LMFDB - Embedded newform 9196.2.a.m.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9196,2,Mod(1,9196)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9196, 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("9196.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9196 = 2^{2} \cdot 11^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9196.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:	$73.4304296988$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.114134848.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 10x^{4} - 6x^{3} + 20x^{2} + 14x - 5 $ x^6 - 10x^4 - 6x^3 + 20x^2 + 14x - 5
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.267039$ of defining polynomial
Character	$\chi$	$=$	9196.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.267039 q^{3} +0.518487 q^{5} +5.09697 q^{7} -2.92869 q^{9} +O(q^{10})$	$q-0.267039 q^{3} +0.518487 q^{5} +5.09697 q^{7} -2.92869 q^{9} -4.11971 q^{13} -0.138456 q^{15} +4.81298 q^{17} -1.00000 q^{19} -1.36109 q^{21} +5.98125 q^{23} -4.73117 q^{25} +1.58319 q^{27} -5.99113 q^{29} +3.95035 q^{31} +2.64271 q^{35} +7.25709 q^{37} +1.10012 q^{39} +5.80963 q^{41} -8.51461 q^{43} -1.51849 q^{45} -10.2394 q^{47} +18.9791 q^{49} -1.28525 q^{51} -2.04548 q^{53} +0.267039 q^{57} +11.0257 q^{59} +3.27039 q^{61} -14.9274 q^{63} -2.13601 q^{65} +9.52141 q^{67} -1.59723 q^{69} -16.5953 q^{71} +3.49710 q^{73} +1.26341 q^{75} -7.72909 q^{79} +8.36330 q^{81} +14.7529 q^{83} +2.49546 q^{85} +1.59987 q^{87} +1.01624 q^{89} -20.9980 q^{91} -1.05490 q^{93} -0.518487 q^{95} +8.98233 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{5} + 4 q^{7} + 2 q^{9}+O(q^{10}) $ 6 * q - 4 * q^5 + 4 * q^7 + 2 * q^9	$ 6 q - 4 q^{5} + 4 q^{7} + 2 q^{9} + 8 q^{13} + 2 q^{15} + 6 q^{17} - 6 q^{19} - 20 q^{21} + 6 q^{23} + 2 q^{25} - 18 q^{27} - 2 q^{29} - 6 q^{31} + 14 q^{35} + 4 q^{37} + 6 q^{39} + 14 q^{41} - 10 q^{43} - 2 q^{45} + 4 q^{47} + 20 q^{49} + 18 q^{51} + 20 q^{59} - 2 q^{61} + 12 q^{63} - 12 q^{65} + 10 q^{67} - 20 q^{69} + 32 q^{73} + 18 q^{75} + 6 q^{79} - 6 q^{81} + 36 q^{83} + 6 q^{85} - 28 q^{87} - 24 q^{89} - 16 q^{91} + 22 q^{93} + 4 q^{95} + 28 q^{97}+O(q^{100}) $ 6 * q - 4 * q^5 + 4 * q^7 + 2 * q^9 + 8 * q^13 + 2 * q^15 + 6 * q^17 - 6 * q^19 - 20 * q^21 + 6 * q^23 + 2 * q^25 - 18 * q^27 - 2 * q^29 - 6 * q^31 + 14 * q^35 + 4 * q^37 + 6 * q^39 + 14 * q^41 - 10 * q^43 - 2 * q^45 + 4 * q^47 + 20 * q^49 + 18 * q^51 + 20 * q^59 - 2 * q^61 + 12 * q^63 - 12 * q^65 + 10 * q^67 - 20 * q^69 + 32 * q^73 + 18 * q^75 + 6 * q^79 - 6 * q^81 + 36 * q^83 + 6 * q^85 - 28 * q^87 - 24 * q^89 - 16 * q^91 + 22 * q^93 + 4 * q^95 + 28 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−0.267039	−0.154175	−0.0770875	−	0.997024i	$-0.524562\pi$
$3$	−0.267039	−0.154175	−0.0770875	+	0.997024i	$0.524562\pi$
$4$	0	0
$5$	0.518487	0.231874	0.115937	−	0.993257i	$-0.463013\pi$
$5$	0.518487	0.231874	0.115937	+	0.993257i	$0.463013\pi$
$6$	0	0
$7$	5.09697	1.92647	0.963237	−	0.268654i	$-0.0865788\pi$
$7$	5.09697	1.92647	0.963237	+	0.268654i	$0.0865788\pi$
$8$	0	0
$9$	−2.92869	−0.976230
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−4.11971	−1.14260	−0.571301	−	0.820741i	$-0.693561\pi$
$13$	−4.11971	−1.14260	−0.571301	+	0.820741i	$0.693561\pi$
$14$	0	0
$15$	−0.138456	−0.0357492
$16$	0	0
$17$	4.81298	1.16732	0.583659	−	0.811999i	$-0.301621\pi$
$17$	4.81298	1.16732	0.583659	+	0.811999i	$0.301621\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−1.36109	−0.297014
$22$	0	0
$23$	5.98125	1.24718	0.623589	−	0.781752i	$-0.285674\pi$
$23$	5.98125	1.24718	0.623589	+	0.781752i	$0.285674\pi$
$24$	0	0
$25$	−4.73117	−0.946234
$26$	0	0
$27$	1.58319	0.304685
$28$	0	0
$29$	−5.99113	−1.11252	−0.556262	−	0.831007i	$-0.687765\pi$
$29$	−5.99113	−1.11252	−0.556262	+	0.831007i	$0.687765\pi$
$30$	0	0
$31$	3.95035	0.709505	0.354752	−	0.934960i	$-0.384565\pi$
$31$	3.95035	0.709505	0.354752	+	0.934960i	$0.384565\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.64271	0.446700
$36$	0	0
$37$	7.25709	1.19306	0.596529	−	0.802591i	$-0.296546\pi$
$37$	7.25709	1.19306	0.596529	+	0.802591i	$0.296546\pi$
$38$	0	0
$39$	1.10012	0.176161
$40$	0	0
$41$	5.80963	0.907312	0.453656	−	0.891177i	$-0.350119\pi$
$41$	5.80963	0.907312	0.453656	+	0.891177i	$0.350119\pi$
$42$	0	0
$43$	−8.51461	−1.29847	−0.649233	−	0.760590i	$-0.724910\pi$
$43$	−8.51461	−1.29847	−0.649233	+	0.760590i	$0.724910\pi$
$44$	0	0
$45$	−1.51849	−0.226363
$46$	0	0
$47$	−10.2394	−1.49357	−0.746787	−	0.665064i	$-0.768404\pi$
$47$	−10.2394	−1.49357	−0.746787	+	0.665064i	$0.768404\pi$
$48$	0	0
$49$	18.9791	2.71130
$50$	0	0
$51$	−1.28525	−0.179971
$52$	0	0
$53$	−2.04548	−0.280969	−0.140484	−	0.990083i	$-0.544866\pi$
$53$	−2.04548	−0.280969	−0.140484	+	0.990083i	$0.544866\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0.267039	0.0353702
$58$	0	0
$59$	11.0257	1.43542	0.717709	−	0.696343i	$-0.245191\pi$
$59$	11.0257	1.43542	0.717709	+	0.696343i	$0.245191\pi$
$60$	0	0
$61$	3.27039	0.418730	0.209365	−	0.977838i	$-0.432860\pi$
$61$	3.27039	0.418730	0.209365	+	0.977838i	$0.432860\pi$
$62$	0	0
$63$	−14.9274	−1.88068
$64$	0	0
$65$	−2.13601	−0.264940
$66$	0	0
$67$	9.52141	1.16322	0.581612	−	0.813466i	$-0.302422\pi$
$67$	9.52141	1.16322	0.581612	+	0.813466i	$0.302422\pi$
$68$	0	0
$69$	−1.59723	−0.192284
$70$	0	0
$71$	−16.5953	−1.96950	−0.984748	−	0.173986i	$-0.944335\pi$
$71$	−16.5953	−1.96950	−0.984748	+	0.173986i	$0.944335\pi$
$72$	0	0
$73$	3.49710	0.409305	0.204653	−	0.978835i	$-0.434394\pi$
$73$	3.49710	0.409305	0.204653	+	0.978835i	$0.434394\pi$
$74$	0	0
$75$	1.26341	0.145886
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−7.72909	−0.869591	−0.434796	−	0.900529i	$-0.643179\pi$
$79$	−7.72909	−0.869591	−0.434796	+	0.900529i	$0.643179\pi$
$80$	0	0
$81$	8.36330	0.929255
$82$	0	0
$83$	14.7529	1.61934	0.809671	−	0.586884i	$-0.199646\pi$
$83$	14.7529	1.61934	0.809671	+	0.586884i	$0.199646\pi$
$84$	0	0
$85$	2.49546	0.270671
$86$	0	0
$87$	1.59987	0.171524
$88$	0	0
$89$	1.01624	0.107722	0.0538608	−	0.998548i	$-0.482847\pi$
$89$	1.01624	0.107722	0.0538608	+	0.998548i	$0.482847\pi$
$90$	0	0
$91$	−20.9980	−2.20119
$92$	0	0
$93$	−1.05490	−0.109388
$94$	0	0
$95$	−0.518487	−0.0531956
$96$	0	0
$97$	8.98233	0.912018	0.456009	−	0.889975i	$-0.349279\pi$
$97$	8.98233	0.912018	0.456009	+	0.889975i	$0.349279\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−0.295216	−0.0293751	−0.0146875	−	0.999892i	$-0.504675\pi$
$101$	−0.295216	−0.0293751	−0.0146875	+	0.999892i	$0.504675\pi$
$102$	0	0
$103$	−7.53355	−0.742302	−0.371151	−	0.928572i	$-0.621037\pi$
$103$	−7.53355	−0.742302	−0.371151	+	0.928572i	$0.621037\pi$
$104$	0	0
$105$	−0.705707	−0.0688699
$106$	0	0
$107$	14.7117	1.42223	0.711117	−	0.703074i	$-0.248190\pi$
$107$	14.7117	1.42223	0.711117	+	0.703074i	$0.248190\pi$
$108$	0	0
$109$	19.4540	1.86336	0.931678	−	0.363284i	$-0.118345\pi$
$109$	19.4540	1.86336	0.931678	+	0.363284i	$0.118345\pi$
$110$	0	0
$111$	−1.93793	−0.183940
$112$	0	0
$113$	19.6846	1.85177	0.925886	−	0.377802i	$-0.123320\pi$
$113$	19.6846	1.85177	0.925886	+	0.377802i	$0.123320\pi$
$114$	0	0
$115$	3.10120	0.289188
$116$	0	0
$117$	12.0654	1.11544
$118$	0	0
$119$	24.5316	2.24881
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−1.55140	−0.139885
$124$	0	0
$125$	−5.04548	−0.451282
$126$	0	0
$127$	−7.32443	−0.649938	−0.324969	−	0.945725i	$-0.605354\pi$
$127$	−7.32443	−0.649938	−0.324969	+	0.945725i	$0.605354\pi$
$128$	0	0
$129$	2.27373	0.200191
$130$	0	0
$131$	−10.6849	−0.933541	−0.466771	−	0.884378i	$-0.654583\pi$
$131$	−10.6849	−0.933541	−0.466771	+	0.884378i	$0.654583\pi$
$132$	0	0
$133$	−5.09697	−0.441963
$134$	0	0
$135$	0.820864	0.0706487
$136$	0	0
$137$	−9.82214	−0.839162	−0.419581	−	0.907718i	$-0.637823\pi$
$137$	−9.82214	−0.839162	−0.419581	+	0.907718i	$0.637823\pi$
$138$	0	0
$139$	−0.816129	−0.0692231	−0.0346116	−	0.999401i	$-0.511019\pi$
$139$	−0.816129	−0.0692231	−0.0346116	+	0.999401i	$0.511019\pi$
$140$	0	0
$141$	2.73433	0.230272
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−3.10632	−0.257966
$146$	0	0
$147$	−5.06816	−0.418015
$148$	0	0
$149$	1.95452	0.160120	0.0800602	−	0.996790i	$-0.474489\pi$
$149$	1.95452	0.160120	0.0800602	+	0.996790i	$0.474489\pi$
$150$	0	0
$151$	12.3481	1.00488	0.502438	−	0.864613i	$-0.332436\pi$
$151$	12.3481	1.00488	0.502438	+	0.864613i	$0.332436\pi$
$152$	0	0
$153$	−14.0957	−1.13957
$154$	0	0
$155$	2.04821	0.164516
$156$	0	0
$157$	18.2012	1.45261	0.726307	−	0.687370i	$-0.241235\pi$
$157$	18.2012	1.45261	0.726307	+	0.687370i	$0.241235\pi$
$158$	0	0
$159$	0.546224	0.0433183
$160$	0	0
$161$	30.4863	2.40266
$162$	0	0
$163$	−19.3252	−1.51366	−0.756832	−	0.653609i	$-0.773254\pi$
$163$	−19.3252	−1.51366	−0.756832	+	0.653609i	$0.773254\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	21.0979	1.63260	0.816301	−	0.577626i	$-0.196021\pi$
$167$	21.0979	1.63260	0.816301	+	0.577626i	$0.196021\pi$
$168$	0	0
$169$	3.97202	0.305540
$170$	0	0
$171$	2.92869	0.223963
$172$	0	0
$173$	−6.51306	−0.495179	−0.247589	−	0.968865i	$-0.579638\pi$
$173$	−6.51306	−0.495179	−0.247589	+	0.968865i	$0.579638\pi$
$174$	0	0
$175$	−24.1146	−1.82290
$176$	0	0
$177$	−2.94428	−0.221306
$178$	0	0
$179$	3.69464	0.276151	0.138075	−	0.990422i	$-0.455908\pi$
$179$	3.69464	0.276151	0.138075	+	0.990422i	$0.455908\pi$
$180$	0	0
$181$	2.44337	0.181614	0.0908072	−	0.995868i	$-0.471055\pi$
$181$	2.44337	0.181614	0.0908072	+	0.995868i	$0.471055\pi$
$182$	0	0
$183$	−0.873321	−0.0645578
$184$	0	0
$185$	3.76270	0.276640
$186$	0	0
$187$	0	0
$188$	0	0
$189$	8.06948	0.586968
$190$	0	0
$191$	−10.3633	−0.749862	−0.374931	−	0.927053i	$-0.622334\pi$
$191$	−10.3633	−0.749862	−0.374931	+	0.927053i	$0.622334\pi$
$192$	0	0
$193$	17.0613	1.22810	0.614049	−	0.789268i	$-0.289540\pi$
$193$	17.0613	1.22810	0.614049	+	0.789268i	$0.289540\pi$
$194$	0	0
$195$	0.570399	0.0408471
$196$	0	0
$197$	−14.6439	−1.04334	−0.521668	−	0.853149i	$-0.674690\pi$
$197$	−14.6439	−1.04334	−0.521668	+	0.853149i	$0.674690\pi$
$198$	0	0
$199$	−7.67961	−0.544393	−0.272197	−	0.962242i	$-0.587750\pi$
$199$	−7.67961	−0.544393	−0.272197	+	0.962242i	$0.587750\pi$
$200$	0	0
$201$	−2.54259	−0.179340
$202$	0	0
$203$	−30.5366	−2.14325
$204$	0	0
$205$	3.01222	0.210382
$206$	0	0
$207$	−17.5172	−1.21753
$208$	0	0
$209$	0	0
$210$	0	0
$211$	19.3484	1.33200	0.665998	−	0.745954i	$-0.268006\pi$
$211$	19.3484	1.33200	0.665998	+	0.745954i	$0.268006\pi$
$212$	0	0
$213$	4.43159	0.303647
$214$	0	0
$215$	−4.41471	−0.301081
$216$	0	0
$217$	20.1348	1.36684
$218$	0	0
$219$	−0.933864	−0.0631046
$220$	0	0
$221$	−19.8281	−1.33378
$222$	0	0
$223$	15.6458	1.04772	0.523861	−	0.851804i	$-0.324491\pi$
$223$	15.6458	1.04772	0.523861	+	0.851804i	$0.324491\pi$
$224$	0	0
$225$	13.8561	0.923742
$226$	0	0
$227$	17.3623	1.15238	0.576188	−	0.817317i	$-0.304539\pi$
$227$	17.3623	1.15238	0.576188	+	0.817317i	$0.304539\pi$
$228$	0	0
$229$	−11.5358	−0.762308	−0.381154	−	0.924512i	$-0.624473\pi$
$229$	−11.5358	−0.762308	−0.381154	+	0.924512i	$0.624473\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−4.19411	−0.274765	−0.137383	−	0.990518i	$-0.543869\pi$
$233$	−4.19411	−0.274765	−0.137383	+	0.990518i	$0.543869\pi$
$234$	0	0
$235$	−5.30900	−0.346321
$236$	0	0
$237$	2.06397	0.134069
$238$	0	0
$239$	21.4307	1.38624	0.693119	−	0.720823i	$-0.256236\pi$
$239$	21.4307	1.38624	0.693119	+	0.720823i	$0.256236\pi$
$240$	0	0
$241$	−19.4248	−1.25126	−0.625632	−	0.780118i	$-0.715159\pi$
$241$	−19.4248	−1.25126	−0.625632	+	0.780118i	$0.715159\pi$
$242$	0	0
$243$	−6.98290	−0.447953
$244$	0	0
$245$	9.84041	0.628681
$246$	0	0
$247$	4.11971	0.262131
$248$	0	0
$249$	−3.93960	−0.249662
$250$	0	0
$251$	−13.1137	−0.827731	−0.413866	−	0.910338i	$-0.635822\pi$
$251$	−13.1137	−0.827731	−0.413866	+	0.910338i	$0.635822\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−0.666386	−0.0417307
$256$	0	0
$257$	5.83037	0.363689	0.181844	−	0.983327i	$-0.441793\pi$
$257$	5.83037	0.363689	0.181844	+	0.983327i	$0.441793\pi$
$258$	0	0
$259$	36.9892	2.29840
$260$	0	0
$261$	17.5462	1.08608
$262$	0	0
$263$	14.9657	0.922825	0.461413	−	0.887186i	$-0.347343\pi$
$263$	14.9657	0.922825	0.461413	+	0.887186i	$0.347343\pi$
$264$	0	0
$265$	−1.06056	−0.0651494
$266$	0	0
$267$	−0.271377	−0.0166080
$268$	0	0
$269$	−4.30265	−0.262337	−0.131169	−	0.991360i	$-0.541873\pi$
$269$	−4.30265	−0.262337	−0.131169	+	0.991360i	$0.541873\pi$
$270$	0	0
$271$	18.9170	1.14913	0.574564	−	0.818460i	$-0.305172\pi$
$271$	18.9170	1.14913	0.574564	+	0.818460i	$0.305172\pi$
$272$	0	0
$273$	5.60730	0.339369
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−25.6733	−1.54256	−0.771280	−	0.636497i	$-0.780383\pi$
$277$	−25.6733	−1.54256	−0.771280	+	0.636497i	$0.780383\pi$
$278$	0	0
$279$	−11.5694	−0.692640
$280$	0	0
$281$	−10.0671	−0.600550	−0.300275	−	0.953853i	$-0.597079\pi$
$281$	−10.0671	−0.600550	−0.300275	+	0.953853i	$0.597079\pi$
$282$	0	0
$283$	−10.2052	−0.606634	−0.303317	−	0.952890i	$-0.598094\pi$
$283$	−10.2052	−0.606634	−0.303317	+	0.952890i	$0.598094\pi$
$284$	0	0
$285$	0.138456	0.00820144
$286$	0	0
$287$	29.6115	1.74791
$288$	0	0
$289$	6.16473	0.362631
$290$	0	0
$291$	−2.39863	−0.140610
$292$	0	0
$293$	7.45349	0.435438	0.217719	−	0.976012i	$-0.430138\pi$
$293$	7.45349	0.435438	0.217719	+	0.976012i	$0.430138\pi$
$294$	0	0
$295$	5.71666	0.332837
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−24.6410	−1.42503
$300$	0	0
$301$	−43.3987	−2.50146
$302$	0	0
$303$	0.0788341	0.00452890
$304$	0	0
$305$	1.69565	0.0970928
$306$	0	0
$307$	19.5117	1.11359	0.556795	−	0.830650i	$-0.312031\pi$
$307$	19.5117	1.11359	0.556795	+	0.830650i	$0.312031\pi$
$308$	0	0
$309$	2.01175	0.114445
$310$	0	0
$311$	17.6341	0.999940	0.499970	−	0.866043i	$-0.333344\pi$
$311$	17.6341	0.999940	0.499970	+	0.866043i	$0.333344\pi$
$312$	0	0
$313$	27.5381	1.55655	0.778273	−	0.627926i	$-0.216096\pi$
$313$	27.5381	1.55655	0.778273	+	0.627926i	$0.216096\pi$
$314$	0	0
$315$	−7.73968	−0.436082
$316$	0	0
$317$	−29.1807	−1.63895	−0.819474	−	0.573116i	$-0.805735\pi$
$317$	−29.1807	−1.63895	−0.819474	+	0.573116i	$0.805735\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−3.92860	−0.219273
$322$	0	0
$323$	−4.81298	−0.267801
$324$	0	0
$325$	19.4911	1.08117
$326$	0	0
$327$	−5.19498	−0.287283
$328$	0	0
$329$	−52.1900	−2.87733
$330$	0	0
$331$	−20.8204	−1.14439	−0.572196	−	0.820117i	$-0.693908\pi$
$331$	−20.8204	−1.14439	−0.572196	+	0.820117i	$0.693908\pi$
$332$	0	0
$333$	−21.2538	−1.16470
$334$	0	0
$335$	4.93672	0.269722
$336$	0	0
$337$	7.39226	0.402682	0.201341	−	0.979521i	$-0.435470\pi$
$337$	7.39226	0.402682	0.201341	+	0.979521i	$0.435470\pi$
$338$	0	0
$339$	−5.25656	−0.285497
$340$	0	0
$341$	0	0
$342$	0	0
$343$	61.0571	3.29677
$344$	0	0
$345$	−0.828142	−0.0445856
$346$	0	0
$347$	−18.7257	−1.00525	−0.502624	−	0.864505i	$-0.667632\pi$
$347$	−18.7257	−1.00525	−0.502624	+	0.864505i	$0.667632\pi$
$348$	0	0
$349$	−3.02466	−0.161906	−0.0809531	−	0.996718i	$-0.525796\pi$
$349$	−3.02466	−0.161906	−0.0809531	+	0.996718i	$0.525796\pi$
$350$	0	0
$351$	−6.52229	−0.348134
$352$	0	0
$353$	13.1701	0.700974	0.350487	−	0.936568i	$-0.386016\pi$
$353$	13.1701	0.700974	0.350487	+	0.936568i	$0.386016\pi$
$354$	0	0
$355$	−8.60443	−0.456676
$356$	0	0
$357$	−6.55089	−0.346710
$358$	0	0
$359$	3.00617	0.158660	0.0793299	−	0.996848i	$-0.474722\pi$
$359$	3.00617	0.158660	0.0793299	+	0.996848i	$0.474722\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	1.81320	0.0949073
$366$	0	0
$367$	−18.3380	−0.957233	−0.478617	−	0.878024i	$-0.658862\pi$
$367$	−18.3380	−0.957233	−0.478617	+	0.878024i	$0.658862\pi$
$368$	0	0
$369$	−17.0146	−0.885745
$370$	0	0
$371$	−10.4258	−0.541279
$372$	0	0
$373$	9.36579	0.484942	0.242471	−	0.970159i	$-0.422042\pi$
$373$	9.36579	0.484942	0.242471	+	0.970159i	$0.422042\pi$
$374$	0	0
$375$	1.34734	0.0695764
$376$	0	0
$377$	24.6817	1.27117
$378$	0	0
$379$	−6.53349	−0.335603	−0.167802	−	0.985821i	$-0.553667\pi$
$379$	−6.53349	−0.335603	−0.167802	+	0.985821i	$0.553667\pi$
$380$	0	0
$381$	1.95591	0.100204
$382$	0	0
$383$	−11.7921	−0.602550	−0.301275	−	0.953537i	$-0.597412\pi$
$383$	−11.7921	−0.602550	−0.301275	+	0.953537i	$0.597412\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	24.9366	1.26760
$388$	0	0
$389$	10.7593	0.545520	0.272760	−	0.962082i	$-0.412064\pi$
$389$	10.7593	0.545520	0.272760	+	0.962082i	$0.412064\pi$
$390$	0	0
$391$	28.7876	1.45585
$392$	0	0
$393$	2.85328	0.143929
$394$	0	0
$395$	−4.00743	−0.201636
$396$	0	0
$397$	−13.9879	−0.702033	−0.351017	−	0.936369i	$-0.614164\pi$
$397$	−13.9879	−0.702033	−0.351017	+	0.936369i	$0.614164\pi$
$398$	0	0
$399$	1.36109	0.0681397
$400$	0	0
$401$	−4.87064	−0.243228	−0.121614	−	0.992577i	$-0.538807\pi$
$401$	−4.87064	−0.243228	−0.121614	+	0.992577i	$0.538807\pi$
$402$	0	0
$403$	−16.2743	−0.810682
$404$	0	0
$405$	4.33626	0.215470
$406$	0	0
$407$	0	0
$408$	0	0
$409$	26.3790	1.30436	0.652179	−	0.758065i	$-0.273855\pi$
$409$	26.3790	1.30436	0.652179	+	0.758065i	$0.273855\pi$
$410$	0	0
$411$	2.62289	0.129378
$412$	0	0
$413$	56.1975	2.76530
$414$	0	0
$415$	7.64918	0.375484
$416$	0	0
$417$	0.217938	0.0106725
$418$	0	0
$419$	−3.11450	−0.152153	−0.0760766	−	0.997102i	$-0.524239\pi$
$419$	−3.11450	−0.152153	−0.0760766	+	0.997102i	$0.524239\pi$
$420$	0	0
$421$	−32.9808	−1.60739	−0.803694	−	0.595043i	$-0.797135\pi$
$421$	−32.9808	−1.60739	−0.803694	+	0.595043i	$0.797135\pi$
$422$	0	0
$423$	29.9881	1.45807
$424$	0	0
$425$	−22.7710	−1.10456
$426$	0	0
$427$	16.6691	0.806673
$428$	0	0
$429$	0	0
$430$	0	0
$431$	11.7394	0.565467	0.282734	−	0.959198i	$-0.408759\pi$
$431$	11.7394	0.565467	0.282734	+	0.959198i	$0.408759\pi$
$432$	0	0
$433$	29.1256	1.39969	0.699843	−	0.714297i	$-0.253253\pi$
$433$	29.1256	1.39969	0.699843	+	0.714297i	$0.253253\pi$
$434$	0	0
$435$	0.829509	0.0397719
$436$	0	0
$437$	−5.98125	−0.286122
$438$	0	0
$439$	−24.4850	−1.16860	−0.584302	−	0.811536i	$-0.698632\pi$
$439$	−24.4850	−1.16860	−0.584302	+	0.811536i	$0.698632\pi$
$440$	0	0
$441$	−55.5839	−2.64685
$442$	0	0
$443$	9.63442	0.457745	0.228873	−	0.973456i	$-0.426496\pi$
$443$	9.63442	0.457745	0.228873	+	0.973456i	$0.426496\pi$
$444$	0	0
$445$	0.526909	0.0249779
$446$	0	0
$447$	−0.521933	−0.0246866
$448$	0	0
$449$	20.6096	0.972626	0.486313	−	0.873785i	$-0.338341\pi$
$449$	20.6096	0.972626	0.486313	+	0.873785i	$0.338341\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−3.29743	−0.154927
$454$	0	0
$455$	−10.8872	−0.510400
$456$	0	0
$457$	−18.5937	−0.869776	−0.434888	−	0.900485i	$-0.643212\pi$
$457$	−18.5937	−0.869776	−0.434888	+	0.900485i	$0.643212\pi$
$458$	0	0
$459$	7.61986	0.355665
$460$	0	0
$461$	−24.1919	−1.12673	−0.563366	−	0.826208i	$-0.690494\pi$
$461$	−24.1919	−1.12673	−0.563366	+	0.826208i	$0.690494\pi$
$462$	0	0
$463$	2.69713	0.125346	0.0626731	−	0.998034i	$-0.480037\pi$
$463$	2.69713	0.125346	0.0626731	+	0.998034i	$0.480037\pi$
$464$	0	0
$465$	−0.546951	−0.0253642
$466$	0	0
$467$	−37.0275	−1.71343	−0.856715	−	0.515791i	$-0.827498\pi$
$467$	−37.0275	−1.71343	−0.856715	+	0.515791i	$0.827498\pi$
$468$	0	0
$469$	48.5303	2.24092
$470$	0	0
$471$	−4.86043	−0.223957
$472$	0	0
$473$	0	0
$474$	0	0
$475$	4.73117	0.217081
$476$	0	0
$477$	5.99058	0.274290
$478$	0	0
$479$	21.7171	0.992281	0.496140	−	0.868242i	$-0.334750\pi$
$479$	21.7171	0.992281	0.496140	+	0.868242i	$0.334750\pi$
$480$	0	0
$481$	−29.8971	−1.36319
$482$	0	0
$483$	−8.14103	−0.370429
$484$	0	0
$485$	4.65722	0.211473
$486$	0	0
$487$	9.16809	0.415446	0.207723	−	0.978188i	$-0.433395\pi$
$487$	9.16809	0.415446	0.207723	+	0.978188i	$0.433395\pi$
$488$	0	0
$489$	5.16058	0.233369
$490$	0	0
$491$	17.0821	0.770905	0.385452	−	0.922728i	$-0.374045\pi$
$491$	17.0821	0.770905	0.385452	+	0.922728i	$0.374045\pi$
$492$	0	0
$493$	−28.8352	−1.29867
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−84.5856	−3.79418
$498$	0	0
$499$	−26.9315	−1.20562	−0.602810	−	0.797885i	$-0.705952\pi$
$499$	−26.9315	−1.20562	−0.602810	+	0.797885i	$0.705952\pi$
$500$	0	0
$501$	−5.63396	−0.251707
$502$	0	0
$503$	25.1348	1.12070	0.560352	−	0.828255i	$-0.310666\pi$
$503$	25.1348	1.12070	0.560352	+	0.828255i	$0.310666\pi$
$504$	0	0
$505$	−0.153065	−0.00681132
$506$	0	0
$507$	−1.06068	−0.0471066
$508$	0	0
$509$	27.1591	1.20381	0.601903	−	0.798569i	$-0.294409\pi$
$509$	27.1591	1.20381	0.601903	+	0.798569i	$0.294409\pi$
$510$	0	0
$511$	17.8246	0.788516
$512$	0	0
$513$	−1.58319	−0.0698996
$514$	0	0
$515$	−3.90604	−0.172121
$516$	0	0
$517$	0	0
$518$	0	0
$519$	1.73924	0.0763442
$520$	0	0
$521$	12.8941	0.564901	0.282450	−	0.959282i	$-0.408853\pi$
$521$	12.8941	0.564901	0.282450	+	0.959282i	$0.408853\pi$
$522$	0	0
$523$	14.2122	0.621456	0.310728	−	0.950499i	$-0.399427\pi$
$523$	14.2122	0.621456	0.310728	+	0.950499i	$0.399427\pi$
$524$	0	0
$525$	6.43955	0.281045
$526$	0	0
$527$	19.0130	0.828217
$528$	0	0
$529$	12.7754	0.555453
$530$	0	0
$531$	−32.2907	−1.40130
$532$	0	0
$533$	−23.9340	−1.03670
$534$	0	0
$535$	7.62782	0.329779
$536$	0	0
$537$	−0.986614	−0.0425755
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−19.9337	−0.857018	−0.428509	−	0.903538i	$-0.640961\pi$
$541$	−19.9337	−0.857018	−0.428509	+	0.903538i	$0.640961\pi$
$542$	0	0
$543$	−0.652475	−0.0280004
$544$	0	0
$545$	10.0866	0.432064
$546$	0	0
$547$	15.1760	0.648878	0.324439	−	0.945907i	$-0.394825\pi$
$547$	15.1760	0.648878	0.324439	+	0.945907i	$0.394825\pi$
$548$	0	0
$549$	−9.57795	−0.408777
$550$	0	0
$551$	5.99113	0.255231
$552$	0	0
$553$	−39.3950	−1.67524
$554$	0	0
$555$	−1.00479	−0.0426509
$556$	0	0
$557$	19.6997	0.834705	0.417353	−	0.908745i	$-0.362958\pi$
$557$	19.6997	0.834705	0.417353	+	0.908745i	$0.362958\pi$
$558$	0	0
$559$	35.0777	1.48363
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−11.9735	−0.504624	−0.252312	−	0.967646i	$-0.581191\pi$
$563$	−11.9735	−0.504624	−0.252312	+	0.967646i	$0.581191\pi$
$564$	0	0
$565$	10.2062	0.429378
$566$	0	0
$567$	42.6275	1.79019
$568$	0	0
$569$	27.2806	1.14366	0.571832	−	0.820371i	$-0.306233\pi$
$569$	27.2806	1.14366	0.571832	+	0.820371i	$0.306233\pi$
$570$	0	0
$571$	−33.9116	−1.41916	−0.709579	−	0.704626i	$-0.751115\pi$
$571$	−33.9116	−1.41916	−0.709579	+	0.704626i	$0.751115\pi$
$572$	0	0
$573$	2.76740	0.115610
$574$	0	0
$575$	−28.2983	−1.18012
$576$	0	0
$577$	1.06078	0.0441609	0.0220805	−	0.999756i	$-0.492971\pi$
$577$	1.06078	0.0441609	0.0220805	+	0.999756i	$0.492971\pi$
$578$	0	0
$579$	−4.55602	−0.189342
$580$	0	0
$581$	75.1951	3.11962
$582$	0	0
$583$	0	0
$584$	0	0
$585$	6.25573	0.258642
$586$	0	0
$587$	41.1871	1.69997	0.849987	−	0.526804i	$-0.176610\pi$
$587$	41.1871	1.69997	0.849987	+	0.526804i	$0.176610\pi$
$588$	0	0
$589$	−3.95035	−0.162772
$590$	0	0
$591$	3.91050	0.160856
$592$	0	0
$593$	30.1699	1.23893	0.619464	−	0.785025i	$-0.287350\pi$
$593$	30.1699	1.23893	0.619464	+	0.785025i	$0.287350\pi$
$594$	0	0
$595$	12.7193	0.521440
$596$	0	0
$597$	2.05076	0.0839318
$598$	0	0
$599$	22.3931	0.914958	0.457479	−	0.889220i	$-0.348752\pi$
$599$	22.3931	0.914958	0.457479	+	0.889220i	$0.348752\pi$
$600$	0	0
$601$	−12.5837	−0.513299	−0.256650	−	0.966504i	$-0.582619\pi$
$601$	−12.5837	−0.513299	−0.256650	+	0.966504i	$0.582619\pi$
$602$	0	0
$603$	−27.8852	−1.13558
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−9.92646	−0.402903	−0.201451	−	0.979499i	$-0.564566\pi$
$607$	−9.92646	−0.402903	−0.201451	+	0.979499i	$0.564566\pi$
$608$	0	0
$609$	8.15446	0.330436
$610$	0	0
$611$	42.1835	1.70656
$612$	0	0
$613$	−32.0150	−1.29307	−0.646536	−	0.762884i	$-0.723783\pi$
$613$	−32.0150	−1.29307	−0.646536	+	0.762884i	$0.723783\pi$
$614$	0	0
$615$	−0.804379	−0.0324357
$616$	0	0
$617$	45.6339	1.83715	0.918575	−	0.395247i	$-0.129341\pi$
$617$	45.6339	1.83715	0.918575	+	0.395247i	$0.129341\pi$
$618$	0	0
$619$	34.6141	1.39126	0.695630	−	0.718400i	$-0.255125\pi$
$619$	34.6141	1.39126	0.695630	+	0.718400i	$0.255125\pi$
$620$	0	0
$621$	9.46947	0.379997
$622$	0	0
$623$	5.17976	0.207523
$624$	0	0
$625$	21.0398	0.841594
$626$	0	0
$627$	0	0
$628$	0	0
$629$	34.9282	1.39268
$630$	0	0
$631$	0.430341	0.0171316	0.00856580	−	0.999963i	$-0.497273\pi$
$631$	0.430341	0.0171316	0.00856580	+	0.999963i	$0.497273\pi$
$632$	0	0
$633$	−5.16677	−0.205361
$634$	0	0
$635$	−3.79762	−0.150704
$636$	0	0
$637$	−78.1884	−3.09794
$638$	0	0
$639$	48.6024	1.92268
$640$	0	0
$641$	−47.4840	−1.87551	−0.937753	−	0.347302i	$-0.887098\pi$
$641$	−47.4840	−1.87551	−0.937753	+	0.347302i	$0.887098\pi$
$642$	0	0
$643$	−11.2346	−0.443051	−0.221525	−	0.975155i	$-0.571104\pi$
$643$	−11.2346	−0.443051	−0.221525	+	0.975155i	$0.571104\pi$
$644$	0	0
$645$	1.17890	0.0464191
$646$	0	0
$647$	24.0322	0.944803	0.472401	−	0.881384i	$-0.343387\pi$
$647$	24.0322	0.944803	0.472401	+	0.881384i	$0.343387\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−5.37679	−0.210733
$652$	0	0
$653$	17.3209	0.677819	0.338910	−	0.940819i	$-0.389942\pi$
$653$	17.3209	0.677819	0.338910	+	0.940819i	$0.389942\pi$
$654$	0	0
$655$	−5.53996	−0.216464
$656$	0	0
$657$	−10.2419	−0.399576
$658$	0	0
$659$	−18.0899	−0.704684	−0.352342	−	0.935871i	$-0.614615\pi$
$659$	−18.0899	−0.704684	−0.352342	+	0.935871i	$0.614615\pi$
$660$	0	0
$661$	0.734767	0.0285791	0.0142896	−	0.999898i	$-0.495451\pi$
$661$	0.734767	0.0285791	0.0142896	+	0.999898i	$0.495451\pi$
$662$	0	0
$663$	5.29487	0.205636
$664$	0	0
$665$	−2.64271	−0.102480
$666$	0	0
$667$	−35.8345	−1.38752
$668$	0	0
$669$	−4.17804	−0.161533
$670$	0	0
$671$	0	0
$672$	0	0
$673$	35.4115	1.36501	0.682507	−	0.730879i	$-0.260890\pi$
$673$	35.4115	1.36501	0.682507	+	0.730879i	$0.260890\pi$
$674$	0	0
$675$	−7.49035	−0.288304
$676$	0	0
$677$	25.8475	0.993400	0.496700	−	0.867922i	$-0.334545\pi$
$677$	25.8475	0.993400	0.496700	+	0.867922i	$0.334545\pi$
$678$	0	0
$679$	45.7827	1.75698
$680$	0	0
$681$	−4.63641	−0.177668
$682$	0	0
$683$	−33.2465	−1.27214	−0.636071	−	0.771630i	$-0.719442\pi$
$683$	−33.2465	−1.27214	−0.636071	+	0.771630i	$0.719442\pi$
$684$	0	0
$685$	−5.09265	−0.194580
$686$	0	0
$687$	3.08051	0.117529
$688$	0	0
$689$	8.42680	0.321035
$690$	0	0
$691$	−27.5007	−1.04617	−0.523087	−	0.852279i	$-0.675220\pi$
$691$	−27.5007	−1.04617	−0.523087	+	0.852279i	$0.675220\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−0.423152	−0.0160511
$696$	0	0
$697$	27.9616	1.05912
$698$	0	0
$699$	1.11999	0.0423619
$700$	0	0
$701$	−34.2830	−1.29485	−0.647425	−	0.762129i	$-0.724154\pi$
$701$	−34.2830	−1.29485	−0.647425	+	0.762129i	$0.724154\pi$
$702$	0	0
$703$	−7.25709	−0.273706
$704$	0	0
$705$	1.41771	0.0533941
$706$	0	0
$707$	−1.50471	−0.0565903
$708$	0	0
$709$	−30.6199	−1.14996	−0.574978	−	0.818169i	$-0.694989\pi$
$709$	−30.6199	−1.14996	−0.574978	+	0.818169i	$0.694989\pi$
$710$	0	0
$711$	22.6361	0.848921
$712$	0	0
$713$	23.6281	0.884878
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−5.72284	−0.213723
$718$	0	0
$719$	10.4753	0.390663	0.195331	−	0.980737i	$-0.437422\pi$
$719$	10.4753	0.390663	0.195331	+	0.980737i	$0.437422\pi$
$720$	0	0
$721$	−38.3983	−1.43003
$722$	0	0
$723$	5.18719	0.192914
$724$	0	0
$725$	28.3451	1.05271
$726$	0	0
$727$	22.7343	0.843168	0.421584	−	0.906789i	$-0.361474\pi$
$727$	22.7343	0.843168	0.421584	+	0.906789i	$0.361474\pi$
$728$	0	0
$729$	−23.2252	−0.860192
$730$	0	0
$731$	−40.9806	−1.51572
$732$	0	0
$733$	46.6306	1.72234	0.861170	−	0.508318i	$-0.169732\pi$
$733$	46.6306	1.72234	0.861170	+	0.508318i	$0.169732\pi$
$734$	0	0
$735$	−2.62777	−0.0969269
$736$	0	0
$737$	0	0
$738$	0	0
$739$	18.1493	0.667633	0.333817	−	0.942638i	$-0.391663\pi$
$739$	18.1493	0.667633	0.333817	+	0.942638i	$0.391663\pi$
$740$	0	0
$741$	−1.10012	−0.0404141
$742$	0	0
$743$	19.1850	0.703831	0.351915	−	0.936032i	$-0.385530\pi$
$743$	19.1850	0.703831	0.351915	+	0.936032i	$0.385530\pi$
$744$	0	0
$745$	1.01339	0.0371278
$746$	0	0
$747$	−43.2067	−1.58085
$748$	0	0
$749$	74.9851	2.73990
$750$	0	0
$751$	−12.5092	−0.456466	−0.228233	−	0.973607i	$-0.573295\pi$
$751$	−12.5092	−0.456466	−0.228233	+	0.973607i	$0.573295\pi$
$752$	0	0
$753$	3.50188	0.127615
$754$	0	0
$755$	6.40234	0.233005
$756$	0	0
$757$	−1.10374	−0.0401162	−0.0200581	−	0.999799i	$-0.506385\pi$
$757$	−1.10374	−0.0401162	−0.0200581	+	0.999799i	$0.506385\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	36.9913	1.34093	0.670466	−	0.741940i	$-0.266094\pi$
$761$	36.9913	1.34093	0.670466	+	0.741940i	$0.266094\pi$
$762$	0	0
$763$	99.1565	3.58971
$764$	0	0
$765$	−7.30844	−0.264237
$766$	0	0
$767$	−45.4225	−1.64011
$768$	0	0
$769$	21.0843	0.760321	0.380160	−	0.924921i	$-0.375869\pi$
$769$	21.0843	0.760321	0.380160	+	0.924921i	$0.375869\pi$
$770$	0	0
$771$	−1.55694	−0.0560717
$772$	0	0
$773$	41.3989	1.48902	0.744508	−	0.667613i	$-0.232684\pi$
$773$	41.3989	1.48902	0.744508	+	0.667613i	$0.232684\pi$
$774$	0	0
$775$	−18.6898	−0.671358
$776$	0	0
$777$	−9.87755	−0.354355
$778$	0	0
$779$	−5.80963	−0.208152
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−9.48510	−0.338970
$784$	0	0
$785$	9.43708	0.336824
$786$	0	0
$787$	10.6169	0.378452	0.189226	−	0.981934i	$-0.439402\pi$
$787$	10.6169	0.378452	0.189226	+	0.981934i	$0.439402\pi$
$788$	0	0
$789$	−3.99643	−0.142277
$790$	0	0
$791$	100.332	3.56739
$792$	0	0
$793$	−13.4731	−0.478442
$794$	0	0
$795$	0.283210	0.0100444
$796$	0	0
$797$	46.2638	1.63875	0.819375	−	0.573258i	$-0.194321\pi$
$797$	46.2638	1.63875	0.819375	+	0.573258i	$0.194321\pi$
$798$	0	0
$799$	−49.2821	−1.74347
$800$	0	0
$801$	−2.97626	−0.105161
$802$	0	0
$803$	0	0
$804$	0	0
$805$	15.8067	0.557114
$806$	0	0
$807$	1.14897	0.0404458
$808$	0	0
$809$	21.0659	0.740637	0.370318	−	0.928905i	$-0.379249\pi$
$809$	21.0659	0.740637	0.370318	+	0.928905i	$0.379249\pi$
$810$	0	0
$811$	−18.5834	−0.652552	−0.326276	−	0.945275i	$-0.605794\pi$
$811$	−18.5834	−0.652552	−0.326276	+	0.945275i	$0.605794\pi$
$812$	0	0
$813$	−5.05159	−0.177167
$814$	0	0
$815$	−10.0198	−0.350980
$816$	0	0
$817$	8.51461	0.297888
$818$	0	0
$819$	61.4968	2.14887
$820$	0	0
$821$	13.4643	0.469908	0.234954	−	0.972006i	$-0.424506\pi$
$821$	13.4643	0.469908	0.234954	+	0.972006i	$0.424506\pi$
$822$	0	0
$823$	−11.6154	−0.404887	−0.202443	−	0.979294i	$-0.564888\pi$
$823$	−11.6154	−0.404887	−0.202443	+	0.979294i	$0.564888\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	33.7319	1.17297	0.586487	−	0.809959i	$-0.300511\pi$
$827$	33.7319	1.17297	0.586487	+	0.809959i	$0.300511\pi$
$828$	0	0
$829$	−46.6617	−1.62063	−0.810314	−	0.585996i	$-0.800703\pi$
$829$	−46.6617	−1.62063	−0.810314	+	0.585996i	$0.800703\pi$
$830$	0	0
$831$	6.85577	0.237824
$832$	0	0
$833$	91.3459	3.16495
$834$	0	0
$835$	10.9390	0.378559
$836$	0	0
$837$	6.25417	0.216176
$838$	0	0
$839$	14.5103	0.500950	0.250475	−	0.968123i	$-0.419413\pi$
$839$	14.5103	0.500950	0.250475	+	0.968123i	$0.419413\pi$
$840$	0	0
$841$	6.89362	0.237711
$842$	0	0
$843$	2.68830	0.0925899
$844$	0	0
$845$	2.05944	0.0708468
$846$	0	0
$847$	0	0
$848$	0	0
$849$	2.72518	0.0935278
$850$	0	0
$851$	43.4065	1.48796
$852$	0	0
$853$	−20.8893	−0.715236	−0.357618	−	0.933868i	$-0.616411\pi$
$853$	−20.8893	−0.715236	−0.357618	+	0.933868i	$0.616411\pi$
$854$	0	0
$855$	1.51849	0.0519311
$856$	0	0
$857$	20.9960	0.717208	0.358604	−	0.933490i	$-0.383253\pi$
$857$	20.9960	0.717208	0.358604	+	0.933490i	$0.383253\pi$
$858$	0	0
$859$	24.5725	0.838403	0.419201	−	0.907893i	$-0.362310\pi$
$859$	24.5725	0.838403	0.419201	+	0.907893i	$0.362310\pi$
$860$	0	0
$861$	−7.90743	−0.269485
$862$	0	0
$863$	14.1989	0.483337	0.241669	−	0.970359i	$-0.422305\pi$
$863$	14.1989	0.483337	0.241669	+	0.970359i	$0.422305\pi$
$864$	0	0
$865$	−3.37693	−0.114819
$866$	0	0
$867$	−1.64622	−0.0559087
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−39.2254	−1.32910
$872$	0	0
$873$	−26.3065	−0.890339
$874$	0	0
$875$	−25.7167	−0.869382
$876$	0	0
$877$	54.9559	1.85573	0.927865	−	0.372917i	$-0.121642\pi$
$877$	54.9559	1.85573	0.927865	+	0.372917i	$0.121642\pi$
$878$	0	0
$879$	−1.99037	−0.0671336
$880$	0	0
$881$	−47.9021	−1.61386	−0.806931	−	0.590646i	$-0.798873\pi$
$881$	−47.9021	−1.61386	−0.806931	+	0.590646i	$0.798873\pi$
$882$	0	0
$883$	6.51500	0.219247	0.109624	−	0.993973i	$-0.465035\pi$
$883$	6.51500	0.219247	0.109624	+	0.993973i	$0.465035\pi$
$884$	0	0
$885$	−1.52657	−0.0513151
$886$	0	0
$887$	−25.0785	−0.842053	−0.421027	−	0.907048i	$-0.638330\pi$
$887$	−25.0785	−0.842053	−0.421027	+	0.907048i	$0.638330\pi$
$888$	0	0
$889$	−37.3324	−1.25209
$890$	0	0
$891$	0	0
$892$	0	0
$893$	10.2394	0.342649
$894$	0	0
$895$	1.91562	0.0640322
$896$	0	0
$897$	6.58012	0.219704
$898$	0	0
$899$	−23.6671	−0.789341
$900$	0	0
$901$	−9.84485	−0.327980
$902$	0	0
$903$	11.5891	0.385663
$904$	0	0
$905$	1.26686	0.0421117
$906$	0	0
$907$	41.3260	1.37221	0.686104	−	0.727504i	$-0.259320\pi$
$907$	41.3260	1.37221	0.686104	+	0.727504i	$0.259320\pi$
$908$	0	0
$909$	0.864596	0.0286768
$910$	0	0
$911$	38.3970	1.27215	0.636075	−	0.771627i	$-0.280557\pi$
$911$	38.3970	1.27215	0.636075	+	0.771627i	$0.280557\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−0.452805	−0.0149693
$916$	0	0
$917$	−54.4605	−1.79844
$918$	0	0
$919$	−17.5139	−0.577729	−0.288865	−	0.957370i	$-0.593278\pi$
$919$	−17.5139	−0.577729	−0.288865	+	0.957370i	$0.593278\pi$
$920$	0	0
$921$	−5.21038	−0.171688
$922$	0	0
$923$	68.3677	2.25035
$924$	0	0
$925$	−34.3345	−1.12891
$926$	0	0
$927$	22.0634	0.724658
$928$	0	0
$929$	−40.3052	−1.32237	−0.661186	−	0.750222i	$-0.729947\pi$
$929$	−40.3052	−1.32237	−0.661186	+	0.750222i	$0.729947\pi$
$930$	0	0
$931$	−18.9791	−0.622015
$932$	0	0
$933$	−4.70900	−0.154166
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−13.3193	−0.435122	−0.217561	−	0.976047i	$-0.569810\pi$
$937$	−13.3193	−0.435122	−0.217561	+	0.976047i	$0.569810\pi$
$938$	0	0
$939$	−7.35375	−0.239981
$940$	0	0
$941$	−14.6329	−0.477018	−0.238509	−	0.971140i	$-0.576659\pi$
$941$	−14.6329	−0.477018	−0.238509	+	0.971140i	$0.576659\pi$
$942$	0	0
$943$	34.7489	1.13158
$944$	0	0
$945$	4.18392	0.136103
$946$	0	0
$947$	−21.6890	−0.704799	−0.352400	−	0.935850i	$-0.614634\pi$
$947$	−21.6890	−0.704799	−0.352400	+	0.935850i	$0.614634\pi$
$948$	0	0
$949$	−14.4071	−0.467673
$950$	0	0
$951$	7.79237	0.252685
$952$	0	0
$953$	19.3512	0.626848	0.313424	−	0.949613i	$-0.398524\pi$
$953$	19.3512	0.626848	0.313424	+	0.949613i	$0.398524\pi$
$954$	0	0
$955$	−5.37323	−0.173874
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−50.0631	−1.61662
$960$	0	0
$961$	−15.3947	−0.496603
$962$	0	0
$963$	−43.0860	−1.38843
$964$	0	0
$965$	8.84604	0.284764
$966$	0	0
$967$	−44.1468	−1.41967	−0.709833	−	0.704370i	$-0.751230\pi$
$967$	−44.1468	−1.41967	−0.709833	+	0.704370i	$0.751230\pi$
$968$	0	0
$969$	1.28525	0.0412882
$970$	0	0
$971$	−15.3821	−0.493635	−0.246818	−	0.969062i	$-0.579385\pi$
$971$	−15.3821	−0.493635	−0.246818	+	0.969062i	$0.579385\pi$
$972$	0	0
$973$	−4.15978	−0.133357
$974$	0	0
$975$	−5.20487	−0.166689
$976$	0	0
$977$	−30.3264	−0.970227	−0.485114	−	0.874451i	$-0.661222\pi$
$977$	−30.3264	−0.970227	−0.485114	+	0.874451i	$0.661222\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−56.9748	−1.81906
$982$	0	0
$983$	−36.2126	−1.15500	−0.577502	−	0.816389i	$-0.695972\pi$
$983$	−36.2126	−1.15500	−0.577502	+	0.816389i	$0.695972\pi$
$984$	0	0
$985$	−7.59268	−0.241923
$986$	0	0
$987$	13.9368	0.443612
$988$	0	0
$989$	−50.9280	−1.61942
$990$	0	0
$991$	−34.3004	−1.08959	−0.544795	−	0.838570i	$-0.683392\pi$
$991$	−34.3004	−1.08959	−0.544795	+	0.838570i	$0.683392\pi$
$992$	0	0
$993$	5.55986	0.176437
$994$	0	0
$995$	−3.98177	−0.126231
$996$	0	0
$997$	1.04897	0.0332212	0.0166106	−	0.999862i	$-0.494712\pi$
$997$	1.04897	0.0332212	0.0166106	+	0.999862i	$0.494712\pi$
$998$	0	0
$999$	11.4894	0.363507

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9196.2.a.m.1.3	yes	6
11.10	odd	2		9196.2.a.l.1.3	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9196.2.a.l.1.3	✓	6	11.10	odd	2
9196.2.a.m.1.3	yes	6	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:	\( N \)	\(=\)	\( 9196 = 2^{2} \cdot 11^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9196.a (trivial)

\(f(q)\)	\(=\)	\(q-0.267039 q^{3} +0.518487 q^{5} +5.09697 q^{7} -2.92869 q^{9} +O(q^{10})\)	\(q-0.267039 q^{3} +0.518487 q^{5} +5.09697 q^{7} -2.92869 q^{9} -4.11971 q^{13} -0.138456 q^{15} +4.81298 q^{17} -1.00000 q^{19} -1.36109 q^{21} +5.98125 q^{23} -4.73117 q^{25} +1.58319 q^{27} -5.99113 q^{29} +3.95035 q^{31} +2.64271 q^{35} +7.25709 q^{37} +1.10012 q^{39} +5.80963 q^{41} -8.51461 q^{43} -1.51849 q^{45} -10.2394 q^{47} +18.9791 q^{49} -1.28525 q^{51} -2.04548 q^{53} +0.267039 q^{57} +11.0257 q^{59} +3.27039 q^{61} -14.9274 q^{63} -2.13601 q^{65} +9.52141 q^{67} -1.59723 q^{69} -16.5953 q^{71} +3.49710 q^{73} +1.26341 q^{75} -7.72909 q^{79} +8.36330 q^{81} +14.7529 q^{83} +2.49546 q^{85} +1.59987 q^{87} +1.01624 q^{89} -20.9980 q^{91} -1.05490 q^{93} -0.518487 q^{95} +8.98233 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{5} + 4 q^{7} + 2 q^{9}+O(q^{10}) \) 6 * q - 4 * q^5 + 4 * q^7 + 2 * q^9	\( 6 q - 4 q^{5} + 4 q^{7} + 2 q^{9} + 8 q^{13} + 2 q^{15} + 6 q^{17} - 6 q^{19} - 20 q^{21} + 6 q^{23} + 2 q^{25} - 18 q^{27} - 2 q^{29} - 6 q^{31} + 14 q^{35} + 4 q^{37} + 6 q^{39} + 14 q^{41} - 10 q^{43} - 2 q^{45} + 4 q^{47} + 20 q^{49} + 18 q^{51} + 20 q^{59} - 2 q^{61} + 12 q^{63} - 12 q^{65} + 10 q^{67} - 20 q^{69} + 32 q^{73} + 18 q^{75} + 6 q^{79} - 6 q^{81} + 36 q^{83} + 6 q^{85} - 28 q^{87} - 24 q^{89} - 16 q^{91} + 22 q^{93} + 4 q^{95} + 28 q^{97}+O(q^{100}) \) 6 * q - 4 * q^5 + 4 * q^7 + 2 * q^9 + 8 * q^13 + 2 * q^15 + 6 * q^17 - 6 * q^19 - 20 * q^21 + 6 * q^23 + 2 * q^25 - 18 * q^27 - 2 * q^29 - 6 * q^31 + 14 * q^35 + 4 * q^37 + 6 * q^39 + 14 * q^41 - 10 * q^43 - 2 * q^45 + 4 * q^47 + 20 * q^49 + 18 * q^51 + 20 * q^59 - 2 * q^61 + 12 * q^63 - 12 * q^65 + 10 * q^67 - 20 * q^69 + 32 * q^73 + 18 * q^75 + 6 * q^79 - 6 * q^81 + 36 * q^83 + 6 * q^85 - 28 * q^87 - 24 * q^89 - 16 * q^91 + 22 * q^93 + 4 * q^95 + 28 * q^97

Introduction

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