LMFDB - Embedded newform 1125.2.b.d.874.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1125,2,Mod(874,1125)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1125.874"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1125, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1125 = 3^{2} \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1125.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,-2,0,0,0,0,0,0,20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$8.98317022739$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{10})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + x^{2} - x + 1 $ x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			874.3
Root			$0.809017 + 0.587785i$ of defining polynomial
Character	$\chi$	$=$	1125.874
Dual form			1125.2.b.d.874.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.17557i q^{2} +0.618034 q^{4} -2.62866i q^{7} +3.07768i q^{8} +5.00000 q^{11} -6.88191i q^{13} +3.09017 q^{14} -2.38197 q^{16} -4.97980i q^{17} -3.47214 q^{19} +5.87785i q^{22} -5.42882i q^{23} +8.09017 q^{26} -1.62460i q^{28} -5.00000 q^{29} -2.85410 q^{31} +3.35520i q^{32} +5.85410 q^{34} +1.62460i q^{37} -4.08174i q^{38} +8.09017 q^{41} -1.00406i q^{43} +3.09017 q^{44} +6.38197 q^{46} +8.33499i q^{47} +0.0901699 q^{49} -4.25325i q^{52} +7.15942i q^{53} +8.09017 q^{56} -5.87785i q^{58} -1.90983 q^{59} +7.32624 q^{61} -3.35520i q^{62} -8.70820 q^{64} -8.50651i q^{67} -3.07768i q^{68} +13.0902 q^{71} +4.25325i q^{73} -1.90983 q^{74} -2.14590 q^{76} -13.1433i q^{77} +8.47214 q^{79} +9.51057i q^{82} +8.22899i q^{83} +1.18034 q^{86} +15.3884i q^{88} -11.1803 q^{89} -18.0902 q^{91} -3.35520i q^{92} -9.79837 q^{94} -14.3844i q^{97} +0.106001i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{4} + 20 q^{11} - 10 q^{14} - 14 q^{16} + 4 q^{19} + 10 q^{26} - 20 q^{29} + 2 q^{31} + 10 q^{34} + 10 q^{41} - 10 q^{44} + 30 q^{46} - 22 q^{49} + 10 q^{56} - 30 q^{59} - 2 q^{61} - 8 q^{64}+ \cdots + 10 q^{94}+O(q^{100}) $ 4 * q - 2 * q^4 + 20 * q^11 - 10 * q^14 - 14 * q^16 + 4 * q^19 + 10 * q^26 - 20 * q^29 + 2 * q^31 + 10 * q^34 + 10 * q^41 - 10 * q^44 + 30 * q^46 - 22 * q^49 + 10 * q^56 - 30 * q^59 - 2 * q^61 - 8 * q^64 + 30 * q^71 - 30 * q^74 - 22 * q^76 + 16 * q^79 - 40 * q^86 - 50 * q^91 + 10 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1125\mathbb{Z}\right)^\times$.

$n$	$127$	$1001$
$\chi(n)$	$-1$	$1$

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 $

$2$	1.17557i	0.831254i	0.909535	+	0.415627i	$0.136438\pi$
$2$	1.17557i	0.831254i	−0.909535	+	0.415627i	$0.863562\pi$
$3$	0	0
$3$	0	0
$4$	0.618034	0.309017
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 2.62866i	− 0.993538i	−0.867883	−	0.496769i	$-0.834520\pi$
$7$	− 2.62866i	− 0.993538i	0.867883	−	0.496769i	$-0.165480\pi$
$8$	3.07768i	1.08813i
$9$	0	0
$10$	0	0
$11$	5.00000	1.50756	0.753778	−	0.657129i	$-0.228229\pi$
$11$	5.00000	1.50756	0.753778	+	0.657129i	$0.228229\pi$
$12$	0	0
$13$	− 6.88191i	− 1.90870i	−0.298693	−	0.954349i	$-0.596551\pi$
$13$	− 6.88191i	− 1.90870i	0.298693	−	0.954349i	$-0.403449\pi$
$14$	3.09017	0.825883
$15$	0	0
$16$	−2.38197	−0.595492
$17$	− 4.97980i	− 1.20778i	−0.797068	−	0.603889i	$-0.793617\pi$
$17$	− 4.97980i	− 1.20778i	0.797068	−	0.603889i	$-0.206383\pi$
$18$	0	0
$19$	−3.47214	−0.796563	−0.398281	−	0.917263i	$-0.630393\pi$
$19$	−3.47214	−0.796563	−0.398281	+	0.917263i	$0.630393\pi$
$20$	0	0
$21$	0	0
$22$	5.87785i	1.25316i
$23$	− 5.42882i	− 1.13199i	−0.824409	−	0.565994i	$-0.808492\pi$
$23$	− 5.42882i	− 1.13199i	0.824409	−	0.565994i	$-0.191508\pi$
$24$	0	0
$25$	0	0
$26$	8.09017	1.58661
$27$	0	0
$28$	− 1.62460i	− 0.307020i
$29$	−5.00000	−0.928477	−0.464238	−	0.885710i	$-0.653672\pi$
$29$	−5.00000	−0.928477	−0.464238	+	0.885710i	$0.653672\pi$
$30$	0	0
$31$	−2.85410	−0.512612	−0.256306	−	0.966596i	$-0.582505\pi$
$31$	−2.85410	−0.512612	−0.256306	+	0.966596i	$0.582505\pi$
$32$	3.35520i	0.593121i
$33$	0	0
$34$	5.85410	1.00397
$35$	0	0
$36$	0	0
$37$	1.62460i	0.267082i	0.991043	+	0.133541i	$0.0426348\pi$
$37$	1.62460i	0.267082i	−0.991043	+	0.133541i	$0.957365\pi$
$38$	− 4.08174i	− 0.662146i
$39$	0	0
$40$	0	0
$41$	8.09017	1.26347	0.631736	−	0.775183i	$-0.282343\pi$
$41$	8.09017	1.26347	0.631736	+	0.775183i	$0.282343\pi$
$42$	0	0
$43$	− 1.00406i	− 0.153117i	−0.997065	−	0.0765586i	$-0.975607\pi$
$43$	− 1.00406i	− 0.153117i	0.997065	−	0.0765586i	$-0.0243932\pi$
$44$	3.09017	0.465861
$45$	0	0
$46$	6.38197	0.940970
$47$	8.33499i	1.21578i	0.794020	+	0.607892i	$0.207985\pi$
$47$	8.33499i	1.21578i	−0.794020	+	0.607892i	$0.792015\pi$
$48$	0	0
$49$	0.0901699	0.0128814
$50$	0	0
$51$	0	0
$52$	− 4.25325i	− 0.589820i
$53$	7.15942i	0.983423i	0.870758	+	0.491711i	$0.163628\pi$
$53$	7.15942i	0.983423i	−0.870758	+	0.491711i	$0.836372\pi$
$54$	0	0
$55$	0	0
$56$	8.09017	1.08109
$57$	0	0
$58$	− 5.87785i	− 0.771800i
$59$	−1.90983	−0.248639	−0.124319	−	0.992242i	$-0.539675\pi$
$59$	−1.90983	−0.248639	−0.124319	+	0.992242i	$0.539675\pi$
$60$	0	0
$61$	7.32624	0.938029	0.469014	−	0.883191i	$-0.344609\pi$
$61$	7.32624	0.938029	0.469014	+	0.883191i	$0.344609\pi$
$62$	− 3.35520i	− 0.426111i
$63$	0	0
$64$	−8.70820	−1.08853
$65$	0	0
$66$	0	0
$67$	− 8.50651i	− 1.03924i	−0.854399	−	0.519618i	$-0.826074\pi$
$67$	− 8.50651i	− 1.03924i	0.854399	−	0.519618i	$-0.173926\pi$
$68$	− 3.07768i	− 0.373224i
$69$	0	0
$70$	0	0
$71$	13.0902	1.55352	0.776759	−	0.629798i	$-0.216862\pi$
$71$	13.0902	1.55352	0.776759	+	0.629798i	$0.216862\pi$
$72$	0	0
$73$	4.25325i	0.497806i	0.968528	+	0.248903i	$0.0800700\pi$
$73$	4.25325i	0.497806i	−0.968528	+	0.248903i	$0.919930\pi$
$74$	−1.90983	−0.222013
$75$	0	0
$76$	−2.14590	−0.246151
$77$	− 13.1433i	− 1.49782i
$78$	0	0
$79$	8.47214	0.953190	0.476595	−	0.879123i	$-0.341871\pi$
$79$	8.47214	0.953190	0.476595	+	0.879123i	$0.341871\pi$
$80$	0	0
$81$	0	0
$82$	9.51057i	1.05027i
$83$	8.22899i	0.903249i	0.892208	+	0.451625i	$0.149155\pi$
$83$	8.22899i	0.903249i	−0.892208	+	0.451625i	$0.850845\pi$
$84$	0	0
$85$	0	0
$86$	1.18034	0.127279
$87$	0	0
$88$	15.3884i	1.64041i
$89$	−11.1803	−1.18511	−0.592557	−	0.805529i	$-0.701881\pi$
$89$	−11.1803	−1.18511	−0.592557	+	0.805529i	$0.701881\pi$
$90$	0	0
$91$	−18.0902	−1.89637
$92$	− 3.35520i	− 0.349804i
$93$	0	0
$94$	−9.79837	−1.01063
$95$	0	0
$96$	0	0
$97$	− 14.3844i	− 1.46051i	−0.683174	−	0.730255i	$-0.739401\pi$
$97$	− 14.3844i	− 1.46051i	0.683174	−	0.730255i	$-0.260599\pi$
$98$	0.106001i	0.0107077i
$99$	0	0
$100$	0	0
$101$	−6.18034	−0.614967	−0.307483	−	0.951553i	$-0.599487\pi$
$101$	−6.18034	−0.614967	−0.307483	+	0.951553i	$0.599487\pi$
$102$	0	0
$103$	12.7598i	1.25726i	0.777706	+	0.628628i	$0.216383\pi$
$103$	12.7598i	1.25726i	−0.777706	+	0.628628i	$0.783617\pi$
$104$	21.1803	2.07690
$105$	0	0
$106$	−8.41641	−0.817474
$107$	8.61251i	0.832603i	0.909227	+	0.416301i	$0.136674\pi$
$107$	8.61251i	0.832603i	−0.909227	+	0.416301i	$0.863326\pi$
$108$	0	0
$109$	10.7082	1.02566	0.512830	−	0.858490i	$-0.328597\pi$
$109$	10.7082	1.02566	0.512830	+	0.858490i	$0.328597\pi$
$110$	0	0
$111$	0	0
$112$	6.26137i	0.591644i
$113$	9.40456i	0.884707i	0.896841	+	0.442353i	$0.145856\pi$
$113$	9.40456i	0.884707i	−0.896841	+	0.442353i	$0.854144\pi$
$114$	0	0
$115$	0	0
$116$	−3.09017	−0.286915
$117$	0	0
$118$	− 2.24514i	− 0.206682i
$119$	−13.0902	−1.19997
$120$	0	0
$121$	14.0000	1.27273
$122$	8.61251i	0.779740i
$123$	0	0
$124$	−1.76393	−0.158406
$125$	0	0
$126$	0	0
$127$	12.1392i	1.07718i	0.842567	+	0.538591i	$0.181043\pi$
$127$	12.1392i	1.07718i	−0.842567	+	0.538591i	$0.818957\pi$
$128$	− 3.52671i	− 0.311720i
$129$	0	0
$130$	0	0
$131$	1.90983	0.166863	0.0834313	−	0.996514i	$-0.473412\pi$
$131$	1.90983	0.166863	0.0834313	+	0.996514i	$0.473412\pi$
$132$	0	0
$133$	9.12705i	0.791416i
$134$	10.0000	0.863868
$135$	0	0
$136$	15.3262	1.31421
$137$	15.3884i	1.31472i	0.753576	+	0.657361i	$0.228327\pi$
$137$	15.3884i	1.31472i	−0.753576	+	0.657361i	$0.771673\pi$
$138$	0	0
$139$	−0.708204	−0.0600691	−0.0300345	−	0.999549i	$-0.509562\pi$
$139$	−0.708204	−0.0600691	−0.0300345	+	0.999549i	$0.509562\pi$
$140$	0	0
$141$	0	0
$142$	15.3884i	1.29137i
$143$	− 34.4095i	− 2.87747i
$144$	0	0
$145$	0	0
$146$	−5.00000	−0.413803
$147$	0	0
$148$	1.00406i	0.0825330i
$149$	5.00000	0.409616	0.204808	−	0.978802i	$-0.434343\pi$
$149$	5.00000	0.409616	0.204808	+	0.978802i	$0.434343\pi$
$150$	0	0
$151$	6.94427	0.565117	0.282558	−	0.959250i	$-0.408817\pi$
$151$	6.94427	0.565117	0.282558	+	0.959250i	$0.408817\pi$
$152$	− 10.6861i	− 0.866760i
$153$	0	0
$154$	15.4508	1.24506
$155$	0	0
$156$	0	0
$157$	− 5.87785i	− 0.469104i	−0.972104	−	0.234552i	$-0.924638\pi$
$157$	− 5.87785i	− 0.469104i	0.972104	−	0.234552i	$-0.0753623\pi$
$158$	9.95959i	0.792343i
$159$	0	0
$160$	0	0
$161$	−14.2705	−1.12467
$162$	0	0
$163$	3.63271i	0.284536i	0.989828	+	0.142268i	$0.0454395\pi$
$163$	3.63271i	0.284536i	−0.989828	+	0.142268i	$0.954561\pi$
$164$	5.00000	0.390434
$165$	0	0
$166$	−9.67376	−0.750830
$167$	− 7.05342i	− 0.545810i	−0.962041	−	0.272905i	$-0.912015\pi$
$167$	− 7.05342i	− 0.545810i	0.962041	−	0.272905i	$-0.0879845\pi$
$168$	0	0
$169$	−34.3607	−2.64313
$170$	0	0
$171$	0	0
$172$	− 0.620541i	− 0.0473158i
$173$	− 5.87785i	− 0.446885i	−0.974717	−	0.223442i	$-0.928271\pi$
$173$	− 5.87785i	− 0.446885i	0.974717	−	0.223442i	$-0.0717295\pi$
$174$	0	0
$175$	0	0
$176$	−11.9098	−0.897737
$177$	0	0
$178$	− 13.1433i	− 0.985130i
$179$	−21.1803	−1.58309	−0.791546	−	0.611109i	$-0.790724\pi$
$179$	−21.1803	−1.58309	−0.791546	+	0.611109i	$0.790724\pi$
$180$	0	0
$181$	−9.23607	−0.686512	−0.343256	−	0.939242i	$-0.611530\pi$
$181$	−9.23607	−0.686512	−0.343256	+	0.939242i	$0.611530\pi$
$182$	− 21.2663i	− 1.57636i
$183$	0	0
$184$	16.7082	1.23175
$185$	0	0
$186$	0	0
$187$	− 24.8990i	− 1.82079i
$188$	5.15131i	0.375698i
$189$	0	0
$190$	0	0
$191$	−11.1803	−0.808981	−0.404491	−	0.914542i	$-0.632551\pi$
$191$	−11.1803	−0.808981	−0.404491	+	0.914542i	$0.632551\pi$
$192$	0	0
$193$	15.3884i	1.10768i	0.832622	+	0.553841i	$0.186839\pi$
$193$	15.3884i	1.10768i	−0.832622	+	0.553841i	$0.813161\pi$
$194$	16.9098	1.21406
$195$	0	0
$196$	0.0557281	0.00398058
$197$	4.53077i	0.322804i	0.986889	+	0.161402i	$0.0516016\pi$
$197$	4.53077i	0.322804i	−0.986889	+	0.161402i	$0.948398\pi$
$198$	0	0
$199$	−5.47214	−0.387909	−0.193955	−	0.981010i	$-0.562131\pi$
$199$	−5.47214	−0.387909	−0.193955	+	0.981010i	$0.562131\pi$
$200$	0	0
$201$	0	0
$202$	− 7.26543i	− 0.511194i
$203$	13.1433i	0.922477i
$204$	0	0
$205$	0	0
$206$	−15.0000	−1.04510
$207$	0	0
$208$	16.3925i	1.13661i
$209$	−17.3607	−1.20086
$210$	0	0
$211$	14.7639	1.01639	0.508195	−	0.861242i	$-0.330313\pi$
$211$	14.7639	1.01639	0.508195	+	0.861242i	$0.330313\pi$
$212$	4.42477i	0.303894i
$213$	0	0
$214$	−10.1246	−0.692104
$215$	0	0
$216$	0	0
$217$	7.50245i	0.509300i
$218$	12.5882i	0.852584i
$219$	0	0
$220$	0	0
$221$	−34.2705	−2.30528
$222$	0	0
$223$	− 25.9030i	− 1.73460i	−0.497789	−	0.867298i	$-0.665855\pi$
$223$	− 25.9030i	− 1.73460i	0.497789	−	0.867298i	$-0.334145\pi$
$224$	8.81966	0.589288
$225$	0	0
$226$	−11.0557	−0.735416
$227$	− 8.16348i	− 0.541829i	−0.962603	−	0.270915i	$-0.912674\pi$
$227$	− 8.16348i	− 0.541829i	0.962603	−	0.270915i	$-0.0873261\pi$
$228$	0	0
$229$	26.6525	1.76125	0.880623	−	0.473818i	$-0.157125\pi$
$229$	26.6525	1.76125	0.880623	+	0.473818i	$0.157125\pi$
$230$	0	0
$231$	0	0
$232$	− 15.3884i	− 1.01030i
$233$	2.24514i	0.147084i	0.997292	+	0.0735420i	$0.0234303\pi$
$233$	2.24514i	0.147084i	−0.997292	+	0.0735420i	$0.976570\pi$
$234$	0	0
$235$	0	0
$236$	−1.18034	−0.0768336
$237$	0	0
$238$	− 15.3884i	− 0.997483i
$239$	14.2705	0.923083	0.461541	−	0.887119i	$-0.347297\pi$
$239$	14.2705	0.923083	0.461541	+	0.887119i	$0.347297\pi$
$240$	0	0
$241$	−27.5967	−1.77766	−0.888831	−	0.458234i	$-0.848482\pi$
$241$	−27.5967	−1.77766	−0.888831	+	0.458234i	$0.848482\pi$
$242$	16.4580i	1.05796i
$243$	0	0
$244$	4.52786	0.289867
$245$	0	0
$246$	0	0
$247$	23.8949i	1.52040i
$248$	− 8.78402i	− 0.557786i
$249$	0	0
$250$	0	0
$251$	3.09017	0.195050	0.0975249	−	0.995233i	$-0.468907\pi$
$251$	3.09017	0.195050	0.0975249	+	0.995233i	$0.468907\pi$
$252$	0	0
$253$	− 27.1441i	− 1.70654i
$254$	−14.2705	−0.895411
$255$	0	0
$256$	−13.2705	−0.829407
$257$	22.4418i	1.39988i	0.714200	+	0.699942i	$0.246791\pi$
$257$	22.4418i	1.39988i	−0.714200	+	0.699942i	$0.753209\pi$
$258$	0	0
$259$	4.27051	0.265357
$260$	0	0
$261$	0	0
$262$	2.24514i	0.138705i
$263$	4.08174i	0.251691i	0.992050	+	0.125845i	$0.0401643\pi$
$263$	4.08174i	0.251691i	−0.992050	+	0.125845i	$0.959836\pi$
$264$	0	0
$265$	0	0
$266$	−10.7295	−0.657867
$267$	0	0
$268$	− 5.25731i	− 0.321141i
$269$	−8.09017	−0.493266	−0.246633	−	0.969109i	$-0.579324\pi$
$269$	−8.09017	−0.493266	−0.246633	+	0.969109i	$0.579324\pi$
$270$	0	0
$271$	17.8541	1.08456	0.542280	−	0.840198i	$-0.317561\pi$
$271$	17.8541	1.08456	0.542280	+	0.840198i	$0.317561\pi$
$272$	11.8617i	0.719222i
$273$	0	0
$274$	−18.0902	−1.09287
$275$	0	0
$276$	0	0
$277$	− 6.26137i	− 0.376209i	−0.982149	−	0.188105i	$-0.939766\pi$
$277$	− 6.26137i	− 0.376209i	0.982149	−	0.188105i	$-0.0602344\pi$
$278$	− 0.832544i	− 0.0499327i
$279$	0	0
$280$	0	0
$281$	−29.2705	−1.74613	−0.873066	−	0.487602i	$-0.837872\pi$
$281$	−29.2705	−1.74613	−0.873066	+	0.487602i	$0.837872\pi$
$282$	0	0
$283$	− 15.3884i	− 0.914746i	−0.889275	−	0.457373i	$-0.848790\pi$
$283$	− 15.3884i	− 0.914746i	0.889275	−	0.457373i	$-0.151210\pi$
$284$	8.09017	0.480063
$285$	0	0
$286$	40.4508	2.39191
$287$	− 21.2663i	− 1.25531i
$288$	0	0
$289$	−7.79837	−0.458728
$290$	0	0
$291$	0	0
$292$	2.62866i	0.153830i
$293$	9.06154i	0.529381i	0.964333	+	0.264690i	$0.0852697\pi$
$293$	9.06154i	0.529381i	−0.964333	+	0.264690i	$0.914730\pi$
$294$	0	0
$295$	0	0
$296$	−5.00000	−0.290619
$297$	0	0
$298$	5.87785i	0.340495i
$299$	−37.3607	−2.16062
$300$	0	0
$301$	−2.63932	−0.152128
$302$	8.16348i	0.469756i
$303$	0	0
$304$	8.27051	0.474346
$305$	0	0
$306$	0	0
$307$	21.8868i	1.24915i	0.780966	+	0.624573i	$0.214727\pi$
$307$	21.8868i	1.24915i	−0.780966	+	0.624573i	$0.785273\pi$
$308$	− 8.12299i	− 0.462850i
$309$	0	0
$310$	0	0
$311$	9.27051	0.525682	0.262841	−	0.964839i	$-0.415340\pi$
$311$	9.27051	0.525682	0.262841	+	0.964839i	$0.415340\pi$
$312$	0	0
$313$	− 7.88597i	− 0.445741i	−0.974848	−	0.222871i	$-0.928457\pi$
$313$	− 7.88597i	− 0.445741i	0.974848	−	0.222871i	$-0.0715427\pi$
$314$	6.90983	0.389944
$315$	0	0
$316$	5.23607	0.294552
$317$	− 27.1441i	− 1.52457i	−0.647244	−	0.762283i	$-0.724079\pi$
$317$	− 27.1441i	− 1.52457i	0.647244	−	0.762283i	$-0.275921\pi$
$318$	0	0
$319$	−25.0000	−1.39973
$320$	0	0
$321$	0	0
$322$	− 16.7760i	− 0.934889i
$323$	17.2905i	0.962071i
$324$	0	0
$325$	0	0
$326$	−4.27051	−0.236522
$327$	0	0
$328$	24.8990i	1.37482i
$329$	21.9098	1.20793
$330$	0	0
$331$	2.32624	0.127862	0.0639308	−	0.997954i	$-0.479636\pi$
$331$	2.32624	0.127862	0.0639308	+	0.997954i	$0.479636\pi$
$332$	5.08580i	0.279119i
$333$	0	0
$334$	8.29180	0.453707
$335$	0	0
$336$	0	0
$337$	22.2703i	1.21314i	0.795029	+	0.606571i	$0.207455\pi$
$337$	22.2703i	1.21314i	−0.795029	+	0.606571i	$0.792545\pi$
$338$	− 40.3934i	− 2.19711i
$339$	0	0
$340$	0	0
$341$	−14.2705	−0.772791
$342$	0	0
$343$	− 18.6376i	− 1.00634i
$344$	3.09017	0.166611
$345$	0	0
$346$	6.90983	0.371475
$347$	− 33.2340i	− 1.78409i	−0.451943	−	0.892047i	$-0.649269\pi$
$347$	− 33.2340i	− 1.78409i	0.451943	−	0.892047i	$-0.350731\pi$
$348$	0	0
$349$	23.5623	1.26126	0.630631	−	0.776083i	$-0.282796\pi$
$349$	23.5623	1.26126	0.630631	+	0.776083i	$0.282796\pi$
$350$	0	0
$351$	0	0
$352$	16.7760i	0.894163i
$353$	− 15.8374i	− 0.842942i	−0.906842	−	0.421471i	$-0.861514\pi$
$353$	− 15.8374i	− 0.842942i	0.906842	−	0.421471i	$-0.138486\pi$
$354$	0	0
$355$	0	0
$356$	−6.90983	−0.366220
$357$	0	0
$358$	− 24.8990i	− 1.31595i
$359$	6.18034	0.326186	0.163093	−	0.986611i	$-0.447853\pi$
$359$	6.18034	0.326186	0.163093	+	0.986611i	$0.447853\pi$
$360$	0	0
$361$	−6.94427	−0.365488
$362$	− 10.8576i	− 0.570665i
$363$	0	0
$364$	−11.1803	−0.586009
$365$	0	0
$366$	0	0
$367$	− 26.5236i	− 1.38452i	−0.721648	−	0.692260i	$-0.756615\pi$
$367$	− 26.5236i	− 1.38452i	0.721648	−	0.692260i	$-0.243385\pi$
$368$	12.9313i	0.674089i
$369$	0	0
$370$	0	0
$371$	18.8197	0.977068
$372$	0	0
$373$	16.7760i	0.868628i	0.900762	+	0.434314i	$0.143009\pi$
$373$	16.7760i	0.868628i	−0.900762	+	0.434314i	$0.856991\pi$
$374$	29.2705	1.51354
$375$	0	0
$376$	−25.6525	−1.32293
$377$	34.4095i	1.77218i
$378$	0	0
$379$	14.0902	0.723763	0.361882	−	0.932224i	$-0.382134\pi$
$379$	14.0902	0.723763	0.361882	+	0.932224i	$0.382134\pi$
$380$	0	0
$381$	0	0
$382$	− 13.1433i	− 0.672469i
$383$	− 14.9394i	− 0.763367i	−0.924293	−	0.381684i	$-0.875344\pi$
$383$	− 14.9394i	− 0.763367i	0.924293	−	0.381684i	$-0.124656\pi$
$384$	0	0
$385$	0	0
$386$	−18.0902	−0.920765
$387$	0	0
$388$	− 8.89002i	− 0.451323i
$389$	−35.4508	−1.79743	−0.898715	−	0.438534i	$-0.855498\pi$
$389$	−35.4508	−1.79743	−0.898715	+	0.438534i	$0.855498\pi$
$390$	0	0
$391$	−27.0344	−1.36719
$392$	0.277515i	0.0140166i
$393$	0	0
$394$	−5.32624	−0.268332
$395$	0	0
$396$	0	0
$397$	− 5.25731i	− 0.263857i	−0.991259	−	0.131928i	$-0.957883\pi$
$397$	− 5.25731i	− 0.263857i	0.991259	−	0.131928i	$-0.0421169\pi$
$398$	− 6.43288i	− 0.322451i
$399$	0	0
$400$	0	0
$401$	5.00000	0.249688	0.124844	−	0.992176i	$-0.460157\pi$
$401$	5.00000	0.249688	0.124844	+	0.992176i	$0.460157\pi$
$402$	0	0
$403$	19.6417i	0.978421i
$404$	−3.81966	−0.190035
$405$	0	0
$406$	−15.4508	−0.766813
$407$	8.12299i	0.402642i
$408$	0	0
$409$	1.52786	0.0755480	0.0377740	−	0.999286i	$-0.487973\pi$
$409$	1.52786	0.0755480	0.0377740	+	0.999286i	$0.487973\pi$
$410$	0	0
$411$	0	0
$412$	7.88597i	0.388514i
$413$	5.02029i	0.247032i
$414$	0	0
$415$	0	0
$416$	23.0902	1.13209
$417$	0	0
$418$	− 20.4087i	− 0.998222i
$419$	19.2705	0.941426	0.470713	−	0.882286i	$-0.343997\pi$
$419$	19.2705	0.941426	0.470713	+	0.882286i	$0.343997\pi$
$420$	0	0
$421$	0.763932	0.0372318	0.0186159	−	0.999827i	$-0.494074\pi$
$421$	0.763932	0.0372318	0.0186159	+	0.999827i	$0.494074\pi$
$422$	17.3560i	0.844879i
$423$	0	0
$424$	−22.0344	−1.07009
$425$	0	0
$426$	0	0
$427$	− 19.2582i	− 0.931967i
$428$	5.32282i	0.257288i
$429$	0	0
$430$	0	0
$431$	41.1803	1.98359	0.991794	−	0.127849i	$-0.0408074\pi$
$431$	41.1803	1.98359	0.991794	+	0.127849i	$0.0408074\pi$
$432$	0	0
$433$	− 6.49839i	− 0.312293i	−0.987734	−	0.156146i	$-0.950093\pi$
$433$	− 6.49839i	− 0.312293i	0.987734	−	0.156146i	$-0.0499072\pi$
$434$	−8.81966	−0.423357
$435$	0	0
$436$	6.61803	0.316946
$437$	18.8496i	0.901699i
$438$	0	0
$439$	27.0902	1.29294	0.646472	−	0.762938i	$-0.276244\pi$
$439$	27.0902	1.29294	0.646472	+	0.762938i	$0.276244\pi$
$440$	0	0
$441$	0	0
$442$	− 40.2874i	− 1.91628i
$443$	27.2501i	1.29469i	0.762196	+	0.647346i	$0.224121\pi$
$443$	27.2501i	1.29469i	−0.762196	+	0.647346i	$0.775879\pi$
$444$	0	0
$445$	0	0
$446$	30.4508	1.44189
$447$	0	0
$448$	22.8909i	1.08149i
$449$	−0.729490	−0.0344268	−0.0172134	−	0.999852i	$-0.505479\pi$
$449$	−0.729490	−0.0344268	−0.0172134	+	0.999852i	$0.505479\pi$
$450$	0	0
$451$	40.4508	1.90476
$452$	5.81234i	0.273389i
$453$	0	0
$454$	9.59675	0.450398
$455$	0	0
$456$	0	0
$457$	18.4006i	0.860743i	0.902652	+	0.430372i	$0.141617\pi$
$457$	18.4006i	0.860743i	−0.902652	+	0.430372i	$0.858383\pi$
$458$	31.3319i	1.46404i
$459$	0	0
$460$	0	0
$461$	−15.0000	−0.698620	−0.349310	−	0.937007i	$-0.613584\pi$
$461$	−15.0000	−0.698620	−0.349310	+	0.937007i	$0.613584\pi$
$462$	0	0
$463$	35.4136i	1.64581i	0.568179	+	0.822905i	$0.307648\pi$
$463$	35.4136i	1.64581i	−0.568179	+	0.822905i	$0.692352\pi$
$464$	11.9098	0.552900
$465$	0	0
$466$	−2.63932	−0.122264
$467$	16.5640i	0.766490i	0.923647	+	0.383245i	$0.125193\pi$
$467$	16.5640i	0.766490i	−0.923647	+	0.383245i	$0.874807\pi$
$468$	0	0
$469$	−22.3607	−1.03252
$470$	0	0
$471$	0	0
$472$	− 5.87785i	− 0.270550i
$473$	− 5.02029i	− 0.230833i
$474$	0	0
$475$	0	0
$476$	−8.09017	−0.370812
$477$	0	0
$478$	16.7760i	0.767316i
$479$	12.3607	0.564774	0.282387	−	0.959301i	$-0.408874\pi$
$479$	12.3607	0.564774	0.282387	+	0.959301i	$0.408874\pi$
$480$	0	0
$481$	11.1803	0.509780
$482$	− 32.4419i	− 1.47769i
$483$	0	0
$484$	8.65248	0.393294
$485$	0	0
$486$	0	0
$487$	19.0211i	0.861930i	0.902369	+	0.430965i	$0.141827\pi$
$487$	19.0211i	0.861930i	−0.902369	+	0.430965i	$0.858173\pi$
$488$	22.5478i	1.02069i
$489$	0	0
$490$	0	0
$491$	23.0902	1.04204	0.521022	−	0.853543i	$-0.325551\pi$
$491$	23.0902	1.04204	0.521022	+	0.853543i	$0.325551\pi$
$492$	0	0
$493$	24.8990i	1.12139i
$494$	−28.0902	−1.26384
$495$	0	0
$496$	6.79837	0.305256
$497$	− 34.4095i	− 1.54348i
$498$	0	0
$499$	−6.52786	−0.292227	−0.146114	−	0.989268i	$-0.546677\pi$
$499$	−6.52786	−0.292227	−0.146114	+	0.989268i	$0.546677\pi$
$500$	0	0
$501$	0	0
$502$	3.63271i	0.162136i
$503$	− 9.51057i	− 0.424055i	−0.977264	−	0.212028i	$-0.931993\pi$
$503$	− 9.51057i	− 0.424055i	0.977264	−	0.212028i	$-0.0680067\pi$
$504$	0	0
$505$	0	0
$506$	31.9098	1.41856
$507$	0	0
$508$	7.50245i	0.332867i
$509$	13.8197	0.612546	0.306273	−	0.951944i	$-0.400918\pi$
$509$	13.8197	0.612546	0.306273	+	0.951944i	$0.400918\pi$
$510$	0	0
$511$	11.1803	0.494589
$512$	− 22.6538i	− 1.00117i
$513$	0	0
$514$	−26.3820	−1.16366
$515$	0	0
$516$	0	0
$517$	41.6750i	1.83286i
$518$	5.02029i	0.220579i
$519$	0	0
$520$	0	0
$521$	−6.90983	−0.302725	−0.151363	−	0.988478i	$-0.548366\pi$
$521$	−6.90983	−0.302725	−0.151363	+	0.988478i	$0.548366\pi$
$522$	0	0
$523$	17.0130i	0.743927i	0.928247	+	0.371964i	$0.121315\pi$
$523$	17.0130i	0.743927i	−0.928247	+	0.371964i	$0.878685\pi$
$524$	1.18034	0.0515634
$525$	0	0
$526$	−4.79837	−0.209219
$527$	14.2128i	0.619121i
$528$	0	0
$529$	−6.47214	−0.281397
$530$	0	0
$531$	0	0
$532$	5.64083i	0.244561i
$533$	− 55.6758i	− 2.41159i
$534$	0	0
$535$	0	0
$536$	26.1803	1.13082
$537$	0	0
$538$	− 9.51057i	− 0.410030i
$539$	0.450850	0.0194195
$540$	0	0
$541$	−9.50658	−0.408720	−0.204360	−	0.978896i	$-0.565511\pi$
$541$	−9.50658	−0.408720	−0.204360	+	0.978896i	$0.565511\pi$
$542$	20.9888i	0.901544i
$543$	0	0
$544$	16.7082	0.716358
$545$	0	0
$546$	0	0
$547$	38.6628i	1.65310i	0.562862	+	0.826551i	$0.309700\pi$
$547$	38.6628i	1.65310i	−0.562862	+	0.826551i	$0.690300\pi$
$548$	9.51057i	0.406271i
$549$	0	0
$550$	0	0
$551$	17.3607	0.739590
$552$	0	0
$553$	− 22.2703i	− 0.947031i
$554$	7.36068	0.312725
$555$	0	0
$556$	−0.437694	−0.0185624
$557$	− 40.2874i	− 1.70703i	−0.521067	−	0.853516i	$-0.674466\pi$
$557$	− 40.2874i	− 1.70703i	0.521067	−	0.853516i	$-0.325534\pi$
$558$	0	0
$559$	−6.90983	−0.292255
$560$	0	0
$561$	0	0
$562$	− 34.4095i	− 1.45148i
$563$	18.5721i	0.782721i	0.920237	+	0.391360i	$0.127995\pi$
$563$	18.5721i	0.782721i	−0.920237	+	0.391360i	$0.872005\pi$
$564$	0	0
$565$	0	0
$566$	18.0902	0.760387
$567$	0	0
$568$	40.2874i	1.69042i
$569$	−17.3607	−0.727798	−0.363899	−	0.931438i	$-0.618555\pi$
$569$	−17.3607	−0.727798	−0.363899	+	0.931438i	$0.618555\pi$
$570$	0	0
$571$	−7.27051	−0.304261	−0.152131	−	0.988360i	$-0.548613\pi$
$571$	−7.27051	−0.304261	−0.152131	+	0.988360i	$0.548613\pi$
$572$	− 21.2663i	− 0.889187i
$573$	0	0
$574$	25.0000	1.04348
$575$	0	0
$576$	0	0
$577$	− 8.89002i	− 0.370097i	−0.982729	−	0.185048i	$-0.940756\pi$
$577$	− 8.89002i	− 0.370097i	0.982729	−	0.185048i	$-0.0592441\pi$
$578$	− 9.16754i	− 0.381319i
$579$	0	0
$580$	0	0
$581$	21.6312	0.897413
$582$	0	0
$583$	35.7971i	1.48257i
$584$	−13.0902	−0.541675
$585$	0	0
$586$	−10.6525	−0.440050
$587$	33.5115i	1.38317i	0.722296	+	0.691584i	$0.243087\pi$
$587$	33.5115i	1.38317i	−0.722296	+	0.691584i	$0.756913\pi$
$588$	0	0
$589$	9.90983	0.408327
$590$	0	0
$591$	0	0
$592$	− 3.86974i	− 0.159045i
$593$	− 13.2493i	− 0.544083i	−0.962286	−	0.272041i	$-0.912301\pi$
$593$	− 13.2493i	− 0.544083i	0.962286	−	0.272041i	$-0.0876987\pi$
$594$	0	0
$595$	0	0
$596$	3.09017	0.126578
$597$	0	0
$598$	− 43.9201i	− 1.79603i
$599$	14.2705	0.583077	0.291539	−	0.956559i	$-0.405833\pi$
$599$	14.2705	0.583077	0.291539	+	0.956559i	$0.405833\pi$
$600$	0	0
$601$	−2.12461	−0.0866647	−0.0433324	−	0.999061i	$-0.513797\pi$
$601$	−2.12461	−0.0866647	−0.0433324	+	0.999061i	$0.513797\pi$
$602$	− 3.10271i	− 0.126457i
$603$	0	0
$604$	4.29180	0.174631
$605$	0	0
$606$	0	0
$607$	− 22.2703i	− 0.903925i	−0.892037	−	0.451962i	$-0.850724\pi$
$607$	− 22.2703i	− 0.903925i	0.892037	−	0.451962i	$-0.149276\pi$
$608$	− 11.6497i	− 0.472458i
$609$	0	0
$610$	0	0
$611$	57.3607	2.32056
$612$	0	0
$613$	− 8.89002i	− 0.359065i	−0.983752	−	0.179532i	$-0.942542\pi$
$613$	− 8.89002i	− 0.359065i	0.983752	−	0.179532i	$-0.0574585\pi$
$614$	−25.7295	−1.03836
$615$	0	0
$616$	40.4508	1.62981
$617$	40.7769i	1.64162i	0.571204	+	0.820808i	$0.306477\pi$
$617$	40.7769i	1.64162i	−0.571204	+	0.820808i	$0.693523\pi$
$618$	0	0
$619$	−6.65248	−0.267386	−0.133693	−	0.991023i	$-0.542684\pi$
$619$	−6.65248	−0.267386	−0.133693	+	0.991023i	$0.542684\pi$
$620$	0	0
$621$	0	0
$622$	10.8981i	0.436976i
$623$	29.3893i	1.17746i
$624$	0	0
$625$	0	0
$626$	9.27051	0.370524
$627$	0	0
$628$	− 3.63271i	− 0.144961i
$629$	8.09017	0.322576
$630$	0	0
$631$	−29.1803	−1.16165	−0.580825	−	0.814028i	$-0.697270\pi$
$631$	−29.1803	−1.16165	−0.580825	+	0.814028i	$0.697270\pi$
$632$	26.0746i	1.03719i
$633$	0	0
$634$	31.9098	1.26730
$635$	0	0
$636$	0	0
$637$	− 0.620541i	− 0.0245867i
$638$	− 29.3893i	− 1.16353i
$639$	0	0
$640$	0	0
$641$	−10.0000	−0.394976	−0.197488	−	0.980305i	$-0.563278\pi$
$641$	−10.0000	−0.394976	−0.197488	+	0.980305i	$0.563278\pi$
$642$	0	0
$643$	− 6.49839i	− 0.256272i	−0.991757	−	0.128136i	$-0.959101\pi$
$643$	− 6.49839i	− 0.256272i	0.991757	−	0.128136i	$-0.0408993\pi$
$644$	−8.81966	−0.347543
$645$	0	0
$646$	−20.3262	−0.799725
$647$	− 1.38757i	− 0.0545511i	−0.999628	−	0.0272756i	$-0.991317\pi$
$647$	− 1.38757i	− 0.0545511i	0.999628	−	0.0272756i	$-0.00868316\pi$
$648$	0	0
$649$	−9.54915	−0.374837
$650$	0	0
$651$	0	0
$652$	2.24514i	0.0879265i
$653$	2.35114i	0.0920073i	0.998941	+	0.0460036i	$0.0146486\pi$
$653$	2.35114i	0.0920073i	−0.998941	+	0.0460036i	$0.985351\pi$
$654$	0	0
$655$	0	0
$656$	−19.2705	−0.752387
$657$	0	0
$658$	25.7565i	1.00409i
$659$	−8.81966	−0.343565	−0.171783	−	0.985135i	$-0.554953\pi$
$659$	−8.81966	−0.343565	−0.171783	+	0.985135i	$0.554953\pi$
$660$	0	0
$661$	−32.8541	−1.27788	−0.638938	−	0.769258i	$-0.720626\pi$
$661$	−32.8541	−1.27788	−0.638938	+	0.769258i	$0.720626\pi$
$662$	2.73466i	0.106285i
$663$	0	0
$664$	−25.3262	−0.982849
$665$	0	0
$666$	0	0
$667$	27.1441i	1.05102i
$668$	− 4.35926i	− 0.168665i
$669$	0	0
$670$	0	0
$671$	36.6312	1.41413
$672$	0	0
$673$	7.26543i	0.280062i	0.990147	+	0.140031i	$0.0447202\pi$
$673$	7.26543i	0.280062i	−0.990147	+	0.140031i	$0.955280\pi$
$674$	−26.1803	−1.00843
$675$	0	0
$676$	−21.2361	−0.816772
$677$	− 4.70228i	− 0.180723i	−0.995909	−	0.0903617i	$-0.971198\pi$
$677$	− 4.70228i	− 0.180723i	0.995909	−	0.0903617i	$-0.0288023\pi$
$678$	0	0
$679$	−37.8115	−1.45107
$680$	0	0
$681$	0	0
$682$	− 16.7760i	− 0.642386i
$683$	− 44.3691i	− 1.69774i	−0.528603	−	0.848869i	$-0.677284\pi$
$683$	− 44.3691i	− 1.69774i	0.528603	−	0.848869i	$-0.322716\pi$
$684$	0	0
$685$	0	0
$686$	21.9098	0.836521
$687$	0	0
$688$	2.39163i	0.0911800i
$689$	49.2705	1.87706
$690$	0	0
$691$	−5.94427	−0.226131	−0.113065	−	0.993588i	$-0.536067\pi$
$691$	−5.94427	−0.226131	−0.113065	+	0.993588i	$0.536067\pi$
$692$	− 3.63271i	− 0.138095i
$693$	0	0
$694$	39.0689	1.48303
$695$	0	0
$696$	0	0
$697$	− 40.2874i	− 1.52599i
$698$	27.6992i	1.04843i
$699$	0	0
$700$	0	0
$701$	10.0000	0.377695	0.188847	−	0.982006i	$-0.439525\pi$
$701$	10.0000	0.377695	0.188847	+	0.982006i	$0.439525\pi$
$702$	0	0
$703$	− 5.64083i	− 0.212748i
$704$	−43.5410	−1.64101
$705$	0	0
$706$	18.6180	0.700699
$707$	16.2460i	0.610993i
$708$	0	0
$709$	−37.2918	−1.40052	−0.700261	−	0.713887i	$-0.746933\pi$
$709$	−37.2918	−1.40052	−0.700261	+	0.713887i	$0.746933\pi$
$710$	0	0
$711$	0	0
$712$	− 34.4095i	− 1.28955i
$713$	15.4944i	0.580271i
$714$	0	0
$715$	0	0
$716$	−13.0902	−0.489203
$717$	0	0
$718$	7.26543i	0.271143i
$719$	37.3607	1.39332	0.696659	−	0.717402i	$-0.254669\pi$
$719$	37.3607	1.39332	0.696659	+	0.717402i	$0.254669\pi$
$720$	0	0
$721$	33.5410	1.24913
$722$	− 8.16348i	− 0.303813i
$723$	0	0
$724$	−5.70820	−0.212144
$725$	0	0
$726$	0	0
$727$	18.6376i	0.691231i	0.938376	+	0.345615i	$0.112330\pi$
$727$	18.6376i	0.691231i	−0.938376	+	0.345615i	$0.887670\pi$
$728$	− 55.6758i	− 2.06348i
$729$	0	0
$730$	0	0
$731$	−5.00000	−0.184932
$732$	0	0
$733$	− 7.26543i	− 0.268355i	−0.990957	−	0.134177i	$-0.957161\pi$
$733$	− 7.26543i	− 0.268355i	0.990957	−	0.134177i	$-0.0428392\pi$
$734$	31.1803	1.15089
$735$	0	0
$736$	18.2148	0.671406
$737$	− 42.5325i	− 1.56671i
$738$	0	0
$739$	37.1033	1.36487	0.682434	−	0.730947i	$-0.260921\pi$
$739$	37.1033	1.36487	0.682434	+	0.730947i	$0.260921\pi$
$740$	0	0
$741$	0	0
$742$	22.1238i	0.812192i
$743$	38.0423i	1.39564i	0.716276	+	0.697818i	$0.245845\pi$
$743$	38.0423i	1.39564i	−0.716276	+	0.697818i	$0.754155\pi$
$744$	0	0
$745$	0	0
$746$	−19.7214	−0.722050
$747$	0	0
$748$	− 15.3884i	− 0.562656i
$749$	22.6393	0.827223
$750$	0	0
$751$	25.0344	0.913520	0.456760	−	0.889590i	$-0.349010\pi$
$751$	25.0344	0.913520	0.456760	+	0.889590i	$0.349010\pi$
$752$	− 19.8537i	− 0.723989i
$753$	0	0
$754$	−40.4508	−1.47313
$755$	0	0
$756$	0	0
$757$	− 37.6587i	− 1.36873i	−0.729139	−	0.684365i	$-0.760079\pi$
$757$	− 37.6587i	− 1.36873i	0.729139	−	0.684365i	$-0.239921\pi$
$758$	16.5640i	0.601631i
$759$	0	0
$760$	0	0
$761$	−43.5410	−1.57836	−0.789180	−	0.614162i	$-0.789494\pi$
$761$	−43.5410	−1.57836	−0.789180	+	0.614162i	$0.789494\pi$
$762$	0	0
$763$	− 28.1482i	− 1.01903i
$764$	−6.90983	−0.249989
$765$	0	0
$766$	17.5623	0.634552
$767$	13.1433i	0.474576i
$768$	0	0
$769$	14.6525	0.528382	0.264191	−	0.964470i	$-0.414895\pi$
$769$	14.6525	0.528382	0.264191	+	0.964470i	$0.414895\pi$
$770$	0	0
$771$	0	0
$772$	9.51057i	0.342293i
$773$	51.0795i	1.83720i	0.395185	+	0.918602i	$0.370681\pi$
$773$	51.0795i	1.83720i	−0.395185	+	0.918602i	$0.629319\pi$
$774$	0	0
$775$	0	0
$776$	44.2705	1.58922
$777$	0	0
$778$	− 41.6750i	− 1.49412i
$779$	−28.0902	−1.00643
$780$	0	0
$781$	65.4508	2.34202
$782$	− 31.7809i	− 1.13648i
$783$	0	0
$784$	−0.214782	−0.00767078
$785$	0	0
$786$	0	0
$787$	− 2.24514i	− 0.0800306i	−0.999199	−	0.0400153i	$-0.987259\pi$
$787$	− 2.24514i	− 0.0800306i	0.999199	−	0.0400153i	$-0.0127407\pi$
$788$	2.80017i	0.0997519i
$789$	0	0
$790$	0	0
$791$	24.7214	0.878990
$792$	0	0
$793$	− 50.4185i	− 1.79041i
$794$	6.18034	0.219332
$795$	0	0
$796$	−3.38197	−0.119871
$797$	− 29.4298i	− 1.04246i	−0.853418	−	0.521228i	$-0.825474\pi$
$797$	− 29.4298i	− 1.04246i	0.853418	−	0.521228i	$-0.174526\pi$
$798$	0	0
$799$	41.5066	1.46840
$800$	0	0
$801$	0	0
$802$	5.87785i	0.207554i
$803$	21.2663i	0.750470i
$804$	0	0
$805$	0	0
$806$	−23.0902	−0.813317
$807$	0	0
$808$	− 19.0211i	− 0.669161i
$809$	−21.9098	−0.770309	−0.385154	−	0.922852i	$-0.625852\pi$
$809$	−21.9098	−0.770309	−0.385154	+	0.922852i	$0.625852\pi$
$810$	0	0
$811$	−54.6869	−1.92032	−0.960159	−	0.279455i	$-0.909846\pi$
$811$	−54.6869	−1.92032	−0.960159	+	0.279455i	$0.909846\pi$
$812$	8.12299i	0.285061i
$813$	0	0
$814$	−9.54915	−0.334698
$815$	0	0
$816$	0	0
$817$	3.48622i	0.121967i
$818$	1.79611i	0.0627996i
$819$	0	0
$820$	0	0
$821$	−33.5410	−1.17059	−0.585295	−	0.810821i	$-0.699021\pi$
$821$	−33.5410	−1.17059	−0.585295	+	0.810821i	$0.699021\pi$
$822$	0	0
$823$	− 29.5358i	− 1.02955i	−0.857325	−	0.514776i	$-0.827875\pi$
$823$	− 29.5358i	− 1.02955i	0.857325	−	0.514776i	$-0.172125\pi$
$824$	−39.2705	−1.36805
$825$	0	0
$826$	−5.90170	−0.205346
$827$	20.1967i	0.702308i	0.936318	+	0.351154i	$0.114211\pi$
$827$	20.1967i	0.702308i	−0.936318	+	0.351154i	$0.885789\pi$
$828$	0	0
$829$	27.7082	0.962346	0.481173	−	0.876626i	$-0.340211\pi$
$829$	27.7082	0.962346	0.481173	+	0.876626i	$0.340211\pi$
$830$	0	0
$831$	0	0
$832$	59.9291i	2.07767i
$833$	− 0.449028i	− 0.0155579i
$834$	0	0
$835$	0	0
$836$	−10.7295	−0.371087
$837$	0	0
$838$	22.6538i	0.782564i
$839$	25.0000	0.863096	0.431548	−	0.902090i	$-0.357968\pi$
$839$	25.0000	0.863096	0.431548	+	0.902090i	$0.357968\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	0.898056i	0.0309491i
$843$	0	0
$844$	9.12461	0.314082
$845$	0	0
$846$	0	0
$847$	− 36.8012i	− 1.26450i
$848$	− 17.0535i	− 0.585620i
$849$	0	0
$850$	0	0
$851$	8.81966	0.302334
$852$	0	0
$853$	27.9112i	0.955660i	0.878452	+	0.477830i	$0.158576\pi$
$853$	27.9112i	0.955660i	−0.878452	+	0.477830i	$0.841424\pi$
$854$	22.6393	0.774702
$855$	0	0
$856$	−26.5066	−0.905976
$857$	14.2128i	0.485502i	0.970089	+	0.242751i	$0.0780497\pi$
$857$	14.2128i	0.485502i	−0.970089	+	0.242751i	$0.921950\pi$
$858$	0	0
$859$	−39.5410	−1.34912	−0.674561	−	0.738219i	$-0.735667\pi$
$859$	−39.5410	−1.34912	−0.674561	+	0.738219i	$0.735667\pi$
$860$	0	0
$861$	0	0
$862$	48.4104i	1.64886i
$863$	16.7760i	0.571061i	0.958370	+	0.285531i	$0.0921698\pi$
$863$	16.7760i	0.571061i	−0.958370	+	0.285531i	$0.907830\pi$
$864$	0	0
$865$	0	0
$866$	7.63932	0.259595
$867$	0	0
$868$	4.63677i	0.157382i
$869$	42.3607	1.43699
$870$	0	0
$871$	−58.5410	−1.98359
$872$	32.9565i	1.11605i
$873$	0	0
$874$	−22.1591	−0.749541
$875$	0	0
$876$	0	0
$877$	38.8998i	1.31355i	0.754085	+	0.656777i	$0.228081\pi$
$877$	38.8998i	1.31355i	−0.754085	+	0.656777i	$0.771919\pi$
$878$	31.8464i	1.07476i
$879$	0	0
$880$	0	0
$881$	−42.3607	−1.42717	−0.713584	−	0.700570i	$-0.752929\pi$
$881$	−42.3607	−1.42717	−0.713584	+	0.700570i	$0.752929\pi$
$882$	0	0
$883$	− 42.9161i	− 1.44424i	−0.691768	−	0.722120i	$-0.743168\pi$
$883$	− 42.9161i	− 1.44424i	0.691768	−	0.722120i	$-0.256832\pi$
$884$	−21.1803	−0.712372
$885$	0	0
$886$	−32.0344	−1.07622
$887$	20.0907i	0.674580i	0.941401	+	0.337290i	$0.109510\pi$
$887$	20.0907i	0.674580i	−0.941401	+	0.337290i	$0.890490\pi$
$888$	0	0
$889$	31.9098	1.07022
$890$	0	0
$891$	0	0
$892$	− 16.0090i	− 0.536020i
$893$	− 28.9402i	− 0.968448i
$894$	0	0
$895$	0	0
$896$	−9.27051	−0.309706
$897$	0	0
$898$	− 0.857567i	− 0.0286174i
$899$	14.2705	0.475948
$900$	0	0
$901$	35.6525	1.18776
$902$	47.5528i	1.58334i
$903$	0	0
$904$	−28.9443	−0.962672
$905$	0	0
$906$	0	0
$907$	29.1522i	0.967984i	0.875072	+	0.483992i	$0.160814\pi$
$907$	29.1522i	0.967984i	−0.875072	+	0.483992i	$0.839186\pi$
$908$	− 5.04531i	− 0.167434i
$909$	0	0
$910$	0	0
$911$	−11.9098	−0.394590	−0.197295	−	0.980344i	$-0.563216\pi$
$911$	−11.9098	−0.394590	−0.197295	+	0.980344i	$0.563216\pi$
$912$	0	0
$913$	41.1450i	1.36170i
$914$	−21.6312	−0.715496
$915$	0	0
$916$	16.4721	0.544255
$917$	− 5.02029i	− 0.165784i
$918$	0	0
$919$	27.3820	0.903248	0.451624	−	0.892208i	$-0.350845\pi$
$919$	27.3820	0.903248	0.451624	+	0.892208i	$0.350845\pi$
$920$	0	0
$921$	0	0
$922$	− 17.6336i	− 0.580730i
$923$	− 90.0854i	− 2.96520i
$924$	0	0
$925$	0	0
$926$	−41.6312	−1.36809
$927$	0	0
$928$	− 16.7760i	− 0.550699i
$929$	11.9098	0.390749	0.195374	−	0.980729i	$-0.437408\pi$
$929$	11.9098	0.390749	0.195374	+	0.980729i	$0.437408\pi$
$930$	0	0
$931$	−0.313082	−0.0102609
$932$	1.38757i	0.0454515i
$933$	0	0
$934$	−19.4721	−0.637148
$935$	0	0
$936$	0	0
$937$	− 12.5227i	− 0.409100i	−0.978856	−	0.204550i	$-0.934427\pi$
$937$	− 12.5227i	− 0.409100i	0.978856	−	0.204550i	$-0.0655731\pi$
$938$	− 26.2866i	− 0.858286i
$939$	0	0
$940$	0	0
$941$	−38.0902	−1.24170	−0.620852	−	0.783928i	$-0.713213\pi$
$941$	−38.0902	−1.24170	−0.620852	+	0.783928i	$0.713213\pi$
$942$	0	0
$943$	− 43.9201i	− 1.43024i
$944$	4.54915	0.148062
$945$	0	0
$946$	5.90170	0.191881
$947$	− 28.0422i	− 0.911248i	−0.890172	−	0.455624i	$-0.849416\pi$
$947$	− 28.0422i	− 0.911248i	0.890172	−	0.455624i	$-0.150584\pi$
$948$	0	0
$949$	29.2705	0.950161
$950$	0	0
$951$	0	0
$952$	− 40.2874i	− 1.30572i
$953$	− 11.3067i	− 0.366259i	−0.983089	−	0.183130i	$-0.941377\pi$
$953$	− 11.3067i	− 0.366259i	0.983089	−	0.183130i	$-0.0586228\pi$
$954$	0	0
$955$	0	0
$956$	8.81966	0.285248
$957$	0	0
$958$	14.5309i	0.469470i
$959$	40.4508	1.30623
$960$	0	0
$961$	−22.8541	−0.737229
$962$	13.1433i	0.423756i
$963$	0	0
$964$	−17.0557	−0.549328
$965$	0	0
$966$	0	0
$967$	− 30.3933i	− 0.977383i	−0.872457	−	0.488692i	$-0.837474\pi$
$967$	− 30.3933i	− 0.977383i	0.872457	−	0.488692i	$-0.162526\pi$
$968$	43.0876i	1.38489i
$969$	0	0
$970$	0	0
$971$	12.3607	0.396673	0.198337	−	0.980134i	$-0.436446\pi$
$971$	12.3607	0.396673	0.198337	+	0.980134i	$0.436446\pi$
$972$	0	0
$973$	1.86162i	0.0596809i
$974$	−22.3607	−0.716482
$975$	0	0
$976$	−17.4508	−0.558588
$977$	− 20.3682i	− 0.651637i	−0.945432	−	0.325818i	$-0.894360\pi$
$977$	− 20.3682i	− 0.651637i	0.945432	−	0.325818i	$-0.105640\pi$
$978$	0	0
$979$	−55.9017	−1.78663
$980$	0	0
$981$	0	0
$982$	27.1441i	0.866204i
$983$	30.6708i	0.978248i	0.872214	+	0.489124i	$0.162683\pi$
$983$	30.6708i	0.978248i	−0.872214	+	0.489124i	$0.837317\pi$
$984$	0	0
$985$	0	0
$986$	−29.2705	−0.932163
$987$	0	0
$988$	14.7679i	0.469829i
$989$	−5.45085	−0.173327
$990$	0	0
$991$	48.5066	1.54086	0.770431	−	0.637523i	$-0.220041\pi$
$991$	48.5066	1.54086	0.770431	+	0.637523i	$0.220041\pi$
$992$	− 9.57608i	− 0.304041i
$993$	0	0
$994$	40.4508	1.28302
$995$	0	0
$996$	0	0
$997$	8.89002i	0.281550i	0.990042	+	0.140775i	$0.0449594\pi$
$997$	8.89002i	0.281550i	−0.990042	+	0.140775i	$0.955041\pi$
$998$	− 7.67396i	− 0.242915i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1125.2.b.d.874.3		4
3.2	odd	2		1125.2.b.c.874.2		4
5.2	odd	4		1125.2.a.j.1.2	yes	4
5.3	odd	4		1125.2.a.j.1.3	yes	4
5.4	even	2	inner	1125.2.b.d.874.2		4
15.2	even	4		1125.2.a.i.1.3	yes	4
15.8	even	4		1125.2.a.i.1.2	✓	4
15.14	odd	2		1125.2.b.c.874.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1125.2.a.i.1.2	✓	4	15.8	even	4
1125.2.a.i.1.3	yes	4	15.2	even	4
1125.2.a.j.1.2	yes	4	5.2	odd	4
1125.2.a.j.1.3	yes	4	5.3	odd	4
1125.2.b.c.874.2		4	3.2	odd	2
1125.2.b.c.874.3		4	15.14	odd	2
1125.2.b.d.874.2		4	5.4	even	2	inner
1125.2.b.d.874.3		4	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1125 = 3^{2} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1125.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.17557i q^{2} +0.618034 q^{4} -2.62866i q^{7} +3.07768i q^{8} +5.00000 q^{11} -6.88191i q^{13} +3.09017 q^{14} -2.38197 q^{16} -4.97980i q^{17} -3.47214 q^{19} +5.87785i q^{22} -5.42882i q^{23} +8.09017 q^{26} -1.62460i q^{28} -5.00000 q^{29} -2.85410 q^{31} +3.35520i q^{32} +5.85410 q^{34} +1.62460i q^{37} -4.08174i q^{38} +8.09017 q^{41} -1.00406i q^{43} +3.09017 q^{44} +6.38197 q^{46} +8.33499i q^{47} +0.0901699 q^{49} -4.25325i q^{52} +7.15942i q^{53} +8.09017 q^{56} -5.87785i q^{58} -1.90983 q^{59} +7.32624 q^{61} -3.35520i q^{62} -8.70820 q^{64} -8.50651i q^{67} -3.07768i q^{68} +13.0902 q^{71} +4.25325i q^{73} -1.90983 q^{74} -2.14590 q^{76} -13.1433i q^{77} +8.47214 q^{79} +9.51057i q^{82} +8.22899i q^{83} +1.18034 q^{86} +15.3884i q^{88} -11.1803 q^{89} -18.0902 q^{91} -3.35520i q^{92} -9.79837 q^{94} -14.3844i q^{97} +0.106001i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{4} + 20 q^{11} - 10 q^{14} - 14 q^{16} + 4 q^{19} + 10 q^{26} - 20 q^{29} + 2 q^{31} + 10 q^{34} + 10 q^{41} - 10 q^{44} + 30 q^{46} - 22 q^{49} + 10 q^{56} - 30 q^{59} - 2 q^{61} - 8 q^{64}+ \cdots + 10 q^{94}+O(q^{100}) \) 4 * q - 2 * q^4 + 20 * q^11 - 10 * q^14 - 14 * q^16 + 4 * q^19 + 10 * q^26 - 20 * q^29 + 2 * q^31 + 10 * q^34 + 10 * q^41 - 10 * q^44 + 30 * q^46 - 22 * q^49 + 10 * q^56 - 30 * q^59 - 2 * q^61 - 8 * q^64 + 30 * q^71 - 30 * q^74 - 22 * q^76 + 16 * q^79 - 40 * q^86 - 50 * q^91 + 10 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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