LMFDB - Embedded newform 414.3.c.b.323.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [414,3,Mod(323,414)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(414, 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("414.323");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 414 = 2 \cdot 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	414.c (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:	$11.2806829445$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 19x^{6} - 88x^{5} + 301x^{4} - 1010x^{3} + 2713x^{2} - 7044x + 9558 $ x^8 - 2x^7 + 19x^6 - 88x^5 + 301x^4 - 1010x^3 + 2713x^2 - 7044*x + 9558
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			323.7
Root			$1.41915 - 2.47946i$ of defining polynomial
Character	$\chi$	$=$	414.323
Dual form			414.3.c.b.323.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{2} -2.00000 q^{4} +4.01397i q^{5} +13.7953 q^{7} -2.82843i q^{8} +O(q^{10})$	\(q+1.41421i q^{2} -2.00000 q^{4} +4.01397i q^{5} +13.7953 q^{7} -2.82843i q^{8} -5.67661 q^{10} -19.5095i q^{11} +17.2413 q^{13} +19.5095i q^{14} +4.00000 q^{16} -9.21348i q^{17} -14.2566 q^{19} -8.02794i q^{20} +27.5906 q^{22} +4.79583i q^{23} +8.88805 q^{25} +24.3828i q^{26} -27.5906 q^{28} +30.9966i q^{29} -0.461260 q^{31} +5.65685i q^{32} +13.0298 q^{34} +55.3739i q^{35} +40.4825 q^{37} -20.1618i q^{38} +11.3532 q^{40} +66.2247i q^{41} -12.2644 q^{43} +39.0190i q^{44} -6.78233 q^{46} -45.6272i q^{47} +141.310 q^{49} +12.5696i q^{50} -34.4825 q^{52} -15.5227i q^{53} +78.3105 q^{55} -39.0190i q^{56} -43.8358 q^{58} +65.1749i q^{59} -19.7839 q^{61} -0.652321i q^{62} -8.00000 q^{64} +69.2059i q^{65} -81.0283 q^{67} +18.4270i q^{68} -78.3105 q^{70} -109.571i q^{71} -100.766 q^{73} +57.2510i q^{74} +28.5131 q^{76} -269.139i q^{77} -150.150 q^{79} +16.0559i q^{80} -93.6559 q^{82} +62.0203i q^{83} +36.9826 q^{85} -17.3444i q^{86} -55.1812 q^{88} +58.8455i q^{89} +237.848 q^{91} -9.59166i q^{92} +64.5266 q^{94} -57.2254i q^{95} +15.3056 q^{97} +199.843i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{4} + 16 q^{7}+O(q^{10}) $ 8 * q - 16 * q^4 + 16 * q^7	$ 8 q - 16 q^{4} + 16 q^{7} - 8 q^{10} - 8 q^{13} + 32 q^{16} - 48 q^{19} + 32 q^{22} - 32 q^{28} - 32 q^{31} - 8 q^{34} + 32 q^{37} + 16 q^{40} + 32 q^{43} + 80 q^{49} + 16 q^{52} + 32 q^{55} + 16 q^{58} + 48 q^{61} - 64 q^{64} - 16 q^{67} - 32 q^{70} - 432 q^{73} + 96 q^{76} - 416 q^{79} - 144 q^{82} + 584 q^{85} - 64 q^{88} + 368 q^{91} + 128 q^{94} - 8 q^{97}+O(q^{100}) $ 8 * q - 16 * q^4 + 16 * q^7 - 8 * q^10 - 8 * q^13 + 32 * q^16 - 48 * q^19 + 32 * q^22 - 32 * q^28 - 32 * q^31 - 8 * q^34 + 32 * q^37 + 16 * q^40 + 32 * q^43 + 80 * q^49 + 16 * q^52 + 32 * q^55 + 16 * q^58 + 48 * q^61 - 64 * q^64 - 16 * q^67 - 32 * q^70 - 432 * q^73 + 96 * q^76 - 416 * q^79 - 144 * q^82 + 584 * q^85 - 64 * q^88 + 368 * q^91 + 128 * q^94 - 8 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/414\mathbb{Z}\right)^\times$.

$n$	$47$	$235$
$\chi(n)$	$-1$	$1$

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
$5$	4.01397i	0.802794i	0.915904	+	0.401397i	$0.131475\pi$
$5$	4.01397i	0.802794i	−0.915904	+	0.401397i	$0.868525\pi$
$6$	0	0
$7$	13.7953	1.97076	0.985378	−	0.170383i	$-0.0545006\pi$
$7$	13.7953	1.97076	0.985378	+	0.170383i	$0.0545006\pi$
$8$	− 2.82843i	− 0.353553i
$9$	0	0
$10$	−5.67661	−0.567661
$11$	− 19.5095i	− 1.77359i	−0.462163	−	0.886795i	$-0.652927\pi$
$11$	− 19.5095i	− 1.77359i	0.462163	−	0.886795i	$-0.347073\pi$
$12$	0	0
$13$	17.2413	1.32625	0.663126	−	0.748508i	$-0.269229\pi$
$13$	17.2413	1.32625	0.663126	+	0.748508i	$0.269229\pi$
$14$	19.5095i	1.39353i
$15$	0	0
$16$	4.00000	0.250000
$17$	− 9.21348i	− 0.541969i	−0.962584	−	0.270985i	$-0.912651\pi$
$17$	− 9.21348i	− 0.541969i	0.962584	−	0.270985i	$-0.0873493\pi$
$18$	0	0
$19$	−14.2566	−0.750345	−0.375172	−	0.926955i	$-0.622416\pi$
$19$	−14.2566	−0.750345	−0.375172	+	0.926955i	$0.622416\pi$
$20$	− 8.02794i	− 0.401397i
$21$	0	0
$22$	27.5906	1.25412
$23$	4.79583i	0.208514i
$23$	4.79583i	0.208514i
$24$	0	0
$25$	8.88805	0.355522
$26$	24.3828i	0.937801i
$27$	0	0
$28$	−27.5906	−0.985378
$29$	30.9966i	1.06885i	0.845217	+	0.534423i	$0.179471\pi$
$29$	30.9966i	1.06885i	−0.845217	+	0.534423i	$0.820529\pi$
$30$	0	0
$31$	−0.461260	−0.0148794	−0.00743968	−	0.999972i	$-0.502368\pi$
$31$	−0.461260	−0.0148794	−0.00743968	+	0.999972i	$0.502368\pi$
$32$	5.65685i	0.176777i
$33$	0	0
$34$	13.0298	0.383230
$35$	55.3739i	1.58211i
$36$	0	0
$37$	40.4825	1.09412	0.547061	−	0.837093i	$-0.315746\pi$
$37$	40.4825	1.09412	0.547061	+	0.837093i	$0.315746\pi$
$38$	− 20.1618i	− 0.530574i
$39$	0	0
$40$	11.3532	0.283830
$41$	66.2247i	1.61524i	0.589705	+	0.807618i	$0.299244\pi$
$41$	66.2247i	1.61524i	−0.589705	+	0.807618i	$0.700756\pi$
$42$	0	0
$43$	−12.2644	−0.285218	−0.142609	−	0.989779i	$-0.545549\pi$
$43$	−12.2644	−0.285218	−0.142609	+	0.989779i	$0.545549\pi$
$44$	39.0190i	0.886795i
$45$	0	0
$46$	−6.78233	−0.147442
$47$	− 45.6272i	− 0.970791i	−0.874295	−	0.485395i	$-0.838676\pi$
$47$	− 45.6272i	− 0.970791i	0.874295	−	0.485395i	$-0.161324\pi$
$48$	0	0
$49$	141.310	2.88388
$50$	12.5696i	0.251392i
$51$	0	0
$52$	−34.4825	−0.663126
$53$	− 15.5227i	− 0.292880i	−0.989220	−	0.146440i	$-0.953218\pi$
$53$	− 15.5227i	− 0.292880i	0.989220	−	0.146440i	$-0.0467816\pi$
$54$	0	0
$55$	78.3105	1.42383
$56$	− 39.0190i	− 0.696767i
$57$	0	0
$58$	−43.8358	−0.755789
$59$	65.1749i	1.10466i	0.833626	+	0.552329i	$0.186261\pi$
$59$	65.1749i	1.10466i	−0.833626	+	0.552329i	$0.813739\pi$
$60$	0	0
$61$	−19.7839	−0.324327	−0.162163	−	0.986764i	$-0.551847\pi$
$61$	−19.7839	−0.324327	−0.162163	+	0.986764i	$0.551847\pi$
$62$	− 0.652321i	− 0.0105213i
$63$	0	0
$64$	−8.00000	−0.125000
$65$	69.2059i	1.06471i
$66$	0	0
$67$	−81.0283	−1.20938	−0.604689	−	0.796462i	$-0.706702\pi$
$67$	−81.0283	−1.20938	−0.604689	+	0.796462i	$0.706702\pi$
$68$	18.4270i	0.270985i
$69$	0	0
$70$	−78.3105	−1.11872
$71$	− 109.571i	− 1.54325i	−0.636078	−	0.771625i	$-0.719444\pi$
$71$	− 109.571i	− 1.54325i	0.636078	−	0.771625i	$-0.280556\pi$
$72$	0	0
$73$	−100.766	−1.38036	−0.690179	−	0.723639i	$-0.742468\pi$
$73$	−100.766	−1.38036	−0.690179	+	0.723639i	$0.742468\pi$
$74$	57.2510i	0.773662i
$75$	0	0
$76$	28.5131	0.375172
$77$	− 269.139i	− 3.49531i
$78$	0	0
$79$	−150.150	−1.90063	−0.950315	−	0.311289i	$-0.899239\pi$
$79$	−150.150	−1.90063	−0.950315	+	0.311289i	$0.899239\pi$
$80$	16.0559i	0.200698i
$81$	0	0
$82$	−93.6559	−1.14214
$83$	62.0203i	0.747232i	0.927583	+	0.373616i	$0.121882\pi$
$83$	62.0203i	0.747232i	−0.927583	+	0.373616i	$0.878118\pi$
$84$	0	0
$85$	36.9826	0.435090
$86$	− 17.3444i	− 0.201680i
$87$	0	0
$88$	−55.1812	−0.627059
$89$	58.8455i	0.661185i	0.943774	+	0.330593i	$0.107249\pi$
$89$	58.8455i	0.661185i	−0.943774	+	0.330593i	$0.892751\pi$
$90$	0	0
$91$	237.848	2.61372
$92$	− 9.59166i	− 0.104257i
$93$	0	0
$94$	64.5266	0.686453
$95$	− 57.2254i	− 0.602372i
$96$	0	0
$97$	15.3056	0.157789	0.0788947	−	0.996883i	$-0.474861\pi$
$97$	15.3056	0.157789	0.0788947	+	0.996883i	$0.474861\pi$
$98$	199.843i	2.03921i
$99$	0	0
$100$	−17.7761	−0.177761
$101$	25.2840i	0.250337i	0.992136	+	0.125168i	$0.0399471\pi$
$101$	25.2840i	0.250337i	−0.992136	+	0.125168i	$0.960053\pi$
$102$	0	0
$103$	49.6907	0.482434	0.241217	−	0.970471i	$-0.422453\pi$
$103$	49.6907	0.482434	0.241217	+	0.970471i	$0.422453\pi$
$104$	− 48.7657i	− 0.468901i
$105$	0	0
$106$	21.9524	0.207098
$107$	4.08689i	0.0381953i	0.999818	+	0.0190976i	$0.00607933\pi$
$107$	4.08689i	0.0381953i	−0.999818	+	0.0190976i	$0.993921\pi$
$108$	0	0
$109$	48.0476	0.440804	0.220402	−	0.975409i	$-0.429263\pi$
$109$	48.0476	0.440804	0.220402	+	0.975409i	$0.429263\pi$
$110$	110.748i	1.00680i
$111$	0	0
$112$	55.1812	0.492689
$113$	− 76.2845i	− 0.675084i	−0.941311	−	0.337542i	$-0.890405\pi$
$113$	− 76.2845i	− 0.675084i	0.941311	−	0.337542i	$-0.109595\pi$
$114$	0	0
$115$	−19.2503	−0.167394
$116$	− 61.9931i	− 0.534423i
$117$	0	0
$118$	−92.1712	−0.781112
$119$	− 127.103i	− 1.06809i
$120$	0	0
$121$	−259.620	−2.14562
$122$	− 27.9787i	− 0.229333i
$123$	0	0
$124$	0.922521	0.00743968
$125$	136.026i	1.08820i
$126$	0	0
$127$	231.325	1.82146	0.910729	−	0.413004i	$-0.135520\pi$
$127$	231.325	1.82146	0.910729	+	0.413004i	$0.135520\pi$
$128$	− 11.3137i	− 0.0883883i
$129$	0	0
$130$	−97.8720	−0.752861
$131$	− 7.76763i	− 0.0592949i	−0.999560	−	0.0296474i	$-0.990562\pi$
$131$	− 7.76763i	− 0.0592949i	0.999560	−	0.0296474i	$-0.00943845\pi$
$132$	0	0
$133$	−196.673	−1.47875
$134$	− 114.591i	− 0.855159i
$135$	0	0
$136$	−26.0597	−0.191615
$137$	− 152.118i	− 1.11035i	−0.831733	−	0.555176i	$-0.812651\pi$
$137$	− 152.118i	− 1.11035i	0.831733	−	0.555176i	$-0.187349\pi$
$138$	0	0
$139$	−61.9125	−0.445414	−0.222707	−	0.974885i	$-0.571489\pi$
$139$	−61.9125	−0.445414	−0.222707	+	0.974885i	$0.571489\pi$
$140$	− 110.748i	− 0.791055i
$141$	0	0
$142$	154.956	1.09124
$143$	− 336.368i	− 2.35223i
$144$	0	0
$145$	−124.419	−0.858064
$146$	− 142.505i	− 0.976060i
$147$	0	0
$148$	−80.9651	−0.547061
$149$	60.1905i	0.403963i	0.979389	+	0.201982i	$0.0647381\pi$
$149$	60.1905i	0.403963i	−0.979389	+	0.201982i	$0.935262\pi$
$150$	0	0
$151$	184.698	1.22316	0.611582	−	0.791181i	$-0.290534\pi$
$151$	184.698	1.22316	0.611582	+	0.791181i	$0.290534\pi$
$152$	40.3236i	0.265287i
$153$	0	0
$154$	380.620	2.47156
$155$	− 1.85149i	− 0.0119451i
$156$	0	0
$157$	−286.887	−1.82731	−0.913654	−	0.406492i	$-0.866752\pi$
$157$	−286.887	−1.82731	−0.913654	+	0.406492i	$0.866752\pi$
$158$	− 212.344i	− 1.34395i
$159$	0	0
$160$	−22.7064	−0.141915
$161$	66.1599i	0.410931i
$162$	0	0
$163$	125.834	0.771985	0.385993	−	0.922502i	$-0.373859\pi$
$163$	125.834	0.771985	0.385993	+	0.922502i	$0.373859\pi$
$164$	− 132.449i	− 0.807618i
$165$	0	0
$166$	−87.7099	−0.528373
$167$	− 168.851i	− 1.01108i	−0.862802	−	0.505542i	$-0.831292\pi$
$167$	− 168.851i	− 1.01108i	0.862802	−	0.505542i	$-0.168708\pi$
$168$	0	0
$169$	128.261	0.758943
$170$	52.3013i	0.307655i
$171$	0	0
$172$	24.5287	0.142609
$173$	− 124.978i	− 0.722415i	−0.932486	−	0.361207i	$-0.882365\pi$
$173$	− 124.978i	− 0.722415i	0.932486	−	0.361207i	$-0.117635\pi$
$174$	0	0
$175$	122.613	0.700647
$176$	− 78.0379i	− 0.443397i
$177$	0	0
$178$	−83.2201	−0.467529
$179$	− 133.843i	− 0.747726i	−0.927484	−	0.373863i	$-0.878033\pi$
$179$	− 133.843i	− 0.747726i	0.927484	−	0.373863i	$-0.121967\pi$
$180$	0	0
$181$	−69.0489	−0.381486	−0.190743	−	0.981640i	$-0.561090\pi$
$181$	−69.0489	−0.381486	−0.190743	+	0.981640i	$0.561090\pi$
$182$	336.368i	1.84818i
$183$	0	0
$184$	13.5647	0.0737210
$185$	162.496i	0.878355i
$186$	0	0
$187$	−179.750	−0.961231
$188$	91.2543i	0.485395i
$189$	0	0
$190$	80.9289	0.425941
$191$	11.1819i	0.0585439i	0.999571	+	0.0292720i	$0.00931889\pi$
$191$	11.1819i	0.0585439i	−0.999571	+	0.0292720i	$0.990681\pi$
$192$	0	0
$193$	63.6882	0.329991	0.164995	−	0.986294i	$-0.447239\pi$
$193$	63.6882	0.329991	0.164995	+	0.986294i	$0.447239\pi$
$194$	21.6453i	0.111574i
$195$	0	0
$196$	−282.620	−1.44194
$197$	273.301i	1.38732i	0.720305	+	0.693658i	$0.244002\pi$
$197$	273.301i	1.38732i	−0.720305	+	0.693658i	$0.755998\pi$
$198$	0	0
$199$	169.046	0.849477	0.424739	−	0.905316i	$-0.360366\pi$
$199$	169.046	0.849477	0.424739	+	0.905316i	$0.360366\pi$
$200$	− 25.1392i	− 0.125696i
$201$	0	0
$202$	−35.7570	−0.177015
$203$	427.607i	2.10644i
$204$	0	0
$205$	−265.824	−1.29670
$206$	70.2733i	0.341132i
$207$	0	0
$208$	68.9651	0.331563
$209$	278.138i	1.33080i
$210$	0	0
$211$	−359.789	−1.70516	−0.852580	−	0.522597i	$-0.824963\pi$
$211$	−359.789	−1.70516	−0.852580	+	0.522597i	$0.824963\pi$
$212$	31.0453i	0.146440i
$213$	0	0
$214$	−5.77974	−0.0270081
$215$	− 49.2288i	− 0.228971i
$216$	0	0
$217$	−6.36322	−0.0293236
$218$	67.9496i	0.311696i
$219$	0	0
$220$	−156.621	−0.711913
$221$	− 158.852i	− 0.718788i
$222$	0	0
$223$	333.851	1.49709	0.748546	−	0.663083i	$-0.230752\pi$
$223$	333.851	1.49709	0.748546	+	0.663083i	$0.230752\pi$
$224$	78.0379i	0.348384i
$225$	0	0
$226$	107.883	0.477356
$227$	− 293.935i	− 1.29487i	−0.762121	−	0.647434i	$-0.775842\pi$
$227$	− 293.935i	− 1.29487i	0.762121	−	0.647434i	$-0.224158\pi$
$228$	0	0
$229$	−81.0240	−0.353817	−0.176908	−	0.984227i	$-0.556610\pi$
$229$	−81.0240	−0.353817	−0.176908	+	0.984227i	$0.556610\pi$
$230$	− 27.2241i	− 0.118365i
$231$	0	0
$232$	87.6715	0.377894
$233$	− 83.8410i	− 0.359833i	−0.983682	−	0.179916i	$-0.942417\pi$
$233$	− 83.8410i	− 0.359833i	0.983682	−	0.179916i	$-0.0575827\pi$
$234$	0	0
$235$	183.146	0.779345
$236$	− 130.350i	− 0.552329i
$237$	0	0
$238$	179.750	0.755253
$239$	12.5141i	0.0523604i	0.999657	+	0.0261802i	$0.00833438\pi$
$239$	12.5141i	0.0523604i	−0.999657	+	0.0261802i	$0.991666\pi$
$240$	0	0
$241$	−58.9056	−0.244422	−0.122211	−	0.992504i	$-0.538998\pi$
$241$	−58.9056	−0.244422	−0.122211	+	0.992504i	$0.538998\pi$
$242$	− 367.158i	− 1.51718i
$243$	0	0
$244$	39.5678	0.162163
$245$	567.214i	2.31516i
$246$	0	0
$247$	−245.801	−0.995146
$248$	1.30464i	0.00526065i
$249$	0	0
$250$	−192.369	−0.769477
$251$	21.9954i	0.0876311i	0.999040	+	0.0438155i	$0.0139514\pi$
$251$	21.9954i	0.0876311i	−0.999040	+	0.0438155i	$0.986049\pi$
$252$	0	0
$253$	93.5642	0.369819
$254$	327.143i	1.28797i
$255$	0	0
$256$	16.0000	0.0625000
$257$	154.734i	0.602079i	0.953612	+	0.301039i	$0.0973335\pi$
$257$	154.734i	0.602079i	−0.953612	+	0.301039i	$0.902666\pi$
$258$	0	0
$259$	558.468	2.15625
$260$	− 138.412i	− 0.532353i
$261$	0	0
$262$	10.9851	0.0419278
$263$	140.716i	0.535043i	0.963552	+	0.267522i	$0.0862047\pi$
$263$	140.716i	0.535043i	−0.963552	+	0.267522i	$0.913795\pi$
$264$	0	0
$265$	62.3075	0.235123
$266$	− 278.138i	− 1.04563i
$267$	0	0
$268$	162.057	0.604689
$269$	226.070i	0.840410i	0.907429	+	0.420205i	$0.138042\pi$
$269$	226.070i	0.840410i	−0.907429	+	0.420205i	$0.861958\pi$
$270$	0	0
$271$	−531.625	−1.96172	−0.980858	−	0.194725i	$-0.937619\pi$
$271$	−531.625	−1.96172	−0.980858	+	0.194725i	$0.937619\pi$
$272$	− 36.8539i	− 0.135492i
$273$	0	0
$274$	215.128	0.785138
$275$	− 173.401i	− 0.630550i
$276$	0	0
$277$	−82.7137	−0.298605	−0.149303	−	0.988792i	$-0.547703\pi$
$277$	−82.7137	−0.298605	−0.149303	+	0.988792i	$0.547703\pi$
$278$	− 87.5575i	− 0.314955i
$279$	0	0
$280$	156.621	0.559361
$281$	167.613i	0.596487i	0.954490	+	0.298244i	$0.0964008\pi$
$281$	167.613i	0.596487i	−0.954490	+	0.298244i	$0.903599\pi$
$282$	0	0
$283$	−237.759	−0.840138	−0.420069	−	0.907492i	$-0.637994\pi$
$283$	−237.759	−0.840138	−0.420069	+	0.907492i	$0.637994\pi$
$284$	219.141i	0.771625i
$285$	0	0
$286$	475.697	1.66327
$287$	913.589i	3.18324i
$288$	0	0
$289$	204.112	0.706269
$290$	− 175.955i	− 0.606743i
$291$	0	0
$292$	201.532	0.690179
$293$	− 88.3611i	− 0.301574i	−0.988566	−	0.150787i	$-0.951819\pi$
$293$	− 88.3611i	− 0.301574i	0.988566	−	0.150787i	$-0.0481807\pi$
$294$	0	0
$295$	−261.610	−0.886813
$296$	− 114.502i	− 0.386831i
$297$	0	0
$298$	−85.1222	−0.285645
$299$	82.6862i	0.276543i
$300$	0	0
$301$	−169.191	−0.562095
$302$	261.202i	0.864907i
$303$	0	0
$304$	−57.0262	−0.187586
$305$	− 79.4120i	− 0.260367i
$306$	0	0
$307$	−237.933	−0.775028	−0.387514	−	0.921864i	$-0.626666\pi$
$307$	−237.933	−0.775028	−0.387514	+	0.921864i	$0.626666\pi$
$308$	538.278i	1.74766i
$309$	0	0
$310$	2.61840	0.00844644
$311$	521.338i	1.67633i	0.545418	+	0.838164i	$0.316371\pi$
$311$	521.338i	1.67633i	−0.545418	+	0.838164i	$0.683629\pi$
$312$	0	0
$313$	−443.016	−1.41539	−0.707694	−	0.706519i	$-0.750264\pi$
$313$	−443.016	−1.41539	−0.707694	+	0.706519i	$0.750264\pi$
$314$	− 405.720i	− 1.29210i
$315$	0	0
$316$	300.300	0.950315
$317$	− 444.609i	− 1.40255i	−0.712890	−	0.701276i	$-0.752614\pi$
$317$	− 444.609i	− 1.40255i	0.712890	−	0.701276i	$-0.247386\pi$
$318$	0	0
$319$	604.727	1.89570
$320$	− 32.1118i	− 0.100349i
$321$	0	0
$322$	−93.5642	−0.290572
$323$	131.352i	0.406664i
$324$	0	0
$325$	153.241	0.471512
$326$	177.956i	0.545876i
$327$	0	0
$328$	187.312	0.571072
$329$	− 629.440i	− 1.91319i
$330$	0	0
$331$	−331.690	−1.00208	−0.501042	−	0.865423i	$-0.667050\pi$
$331$	−331.690	−1.00208	−0.501042	+	0.865423i	$0.667050\pi$
$332$	− 124.041i	− 0.373616i
$333$	0	0
$334$	238.792	0.714945
$335$	− 325.245i	− 0.970881i
$336$	0	0
$337$	260.727	0.773670	0.386835	−	0.922149i	$-0.373568\pi$
$337$	260.727	0.773670	0.386835	+	0.922149i	$0.373568\pi$
$338$	181.389i	0.536654i
$339$	0	0
$340$	−73.9653	−0.217545
$341$	8.99895i	0.0263899i
$342$	0	0
$343$	1273.44	3.71266
$344$	34.6889i	0.100840i
$345$	0	0
$346$	176.745	0.510824
$347$	477.334i	1.37560i	0.725899	+	0.687801i	$0.241424\pi$
$347$	477.334i	1.37560i	−0.725899	+	0.687801i	$0.758576\pi$
$348$	0	0
$349$	−585.243	−1.67691	−0.838457	−	0.544968i	$-0.816542\pi$
$349$	−585.243	−1.67691	−0.838457	+	0.544968i	$0.816542\pi$
$350$	173.401i	0.495432i
$351$	0	0
$352$	110.362	0.313529
$353$	270.001i	0.764874i	0.923982	+	0.382437i	$0.124915\pi$
$353$	270.001i	0.764874i	−0.923982	+	0.382437i	$0.875085\pi$
$354$	0	0
$355$	439.813	1.23891
$356$	− 117.691i	− 0.330593i
$357$	0	0
$358$	189.282	0.528722
$359$	− 109.266i	− 0.304361i	−0.988353	−	0.152181i	$-0.951370\pi$
$359$	− 109.266i	− 0.304361i	0.988353	−	0.152181i	$-0.0486296\pi$
$360$	0	0
$361$	−157.751	−0.436983
$362$	− 97.6499i	− 0.269751i
$363$	0	0
$364$	−475.697	−1.30686
$365$	− 404.472i	− 1.10814i
$366$	0	0
$367$	−546.847	−1.49005	−0.745024	−	0.667038i	$-0.767562\pi$
$367$	−546.847	−1.49005	−0.745024	+	0.667038i	$0.767562\pi$
$368$	19.1833i	0.0521286i
$369$	0	0
$370$	−229.804	−0.621091
$371$	− 214.140i	− 0.577196i
$372$	0	0
$373$	−14.7430	−0.0395254	−0.0197627	−	0.999805i	$-0.506291\pi$
$373$	−14.7430	−0.0395254	−0.0197627	+	0.999805i	$0.506291\pi$
$374$	− 254.205i	− 0.679693i
$375$	0	0
$376$	−129.053	−0.343226
$377$	534.420i	1.41756i
$378$	0	0
$379$	−446.501	−1.17810	−0.589052	−	0.808095i	$-0.700499\pi$
$379$	−446.501	−1.17810	−0.589052	+	0.808095i	$0.700499\pi$
$380$	114.451i	0.301186i
$381$	0	0
$382$	−15.8136	−0.0413968
$383$	− 78.0057i	− 0.203670i	−0.994801	−	0.101835i	$-0.967529\pi$
$383$	− 78.0057i	− 0.203670i	0.994801	−	0.101835i	$-0.0324714\pi$
$384$	0	0
$385$	1080.32	2.80602
$386$	90.0687i	0.233339i
$387$	0	0
$388$	−30.6111	−0.0788947
$389$	91.1103i	0.234217i	0.993119	+	0.117108i	$0.0373625\pi$
$389$	91.1103i	0.234217i	−0.993119	+	0.117108i	$0.962638\pi$
$390$	0	0
$391$	44.1863	0.113008
$392$	− 399.685i	− 1.01960i
$393$	0	0
$394$	−386.506	−0.980980
$395$	− 602.697i	− 1.52581i
$396$	0	0
$397$	−155.571	−0.391867	−0.195934	−	0.980617i	$-0.562774\pi$
$397$	−155.571	−0.391867	−0.195934	+	0.980617i	$0.562774\pi$
$398$	239.067i	0.600671i
$399$	0	0
$400$	35.5522	0.0888805
$401$	− 434.586i	− 1.08376i	−0.840457	−	0.541878i	$-0.817714\pi$
$401$	− 434.586i	− 1.08376i	0.840457	−	0.541878i	$-0.182286\pi$
$402$	0	0
$403$	−7.95271	−0.0197338
$404$	− 50.5680i	− 0.125168i
$405$	0	0
$406$	−604.727	−1.48948
$407$	− 789.794i	− 1.94052i
$408$	0	0
$409$	187.098	0.457452	0.228726	−	0.973491i	$-0.426544\pi$
$409$	187.098	0.457452	0.228726	+	0.973491i	$0.426544\pi$
$410$	− 375.932i	− 0.916907i
$411$	0	0
$412$	−99.3814	−0.241217
$413$	899.106i	2.17701i
$414$	0	0
$415$	−248.947	−0.599873
$416$	97.5314i	0.234450i
$417$	0	0
$418$	−393.347	−0.941020
$419$	− 282.354i	− 0.673877i	−0.941527	−	0.336938i	$-0.890609\pi$
$419$	− 282.354i	− 0.673877i	0.941527	−	0.336938i	$-0.109391\pi$
$420$	0	0
$421$	−543.715	−1.29149	−0.645743	−	0.763555i	$-0.723452\pi$
$421$	−543.715	−1.29149	−0.645743	+	0.763555i	$0.723452\pi$
$422$	− 508.818i	− 1.20573i
$423$	0	0
$424$	−43.9047	−0.103549
$425$	− 81.8899i	− 0.192682i
$426$	0	0
$427$	−272.925	−0.639168
$428$	− 8.17379i	− 0.0190976i
$429$	0	0
$430$	69.6200	0.161907
$431$	− 253.221i	− 0.587521i	−0.955879	−	0.293760i	$-0.905093\pi$
$431$	− 253.221i	− 0.587521i	0.955879	−	0.293760i	$-0.0949068\pi$
$432$	0	0
$433$	603.335	1.39338	0.696691	−	0.717371i	$-0.254655\pi$
$433$	603.335	1.39338	0.696691	+	0.717371i	$0.254655\pi$
$434$	− 8.99895i	− 0.0207349i
$435$	0	0
$436$	−96.0953	−0.220402
$437$	− 68.3720i	− 0.156458i
$438$	0	0
$439$	−42.1512	−0.0960163	−0.0480082	−	0.998847i	$-0.515287\pi$
$439$	−42.1512	−0.0960163	−0.0480082	+	0.998847i	$0.515287\pi$
$440$	− 221.495i	− 0.503399i
$441$	0	0
$442$	224.651	0.508260
$443$	− 553.720i	− 1.24993i	−0.780652	−	0.624966i	$-0.785113\pi$
$443$	− 553.720i	− 1.24993i	0.780652	−	0.624966i	$-0.214887\pi$
$444$	0	0
$445$	−236.204	−0.530796
$446$	472.137i	1.05860i
$447$	0	0
$448$	−110.362	−0.246344
$449$	266.907i	0.594447i	0.954808	+	0.297224i	$0.0960607\pi$
$449$	266.907i	0.594447i	−0.954808	+	0.297224i	$0.903939\pi$
$450$	0	0
$451$	1292.01	2.86477
$452$	152.569i	0.337542i
$453$	0	0
$454$	415.687	0.915610
$455$	954.716i	2.09828i
$456$	0	0
$457$	−384.602	−0.841580	−0.420790	−	0.907158i	$-0.638247\pi$
$457$	−384.602	−0.841580	−0.420790	+	0.907158i	$0.638247\pi$
$458$	− 114.585i	− 0.250186i
$459$	0	0
$460$	38.5006	0.0836970
$461$	− 59.7324i	− 0.129571i	−0.997899	−	0.0647857i	$-0.979364\pi$
$461$	− 59.7324i	− 0.129571i	0.997899	−	0.0647857i	$-0.0206364\pi$
$462$	0	0
$463$	550.909	1.18987	0.594934	−	0.803774i	$-0.297178\pi$
$463$	550.909	1.18987	0.594934	+	0.803774i	$0.297178\pi$
$464$	123.986i	0.267212i
$465$	0	0
$466$	118.569	0.254440
$467$	350.044i	0.749558i	0.927114	+	0.374779i	$0.122281\pi$
$467$	350.044i	0.749558i	−0.927114	+	0.374779i	$0.877719\pi$
$468$	0	0
$469$	−1117.81	−2.38339
$470$	259.008i	0.551080i
$471$	0	0
$472$	184.342	0.390556
$473$	239.272i	0.505860i
$474$	0	0
$475$	−126.713	−0.266764
$476$	254.205i	0.534045i
$477$	0	0
$478$	−17.6977	−0.0370244
$479$	− 881.460i	− 1.84021i	−0.391674	−	0.920104i	$-0.628104\pi$
$479$	− 881.460i	− 1.84021i	0.391674	−	0.920104i	$-0.371896\pi$
$480$	0	0
$481$	697.970	1.45108
$482$	− 83.3051i	− 0.172832i
$483$	0	0
$484$	519.240	1.07281
$485$	61.4361i	0.126672i
$486$	0	0
$487$	297.985	0.611879	0.305940	−	0.952051i	$-0.401029\pi$
$487$	297.985	0.611879	0.305940	+	0.952051i	$0.401029\pi$
$488$	55.9574i	0.114667i
$489$	0	0
$490$	−802.162	−1.63707
$491$	− 430.526i	− 0.876835i	−0.898771	−	0.438418i	$-0.855539\pi$
$491$	− 430.526i	− 0.876835i	0.898771	−	0.438418i	$-0.144461\pi$
$492$	0	0
$493$	285.586	0.579282
$494$	− 347.615i	− 0.703674i
$495$	0	0
$496$	−1.84504	−0.00371984
$497$	− 1511.56i	− 3.04137i
$498$	0	0
$499$	−246.893	−0.494776	−0.247388	−	0.968916i	$-0.579572\pi$
$499$	−246.893	−0.494776	−0.247388	+	0.968916i	$0.579572\pi$
$500$	− 272.051i	− 0.544102i
$501$	0	0
$502$	−31.1062	−0.0619645
$503$	219.501i	0.436384i	0.975906	+	0.218192i	$0.0700158\pi$
$503$	219.501i	0.436384i	−0.975906	+	0.218192i	$0.929984\pi$
$504$	0	0
$505$	−101.489	−0.200969
$506$	132.320i	0.261502i
$507$	0	0
$508$	−462.650	−0.910729
$509$	783.246i	1.53879i	0.638771	+	0.769397i	$0.279443\pi$
$509$	783.246i	1.53879i	−0.638771	+	0.769397i	$0.720557\pi$
$510$	0	0
$511$	−1390.10	−2.72035
$512$	22.6274i	0.0441942i
$513$	0	0
$514$	−218.827	−0.425734
$515$	199.457i	0.387295i
$516$	0	0
$517$	−890.163	−1.72178
$518$	789.794i	1.52470i
$519$	0	0
$520$	195.744	0.376431
$521$	138.865i	0.266535i	0.991080	+	0.133267i	$0.0425469\pi$
$521$	138.865i	0.266535i	−0.991080	+	0.133267i	$0.957453\pi$
$522$	0	0
$523$	390.139	0.745963	0.372982	−	0.927839i	$-0.378335\pi$
$523$	390.139	0.745963	0.372982	+	0.927839i	$0.378335\pi$
$524$	15.5353i	0.0296474i
$525$	0	0
$526$	−199.003	−0.378333
$527$	4.24981i	0.00806416i
$528$	0	0
$529$	−23.0000	−0.0434783
$530$	88.1161i	0.166257i
$531$	0	0
$532$	393.347	0.739373
$533$	1141.80i	2.14221i
$534$	0	0
$535$	−16.4047	−0.0306629
$536$	229.183i	0.427580i
$537$	0	0
$538$	−319.712	−0.594260
$539$	− 2756.89i	− 5.11482i
$540$	0	0
$541$	210.330	0.388780	0.194390	−	0.980924i	$-0.437727\pi$
$541$	210.330	0.388780	0.194390	+	0.980924i	$0.437727\pi$
$542$	− 751.831i	− 1.38714i
$543$	0	0
$544$	52.1193	0.0958076
$545$	192.862i	0.353875i
$546$	0	0
$547$	132.110	0.241517	0.120758	−	0.992682i	$-0.461467\pi$
$547$	132.110	0.241517	0.120758	+	0.992682i	$0.461467\pi$
$548$	304.237i	0.555176i
$549$	0	0
$550$	245.226	0.445866
$551$	− 441.904i	− 0.802004i
$552$	0	0
$553$	−2071.36	−3.74568
$554$	− 116.975i	− 0.211146i
$555$	0	0
$556$	123.825	0.222707
$557$	− 732.730i	− 1.31549i	−0.753239	−	0.657746i	$-0.771510\pi$
$557$	− 732.730i	− 1.31549i	0.753239	−	0.657746i	$-0.228490\pi$
$558$	0	0
$559$	−211.453	−0.378271
$560$	221.495i	0.395528i
$561$	0	0
$562$	−237.040	−0.421780
$563$	177.675i	0.315586i	0.987472	+	0.157793i	$0.0504379\pi$
$563$	177.675i	0.315586i	−0.987472	+	0.157793i	$0.949562\pi$
$564$	0	0
$565$	306.203	0.541953
$566$	− 336.242i	− 0.594067i
$567$	0	0
$568$	−309.913	−0.545621
$569$	− 339.224i	− 0.596175i	−0.954539	−	0.298088i	$-0.903651\pi$
$569$	− 339.224i	− 0.596175i	0.954539	−	0.298088i	$-0.0963488\pi$
$570$	0	0
$571$	−108.901	−0.190719	−0.0953595	−	0.995443i	$-0.530400\pi$
$571$	−108.901	−0.190719	−0.0953595	+	0.995443i	$0.530400\pi$
$572$	672.737i	1.17611i
$573$	0	0
$574$	−1292.01	−2.25089
$575$	42.6256i	0.0741315i
$576$	0	0
$577$	−22.0105	−0.0381465	−0.0190732	−	0.999818i	$-0.506072\pi$
$577$	−22.0105	−0.0381465	−0.0190732	+	0.999818i	$0.506072\pi$
$578$	288.658i	0.499408i
$579$	0	0
$580$	248.839	0.429032
$581$	855.588i	1.47261i
$582$	0	0
$583$	−302.839	−0.519449
$584$	285.010i	0.488030i
$585$	0	0
$586$	124.961	0.213245
$587$	294.308i	0.501376i	0.968068	+	0.250688i	$0.0806569\pi$
$587$	294.308i	0.501376i	−0.968068	+	0.250688i	$0.919343\pi$
$588$	0	0
$589$	6.57598	0.0111647
$590$	− 369.972i	− 0.627072i
$591$	0	0
$592$	161.930	0.273531
$593$	768.299i	1.29561i	0.761805	+	0.647807i	$0.224314\pi$
$593$	768.299i	1.29561i	−0.761805	+	0.647807i	$0.775686\pi$
$594$	0	0
$595$	510.186	0.857456
$596$	− 120.381i	− 0.201982i
$597$	0	0
$598$	−116.936	−0.195545
$599$	− 132.464i	− 0.221142i	−0.993868	−	0.110571i	$-0.964732\pi$
$599$	− 132.464i	− 0.221142i	0.993868	−	0.110571i	$-0.0352679\pi$
$600$	0	0
$601$	220.061	0.366157	0.183079	−	0.983098i	$-0.441394\pi$
$601$	220.061	0.366157	0.183079	+	0.983098i	$0.441394\pi$
$602$	− 239.272i	− 0.397461i
$603$	0	0
$604$	−369.395	−0.611582
$605$	− 1042.11i	− 1.72249i
$606$	0	0
$607$	4.70721	0.00775487	0.00387743	−	0.999992i	$-0.498766\pi$
$607$	4.70721	0.00775487	0.00387743	+	0.999992i	$0.498766\pi$
$608$	− 80.6472i	− 0.132643i
$609$	0	0
$610$	112.306	0.184108
$611$	− 786.670i	− 1.28751i
$612$	0	0
$613$	205.778	0.335690	0.167845	−	0.985813i	$-0.446319\pi$
$613$	205.778	0.335690	0.167845	+	0.985813i	$0.446319\pi$
$614$	− 336.489i	− 0.548027i
$615$	0	0
$616$	−761.240	−1.23578
$617$	623.417i	1.01040i	0.863002	+	0.505200i	$0.168581\pi$
$617$	623.417i	1.01040i	−0.863002	+	0.505200i	$0.831419\pi$
$618$	0	0
$619$	511.447	0.826247	0.413124	−	0.910675i	$-0.364438\pi$
$619$	511.447	0.826247	0.413124	+	0.910675i	$0.364438\pi$
$620$	3.70297i	0.00597253i
$621$	0	0
$622$	−737.284	−1.18534
$623$	811.791i	1.30303i
$624$	0	0
$625$	−323.801	−0.518082
$626$	− 626.520i	− 1.00083i
$627$	0	0
$628$	573.775	0.913654
$629$	− 372.985i	− 0.592981i
$630$	0	0
$631$	388.506	0.615699	0.307849	−	0.951435i	$-0.400391\pi$
$631$	388.506	0.615699	0.307849	+	0.951435i	$0.400391\pi$
$632$	424.688i	0.671974i
$633$	0	0
$634$	628.772	0.991754
$635$	928.532i	1.46226i
$636$	0	0
$637$	2436.36	3.82475
$638$	855.213i	1.34046i
$639$	0	0
$640$	45.4129	0.0709576
$641$	− 1049.39i	− 1.63712i	−0.574420	−	0.818561i	$-0.694772\pi$
$641$	− 1049.39i	− 1.63712i	0.574420	−	0.818561i	$-0.305228\pi$
$642$	0	0
$643$	532.084	0.827503	0.413751	−	0.910390i	$-0.364218\pi$
$643$	532.084	0.827503	0.413751	+	0.910390i	$0.364218\pi$
$644$	− 132.320i	− 0.205465i
$645$	0	0
$646$	−185.760	−0.287555
$647$	− 168.248i	− 0.260043i	−0.991511	−	0.130022i	$-0.958495\pi$
$647$	− 168.248i	− 0.260043i	0.991511	−	0.130022i	$-0.0415047\pi$
$648$	0	0
$649$	1271.53	1.95921
$650$	216.716i	0.333409i
$651$	0	0
$652$	−251.667	−0.385993
$653$	773.557i	1.18462i	0.805710	+	0.592310i	$0.201784\pi$
$653$	773.557i	1.18462i	−0.805710	+	0.592310i	$0.798216\pi$
$654$	0	0
$655$	31.1790	0.0476015
$656$	264.899i	0.403809i
$657$	0	0
$658$	890.163	1.35283
$659$	316.884i	0.480856i	0.970667	+	0.240428i	$0.0772878\pi$
$659$	316.884i	0.480856i	−0.970667	+	0.240428i	$0.922712\pi$
$660$	0	0
$661$	1096.64	1.65907	0.829533	−	0.558458i	$-0.188607\pi$
$661$	1096.64	1.65907	0.829533	+	0.558458i	$0.188607\pi$
$662$	− 469.080i	− 0.708581i
$663$	0	0
$664$	175.420	0.264186
$665$	− 789.440i	− 1.18713i
$666$	0	0
$667$	−148.654	−0.222870
$668$	337.702i	0.505542i
$669$	0	0
$670$	459.966	0.686516
$671$	385.974i	0.575222i
$672$	0	0
$673$	−574.668	−0.853890	−0.426945	−	0.904278i	$-0.640410\pi$
$673$	−574.668	−0.853890	−0.426945	+	0.904278i	$0.640410\pi$
$674$	368.723i	0.547067i
$675$	0	0
$676$	−256.523	−0.379472
$677$	309.184i	0.456698i	0.973579	+	0.228349i	$0.0733326\pi$
$677$	309.184i	0.456698i	−0.973579	+	0.228349i	$0.926667\pi$
$678$	0	0
$679$	211.145	0.310964
$680$	− 104.603i	− 0.153827i
$681$	0	0
$682$	−12.7264	−0.0186605
$683$	352.724i	0.516434i	0.966087	+	0.258217i	$0.0831349\pi$
$683$	352.724i	0.516434i	−0.966087	+	0.258217i	$0.916865\pi$
$684$	0	0
$685$	610.598	0.891384
$686$	1800.92i	2.62525i
$687$	0	0
$688$	−49.0575	−0.0713045
$689$	− 267.630i	− 0.388433i
$690$	0	0
$691$	230.898	0.334150	0.167075	−	0.985944i	$-0.446568\pi$
$691$	230.898	0.334150	0.167075	+	0.985944i	$0.446568\pi$
$692$	249.955i	0.361207i
$693$	0	0
$694$	−675.052	−0.972697
$695$	− 248.515i	− 0.357575i
$696$	0	0
$697$	610.160	0.875409
$698$	− 827.658i	− 1.18576i
$699$	0	0
$700$	−245.226	−0.350324
$701$	− 1372.09i	− 1.95733i	−0.205466	−	0.978664i	$-0.565871\pi$
$701$	− 1372.09i	− 1.95733i	0.205466	−	0.978664i	$-0.434129\pi$
$702$	0	0
$703$	−577.141	−0.820969
$704$	156.076i	0.221699i
$705$	0	0
$706$	−381.839	−0.540848
$707$	348.800i	0.493353i
$708$	0	0
$709$	−1027.99	−1.44992	−0.724961	−	0.688790i	$-0.758142\pi$
$709$	−1027.99	−1.44992	−0.724961	+	0.688790i	$0.758142\pi$
$710$	621.990i	0.876042i
$711$	0	0
$712$	166.440	0.233764
$713$	− 2.21213i	− 0.00310256i
$714$	0	0
$715$	1350.17	1.88835
$716$	267.686i	0.373863i
$717$	0	0
$718$	154.525	0.215216
$719$	733.527i	1.02020i	0.860114	+	0.510102i	$0.170392\pi$
$719$	733.527i	1.02020i	−0.860114	+	0.510102i	$0.829608\pi$
$720$	0	0
$721$	685.498	0.950760
$722$	− 223.093i	− 0.308993i
$723$	0	0
$724$	138.098	0.190743
$725$	275.499i	0.379999i
$726$	0	0
$727$	735.082	1.01112	0.505559	−	0.862792i	$-0.331286\pi$
$727$	735.082	1.01112	0.505559	+	0.862792i	$0.331286\pi$
$728$	− 672.737i	− 0.924089i
$729$	0	0
$730$	572.010	0.783575
$731$	112.998i	0.154579i
$732$	0	0
$733$	122.777	0.167500	0.0837498	−	0.996487i	$-0.473310\pi$
$733$	122.777	0.167500	0.0837498	+	0.996487i	$0.473310\pi$
$734$	− 773.359i	− 1.05362i
$735$	0	0
$736$	−27.1293	−0.0368605
$737$	1580.82i	2.14494i
$738$	0	0
$739$	−653.661	−0.884520	−0.442260	−	0.896887i	$-0.645823\pi$
$739$	−653.661	−0.884520	−0.442260	+	0.896887i	$0.645823\pi$
$740$	− 324.991i	− 0.439177i
$741$	0	0
$742$	302.839	0.408139
$743$	1282.49i	1.72610i	0.505122	+	0.863048i	$0.331447\pi$
$743$	1282.49i	1.72610i	−0.505122	+	0.863048i	$0.668553\pi$
$744$	0	0
$745$	−241.603	−0.324299
$746$	− 20.8497i	− 0.0279487i
$747$	0	0
$748$	359.501	0.480616
$749$	56.3799i	0.0752735i
$750$	0	0
$751$	46.0106	0.0612658	0.0306329	−	0.999531i	$-0.490248\pi$
$751$	46.0106	0.0612658	0.0306329	+	0.999531i	$0.490248\pi$
$752$	− 182.509i	− 0.242698i
$753$	0	0
$754$	−755.784	−1.00237
$755$	741.371i	0.981948i
$756$	0	0
$757$	−1062.94	−1.40414	−0.702071	−	0.712106i	$-0.747741\pi$
$757$	−1062.94	−1.40414	−0.702071	+	0.712106i	$0.747741\pi$
$758$	− 631.448i	− 0.833045i
$759$	0	0
$760$	−161.858	−0.212971
$761$	− 812.603i	− 1.06781i	−0.845544	−	0.533905i	$-0.820724\pi$
$761$	− 812.603i	− 1.06781i	0.845544	−	0.533905i	$-0.179276\pi$
$762$	0	0
$763$	662.831	0.868717
$764$	− 22.3638i	− 0.0292720i
$765$	0	0
$766$	110.317	0.144017
$767$	1123.70i	1.46506i
$768$	0	0
$769$	752.268	0.978242	0.489121	−	0.872216i	$-0.337318\pi$
$769$	752.268	0.978242	0.489121	+	0.872216i	$0.337318\pi$
$770$	1527.80i	1.98415i
$771$	0	0
$772$	−127.376	−0.164995
$773$	− 567.716i	− 0.734432i	−0.930136	−	0.367216i	$-0.880311\pi$
$773$	− 567.716i	− 0.734432i	0.930136	−	0.367216i	$-0.119689\pi$
$774$	0	0
$775$	−4.09971	−0.00528994
$776$	− 43.2907i	− 0.0557870i
$777$	0	0
$778$	−128.849	−0.165616
$779$	− 944.136i	− 1.21198i
$780$	0	0
$781$	−2137.67	−2.73709
$782$	62.4889i	0.0799090i
$783$	0	0
$784$	565.240	0.720970
$785$	− 1151.56i	− 1.46695i
$786$	0	0
$787$	−546.593	−0.694528	−0.347264	−	0.937767i	$-0.612889\pi$
$787$	−546.593	−0.694528	−0.347264	+	0.937767i	$0.612889\pi$
$788$	− 546.602i	− 0.693658i
$789$	0	0
$790$	852.342	1.07891
$791$	− 1052.37i	− 1.33043i
$792$	0	0
$793$	−341.100	−0.430139
$794$	− 220.011i	− 0.277092i
$795$	0	0
$796$	−338.092	−0.424739
$797$	− 1146.18i	− 1.43812i	−0.694949	−	0.719059i	$-0.744573\pi$
$797$	− 1146.18i	− 1.43812i	0.694949	−	0.719059i	$-0.255427\pi$
$798$	0	0
$799$	−420.385	−0.526139
$800$	50.2784i	0.0628480i
$801$	0	0
$802$	614.597	0.766331
$803$	1965.89i	2.44819i
$804$	0	0
$805$	−265.564	−0.329893
$806$	− 11.2468i	− 0.0139539i
$807$	0	0
$808$	71.5140	0.0885074
$809$	1237.02i	1.52907i	0.644582	+	0.764535i	$0.277031\pi$
$809$	1237.02i	1.52907i	−0.644582	+	0.764535i	$0.722969\pi$
$810$	0	0
$811$	263.725	0.325185	0.162593	−	0.986693i	$-0.448014\pi$
$811$	263.725	0.325185	0.162593	+	0.986693i	$0.448014\pi$
$812$	− 855.213i	− 1.05322i
$813$	0	0
$814$	1116.94	1.37216
$815$	505.092i	0.619745i
$816$	0	0
$817$	174.848	0.214012
$818$	264.596i	0.323467i
$819$	0	0
$820$	531.648	0.648351
$821$	− 276.765i	− 0.337107i	−0.985692	−	0.168554i	$-0.946090\pi$
$821$	− 276.765i	− 0.337107i	0.985692	−	0.168554i	$-0.0539096\pi$
$822$	0	0
$823$	−832.583	−1.01164	−0.505822	−	0.862638i	$-0.668811\pi$
$823$	−832.583	−1.01164	−0.505822	+	0.862638i	$0.668811\pi$
$824$	− 140.547i	− 0.170566i
$825$	0	0
$826$	−1271.53	−1.53938
$827$	1505.46i	1.82038i	0.414187	+	0.910192i	$0.364066\pi$
$827$	1505.46i	1.82038i	−0.414187	+	0.910192i	$0.635934\pi$
$828$	0	0
$829$	137.703	0.166107	0.0830536	−	0.996545i	$-0.473533\pi$
$829$	137.703	0.166107	0.0830536	+	0.996545i	$0.473533\pi$
$830$	− 352.065i	− 0.424175i
$831$	0	0
$832$	−137.930	−0.165781
$833$	− 1301.96i	− 1.56297i
$834$	0	0
$835$	677.763	0.811693
$836$	− 556.276i	− 0.665402i
$837$	0	0
$838$	399.309	0.476503
$839$	669.877i	0.798423i	0.916859	+	0.399211i	$0.130716\pi$
$839$	669.877i	0.798423i	−0.916859	+	0.399211i	$0.869284\pi$
$840$	0	0
$841$	−119.787	−0.142434
$842$	− 768.930i	− 0.913218i
$843$	0	0
$844$	719.577	0.852580
$845$	514.837i	0.609275i
$846$	0	0
$847$	−3581.53	−4.22849
$848$	− 62.0906i	− 0.0732201i
$849$	0	0
$850$	115.810	0.136247
$851$	194.147i	0.228140i
$852$	0	0
$853$	962.040	1.12783	0.563915	−	0.825833i	$-0.309294\pi$
$853$	962.040	1.12783	0.563915	+	0.825833i	$0.309294\pi$
$854$	− 385.974i	− 0.451960i
$855$	0	0
$856$	11.5595	0.0135041
$857$	− 841.887i	− 0.982365i	−0.871057	−	0.491183i	$-0.836565\pi$
$857$	− 841.887i	− 0.982365i	0.871057	−	0.491183i	$-0.163435\pi$
$858$	0	0
$859$	671.416	0.781625	0.390812	−	0.920470i	$-0.372194\pi$
$859$	671.416	0.781625	0.390812	+	0.920470i	$0.372194\pi$
$860$	98.4576i	0.114486i
$861$	0	0
$862$	358.109	0.415440
$863$	− 145.440i	− 0.168529i	−0.996443	−	0.0842644i	$-0.973146\pi$
$863$	− 145.440i	− 0.168529i	0.996443	−	0.0842644i	$-0.0268540\pi$
$864$	0	0
$865$	501.657	0.579950
$866$	853.244i	0.985270i
$867$	0	0
$868$	12.7264	0.0146618
$869$	2929.35i	3.37094i
$870$	0	0
$871$	−1397.03	−1.60394
$872$	− 135.899i	− 0.155848i
$873$	0	0
$874$	96.6926	0.110632
$875$	1876.51i	2.14459i
$876$	0	0
$877$	471.691	0.537846	0.268923	−	0.963162i	$-0.413332\pi$
$877$	471.691	0.537846	0.268923	+	0.963162i	$0.413332\pi$
$878$	− 59.6107i	− 0.0678938i
$879$	0	0
$880$	313.242	0.355957
$881$	− 1448.19i	− 1.64380i	−0.569634	−	0.821898i	$-0.692915\pi$
$881$	− 1448.19i	− 1.64380i	0.569634	−	0.821898i	$-0.307085\pi$
$882$	0	0
$883$	−1237.51	−1.40148	−0.700740	−	0.713417i	$-0.747147\pi$
$883$	−1237.51	−1.40148	−0.700740	+	0.713417i	$0.747147\pi$
$884$	317.704i	0.359394i
$885$	0	0
$886$	783.078	0.883835
$887$	965.839i	1.08888i	0.838799	+	0.544442i	$0.183258\pi$
$887$	965.839i	1.08888i	−0.838799	+	0.544442i	$0.816742\pi$
$888$	0	0
$889$	3191.20	3.58965
$890$	− 334.043i	− 0.375329i
$891$	0	0
$892$	−667.703	−0.748546
$893$	650.486i	0.728428i
$894$	0	0
$895$	537.241	0.600270
$896$	− 156.076i	− 0.174192i
$897$	0	0
$898$	−377.463	−0.420338
$899$	− 14.2975i	− 0.0159038i
$900$	0	0
$901$	−143.018	−0.158732
$902$	1827.18i	2.02570i
$903$	0	0
$904$	−215.765	−0.238678
$905$	− 277.160i	− 0.306254i
$906$	0	0
$907$	−446.706	−0.492510	−0.246255	−	0.969205i	$-0.579200\pi$
$907$	−446.706	−0.492510	−0.246255	+	0.969205i	$0.579200\pi$
$908$	587.870i	0.647434i
$909$	0	0
$910$	−1350.17	−1.48371
$911$	− 1066.10i	− 1.17025i	−0.810942	−	0.585127i	$-0.801045\pi$
$911$	− 1066.10i	− 1.17025i	0.810942	−	0.585127i	$-0.198955\pi$
$912$	0	0
$913$	1209.98	1.32528
$914$	− 543.910i	− 0.595087i
$915$	0	0
$916$	162.048	0.176908
$917$	− 107.157i	− 0.116856i
$918$	0	0
$919$	355.173	0.386478	0.193239	−	0.981152i	$-0.438101\pi$
$919$	355.173	0.386478	0.193239	+	0.981152i	$0.438101\pi$
$920$	54.4481i	0.0591827i
$921$	0	0
$922$	84.4743	0.0916208
$923$	− 1889.14i	− 2.04674i
$924$	0	0
$925$	359.811	0.388985
$926$	779.103i	0.841364i
$927$	0	0
$928$	−175.343	−0.188947
$929$	940.650i	1.01254i	0.862375	+	0.506270i	$0.168976\pi$
$929$	940.650i	1.01254i	−0.862375	+	0.506270i	$0.831024\pi$
$930$	0	0
$931$	−2014.59	−2.16390
$932$	167.682i	0.179916i
$933$	0	0
$934$	−495.036	−0.530018
$935$	− 721.512i	− 0.771671i
$936$	0	0
$937$	−766.421	−0.817952	−0.408976	−	0.912545i	$-0.634114\pi$
$937$	−766.421	−0.817952	−0.408976	+	0.912545i	$0.634114\pi$
$938$	− 1580.82i	− 1.68531i
$939$	0	0
$940$	−366.292	−0.389672
$941$	658.037i	0.699296i	0.936881	+	0.349648i	$0.113699\pi$
$941$	658.037i	0.699296i	−0.936881	+	0.349648i	$0.886301\pi$
$942$	0	0
$943$	−317.603	−0.336800
$944$	260.699i	0.276165i
$945$	0	0
$946$	−338.381	−0.357697
$947$	− 1656.18i	− 1.74888i	−0.485138	−	0.874438i	$-0.661231\pi$
$947$	− 1656.18i	− 1.74888i	0.485138	−	0.874438i	$-0.338769\pi$
$948$	0	0
$949$	−1737.34	−1.83070
$950$	− 179.199i	− 0.188631i
$951$	0	0
$952$	−359.501	−0.377627
$953$	920.556i	0.965956i	0.875632	+	0.482978i	$0.160445\pi$
$953$	920.556i	0.965956i	−0.875632	+	0.482978i	$0.839555\pi$
$954$	0	0
$955$	−44.8838	−0.0469987
$956$	− 25.0283i	− 0.0261802i
$957$	0	0
$958$	1246.57	1.30122
$959$	− 2098.52i	− 2.18823i
$960$	0	0
$961$	−960.787	−0.999779
$962$	987.079i	1.02607i
$963$	0	0
$964$	117.811	0.122211
$965$	255.642i	0.264914i
$966$	0	0
$967$	−56.2259	−0.0581446	−0.0290723	−	0.999577i	$-0.509255\pi$
$967$	−56.2259	−0.0581446	−0.0290723	+	0.999577i	$0.509255\pi$
$968$	734.316i	0.758591i
$969$	0	0
$970$	−86.8838	−0.0895709
$971$	− 428.609i	− 0.441410i	−0.975341	−	0.220705i	$-0.929164\pi$
$971$	− 428.609i	− 0.441410i	0.975341	−	0.220705i	$-0.0708358\pi$
$972$	0	0
$973$	−854.101	−0.877802
$974$	421.415i	0.432664i
$975$	0	0
$976$	−79.1357	−0.0810816
$977$	− 1886.22i	− 1.93063i	−0.261095	−	0.965313i	$-0.584084\pi$
$977$	− 1886.22i	− 1.93063i	0.261095	−	0.965313i	$-0.415916\pi$
$978$	0	0
$979$	1148.05	1.17267
$980$	− 1134.43i	− 1.15758i
$981$	0	0
$982$	608.856	0.620016
$983$	− 958.012i	− 0.974580i	−0.873240	−	0.487290i	$-0.837985\pi$
$983$	− 958.012i	− 0.974580i	0.873240	−	0.487290i	$-0.162015\pi$
$984$	0	0
$985$	−1097.02	−1.11373
$986$	403.880i	0.409615i
$987$	0	0
$988$	491.602	0.497573
$989$	− 58.8178i	− 0.0594720i
$990$	0	0
$991$	871.718	0.879634	0.439817	−	0.898087i	$-0.355043\pi$
$991$	871.718	0.879634	0.439817	+	0.898087i	$0.355043\pi$
$992$	− 2.60928i	− 0.00263033i
$993$	0	0
$994$	2137.67	2.15057
$995$	678.545i	0.681955i
$996$	0	0
$997$	1073.82	1.07706	0.538528	−	0.842608i	$-0.318981\pi$
$997$	1073.82	1.07706	0.538528	+	0.842608i	$0.318981\pi$
$998$	− 349.160i	− 0.349860i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	414.3.c.b.323.7	yes	8
3.2	odd	2	inner	414.3.c.b.323.2	✓	8
4.3	odd	2		3312.3.g.b.737.6		8
12.11	even	2		3312.3.g.b.737.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
414.3.c.b.323.2	✓	8	3.2	odd	2	inner
414.3.c.b.323.7	yes	8	1.1	even	1	trivial
3312.3.g.b.737.3		8	12.11	even	2
3312.3.g.b.737.6		8	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	414.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} -2.00000 q^{4} +4.01397i q^{5} +13.7953 q^{7} -2.82843i q^{8} +O(q^{10})\)	\(q+1.41421i q^{2} -2.00000 q^{4} +4.01397i q^{5} +13.7953 q^{7} -2.82843i q^{8} -5.67661 q^{10} -19.5095i q^{11} +17.2413 q^{13} +19.5095i q^{14} +4.00000 q^{16} -9.21348i q^{17} -14.2566 q^{19} -8.02794i q^{20} +27.5906 q^{22} +4.79583i q^{23} +8.88805 q^{25} +24.3828i q^{26} -27.5906 q^{28} +30.9966i q^{29} -0.461260 q^{31} +5.65685i q^{32} +13.0298 q^{34} +55.3739i q^{35} +40.4825 q^{37} -20.1618i q^{38} +11.3532 q^{40} +66.2247i q^{41} -12.2644 q^{43} +39.0190i q^{44} -6.78233 q^{46} -45.6272i q^{47} +141.310 q^{49} +12.5696i q^{50} -34.4825 q^{52} -15.5227i q^{53} +78.3105 q^{55} -39.0190i q^{56} -43.8358 q^{58} +65.1749i q^{59} -19.7839 q^{61} -0.652321i q^{62} -8.00000 q^{64} +69.2059i q^{65} -81.0283 q^{67} +18.4270i q^{68} -78.3105 q^{70} -109.571i q^{71} -100.766 q^{73} +57.2510i q^{74} +28.5131 q^{76} -269.139i q^{77} -150.150 q^{79} +16.0559i q^{80} -93.6559 q^{82} +62.0203i q^{83} +36.9826 q^{85} -17.3444i q^{86} -55.1812 q^{88} +58.8455i q^{89} +237.848 q^{91} -9.59166i q^{92} +64.5266 q^{94} -57.2254i q^{95} +15.3056 q^{97} +199.843i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{4} + 16 q^{7}+O(q^{10}) \) 8 * q - 16 * q^4 + 16 * q^7	\( 8 q - 16 q^{4} + 16 q^{7} - 8 q^{10} - 8 q^{13} + 32 q^{16} - 48 q^{19} + 32 q^{22} - 32 q^{28} - 32 q^{31} - 8 q^{34} + 32 q^{37} + 16 q^{40} + 32 q^{43} + 80 q^{49} + 16 q^{52} + 32 q^{55} + 16 q^{58} + 48 q^{61} - 64 q^{64} - 16 q^{67} - 32 q^{70} - 432 q^{73} + 96 q^{76} - 416 q^{79} - 144 q^{82} + 584 q^{85} - 64 q^{88} + 368 q^{91} + 128 q^{94} - 8 q^{97}+O(q^{100}) \) 8 * q - 16 * q^4 + 16 * q^7 - 8 * q^10 - 8 * q^13 + 32 * q^16 - 48 * q^19 + 32 * q^22 - 32 * q^28 - 32 * q^31 - 8 * q^34 + 32 * q^37 + 16 * q^40 + 32 * q^43 + 80 * q^49 + 16 * q^52 + 32 * q^55 + 16 * q^58 + 48 * q^61 - 64 * q^64 - 16 * q^67 - 32 * q^70 - 432 * q^73 + 96 * q^76 - 416 * q^79 - 144 * q^82 + 584 * q^85 - 64 * q^88 + 368 * q^91 + 128 * q^94 - 8 * q^97

Introduction

L-functions

Modular forms

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