LMFDB - Embedded newform 9196.2.a.x.1.19

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:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 6 x^{19} - 26 x^{18} + 208 x^{17} + 185 x^{16} - 2910 x^{15} + 687 x^{14} + 21067 x^{13} + \cdots - 3520 $ x^20 - 6x^19 - 26x^18 + 208x^17 + 185x^16 - 2910x^15 + 687x^14 + 21067x^13 - 16530x^12 - 83962x^11 + 93139x^10 + 180932x^9 - 242362x^8 - 194923x^7 + 303963x^6 + 96332x^5 - 176108x^4 - 24984x^3 + 44976x^2 + 3840*x - 3520
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 836)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Root			$3.12040$ 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)

$f(q)$	$=$	$q+3.12040 q^{3} -2.72877 q^{5} +5.12972 q^{7} +6.73690 q^{9} -2.22565 q^{13} -8.51486 q^{15} +7.31671 q^{17} -1.00000 q^{19} +16.0068 q^{21} -1.47420 q^{23} +2.44619 q^{25} +11.6606 q^{27} +3.39993 q^{29} -3.33744 q^{31} -13.9978 q^{35} +1.98551 q^{37} -6.94492 q^{39} -8.50050 q^{41} -2.36107 q^{43} -18.3835 q^{45} +0.985870 q^{47} +19.3141 q^{49} +22.8311 q^{51} -2.14192 q^{53} -3.12040 q^{57} +7.58232 q^{59} -3.63177 q^{61} +34.5584 q^{63} +6.07329 q^{65} +13.8608 q^{67} -4.60009 q^{69} +14.3974 q^{71} +2.93658 q^{73} +7.63310 q^{75} +10.9359 q^{79} +16.1751 q^{81} +9.69263 q^{83} -19.9656 q^{85} +10.6091 q^{87} +6.94960 q^{89} -11.4170 q^{91} -10.4142 q^{93} +2.72877 q^{95} -1.66145 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 6 q^{3} + 2 q^{5} + 4 q^{7} + 28 q^{9} - 5 q^{13} + 14 q^{15} - 6 q^{17} - 20 q^{19} - q^{21} + 18 q^{23} + 44 q^{25} + 24 q^{27} + q^{29} + 23 q^{31} - 36 q^{35} + q^{37} + 21 q^{39} + 6 q^{41}+ \cdots + 9 q^{97}+O(q^{100}) $ 20 * q + 6 * q^3 + 2 * q^5 + 4 * q^7 + 28 * q^9 - 5 * q^13 + 14 * q^15 - 6 * q^17 - 20 * q^19 - q^21 + 18 * q^23 + 44 * q^25 + 24 * q^27 + q^29 + 23 * q^31 - 36 * q^35 + q^37 + 21 * q^39 + 6 * q^41 - 8 * q^43 - 15 * q^45 + 34 * q^47 + 42 * q^49 + 14 * q^51 - 10 * q^53 - 6 * q^57 + 52 * q^59 - 2 * q^61 + 59 * q^63 + 25 * q^65 + 14 * q^67 + 15 * q^69 + 88 * q^71 - 13 * q^73 + 15 * q^75 - 2 * q^79 + 48 * q^81 - 2 * q^83 + 11 * q^85 + 37 * q^87 + 46 * q^89 - 4 * q^91 + 2 * q^93 - 2 * q^95 + 9 * q^97

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	3.12040	1.80156	0.900782	−	0.434271i	$-0.142994\pi$
$3$	3.12040	1.80156	0.900782	+	0.434271i	$0.142994\pi$
$4$	0	0
$5$	−2.72877	−1.22034	−0.610172	−	0.792269i	$-0.708900\pi$
$5$	−2.72877	−1.22034	−0.610172	+	0.792269i	$0.708900\pi$
$6$	0	0
$7$	5.12972	1.93885	0.969427	−	0.245380i	$-0.0789129\pi$
$7$	5.12972	1.93885	0.969427	+	0.245380i	$0.0789129\pi$
$8$	0	0
$9$	6.73690	2.24563
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−2.22565	−0.617284	−0.308642	−	0.951178i	$-0.599874\pi$
$13$	−2.22565	−0.617284	−0.308642	+	0.951178i	$0.599874\pi$
$14$	0	0
$15$	−8.51486	−2.19853
$16$	0	0
$17$	7.31671	1.77456	0.887282	−	0.461228i	$-0.152591\pi$
$17$	7.31671	1.77456	0.887282	+	0.461228i	$0.152591\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	16.0068	3.49297
$22$	0	0
$23$	−1.47420	−0.307392	−0.153696	−	0.988118i	$-0.549118\pi$
$23$	−1.47420	−0.307392	−0.153696	+	0.988118i	$0.549118\pi$
$24$	0	0
$25$	2.44619	0.489238
$26$	0	0
$27$	11.6606	2.24409
$28$	0	0
$29$	3.39993	0.631351	0.315676	−	0.948867i	$-0.397769\pi$
$29$	3.39993	0.631351	0.315676	+	0.948867i	$0.397769\pi$
$30$	0	0
$31$	−3.33744	−0.599423	−0.299711	−	0.954030i	$-0.596890\pi$
$31$	−3.33744	−0.599423	−0.299711	+	0.954030i	$0.596890\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−13.9978	−2.36607
$36$	0	0
$37$	1.98551	0.326415	0.163208	−	0.986592i	$-0.447816\pi$
$37$	1.98551	0.326415	0.163208	+	0.986592i	$0.447816\pi$
$38$	0	0
$39$	−6.94492	−1.11208
$40$	0	0
$41$	−8.50050	−1.32756	−0.663778	−	0.747930i	$-0.731048\pi$
$41$	−8.50050	−1.32756	−0.663778	+	0.747930i	$0.731048\pi$
$42$	0	0
$43$	−2.36107	−0.360060	−0.180030	−	0.983661i	$-0.557619\pi$
$43$	−2.36107	−0.360060	−0.180030	+	0.983661i	$0.557619\pi$
$44$	0	0
$45$	−18.3835	−2.74044
$46$	0	0
$47$	0.985870	0.143804	0.0719020	−	0.997412i	$-0.477093\pi$
$47$	0.985870	0.143804	0.0719020	+	0.997412i	$0.477093\pi$
$48$	0	0
$49$	19.3141	2.75915
$50$	0	0
$51$	22.8311	3.19699
$52$	0	0
$53$	−2.14192	−0.294215	−0.147108	−	0.989121i	$-0.546996\pi$
$53$	−2.14192	−0.294215	−0.147108	+	0.989121i	$0.546996\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−3.12040	−0.413307
$58$	0	0
$59$	7.58232	0.987134	0.493567	−	0.869708i	$-0.335693\pi$
$59$	7.58232	0.987134	0.493567	+	0.869708i	$0.335693\pi$
$60$	0	0
$61$	−3.63177	−0.465001	−0.232500	−	0.972596i	$-0.574691\pi$
$61$	−3.63177	−0.465001	−0.232500	+	0.972596i	$0.574691\pi$
$62$	0	0
$63$	34.5584	4.35395
$64$	0	0
$65$	6.07329	0.753298
$66$	0	0
$67$	13.8608	1.69337	0.846683	−	0.532097i	$-0.178596\pi$
$67$	13.8608	1.69337	0.846683	+	0.532097i	$0.178596\pi$
$68$	0	0
$69$	−4.60009	−0.553786
$70$	0	0
$71$	14.3974	1.70865	0.854327	−	0.519736i	$-0.173970\pi$
$71$	14.3974	1.70865	0.854327	+	0.519736i	$0.173970\pi$
$72$	0	0
$73$	2.93658	0.343701	0.171851	−	0.985123i	$-0.445025\pi$
$73$	2.93658	0.343701	0.171851	+	0.985123i	$0.445025\pi$
$74$	0	0
$75$	7.63310	0.881394
$76$	0	0
$77$	0	0
$78$	0	0
$79$	10.9359	1.23039	0.615194	−	0.788375i	$-0.289078\pi$
$79$	10.9359	1.23039	0.615194	+	0.788375i	$0.289078\pi$
$80$	0	0
$81$	16.1751	1.79724
$82$	0	0
$83$	9.69263	1.06390	0.531952	−	0.846774i	$-0.321459\pi$
$83$	9.69263	1.06390	0.531952	+	0.846774i	$0.321459\pi$
$84$	0	0
$85$	−19.9656	−2.16558
$86$	0	0
$87$	10.6091	1.13742
$88$	0	0
$89$	6.94960	0.736656	0.368328	−	0.929696i	$-0.379930\pi$
$89$	6.94960	0.736656	0.368328	+	0.929696i	$0.379930\pi$
$90$	0	0
$91$	−11.4170	−1.19682
$92$	0	0
$93$	−10.4142	−1.07990
$94$	0	0
$95$	2.72877	0.279966
$96$	0	0
$97$	−1.66145	−0.168694	−0.0843472	−	0.996436i	$-0.526880\pi$
$97$	−1.66145	−0.168694	−0.0843472	+	0.996436i	$0.526880\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−9.33794	−0.929160	−0.464580	−	0.885531i	$-0.653795\pi$
$101$	−9.33794	−0.929160	−0.464580	+	0.885531i	$0.653795\pi$
$102$	0	0
$103$	−14.9154	−1.46966	−0.734829	−	0.678253i	$-0.762738\pi$
$103$	−14.9154	−1.46966	−0.734829	+	0.678253i	$0.762738\pi$
$104$	0	0
$105$	−43.6789	−4.26262
$106$	0	0
$107$	−7.06895	−0.683381	−0.341690	−	0.939813i	$-0.610999\pi$
$107$	−7.06895	−0.683381	−0.341690	+	0.939813i	$0.610999\pi$
$108$	0	0
$109$	−12.9175	−1.23727	−0.618634	−	0.785680i	$-0.712313\pi$
$109$	−12.9175	−1.23727	−0.618634	+	0.785680i	$0.712313\pi$
$110$	0	0
$111$	6.19558	0.588058
$112$	0	0
$113$	16.1876	1.52280	0.761400	−	0.648282i	$-0.224512\pi$
$113$	16.1876	1.52280	0.761400	+	0.648282i	$0.224512\pi$
$114$	0	0
$115$	4.02275	0.375124
$116$	0	0
$117$	−14.9940	−1.38619
$118$	0	0
$119$	37.5327	3.44062
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−26.5250	−2.39168
$124$	0	0
$125$	6.96876	0.623305
$126$	0	0
$127$	3.09073	0.274258	0.137129	−	0.990553i	$-0.456213\pi$
$127$	3.09073	0.274258	0.137129	+	0.990553i	$0.456213\pi$
$128$	0	0
$129$	−7.36749	−0.648671
$130$	0	0
$131$	−9.54201	−0.833689	−0.416845	−	0.908978i	$-0.636864\pi$
$131$	−9.54201	−0.833689	−0.416845	+	0.908978i	$0.636864\pi$
$132$	0	0
$133$	−5.12972	−0.444804
$134$	0	0
$135$	−31.8192	−2.73856
$136$	0	0
$137$	−8.89967	−0.760350	−0.380175	−	0.924915i	$-0.624136\pi$
$137$	−8.89967	−0.760350	−0.380175	+	0.924915i	$0.624136\pi$
$138$	0	0
$139$	−10.6310	−0.901707	−0.450853	−	0.892598i	$-0.648880\pi$
$139$	−10.6310	−0.901707	−0.450853	+	0.892598i	$0.648880\pi$
$140$	0	0
$141$	3.07631	0.259072
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−9.27763	−0.770466
$146$	0	0
$147$	60.2677	4.97079
$148$	0	0
$149$	−9.60713	−0.787047	−0.393524	−	0.919315i	$-0.628744\pi$
$149$	−9.60713	−0.787047	−0.393524	+	0.919315i	$0.628744\pi$
$150$	0	0
$151$	−6.33174	−0.515269	−0.257635	−	0.966242i	$-0.582943\pi$
$151$	−6.33174	−0.515269	−0.257635	+	0.966242i	$0.582943\pi$
$152$	0	0
$153$	49.2920	3.98502
$154$	0	0
$155$	9.10712	0.731501
$156$	0	0
$157$	−9.73160	−0.776667	−0.388333	−	0.921519i	$-0.626949\pi$
$157$	−9.73160	−0.776667	−0.388333	+	0.921519i	$0.626949\pi$
$158$	0	0
$159$	−6.68364	−0.530047
$160$	0	0
$161$	−7.56224	−0.595988
$162$	0	0
$163$	−15.5100	−1.21484	−0.607418	−	0.794382i	$-0.707795\pi$
$163$	−15.5100	−1.21484	−0.607418	+	0.794382i	$0.707795\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−5.30159	−0.410249	−0.205125	−	0.978736i	$-0.565760\pi$
$167$	−5.30159	−0.410249	−0.205125	+	0.978736i	$0.565760\pi$
$168$	0	0
$169$	−8.04649	−0.618961
$170$	0	0
$171$	−6.73690	−0.515184
$172$	0	0
$173$	16.2736	1.23726	0.618631	−	0.785682i	$-0.287688\pi$
$173$	16.2736	1.23726	0.618631	+	0.785682i	$0.287688\pi$
$174$	0	0
$175$	12.5483	0.948561
$176$	0	0
$177$	23.6599	1.77839
$178$	0	0
$179$	0.762189	0.0569687	0.0284843	−	0.999594i	$-0.490932\pi$
$179$	0.762189	0.0569687	0.0284843	+	0.999594i	$0.490932\pi$
$180$	0	0
$181$	−13.9838	−1.03940	−0.519702	−	0.854348i	$-0.673957\pi$
$181$	−13.9838	−1.03940	−0.519702	+	0.854348i	$0.673957\pi$
$182$	0	0
$183$	−11.3326	−0.837729
$184$	0	0
$185$	−5.41799	−0.398339
$186$	0	0
$187$	0	0
$188$	0	0
$189$	59.8158	4.35096
$190$	0	0
$191$	24.1936	1.75059	0.875294	−	0.483592i	$-0.160668\pi$
$191$	24.1936	1.75059	0.875294	+	0.483592i	$0.160668\pi$
$192$	0	0
$193$	7.88369	0.567480	0.283740	−	0.958901i	$-0.408425\pi$
$193$	7.88369	0.567480	0.283740	+	0.958901i	$0.408425\pi$
$194$	0	0
$195$	18.9511	1.35712
$196$	0	0
$197$	−0.633797	−0.0451561	−0.0225781	−	0.999745i	$-0.507187\pi$
$197$	−0.633797	−0.0451561	−0.0225781	+	0.999745i	$0.507187\pi$
$198$	0	0
$199$	22.5019	1.59512	0.797558	−	0.603242i	$-0.206125\pi$
$199$	22.5019	1.59512	0.797558	+	0.603242i	$0.206125\pi$
$200$	0	0
$201$	43.2513	3.05071
$202$	0	0
$203$	17.4407	1.22410
$204$	0	0
$205$	23.1959	1.62007
$206$	0	0
$207$	−9.93154	−0.690290
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−25.3248	−1.74343	−0.871714	−	0.490014i	$-0.836992\pi$
$211$	−25.3248	−1.74343	−0.871714	+	0.490014i	$0.836992\pi$
$212$	0	0
$213$	44.9256	3.07825
$214$	0	0
$215$	6.44282	0.439397
$216$	0	0
$217$	−17.1202	−1.16219
$218$	0	0
$219$	9.16332	0.619200
$220$	0	0
$221$	−16.2844	−1.09541
$222$	0	0
$223$	−27.2163	−1.82254	−0.911269	−	0.411811i	$-0.864897\pi$
$223$	−27.2163	−1.82254	−0.911269	+	0.411811i	$0.864897\pi$
$224$	0	0
$225$	16.4797	1.09865
$226$	0	0
$227$	−1.07037	−0.0710432	−0.0355216	−	0.999369i	$-0.511309\pi$
$227$	−1.07037	−0.0710432	−0.0355216	+	0.999369i	$0.511309\pi$
$228$	0	0
$229$	3.81557	0.252140	0.126070	−	0.992021i	$-0.459764\pi$
$229$	3.81557	0.252140	0.126070	+	0.992021i	$0.459764\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	11.0501	0.723919	0.361959	−	0.932194i	$-0.382108\pi$
$233$	11.0501	0.723919	0.361959	+	0.932194i	$0.382108\pi$
$234$	0	0
$235$	−2.69021	−0.175490
$236$	0	0
$237$	34.1245	2.21662
$238$	0	0
$239$	10.9977	0.711385	0.355692	−	0.934603i	$-0.384245\pi$
$239$	10.9977	0.711385	0.355692	+	0.934603i	$0.384245\pi$
$240$	0	0
$241$	6.17226	0.397590	0.198795	−	0.980041i	$-0.436297\pi$
$241$	6.17226	0.397590	0.198795	+	0.980041i	$0.436297\pi$
$242$	0	0
$243$	15.4910	0.993747
$244$	0	0
$245$	−52.7037	−3.36712
$246$	0	0
$247$	2.22565	0.141615
$248$	0	0
$249$	30.2449	1.91669
$250$	0	0
$251$	5.28856	0.333811	0.166906	−	0.985973i	$-0.446622\pi$
$251$	5.28856	0.333811	0.166906	+	0.985973i	$0.446622\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−62.3008	−3.90143
$256$	0	0
$257$	5.04237	0.314535	0.157267	−	0.987556i	$-0.449732\pi$
$257$	5.04237	0.314535	0.157267	+	0.987556i	$0.449732\pi$
$258$	0	0
$259$	10.1851	0.632872
$260$	0	0
$261$	22.9050	1.41778
$262$	0	0
$263$	−24.8686	−1.53346	−0.766731	−	0.641968i	$-0.778118\pi$
$263$	−24.8686	−1.53346	−0.766731	+	0.641968i	$0.778118\pi$
$264$	0	0
$265$	5.84480	0.359043
$266$	0	0
$267$	21.6855	1.32713
$268$	0	0
$269$	5.24639	0.319878	0.159939	−	0.987127i	$-0.448870\pi$
$269$	5.24639	0.319878	0.159939	+	0.987127i	$0.448870\pi$
$270$	0	0
$271$	−16.5708	−1.00660	−0.503301	−	0.864111i	$-0.667881\pi$
$271$	−16.5708	−1.00660	−0.503301	+	0.864111i	$0.667881\pi$
$272$	0	0
$273$	−35.6255	−2.15615
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−15.8505	−0.952365	−0.476183	−	0.879346i	$-0.657980\pi$
$277$	−15.8505	−0.952365	−0.476183	+	0.879346i	$0.657980\pi$
$278$	0	0
$279$	−22.4840	−1.34608
$280$	0	0
$281$	18.9786	1.13217	0.566085	−	0.824347i	$-0.308458\pi$
$281$	18.9786	1.13217	0.566085	+	0.824347i	$0.308458\pi$
$282$	0	0
$283$	22.6770	1.34801	0.674003	−	0.738729i	$-0.264574\pi$
$283$	22.6770	1.34801	0.674003	+	0.738729i	$0.264574\pi$
$284$	0	0
$285$	8.51486	0.504377
$286$	0	0
$287$	−43.6052	−2.57394
$288$	0	0
$289$	36.5343	2.14907
$290$	0	0
$291$	−5.18438	−0.303914
$292$	0	0
$293$	5.60276	0.327317	0.163658	−	0.986517i	$-0.447671\pi$
$293$	5.60276	0.327317	0.163658	+	0.986517i	$0.447671\pi$
$294$	0	0
$295$	−20.6904	−1.20464
$296$	0	0
$297$	0	0
$298$	0	0
$299$	3.28105	0.189748
$300$	0	0
$301$	−12.1116	−0.698104
$302$	0	0
$303$	−29.1381	−1.67394
$304$	0	0
$305$	9.91027	0.567461
$306$	0	0
$307$	8.93573	0.509989	0.254994	−	0.966943i	$-0.417926\pi$
$307$	8.93573	0.509989	0.254994	+	0.966943i	$0.417926\pi$
$308$	0	0
$309$	−46.5420	−2.64768
$310$	0	0
$311$	−8.19687	−0.464802	−0.232401	−	0.972620i	$-0.574658\pi$
$311$	−8.19687	−0.464802	−0.232401	+	0.972620i	$0.574658\pi$
$312$	0	0
$313$	−8.52611	−0.481924	−0.240962	−	0.970534i	$-0.577463\pi$
$313$	−8.52611	−0.481924	−0.240962	+	0.970534i	$0.577463\pi$
$314$	0	0
$315$	−94.3021	−5.31332
$316$	0	0
$317$	5.83246	0.327583	0.163792	−	0.986495i	$-0.447628\pi$
$317$	5.83246	0.327583	0.163792	+	0.986495i	$0.447628\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−22.0579	−1.23115
$322$	0	0
$323$	−7.31671	−0.407113
$324$	0	0
$325$	−5.44436	−0.301999
$326$	0	0
$327$	−40.3076	−2.22902
$328$	0	0
$329$	5.05724	0.278815
$330$	0	0
$331$	13.6180	0.748513	0.374256	−	0.927325i	$-0.377898\pi$
$331$	13.6180	0.748513	0.374256	+	0.927325i	$0.377898\pi$
$332$	0	0
$333$	13.3762	0.733009
$334$	0	0
$335$	−37.8230	−2.06649
$336$	0	0
$337$	3.95006	0.215173	0.107587	−	0.994196i	$-0.465688\pi$
$337$	3.95006	0.215173	0.107587	+	0.994196i	$0.465688\pi$
$338$	0	0
$339$	50.5118	2.74342
$340$	0	0
$341$	0	0
$342$	0	0
$343$	63.1678	3.41074
$344$	0	0
$345$	12.5526	0.675809
$346$	0	0
$347$	−14.7101	−0.789681	−0.394841	−	0.918750i	$-0.629200\pi$
$347$	−14.7101	−0.789681	−0.394841	+	0.918750i	$0.629200\pi$
$348$	0	0
$349$	10.4894	0.561483	0.280741	−	0.959783i	$-0.409420\pi$
$349$	10.4894	0.561483	0.280741	+	0.959783i	$0.409420\pi$
$350$	0	0
$351$	−25.9525	−1.38524
$352$	0	0
$353$	17.3929	0.925733	0.462867	−	0.886428i	$-0.346821\pi$
$353$	17.3929	0.925733	0.462867	+	0.886428i	$0.346821\pi$
$354$	0	0
$355$	−39.2871	−2.08514
$356$	0	0
$357$	117.117	6.19849
$358$	0	0
$359$	18.4006	0.971144	0.485572	−	0.874197i	$-0.338611\pi$
$359$	18.4006	0.971144	0.485572	+	0.874197i	$0.338611\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−8.01327	−0.419433
$366$	0	0
$367$	−14.0611	−0.733982	−0.366991	−	0.930224i	$-0.619612\pi$
$367$	−14.0611	−0.733982	−0.366991	+	0.930224i	$0.619612\pi$
$368$	0	0
$369$	−57.2670	−2.98120
$370$	0	0
$371$	−10.9875	−0.570440
$372$	0	0
$373$	−36.3515	−1.88221	−0.941106	−	0.338112i	$-0.890212\pi$
$373$	−36.3515	−1.88221	−0.941106	+	0.338112i	$0.890212\pi$
$374$	0	0
$375$	21.7453	1.12292
$376$	0	0
$377$	−7.56705	−0.389723
$378$	0	0
$379$	24.5666	1.26190	0.630952	−	0.775822i	$-0.282665\pi$
$379$	24.5666	1.26190	0.630952	+	0.775822i	$0.282665\pi$
$380$	0	0
$381$	9.64431	0.494093
$382$	0	0
$383$	−16.7080	−0.853739	−0.426870	−	0.904313i	$-0.640384\pi$
$383$	−16.7080	−0.853739	−0.426870	+	0.904313i	$0.640384\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−15.9063	−0.808563
$388$	0	0
$389$	3.83619	0.194502	0.0972512	−	0.995260i	$-0.468995\pi$
$389$	3.83619	0.194502	0.0972512	+	0.995260i	$0.468995\pi$
$390$	0	0
$391$	−10.7863	−0.545486
$392$	0	0
$393$	−29.7749	−1.50194
$394$	0	0
$395$	−29.8417	−1.50150
$396$	0	0
$397$	36.1233	1.81297	0.906487	−	0.422233i	$-0.138753\pi$
$397$	36.1233	1.81297	0.906487	+	0.422233i	$0.138753\pi$
$398$	0	0
$399$	−16.0068	−0.801342
$400$	0	0
$401$	16.1216	0.805072	0.402536	−	0.915404i	$-0.368129\pi$
$401$	16.1216	0.805072	0.402536	+	0.915404i	$0.368129\pi$
$402$	0	0
$403$	7.42798	0.370014
$404$	0	0
$405$	−44.1382	−2.19324
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−23.1921	−1.14678	−0.573389	−	0.819283i	$-0.694372\pi$
$409$	−23.1921	−1.14678	−0.573389	+	0.819283i	$0.694372\pi$
$410$	0	0
$411$	−27.7705	−1.36982
$412$	0	0
$413$	38.8952	1.91391
$414$	0	0
$415$	−26.4490	−1.29833
$416$	0	0
$417$	−33.1729	−1.62448
$418$	0	0
$419$	−2.05703	−0.100492	−0.0502462	−	0.998737i	$-0.516001\pi$
$419$	−2.05703	−0.100492	−0.0502462	+	0.998737i	$0.516001\pi$
$420$	0	0
$421$	1.40967	0.0687030	0.0343515	−	0.999410i	$-0.489063\pi$
$421$	1.40967	0.0687030	0.0343515	+	0.999410i	$0.489063\pi$
$422$	0	0
$423$	6.64171	0.322931
$424$	0	0
$425$	17.8981	0.868184
$426$	0	0
$427$	−18.6300	−0.901568
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−5.49061	−0.264473	−0.132237	−	0.991218i	$-0.542216\pi$
$431$	−5.49061	−0.264473	−0.132237	+	0.991218i	$0.542216\pi$
$432$	0	0
$433$	−14.5878	−0.701046	−0.350523	−	0.936554i	$-0.613996\pi$
$433$	−14.5878	−0.701046	−0.350523	+	0.936554i	$0.613996\pi$
$434$	0	0
$435$	−28.9499	−1.38804
$436$	0	0
$437$	1.47420	0.0705205
$438$	0	0
$439$	6.27926	0.299693	0.149846	−	0.988709i	$-0.452122\pi$
$439$	6.27926	0.299693	0.149846	+	0.988709i	$0.452122\pi$
$440$	0	0
$441$	130.117	6.19605
$442$	0	0
$443$	8.62815	0.409936	0.204968	−	0.978769i	$-0.434291\pi$
$443$	8.62815	0.409936	0.204968	+	0.978769i	$0.434291\pi$
$444$	0	0
$445$	−18.9639	−0.898973
$446$	0	0
$447$	−29.9781	−1.41792
$448$	0	0
$449$	21.9377	1.03531	0.517653	−	0.855591i	$-0.326806\pi$
$449$	21.9377	1.03531	0.517653	+	0.855591i	$0.326806\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−19.7576	−0.928291
$454$	0	0
$455$	31.1543	1.46054
$456$	0	0
$457$	12.0124	0.561918	0.280959	−	0.959720i	$-0.409348\pi$
$457$	12.0124	0.561918	0.280959	+	0.959720i	$0.409348\pi$
$458$	0	0
$459$	85.3174	3.98228
$460$	0	0
$461$	−22.7493	−1.05954	−0.529771	−	0.848141i	$-0.677722\pi$
$461$	−22.7493	−1.05954	−0.529771	+	0.848141i	$0.677722\pi$
$462$	0	0
$463$	19.1300	0.889049	0.444524	−	0.895767i	$-0.353373\pi$
$463$	19.1300	0.889049	0.444524	+	0.895767i	$0.353373\pi$
$464$	0	0
$465$	28.4179	1.31785
$466$	0	0
$467$	38.9724	1.80343	0.901714	−	0.432334i	$-0.142310\pi$
$467$	38.9724	1.80343	0.901714	+	0.432334i	$0.142310\pi$
$468$	0	0
$469$	71.1021	3.28319
$470$	0	0
$471$	−30.3665	−1.39921
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−2.44619	−0.112239
$476$	0	0
$477$	−14.4299	−0.660699
$478$	0	0
$479$	−19.8793	−0.908307	−0.454153	−	0.890923i	$-0.650058\pi$
$479$	−19.8793	−0.908307	−0.454153	+	0.890923i	$0.650058\pi$
$480$	0	0
$481$	−4.41904	−0.201491
$482$	0	0
$483$	−23.5972	−1.07371
$484$	0	0
$485$	4.53371	0.205865
$486$	0	0
$487$	30.2953	1.37281	0.686407	−	0.727218i	$-0.259187\pi$
$487$	30.2953	1.37281	0.686407	+	0.727218i	$0.259187\pi$
$488$	0	0
$489$	−48.3974	−2.18861
$490$	0	0
$491$	−22.3023	−1.00649	−0.503245	−	0.864144i	$-0.667861\pi$
$491$	−22.3023	−1.00649	−0.503245	+	0.864144i	$0.667861\pi$
$492$	0	0
$493$	24.8763	1.12037
$494$	0	0
$495$	0	0
$496$	0	0
$497$	73.8545	3.31283
$498$	0	0
$499$	10.4149	0.466234	0.233117	−	0.972449i	$-0.425108\pi$
$499$	10.4149	0.466234	0.233117	+	0.972449i	$0.425108\pi$
$500$	0	0
$501$	−16.5431	−0.739090
$502$	0	0
$503$	28.6298	1.27654	0.638270	−	0.769813i	$-0.279650\pi$
$503$	28.6298	1.27654	0.638270	+	0.769813i	$0.279650\pi$
$504$	0	0
$505$	25.4811	1.13389
$506$	0	0
$507$	−25.1083	−1.11510
$508$	0	0
$509$	−22.0977	−0.979465	−0.489733	−	0.871873i	$-0.662906\pi$
$509$	−22.0977	−0.979465	−0.489733	+	0.871873i	$0.662906\pi$
$510$	0	0
$511$	15.0639	0.666386
$512$	0	0
$513$	−11.6606	−0.514829
$514$	0	0
$515$	40.7007	1.79349
$516$	0	0
$517$	0	0
$518$	0	0
$519$	50.7803	2.22901
$520$	0	0
$521$	−26.1243	−1.14453	−0.572263	−	0.820070i	$-0.693934\pi$
$521$	−26.1243	−1.14453	−0.572263	+	0.820070i	$0.693934\pi$
$522$	0	0
$523$	−9.16845	−0.400909	−0.200454	−	0.979703i	$-0.564242\pi$
$523$	−9.16845	−0.400909	−0.200454	+	0.979703i	$0.564242\pi$
$524$	0	0
$525$	39.1557	1.70889
$526$	0	0
$527$	−24.4191	−1.06371
$528$	0	0
$529$	−20.8267	−0.905510
$530$	0	0
$531$	51.0813	2.21674
$532$	0	0
$533$	18.9191	0.819479
$534$	0	0
$535$	19.2895	0.833959
$536$	0	0
$537$	2.37834	0.102633
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−14.2689	−0.613469	−0.306735	−	0.951795i	$-0.599236\pi$
$541$	−14.2689	−0.613469	−0.306735	+	0.951795i	$0.599236\pi$
$542$	0	0
$543$	−43.6349	−1.87255
$544$	0	0
$545$	35.2488	1.50989
$546$	0	0
$547$	14.0347	0.600078	0.300039	−	0.953927i	$-0.403000\pi$
$547$	14.0347	0.600078	0.300039	+	0.953927i	$0.403000\pi$
$548$	0	0
$549$	−24.4669	−1.04422
$550$	0	0
$551$	−3.39993	−0.144842
$552$	0	0
$553$	56.0983	2.38554
$554$	0	0
$555$	−16.9063	−0.717633
$556$	0	0
$557$	−31.6373	−1.34052	−0.670259	−	0.742128i	$-0.733817\pi$
$557$	−31.6373	−1.34052	−0.670259	+	0.742128i	$0.733817\pi$
$558$	0	0
$559$	5.25492	0.222259
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−39.7092	−1.67355	−0.836773	−	0.547551i	$-0.815560\pi$
$563$	−39.7092	−1.67355	−0.836773	+	0.547551i	$0.815560\pi$
$564$	0	0
$565$	−44.1722	−1.85834
$566$	0	0
$567$	82.9739	3.48458
$568$	0	0
$569$	27.2590	1.14276	0.571378	−	0.820687i	$-0.306409\pi$
$569$	27.2590	1.14276	0.571378	+	0.820687i	$0.306409\pi$
$570$	0	0
$571$	17.1231	0.716579	0.358290	−	0.933610i	$-0.383360\pi$
$571$	17.1231	0.716579	0.358290	+	0.933610i	$0.383360\pi$
$572$	0	0
$573$	75.4937	3.15379
$574$	0	0
$575$	−3.60617	−0.150388
$576$	0	0
$577$	9.01665	0.375368	0.187684	−	0.982229i	$-0.439902\pi$
$577$	9.01665	0.375368	0.187684	+	0.982229i	$0.439902\pi$
$578$	0	0
$579$	24.6003	1.02235
$580$	0	0
$581$	49.7205	2.06276
$582$	0	0
$583$	0	0
$584$	0	0
$585$	40.9151	1.69163
$586$	0	0
$587$	46.6777	1.92659	0.963297	−	0.268438i	$-0.0865074\pi$
$587$	46.6777	1.92659	0.963297	+	0.268438i	$0.0865074\pi$
$588$	0	0
$589$	3.33744	0.137517
$590$	0	0
$591$	−1.97770	−0.0813517
$592$	0	0
$593$	−44.7414	−1.83731	−0.918655	−	0.395060i	$-0.870724\pi$
$593$	−44.7414	−1.83731	−0.918655	+	0.395060i	$0.870724\pi$
$594$	0	0
$595$	−102.418	−4.19874
$596$	0	0
$597$	70.2149	2.87370
$598$	0	0
$599$	−37.2767	−1.52308	−0.761542	−	0.648115i	$-0.775557\pi$
$599$	−37.2767	−1.52308	−0.761542	+	0.648115i	$0.775557\pi$
$600$	0	0
$601$	−10.2258	−0.417119	−0.208560	−	0.978010i	$-0.566878\pi$
$601$	−10.2258	−0.417119	−0.208560	+	0.978010i	$0.566878\pi$
$602$	0	0
$603$	93.3788	3.80268
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−2.88075	−0.116926	−0.0584630	−	0.998290i	$-0.518620\pi$
$607$	−2.88075	−0.116926	−0.0584630	+	0.998290i	$0.518620\pi$
$608$	0	0
$609$	54.4220	2.20529
$610$	0	0
$611$	−2.19420	−0.0887679
$612$	0	0
$613$	−13.5527	−0.547390	−0.273695	−	0.961817i	$-0.588246\pi$
$613$	−13.5527	−0.547390	−0.273695	+	0.961817i	$0.588246\pi$
$614$	0	0
$615$	72.3806	2.91867
$616$	0	0
$617$	−13.4888	−0.543039	−0.271519	−	0.962433i	$-0.587526\pi$
$617$	−13.4888	−0.543039	−0.271519	+	0.962433i	$0.587526\pi$
$618$	0	0
$619$	19.5451	0.785586	0.392793	−	0.919627i	$-0.371509\pi$
$619$	19.5451	0.785586	0.392793	+	0.919627i	$0.371509\pi$
$620$	0	0
$621$	−17.1901	−0.689815
$622$	0	0
$623$	35.6495	1.42827
$624$	0	0
$625$	−31.2471	−1.24988
$626$	0	0
$627$	0	0
$628$	0	0
$629$	14.5274	0.579245
$630$	0	0
$631$	0.157148	0.00625597	0.00312799	−	0.999995i	$-0.499004\pi$
$631$	0.157148	0.00625597	0.00312799	+	0.999995i	$0.499004\pi$
$632$	0	0
$633$	−79.0234	−3.14090
$634$	0	0
$635$	−8.43389	−0.334689
$636$	0	0
$637$	−42.9864	−1.70318
$638$	0	0
$639$	96.9936	3.83701
$640$	0	0
$641$	0.492765	0.0194631	0.00973153	−	0.999953i	$-0.496902\pi$
$641$	0.492765	0.0194631	0.00973153	+	0.999953i	$0.496902\pi$
$642$	0	0
$643$	−43.3127	−1.70809	−0.854044	−	0.520201i	$-0.825857\pi$
$643$	−43.3127	−1.70809	−0.854044	+	0.520201i	$0.825857\pi$
$644$	0	0
$645$	20.1042	0.791602
$646$	0	0
$647$	−43.4075	−1.70652	−0.853262	−	0.521483i	$-0.825379\pi$
$647$	−43.4075	−1.70652	−0.853262	+	0.521483i	$0.825379\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−53.4218	−2.09376
$652$	0	0
$653$	−15.5859	−0.609925	−0.304962	−	0.952364i	$-0.598644\pi$
$653$	−15.5859	−0.609925	−0.304962	+	0.952364i	$0.598644\pi$
$654$	0	0
$655$	26.0380	1.01739
$656$	0	0
$657$	19.7835	0.771827
$658$	0	0
$659$	14.2499	0.555099	0.277549	−	0.960711i	$-0.410478\pi$
$659$	14.2499	0.555099	0.277549	+	0.960711i	$0.410478\pi$
$660$	0	0
$661$	16.8837	0.656699	0.328350	−	0.944556i	$-0.393508\pi$
$661$	16.8837	0.656699	0.328350	+	0.944556i	$0.393508\pi$
$662$	0	0
$663$	−50.8139	−1.97345
$664$	0	0
$665$	13.9978	0.542813
$666$	0	0
$667$	−5.01218	−0.194072
$668$	0	0
$669$	−84.9258	−3.28342
$670$	0	0
$671$	0	0
$672$	0	0
$673$	39.1449	1.50892	0.754462	−	0.656343i	$-0.227898\pi$
$673$	39.1449	1.50892	0.754462	+	0.656343i	$0.227898\pi$
$674$	0	0
$675$	28.5241	1.09789
$676$	0	0
$677$	36.6870	1.41000	0.704998	−	0.709209i	$-0.250948\pi$
$677$	36.6870	1.41000	0.704998	+	0.709209i	$0.250948\pi$
$678$	0	0
$679$	−8.52276	−0.327074
$680$	0	0
$681$	−3.34000	−0.127989
$682$	0	0
$683$	−9.25467	−0.354120	−0.177060	−	0.984200i	$-0.556659\pi$
$683$	−9.25467	−0.354120	−0.177060	+	0.984200i	$0.556659\pi$
$684$	0	0
$685$	24.2852	0.927888
$686$	0	0
$687$	11.9061	0.454247
$688$	0	0
$689$	4.76716	0.181614
$690$	0	0
$691$	−43.3418	−1.64880	−0.824400	−	0.566008i	$-0.808487\pi$
$691$	−43.3418	−1.64880	−0.824400	+	0.566008i	$0.808487\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	29.0095	1.10039
$696$	0	0
$697$	−62.1957	−2.35583
$698$	0	0
$699$	34.4809	1.30419
$700$	0	0
$701$	17.6127	0.665223	0.332611	−	0.943064i	$-0.392070\pi$
$701$	17.6127	0.665223	0.332611	+	0.943064i	$0.392070\pi$
$702$	0	0
$703$	−1.98551	−0.0748848
$704$	0	0
$705$	−8.39455	−0.316157
$706$	0	0
$707$	−47.9011	−1.80151
$708$	0	0
$709$	−21.2610	−0.798473	−0.399237	−	0.916848i	$-0.630725\pi$
$709$	−21.2610	−0.798473	−0.399237	+	0.916848i	$0.630725\pi$
$710$	0	0
$711$	73.6743	2.76300
$712$	0	0
$713$	4.92006	0.184258
$714$	0	0
$715$	0	0
$716$	0	0
$717$	34.3174	1.28161
$718$	0	0
$719$	−34.6096	−1.29072	−0.645359	−	0.763879i	$-0.723292\pi$
$719$	−34.6096	−1.29072	−0.645359	+	0.763879i	$0.723292\pi$
$720$	0	0
$721$	−76.5119	−2.84945
$722$	0	0
$723$	19.2599	0.716284
$724$	0	0
$725$	8.31688	0.308881
$726$	0	0
$727$	−34.3480	−1.27390	−0.636948	−	0.770906i	$-0.719804\pi$
$727$	−34.3480	−1.27390	−0.636948	+	0.770906i	$0.719804\pi$
$728$	0	0
$729$	−0.187287	−0.00693657
$730$	0	0
$731$	−17.2753	−0.638949
$732$	0	0
$733$	−22.9284	−0.846879	−0.423439	−	0.905924i	$-0.639177\pi$
$733$	−22.9284	−0.846879	−0.423439	+	0.905924i	$0.639177\pi$
$734$	0	0
$735$	−164.457	−6.06607
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0.615346	0.0226359	0.0113179	−	0.999936i	$-0.496397\pi$
$739$	0.615346	0.0226359	0.0113179	+	0.999936i	$0.496397\pi$
$740$	0	0
$741$	6.94492	0.255128
$742$	0	0
$743$	−31.4662	−1.15438	−0.577191	−	0.816609i	$-0.695851\pi$
$743$	−31.4662	−1.15438	−0.577191	+	0.816609i	$0.695851\pi$
$744$	0	0
$745$	26.2157	0.960468
$746$	0	0
$747$	65.2983	2.38914
$748$	0	0
$749$	−36.2618	−1.32498
$750$	0	0
$751$	−45.8699	−1.67382	−0.836908	−	0.547344i	$-0.815639\pi$
$751$	−45.8699	−1.67382	−0.836908	+	0.547344i	$0.815639\pi$
$752$	0	0
$753$	16.5024	0.601382
$754$	0	0
$755$	17.2779	0.628806
$756$	0	0
$757$	27.8578	1.01251	0.506255	−	0.862384i	$-0.331030\pi$
$757$	27.8578	1.01251	0.506255	+	0.862384i	$0.331030\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−35.7692	−1.29663	−0.648317	−	0.761370i	$-0.724527\pi$
$761$	−35.7692	−1.29663	−0.648317	+	0.761370i	$0.724527\pi$
$762$	0	0
$763$	−66.2630	−2.39888
$764$	0	0
$765$	−134.506	−4.86309
$766$	0	0
$767$	−16.8756	−0.609342
$768$	0	0
$769$	36.8627	1.32930	0.664651	−	0.747154i	$-0.268580\pi$
$769$	36.8627	1.32930	0.664651	+	0.747154i	$0.268580\pi$
$770$	0	0
$771$	15.7342	0.566654
$772$	0	0
$773$	17.5177	0.630068	0.315034	−	0.949080i	$-0.397984\pi$
$773$	17.5177	0.630068	0.315034	+	0.949080i	$0.397984\pi$
$774$	0	0
$775$	−8.16403	−0.293260
$776$	0	0
$777$	31.7816	1.14016
$778$	0	0
$779$	8.50050	0.304562
$780$	0	0
$781$	0	0
$782$	0	0
$783$	39.6453	1.41681
$784$	0	0
$785$	26.5553	0.947800
$786$	0	0
$787$	−54.4756	−1.94184	−0.970922	−	0.239396i	$-0.923051\pi$
$787$	−54.4756	−1.94184	−0.970922	+	0.239396i	$0.923051\pi$
$788$	0	0
$789$	−77.6000	−2.76263
$790$	0	0
$791$	83.0379	2.95249
$792$	0	0
$793$	8.08305	0.287037
$794$	0	0
$795$	18.2381	0.646840
$796$	0	0
$797$	27.5174	0.974715	0.487358	−	0.873202i	$-0.337961\pi$
$797$	27.5174	0.974715	0.487358	+	0.873202i	$0.337961\pi$
$798$	0	0
$799$	7.21333	0.255189
$800$	0	0
$801$	46.8187	1.65426
$802$	0	0
$803$	0	0
$804$	0	0
$805$	20.6356	0.727310
$806$	0	0
$807$	16.3708	0.576281
$808$	0	0
$809$	14.7423	0.518310	0.259155	−	0.965836i	$-0.416556\pi$
$809$	14.7423	0.518310	0.259155	+	0.965836i	$0.416556\pi$
$810$	0	0
$811$	−25.6478	−0.900615	−0.450308	−	0.892873i	$-0.648686\pi$
$811$	−25.6478	−0.900615	−0.450308	+	0.892873i	$0.648686\pi$
$812$	0	0
$813$	−51.7074	−1.81346
$814$	0	0
$815$	42.3232	1.48252
$816$	0	0
$817$	2.36107	0.0826034
$818$	0	0
$819$	−76.9150	−2.68763
$820$	0	0
$821$	−35.9928	−1.25616	−0.628078	−	0.778150i	$-0.716158\pi$
$821$	−35.9928	−1.25616	−0.628078	+	0.778150i	$0.716158\pi$
$822$	0	0
$823$	−18.6406	−0.649770	−0.324885	−	0.945754i	$-0.605326\pi$
$823$	−18.6406	−0.649770	−0.324885	+	0.945754i	$0.605326\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0.277915	0.00966406	0.00483203	−	0.999988i	$-0.498462\pi$
$827$	0.277915	0.00966406	0.00483203	+	0.999988i	$0.498462\pi$
$828$	0	0
$829$	3.44865	0.119777	0.0598883	−	0.998205i	$-0.480926\pi$
$829$	3.44865	0.119777	0.0598883	+	0.998205i	$0.480926\pi$
$830$	0	0
$831$	−49.4600	−1.71575
$832$	0	0
$833$	141.316	4.89629
$834$	0	0
$835$	14.4668	0.500645
$836$	0	0
$837$	−38.9167	−1.34516
$838$	0	0
$839$	11.4613	0.395687	0.197843	−	0.980234i	$-0.436606\pi$
$839$	11.4613	0.395687	0.197843	+	0.980234i	$0.436606\pi$
$840$	0	0
$841$	−17.4405	−0.601395
$842$	0	0
$843$	59.2209	2.03968
$844$	0	0
$845$	21.9570	0.755344
$846$	0	0
$847$	0	0
$848$	0	0
$849$	70.7612	2.42852
$850$	0	0
$851$	−2.92703	−0.100337
$852$	0	0
$853$	−13.3326	−0.456500	−0.228250	−	0.973603i	$-0.573300\pi$
$853$	−13.3326	−0.456500	−0.228250	+	0.973603i	$0.573300\pi$
$854$	0	0
$855$	18.3835	0.628701
$856$	0	0
$857$	6.22828	0.212754	0.106377	−	0.994326i	$-0.466075\pi$
$857$	6.22828	0.212754	0.106377	+	0.994326i	$0.466075\pi$
$858$	0	0
$859$	−11.5705	−0.394782	−0.197391	−	0.980325i	$-0.563247\pi$
$859$	−11.5705	−0.394782	−0.197391	+	0.980325i	$0.563247\pi$
$860$	0	0
$861$	−136.066	−4.63711
$862$	0	0
$863$	52.8830	1.80016	0.900079	−	0.435727i	$-0.143509\pi$
$863$	52.8830	1.80016	0.900079	+	0.435727i	$0.143509\pi$
$864$	0	0
$865$	−44.4070	−1.50988
$866$	0	0
$867$	114.002	3.87170
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−30.8493	−1.04529
$872$	0	0
$873$	−11.1930	−0.378826
$874$	0	0
$875$	35.7478	1.20850
$876$	0	0
$877$	31.9765	1.07977	0.539884	−	0.841739i	$-0.318468\pi$
$877$	31.9765	1.07977	0.539884	+	0.841739i	$0.318468\pi$
$878$	0	0
$879$	17.4828	0.589682
$880$	0	0
$881$	−5.86371	−0.197553	−0.0987767	−	0.995110i	$-0.531493\pi$
$881$	−5.86371	−0.197553	−0.0987767	+	0.995110i	$0.531493\pi$
$882$	0	0
$883$	−45.3608	−1.52651	−0.763256	−	0.646096i	$-0.776401\pi$
$883$	−45.3608	−1.52651	−0.763256	+	0.646096i	$0.776401\pi$
$884$	0	0
$885$	−64.5624	−2.17024
$886$	0	0
$887$	4.84138	0.162557	0.0812787	−	0.996691i	$-0.474100\pi$
$887$	4.84138	0.162557	0.0812787	+	0.996691i	$0.474100\pi$
$888$	0	0
$889$	15.8546	0.531746
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−0.985870	−0.0329909
$894$	0	0
$895$	−2.07984	−0.0695214
$896$	0	0
$897$	10.2382	0.341843
$898$	0	0
$899$	−11.3471	−0.378446
$900$	0	0
$901$	−15.6718	−0.522103
$902$	0	0
$903$	−37.7932	−1.25768
$904$	0	0
$905$	38.1585	1.26843
$906$	0	0
$907$	2.81871	0.0935937	0.0467968	−	0.998904i	$-0.485099\pi$
$907$	2.81871	0.0935937	0.0467968	+	0.998904i	$0.485099\pi$
$908$	0	0
$909$	−62.9088	−2.08655
$910$	0	0
$911$	11.4181	0.378299	0.189150	−	0.981948i	$-0.439427\pi$
$911$	11.4181	0.378299	0.189150	+	0.981948i	$0.439427\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	30.9240	1.02232
$916$	0	0
$917$	−48.9479	−1.61640
$918$	0	0
$919$	−20.3585	−0.671564	−0.335782	−	0.941940i	$-0.609001\pi$
$919$	−20.3585	−0.671564	−0.335782	+	0.941940i	$0.609001\pi$
$920$	0	0
$921$	27.8831	0.918778
$922$	0	0
$923$	−32.0435	−1.05472
$924$	0	0
$925$	4.85693	0.159695
$926$	0	0
$927$	−100.484	−3.30031
$928$	0	0
$929$	−2.44257	−0.0801381	−0.0400690	−	0.999197i	$-0.512758\pi$
$929$	−2.44257	−0.0801381	−0.0400690	+	0.999197i	$0.512758\pi$
$930$	0	0
$931$	−19.3141	−0.632993
$932$	0	0
$933$	−25.5775	−0.837371
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−44.5993	−1.45700	−0.728498	−	0.685048i	$-0.759781\pi$
$937$	−44.5993	−1.45700	−0.728498	+	0.685048i	$0.759781\pi$
$938$	0	0
$939$	−26.6049	−0.868218
$940$	0	0
$941$	10.8386	0.353327	0.176664	−	0.984271i	$-0.443469\pi$
$941$	10.8386	0.353327	0.176664	+	0.984271i	$0.443469\pi$
$942$	0	0
$943$	12.5314	0.408080
$944$	0	0
$945$	−163.224	−5.30966
$946$	0	0
$947$	0.142195	0.00462071	0.00231036	−	0.999997i	$-0.499265\pi$
$947$	0.142195	0.00462071	0.00231036	+	0.999997i	$0.499265\pi$
$948$	0	0
$949$	−6.53581	−0.212161
$950$	0	0
$951$	18.1996	0.590162
$952$	0	0
$953$	38.6299	1.25135	0.625673	−	0.780085i	$-0.284824\pi$
$953$	38.6299	1.25135	0.625673	+	0.780085i	$0.284824\pi$
$954$	0	0
$955$	−66.0188	−2.13632
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−45.6529	−1.47421
$960$	0	0
$961$	−19.8615	−0.640693
$962$	0	0
$963$	−47.6228	−1.53462
$964$	0	0
$965$	−21.5128	−0.692521
$966$	0	0
$967$	−16.4135	−0.527824	−0.263912	−	0.964547i	$-0.585013\pi$
$967$	−16.4135	−0.527824	−0.263912	+	0.964547i	$0.585013\pi$
$968$	0	0
$969$	−22.8311	−0.733440
$970$	0	0
$971$	−50.3184	−1.61479	−0.807397	−	0.590009i	$-0.799124\pi$
$971$	−50.3184	−1.61479	−0.807397	+	0.590009i	$0.799124\pi$
$972$	0	0
$973$	−54.5339	−1.74828
$974$	0	0
$975$	−16.9886	−0.544070
$976$	0	0
$977$	11.4858	0.367463	0.183731	−	0.982976i	$-0.441182\pi$
$977$	11.4858	0.367463	0.183731	+	0.982976i	$0.441182\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−87.0236	−2.77845
$982$	0	0
$983$	−18.2947	−0.583510	−0.291755	−	0.956493i	$-0.594239\pi$
$983$	−18.2947	−0.583510	−0.291755	+	0.956493i	$0.594239\pi$
$984$	0	0
$985$	1.72949	0.0551060
$986$	0	0
$987$	15.7806	0.502303
$988$	0	0
$989$	3.48069	0.110680
$990$	0	0
$991$	10.8582	0.344922	0.172461	−	0.985016i	$-0.444828\pi$
$991$	10.8582	0.344922	0.172461	+	0.985016i	$0.444828\pi$
$992$	0	0
$993$	42.4936	1.34849
$994$	0	0
$995$	−61.4025	−1.94659
$996$	0	0
$997$	11.7038	0.370662	0.185331	−	0.982676i	$-0.440664\pi$
$997$	11.7038	0.370662	0.185331	+	0.982676i	$0.440664\pi$
$998$	0	0
$999$	23.1523	0.732505

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9196.2.a.x.1.19		20
11.3	even	5		836.2.j.c.229.2	✓	40
11.4	even	5		836.2.j.c.533.2	yes	40
11.10	odd	2		9196.2.a.w.1.19		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
836.2.j.c.229.2	✓	40	11.3	even	5
836.2.j.c.533.2	yes	40	11.4	even	5
9196.2.a.w.1.19		20	11.10	odd	2
9196.2.a.x.1.19		20	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)

\(f(q)\)	\(=\)	\(q+3.12040 q^{3} -2.72877 q^{5} +5.12972 q^{7} +6.73690 q^{9} -2.22565 q^{13} -8.51486 q^{15} +7.31671 q^{17} -1.00000 q^{19} +16.0068 q^{21} -1.47420 q^{23} +2.44619 q^{25} +11.6606 q^{27} +3.39993 q^{29} -3.33744 q^{31} -13.9978 q^{35} +1.98551 q^{37} -6.94492 q^{39} -8.50050 q^{41} -2.36107 q^{43} -18.3835 q^{45} +0.985870 q^{47} +19.3141 q^{49} +22.8311 q^{51} -2.14192 q^{53} -3.12040 q^{57} +7.58232 q^{59} -3.63177 q^{61} +34.5584 q^{63} +6.07329 q^{65} +13.8608 q^{67} -4.60009 q^{69} +14.3974 q^{71} +2.93658 q^{73} +7.63310 q^{75} +10.9359 q^{79} +16.1751 q^{81} +9.69263 q^{83} -19.9656 q^{85} +10.6091 q^{87} +6.94960 q^{89} -11.4170 q^{91} -10.4142 q^{93} +2.72877 q^{95} -1.66145 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 6 q^{3} + 2 q^{5} + 4 q^{7} + 28 q^{9} - 5 q^{13} + 14 q^{15} - 6 q^{17} - 20 q^{19} - q^{21} + 18 q^{23} + 44 q^{25} + 24 q^{27} + q^{29} + 23 q^{31} - 36 q^{35} + q^{37} + 21 q^{39} + 6 q^{41}+ \cdots + 9 q^{97}+O(q^{100}) \) 20 * q + 6 * q^3 + 2 * q^5 + 4 * q^7 + 28 * q^9 - 5 * q^13 + 14 * q^15 - 6 * q^17 - 20 * q^19 - q^21 + 18 * q^23 + 44 * q^25 + 24 * q^27 + q^29 + 23 * q^31 - 36 * q^35 + q^37 + 21 * q^39 + 6 * q^41 - 8 * q^43 - 15 * q^45 + 34 * q^47 + 42 * q^49 + 14 * q^51 - 10 * q^53 - 6 * q^57 + 52 * q^59 - 2 * q^61 + 59 * q^63 + 25 * q^65 + 14 * q^67 + 15 * q^69 + 88 * q^71 - 13 * q^73 + 15 * q^75 - 2 * q^79 + 48 * q^81 - 2 * q^83 + 11 * q^85 + 37 * q^87 + 46 * q^89 - 4 * q^91 + 2 * q^93 - 2 * q^95 + 9 * q^97

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