q^94 + 6 * q^95 - 9 * q^97 + 12 * 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 $

$2$	−1.80194	−1.27416	−0.637081	−	0.770797i	$-0.719858\pi$
$2$	−1.80194	−1.27416	−0.637081	+	0.770797i	$0.719858\pi$
$3$	0	0
$3$	0	0
$4$	1.24698	0.623490
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0.801938	0.303104	0.151552	−	0.988449i	$-0.451573\pi$
$7$	0.801938	0.303104	0.151552	+	0.988449i	$0.451573\pi$
$8$	1.35690	0.479735
$9$	0	0
$10$	−1.80194	−0.569823
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−1.44504	−0.386204
$15$	0	0
$16$	−4.93900	−1.23475
$17$	0.396125	0.0960743	0.0480372	−	0.998846i	$-0.484703\pi$
$17$	0.396125	0.0960743	0.0480372	+	0.998846i	$0.484703\pi$
$18$	0	0
$19$	−2.74094	−0.628814	−0.314407	−	0.949288i	$-0.601806\pi$
$19$	−2.74094	−0.628814	−0.314407	+	0.949288i	$0.601806\pi$
$20$	1.24698	0.278833
$21$	0	0
$22$	1.80194	0.384174
$23$	−1.55496	−0.324231	−0.162116	−	0.986772i	$-0.551832\pi$
$23$	−1.55496	−0.324231	−0.162116	+	0.986772i	$0.551832\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	1.00000	0.188982
$29$	7.34481	1.36390	0.681949	−	0.731400i	$-0.261133\pi$
$29$	7.34481	1.36390	0.681949	+	0.731400i	$0.261133\pi$
$30$	0	0
$31$	2.38404	0.428187	0.214093	−	0.976813i	$-0.431320\pi$
$31$	2.38404	0.428187	0.214093	+	0.976813i	$0.431320\pi$
$32$	6.18598	1.09354
$33$	0	0
$34$	−0.713792	−0.122414
$35$	0.801938	0.135552
$36$	0	0
$37$	−5.96077	−0.979945	−0.489972	−	0.871738i	$-0.662993\pi$
$37$	−5.96077	−0.979945	−0.489972	+	0.871738i	$0.662993\pi$
$38$	4.93900	0.801212
$39$	0	0
$40$	1.35690	0.214544
$41$	−8.80194	−1.37463	−0.687316	−	0.726359i	$-0.741211\pi$
$41$	−8.80194	−1.37463	−0.687316	+	0.726359i	$0.741211\pi$
$42$	0	0
$43$	−6.80194	−1.03729	−0.518643	−	0.854991i	$-0.673563\pi$
$43$	−6.80194	−1.03729	−0.518643	+	0.854991i	$0.673563\pi$
$44$	−1.24698	−0.187989
$45$	0	0
$46$	2.80194	0.413123
$47$	11.7899	1.71973	0.859864	−	0.510524i	$-0.170548\pi$
$47$	11.7899	1.71973	0.859864	+	0.510524i	$0.170548\pi$
$48$	0	0
$49$	−6.35690	−0.908128
$50$	−1.80194	−0.254832
$51$	0	0
$52$	0	0
$53$	−0.868313	−0.119272	−0.0596360	−	0.998220i	$-0.518994\pi$
$53$	−0.868313	−0.119272	−0.0596360	+	0.998220i	$0.518994\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	1.08815	0.145410
$57$	0	0
$58$	−13.2349	−1.73783
$59$	8.11960	1.05708	0.528541	−	0.848908i	$-0.322739\pi$
$59$	8.11960	1.05708	0.528541	+	0.848908i	$0.322739\pi$
$60$	0	0
$61$	−4.11529	−0.526909	−0.263455	−	0.964672i	$-0.584862\pi$
$61$	−4.11529	−0.526909	−0.263455	+	0.964672i	$0.584862\pi$
$62$	−4.29590	−0.545579
$63$	0	0
$64$	−1.26875	−0.158594
$65$	0	0
$66$	0	0
$67$	9.93900	1.21424	0.607121	−	0.794609i	$-0.292324\pi$
$67$	9.93900	1.21424	0.607121	+	0.794609i	$0.292324\pi$
$68$	0.493959	0.0599014
$69$	0	0
$70$	−1.44504	−0.172716
$71$	6.07069	0.720458	0.360229	−	0.932864i	$-0.382699\pi$
$71$	6.07069	0.720458	0.360229	+	0.932864i	$0.382699\pi$
$72$	0	0
$73$	−4.29052	−0.502167	−0.251084	−	0.967965i	$-0.580787\pi$
$73$	−4.29052	−0.502167	−0.251084	+	0.967965i	$0.580787\pi$
$74$	10.7409	1.24861
$75$	0	0
$76$	−3.41789	−0.392059
$77$	−0.801938	−0.0913893
$78$	0	0
$79$	−14.1075	−1.58722	−0.793610	−	0.608427i	$-0.791801\pi$
$79$	−14.1075	−1.58722	−0.793610	+	0.608427i	$0.791801\pi$
$80$	−4.93900	−0.552197
$81$	0	0
$82$	15.8605	1.75150
$83$	−1.83877	−0.201832	−0.100916	−	0.994895i	$-0.532177\pi$
$83$	−1.83877	−0.201832	−0.100916	+	0.994895i	$0.532177\pi$
$84$	0	0
$85$	0.396125	0.0429657
$86$	12.2567	1.32167
$87$	0	0
$88$	−1.35690	−0.144646
$89$	6.52111	0.691236	0.345618	−	0.938375i	$-0.387669\pi$
$89$	6.52111	0.691236	0.345618	+	0.938375i	$0.387669\pi$
$90$	0	0
$91$	0	0
$92$	−1.93900	−0.202155
$93$	0	0
$94$	−21.2446	−2.19121
$95$	−2.74094	−0.281214
$96$	0	0
$97$	−13.1521	−1.33540	−0.667698	−	0.744432i	$-0.732720\pi$
$97$	−13.1521	−1.33540	−0.667698	+	0.744432i	$0.732720\pi$
$98$	11.4547	1.15710
$99$	0	0
$100$	1.24698	0.124698
$101$	−17.8756	−1.77869	−0.889345	−	0.457237i	$-0.848839\pi$
$101$	−17.8756	−1.77869	−0.889345	+	0.457237i	$0.848839\pi$
$102$	0	0
$103$	6.39373	0.629993	0.314997	−	0.949093i	$-0.397997\pi$
$103$	6.39373	0.629993	0.314997	+	0.949093i	$0.397997\pi$
$104$	0	0
$105$	0	0
$106$	1.56465	0.151972
$107$	−15.6353	−1.51152	−0.755762	−	0.654846i	$-0.772734\pi$
$107$	−15.6353	−1.51152	−0.755762	+	0.654846i	$0.772734\pi$
$108$	0	0
$109$	7.42758	0.711433	0.355717	−	0.934594i	$-0.384237\pi$
$109$	7.42758	0.711433	0.355717	+	0.934594i	$0.384237\pi$
$110$	1.80194	0.171808
$111$	0	0
$112$	−3.96077	−0.374258
$113$	15.4765	1.45591	0.727953	−	0.685627i	$-0.240472\pi$
$113$	15.4765	1.45591	0.727953	+	0.685627i	$0.240472\pi$
$114$	0	0
$115$	−1.55496	−0.145001
$116$	9.15883	0.850376
$117$	0	0
$118$	−14.6310	−1.34689
$119$	0.317667	0.0291205
$120$	0	0
$121$	−10.0000	−0.909091
$122$	7.41550	0.671368
$123$	0	0
$124$	2.97285	0.266970
$125$	1.00000	0.0894427
$126$	0	0
$127$	−9.97823	−0.885425	−0.442712	−	0.896664i	$-0.645984\pi$
$127$	−9.97823	−0.885425	−0.442712	+	0.896664i	$0.645984\pi$
$128$	−10.0858	−0.891463
$129$	0	0
$130$	0	0
$131$	−4.71917	−0.412316	−0.206158	−	0.978519i	$-0.566096\pi$
$131$	−4.71917	−0.412316	−0.206158	+	0.978519i	$0.566096\pi$
$132$	0	0
$133$	−2.19806	−0.190596
$134$	−17.9095	−1.54714
$135$	0	0
$136$	0.537500	0.0460902
$137$	3.97046	0.339219	0.169610	−	0.985511i	$-0.445749\pi$
$137$	3.97046	0.339219	0.169610	+	0.985511i	$0.445749\pi$
$138$	0	0
$139$	−15.3351	−1.30071	−0.650354	−	0.759631i	$-0.725380\pi$
$139$	−15.3351	−1.30071	−0.650354	+	0.759631i	$0.725380\pi$
$140$	1.00000	0.0845154
$141$	0	0
$142$	−10.9390	−0.917981
$143$	0	0
$144$	0	0
$145$	7.34481	0.609954
$146$	7.73125	0.639843
$147$	0	0
$148$	−7.43296	−0.610986
$149$	−15.1860	−1.24408	−0.622042	−	0.782984i	$-0.713697\pi$
$149$	−15.1860	−1.24408	−0.622042	+	0.782984i	$0.713697\pi$
$150$	0	0
$151$	17.3448	1.41150	0.705750	−	0.708460i	$-0.250610\pi$
$151$	17.3448	1.41150	0.705750	+	0.708460i	$0.250610\pi$
$152$	−3.71917	−0.301664
$153$	0	0
$154$	1.44504	0.116445
$155$	2.38404	0.191491
$156$	0	0
$157$	−20.0489	−1.60008	−0.800039	−	0.599948i	$-0.795188\pi$
$157$	−20.0489	−1.60008	−0.800039	+	0.599948i	$0.795188\pi$
$158$	25.4209	2.02238
$159$	0	0
$160$	6.18598	0.489045
$161$	−1.24698	−0.0982758
$162$	0	0
$163$	−4.68233	−0.366749	−0.183374	−	0.983043i	$-0.558702\pi$
$163$	−4.68233	−0.366749	−0.183374	+	0.983043i	$0.558702\pi$
$164$	−10.9758	−0.857069
$165$	0	0
$166$	3.31336	0.257166
$167$	10.5797	0.818683	0.409341	−	0.912381i	$-0.365759\pi$
$167$	10.5797	0.818683	0.409341	+	0.912381i	$0.365759\pi$
$168$	0	0
$169$	0	0
$170$	−0.713792	−0.0547453
$171$	0	0
$172$	−8.48188	−0.646737
$173$	4.97823	0.378488	0.189244	−	0.981930i	$-0.439396\pi$
$173$	4.97823	0.378488	0.189244	+	0.981930i	$0.439396\pi$
$174$	0	0
$175$	0.801938	0.0606208
$176$	4.93900	0.372291
$177$	0	0
$178$	−11.7506	−0.880747
$179$	−13.5754	−1.01467	−0.507337	−	0.861748i	$-0.669370\pi$
$179$	−13.5754	−1.01467	−0.507337	+	0.861748i	$0.669370\pi$
$180$	0	0
$181$	9.93661	0.738582	0.369291	−	0.929314i	$-0.379601\pi$
$181$	9.93661	0.738582	0.369291	+	0.929314i	$0.379601\pi$
$182$	0	0
$183$	0	0
$184$	−2.10992	−0.155545
$185$	−5.96077	−0.438245
$186$	0	0
$187$	−0.396125	−0.0289675
$188$	14.7017	1.07223
$189$	0	0
$190$	4.93900	0.358313
$191$	18.8756	1.36579	0.682896	−	0.730516i	$-0.260720\pi$
$191$	18.8756	1.36579	0.682896	+	0.730516i	$0.260720\pi$
$192$	0	0
$193$	11.3817	0.819269	0.409635	−	0.912250i	$-0.365656\pi$
$193$	11.3817	0.819269	0.409635	+	0.912250i	$0.365656\pi$
$194$	23.6993	1.70151
$195$	0	0
$196$	−7.92692	−0.566209
$197$	−26.5840	−1.89403	−0.947017	−	0.321184i	$-0.895919\pi$
$197$	−26.5840	−1.89403	−0.947017	+	0.321184i	$0.895919\pi$
$198$	0	0
$199$	−8.17523	−0.579526	−0.289763	−	0.957098i	$-0.593577\pi$
$199$	−8.17523	−0.579526	−0.289763	+	0.957098i	$0.593577\pi$
$200$	1.35690	0.0959470
$201$	0	0
$202$	32.2107	2.26634
$203$	5.89008	0.413403
$204$	0	0
$205$	−8.80194	−0.614754
$206$	−11.5211	−0.802714
$207$	0	0
$208$	0	0
$209$	2.74094	0.189595
$210$	0	0
$211$	14.6286	1.00708	0.503538	−	0.863973i	$-0.332031\pi$
$211$	14.6286	1.00708	0.503538	+	0.863973i	$0.332031\pi$
$212$	−1.08277	−0.0743649
$213$	0	0
$214$	28.1739	1.92593
$215$	−6.80194	−0.463888
$216$	0	0
$217$	1.91185	0.129785
$218$	−13.3840	−0.906482
$219$	0	0
$220$	−1.24698	−0.0840713
$221$	0	0
$222$	0	0
$223$	−10.2795	−0.688366	−0.344183	−	0.938902i	$-0.611844\pi$
$223$	−10.2795	−0.688366	−0.344183	+	0.938902i	$0.611844\pi$
$224$	4.96077	0.331455
$225$	0	0
$226$	−27.8877	−1.85506
$227$	−5.43296	−0.360598	−0.180299	−	0.983612i	$-0.557707\pi$
$227$	−5.43296	−0.360598	−0.180299	+	0.983612i	$0.557707\pi$
$228$	0	0
$229$	19.8998	1.31501	0.657507	−	0.753448i	$-0.271611\pi$
$229$	19.8998	1.31501	0.657507	+	0.753448i	$0.271611\pi$
$230$	2.80194	0.184754
$231$	0	0
$232$	9.96615	0.654310
$233$	11.4940	0.752994	0.376497	−	0.926418i	$-0.377128\pi$
$233$	11.4940	0.752994	0.376497	+	0.926418i	$0.377128\pi$
$234$	0	0
$235$	11.7899	0.769085
$236$	10.1250	0.659080
$237$	0	0
$238$	−0.572417	−0.0371043
$239$	−22.0804	−1.42826	−0.714130	−	0.700013i	$-0.753178\pi$
$239$	−22.0804	−1.42826	−0.714130	+	0.700013i	$0.753178\pi$
$240$	0	0
$241$	21.3817	1.37731	0.688657	−	0.725088i	$-0.258201\pi$
$241$	21.3817	1.37731	0.688657	+	0.725088i	$0.258201\pi$
$242$	18.0194	1.15833
$243$	0	0
$244$	−5.13169	−0.328523
$245$	−6.35690	−0.406127
$246$	0	0
$247$	0	0
$248$	3.23490	0.205416
$249$	0	0
$250$	−1.80194	−0.113965
$251$	−0.381059	−0.0240522	−0.0120261	−	0.999928i	$-0.503828\pi$
$251$	−0.381059	−0.0240522	−0.0120261	+	0.999928i	$0.503828\pi$
$252$	0	0
$253$	1.55496	0.0977594
$254$	17.9801	1.12817
$255$	0	0
$256$	20.7114	1.29446
$257$	17.3207	1.08043	0.540216	−	0.841526i	$-0.318342\pi$
$257$	17.3207	1.08043	0.540216	+	0.841526i	$0.318342\pi$
$258$	0	0
$259$	−4.78017	−0.297025
$260$	0	0
$261$	0	0
$262$	8.50365	0.525357
$263$	−23.3545	−1.44010	−0.720050	−	0.693922i	$-0.755881\pi$
$263$	−23.3545	−1.44010	−0.720050	+	0.693922i	$0.755881\pi$
$264$	0	0
$265$	−0.868313	−0.0533401
$266$	3.96077	0.242850
$267$	0	0
$268$	12.3937	0.757068
$269$	20.6310	1.25790	0.628948	−	0.777448i	$-0.283486\pi$
$269$	20.6310	1.25790	0.628948	+	0.777448i	$0.283486\pi$
$270$	0	0
$271$	0.356896	0.0216799	0.0108399	−	0.999941i	$-0.496549\pi$
$271$	0.356896	0.0216799	0.0108399	+	0.999941i	$0.496549\pi$
$272$	−1.95646	−0.118628
$273$	0	0
$274$	−7.15452	−0.432220
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	16.6025	0.997550	0.498775	−	0.866731i	$-0.333783\pi$
$277$	16.6025	0.997550	0.498775	+	0.866731i	$0.333783\pi$
$278$	27.6329	1.65731
$279$	0	0
$280$	1.08815	0.0650292
$281$	8.12737	0.484839	0.242419	−	0.970172i	$-0.422059\pi$
$281$	8.12737	0.484839	0.242419	+	0.970172i	$0.422059\pi$
$282$	0	0
$283$	−14.5851	−0.866994	−0.433497	−	0.901155i	$-0.642720\pi$
$283$	−14.5851	−0.866994	−0.433497	+	0.901155i	$0.642720\pi$
$284$	7.57002	0.449198
$285$	0	0
$286$	0	0
$287$	−7.05861	−0.416656
$288$	0	0
$289$	−16.8431	−0.990770
$290$	−13.2349	−0.777180
$291$	0	0
$292$	−5.35019	−0.313096
$293$	−25.4456	−1.48655	−0.743275	−	0.668986i	$-0.766729\pi$
$293$	−25.4456	−1.48655	−0.743275	+	0.668986i	$0.766729\pi$
$294$	0	0
$295$	8.11960	0.472742
$296$	−8.08815	−0.470114
$297$	0	0
$298$	27.3642	1.58517
$299$	0	0
$300$	0	0
$301$	−5.45473	−0.314405
$302$	−31.2543	−1.79848
$303$	0	0
$304$	13.5375	0.776429
$305$	−4.11529	−0.235641
$306$	0	0
$307$	−14.8955	−0.850129	−0.425064	−	0.905163i	$-0.639749\pi$
$307$	−14.8955	−0.850129	−0.425064	+	0.905163i	$0.639749\pi$
$308$	−1.00000	−0.0569803
$309$	0	0
$310$	−4.29590	−0.243991
$311$	6.21313	0.352314	0.176157	−	0.984362i	$-0.443633\pi$
$311$	6.21313	0.352314	0.176157	+	0.984362i	$0.443633\pi$
$312$	0	0
$313$	−10.7694	−0.608723	−0.304362	−	0.952557i	$-0.598443\pi$
$313$	−10.7694	−0.608723	−0.304362	+	0.952557i	$0.598443\pi$
$314$	36.1269	2.03876
$315$	0	0
$316$	−17.5918	−0.989616
$317$	−12.6504	−0.710517	−0.355259	−	0.934768i	$-0.615607\pi$
$317$	−12.6504	−0.710517	−0.355259	+	0.934768i	$0.615607\pi$
$318$	0	0
$319$	−7.34481	−0.411231
$320$	−1.26875	−0.0709253
$321$	0	0
$322$	2.24698	0.125219
$323$	−1.08575	−0.0604129
$324$	0	0
$325$	0	0
$326$	8.43727	0.467297
$327$	0	0
$328$	−11.9433	−0.659459
$329$	9.45473	0.521256
$330$	0	0
$331$	35.7972	1.96759	0.983795	−	0.179299i	$-0.0573828\pi$
$331$	35.7972	1.96759	0.983795	+	0.179299i	$0.0573828\pi$
$332$	−2.29291	−0.125840
$333$	0	0
$334$	−19.0640	−1.04313
$335$	9.93900	0.543026
$336$	0	0
$337$	−5.51142	−0.300226	−0.150113	−	0.988669i	$-0.547964\pi$
$337$	−5.51142	−0.300226	−0.150113	+	0.988669i	$0.547964\pi$
$338$	0	0
$339$	0	0
$340$	0.493959	0.0267887
$341$	−2.38404	−0.129103
$342$	0	0
$343$	−10.7114	−0.578361
$344$	−9.22952	−0.497622
$345$	0	0
$346$	−8.97046	−0.482255
$347$	−18.6770	−1.00263	−0.501316	−	0.865264i	$-0.667150\pi$
$347$	−18.6770	−1.00263	−0.501316	+	0.865264i	$0.667150\pi$
$348$	0	0
$349$	16.7084	0.894381	0.447190	−	0.894439i	$-0.352425\pi$
$349$	16.7084	0.894381	0.447190	+	0.894439i	$0.352425\pi$
$350$	−1.44504	−0.0772407
$351$	0	0
$352$	−6.18598	−0.329714
$353$	16.9879	0.904176	0.452088	−	0.891973i	$-0.350679\pi$
$353$	16.9879	0.904176	0.452088	+	0.891973i	$0.350679\pi$
$354$	0	0
$355$	6.07069	0.322199
$356$	8.13169	0.430979
$357$	0	0
$358$	24.4620	1.29286
$359$	−24.8213	−1.31002	−0.655009	−	0.755621i	$-0.727335\pi$
$359$	−24.8213	−1.31002	−0.655009	+	0.755621i	$0.727335\pi$
$360$	0	0
$361$	−11.4873	−0.604592
$362$	−17.9051	−0.941074
$363$	0	0
$364$	0	0
$365$	−4.29052	−0.224576
$366$	0	0
$367$	12.5459	0.654889	0.327444	−	0.944870i	$-0.393813\pi$
$367$	12.5459	0.654889	0.327444	+	0.944870i	$0.393813\pi$
$368$	7.67994	0.400345
$369$	0	0
$370$	10.7409	0.558395
$371$	−0.696333	−0.0361518
$372$	0	0
$373$	−4.19136	−0.217020	−0.108510	−	0.994095i	$-0.534608\pi$
$373$	−4.19136	−0.217020	−0.108510	+	0.994095i	$0.534608\pi$
$374$	0.713792	0.0369093
$375$	0	0
$376$	15.9976	0.825014
$377$	0	0
$378$	0	0
$379$	4.24027	0.217808	0.108904	−	0.994052i	$-0.465266\pi$
$379$	4.24027	0.217808	0.108904	+	0.994052i	$0.465266\pi$
$380$	−3.41789	−0.175334
$381$	0	0
$382$	−34.0127	−1.74024
$383$	−26.6437	−1.36143	−0.680715	−	0.732549i	$-0.738331\pi$
$383$	−26.6437	−1.36143	−0.680715	+	0.732549i	$0.738331\pi$
$384$	0	0
$385$	−0.801938	−0.0408705
$386$	−20.5090	−1.04388
$387$	0	0
$388$	−16.4004	−0.832606
$389$	2.76377	0.140129	0.0700645	−	0.997542i	$-0.477680\pi$
$389$	2.76377	0.140129	0.0700645	+	0.997542i	$0.477680\pi$
$390$	0	0
$391$	−0.615957	−0.0311503
$392$	−8.62565	−0.435661
$393$	0	0
$394$	47.9028	2.41331
$395$	−14.1075	−0.709827
$396$	0	0
$397$	−7.72156	−0.387534	−0.193767	−	0.981048i	$-0.562071\pi$
$397$	−7.72156	−0.387534	−0.193767	+	0.981048i	$0.562071\pi$
$398$	14.7313	0.738411
$399$	0	0
$400$	−4.93900	−0.246950
$401$	21.8659	1.09193	0.545966	−	0.837807i	$-0.316163\pi$
$401$	21.8659	1.09193	0.545966	+	0.837807i	$0.316163\pi$
$402$	0	0
$403$	0	0
$404$	−22.2905	−1.10899
$405$	0	0
$406$	−10.6136	−0.526742
$407$	5.96077	0.295464
$408$	0	0
$409$	2.50173	0.123703	0.0618513	−	0.998085i	$-0.480300\pi$
$409$	2.50173	0.123703	0.0618513	+	0.998085i	$0.480300\pi$
$410$	15.8605	0.783296
$411$	0	0
$412$	7.97285	0.392794
$413$	6.51142	0.320406
$414$	0	0
$415$	−1.83877	−0.0902618
$416$	0	0
$417$	0	0
$418$	−4.93900	−0.241574
$419$	−3.97392	−0.194139	−0.0970693	−	0.995278i	$-0.530947\pi$
$419$	−3.97392	−0.194139	−0.0970693	+	0.995278i	$0.530947\pi$
$420$	0	0
$421$	−25.1105	−1.22381	−0.611906	−	0.790931i	$-0.709597\pi$
$421$	−25.1105	−1.22381	−0.611906	+	0.790931i	$0.709597\pi$
$422$	−26.3599	−1.28318
$423$	0	0
$424$	−1.17821	−0.0572190
$425$	0.396125	0.0192149
$426$	0	0
$427$	−3.30021	−0.159708
$428$	−19.4969	−0.942420
$429$	0	0
$430$	12.2567	0.591069
$431$	−7.01746	−0.338019	−0.169010	−	0.985614i	$-0.554057\pi$
$431$	−7.01746	−0.338019	−0.169010	+	0.985614i	$0.554057\pi$
$432$	0	0
$433$	−19.8334	−0.953132	−0.476566	−	0.879139i	$-0.658119\pi$
$433$	−19.8334	−0.953132	−0.476566	+	0.879139i	$0.658119\pi$
$434$	−3.44504	−0.165367
$435$	0	0
$436$	9.26205	0.443572
$437$	4.26205	0.203881
$438$	0	0
$439$	−17.2392	−0.822783	−0.411391	−	0.911459i	$-0.634957\pi$
$439$	−17.2392	−0.822783	−0.411391	+	0.911459i	$0.634957\pi$
$440$	−1.35690	−0.0646875
$441$	0	0
$442$	0	0
$443$	2.04115	0.0969779	0.0484889	−	0.998824i	$-0.484559\pi$
$443$	2.04115	0.0969779	0.0484889	+	0.998824i	$0.484559\pi$
$444$	0	0
$445$	6.52111	0.309130
$446$	18.5230	0.877091
$447$	0	0
$448$	−1.01746	−0.0480704
$449$	14.3502	0.677227	0.338614	−	0.940925i	$-0.390042\pi$
$449$	14.3502	0.677227	0.338614	+	0.940925i	$0.390042\pi$
$450$	0	0
$451$	8.80194	0.414467
$452$	19.2989	0.907743
$453$	0	0
$454$	9.78986	0.459461
$455$	0	0
$456$	0	0
$457$	24.8122	1.16067	0.580333	−	0.814379i	$-0.302922\pi$
$457$	24.8122	1.16067	0.580333	+	0.814379i	$0.302922\pi$
$458$	−35.8582	−1.67554
$459$	0	0
$460$	−1.93900	−0.0904064
$461$	19.1739	0.893018	0.446509	−	0.894779i	$-0.352667\pi$
$461$	19.1739	0.893018	0.446509	+	0.894779i	$0.352667\pi$
$462$	0	0
$463$	−15.8538	−0.736790	−0.368395	−	0.929669i	$-0.620093\pi$
$463$	−15.8538	−0.736790	−0.368395	+	0.929669i	$0.620093\pi$
$464$	−36.2760	−1.68407
$465$	0	0
$466$	−20.7114	−0.959437
$467$	−20.2741	−0.938175	−0.469087	−	0.883152i	$-0.655417\pi$
$467$	−20.2741	−0.938175	−0.469087	+	0.883152i	$0.655417\pi$
$468$	0	0
$469$	7.97046	0.368042
$470$	−21.2446	−0.979940
$471$	0	0
$472$	11.0175	0.507120
$473$	6.80194	0.312753
$474$	0	0
$475$	−2.74094	−0.125763
$476$	0.396125	0.0181563
$477$	0	0
$478$	39.7875	1.81984
$479$	14.4316	0.659398	0.329699	−	0.944086i	$-0.393053\pi$
$479$	14.4316	0.659398	0.329699	+	0.944086i	$0.393053\pi$
$480$	0	0
$481$	0	0
$482$	−38.5284	−1.75492
$483$	0	0
$484$	−12.4698	−0.566809
$485$	−13.1521	−0.597207
$486$	0	0
$487$	−35.0901	−1.59008	−0.795041	−	0.606555i	$-0.792551\pi$
$487$	−35.0901	−1.59008	−0.795041	+	0.606555i	$0.792551\pi$
$488$	−5.58402	−0.252777
$489$	0	0
$490$	11.4547	0.517472
$491$	−2.75973	−0.124545	−0.0622723	−	0.998059i	$-0.519835\pi$
$491$	−2.75973	−0.124545	−0.0622723	+	0.998059i	$0.519835\pi$
$492$	0	0
$493$	2.90946	0.131036
$494$	0	0
$495$	0	0
$496$	−11.7748	−0.528704
$497$	4.86831	0.218374
$498$	0	0
$499$	17.3840	0.778217	0.389108	−	0.921192i	$-0.372783\pi$
$499$	17.3840	0.778217	0.389108	+	0.921192i	$0.372783\pi$
$500$	1.24698	0.0557666
$501$	0	0
$502$	0.686645	0.0306465
$503$	−12.9987	−0.579582	−0.289791	−	0.957090i	$-0.593586\pi$
$503$	−12.9987	−0.579582	−0.289791	+	0.957090i	$0.593586\pi$
$504$	0	0
$505$	−17.8756	−0.795454
$506$	−2.80194	−0.124561
$507$	0	0
$508$	−12.4426	−0.552053
$509$	−0.839838	−0.0372252	−0.0186126	−	0.999827i	$-0.505925\pi$
$509$	−0.839838	−0.0372252	−0.0186126	+	0.999827i	$0.505925\pi$
$510$	0	0
$511$	−3.44073	−0.152209
$512$	−17.1491	−0.757892
$513$	0	0
$514$	−31.2107	−1.37665
$515$	6.39373	0.281741
$516$	0	0
$517$	−11.7899	−0.518517
$518$	8.61356	0.378458
$519$	0	0
$520$	0	0
$521$	26.2416	1.14967	0.574833	−	0.818271i	$-0.305067\pi$
$521$	26.2416	1.14967	0.574833	+	0.818271i	$0.305067\pi$
$522$	0	0
$523$	−29.2349	−1.27835	−0.639176	−	0.769060i	$-0.720725\pi$
$523$	−29.2349	−1.27835	−0.639176	+	0.769060i	$0.720725\pi$
$524$	−5.88471	−0.257075
$525$	0	0
$526$	42.0834	1.83492
$527$	0.944378	0.0411377
$528$	0	0
$529$	−20.5821	−0.894874
$530$	1.56465	0.0679639
$531$	0	0
$532$	−2.74094	−0.118835
$533$	0	0
$534$	0	0
$535$	−15.6353	−0.675974
$536$	13.4862	0.582515
$537$	0	0
$538$	−37.1758	−1.60276
$539$	6.35690	0.273811
$540$	0	0
$541$	18.4746	0.794284	0.397142	−	0.917757i	$-0.370002\pi$
$541$	18.4746	0.794284	0.397142	+	0.917757i	$0.370002\pi$
$542$	−0.643104	−0.0276237
$543$	0	0
$544$	2.45042	0.105061
$545$	7.42758	0.318163
$546$	0	0
$547$	−0.450419	−0.0192585	−0.00962926	−	0.999954i	$-0.503065\pi$
$547$	−0.450419	−0.0192585	−0.00962926	+	0.999954i	$0.503065\pi$
$548$	4.95108	0.211500
$549$	0	0
$550$	1.80194	0.0768349
$551$	−20.1317	−0.857639
$552$	0	0
$553$	−11.3134	−0.481093
$554$	−29.9168	−1.27104
$555$	0	0
$556$	−19.1226	−0.810978
$557$	30.4058	1.28834	0.644168	−	0.764884i	$-0.277204\pi$
$557$	30.4058	1.28834	0.644168	+	0.764884i	$0.277204\pi$
$558$	0	0
$559$	0	0
$560$	−3.96077	−0.167373
$561$	0	0
$562$	−14.6450	−0.617763
$563$	45.0471	1.89851	0.949255	−	0.314508i	$-0.101840\pi$
$563$	45.0471	1.89851	0.949255	+	0.314508i	$0.101840\pi$
$564$	0	0
$565$	15.4765	0.651101
$566$	26.2814	1.10469
$567$	0	0
$568$	8.23729	0.345629
$569$	−16.4892	−0.691262	−0.345631	−	0.938370i	$-0.612335\pi$
$569$	−16.4892	−0.691262	−0.345631	+	0.938370i	$0.612335\pi$
$570$	0	0
$571$	42.9124	1.79583	0.897915	−	0.440169i	$-0.145081\pi$
$571$	42.9124	1.79583	0.897915	+	0.440169i	$0.145081\pi$
$572$	0	0
$573$	0	0
$574$	12.7192	0.530888
$575$	−1.55496	−0.0648462
$576$	0	0
$577$	−36.8377	−1.53357	−0.766787	−	0.641902i	$-0.778146\pi$
$577$	−36.8377	−1.53357	−0.766787	+	0.641902i	$0.778146\pi$
$578$	30.3502	1.26240
$579$	0	0
$580$	9.15883	0.380300
$581$	−1.47458	−0.0611760
$582$	0	0
$583$	0.868313	0.0359619
$584$	−5.82179	−0.240907
$585$	0	0
$586$	45.8514	1.89411
$587$	−25.1860	−1.03954	−0.519768	−	0.854307i	$-0.673982\pi$
$587$	−25.1860	−1.03954	−0.519768	+	0.854307i	$0.673982\pi$
$588$	0	0
$589$	−6.53452	−0.269250
$590$	−14.6310	−0.602350
$591$	0	0
$592$	29.4403	1.20999
$593$	−19.1263	−0.785423	−0.392712	−	0.919662i	$-0.628463\pi$
$593$	−19.1263	−0.785423	−0.392712	+	0.919662i	$0.628463\pi$
$594$	0	0
$595$	0.317667	0.0130231
$596$	−18.9366	−0.775674
$597$	0	0
$598$	0	0
$599$	−42.1825	−1.72353	−0.861766	−	0.507307i	$-0.830641\pi$
$599$	−42.1825	−1.72353	−0.861766	+	0.507307i	$0.830641\pi$
$600$	0	0
$601$	42.2277	1.72250	0.861252	−	0.508178i	$-0.169681\pi$
$601$	42.2277	1.72250	0.861252	+	0.508178i	$0.169681\pi$
$602$	9.82908	0.400604
$603$	0	0
$604$	21.6286	0.880056
$605$	−10.0000	−0.406558
$606$	0	0
$607$	−0.0284750	−0.00115577	−0.000577883	−	1.00000i	$-0.500184\pi$
$607$	−0.0284750	−0.00115577	−0.000577883		1.00000i	$0.500184\pi$
$608$	−16.9554	−0.687632
$609$	0	0
$610$	7.41550	0.300245
$611$	0	0
$612$	0	0
$613$	22.2064	0.896909	0.448454	−	0.893806i	$-0.351975\pi$
$613$	22.2064	0.896909	0.448454	+	0.893806i	$0.351975\pi$
$614$	26.8407	1.08320
$615$	0	0
$616$	−1.08815	−0.0438427
$617$	−12.3653	−0.497806	−0.248903	−	0.968528i	$-0.580070\pi$
$617$	−12.3653	−0.497806	−0.248903	+	0.968528i	$0.580070\pi$
$618$	0	0
$619$	23.2586	0.934842	0.467421	−	0.884035i	$-0.345183\pi$
$619$	23.2586	0.934842	0.467421	+	0.884035i	$0.345183\pi$
$620$	2.97285	0.119393
$621$	0	0
$622$	−11.1957	−0.448905
$623$	5.22952	0.209516
$624$	0	0
$625$	1.00000	0.0400000
$626$	19.4058	0.775612
$627$	0	0
$628$	−25.0006	−0.997632
$629$	−2.36121	−0.0941475
$630$	0	0
$631$	−45.3817	−1.80661	−0.903307	−	0.428994i	$-0.858868\pi$
$631$	−45.3817	−1.80661	−0.903307	+	0.428994i	$0.858868\pi$
$632$	−19.1424	−0.761445
$633$	0	0
$634$	22.7952	0.905314
$635$	−9.97823	−0.395974
$636$	0	0
$637$	0	0
$638$	13.2349	0.523975
$639$	0	0
$640$	−10.0858	−0.398674
$641$	0.511418	0.0201998	0.0100999	−	0.999949i	$-0.496785\pi$
$641$	0.511418	0.0201998	0.0100999	+	0.999949i	$0.496785\pi$
$642$	0	0
$643$	−20.6668	−0.815019	−0.407509	−	0.913201i	$-0.633603\pi$
$643$	−20.6668	−0.815019	−0.407509	+	0.913201i	$0.633603\pi$
$644$	−1.55496	−0.0612739
$645$	0	0
$646$	1.95646	0.0769759
$647$	−28.3491	−1.11452	−0.557260	−	0.830338i	$-0.688147\pi$
$647$	−28.3491	−1.11452	−0.557260	+	0.830338i	$0.688147\pi$
$648$	0	0
$649$	−8.11960	−0.318722
$650$	0	0
$651$	0	0
$652$	−5.83877	−0.228664
$653$	38.4959	1.50646	0.753230	−	0.657757i	$-0.228495\pi$
$653$	38.4959	1.50646	0.753230	+	0.657757i	$0.228495\pi$
$654$	0	0
$655$	−4.71917	−0.184393
$656$	43.4728	1.69733
$657$	0	0
$658$	−17.0368	−0.664165
$659$	−36.4432	−1.41963	−0.709814	−	0.704390i	$-0.751221\pi$
$659$	−36.4432	−1.41963	−0.709814	+	0.704390i	$0.751221\pi$
$660$	0	0
$661$	−24.4198	−0.949821	−0.474910	−	0.880034i	$-0.657520\pi$
$661$	−24.4198	−0.949821	−0.474910	+	0.880034i	$0.657520\pi$
$662$	−64.5042	−2.50703
$663$	0	0
$664$	−2.49502	−0.0968257
$665$	−2.19806	−0.0852372
$666$	0	0
$667$	−11.4209	−0.442218
$668$	13.1927	0.510440
$669$	0	0
$670$	−17.9095	−0.691903
$671$	4.11529	0.158869
$672$	0	0
$673$	−22.9366	−0.884141	−0.442071	−	0.896980i	$-0.645756\pi$
$673$	−22.9366	−0.884141	−0.442071	+	0.896980i	$0.645756\pi$
$674$	9.93123	0.382537
$675$	0	0
$676$	0	0
$677$	−12.9530	−0.497824	−0.248912	−	0.968526i	$-0.580073\pi$
$677$	−12.9530	−0.497824	−0.248912	+	0.968526i	$0.580073\pi$
$678$	0	0
$679$	−10.5472	−0.404764
$680$	0.537500	0.0206122
$681$	0	0
$682$	4.29590	0.164498
$683$	−38.6316	−1.47820	−0.739099	−	0.673597i	$-0.764748\pi$
$683$	−38.6316	−1.47820	−0.739099	+	0.673597i	$0.764748\pi$
$684$	0	0
$685$	3.97046	0.151703
$686$	19.3013	0.736926
$687$	0	0
$688$	33.5948	1.28079
$689$	0	0
$690$	0	0
$691$	−35.1347	−1.33659	−0.668293	−	0.743898i	$-0.732975\pi$
$691$	−35.1347	−1.33659	−0.668293	+	0.743898i	$0.732975\pi$
$692$	6.20775	0.235983
$693$	0	0
$694$	33.6547	1.27752
$695$	−15.3351	−0.581694
$696$	0	0
$697$	−3.48666	−0.132067
$698$	−30.1075	−1.13959
$699$	0	0
$700$	1.00000	0.0377964
$701$	−19.6286	−0.741363	−0.370682	−	0.928760i	$-0.620876\pi$
$701$	−19.6286	−0.741363	−0.370682	+	0.928760i	$0.620876\pi$
$702$	0	0
$703$	16.3381	0.616203
$704$	1.26875	0.0478178
$705$	0	0
$706$	−30.6112	−1.15207
$707$	−14.3351	−0.539128
$708$	0	0
$709$	−3.88769	−0.146005	−0.0730026	−	0.997332i	$-0.523258\pi$
$709$	−3.88769	−0.146005	−0.0730026	+	0.997332i	$0.523258\pi$
$710$	−10.9390	−0.410533
$711$	0	0
$712$	8.84846	0.331610
$713$	−3.70709	−0.138831
$714$	0	0
$715$	0	0
$716$	−16.9282	−0.632638
$717$	0	0
$718$	44.7265	1.66918
$719$	−7.42998	−0.277091	−0.138546	−	0.990356i	$-0.544243\pi$
$719$	−7.42998	−0.277091	−0.138546	+	0.990356i	$0.544243\pi$
$720$	0	0
$721$	5.12737	0.190953
$722$	20.6993	0.770349
$723$	0	0
$724$	12.3907	0.460499
$725$	7.34481	0.272780
$726$	0	0
$727$	−47.0127	−1.74360	−0.871802	−	0.489859i	$-0.837048\pi$
$727$	−47.0127	−1.74360	−0.871802	+	0.489859i	$0.837048\pi$
$728$	0	0
$729$	0	0
$730$	7.73125	0.286146
$731$	−2.69441	−0.0996565
$732$	0	0
$733$	−24.8683	−0.918532	−0.459266	−	0.888299i	$-0.651888\pi$
$733$	−24.8683	−0.918532	−0.459266	+	0.888299i	$0.651888\pi$
$734$	−22.6069	−0.834434
$735$	0	0
$736$	−9.61894	−0.354559
$737$	−9.93900	−0.366108
$738$	0	0
$739$	−36.8437	−1.35532	−0.677658	−	0.735377i	$-0.737005\pi$
$739$	−36.8437	−1.35532	−0.677658	+	0.735377i	$0.737005\pi$
$740$	−7.43296	−0.273241
$741$	0	0
$742$	1.25475	0.0460633
$743$	20.9028	0.766848	0.383424	−	0.923572i	$-0.374745\pi$
$743$	20.9028	0.766848	0.383424	+	0.923572i	$0.374745\pi$
$744$	0	0
$745$	−15.1860	−0.556371
$746$	7.55257	0.276519
$747$	0	0
$748$	−0.493959	−0.0180609
$749$	−12.5386	−0.458149
$750$	0	0
$751$	3.72481	0.135920	0.0679601	−	0.997688i	$-0.478351\pi$
$751$	3.72481	0.135920	0.0679601	+	0.997688i	$0.478351\pi$
$752$	−58.2301	−2.12343
$753$	0	0
$754$	0	0
$755$	17.3448	0.631242
$756$	0	0
$757$	−39.4446	−1.43364	−0.716819	−	0.697260i	$-0.754402\pi$
$757$	−39.4446	−1.43364	−0.716819	+	0.697260i	$0.754402\pi$
$758$	−7.64071	−0.277523
$759$	0	0
$760$	−3.71917	−0.134908
$761$	−28.1396	−1.02006	−0.510029	−	0.860157i	$-0.670365\pi$
$761$	−28.1396	−1.02006	−0.510029	+	0.860157i	$0.670365\pi$
$762$	0	0
$763$	5.95646	0.215638
$764$	23.5375	0.851557
$765$	0	0
$766$	48.0103	1.73468
$767$	0	0
$768$	0	0
$769$	46.9909	1.69454	0.847268	−	0.531166i	$-0.178246\pi$
$769$	46.9909	1.69454	0.847268	+	0.531166i	$0.178246\pi$
$770$	1.44504	0.0520757
$771$	0	0
$772$	14.1927	0.510806
$773$	15.9594	0.574021	0.287011	−	0.957927i	$-0.407338\pi$
$773$	15.9594	0.574021	0.287011	+	0.957927i	$0.407338\pi$
$774$	0	0
$775$	2.38404	0.0856374
$776$	−17.8461	−0.640637
$777$	0	0
$778$	−4.98015	−0.178547
$779$	24.1256	0.864388
$780$	0	0
$781$	−6.07069	−0.217226
$782$	1.10992	0.0396905
$783$	0	0
$784$	31.3967	1.12131
$785$	−20.0489	−0.715577
$786$	0	0
$787$	−43.0465	−1.53444	−0.767221	−	0.641382i	$-0.778361\pi$
$787$	−43.0465	−1.53444	−0.767221	+	0.641382i	$0.778361\pi$
$788$	−33.1497	−1.18091
$789$	0	0
$790$	25.4209	0.904434
$791$	12.4112	0.441291
$792$	0	0
$793$	0	0
$794$	13.9138	0.493781
$795$	0	0
$796$	−10.1943	−0.361329
$797$	44.7646	1.58564	0.792822	−	0.609453i	$-0.208611\pi$
$797$	44.7646	1.58564	0.792822	+	0.609453i	$0.208611\pi$
$798$	0	0
$799$	4.67025	0.165222
$800$	6.18598	0.218707
$801$	0	0
$802$	−39.4010	−1.39130
$803$	4.29052	0.151409
$804$	0	0
$805$	−1.24698	−0.0439503
$806$	0	0
$807$	0	0
$808$	−24.2553	−0.853300
$809$	37.5918	1.32166	0.660829	−	0.750537i	$-0.270205\pi$
$809$	37.5918	1.32166	0.660829	+	0.750537i	$0.270205\pi$
$810$	0	0
$811$	−16.5797	−0.582192	−0.291096	−	0.956694i	$-0.594020\pi$
$811$	−16.5797	−0.582192	−0.291096	+	0.956694i	$0.594020\pi$
$812$	7.34481	0.257752
$813$	0	0
$814$	−10.7409	−0.376470
$815$	−4.68233	−0.164015
$816$	0	0
$817$	18.6437	0.652260
$818$	−4.50796	−0.157617
$819$	0	0
$820$	−10.9758	−0.383293
$821$	3.49263	0.121894	0.0609468	−	0.998141i	$-0.480588\pi$
$821$	3.49263	0.121894	0.0609468	+	0.998141i	$0.480588\pi$
$822$	0	0
$823$	−35.9318	−1.25250	−0.626252	−	0.779620i	$-0.715412\pi$
$823$	−35.9318	−1.25250	−0.626252	+	0.779620i	$0.715412\pi$
$824$	8.67563	0.302230
$825$	0	0
$826$	−11.7332	−0.408249
$827$	−23.2319	−0.807853	−0.403926	−	0.914791i	$-0.632355\pi$
$827$	−23.2319	−0.807853	−0.403926	+	0.914791i	$0.632355\pi$
$828$	0	0
$829$	−31.9028	−1.10803	−0.554014	−	0.832507i	$-0.686905\pi$
$829$	−31.9028	−1.10803	−0.554014	+	0.832507i	$0.686905\pi$
$830$	3.31336	0.115008
$831$	0	0
$832$	0	0
$833$	−2.51812	−0.0872478
$834$	0	0
$835$	10.5797	0.366126
$836$	3.41789	0.118210
$837$	0	0
$838$	7.16075	0.247364
$839$	−34.3978	−1.18754	−0.593772	−	0.804634i	$-0.702362\pi$
$839$	−34.3978	−1.18754	−0.593772	+	0.804634i	$0.702362\pi$
$840$	0	0
$841$	24.9463	0.860217
$842$	45.2476	1.55933
$843$	0	0
$844$	18.2416	0.627902
$845$	0	0
$846$	0	0
$847$	−8.01938	−0.275549
$848$	4.28860	0.147271
$849$	0	0
$850$	−0.713792	−0.0244829
$851$	9.26875	0.317729
$852$	0	0
$853$	−49.5991	−1.69824	−0.849120	−	0.528200i	$-0.822867\pi$
$853$	−49.5991	−1.69824	−0.849120	+	0.528200i	$0.822867\pi$
$854$	5.94677	0.203494
$855$	0	0
$856$	−21.2155	−0.725132
$857$	−30.5042	−1.04200	−0.521002	−	0.853555i	$-0.674442\pi$
$857$	−30.5042	−1.04200	−0.521002	+	0.853555i	$0.674442\pi$
$858$	0	0
$859$	27.8595	0.950553	0.475277	−	0.879836i	$-0.342348\pi$
$859$	27.8595	0.950553	0.475277	+	0.879836i	$0.342348\pi$
$860$	−8.48188	−0.289230
$861$	0	0
$862$	12.6450	0.430691
$863$	−18.4413	−0.627750	−0.313875	−	0.949464i	$-0.601627\pi$
$863$	−18.4413	−0.627750	−0.313875	+	0.949464i	$0.601627\pi$
$864$	0	0
$865$	4.97823	0.169265
$866$	35.7385	1.21445
$867$	0	0
$868$	2.38404	0.0809197
$869$	14.1075	0.478565
$870$	0	0
$871$	0	0
$872$	10.0785	0.341300
$873$	0	0
$874$	−7.67994	−0.259778
$875$	0.801938	0.0271104
$876$	0	0
$877$	−36.5163	−1.23307	−0.616534	−	0.787328i	$-0.711464\pi$
$877$	−36.5163	−1.23307	−0.616534	+	0.787328i	$0.711464\pi$
$878$	31.0640	1.04836
$879$	0	0
$880$	4.93900	0.166494
$881$	−12.0562	−0.406184	−0.203092	−	0.979160i	$-0.565099\pi$
$881$	−12.0562	−0.406184	−0.203092	+	0.979160i	$0.565099\pi$
$882$	0	0
$883$	−54.3414	−1.82873	−0.914366	−	0.404888i	$-0.867310\pi$
$883$	−54.3414	−1.82873	−0.914366	+	0.404888i	$0.867310\pi$
$884$	0	0
$885$	0	0
$886$	−3.67802	−0.123566
$887$	5.92500	0.198942	0.0994710	−	0.995040i	$-0.468285\pi$
$887$	5.92500	0.198942	0.0994710	+	0.995040i	$0.468285\pi$
$888$	0	0
$889$	−8.00192	−0.268376
$890$	−11.7506	−0.393882
$891$	0	0
$892$	−12.8183	−0.429189
$893$	−32.3153	−1.08139
$894$	0	0
$895$	−13.5754	−0.453776
$896$	−8.08815	−0.270206
$897$	0	0
$898$	−25.8582	−0.862898
$899$	17.5104	0.584003
$900$	0	0
$901$	−0.343960	−0.0114590
$902$	−15.8605	−0.528098
$903$	0	0
$904$	21.0000	0.698450
$905$	9.93661	0.330304
$906$	0	0
$907$	27.5784	0.915725	0.457863	−	0.889023i	$-0.348615\pi$
$907$	27.5784	0.915725	0.457863	+	0.889023i	$0.348615\pi$
$908$	−6.77479	−0.224829
$909$	0	0
$910$	0	0
$911$	19.5670	0.648285	0.324142	−	0.946008i	$-0.394924\pi$
$911$	19.5670	0.648285	0.324142	+	0.946008i	$0.394924\pi$
$912$	0	0
$913$	1.83877	0.0608545
$914$	−44.7101	−1.47888
$915$	0	0
$916$	24.8146	0.819898
$917$	−3.78448	−0.124975
$918$	0	0
$919$	−18.3739	−0.606098	−0.303049	−	0.952975i	$-0.598005\pi$
$919$	−18.3739	−0.606098	−0.303049	+	0.952975i	$0.598005\pi$
$920$	−2.10992	−0.0695619
$921$	0	0
$922$	−34.5502	−1.13785
$923$	0	0
$924$	0	0
$925$	−5.96077	−0.195989
$926$	28.5676	0.938791
$927$	0	0
$928$	45.4349	1.49147
$929$	8.94305	0.293412	0.146706	−	0.989180i	$-0.453133\pi$
$929$	8.94305	0.293412	0.146706	+	0.989180i	$0.453133\pi$
$930$	0	0
$931$	17.4239	0.571044
$932$	14.3327	0.469484
$933$	0	0
$934$	36.5327	1.19539
$935$	−0.396125	−0.0129547
$936$	0	0
$937$	−35.7047	−1.16642	−0.583211	−	0.812321i	$-0.698204\pi$
$937$	−35.7047	−1.16642	−0.583211	+	0.812321i	$0.698204\pi$
$938$	−14.3623	−0.468945
$939$	0	0
$940$	14.7017	0.479517
$941$	1.55257	0.0506122	0.0253061	−	0.999680i	$-0.491944\pi$
$941$	1.55257	0.0506122	0.0253061	+	0.999680i	$0.491944\pi$
$942$	0	0
$943$	13.6866	0.445698
$944$	−40.1027	−1.30523
$945$	0	0
$946$	−12.2567	−0.398499
$947$	−5.96940	−0.193979	−0.0969896	−	0.995285i	$-0.530921\pi$
$947$	−5.96940	−0.193979	−0.0969896	+	0.995285i	$0.530921\pi$
$948$	0	0
$949$	0	0
$950$	4.93900	0.160242
$951$	0	0
$952$	0.431041	0.0139701
$953$	2.24459	0.0727093	0.0363546	−	0.999339i	$-0.488425\pi$
$953$	2.24459	0.0727093	0.0363546	+	0.999339i	$0.488425\pi$
$954$	0	0
$955$	18.8756	0.610800
$956$	−27.5338	−0.890506
$957$	0	0
$958$	−26.0049	−0.840180
$959$	3.18406	0.102819
$960$	0	0
$961$	−25.3163	−0.816656
$962$	0	0
$963$	0	0
$964$	26.6625	0.858741
$965$	11.3817	0.366388
$966$	0	0
$967$	57.1624	1.83822	0.919110	−	0.394002i	$-0.128910\pi$
$967$	57.1624	1.83822	0.919110	+	0.394002i	$0.128910\pi$
$968$	−13.5690	−0.436123
$969$	0	0
$970$	23.6993	0.760939
$971$	13.8474	0.444384	0.222192	−	0.975003i	$-0.428679\pi$
$971$	13.8474	0.444384	0.222192	+	0.975003i	$0.428679\pi$
$972$	0	0
$973$	−12.2978	−0.394250
$974$	63.2301	2.02602
$975$	0	0
$976$	20.3254	0.650601
$977$	40.7493	1.30369	0.651843	−	0.758354i	$-0.273996\pi$
$977$	40.7493	1.30369	0.651843	+	0.758354i	$0.273996\pi$
$978$	0	0
$979$	−6.52111	−0.208415
$980$	−7.92692	−0.253216
$981$	0	0
$982$	4.97285	0.158690
$983$	21.4644	0.684609	0.342304	−	0.939589i	$-0.388793\pi$
$983$	21.4644	0.684609	0.342304	+	0.939589i	$0.388793\pi$
$984$	0	0
$985$	−26.5840	−0.847037
$986$	−5.24267	−0.166961
$987$	0	0
$988$	0	0
$989$	10.5767	0.336320
$990$	0	0
$991$	44.1584	1.40274	0.701368	−	0.712799i	$-0.252573\pi$
$991$	44.1584	1.40274	0.701368	+	0.712799i	$0.252573\pi$
$992$	14.7476	0.468238
$993$	0	0
$994$	−8.77240	−0.278244
$995$	−8.17523	−0.259172
$996$	0	0
$997$	−56.7426	−1.79706	−0.898528	−	0.438916i	$-0.855362\pi$
$997$	−56.7426	−1.79706	−0.898528	+	0.438916i	$0.855362\pi$
$998$	−31.3250	−0.991574
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7605.2.a.bq.1.1		3
3.2	odd	2		2535.2.a.bd.1.3	yes	3
13.12	even	2		7605.2.a.bz.1.3		3
39.38	odd	2		2535.2.a.v.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2535.2.a.v.1.1	✓	3	39.38	odd	2
2535.2.a.bd.1.3	yes	3	3.2	odd	2
7605.2.a.bq.1.1		3	1.1	even	1	trivial
7605.2.a.bz.1.3		3	13.12	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:	\( N \)	\(=\)	\( 7605 = 3^{2} \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7605.a (trivial)

\(f(q)\)	\(=\)	\(q-1.80194 q^{2} +1.24698 q^{4} +1.00000 q^{5} +0.801938 q^{7} +1.35690 q^{8} +O(q^{10})\)	\(q-1.80194 q^{2} +1.24698 q^{4} +1.00000 q^{5} +0.801938 q^{7} +1.35690 q^{8} -1.80194 q^{10} -1.00000 q^{11} -1.44504 q^{14} -4.93900 q^{16} +0.396125 q^{17} -2.74094 q^{19} +1.24698 q^{20} +1.80194 q^{22} -1.55496 q^{23} +1.00000 q^{25} +1.00000 q^{28} +7.34481 q^{29} +2.38404 q^{31} +6.18598 q^{32} -0.713792 q^{34} +0.801938 q^{35} -5.96077 q^{37} +4.93900 q^{38} +1.35690 q^{40} -8.80194 q^{41} -6.80194 q^{43} -1.24698 q^{44} +2.80194 q^{46} +11.7899 q^{47} -6.35690 q^{49} -1.80194 q^{50} -0.868313 q^{53} -1.00000 q^{55} +1.08815 q^{56} -13.2349 q^{58} +8.11960 q^{59} -4.11529 q^{61} -4.29590 q^{62} -1.26875 q^{64} +9.93900 q^{67} +0.493959 q^{68} -1.44504 q^{70} +6.07069 q^{71} -4.29052 q^{73} +10.7409 q^{74} -3.41789 q^{76} -0.801938 q^{77} -14.1075 q^{79} -4.93900 q^{80} +15.8605 q^{82} -1.83877 q^{83} +0.396125 q^{85} +12.2567 q^{86} -1.35690 q^{88} +6.52111 q^{89} -1.93900 q^{92} -21.2446 q^{94} -2.74094 q^{95} -13.1521 q^{97} +11.4547 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} - q^{4} + 3 q^{5} - 2 q^{7}+O(q^{10}) \) 3 * q - q^2 - q^4 + 3 * q^5 - 2 * q^7	\( 3 q - q^{2} - q^{4} + 3 q^{5} - 2 q^{7} - q^{10} - 3 q^{11} - 4 q^{14} - 5 q^{16} + 10 q^{17} + 6 q^{19} - q^{20} + q^{22} - 5 q^{23} + 3 q^{25} + 3 q^{28} - q^{29} - 3 q^{31} + 4 q^{32} + 6 q^{34} - 2 q^{35} - 5 q^{37} + 5 q^{38} - 22 q^{41} - 16 q^{43} + q^{44} + 4 q^{46} + 12 q^{47} - 15 q^{49} - q^{50} - 5 q^{53} - 3 q^{55} + 7 q^{56} - 16 q^{58} + 3 q^{59} - 10 q^{61} + q^{62} + 4 q^{64} + 20 q^{67} - 8 q^{68} - 4 q^{70} + 6 q^{71} - 2 q^{73} + 18 q^{74} - 16 q^{76} + 2 q^{77} - 2 q^{79} - 5 q^{80} + 12 q^{82} + 27 q^{83} + 10 q^{85} + 10 q^{86} + 4 q^{89} + 4 q^{92} - 18 q^{94} + 6 q^{95} - 9 q^{97} + 12 q^{98}+O(q^{100}) \) 3 * q - q^2 - q^4 + 3 * q^5 - 2 * q^7 - q^10 - 3 * q^11 - 4 * q^14 - 5 * q^16 + 10 * q^17 + 6 * q^19 - q^20 + q^22 - 5 * q^23 + 3 * q^25 + 3 * q^28 - q^29 - 3 * q^31 + 4 * q^32 + 6 * q^34 - 2 * q^35 - 5 * q^37 + 5 * q^38 - 22 * q^41 - 16 * q^43 + q^44 + 4 * q^46 + 12 * q^47 - 15 * q^49 - q^50 - 5 * q^53 - 3 * q^55 + 7 * q^56 - 16 * q^58 + 3 * q^59 - 10 * q^61 + q^62 + 4 * q^64 + 20 * q^67 - 8 * q^68 - 4 * q^70 + 6 * q^71 - 2 * q^73 + 18 * q^74 - 16 * q^76 + 2 * q^77 - 2 * q^79 - 5 * q^80 + 12 * q^82 + 27 * q^83 + 10 * q^85 + 10 * q^86 + 4 * q^89 + 4 * q^92 - 18 * q^94 + 6 * q^95 - 9 * q^97 + 12 * q^98

Introduction

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