LMFDB - Embedded newform 9675.2.a.ch.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9675,2,Mod(1,9675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9675, 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("9675.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9675 = 3^{2} \cdot 5^{2} \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9675.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:	$77.2552639556$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.1933097.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 7x^{3} + 13x^{2} + 5x - 4 $ x^5 - 2x^4 - 7x^3 + 13x^2 + 5x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 215)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.434772$ of defining polynomial
Character	$\chi$	$=$	9675.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.434772 q^{2} -1.81097 q^{4} -3.42802 q^{7} -1.65691 q^{8} +O(q^{10})$	$q+0.434772 q^{2} -1.81097 q^{4} -3.42802 q^{7} -1.65691 q^{8} +3.52645 q^{11} -4.34418 q^{13} -1.49041 q^{14} +2.90157 q^{16} +0.147314 q^{17} -8.29463 q^{19} +1.53320 q^{22} +3.47463 q^{23} -1.88873 q^{26} +6.20805 q^{28} +10.4780 q^{29} +2.52243 q^{31} +4.57533 q^{32} +0.0640482 q^{34} +4.80422 q^{37} -3.60628 q^{38} +0.513408 q^{41} +1.00000 q^{43} -6.38631 q^{44} +1.51067 q^{46} +6.75240 q^{47} +4.75132 q^{49} +7.86719 q^{52} -8.14731 q^{53} +5.67991 q^{56} +4.55554 q^{58} -2.09168 q^{59} +10.6884 q^{61} +1.09668 q^{62} -3.81391 q^{64} +4.06976 q^{67} -0.266782 q^{68} +11.1507 q^{71} -5.53027 q^{73} +2.08874 q^{74} +15.0213 q^{76} -12.0887 q^{77} +2.32623 q^{79} +0.223216 q^{82} -4.60509 q^{83} +0.434772 q^{86} -5.84300 q^{88} -15.6084 q^{89} +14.8919 q^{91} -6.29246 q^{92} +2.93576 q^{94} -17.7693 q^{97} +2.06574 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 2 q^{2} + 8 q^{4} - 5 q^{7} + 3 q^{8}+O(q^{10}) $ 5 * q + 2 * q^2 + 8 * q^4 - 5 * q^7 + 3 * q^8	$ 5 q + 2 q^{2} + 8 q^{4} - 5 q^{7} + 3 q^{8} + 6 q^{11} - 5 q^{13} - q^{14} + 14 q^{16} - 17 q^{17} - 6 q^{19} + 8 q^{22} + q^{23} - 22 q^{26} - 26 q^{28} - 6 q^{29} + 6 q^{31} - 7 q^{32} - 5 q^{37} - 16 q^{38} - 2 q^{41} + 5 q^{43} + 15 q^{44} - 14 q^{46} + 18 q^{49} + 38 q^{52} - 23 q^{53} + 19 q^{56} - 12 q^{58} + q^{59} + 20 q^{61} - 3 q^{62} - 25 q^{64} - 21 q^{67} - 48 q^{68} - 4 q^{71} - 5 q^{73} - 24 q^{74} + 32 q^{76} - 26 q^{77} + 41 q^{79} - 38 q^{82} - 7 q^{83} + 2 q^{86} - 12 q^{88} - 20 q^{89} - 42 q^{91} - 52 q^{92} - 42 q^{94} - 37 q^{97} - 26 q^{98}+O(q^{100}) $ 5 * q + 2 * q^2 + 8 * q^4 - 5 * q^7 + 3 * q^8 + 6 * q^11 - 5 * q^13 - q^14 + 14 * q^16 - 17 * q^17 - 6 * q^19 + 8 * q^22 + q^23 - 22 * q^26 - 26 * q^28 - 6 * q^29 + 6 * q^31 - 7 * q^32 - 5 * q^37 - 16 * q^38 - 2 * q^41 + 5 * q^43 + 15 * q^44 - 14 * q^46 + 18 * q^49 + 38 * q^52 - 23 * q^53 + 19 * q^56 - 12 * q^58 + q^59 + 20 * q^61 - 3 * q^62 - 25 * q^64 - 21 * q^67 - 48 * q^68 - 4 * q^71 - 5 * q^73 - 24 * q^74 + 32 * q^76 - 26 * q^77 + 41 * q^79 - 38 * q^82 - 7 * q^83 + 2 * q^86 - 12 * q^88 - 20 * q^89 - 42 * q^91 - 52 * q^92 - 42 * q^94 - 37 * q^97 - 26 * 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 $

$2$	0.434772	0.307431	0.153715	−	0.988115i	$-0.450876\pi$
$2$	0.434772	0.307431	0.153715	+	0.988115i	$0.450876\pi$
$3$	0	0
$3$	0	0
$4$	−1.81097	−0.905486
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.42802	−1.29567	−0.647835	−	0.761781i	$-0.724325\pi$
$7$	−3.42802	−1.29567	−0.647835	+	0.761781i	$0.724325\pi$
$8$	−1.65691	−0.585805
$9$	0	0
$10$	0	0
$11$	3.52645	1.06326	0.531632	−	0.846975i	$-0.321579\pi$
$11$	3.52645	1.06326	0.531632	+	0.846975i	$0.321579\pi$
$12$	0	0
$13$	−4.34418	−1.20486	−0.602429	−	0.798173i	$-0.705800\pi$
$13$	−4.34418	−1.20486	−0.602429	+	0.798173i	$0.705800\pi$
$14$	−1.49041	−0.398328
$15$	0	0
$16$	2.90157	0.725392
$17$	0.147314	0.0357290	0.0178645	−	0.999840i	$-0.494313\pi$
$17$	0.147314	0.0357290	0.0178645	+	0.999840i	$0.494313\pi$
$18$	0	0
$19$	−8.29463	−1.90292	−0.951459	−	0.307775i	$-0.900416\pi$
$19$	−8.29463	−1.90292	−0.951459	+	0.307775i	$0.900416\pi$
$20$	0	0
$21$	0	0
$22$	1.53320	0.326880
$23$	3.47463	0.724511	0.362255	−	0.932079i	$-0.382007\pi$
$23$	3.47463	0.724511	0.362255	+	0.932079i	$0.382007\pi$
$24$	0	0
$25$	0	0
$26$	−1.88873	−0.370410
$27$	0	0
$28$	6.20805	1.17321
$29$	10.4780	1.94571	0.972857	−	0.231409i	$-0.0743336\pi$
$29$	10.4780	1.94571	0.972857	+	0.231409i	$0.0743336\pi$
$30$	0	0
$31$	2.52243	0.453042	0.226521	−	0.974006i	$-0.427265\pi$
$31$	2.52243	0.453042	0.226521	+	0.974006i	$0.427265\pi$
$32$	4.57533	0.808812
$33$	0	0
$34$	0.0640482	0.0109842
$35$	0	0
$36$	0	0
$37$	4.80422	0.789809	0.394904	−	0.918722i	$-0.370778\pi$
$37$	4.80422	0.789809	0.394904	+	0.918722i	$0.370778\pi$
$38$	−3.60628	−0.585015
$39$	0	0
$40$	0	0
$41$	0.513408	0.0801809	0.0400905	−	0.999196i	$-0.487235\pi$
$41$	0.513408	0.0801809	0.0400905	+	0.999196i	$0.487235\pi$
$42$	0	0
$43$	1.00000	0.152499
$43$	1.00000	0.152499
$44$	−6.38631	−0.962772
$45$	0	0
$46$	1.51067	0.222737
$47$	6.75240	0.984939	0.492469	−	0.870330i	$-0.336094\pi$
$47$	6.75240	0.984939	0.492469	+	0.870330i	$0.336094\pi$
$48$	0	0
$49$	4.75132	0.678760
$50$	0	0
$51$	0	0
$52$	7.86719	1.09098
$53$	−8.14731	−1.11912	−0.559560	−	0.828790i	$-0.689030\pi$
$53$	−8.14731	−1.11912	−0.559560	+	0.828790i	$0.689030\pi$
$54$	0	0
$55$	0	0
$56$	5.67991	0.759009
$57$	0	0
$58$	4.55554	0.598172
$59$	−2.09168	−0.272313	−0.136157	−	0.990687i	$-0.543475\pi$
$59$	−2.09168	−0.272313	−0.136157	+	0.990687i	$0.543475\pi$
$60$	0	0
$61$	10.6884	1.36850	0.684252	−	0.729246i	$-0.260129\pi$
$61$	10.6884	1.36850	0.684252	+	0.729246i	$0.260129\pi$
$62$	1.09668	0.139279
$63$	0	0
$64$	−3.81391	−0.476739
$65$	0	0
$66$	0	0
$67$	4.06976	0.497200	0.248600	−	0.968606i	$-0.420029\pi$
$67$	4.06976	0.497200	0.248600	+	0.968606i	$0.420029\pi$
$68$	−0.266782	−0.0323521
$69$	0	0
$70$	0	0
$71$	11.1507	1.32334	0.661670	−	0.749795i	$-0.269848\pi$
$71$	11.1507	1.32334	0.661670	+	0.749795i	$0.269848\pi$
$72$	0	0
$73$	−5.53027	−0.647269	−0.323634	−	0.946182i	$-0.604905\pi$
$73$	−5.53027	−0.647269	−0.323634	+	0.946182i	$0.604905\pi$
$74$	2.08874	0.242811
$75$	0	0
$76$	15.0213	1.72307
$77$	−12.0887	−1.37764
$78$	0	0
$79$	2.32623	0.261722	0.130861	−	0.991401i	$-0.458226\pi$
$79$	2.32623	0.261722	0.130861	+	0.991401i	$0.458226\pi$
$80$	0	0
$81$	0	0
$82$	0.223216	0.0246501
$83$	−4.60509	−0.505474	−0.252737	−	0.967535i	$-0.581331\pi$
$83$	−4.60509	−0.505474	−0.252737	+	0.967535i	$0.581331\pi$
$84$	0	0
$85$	0	0
$86$	0.434772	0.0468827
$87$	0	0
$88$	−5.84300	−0.622866
$89$	−15.6084	−1.65449	−0.827246	−	0.561840i	$-0.810094\pi$
$89$	−15.6084	−1.65449	−0.827246	+	0.561840i	$0.810094\pi$
$90$	0	0
$91$	14.8919	1.56110
$92$	−6.29246	−0.656035
$93$	0	0
$94$	2.93576	0.302800
$95$	0	0
$96$	0	0
$97$	−17.7693	−1.80420	−0.902098	−	0.431532i	$-0.857973\pi$
$97$	−17.7693	−1.80420	−0.902098	+	0.431532i	$0.857973\pi$
$98$	2.06574	0.208671
$99$	0	0
$100$	0	0
$101$	−7.57533	−0.753774	−0.376887	−	0.926259i	$-0.623005\pi$
$101$	−7.57533	−0.753774	−0.376887	+	0.926259i	$0.623005\pi$
$102$	0	0
$103$	−10.4419	−1.02888	−0.514438	−	0.857528i	$-0.671999\pi$
$103$	−10.4419	−1.02888	−0.514438	+	0.857528i	$0.671999\pi$
$104$	7.19789	0.705811
$105$	0	0
$106$	−3.54223	−0.344052
$107$	1.24173	0.120042	0.0600211	−	0.998197i	$-0.480883\pi$
$107$	1.24173	0.120042	0.0600211	+	0.998197i	$0.480883\pi$
$108$	0	0
$109$	19.4202	1.86012	0.930060	−	0.367407i	$-0.119754\pi$
$109$	19.4202	1.86012	0.930060	+	0.367407i	$0.119754\pi$
$110$	0	0
$111$	0	0
$112$	−9.94664	−0.939869
$113$	0.936529	0.0881012	0.0440506	−	0.999029i	$-0.485974\pi$
$113$	0.936529	0.0881012	0.0440506	+	0.999029i	$0.485974\pi$
$114$	0	0
$115$	0	0
$116$	−18.9753	−1.76182
$117$	0	0
$118$	−0.909404	−0.0837174
$119$	−0.504996	−0.0462929
$120$	0	0
$121$	1.43585	0.130532
$122$	4.64700	0.420720
$123$	0	0
$124$	−4.56806	−0.410224
$125$	0	0
$126$	0	0
$127$	−15.8729	−1.40849	−0.704246	−	0.709956i	$-0.748715\pi$
$127$	−15.8729	−1.40849	−0.704246	+	0.709956i	$0.748715\pi$
$128$	−10.8089	−0.955376
$129$	0	0
$130$	0	0
$131$	12.6748	1.10741	0.553703	−	0.832714i	$-0.313214\pi$
$131$	12.6748	1.10741	0.553703	+	0.832714i	$0.313214\pi$
$132$	0	0
$133$	28.4341	2.46555
$134$	1.76942	0.152855
$135$	0	0
$136$	−0.244086	−0.0209302
$137$	3.49655	0.298730	0.149365	−	0.988782i	$-0.452277\pi$
$137$	3.49655	0.298730	0.149365	+	0.988782i	$0.452277\pi$
$138$	0	0
$139$	2.99727	0.254225	0.127112	−	0.991888i	$-0.459429\pi$
$139$	2.99727	0.254225	0.127112	+	0.991888i	$0.459429\pi$
$140$	0	0
$141$	0	0
$142$	4.84800	0.406835
$143$	−15.3195	−1.28108
$144$	0	0
$145$	0	0
$146$	−2.40441	−0.198990
$147$	0	0
$148$	−8.70031	−0.715161
$149$	8.86367	0.726140	0.363070	−	0.931762i	$-0.381729\pi$
$149$	8.86367	0.726140	0.363070	+	0.931762i	$0.381729\pi$
$150$	0	0
$151$	−16.8717	−1.37300	−0.686500	−	0.727130i	$-0.740854\pi$
$151$	−16.8717	−1.37300	−0.686500	+	0.727130i	$0.740854\pi$
$152$	13.7434	1.11474
$153$	0	0
$154$	−5.25585	−0.423529
$155$	0	0
$156$	0	0
$157$	−0.686807	−0.0548132	−0.0274066	−	0.999624i	$-0.508725\pi$
$157$	−0.686807	−0.0548132	−0.0274066	+	0.999624i	$0.508725\pi$
$158$	1.01138	0.0804613
$159$	0	0
$160$	0	0
$161$	−11.9111	−0.938727
$162$	0	0
$163$	−6.91549	−0.541663	−0.270832	−	0.962627i	$-0.587299\pi$
$163$	−6.91549	−0.541663	−0.270832	+	0.962627i	$0.587299\pi$
$164$	−0.929769	−0.0726027
$165$	0	0
$166$	−2.00216	−0.155398
$167$	−7.18000	−0.555605	−0.277803	−	0.960638i	$-0.589606\pi$
$167$	−7.18000	−0.555605	−0.277803	+	0.960638i	$0.589606\pi$
$168$	0	0
$169$	5.87187	0.451682
$170$	0	0
$171$	0	0
$172$	−1.81097	−0.138085
$173$	14.2811	1.08577	0.542887	−	0.839806i	$-0.317331\pi$
$173$	14.2811	1.08577	0.542887	+	0.839806i	$0.317331\pi$
$174$	0	0
$175$	0	0
$176$	10.2322	0.771284
$177$	0	0
$178$	−6.78612	−0.508641
$179$	−10.1134	−0.755914	−0.377957	−	0.925823i	$-0.623373\pi$
$179$	−10.1134	−0.755914	−0.377957	+	0.925823i	$0.623373\pi$
$180$	0	0
$181$	−1.20863	−0.0898366	−0.0449183	−	0.998991i	$-0.514303\pi$
$181$	−1.20863	−0.0898366	−0.0449183	+	0.998991i	$0.514303\pi$
$182$	6.47460	0.479929
$183$	0	0
$184$	−5.75714	−0.424422
$185$	0	0
$186$	0	0
$187$	0.519497	0.0379894
$188$	−12.2284	−0.891849
$189$	0	0
$190$	0	0
$191$	11.7896	0.853068	0.426534	−	0.904472i	$-0.359734\pi$
$191$	11.7896	0.853068	0.426534	+	0.904472i	$0.359734\pi$
$192$	0	0
$193$	−8.78628	−0.632450	−0.316225	−	0.948684i	$-0.602415\pi$
$193$	−8.78628	−0.632450	−0.316225	+	0.948684i	$0.602415\pi$
$194$	−7.72558	−0.554665
$195$	0	0
$196$	−8.60451	−0.614608
$197$	−20.7009	−1.47488	−0.737440	−	0.675412i	$-0.763966\pi$
$197$	−20.7009	−1.47488	−0.737440	+	0.675412i	$0.763966\pi$
$198$	0	0
$199$	−7.88873	−0.559217	−0.279609	−	0.960114i	$-0.590205\pi$
$199$	−7.88873	−0.559217	−0.279609	+	0.960114i	$0.590205\pi$
$200$	0	0
$201$	0	0
$202$	−3.29355	−0.231733
$203$	−35.9187	−2.52100
$204$	0	0
$205$	0	0
$206$	−4.53987	−0.316308
$207$	0	0
$208$	−12.6049	−0.873994
$209$	−29.2506	−2.02331
$210$	0	0
$211$	−18.7883	−1.29344	−0.646719	−	0.762728i	$-0.723860\pi$
$211$	−18.7883	−1.29344	−0.646719	+	0.762728i	$0.723860\pi$
$212$	14.7546	1.01335
$213$	0	0
$214$	0.539869	0.0369047
$215$	0	0
$216$	0	0
$217$	−8.64695	−0.586993
$218$	8.44338	0.571858
$219$	0	0
$220$	0	0
$221$	−0.639959	−0.0430483
$222$	0	0
$223$	13.0551	0.874232	0.437116	−	0.899405i	$-0.356000\pi$
$223$	13.0551	0.874232	0.437116	+	0.899405i	$0.356000\pi$
$224$	−15.6843	−1.04795
$225$	0	0
$226$	0.407177	0.0270850
$227$	21.6000	1.43364	0.716822	−	0.697256i	$-0.245596\pi$
$227$	21.6000	1.43364	0.716822	+	0.697256i	$0.245596\pi$
$228$	0	0
$229$	2.87945	0.190279	0.0951397	−	0.995464i	$-0.469670\pi$
$229$	2.87945	0.190279	0.0951397	+	0.995464i	$0.469670\pi$
$230$	0	0
$231$	0	0
$232$	−17.3610	−1.13981
$233$	22.7687	1.49163	0.745813	−	0.666155i	$-0.232061\pi$
$233$	22.7687	1.49163	0.745813	+	0.666155i	$0.232061\pi$
$234$	0	0
$235$	0	0
$236$	3.78797	0.246576
$237$	0	0
$238$	−0.219558	−0.0142319
$239$	21.4719	1.38890	0.694452	−	0.719539i	$-0.255647\pi$
$239$	21.4719	1.38890	0.694452	+	0.719539i	$0.255647\pi$
$240$	0	0
$241$	−2.39827	−0.154486	−0.0772430	−	0.997012i	$-0.524612\pi$
$241$	−2.39827	−0.154486	−0.0772430	+	0.997012i	$0.524612\pi$
$242$	0.624270	0.0401296
$243$	0	0
$244$	−19.3563	−1.23916
$245$	0	0
$246$	0	0
$247$	36.0333	2.29275
$248$	−4.17943	−0.265394
$249$	0	0
$250$	0	0
$251$	−3.46113	−0.218464	−0.109232	−	0.994016i	$-0.534839\pi$
$251$	−3.46113	−0.218464	−0.109232	+	0.994016i	$0.534839\pi$
$252$	0	0
$253$	12.2531	0.770347
$254$	−6.90110	−0.433013
$255$	0	0
$256$	2.92843	0.183027
$257$	−13.9899	−0.872668	−0.436334	−	0.899785i	$-0.643723\pi$
$257$	−13.9899	−0.872668	−0.436334	+	0.899785i	$0.643723\pi$
$258$	0	0
$259$	−16.4690	−1.02333
$260$	0	0
$261$	0	0
$262$	5.51067	0.340451
$263$	−26.2315	−1.61750	−0.808751	−	0.588151i	$-0.799856\pi$
$263$	−26.2315	−1.61750	−0.808751	+	0.588151i	$0.799856\pi$
$264$	0	0
$265$	0	0
$266$	12.3624	0.757986
$267$	0	0
$268$	−7.37023	−0.450208
$269$	−6.72899	−0.410274	−0.205137	−	0.978733i	$-0.565764\pi$
$269$	−6.72899	−0.410274	−0.205137	+	0.978733i	$0.565764\pi$
$270$	0	0
$271$	5.29736	0.321792	0.160896	−	0.986971i	$-0.448562\pi$
$271$	5.29736	0.321792	0.160896	+	0.986971i	$0.448562\pi$
$272$	0.427443	0.0259175
$273$	0	0
$274$	1.52020	0.0918388
$275$	0	0
$276$	0	0
$277$	−6.76297	−0.406348	−0.203174	−	0.979143i	$-0.565126\pi$
$277$	−6.76297	−0.406348	−0.203174	+	0.979143i	$0.565126\pi$
$278$	1.30313	0.0781564
$279$	0	0
$280$	0	0
$281$	16.3690	0.976492	0.488246	−	0.872706i	$-0.337637\pi$
$281$	16.3690	0.976492	0.488246	+	0.872706i	$0.337637\pi$
$282$	0	0
$283$	28.2202	1.67752	0.838759	−	0.544502i	$-0.183281\pi$
$283$	28.2202	1.67752	0.838759	+	0.544502i	$0.183281\pi$
$284$	−20.1936	−1.19827
$285$	0	0
$286$	−6.66051	−0.393844
$287$	−1.75997	−0.103888
$288$	0	0
$289$	−16.9783	−0.998723
$290$	0	0
$291$	0	0
$292$	10.0152	0.586093
$293$	9.03940	0.528087	0.264044	−	0.964511i	$-0.414944\pi$
$293$	9.03940	0.528087	0.264044	+	0.964511i	$0.414944\pi$
$294$	0	0
$295$	0	0
$296$	−7.96014	−0.462674
$297$	0	0
$298$	3.85368	0.223238
$299$	−15.0944	−0.872932
$300$	0	0
$301$	−3.42802	−0.197588
$302$	−7.33535	−0.422102
$303$	0	0
$304$	−24.0674	−1.38036
$305$	0	0
$306$	0	0
$307$	−13.2061	−0.753711	−0.376856	−	0.926272i	$-0.622995\pi$
$307$	−13.2061	−0.753711	−0.376856	+	0.926272i	$0.622995\pi$
$308$	21.8924	1.24743
$309$	0	0
$310$	0	0
$311$	1.00273	0.0568599	0.0284299	−	0.999596i	$-0.490949\pi$
$311$	1.00273	0.0568599	0.0284299	+	0.999596i	$0.490949\pi$
$312$	0	0
$313$	−5.10184	−0.288373	−0.144186	−	0.989551i	$-0.546057\pi$
$313$	−5.10184	−0.288373	−0.144186	+	0.989551i	$0.546057\pi$
$314$	−0.298605	−0.0168512
$315$	0	0
$316$	−4.21275	−0.236986
$317$	−1.09441	−0.0614683	−0.0307342	−	0.999528i	$-0.509785\pi$
$317$	−1.09441	−0.0614683	−0.0307342	+	0.999528i	$0.509785\pi$
$318$	0	0
$319$	36.9501	2.06881
$320$	0	0
$321$	0	0
$322$	−5.17862	−0.288593
$323$	−1.22192	−0.0679893
$324$	0	0
$325$	0	0
$326$	−3.00667	−0.166524
$327$	0	0
$328$	−0.850670	−0.0469704
$329$	−23.1474	−1.27616
$330$	0	0
$331$	17.1663	0.943547	0.471774	−	0.881720i	$-0.343614\pi$
$331$	17.1663	0.943547	0.471774	+	0.881720i	$0.343614\pi$
$332$	8.33969	0.457700
$333$	0	0
$334$	−3.12167	−0.170810
$335$	0	0
$336$	0	0
$337$	23.0000	1.25289	0.626446	−	0.779465i	$-0.284509\pi$
$337$	23.0000	1.25289	0.626446	+	0.779465i	$0.284509\pi$
$338$	2.55293	0.138861
$339$	0	0
$340$	0	0
$341$	8.89523	0.481704
$342$	0	0
$343$	7.70852	0.416221
$344$	−1.65691	−0.0893344
$345$	0	0
$346$	6.20904	0.333800
$347$	−13.8648	−0.744299	−0.372150	−	0.928173i	$-0.621379\pi$
$347$	−13.8648	−0.744299	−0.372150	+	0.928173i	$0.621379\pi$
$348$	0	0
$349$	−1.71755	−0.0919382	−0.0459691	−	0.998943i	$-0.514638\pi$
$349$	−1.71755	−0.0919382	−0.0459691	+	0.998943i	$0.514638\pi$
$350$	0	0
$351$	0	0
$352$	16.1347	0.859982
$353$	−13.3575	−0.710947	−0.355474	−	0.934686i	$-0.615680\pi$
$353$	−13.3575	−0.710947	−0.355474	+	0.934686i	$0.615680\pi$
$354$	0	0
$355$	0	0
$356$	28.2665	1.49812
$357$	0	0
$358$	−4.39704	−0.232391
$359$	−30.7771	−1.62435	−0.812177	−	0.583412i	$-0.801717\pi$
$359$	−30.7771	−1.62435	−0.812177	+	0.583412i	$0.801717\pi$
$360$	0	0
$361$	49.8009	2.62110
$362$	−0.525478	−0.0276185
$363$	0	0
$364$	−26.9689	−1.41355
$365$	0	0
$366$	0	0
$367$	−17.0314	−0.889030	−0.444515	−	0.895771i	$-0.646624\pi$
$367$	−17.0314	−0.889030	−0.444515	+	0.895771i	$0.646624\pi$
$368$	10.0819	0.525554
$369$	0	0
$370$	0	0
$371$	27.9292	1.45001
$372$	0	0
$373$	−33.5031	−1.73472	−0.867362	−	0.497677i	$-0.834187\pi$
$373$	−33.5031	−1.73472	−0.867362	+	0.497677i	$0.834187\pi$
$374$	0.225863	0.0116791
$375$	0	0
$376$	−11.1881	−0.576982
$377$	−45.5182	−2.34431
$378$	0	0
$379$	−1.63225	−0.0838432	−0.0419216	−	0.999121i	$-0.513348\pi$
$379$	−1.63225	−0.0838432	−0.0419216	+	0.999121i	$0.513348\pi$
$380$	0	0
$381$	0	0
$382$	5.12581	0.262259
$383$	−18.8360	−0.962477	−0.481238	−	0.876590i	$-0.659813\pi$
$383$	−18.8360	−0.962477	−0.481238	+	0.876590i	$0.659813\pi$
$384$	0	0
$385$	0	0
$386$	−3.82003	−0.194434
$387$	0	0
$388$	32.1796	1.63367
$389$	−20.0472	−1.01643	−0.508217	−	0.861229i	$-0.669695\pi$
$389$	−20.0472	−1.01643	−0.508217	+	0.861229i	$0.669695\pi$
$390$	0	0
$391$	0.511863	0.0258860
$392$	−7.87249	−0.397621
$393$	0	0
$394$	−9.00019	−0.453423
$395$	0	0
$396$	0	0
$397$	−9.36558	−0.470045	−0.235022	−	0.971990i	$-0.575516\pi$
$397$	−9.36558	−0.470045	−0.235022	+	0.971990i	$0.575516\pi$
$398$	−3.42980	−0.171920
$399$	0	0
$400$	0	0
$401$	−27.6415	−1.38035	−0.690176	−	0.723641i	$-0.742467\pi$
$401$	−27.6415	−1.38035	−0.690176	+	0.723641i	$0.742467\pi$
$402$	0	0
$403$	−10.9579	−0.545851
$404$	13.7187	0.682532
$405$	0	0
$406$	−15.6165	−0.775033
$407$	16.9418	0.839776
$408$	0	0
$409$	8.75240	0.432778	0.216389	−	0.976307i	$-0.430572\pi$
$409$	8.75240	0.432778	0.216389	+	0.976307i	$0.430572\pi$
$410$	0	0
$411$	0	0
$412$	18.9101	0.931633
$413$	7.17031	0.352828
$414$	0	0
$415$	0	0
$416$	−19.8761	−0.974504
$417$	0	0
$418$	−12.7174	−0.622026
$419$	−5.38920	−0.263280	−0.131640	−	0.991298i	$-0.542024\pi$
$419$	−5.38920	−0.263280	−0.131640	+	0.991298i	$0.542024\pi$
$420$	0	0
$421$	−34.4283	−1.67793	−0.838966	−	0.544184i	$-0.816839\pi$
$421$	−34.4283	−1.67793	−0.838966	+	0.544184i	$0.816839\pi$
$422$	−8.16863	−0.397643
$423$	0	0
$424$	13.4993	0.655586
$425$	0	0
$426$	0	0
$427$	−36.6399	−1.77313
$428$	−2.24873	−0.108697
$429$	0	0
$430$	0	0
$431$	15.4470	0.744056	0.372028	−	0.928222i	$-0.378663\pi$
$431$	15.4470	0.744056	0.372028	+	0.928222i	$0.378663\pi$
$432$	0	0
$433$	−16.5096	−0.793400	−0.396700	−	0.917948i	$-0.629845\pi$
$433$	−16.5096	−0.793400	−0.396700	+	0.917948i	$0.629845\pi$
$434$	−3.75945	−0.180460
$435$	0	0
$436$	−35.1695	−1.68431
$437$	−28.8208	−1.37868
$438$	0	0
$439$	−27.0405	−1.29057	−0.645287	−	0.763941i	$-0.723262\pi$
$439$	−27.0405	−1.29057	−0.645287	+	0.763941i	$0.723262\pi$
$440$	0	0
$441$	0	0
$442$	−0.278237	−0.0132344
$443$	−1.84327	−0.0875762	−0.0437881	−	0.999041i	$-0.513943\pi$
$443$	−1.84327	−0.0875762	−0.0437881	+	0.999041i	$0.513943\pi$
$444$	0	0
$445$	0	0
$446$	5.67598	0.268766
$447$	0	0
$448$	13.0742	0.617696
$449$	12.2474	0.577991	0.288995	−	0.957330i	$-0.406679\pi$
$449$	12.2474	0.577991	0.288995	+	0.957330i	$0.406679\pi$
$450$	0	0
$451$	1.81051	0.0852536
$452$	−1.69603	−0.0797745
$453$	0	0
$454$	9.39110	0.440746
$455$	0	0
$456$	0	0
$457$	−32.9544	−1.54154	−0.770771	−	0.637112i	$-0.780129\pi$
$457$	−32.9544	−1.54154	−0.770771	+	0.637112i	$0.780129\pi$
$458$	1.25191	0.0584977
$459$	0	0
$460$	0	0
$461$	−21.0988	−0.982671	−0.491336	−	0.870970i	$-0.663491\pi$
$461$	−21.0988	−0.982671	−0.491336	+	0.870970i	$0.663491\pi$
$462$	0	0
$463$	−23.6686	−1.09997	−0.549986	−	0.835174i	$-0.685367\pi$
$463$	−23.6686	−1.09997	−0.549986	+	0.835174i	$0.685367\pi$
$464$	30.4026	1.41141
$465$	0	0
$466$	9.89920	0.458572
$467$	17.8786	0.827323	0.413662	−	0.910431i	$-0.364250\pi$
$467$	17.8786	0.827323	0.413662	+	0.910431i	$0.364250\pi$
$468$	0	0
$469$	−13.9512	−0.644207
$470$	0	0
$471$	0	0
$472$	3.46571	0.159522
$473$	3.52645	0.162146
$474$	0	0
$475$	0	0
$476$	0.914535	0.0419176
$477$	0	0
$478$	9.33541	0.426992
$479$	12.7833	0.584086	0.292043	−	0.956405i	$-0.405665\pi$
$479$	12.7833	0.584086	0.292043	+	0.956405i	$0.405665\pi$
$480$	0	0
$481$	−20.8704	−0.951607
$482$	−1.04270	−0.0474937
$483$	0	0
$484$	−2.60029	−0.118195
$485$	0	0
$486$	0	0
$487$	3.29227	0.149187	0.0745935	−	0.997214i	$-0.476234\pi$
$487$	3.29227	0.149187	0.0745935	+	0.997214i	$0.476234\pi$
$488$	−17.7096	−0.801676
$489$	0	0
$490$	0	0
$491$	11.2678	0.508509	0.254255	−	0.967137i	$-0.418170\pi$
$491$	11.2678	0.508509	0.254255	+	0.967137i	$0.418170\pi$
$492$	0	0
$493$	1.54356	0.0695183
$494$	15.6663	0.704860
$495$	0	0
$496$	7.31901	0.328633
$497$	−38.2247	−1.71461
$498$	0	0
$499$	−28.7140	−1.28542	−0.642709	−	0.766111i	$-0.722189\pi$
$499$	−28.7140	−1.28542	−0.642709	+	0.766111i	$0.722189\pi$
$500$	0	0
$501$	0	0
$502$	−1.50480	−0.0671626
$503$	−12.4110	−0.553379	−0.276690	−	0.960959i	$-0.589237\pi$
$503$	−12.4110	−0.553379	−0.276690	+	0.960959i	$0.589237\pi$
$504$	0	0
$505$	0	0
$506$	5.32732	0.236828
$507$	0	0
$508$	28.7454	1.27537
$509$	−20.4592	−0.906839	−0.453419	−	0.891297i	$-0.649796\pi$
$509$	−20.4592	−0.906839	−0.453419	+	0.891297i	$0.649796\pi$
$510$	0	0
$511$	18.9579	0.838647
$512$	22.8909	1.01164
$513$	0	0
$514$	−6.08244	−0.268285
$515$	0	0
$516$	0	0
$517$	23.8120	1.04725
$518$	−7.16025	−0.314603
$519$	0	0
$520$	0	0
$521$	−43.3303	−1.89834	−0.949168	−	0.314770i	$-0.898072\pi$
$521$	−43.3303	−1.89834	−0.949168	+	0.314770i	$0.898072\pi$
$522$	0	0
$523$	9.83341	0.429985	0.214993	−	0.976616i	$-0.431027\pi$
$523$	9.83341	0.429985	0.214993	+	0.976616i	$0.431027\pi$
$524$	−22.9538	−1.00274
$525$	0	0
$526$	−11.4047	−0.497269
$527$	0.371590	0.0161867
$528$	0	0
$529$	−10.9269	−0.475084
$530$	0	0
$531$	0	0
$532$	−51.4935	−2.23253
$533$	−2.23034	−0.0966066
$534$	0	0
$535$	0	0
$536$	−6.74321	−0.291262
$537$	0	0
$538$	−2.92558	−0.126131
$539$	16.7553	0.721701
$540$	0	0
$541$	22.6054	0.971884	0.485942	−	0.873991i	$-0.338477\pi$
$541$	22.6054	0.971884	0.485942	+	0.873991i	$0.338477\pi$
$542$	2.30315	0.0989286
$543$	0	0
$544$	0.674012	0.0288980
$545$	0	0
$546$	0	0
$547$	32.7864	1.40184	0.700922	−	0.713238i	$-0.252772\pi$
$547$	32.7864	1.40184	0.700922	+	0.713238i	$0.252772\pi$
$548$	−6.33216	−0.270496
$549$	0	0
$550$	0	0
$551$	−86.9110	−3.70253
$552$	0	0
$553$	−7.97438	−0.339105
$554$	−2.94035	−0.124924
$555$	0	0
$556$	−5.42797	−0.230197
$557$	−21.8974	−0.927821	−0.463910	−	0.885882i	$-0.653554\pi$
$557$	−21.8974	−0.927821	−0.463910	+	0.885882i	$0.653554\pi$
$558$	0	0
$559$	−4.34418	−0.183739
$560$	0	0
$561$	0	0
$562$	7.11678	0.300203
$563$	−5.29931	−0.223339	−0.111670	−	0.993745i	$-0.535620\pi$
$563$	−5.29931	−0.223339	−0.111670	+	0.993745i	$0.535620\pi$
$564$	0	0
$565$	0	0
$566$	12.2694	0.515720
$567$	0	0
$568$	−18.4756	−0.775219
$569$	9.09310	0.381203	0.190601	−	0.981668i	$-0.438956\pi$
$569$	9.09310	0.381203	0.190601	+	0.981668i	$0.438956\pi$
$570$	0	0
$571$	4.29009	0.179535	0.0897673	−	0.995963i	$-0.471388\pi$
$571$	4.29009	0.179535	0.0897673	+	0.995963i	$0.471388\pi$
$572$	27.7432	1.16000
$573$	0	0
$574$	−0.765188	−0.0319383
$575$	0	0
$576$	0	0
$577$	3.56791	0.148534	0.0742670	−	0.997238i	$-0.476338\pi$
$577$	3.56791	0.148534	0.0742670	+	0.997238i	$0.476338\pi$
$578$	−7.38170	−0.307038
$579$	0	0
$580$	0	0
$581$	15.7863	0.654927
$582$	0	0
$583$	−28.7311	−1.18992
$584$	9.16313	0.379173
$585$	0	0
$586$	3.93008	0.162350
$587$	42.8506	1.76863	0.884317	−	0.466886i	$-0.154624\pi$
$587$	42.8506	1.76863	0.884317	+	0.466886i	$0.154624\pi$
$588$	0	0
$589$	−20.9226	−0.862102
$590$	0	0
$591$	0	0
$592$	13.9398	0.572921
$593$	−22.3566	−0.918078	−0.459039	−	0.888416i	$-0.651806\pi$
$593$	−22.3566	−0.918078	−0.459039	+	0.888416i	$0.651806\pi$
$594$	0	0
$595$	0	0
$596$	−16.0519	−0.657510
$597$	0	0
$598$	−6.56263	−0.268366
$599$	−32.1107	−1.31201	−0.656005	−	0.754757i	$-0.727755\pi$
$599$	−32.1107	−1.31201	−0.656005	+	0.754757i	$0.727755\pi$
$600$	0	0
$601$	−38.4026	−1.56647	−0.783237	−	0.621723i	$-0.786433\pi$
$601$	−38.4026	−1.56647	−0.783237	+	0.621723i	$0.786433\pi$
$602$	−1.49041	−0.0607445
$603$	0	0
$604$	30.5542	1.24323
$605$	0	0
$606$	0	0
$607$	14.6539	0.594784	0.297392	−	0.954755i	$-0.403883\pi$
$607$	14.6539	0.594784	0.297392	+	0.954755i	$0.403883\pi$
$608$	−37.9507	−1.53910
$609$	0	0
$610$	0	0
$611$	−29.3336	−1.18671
$612$	0	0
$613$	43.2473	1.74674	0.873371	−	0.487056i	$-0.161929\pi$
$613$	43.2473	1.74674	0.873371	+	0.487056i	$0.161929\pi$
$614$	−5.74164	−0.231714
$615$	0	0
$616$	20.0299	0.807028
$617$	11.2697	0.453702	0.226851	−	0.973929i	$-0.427157\pi$
$617$	11.2697	0.453702	0.226851	+	0.973929i	$0.427157\pi$
$618$	0	0
$619$	14.7440	0.592611	0.296306	−	0.955093i	$-0.404245\pi$
$619$	14.7440	0.592611	0.296306	+	0.955093i	$0.404245\pi$
$620$	0	0
$621$	0	0
$622$	0.435961	0.0174805
$623$	53.5060	2.14367
$624$	0	0
$625$	0	0
$626$	−2.21814	−0.0886546
$627$	0	0
$628$	1.24379	0.0496326
$629$	0.707730	0.0282191
$630$	0	0
$631$	28.9244	1.15146	0.575731	−	0.817639i	$-0.304717\pi$
$631$	28.9244	1.15146	0.575731	+	0.817639i	$0.304717\pi$
$632$	−3.85435	−0.153318
$633$	0	0
$634$	−0.475821	−0.0188972
$635$	0	0
$636$	0	0
$637$	−20.6406	−0.817809
$638$	16.0649	0.636015
$639$	0	0
$640$	0	0
$641$	−19.0280	−0.751562	−0.375781	−	0.926708i	$-0.622626\pi$
$641$	−19.0280	−0.751562	−0.375781	+	0.926708i	$0.622626\pi$
$642$	0	0
$643$	43.1807	1.70288	0.851440	−	0.524452i	$-0.175730\pi$
$643$	43.1807	1.70288	0.851440	+	0.524452i	$0.175730\pi$
$644$	21.5707	0.850004
$645$	0	0
$646$	−0.531256	−0.0209020
$647$	−18.7593	−0.737503	−0.368751	−	0.929528i	$-0.620215\pi$
$647$	−18.7593	−0.737503	−0.368751	+	0.929528i	$0.620215\pi$
$648$	0	0
$649$	−7.37620	−0.289541
$650$	0	0
$651$	0	0
$652$	12.5238	0.490469
$653$	22.0692	0.863636	0.431818	−	0.901961i	$-0.357872\pi$
$653$	22.0692	0.863636	0.431818	+	0.901961i	$0.357872\pi$
$654$	0	0
$655$	0	0
$656$	1.48969	0.0581626
$657$	0	0
$658$	−10.0638	−0.392329
$659$	−11.3232	−0.441091	−0.220545	−	0.975377i	$-0.570784\pi$
$659$	−11.3232	−0.441091	−0.220545	+	0.975377i	$0.570784\pi$
$660$	0	0
$661$	−16.2409	−0.631697	−0.315849	−	0.948810i	$-0.602289\pi$
$661$	−16.2409	−0.631697	−0.315849	+	0.948810i	$0.602289\pi$
$662$	7.46345	0.290075
$663$	0	0
$664$	7.63020	0.296109
$665$	0	0
$666$	0	0
$667$	36.4071	1.40969
$668$	13.0028	0.503093
$669$	0	0
$670$	0	0
$671$	37.6919	1.45508
$672$	0	0
$673$	−20.3835	−0.785726	−0.392863	−	0.919597i	$-0.628515\pi$
$673$	−20.3835	−0.785726	−0.392863	+	0.919597i	$0.628515\pi$
$674$	9.99979	0.385177
$675$	0	0
$676$	−10.6338	−0.408992
$677$	−39.7020	−1.52587	−0.762936	−	0.646474i	$-0.776243\pi$
$677$	−39.7020	−1.52587	−0.762936	+	0.646474i	$0.776243\pi$
$678$	0	0
$679$	60.9134	2.33764
$680$	0	0
$681$	0	0
$682$	3.86740	0.148090
$683$	−3.69595	−0.141422	−0.0707108	−	0.997497i	$-0.522527\pi$
$683$	−3.69595	−0.141422	−0.0707108	+	0.997497i	$0.522527\pi$
$684$	0	0
$685$	0	0
$686$	3.35145	0.127959
$687$	0	0
$688$	2.90157	0.110621
$689$	35.3934	1.34838
$690$	0	0
$691$	7.30957	0.278069	0.139035	−	0.990288i	$-0.455600\pi$
$691$	7.30957	0.278069	0.139035	+	0.990288i	$0.455600\pi$
$692$	−25.8627	−0.983153
$693$	0	0
$694$	−6.02801	−0.228820
$695$	0	0
$696$	0	0
$697$	0.0756324	0.00286478
$698$	−0.746742	−0.0282646
$699$	0	0
$700$	0	0
$701$	28.8131	1.08826	0.544128	−	0.839002i	$-0.316861\pi$
$701$	28.8131	1.08826	0.544128	+	0.839002i	$0.316861\pi$
$702$	0	0
$703$	−39.8492	−1.50294
$704$	−13.4496	−0.506899
$705$	0	0
$706$	−5.80747	−0.218567
$707$	25.9684	0.976642
$708$	0	0
$709$	8.59692	0.322864	0.161432	−	0.986884i	$-0.448389\pi$
$709$	8.59692	0.322864	0.161432	+	0.986884i	$0.448389\pi$
$710$	0	0
$711$	0	0
$712$	25.8617	0.969209
$713$	8.76452	0.328234
$714$	0	0
$715$	0	0
$716$	18.3152	0.684470
$717$	0	0
$718$	−13.3810	−0.499376
$719$	−36.4995	−1.36120	−0.680600	−	0.732655i	$-0.738281\pi$
$719$	−36.4995	−1.36120	−0.680600	+	0.732655i	$0.738281\pi$
$720$	0	0
$721$	35.7952	1.33308
$722$	21.6520	0.805806
$723$	0	0
$724$	2.18879	0.0813458
$725$	0	0
$726$	0	0
$727$	26.8954	0.997494	0.498747	−	0.866748i	$-0.333794\pi$
$727$	26.8954	0.997494	0.498747	+	0.866748i	$0.333794\pi$
$728$	−24.6745	−0.914498
$729$	0	0
$730$	0	0
$731$	0.147314	0.00544862
$732$	0	0
$733$	−25.7501	−0.951103	−0.475551	−	0.879688i	$-0.657751\pi$
$733$	−25.7501	−0.951103	−0.475551	+	0.879688i	$0.657751\pi$
$734$	−7.40477	−0.273315
$735$	0	0
$736$	15.8976	0.585993
$737$	14.3518	0.528656
$738$	0	0
$739$	−28.5152	−1.04895	−0.524474	−	0.851426i	$-0.675738\pi$
$739$	−28.5152	−1.04895	−0.524474	+	0.851426i	$0.675738\pi$
$740$	0	0
$741$	0	0
$742$	12.1428	0.445777
$743$	17.1761	0.630131	0.315066	−	0.949070i	$-0.397973\pi$
$743$	17.1761	0.630131	0.315066	+	0.949070i	$0.397973\pi$
$744$	0	0
$745$	0	0
$746$	−14.5662	−0.533307
$747$	0	0
$748$	−0.940794	−0.0343988
$749$	−4.25666	−0.155535
$750$	0	0
$751$	−43.6085	−1.59130	−0.795649	−	0.605758i	$-0.792870\pi$
$751$	−43.6085	−1.59130	−0.795649	+	0.605758i	$0.792870\pi$
$752$	19.5926	0.714467
$753$	0	0
$754$	−19.7901	−0.720712
$755$	0	0
$756$	0	0
$757$	17.3185	0.629452	0.314726	−	0.949183i	$-0.398087\pi$
$757$	17.3185	0.629452	0.314726	+	0.949183i	$0.398087\pi$
$758$	−0.709659	−0.0257760
$759$	0	0
$760$	0	0
$761$	37.9806	1.37680	0.688398	−	0.725333i	$-0.258314\pi$
$761$	37.9806	1.37680	0.688398	+	0.725333i	$0.258314\pi$
$762$	0	0
$763$	−66.5729	−2.41010
$764$	−21.3507	−0.772441
$765$	0	0
$766$	−8.18939	−0.295895
$767$	9.08662	0.328099
$768$	0	0
$769$	−13.9857	−0.504338	−0.252169	−	0.967683i	$-0.581144\pi$
$769$	−13.9857	−0.504338	−0.252169	+	0.967683i	$0.581144\pi$
$770$	0	0
$771$	0	0
$772$	15.9117	0.572675
$773$	13.4580	0.484052	0.242026	−	0.970270i	$-0.422188\pi$
$773$	13.4580	0.484052	0.242026	+	0.970270i	$0.422188\pi$
$774$	0	0
$775$	0	0
$776$	29.4420	1.05691
$777$	0	0
$778$	−8.71598	−0.312483
$779$	−4.25853	−0.152578
$780$	0	0
$781$	39.3223	1.40706
$782$	0.222544	0.00795815
$783$	0	0
$784$	13.7863	0.492367
$785$	0	0
$786$	0	0
$787$	−23.8454	−0.849998	−0.424999	−	0.905194i	$-0.639726\pi$
$787$	−23.8454	−0.849998	−0.424999	+	0.905194i	$0.639726\pi$
$788$	37.4888	1.33548
$789$	0	0
$790$	0	0
$791$	−3.21044	−0.114150
$792$	0	0
$793$	−46.4321	−1.64885
$794$	−4.07189	−0.144506
$795$	0	0
$796$	14.2863	0.506364
$797$	−30.5648	−1.08266	−0.541330	−	0.840810i	$-0.682079\pi$
$797$	−30.5648	−1.08266	−0.541330	+	0.840810i	$0.682079\pi$
$798$	0	0
$799$	0.994725	0.0351908
$800$	0	0
$801$	0	0
$802$	−12.0178	−0.424363
$803$	−19.5022	−0.688218
$804$	0	0
$805$	0	0
$806$	−4.76419	−0.167811
$807$	0	0
$808$	12.5516	0.441564
$809$	48.7186	1.71285	0.856427	−	0.516268i	$-0.172679\pi$
$809$	48.7186	1.71285	0.856427	+	0.516268i	$0.172679\pi$
$810$	0	0
$811$	−53.9242	−1.89354	−0.946768	−	0.321917i	$-0.895673\pi$
$811$	−53.9242	−1.89354	−0.946768	+	0.321917i	$0.895673\pi$
$812$	65.0479	2.28273
$813$	0	0
$814$	7.36585	0.258173
$815$	0	0
$816$	0	0
$817$	−8.29463	−0.290192
$818$	3.80530	0.133049
$819$	0	0
$820$	0	0
$821$	40.9746	1.43002	0.715011	−	0.699113i	$-0.246422\pi$
$821$	40.9746	1.43002	0.715011	+	0.699113i	$0.246422\pi$
$822$	0	0
$823$	15.9332	0.555397	0.277699	−	0.960668i	$-0.410428\pi$
$823$	15.9332	0.555397	0.277699	+	0.960668i	$0.410428\pi$
$824$	17.3013	0.602720
$825$	0	0
$826$	3.11745	0.108470
$827$	−3.73538	−0.129892	−0.0649460	−	0.997889i	$-0.520688\pi$
$827$	−3.73538	−0.129892	−0.0649460	+	0.997889i	$0.520688\pi$
$828$	0	0
$829$	14.0809	0.489051	0.244526	−	0.969643i	$-0.421368\pi$
$829$	14.0809	0.489051	0.244526	+	0.969643i	$0.421368\pi$
$830$	0	0
$831$	0	0
$832$	16.5683	0.574402
$833$	0.699937	0.0242514
$834$	0	0
$835$	0	0
$836$	52.9720	1.83208
$837$	0	0
$838$	−2.34308	−0.0809402
$839$	21.4882	0.741855	0.370928	−	0.928662i	$-0.379040\pi$
$839$	21.4882	0.741855	0.370928	+	0.928662i	$0.379040\pi$
$840$	0	0
$841$	80.7882	2.78580
$842$	−14.9685	−0.515847
$843$	0	0
$844$	34.0251	1.17119
$845$	0	0
$846$	0	0
$847$	−4.92214	−0.169127
$848$	−23.6400	−0.811801
$849$	0	0
$850$	0	0
$851$	16.6929	0.572225
$852$	0	0
$853$	−2.14185	−0.0733354	−0.0366677	−	0.999328i	$-0.511674\pi$
$853$	−2.14185	−0.0733354	−0.0366677	+	0.999328i	$0.511674\pi$
$854$	−15.9300	−0.545114
$855$	0	0
$856$	−2.05742	−0.0703213
$857$	18.4102	0.628881	0.314441	−	0.949277i	$-0.398183\pi$
$857$	18.4102	0.628881	0.314441	+	0.949277i	$0.398183\pi$
$858$	0	0
$859$	−17.0069	−0.580268	−0.290134	−	0.956986i	$-0.593700\pi$
$859$	−17.0069	−0.580268	−0.290134	+	0.956986i	$0.593700\pi$
$860$	0	0
$861$	0	0
$862$	6.71593	0.228745
$863$	54.0541	1.84002	0.920011	−	0.391891i	$-0.128179\pi$
$863$	54.0541	1.84002	0.920011	+	0.391891i	$0.128179\pi$
$864$	0	0
$865$	0	0
$866$	−7.17791	−0.243915
$867$	0	0
$868$	15.6594	0.531514
$869$	8.20335	0.278280
$870$	0	0
$871$	−17.6798	−0.599056
$872$	−32.1775	−1.08967
$873$	0	0
$874$	−12.5305	−0.423850
$875$	0	0
$876$	0	0
$877$	−18.8141	−0.635308	−0.317654	−	0.948207i	$-0.602895\pi$
$877$	−18.8141	−0.635308	−0.317654	+	0.948207i	$0.602895\pi$
$878$	−11.7565	−0.396762
$879$	0	0
$880$	0	0
$881$	16.0913	0.542129	0.271065	−	0.962561i	$-0.412624\pi$
$881$	16.0913	0.542129	0.271065	+	0.962561i	$0.412624\pi$
$882$	0	0
$883$	50.3274	1.69365	0.846825	−	0.531871i	$-0.178511\pi$
$883$	50.3274	1.69365	0.846825	+	0.531871i	$0.178511\pi$
$884$	1.15895	0.0389797
$885$	0	0
$886$	−0.801401	−0.0269236
$887$	−50.7671	−1.70459	−0.852297	−	0.523058i	$-0.824791\pi$
$887$	−50.7671	−1.70459	−0.852297	+	0.523058i	$0.824791\pi$
$888$	0	0
$889$	54.4126	1.82494
$890$	0	0
$891$	0	0
$892$	−23.6424	−0.791605
$893$	−56.0087	−1.87426
$894$	0	0
$895$	0	0
$896$	37.0530	1.23785
$897$	0	0
$898$	5.32483	0.177692
$899$	26.4300	0.881490
$900$	0	0
$901$	−1.20022	−0.0399850
$902$	0.787160	0.0262095
$903$	0	0
$904$	−1.55174	−0.0516101
$905$	0	0
$906$	0	0
$907$	−19.1876	−0.637115	−0.318558	−	0.947904i	$-0.603198\pi$
$907$	−19.1876	−0.637115	−0.318558	+	0.947904i	$0.603198\pi$
$908$	−39.1171	−1.29815
$909$	0	0
$910$	0	0
$911$	−8.14924	−0.269996	−0.134998	−	0.990846i	$-0.543103\pi$
$911$	−8.14924	−0.269996	−0.134998	+	0.990846i	$0.543103\pi$
$912$	0	0
$913$	−16.2396	−0.537453
$914$	−14.3277	−0.473917
$915$	0	0
$916$	−5.21461	−0.172295
$917$	−43.4496	−1.43483
$918$	0	0
$919$	12.3979	0.408969	0.204484	−	0.978870i	$-0.434448\pi$
$919$	12.3979	0.408969	0.204484	+	0.978870i	$0.434448\pi$
$920$	0	0
$921$	0	0
$922$	−9.17320	−0.302103
$923$	−48.4405	−1.59444
$924$	0	0
$925$	0	0
$926$	−10.2904	−0.338165
$927$	0	0
$928$	47.9403	1.57372
$929$	13.5123	0.443324	0.221662	−	0.975124i	$-0.428852\pi$
$929$	13.5123	0.443324	0.221662	+	0.975124i	$0.428852\pi$
$930$	0	0
$931$	−39.4104	−1.29162
$932$	−41.2335	−1.35065
$933$	0	0
$934$	7.77313	0.254344
$935$	0	0
$936$	0	0
$937$	12.6732	0.414017	0.207008	−	0.978339i	$-0.433627\pi$
$937$	12.6732	0.414017	0.207008	+	0.978339i	$0.433627\pi$
$938$	−6.06561	−0.198049
$939$	0	0
$940$	0	0
$941$	−43.4722	−1.41715	−0.708576	−	0.705635i	$-0.750662\pi$
$941$	−43.4722	−1.41715	−0.708576	+	0.705635i	$0.750662\pi$
$942$	0	0
$943$	1.78391	0.0580919
$944$	−6.06915	−0.197534
$945$	0	0
$946$	1.53320	0.0498487
$947$	−13.6061	−0.442139	−0.221070	−	0.975258i	$-0.570955\pi$
$947$	−13.6061	−0.442139	−0.221070	+	0.975258i	$0.570955\pi$
$948$	0	0
$949$	24.0245	0.779867
$950$	0	0
$951$	0	0
$952$	0.836731	0.0271186
$953$	−15.7134	−0.509007	−0.254503	−	0.967072i	$-0.581912\pi$
$953$	−15.7134	−0.509007	−0.254503	+	0.967072i	$0.581912\pi$
$954$	0	0
$955$	0	0
$956$	−38.8851	−1.25763
$957$	0	0
$958$	5.55784	0.179566
$959$	−11.9862	−0.387056
$960$	0	0
$961$	−24.6373	−0.794753
$962$	−9.07387	−0.292553
$963$	0	0
$964$	4.34320	0.139885
$965$	0	0
$966$	0	0
$967$	−32.8964	−1.05788	−0.528939	−	0.848660i	$-0.677410\pi$
$967$	−32.8964	−1.05788	−0.528939	+	0.848660i	$0.677410\pi$
$968$	−2.37908	−0.0764664
$969$	0	0
$970$	0	0
$971$	−20.8625	−0.669510	−0.334755	−	0.942305i	$-0.608654\pi$
$971$	−20.8625	−0.669510	−0.334755	+	0.942305i	$0.608654\pi$
$972$	0	0
$973$	−10.2747	−0.329391
$974$	1.43139	0.0458646
$975$	0	0
$976$	31.0130	0.992702
$977$	56.8435	1.81858	0.909292	−	0.416160i	$-0.136624\pi$
$977$	56.8435	1.81858	0.909292	+	0.416160i	$0.136624\pi$
$978$	0	0
$979$	−55.0424	−1.75916
$980$	0	0
$981$	0	0
$982$	4.89893	0.156331
$983$	58.0603	1.85184	0.925918	−	0.377725i	$-0.123294\pi$
$983$	58.0603	1.85184	0.925918	+	0.377725i	$0.123294\pi$
$984$	0	0
$985$	0	0
$986$	0.671096	0.0213720
$987$	0	0
$988$	−65.2554	−2.07605
$989$	3.47463	0.110487
$990$	0	0
$991$	−30.1239	−0.956916	−0.478458	−	0.878111i	$-0.658804\pi$
$991$	−30.1239	−0.956916	−0.478458	+	0.878111i	$0.658804\pi$
$992$	11.5410	0.366426
$993$	0	0
$994$	−16.6190	−0.527124
$995$	0	0
$996$	0	0
$997$	−30.3566	−0.961405	−0.480702	−	0.876884i	$-0.659618\pi$
$997$	−30.3566	−0.961405	−0.480702	+	0.876884i	$0.659618\pi$
$998$	−12.4841	−0.395177
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9675.2.a.ch.1.3		5
3.2	odd	2		1075.2.a.m.1.3		5
5.4	even	2		1935.2.a.u.1.3		5
15.2	even	4		1075.2.b.h.474.5		10
15.8	even	4		1075.2.b.h.474.6		10
15.14	odd	2		215.2.a.c.1.3	✓	5
60.59	even	2		3440.2.a.w.1.2		5
645.644	even	2		9245.2.a.l.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
215.2.a.c.1.3	✓	5	15.14	odd	2
1075.2.a.m.1.3		5	3.2	odd	2
1075.2.b.h.474.5		10	15.2	even	4
1075.2.b.h.474.6		10	15.8	even	4
1935.2.a.u.1.3		5	5.4	even	2
3440.2.a.w.1.2		5	60.59	even	2
9245.2.a.l.1.3		5	645.644	even	2
9675.2.a.ch.1.3		5	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 9675 = 3^{2} \cdot 5^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9675.a (trivial)

\(f(q)\)	\(=\)	\(q+0.434772 q^{2} -1.81097 q^{4} -3.42802 q^{7} -1.65691 q^{8} +O(q^{10})\)	\(q+0.434772 q^{2} -1.81097 q^{4} -3.42802 q^{7} -1.65691 q^{8} +3.52645 q^{11} -4.34418 q^{13} -1.49041 q^{14} +2.90157 q^{16} +0.147314 q^{17} -8.29463 q^{19} +1.53320 q^{22} +3.47463 q^{23} -1.88873 q^{26} +6.20805 q^{28} +10.4780 q^{29} +2.52243 q^{31} +4.57533 q^{32} +0.0640482 q^{34} +4.80422 q^{37} -3.60628 q^{38} +0.513408 q^{41} +1.00000 q^{43} -6.38631 q^{44} +1.51067 q^{46} +6.75240 q^{47} +4.75132 q^{49} +7.86719 q^{52} -8.14731 q^{53} +5.67991 q^{56} +4.55554 q^{58} -2.09168 q^{59} +10.6884 q^{61} +1.09668 q^{62} -3.81391 q^{64} +4.06976 q^{67} -0.266782 q^{68} +11.1507 q^{71} -5.53027 q^{73} +2.08874 q^{74} +15.0213 q^{76} -12.0887 q^{77} +2.32623 q^{79} +0.223216 q^{82} -4.60509 q^{83} +0.434772 q^{86} -5.84300 q^{88} -15.6084 q^{89} +14.8919 q^{91} -6.29246 q^{92} +2.93576 q^{94} -17.7693 q^{97} +2.06574 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 2 q^{2} + 8 q^{4} - 5 q^{7} + 3 q^{8}+O(q^{10}) \) 5 * q + 2 * q^2 + 8 * q^4 - 5 * q^7 + 3 * q^8	\( 5 q + 2 q^{2} + 8 q^{4} - 5 q^{7} + 3 q^{8} + 6 q^{11} - 5 q^{13} - q^{14} + 14 q^{16} - 17 q^{17} - 6 q^{19} + 8 q^{22} + q^{23} - 22 q^{26} - 26 q^{28} - 6 q^{29} + 6 q^{31} - 7 q^{32} - 5 q^{37} - 16 q^{38} - 2 q^{41} + 5 q^{43} + 15 q^{44} - 14 q^{46} + 18 q^{49} + 38 q^{52} - 23 q^{53} + 19 q^{56} - 12 q^{58} + q^{59} + 20 q^{61} - 3 q^{62} - 25 q^{64} - 21 q^{67} - 48 q^{68} - 4 q^{71} - 5 q^{73} - 24 q^{74} + 32 q^{76} - 26 q^{77} + 41 q^{79} - 38 q^{82} - 7 q^{83} + 2 q^{86} - 12 q^{88} - 20 q^{89} - 42 q^{91} - 52 q^{92} - 42 q^{94} - 37 q^{97} - 26 q^{98}+O(q^{100}) \) 5 * q + 2 * q^2 + 8 * q^4 - 5 * q^7 + 3 * q^8 + 6 * q^11 - 5 * q^13 - q^14 + 14 * q^16 - 17 * q^17 - 6 * q^19 + 8 * q^22 + q^23 - 22 * q^26 - 26 * q^28 - 6 * q^29 + 6 * q^31 - 7 * q^32 - 5 * q^37 - 16 * q^38 - 2 * q^41 + 5 * q^43 + 15 * q^44 - 14 * q^46 + 18 * q^49 + 38 * q^52 - 23 * q^53 + 19 * q^56 - 12 * q^58 + q^59 + 20 * q^61 - 3 * q^62 - 25 * q^64 - 21 * q^67 - 48 * q^68 - 4 * q^71 - 5 * q^73 - 24 * q^74 + 32 * q^76 - 26 * q^77 + 41 * q^79 - 38 * q^82 - 7 * q^83 + 2 * q^86 - 12 * q^88 - 20 * q^89 - 42 * q^91 - 52 * q^92 - 42 * q^94 - 37 * q^97 - 26 * q^98

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