LMFDB - Embedded newform 882.3.b.g.197.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [882,3,Mod(197,882)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(882, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("882.197");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 882 = 2 \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	882.b (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$24.0327593166$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{37})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 13x^{2} + 14x + 123 $ x^4 - 2x^3 - 13x^2 + 14*x + 123
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			197.1
Root			$3.54138 + 1.41421i$ of defining polynomial
Character	$\chi$	$=$	882.197
Dual form			882.3.b.g.197.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{2} -2.00000 q^{4} -8.60233i q^{5} +2.82843i q^{8} +O(q^{10})$	$q-1.41421i q^{2} -2.00000 q^{4} -8.60233i q^{5} +2.82843i q^{8} -12.1655 q^{10} +2.82843i q^{11} +12.1655 q^{13} +4.00000 q^{16} -25.8070i q^{17} -24.3311 q^{19} +17.2047i q^{20} +4.00000 q^{22} -42.4264i q^{23} -49.0000 q^{25} -17.2047i q^{26} +15.5563i q^{29} +24.3311 q^{31} -5.65685i q^{32} -36.4966 q^{34} -6.00000 q^{37} +34.4093i q^{38} +24.3311 q^{40} -25.8070i q^{41} -68.0000 q^{43} -5.65685i q^{44} -60.0000 q^{46} +68.8186i q^{47} +69.2965i q^{50} -24.3311 q^{52} +41.0122i q^{53} +24.3311 q^{55} +22.0000 q^{58} +68.8186i q^{59} +97.3242 q^{61} -34.4093i q^{62} -8.00000 q^{64} -104.652i q^{65} -104.000 q^{67} +51.6140i q^{68} +70.7107i q^{71} +60.8276 q^{73} +8.48528i q^{74} +48.6621 q^{76} -20.0000 q^{79} -34.4093i q^{80} -36.4966 q^{82} -222.000 q^{85} +96.1665i q^{86} -8.00000 q^{88} -94.6256i q^{89} +84.8528i q^{92} +97.3242 q^{94} +209.304i q^{95} -158.152 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{4}+O(q^{10}) $ 4 * q - 8 * q^4	$ 4 q - 8 q^{4} + 16 q^{16} + 16 q^{22} - 196 q^{25} - 24 q^{37} - 272 q^{43} - 240 q^{46} + 88 q^{58} - 32 q^{64} - 416 q^{67} - 80 q^{79} - 888 q^{85} - 32 q^{88}+O(q^{100}) $ 4 * q - 8 * q^4 + 16 * q^16 + 16 * q^22 - 196 * q^25 - 24 * q^37 - 272 * q^43 - 240 * q^46 + 88 * q^58 - 32 * q^64 - 416 * q^67 - 80 * q^79 - 888 * q^85 - 32 * q^88

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/882\mathbb{Z}\right)^\times$.

$n$	$199$	$785$
$\chi(n)$	$1$	$-1$

Coefficient data

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

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

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

$2$	− 1.41421i	− 0.707107i
$2$	− 1.41421i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−2.00000	−0.500000
$5$	− 8.60233i	− 1.72047i	−0.509902	−	0.860233i	$-0.670318\pi$
$5$	− 8.60233i	− 1.72047i	0.509902	−	0.860233i	$-0.329682\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	2.82843i	0.353553i
$9$	0	0
$10$	−12.1655	−1.21655
$11$	2.82843i	0.257130i	0.991701	+	0.128565i	$0.0410371\pi$
$11$	2.82843i	0.257130i	−0.991701	+	0.128565i	$0.958963\pi$
$12$	0	0
$13$	12.1655	0.935810	0.467905	−	0.883779i	$-0.345009\pi$
$13$	12.1655	0.935810	0.467905	+	0.883779i	$0.345009\pi$
$14$	0	0
$15$	0	0
$16$	4.00000	0.250000
$17$	− 25.8070i	− 1.51806i	−0.651057	−	0.759029i	$-0.725674\pi$
$17$	− 25.8070i	− 1.51806i	0.651057	−	0.759029i	$-0.274326\pi$
$18$	0	0
$19$	−24.3311	−1.28058	−0.640291	−	0.768133i	$-0.721186\pi$
$19$	−24.3311	−1.28058	−0.640291	+	0.768133i	$0.721186\pi$
$20$	17.2047i	0.860233i
$21$	0	0
$22$	4.00000	0.181818
$23$	− 42.4264i	− 1.84463i	−0.386443	−	0.922313i	$-0.626296\pi$
$23$	− 42.4264i	− 1.84463i	0.386443	−	0.922313i	$-0.373704\pi$
$24$	0	0
$25$	−49.0000	−1.96000
$26$	− 17.2047i	− 0.661717i
$27$	0	0
$28$	0	0
$29$	15.5563i	0.536426i	0.963360	+	0.268213i	$0.0864331\pi$
$29$	15.5563i	0.536426i	−0.963360	+	0.268213i	$0.913567\pi$
$30$	0	0
$31$	24.3311	0.784873	0.392436	−	0.919779i	$-0.371632\pi$
$31$	24.3311	0.784873	0.392436	+	0.919779i	$0.371632\pi$
$32$	− 5.65685i	− 0.176777i
$33$	0	0
$34$	−36.4966	−1.07343
$35$	0	0
$36$	0	0
$37$	−6.00000	−0.162162	−0.0810811	−	0.996708i	$-0.525837\pi$
$37$	−6.00000	−0.162162	−0.0810811	+	0.996708i	$0.525837\pi$
$38$	34.4093i	0.905508i
$39$	0	0
$40$	24.3311	0.608276
$41$	− 25.8070i	− 0.629438i	−0.949185	−	0.314719i	$-0.898090\pi$
$41$	− 25.8070i	− 0.629438i	0.949185	−	0.314719i	$-0.101910\pi$
$42$	0	0
$43$	−68.0000	−1.58140	−0.790698	−	0.612207i	$-0.790282\pi$
$43$	−68.0000	−1.58140	−0.790698	+	0.612207i	$0.790282\pi$
$44$	− 5.65685i	− 0.128565i
$45$	0	0
$46$	−60.0000	−1.30435
$47$	68.8186i	1.46423i	0.681183	+	0.732113i	$0.261466\pi$
$47$	68.8186i	1.46423i	−0.681183	+	0.732113i	$0.738534\pi$
$48$	0	0
$49$	0	0
$50$	69.2965i	1.38593i
$51$	0	0
$52$	−24.3311	−0.467905
$53$	41.0122i	0.773815i	0.922119	+	0.386907i	$0.126457\pi$
$53$	41.0122i	0.773815i	−0.922119	+	0.386907i	$0.873543\pi$
$54$	0	0
$55$	24.3311	0.442383
$56$	0	0
$57$	0	0
$58$	22.0000	0.379310
$59$	68.8186i	1.16642i	0.812323	+	0.583208i	$0.198203\pi$
$59$	68.8186i	1.16642i	−0.812323	+	0.583208i	$0.801797\pi$
$60$	0	0
$61$	97.3242	1.59548	0.797739	−	0.603002i	$-0.206029\pi$
$61$	97.3242	1.59548	0.797739	+	0.603002i	$0.206029\pi$
$62$	− 34.4093i	− 0.554989i
$63$	0	0
$64$	−8.00000	−0.125000
$65$	− 104.652i	− 1.61003i
$66$	0	0
$67$	−104.000	−1.55224	−0.776119	−	0.630586i	$-0.782815\pi$
$67$	−104.000	−1.55224	−0.776119	+	0.630586i	$0.782815\pi$
$68$	51.6140i	0.759029i
$69$	0	0
$70$	0	0
$71$	70.7107i	0.995925i	0.867199	+	0.497963i	$0.165918\pi$
$71$	70.7107i	0.995925i	−0.867199	+	0.497963i	$0.834082\pi$
$72$	0	0
$73$	60.8276	0.833255	0.416628	−	0.909077i	$-0.363212\pi$
$73$	60.8276	0.833255	0.416628	+	0.909077i	$0.363212\pi$
$74$	8.48528i	0.114666i
$75$	0	0
$76$	48.6621	0.640291
$77$	0	0
$78$	0	0
$79$	−20.0000	−0.253165	−0.126582	−	0.991956i	$-0.540401\pi$
$79$	−20.0000	−0.253165	−0.126582	+	0.991956i	$0.540401\pi$
$80$	− 34.4093i	− 0.430116i
$81$	0	0
$82$	−36.4966	−0.445080
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	−222.000	−2.61176
$86$	96.1665i	1.11822i
$87$	0	0
$88$	−8.00000	−0.0909091
$89$	− 94.6256i	− 1.06321i	−0.846993	−	0.531604i	$-0.821589\pi$
$89$	− 94.6256i	− 1.06321i	0.846993	−	0.531604i	$-0.178411\pi$
$90$	0	0
$91$	0	0
$92$	84.8528i	0.922313i
$93$	0	0
$94$	97.3242	1.03536
$95$	209.304i	2.20320i
$96$	0	0
$97$	−158.152	−1.63043	−0.815216	−	0.579158i	$-0.803382\pi$
$97$	−158.152	−1.63043	−0.815216	+	0.579158i	$0.803382\pi$
$98$	0	0
$99$	0	0
$100$	98.0000	0.980000
$101$	− 60.2163i	− 0.596201i	−0.954535	−	0.298100i	$-0.903647\pi$
$101$	− 60.2163i	− 0.596201i	0.954535	−	0.298100i	$-0.0963530\pi$
$102$	0	0
$103$	−170.317	−1.65357	−0.826783	−	0.562521i	$-0.809832\pi$
$103$	−170.317	−1.65357	−0.826783	+	0.562521i	$0.809832\pi$
$104$	34.4093i	0.330859i
$105$	0	0
$106$	58.0000	0.547170
$107$	− 19.7990i	− 0.185037i	−0.995711	−	0.0925186i	$-0.970508\pi$
$107$	− 19.7990i	− 0.185037i	0.995711	−	0.0925186i	$-0.0294918\pi$
$108$	0	0
$109$	−56.0000	−0.513761	−0.256881	−	0.966443i	$-0.582695\pi$
$109$	−56.0000	−0.513761	−0.256881	+	0.966443i	$0.582695\pi$
$110$	− 34.4093i	− 0.312812i
$111$	0	0
$112$	0	0
$113$	− 159.806i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$113$	− 159.806i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$114$	0	0
$115$	−364.966	−3.17362
$116$	− 31.1127i	− 0.268213i
$117$	0	0
$118$	97.3242	0.824781
$119$	0	0
$120$	0	0
$121$	113.000	0.933884
$122$	− 137.637i	− 1.12817i
$123$	0	0
$124$	−48.6621	−0.392436
$125$	206.456i	1.65165i
$126$	0	0
$127$	−76.0000	−0.598425	−0.299213	−	0.954186i	$-0.596724\pi$
$127$	−76.0000	−0.598425	−0.299213	+	0.954186i	$0.596724\pi$
$128$	11.3137i	0.0883883i
$129$	0	0
$130$	−148.000	−1.13846
$131$	− 103.228i	− 0.787999i	−0.919111	−	0.394000i	$-0.871091\pi$
$131$	− 103.228i	− 0.787999i	0.919111	−	0.394000i	$-0.128909\pi$
$132$	0	0
$133$	0	0
$134$	147.078i	1.09760i
$135$	0	0
$136$	72.9932	0.536714
$137$	100.409i	0.732914i	0.930435	+	0.366457i	$0.119429\pi$
$137$	100.409i	0.732914i	−0.930435	+	0.366457i	$0.880571\pi$
$138$	0	0
$139$	145.986	1.05026	0.525131	−	0.851022i	$-0.324016\pi$
$139$	145.986	1.05026	0.525131	+	0.851022i	$0.324016\pi$
$140$	0	0
$141$	0	0
$142$	100.000	0.704225
$143$	34.4093i	0.240624i
$144$	0	0
$145$	133.821	0.922902
$146$	− 86.0233i	− 0.589200i
$147$	0	0
$148$	12.0000	0.0810811
$149$	41.0122i	0.275250i	0.990484	+	0.137625i	$0.0439468\pi$
$149$	41.0122i	0.275250i	−0.990484	+	0.137625i	$0.956053\pi$
$150$	0	0
$151$	160.000	1.05960	0.529801	−	0.848122i	$-0.322266\pi$
$151$	160.000	1.05960	0.529801	+	0.848122i	$0.322266\pi$
$152$	− 68.8186i	− 0.452754i
$153$	0	0
$154$	0	0
$155$	− 209.304i	− 1.35035i
$156$	0	0
$157$	194.648	1.23980	0.619899	−	0.784681i	$-0.287173\pi$
$157$	194.648	1.23980	0.619899	+	0.784681i	$0.287173\pi$
$158$	28.2843i	0.179014i
$159$	0	0
$160$	−48.6621	−0.304138
$161$	0	0
$162$	0	0
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	51.6140i	0.314719i
$165$	0	0
$166$	0	0
$167$	− 172.047i	− 1.03022i	−0.857125	−	0.515109i	$-0.827751\pi$
$167$	− 172.047i	− 1.03022i	0.857125	−	0.515109i	$-0.172249\pi$
$168$	0	0
$169$	−21.0000	−0.124260
$170$	313.955i	1.84680i
$171$	0	0
$172$	136.000	0.790698
$173$	8.60233i	0.0497244i	0.999691	+	0.0248622i	$0.00791470\pi$
$173$	8.60233i	0.0497244i	−0.999691	+	0.0248622i	$0.992085\pi$
$174$	0	0
$175$	0	0
$176$	11.3137i	0.0642824i
$177$	0	0
$178$	−133.821	−0.751802
$179$	144.250i	0.805865i	0.915230	+	0.402932i	$0.132009\pi$
$179$	144.250i	0.805865i	−0.915230	+	0.402932i	$0.867991\pi$
$180$	0	0
$181$	−12.1655	−0.0672128	−0.0336064	−	0.999435i	$-0.510699\pi$
$181$	−12.1655	−0.0672128	−0.0336064	+	0.999435i	$0.510699\pi$
$182$	0	0
$183$	0	0
$184$	120.000	0.652174
$185$	51.6140i	0.278994i
$186$	0	0
$187$	72.9932	0.390338
$188$	− 137.637i	− 0.732113i
$189$	0	0
$190$	296.000	1.55789
$191$	− 263.044i	− 1.37719i	−0.725145	−	0.688596i	$-0.758227\pi$
$191$	− 263.044i	− 1.37719i	0.725145	−	0.688596i	$-0.241773\pi$
$192$	0	0
$193$	38.0000	0.196891	0.0984456	−	0.995142i	$-0.468613\pi$
$193$	38.0000	0.196891	0.0984456	+	0.995142i	$0.468613\pi$
$194$	223.660i	1.15289i
$195$	0	0
$196$	0	0
$197$	− 165.463i	− 0.839914i	−0.907544	−	0.419957i	$-0.862045\pi$
$197$	− 165.463i	− 0.839914i	0.907544	−	0.419957i	$-0.137955\pi$
$198$	0	0
$199$	243.311	1.22267	0.611333	−	0.791374i	$-0.290634\pi$
$199$	243.311	1.22267	0.611333	+	0.791374i	$0.290634\pi$
$200$	− 138.593i	− 0.692965i
$201$	0	0
$202$	−85.1587	−0.421578
$203$	0	0
$204$	0	0
$205$	−222.000	−1.08293
$206$	240.865i	1.16925i
$207$	0	0
$208$	48.6621	0.233952
$209$	− 68.8186i	− 0.329276i
$210$	0	0
$211$	44.0000	0.208531	0.104265	−	0.994550i	$-0.466751\pi$
$211$	44.0000	0.208531	0.104265	+	0.994550i	$0.466751\pi$
$212$	− 82.0244i	− 0.386907i
$213$	0	0
$214$	−28.0000	−0.130841
$215$	584.958i	2.72074i
$216$	0	0
$217$	0	0
$218$	79.1960i	0.363284i
$219$	0	0
$220$	−48.6621	−0.221191
$221$	− 313.955i	− 1.42061i
$222$	0	0
$223$	−194.648	−0.872863	−0.436431	−	0.899738i	$-0.643758\pi$
$223$	−194.648	−0.872863	−0.436431	+	0.899738i	$0.643758\pi$
$224$	0	0
$225$	0	0
$226$	−226.000	−1.00000
$227$	34.4093i	0.151583i	0.997124	+	0.0757914i	$0.0241483\pi$
$227$	34.4093i	0.151583i	−0.997124	+	0.0757914i	$0.975852\pi$
$228$	0	0
$229$	12.1655	0.0531246	0.0265623	−	0.999647i	$-0.491544\pi$
$229$	12.1655	0.0531246	0.0265623	+	0.999647i	$0.491544\pi$
$230$	516.140i	2.24408i
$231$	0	0
$232$	−44.0000	−0.189655
$233$	270.115i	1.15929i	0.814869	+	0.579645i	$0.196809\pi$
$233$	270.115i	1.15929i	−0.814869	+	0.579645i	$0.803191\pi$
$234$	0	0
$235$	592.000	2.51915
$236$	− 137.637i	− 0.583208i
$237$	0	0
$238$	0	0
$239$	− 76.3675i	− 0.319529i	−0.987155	−	0.159765i	$-0.948926\pi$
$239$	− 76.3675i	− 0.319529i	0.987155	−	0.159765i	$-0.0510736\pi$
$240$	0	0
$241$	133.821	0.555273	0.277636	−	0.960686i	$-0.410449\pi$
$241$	133.821	0.555273	0.277636	+	0.960686i	$0.410449\pi$
$242$	− 159.806i	− 0.660356i
$243$	0	0
$244$	−194.648	−0.797739
$245$	0	0
$246$	0	0
$247$	−296.000	−1.19838
$248$	68.8186i	0.277494i
$249$	0	0
$250$	291.973	1.16789
$251$	− 137.637i	− 0.548355i	−0.961679	−	0.274178i	$-0.911594\pi$
$251$	− 137.637i	− 0.548355i	0.961679	−	0.274178i	$-0.0884056\pi$
$252$	0	0
$253$	120.000	0.474308
$254$	107.480i	0.423151i
$255$	0	0
$256$	16.0000	0.0625000
$257$	− 249.467i	− 0.970690i	−0.874323	−	0.485345i	$-0.838694\pi$
$257$	− 249.467i	− 0.970690i	0.874323	−	0.485345i	$-0.161306\pi$
$258$	0	0
$259$	0	0
$260$	209.304i	0.805014i
$261$	0	0
$262$	−145.986	−0.557200
$263$	− 234.759i	− 0.892621i	−0.894878	−	0.446311i	$-0.852738\pi$
$263$	− 234.759i	− 0.892621i	0.894878	−	0.446311i	$-0.147262\pi$
$264$	0	0
$265$	352.800	1.33132
$266$	0	0
$267$	0	0
$268$	208.000	0.776119
$269$	− 215.058i	− 0.799473i	−0.916630	−	0.399736i	$-0.869102\pi$
$269$	− 215.058i	− 0.799473i	0.916630	−	0.399736i	$-0.130898\pi$
$270$	0	0
$271$	−316.304	−1.16717	−0.583586	−	0.812051i	$-0.698351\pi$
$271$	−316.304	−1.16717	−0.583586	+	0.812051i	$0.698351\pi$
$272$	− 103.228i	− 0.379514i
$273$	0	0
$274$	142.000	0.518248
$275$	− 138.593i	− 0.503974i
$276$	0	0
$277$	64.0000	0.231047	0.115523	−	0.993305i	$-0.463145\pi$
$277$	64.0000	0.231047	0.115523	+	0.993305i	$0.463145\pi$
$278$	− 206.456i	− 0.742647i
$279$	0	0
$280$	0	0
$281$	− 462.448i	− 1.64572i	−0.568243	−	0.822861i	$-0.692377\pi$
$281$	− 462.448i	− 1.64572i	0.568243	−	0.822861i	$-0.307623\pi$
$282$	0	0
$283$	24.3311	0.0859754	0.0429877	−	0.999076i	$-0.486312\pi$
$283$	24.3311	0.0859754	0.0429877	+	0.999076i	$0.486312\pi$
$284$	− 141.421i	− 0.497963i
$285$	0	0
$286$	48.6621	0.170147
$287$	0	0
$288$	0	0
$289$	−377.000	−1.30450
$290$	− 189.251i	− 0.652590i
$291$	0	0
$292$	−121.655	−0.416628
$293$	− 283.877i	− 0.968863i	−0.874829	−	0.484431i	$-0.839027\pi$
$293$	− 283.877i	− 0.968863i	0.874829	−	0.484431i	$-0.160973\pi$
$294$	0	0
$295$	592.000	2.00678
$296$	− 16.9706i	− 0.0573330i
$297$	0	0
$298$	58.0000	0.194631
$299$	− 516.140i	− 1.72622i
$300$	0	0
$301$	0	0
$302$	− 226.274i	− 0.749252i
$303$	0	0
$304$	−97.3242	−0.320145
$305$	− 837.214i	− 2.74497i
$306$	0	0
$307$	316.304	1.03031	0.515153	−	0.857099i	$-0.327735\pi$
$307$	316.304	1.03031	0.515153	+	0.857099i	$0.327735\pi$
$308$	0	0
$309$	0	0
$310$	−296.000	−0.954839
$311$	− 275.274i	− 0.885127i	−0.896737	−	0.442563i	$-0.854069\pi$
$311$	− 275.274i	− 0.885127i	0.896737	−	0.442563i	$-0.145931\pi$
$312$	0	0
$313$	−291.973	−0.932820	−0.466410	−	0.884569i	$-0.654453\pi$
$313$	−291.973	−0.932820	−0.466410	+	0.884569i	$0.654453\pi$
$314$	− 275.274i	− 0.876670i
$315$	0	0
$316$	40.0000	0.126582
$317$	− 335.169i	− 1.05731i	−0.848835	−	0.528657i	$-0.822696\pi$
$317$	− 335.169i	− 1.05731i	0.848835	−	0.528657i	$-0.177304\pi$
$318$	0	0
$319$	−44.0000	−0.137931
$320$	68.8186i	0.215058i
$321$	0	0
$322$	0	0
$323$	627.911i	1.94400i
$324$	0	0
$325$	−596.111	−1.83419
$326$	0	0
$327$	0	0
$328$	72.9932	0.222540
$329$	0	0
$330$	0	0
$331$	−388.000	−1.17221	−0.586103	−	0.810237i	$-0.699339\pi$
$331$	−388.000	−1.17221	−0.586103	+	0.810237i	$0.699339\pi$
$332$	0	0
$333$	0	0
$334$	−243.311	−0.728475
$335$	894.642i	2.67057i
$336$	0	0
$337$	−208.000	−0.617211	−0.308605	−	0.951190i	$-0.599862\pi$
$337$	−208.000	−0.617211	−0.308605	+	0.951190i	$0.599862\pi$
$338$	29.6985i	0.0878653i
$339$	0	0
$340$	444.000	1.30588
$341$	68.8186i	0.201814i
$342$	0	0
$343$	0	0
$344$	− 192.333i	− 0.559108i
$345$	0	0
$346$	12.1655	0.0351605
$347$	364.867i	1.05149i	0.850642	+	0.525745i	$0.176213\pi$
$347$	364.867i	1.05149i	−0.850642	+	0.525745i	$0.823787\pi$
$348$	0	0
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	0	0
$351$	0	0
$352$	16.0000	0.0454545
$353$	25.8070i	0.0731076i	0.999332	+	0.0365538i	$0.0116380\pi$
$353$	25.8070i	0.0731076i	−0.999332	+	0.0365538i	$0.988362\pi$
$354$	0	0
$355$	608.276	1.71345
$356$	189.251i	0.531604i
$357$	0	0
$358$	204.000	0.569832
$359$	302.642i	0.843013i	0.906825	+	0.421507i	$0.138499\pi$
$359$	302.642i	0.843013i	−0.906825	+	0.421507i	$0.861501\pi$
$360$	0	0
$361$	231.000	0.639889
$362$	17.2047i	0.0475267i
$363$	0	0
$364$	0	0
$365$	− 523.259i	− 1.43359i
$366$	0	0
$367$	389.297	1.06075	0.530377	−	0.847762i	$-0.322050\pi$
$367$	389.297	1.06075	0.530377	+	0.847762i	$0.322050\pi$
$368$	− 169.706i	− 0.461157i
$369$	0	0
$370$	72.9932	0.197279
$371$	0	0
$372$	0	0
$373$	706.000	1.89276	0.946381	−	0.323054i	$-0.104709\pi$
$373$	706.000	1.89276	0.946381	+	0.323054i	$0.104709\pi$
$374$	− 103.228i	− 0.276010i
$375$	0	0
$376$	−194.648	−0.517682
$377$	189.251i	0.501992i
$378$	0	0
$379$	708.000	1.86807	0.934037	−	0.357176i	$-0.116261\pi$
$379$	708.000	1.86807	0.934037	+	0.357176i	$0.116261\pi$
$380$	− 418.607i	− 1.10160i
$381$	0	0
$382$	−372.000	−0.973822
$383$	− 309.684i	− 0.808574i	−0.914632	−	0.404287i	$-0.867520\pi$
$383$	− 309.684i	− 0.808574i	0.914632	−	0.404287i	$-0.132480\pi$
$384$	0	0
$385$	0	0
$386$	− 53.7401i	− 0.139223i
$387$	0	0
$388$	316.304	0.815216
$389$	− 131.522i	− 0.338102i	−0.985607	−	0.169051i	$-0.945930\pi$
$389$	− 131.522i	− 0.338102i	0.985607	−	0.169051i	$-0.0540703\pi$
$390$	0	0
$391$	−1094.90	−2.80025
$392$	0	0
$393$	0	0
$394$	−234.000	−0.593909
$395$	172.047i	0.435561i
$396$	0	0
$397$	−97.3242	−0.245149	−0.122575	−	0.992459i	$-0.539115\pi$
$397$	−97.3242	−0.245149	−0.122575	+	0.992459i	$0.539115\pi$
$398$	− 344.093i	− 0.864555i
$399$	0	0
$400$	−196.000	−0.490000
$401$	89.0955i	0.222183i	0.993810	+	0.111092i	$0.0354347\pi$
$401$	89.0955i	0.222183i	−0.993810	+	0.111092i	$0.964565\pi$
$402$	0	0
$403$	296.000	0.734491
$404$	120.433i	0.298100i
$405$	0	0
$406$	0	0
$407$	− 16.9706i	− 0.0416967i
$408$	0	0
$409$	−547.449	−1.33851	−0.669253	−	0.743035i	$-0.733386\pi$
$409$	−547.449	−1.33851	−0.669253	+	0.743035i	$0.733386\pi$
$410$	313.955i	0.765745i
$411$	0	0
$412$	340.635	0.826783
$413$	0	0
$414$	0	0
$415$	0	0
$416$	− 68.8186i	− 0.165429i
$417$	0	0
$418$	−97.3242	−0.232833
$419$	− 653.777i	− 1.56033i	−0.625576	−	0.780163i	$-0.715136\pi$
$419$	− 653.777i	− 1.56033i	0.625576	−	0.780163i	$-0.284864\pi$
$420$	0	0
$421$	240.000	0.570071	0.285036	−	0.958517i	$-0.407995\pi$
$421$	240.000	0.570071	0.285036	+	0.958517i	$0.407995\pi$
$422$	− 62.2254i	− 0.147454i
$423$	0	0
$424$	−116.000	−0.273585
$425$	1264.54i	2.97539i
$426$	0	0
$427$	0	0
$428$	39.5980i	0.0925186i
$429$	0	0
$430$	827.256	1.92385
$431$	14.1421i	0.0328124i	0.999865	+	0.0164062i	$0.00522249\pi$
$431$	14.1421i	0.0328124i	−0.999865	+	0.0164062i	$0.994778\pi$
$432$	0	0
$433$	−389.297	−0.899069	−0.449534	−	0.893263i	$-0.648410\pi$
$433$	−389.297	−0.899069	−0.449534	+	0.893263i	$0.648410\pi$
$434$	0	0
$435$	0	0
$436$	112.000	0.256881
$437$	1032.28i	2.36219i
$438$	0	0
$439$	291.973	0.665086	0.332543	−	0.943088i	$-0.392093\pi$
$439$	291.973	0.665086	0.332543	+	0.943088i	$0.392093\pi$
$440$	68.8186i	0.156406i
$441$	0	0
$442$	−444.000	−1.00452
$443$	− 25.4558i	− 0.0574624i	−0.999587	−	0.0287312i	$-0.990853\pi$
$443$	− 25.4558i	− 0.0574624i	0.999587	−	0.0287312i	$-0.00914668\pi$
$444$	0	0
$445$	−814.000	−1.82921
$446$	275.274i	0.617207i
$447$	0	0
$448$	0	0
$449$	733.977i	1.63469i	0.576147	+	0.817346i	$0.304556\pi$
$449$	733.977i	1.63469i	−0.576147	+	0.817346i	$0.695444\pi$
$450$	0	0
$451$	72.9932	0.161847
$452$	319.612i	0.707107i
$453$	0	0
$454$	48.6621	0.107185
$455$	0	0
$456$	0	0
$457$	272.000	0.595186	0.297593	−	0.954693i	$-0.403816\pi$
$457$	272.000	0.595186	0.297593	+	0.954693i	$0.403816\pi$
$458$	− 17.2047i	− 0.0375647i
$459$	0	0
$460$	729.932	1.58681
$461$	− 490.333i	− 1.06363i	−0.846861	−	0.531814i	$-0.821511\pi$
$461$	− 490.333i	− 1.06363i	0.846861	−	0.531814i	$-0.178489\pi$
$462$	0	0
$463$	−436.000	−0.941685	−0.470842	−	0.882217i	$-0.656050\pi$
$463$	−436.000	−0.941685	−0.470842	+	0.882217i	$0.656050\pi$
$464$	62.2254i	0.134106i
$465$	0	0
$466$	382.000	0.819742
$467$	− 481.730i	− 1.03154i	−0.856726	−	0.515771i	$-0.827505\pi$
$467$	− 481.730i	− 1.03154i	0.856726	−	0.515771i	$-0.172495\pi$
$468$	0	0
$469$	0	0
$470$	− 837.214i	− 1.78131i
$471$	0	0
$472$	−194.648	−0.412391
$473$	− 192.333i	− 0.406624i
$474$	0	0
$475$	1192.22	2.50994
$476$	0	0
$477$	0	0
$478$	−108.000	−0.225941
$479$	103.228i	0.215507i	0.994178	+	0.107754i	$0.0343657\pi$
$479$	103.228i	0.215507i	−0.994178	+	0.107754i	$0.965634\pi$
$480$	0	0
$481$	−72.9932	−0.151753
$482$	− 189.251i	− 0.392637i
$483$	0	0
$484$	−226.000	−0.466942
$485$	1360.47i	2.80510i
$486$	0	0
$487$	232.000	0.476386	0.238193	−	0.971218i	$-0.423445\pi$
$487$	232.000	0.476386	0.238193	+	0.971218i	$0.423445\pi$
$488$	275.274i	0.564087i
$489$	0	0
$490$	0	0
$491$	449.720i	0.915927i	0.888971	+	0.457963i	$0.151421\pi$
$491$	449.720i	0.915927i	−0.888971	+	0.457963i	$0.848579\pi$
$492$	0	0
$493$	401.462	0.814325
$494$	418.607i	0.847383i
$495$	0	0
$496$	97.3242	0.196218
$497$	0	0
$498$	0	0
$499$	756.000	1.51503	0.757515	−	0.652818i	$-0.226413\pi$
$499$	756.000	1.51503	0.757515	+	0.652818i	$0.226413\pi$
$500$	− 412.912i	− 0.825823i
$501$	0	0
$502$	−194.648	−0.387746
$503$	550.549i	1.09453i	0.836959	+	0.547265i	$0.184331\pi$
$503$	550.549i	1.09453i	−0.836959	+	0.547265i	$0.815669\pi$
$504$	0	0
$505$	−518.000	−1.02574
$506$	− 169.706i	− 0.335387i
$507$	0	0
$508$	152.000	0.299213
$509$	8.60233i	0.0169004i	0.999964	+	0.00845022i	$0.00268982\pi$
$509$	8.60233i	0.0169004i	−0.999964	+	0.00845022i	$0.997310\pi$
$510$	0	0
$511$	0	0
$512$	− 22.6274i	− 0.0441942i
$513$	0	0
$514$	−352.800	−0.686382
$515$	1465.13i	2.84490i
$516$	0	0
$517$	−194.648	−0.376496
$518$	0	0
$519$	0	0
$520$	296.000	0.569231
$521$	− 94.6256i	− 0.181623i	−0.995868	−	0.0908115i	$-0.971054\pi$
$521$	− 94.6256i	− 0.181623i	0.995868	−	0.0908115i	$-0.0289461\pi$
$522$	0	0
$523$	924.580	1.76784	0.883920	−	0.467639i	$-0.154895\pi$
$523$	924.580	1.76784	0.883920	+	0.467639i	$0.154895\pi$
$524$	206.456i	0.394000i
$525$	0	0
$526$	−332.000	−0.631179
$527$	− 627.911i	− 1.19148i
$528$	0	0
$529$	−1271.00	−2.40265
$530$	− 498.935i	− 0.941387i
$531$	0	0
$532$	0	0
$533$	− 313.955i	− 0.589035i
$534$	0	0
$535$	−170.317	−0.318350
$536$	− 294.156i	− 0.548799i
$537$	0	0
$538$	−304.138	−0.565313
$539$	0	0
$540$	0	0
$541$	592.000	1.09427	0.547135	−	0.837044i	$-0.315718\pi$
$541$	592.000	1.09427	0.547135	+	0.837044i	$0.315718\pi$
$542$	447.321i	0.825315i
$543$	0	0
$544$	−145.986	−0.268357
$545$	481.730i	0.883909i
$546$	0	0
$547$	−416.000	−0.760512	−0.380256	−	0.924881i	$-0.624164\pi$
$547$	−416.000	−0.760512	−0.380256	+	0.924881i	$0.624164\pi$
$548$	− 200.818i	− 0.366457i
$549$	0	0
$550$	−196.000	−0.356364
$551$	− 378.502i	− 0.686937i
$552$	0	0
$553$	0	0
$554$	− 90.5097i	− 0.163375i
$555$	0	0
$556$	−291.973	−0.525131
$557$	− 309.713i	− 0.556037i	−0.960576	−	0.278019i	$-0.910322\pi$
$557$	− 309.713i	− 0.556037i	0.960576	−	0.278019i	$-0.0896777\pi$
$558$	0	0
$559$	−827.256	−1.47988
$560$	0	0
$561$	0	0
$562$	−654.000	−1.16370
$563$	− 894.642i	− 1.58906i	−0.607224	−	0.794531i	$-0.707717\pi$
$563$	− 894.642i	− 1.58906i	0.607224	−	0.794531i	$-0.292283\pi$
$564$	0	0
$565$	−1374.70	−2.43311
$566$	− 34.4093i	− 0.0607938i
$567$	0	0
$568$	−200.000	−0.352113
$569$	733.977i	1.28994i	0.764207	+	0.644971i	$0.223131\pi$
$569$	733.977i	1.28994i	−0.764207	+	0.644971i	$0.776869\pi$
$570$	0	0
$571$	312.000	0.546410	0.273205	−	0.961956i	$-0.411916\pi$
$571$	312.000	0.546410	0.273205	+	0.961956i	$0.411916\pi$
$572$	− 68.8186i	− 0.120312i
$573$	0	0
$574$	0	0
$575$	2078.89i	3.61547i
$576$	0	0
$577$	−194.648	−0.337346	−0.168673	−	0.985672i	$-0.553948\pi$
$577$	−194.648	−0.337346	−0.168673	+	0.985672i	$0.553948\pi$
$578$	533.159i	0.922420i
$579$	0	0
$580$	−267.642	−0.461451
$581$	0	0
$582$	0	0
$583$	−116.000	−0.198971
$584$	172.047i	0.294600i
$585$	0	0
$586$	−401.462	−0.685089
$587$	− 447.321i	− 0.762046i	−0.924566	−	0.381023i	$-0.875572\pi$
$587$	− 447.321i	− 0.762046i	0.924566	−	0.381023i	$-0.124428\pi$
$588$	0	0
$589$	−592.000	−1.00509
$590$	− 837.214i	− 1.41901i
$591$	0	0
$592$	−24.0000	−0.0405405
$593$	− 645.174i	− 1.08798i	−0.839090	−	0.543992i	$-0.816912\pi$
$593$	− 645.174i	− 1.08798i	0.839090	−	0.543992i	$-0.183088\pi$
$594$	0	0
$595$	0	0
$596$	− 82.0244i	− 0.137625i
$597$	0	0
$598$	−729.932	−1.22062
$599$	687.308i	1.14743i	0.819057	+	0.573713i	$0.194497\pi$
$599$	687.308i	1.14743i	−0.819057	+	0.573713i	$0.805503\pi$
$600$	0	0
$601$	1070.57	1.78131	0.890654	−	0.454682i	$-0.150247\pi$
$601$	1070.57	1.78131	0.890654	+	0.454682i	$0.150247\pi$
$602$	0	0
$603$	0	0
$604$	−320.000	−0.529801
$605$	− 972.063i	− 1.60672i
$606$	0	0
$607$	−729.932	−1.20252	−0.601262	−	0.799052i	$-0.705335\pi$
$607$	−729.932	−1.20252	−0.601262	+	0.799052i	$0.705335\pi$
$608$	137.637i	0.226377i
$609$	0	0
$610$	−1184.00	−1.94098
$611$	837.214i	1.37024i
$612$	0	0
$613$	−514.000	−0.838499	−0.419250	−	0.907871i	$-0.637707\pi$
$613$	−514.000	−0.838499	−0.419250	+	0.907871i	$0.637707\pi$
$614$	− 447.321i	− 0.728536i
$615$	0	0
$616$	0	0
$617$	− 202.233i	− 0.327767i	−0.986480	−	0.163884i	$-0.947598\pi$
$617$	− 202.233i	− 0.327767i	0.986480	−	0.163884i	$-0.0524022\pi$
$618$	0	0
$619$	681.269	1.10060	0.550298	−	0.834968i	$-0.314514\pi$
$619$	681.269	1.10060	0.550298	+	0.834968i	$0.314514\pi$
$620$	418.607i	0.675173i
$621$	0	0
$622$	−389.297	−0.625879
$623$	0	0
$624$	0	0
$625$	551.000	0.881600
$626$	412.912i	0.659603i
$627$	0	0
$628$	−389.297	−0.619899
$629$	154.842i	0.246171i
$630$	0	0
$631$	636.000	1.00792	0.503962	−	0.863726i	$-0.331875\pi$
$631$	636.000	1.00792	0.503962	+	0.863726i	$0.331875\pi$
$632$	− 56.5685i	− 0.0895072i
$633$	0	0
$634$	−474.000	−0.747634
$635$	653.777i	1.02957i
$636$	0	0
$637$	0	0
$638$	62.2254i	0.0975320i
$639$	0	0
$640$	97.3242	0.152069
$641$	77.7817i	0.121344i	0.998158	+	0.0606722i	$0.0193244\pi$
$641$	77.7817i	0.121344i	−0.998158	+	0.0606722i	$0.980676\pi$
$642$	0	0
$643$	510.952	0.794638	0.397319	−	0.917681i	$-0.369941\pi$
$643$	510.952	0.794638	0.397319	+	0.917681i	$0.369941\pi$
$644$	0	0
$645$	0	0
$646$	888.000	1.37461
$647$	− 929.051i	− 1.43594i	−0.696076	−	0.717968i	$-0.745072\pi$
$647$	− 929.051i	− 1.43594i	0.696076	−	0.717968i	$-0.254928\pi$
$648$	0	0
$649$	−194.648	−0.299920
$650$	843.028i	1.29697i
$651$	0	0
$652$	0	0
$653$	− 637.810i	− 0.976739i	−0.872637	−	0.488369i	$-0.837592\pi$
$653$	− 637.810i	− 0.976739i	0.872637	−	0.488369i	$-0.162408\pi$
$654$	0	0
$655$	−888.000	−1.35573
$656$	− 103.228i	− 0.157360i
$657$	0	0
$658$	0	0
$659$	200.818i	0.304732i	0.988324	+	0.152366i	$0.0486892\pi$
$659$	200.818i	0.304732i	−0.988324	+	0.152366i	$0.951311\pi$
$660$	0	0
$661$	−97.3242	−0.147238	−0.0736189	−	0.997286i	$-0.523455\pi$
$661$	−97.3242	−0.147238	−0.0736189	+	0.997286i	$0.523455\pi$
$662$	548.715i	0.828874i
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	660.000	0.989505
$668$	344.093i	0.515109i
$669$	0	0
$670$	1265.21	1.88838
$671$	275.274i	0.410245i
$672$	0	0
$673$	−250.000	−0.371471	−0.185736	−	0.982600i	$-0.559467\pi$
$673$	−250.000	−0.371471	−0.185736	+	0.982600i	$0.559467\pi$
$674$	294.156i	0.436434i
$675$	0	0
$676$	42.0000	0.0621302
$677$	− 335.491i	− 0.495555i	−0.968817	−	0.247777i	$-0.920300\pi$
$677$	− 335.491i	− 0.495555i	0.968817	−	0.247777i	$-0.0797002\pi$
$678$	0	0
$679$	0	0
$680$	− 627.911i	− 0.923398i
$681$	0	0
$682$	97.3242	0.142704
$683$	− 291.328i	− 0.426542i	−0.976993	−	0.213271i	$-0.931588\pi$
$683$	− 291.328i	− 0.426542i	0.976993	−	0.213271i	$-0.0684117\pi$
$684$	0	0
$685$	863.752	1.26095
$686$	0	0
$687$	0	0
$688$	−272.000	−0.395349
$689$	498.935i	0.724143i
$690$	0	0
$691$	535.283	0.774650	0.387325	−	0.921943i	$-0.373399\pi$
$691$	535.283	0.774650	0.387325	+	0.921943i	$0.373399\pi$
$692$	− 17.2047i	− 0.0248622i
$693$	0	0
$694$	516.000	0.743516
$695$	− 1255.82i	− 1.80694i
$696$	0	0
$697$	−666.000	−0.955524
$698$	0	0
$699$	0	0
$700$	0	0
$701$	1045.10i	1.49088i	0.666575	+	0.745438i	$0.267760\pi$
$701$	1045.10i	1.49088i	−0.666575	+	0.745438i	$0.732240\pi$
$702$	0	0
$703$	145.986	0.207662
$704$	− 22.6274i	− 0.0321412i
$705$	0	0
$706$	36.4966	0.0516949
$707$	0	0
$708$	0	0
$709$	−488.000	−0.688293	−0.344147	−	0.938916i	$-0.611832\pi$
$709$	−488.000	−0.688293	−0.344147	+	0.938916i	$0.611832\pi$
$710$	− 860.233i	− 1.21160i
$711$	0	0
$712$	267.642	0.375901
$713$	− 1032.28i	− 1.44780i
$714$	0	0
$715$	296.000	0.413986
$716$	− 288.500i	− 0.402932i
$717$	0	0
$718$	428.000	0.596100
$719$	929.051i	1.29214i	0.763277	+	0.646072i	$0.223589\pi$
$719$	929.051i	1.29214i	−0.763277	+	0.646072i	$0.776411\pi$
$720$	0	0
$721$	0	0
$722$	− 326.683i	− 0.452470i
$723$	0	0
$724$	24.3311	0.0336064
$725$	− 762.261i	− 1.05139i
$726$	0	0
$727$	−900.249	−1.23831	−0.619153	−	0.785270i	$-0.712524\pi$
$727$	−900.249	−1.23831	−0.619153	+	0.785270i	$0.712524\pi$
$728$	0	0
$729$	0	0
$730$	−740.000	−1.01370
$731$	1754.87i	2.40065i
$732$	0	0
$733$	1155.72	1.57671	0.788353	−	0.615224i	$-0.210934\pi$
$733$	1155.72	1.57671	0.788353	+	0.615224i	$0.210934\pi$
$734$	− 550.549i	− 0.750067i
$735$	0	0
$736$	−240.000	−0.326087
$737$	− 294.156i	− 0.399127i
$738$	0	0
$739$	−920.000	−1.24493	−0.622463	−	0.782649i	$-0.713868\pi$
$739$	−920.000	−1.24493	−0.622463	+	0.782649i	$0.713868\pi$
$740$	− 103.228i	− 0.139497i
$741$	0	0
$742$	0	0
$743$	755.190i	1.01641i	0.861237	+	0.508203i	$0.169690\pi$
$743$	755.190i	1.01641i	−0.861237	+	0.508203i	$0.830310\pi$
$744$	0	0
$745$	352.800	0.473557
$746$	− 998.435i	− 1.33838i
$747$	0	0
$748$	−145.986	−0.195169
$749$	0	0
$750$	0	0
$751$	−120.000	−0.159787	−0.0798935	−	0.996803i	$-0.525458\pi$
$751$	−120.000	−0.159787	−0.0798935	+	0.996803i	$0.525458\pi$
$752$	275.274i	0.366056i
$753$	0	0
$754$	267.642	0.354962
$755$	− 1376.37i	− 1.82301i
$756$	0	0
$757$	−1016.00	−1.34214	−0.671070	−	0.741394i	$-0.734165\pi$
$757$	−1016.00	−1.34214	−0.671070	+	0.741394i	$0.734165\pi$
$758$	− 1001.26i	− 1.32093i
$759$	0	0
$760$	−592.000	−0.778947
$761$	43.0116i	0.0565199i	0.999601	+	0.0282599i	$0.00899662\pi$
$761$	43.0116i	0.0565199i	−0.999601	+	0.0282599i	$0.991003\pi$
$762$	0	0
$763$	0	0
$764$	526.087i	0.688596i
$765$	0	0
$766$	−437.959	−0.571748
$767$	837.214i	1.09154i
$768$	0	0
$769$	−681.269	−0.885916	−0.442958	−	0.896542i	$-0.646071\pi$
$769$	−681.269	−0.885916	−0.442958	+	0.896542i	$0.646071\pi$
$770$	0	0
$771$	0	0
$772$	−76.0000	−0.0984456
$773$	1161.31i	1.50235i	0.660105	+	0.751173i	$0.270512\pi$
$773$	1161.31i	1.50235i	−0.660105	+	0.751173i	$0.729488\pi$
$774$	0	0
$775$	−1192.22	−1.53835
$776$	− 447.321i	− 0.576444i
$777$	0	0
$778$	−186.000	−0.239075
$779$	627.911i	0.806047i
$780$	0	0
$781$	−200.000	−0.256082
$782$	1548.42i	1.98007i
$783$	0	0
$784$	0	0
$785$	− 1674.43i	− 2.13303i
$786$	0	0
$787$	−194.648	−0.247330	−0.123665	−	0.992324i	$-0.539465\pi$
$787$	−194.648	−0.247330	−0.123665	+	0.992324i	$0.539465\pi$
$788$	330.926i	0.419957i
$789$	0	0
$790$	243.311	0.307988
$791$	0	0
$792$	0	0
$793$	1184.00	1.49306
$794$	137.637i	0.173347i
$795$	0	0
$796$	−486.621	−0.611333
$797$	1109.70i	1.39235i	0.717874	+	0.696173i	$0.245115\pi$
$797$	1109.70i	1.39235i	−0.717874	+	0.696173i	$0.754885\pi$
$798$	0	0
$799$	1776.00	2.22278
$800$	277.186i	0.346482i
$801$	0	0
$802$	126.000	0.157107
$803$	172.047i	0.214255i
$804$	0	0
$805$	0	0
$806$	− 418.607i	− 0.519364i
$807$	0	0
$808$	170.317	0.210789
$809$	− 1030.96i	− 1.27437i	−0.770713	−	0.637183i	$-0.780100\pi$
$809$	− 1030.96i	− 1.27437i	0.770713	−	0.637183i	$-0.219900\pi$
$810$	0	0
$811$	−243.311	−0.300013	−0.150006	−	0.988685i	$-0.547929\pi$
$811$	−243.311	−0.300013	−0.150006	+	0.988685i	$0.547929\pi$
$812$	0	0
$813$	0	0
$814$	−24.0000	−0.0294840
$815$	0	0
$816$	0	0
$817$	1654.51	2.02511
$818$	774.209i	0.946466i
$819$	0	0
$820$	444.000	0.541463
$821$	− 357.796i	− 0.435805i	−0.975971	−	0.217903i	$-0.930078\pi$
$821$	− 357.796i	− 0.435805i	0.975971	−	0.217903i	$-0.0699215\pi$
$822$	0	0
$823$	1228.00	1.49210	0.746051	−	0.665889i	$-0.231947\pi$
$823$	1228.00	1.49210	0.746051	+	0.665889i	$0.231947\pi$
$824$	− 481.730i	− 0.584624i
$825$	0	0
$826$	0	0
$827$	195.161i	0.235987i	0.993014	+	0.117994i	$0.0376462\pi$
$827$	195.161i	0.235987i	−0.993014	+	0.117994i	$0.962354\pi$
$828$	0	0
$829$	790.759	0.953871	0.476936	−	0.878938i	$-0.341748\pi$
$829$	790.759	0.953871	0.476936	+	0.878938i	$0.341748\pi$
$830$	0	0
$831$	0	0
$832$	−97.3242	−0.116976
$833$	0	0
$834$	0	0
$835$	−1480.00	−1.77246
$836$	137.637i	0.164638i
$837$	0	0
$838$	−924.580	−1.10332
$839$	1238.73i	1.47644i	0.674559	+	0.738221i	$0.264334\pi$
$839$	1238.73i	1.47644i	−0.674559	+	0.738221i	$0.735666\pi$
$840$	0	0
$841$	599.000	0.712247
$842$	− 339.411i	− 0.403101i
$843$	0	0
$844$	−88.0000	−0.104265
$845$	180.649i	0.213786i
$846$	0	0
$847$	0	0
$848$	164.049i	0.193454i
$849$	0	0
$850$	1788.33	2.10392
$851$	254.558i	0.299129i
$852$	0	0
$853$	−1557.19	−1.82554	−0.912771	−	0.408472i	$-0.866062\pi$
$853$	−1557.19	−1.82554	−0.912771	+	0.408472i	$0.866062\pi$
$854$	0	0
$855$	0	0
$856$	56.0000	0.0654206
$857$	− 1471.00i	− 1.71645i	−0.513274	−	0.858225i	$-0.671567\pi$
$857$	− 1471.00i	− 1.71645i	0.513274	−	0.858225i	$-0.328433\pi$
$858$	0	0
$859$	−510.952	−0.594822	−0.297411	−	0.954750i	$-0.596123\pi$
$859$	−510.952	−0.594822	−0.297411	+	0.954750i	$0.596123\pi$
$860$	− 1169.92i	− 1.36037i
$861$	0	0
$862$	20.0000	0.0232019
$863$	− 568.514i	− 0.658765i	−0.944197	−	0.329382i	$-0.893160\pi$
$863$	− 568.514i	− 0.658765i	0.944197	−	0.329382i	$-0.106840\pi$
$864$	0	0
$865$	74.0000	0.0855491
$866$	550.549i	0.635738i
$867$	0	0
$868$	0	0
$869$	− 56.5685i	− 0.0650961i
$870$	0	0
$871$	−1265.21	−1.45260
$872$	− 158.392i	− 0.181642i
$873$	0	0
$874$	1459.86	1.67032
$875$	0	0
$876$	0	0
$877$	−882.000	−1.00570	−0.502851	−	0.864373i	$-0.667715\pi$
$877$	−882.000	−1.00570	−0.502851	+	0.864373i	$0.667715\pi$
$878$	− 412.912i	− 0.470287i
$879$	0	0
$880$	97.3242	0.110596
$881$	43.0116i	0.0488214i	0.999702	+	0.0244107i	$0.00777093\pi$
$881$	43.0116i	0.0488214i	−0.999702	+	0.0244107i	$0.992229\pi$
$882$	0	0
$883$	−140.000	−0.158550	−0.0792752	−	0.996853i	$-0.525261\pi$
$883$	−140.000	−0.158550	−0.0792752	+	0.996853i	$0.525261\pi$
$884$	627.911i	0.710306i
$885$	0	0
$886$	−36.0000	−0.0406321
$887$	309.684i	0.349136i	0.984645	+	0.174568i	$0.0558529\pi$
$887$	309.684i	0.349136i	−0.984645	+	0.174568i	$0.944147\pi$
$888$	0	0
$889$	0	0
$890$	1151.17i	1.29345i
$891$	0	0
$892$	389.297	0.436431
$893$	− 1674.43i	− 1.87506i
$894$	0	0
$895$	1240.88	1.38646
$896$	0	0
$897$	0	0
$898$	1038.00	1.15590
$899$	378.502i	0.421026i
$900$	0	0
$901$	1058.40	1.17470
$902$	− 103.228i	− 0.114443i
$903$	0	0
$904$	452.000	0.500000
$905$	104.652i	0.115637i
$906$	0	0
$907$	1228.00	1.35391	0.676957	−	0.736023i	$-0.263298\pi$
$907$	1228.00	1.35391	0.676957	+	0.736023i	$0.263298\pi$
$908$	− 68.8186i	− 0.0757914i
$909$	0	0
$910$	0	0
$911$	− 138.593i	− 0.152133i	−0.997103	−	0.0760664i	$-0.975764\pi$
$911$	− 138.593i	− 0.152133i	0.997103	−	0.0760664i	$-0.0242361\pi$
$912$	0	0
$913$	0	0
$914$	− 384.666i	− 0.420860i
$915$	0	0
$916$	−24.3311	−0.0265623
$917$	0	0
$918$	0	0
$919$	−1608.00	−1.74973	−0.874864	−	0.484369i	$-0.839049\pi$
$919$	−1608.00	−1.74973	−0.874864	+	0.484369i	$0.839049\pi$
$920$	− 1032.28i	− 1.12204i
$921$	0	0
$922$	−693.435	−0.752099
$923$	860.233i	0.931996i
$924$	0	0
$925$	294.000	0.317838
$926$	616.597i	0.665872i
$927$	0	0
$928$	88.0000	0.0948276
$929$	− 1419.38i	− 1.52786i	−0.645298	−	0.763931i	$-0.723267\pi$
$929$	− 1419.38i	− 1.52786i	0.645298	−	0.763931i	$-0.276733\pi$
$930$	0	0
$931$	0	0
$932$	− 540.230i	− 0.579645i
$933$	0	0
$934$	−681.269	−0.729410
$935$	− 627.911i	− 0.671562i
$936$	0	0
$937$	−291.973	−0.311604	−0.155802	−	0.987788i	$-0.549796\pi$
$937$	−291.973	−0.311604	−0.155802	+	0.987788i	$0.549796\pi$
$938$	0	0
$939$	0	0
$940$	−1184.00	−1.25957
$941$	− 1367.77i	− 1.45353i	−0.686887	−	0.726764i	$-0.741023\pi$
$941$	− 1367.77i	− 1.45353i	0.686887	−	0.726764i	$-0.258977\pi$
$942$	0	0
$943$	−1094.90	−1.16108
$944$	275.274i	0.291604i
$945$	0	0
$946$	−272.000	−0.287526
$947$	− 591.141i	− 0.624225i	−0.950045	−	0.312113i	$-0.898963\pi$
$947$	− 591.141i	− 0.624225i	0.950045	−	0.312113i	$-0.101037\pi$
$948$	0	0
$949$	740.000	0.779768
$950$	− 1686.06i	− 1.77480i
$951$	0	0
$952$	0	0
$953$	352.139i	0.369506i	0.982785	+	0.184753i	$0.0591485\pi$
$953$	352.139i	0.369506i	−0.982785	+	0.184753i	$0.940851\pi$
$954$	0	0
$955$	−2262.79	−2.36941
$956$	152.735i	0.159765i
$957$	0	0
$958$	145.986	0.152387
$959$	0	0
$960$	0	0
$961$	−369.000	−0.383975
$962$	103.228i	0.107306i
$963$	0	0
$964$	−267.642	−0.277636
$965$	− 326.888i	− 0.338744i
$966$	0	0
$967$	−260.000	−0.268873	−0.134436	−	0.990922i	$-0.542922\pi$
$967$	−260.000	−0.268873	−0.134436	+	0.990922i	$0.542922\pi$
$968$	319.612i	0.330178i
$969$	0	0
$970$	1924.00	1.98351
$971$	− 1479.60i	− 1.52379i	−0.647701	−	0.761895i	$-0.724269\pi$
$971$	− 1479.60i	− 1.52379i	0.647701	−	0.761895i	$-0.275731\pi$
$972$	0	0
$973$	0	0
$974$	− 328.098i	− 0.336856i
$975$	0	0
$976$	389.297	0.398870
$977$	329.512i	0.337269i	0.985679	+	0.168634i	$0.0539357\pi$
$977$	329.512i	0.337269i	−0.985679	+	0.168634i	$0.946064\pi$
$978$	0	0
$979$	267.642	0.273383
$980$	0	0
$981$	0	0
$982$	636.000	0.647658
$983$	997.870i	1.01513i	0.861614	+	0.507563i	$0.169454\pi$
$983$	997.870i	1.01513i	−0.861614	+	0.507563i	$0.830546\pi$
$984$	0	0
$985$	−1423.37	−1.44504
$986$	− 567.753i	− 0.575815i
$987$	0	0
$988$	592.000	0.599190
$989$	2885.00i	2.91708i
$990$	0	0
$991$	1496.00	1.50959	0.754793	−	0.655963i	$-0.227737\pi$
$991$	1496.00	1.50959	0.754793	+	0.655963i	$0.227737\pi$
$992$	− 137.637i	− 0.138747i
$993$	0	0
$994$	0	0
$995$	− 2093.04i	− 2.10355i
$996$	0	0
$997$	−1557.19	−1.56187	−0.780936	−	0.624611i	$-0.785258\pi$
$997$	−1557.19	−1.56187	−0.780936	+	0.624611i	$0.785258\pi$
$998$	− 1069.15i	− 1.07129i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.3.b.g.197.1	✓	4
3.2	odd	2	inner	882.3.b.g.197.4	yes	4
7.2	even	3		882.3.s.h.557.1		8
7.3	odd	6		882.3.s.h.863.3		8
7.4	even	3		882.3.s.h.863.4		8
7.5	odd	6		882.3.s.h.557.2		8
7.6	odd	2	inner	882.3.b.g.197.2	yes	4
21.2	odd	6		882.3.s.h.557.4		8
21.5	even	6		882.3.s.h.557.3		8
21.11	odd	6		882.3.s.h.863.1		8
21.17	even	6		882.3.s.h.863.2		8
21.20	even	2	inner	882.3.b.g.197.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
882.3.b.g.197.1	✓	4	1.1	even	1	trivial
882.3.b.g.197.2	yes	4	7.6	odd	2	inner
882.3.b.g.197.3	yes	4	21.20	even	2	inner
882.3.b.g.197.4	yes	4	3.2	odd	2	inner
882.3.s.h.557.1		8	7.2	even	3
882.3.s.h.557.2		8	7.5	odd	6
882.3.s.h.557.3		8	21.5	even	6
882.3.s.h.557.4		8	21.2	odd	6
882.3.s.h.863.1		8	21.11	odd	6
882.3.s.h.863.2		8	21.17	even	6
882.3.s.h.863.3		8	7.3	odd	6
882.3.s.h.863.4		8	7.4	even	3

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:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	882.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} -2.00000 q^{4} -8.60233i q^{5} +2.82843i q^{8} +O(q^{10})\)	\(q-1.41421i q^{2} -2.00000 q^{4} -8.60233i q^{5} +2.82843i q^{8} -12.1655 q^{10} +2.82843i q^{11} +12.1655 q^{13} +4.00000 q^{16} -25.8070i q^{17} -24.3311 q^{19} +17.2047i q^{20} +4.00000 q^{22} -42.4264i q^{23} -49.0000 q^{25} -17.2047i q^{26} +15.5563i q^{29} +24.3311 q^{31} -5.65685i q^{32} -36.4966 q^{34} -6.00000 q^{37} +34.4093i q^{38} +24.3311 q^{40} -25.8070i q^{41} -68.0000 q^{43} -5.65685i q^{44} -60.0000 q^{46} +68.8186i q^{47} +69.2965i q^{50} -24.3311 q^{52} +41.0122i q^{53} +24.3311 q^{55} +22.0000 q^{58} +68.8186i q^{59} +97.3242 q^{61} -34.4093i q^{62} -8.00000 q^{64} -104.652i q^{65} -104.000 q^{67} +51.6140i q^{68} +70.7107i q^{71} +60.8276 q^{73} +8.48528i q^{74} +48.6621 q^{76} -20.0000 q^{79} -34.4093i q^{80} -36.4966 q^{82} -222.000 q^{85} +96.1665i q^{86} -8.00000 q^{88} -94.6256i q^{89} +84.8528i q^{92} +97.3242 q^{94} +209.304i q^{95} -158.152 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4}+O(q^{10}) \) 4 * q - 8 * q^4	\( 4 q - 8 q^{4} + 16 q^{16} + 16 q^{22} - 196 q^{25} - 24 q^{37} - 272 q^{43} - 240 q^{46} + 88 q^{58} - 32 q^{64} - 416 q^{67} - 80 q^{79} - 888 q^{85} - 32 q^{88}+O(q^{100}) \) 4 * q - 8 * q^4 + 16 * q^16 + 16 * q^22 - 196 * q^25 - 24 * q^37 - 272 * q^43 - 240 * q^46 + 88 * q^58 - 32 * q^64 - 416 * q^67 - 80 * q^79 - 888 * q^85 - 32 * q^88

Introduction

L-functions

Modular forms

Varieties

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