LMFDB - Embedded newform 1350.3.b.h.1349.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1350,3,Mod(1349,1350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1350, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1350.1349");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1350 = 2 \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1350.b (of order $2$, degree $1$, not 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:	$36.7848356886$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.40960000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 7x^{4} + 1 $ x^8 + 7*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 270)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1349.5
Root			$1.14412 - 1.14412i$ of defining polynomial
Character	$\chi$	$=$	1350.1349
Dual form			1350.3.b.h.1349.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{2} +2.00000 q^{4} -1.00000i q^{7} +2.82843 q^{8} -13.4164i q^{11} +13.9737i q^{13} -1.41421i q^{14} +4.00000 q^{16} +8.48528 q^{17} +25.9737 q^{19} -18.9737i q^{22} +3.55415 q^{23} +19.7617i q^{26} -2.00000i q^{28} +4.93113i q^{29} -35.9473 q^{31} +5.65685 q^{32} +12.0000 q^{34} -13.9737i q^{37} +36.7323 q^{38} -29.0100i q^{41} -75.9473i q^{43} -26.8328i q^{44} +5.02633 q^{46} +69.2592 q^{47} +48.0000 q^{49} +27.9473i q^{52} +89.7839 q^{53} -2.82843i q^{56} +6.97367i q^{58} -8.48528i q^{59} +50.8947 q^{61} -50.8372 q^{62} +8.00000 q^{64} +71.0000i q^{67} +16.9706 q^{68} -126.479i q^{71} -22.0263i q^{73} -19.7617i q^{74} +51.9473 q^{76} -13.4164 q^{77} -75.8683 q^{79} -41.0263i q^{82} -34.5179 q^{83} -107.406i q^{86} -37.9473i q^{88} +147.004i q^{89} +13.9737 q^{91} +7.10831 q^{92} +97.9473 q^{94} +99.8683i q^{97} +67.8823 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{4} + 32 q^{16} + 56 q^{19} + 16 q^{31} + 96 q^{34} + 192 q^{46} + 384 q^{49} - 200 q^{61} + 64 q^{64} + 112 q^{76} + 152 q^{79} - 40 q^{91} + 480 q^{94}+O(q^{100}) $ 8 * q + 16 * q^4 + 32 * q^16 + 56 * q^19 + 16 * q^31 + 96 * q^34 + 192 * q^46 + 384 * q^49 - 200 * q^61 + 64 * q^64 + 112 * q^76 + 152 * q^79 - 40 * q^91 + 480 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times$.

$n$	$1001$	$1027$
$\chi(n)$	$-1$	$-1$

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 $

$2$	1.41421	0.707107
$2$	1.41421	0.707107
$3$	0	0
$3$	0	0
$4$	2.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 1.00000i	− 0.142857i	−0.997446	−	0.0714286i	$-0.977244\pi$
$7$	− 1.00000i	− 0.142857i	0.997446	−	0.0714286i	$-0.0227558\pi$
$8$	2.82843	0.353553
$9$	0	0
$10$	0	0
$11$	− 13.4164i	− 1.21967i	−0.792527	−	0.609837i	$-0.791235\pi$
$11$	− 13.4164i	− 1.21967i	0.792527	−	0.609837i	$-0.208765\pi$
$12$	0	0
$13$	13.9737i	1.07490i	0.843296	+	0.537449i	$0.180612\pi$
$13$	13.9737i	1.07490i	−0.843296	+	0.537449i	$0.819388\pi$
$14$	− 1.41421i	− 0.101015i
$15$	0	0
$16$	4.00000	0.250000
$17$	8.48528	0.499134	0.249567	−	0.968358i	$-0.419712\pi$
$17$	8.48528	0.499134	0.249567	+	0.968358i	$0.419712\pi$
$18$	0	0
$19$	25.9737	1.36704	0.683518	−	0.729934i	$-0.260449\pi$
$19$	25.9737	1.36704	0.683518	+	0.729934i	$0.260449\pi$
$20$	0	0
$21$	0	0
$22$	− 18.9737i	− 0.862439i
$23$	3.55415	0.154528	0.0772642	−	0.997011i	$-0.475381\pi$
$23$	3.55415	0.154528	0.0772642	+	0.997011i	$0.475381\pi$
$24$	0	0
$25$	0	0
$26$	19.7617i	0.760067i
$27$	0	0
$28$	− 2.00000i	− 0.0714286i
$29$	4.93113i	0.170039i	0.996379	+	0.0850194i	$0.0270952\pi$
$29$	4.93113i	0.170039i	−0.996379	+	0.0850194i	$0.972905\pi$
$30$	0	0
$31$	−35.9473	−1.15959	−0.579796	−	0.814762i	$-0.696868\pi$
$31$	−35.9473	−1.15959	−0.579796	+	0.814762i	$0.696868\pi$
$32$	5.65685	0.176777
$33$	0	0
$34$	12.0000	0.352941
$35$	0	0
$36$	0	0
$37$	− 13.9737i	− 0.377667i	−0.982009	−	0.188833i	$-0.939529\pi$
$37$	− 13.9737i	− 0.377667i	0.982009	−	0.188833i	$-0.0604706\pi$
$38$	36.7323	0.966640
$39$	0	0
$40$	0	0
$41$	− 29.0100i	− 0.707561i	−0.935328	−	0.353780i	$-0.884896\pi$
$41$	− 29.0100i	− 0.707561i	0.935328	−	0.353780i	$-0.115104\pi$
$42$	0	0
$43$	− 75.9473i	− 1.76622i	−0.469169	−	0.883109i	$-0.655446\pi$
$43$	− 75.9473i	− 1.76622i	0.469169	−	0.883109i	$-0.344554\pi$
$44$	− 26.8328i	− 0.609837i
$45$	0	0
$46$	5.02633	0.109268
$47$	69.2592	1.47360	0.736800	−	0.676110i	$-0.236336\pi$
$47$	69.2592	1.47360	0.736800	+	0.676110i	$0.236336\pi$
$48$	0	0
$49$	48.0000	0.979592
$50$	0	0
$51$	0	0
$52$	27.9473i	0.537449i
$53$	89.7839	1.69404	0.847018	−	0.531564i	$-0.178395\pi$
$53$	89.7839	1.69404	0.847018	+	0.531564i	$0.178395\pi$
$54$	0	0
$55$	0	0
$56$	− 2.82843i	− 0.0505076i
$57$	0	0
$58$	6.97367i	0.120236i
$59$	− 8.48528i	− 0.143818i	−0.997411	−	0.0719092i	$-0.977091\pi$
$59$	− 8.48528i	− 0.143818i	0.997411	−	0.0719092i	$-0.0229092\pi$
$60$	0	0
$61$	50.8947	0.834339	0.417169	−	0.908829i	$-0.363022\pi$
$61$	50.8947	0.834339	0.417169	+	0.908829i	$0.363022\pi$
$62$	−50.8372	−0.819955
$63$	0	0
$64$	8.00000	0.125000
$65$	0	0
$66$	0	0
$67$	71.0000i	1.05970i	0.848091	+	0.529851i	$0.177752\pi$
$67$	71.0000i	1.05970i	−0.848091	+	0.529851i	$0.822248\pi$
$68$	16.9706	0.249567
$69$	0	0
$70$	0	0
$71$	− 126.479i	− 1.78139i	−0.454597	−	0.890697i	$-0.650217\pi$
$71$	− 126.479i	− 1.78139i	0.454597	−	0.890697i	$-0.349783\pi$
$72$	0	0
$73$	− 22.0263i	− 0.301731i	−0.988554	−	0.150865i	$-0.951794\pi$
$73$	− 22.0263i	− 0.301731i	0.988554	−	0.150865i	$-0.0482060\pi$
$74$	− 19.7617i	− 0.267051i
$75$	0	0
$76$	51.9473	0.683518
$77$	−13.4164	−0.174239
$78$	0	0
$79$	−75.8683	−0.960359	−0.480179	−	0.877170i	$-0.659428\pi$
$79$	−75.8683	−0.960359	−0.480179	+	0.877170i	$0.659428\pi$
$80$	0	0
$81$	0	0
$82$	− 41.0263i	− 0.500321i
$83$	−34.5179	−0.415878	−0.207939	−	0.978142i	$-0.566676\pi$
$83$	−34.5179	−0.415878	−0.207939	+	0.978142i	$0.566676\pi$
$84$	0	0
$85$	0	0
$86$	− 107.406i	− 1.24890i
$87$	0	0
$88$	− 37.9473i	− 0.431220i
$89$	147.004i	1.65173i	0.563870	+	0.825864i	$0.309312\pi$
$89$	147.004i	1.65173i	−0.563870	+	0.825864i	$0.690688\pi$
$90$	0	0
$91$	13.9737	0.153557
$92$	7.10831	0.0772642
$93$	0	0
$94$	97.9473	1.04199
$95$	0	0
$96$	0	0
$97$	99.8683i	1.02957i	0.857319	+	0.514785i	$0.172128\pi$
$97$	99.8683i	1.02957i	−0.857319	+	0.514785i	$0.827872\pi$
$98$	67.8823	0.692676
$99$	0	0
$100$	0	0
$101$	− 4.35437i	− 0.0431125i	−0.999768	−	0.0215563i	$-0.993138\pi$
$101$	− 4.35437i	− 0.0431125i	0.999768	−	0.0215563i	$-0.00686211\pi$
$102$	0	0
$103$	− 12.9473i	− 0.125702i	−0.998023	−	0.0628511i	$-0.979981\pi$
$103$	− 12.9473i	− 0.125702i	0.998023	−	0.0628511i	$-0.0200193\pi$
$104$	39.5235i	0.380034i
$105$	0	0
$106$	126.974	1.19786
$107$	103.777	0.969880	0.484940	−	0.874548i	$-0.338842\pi$
$107$	103.777	0.969880	0.484940	+	0.874548i	$0.338842\pi$
$108$	0	0
$109$	−99.9473	−0.916948	−0.458474	−	0.888708i	$-0.651604\pi$
$109$	−99.9473	−0.916948	−0.458474	+	0.888708i	$0.651604\pi$
$110$	0	0
$111$	0	0
$112$	− 4.00000i	− 0.0357143i
$113$	205.600	1.81947	0.909737	−	0.415186i	$-0.136283\pi$
$113$	205.600	1.81947	0.909737	+	0.415186i	$0.136283\pi$
$114$	0	0
$115$	0	0
$116$	9.86225i	0.0850194i
$117$	0	0
$118$	− 12.0000i	− 0.101695i
$119$	− 8.48528i	− 0.0713049i
$120$	0	0
$121$	−59.0000	−0.487603
$122$	71.9759	0.589967
$123$	0	0
$124$	−71.8947	−0.579796
$125$	0	0
$126$	0	0
$127$	− 34.0000i	− 0.267717i	−0.991000	−	0.133858i	$-0.957263\pi$
$127$	− 34.0000i	− 0.267717i	0.991000	−	0.133858i	$-0.0427367\pi$
$128$	11.3137	0.0883883
$129$	0	0
$130$	0	0
$131$	− 99.0694i	− 0.756255i	−0.925753	−	0.378128i	$-0.876568\pi$
$131$	− 99.0694i	− 0.756255i	0.925753	−	0.378128i	$-0.123432\pi$
$132$	0	0
$133$	− 25.9737i	− 0.195291i
$134$	100.409i	0.749322i
$135$	0	0
$136$	24.0000	0.176471
$137$	−52.8654	−0.385879	−0.192939	−	0.981211i	$-0.561802\pi$
$137$	−52.8654	−0.385879	−0.192939	+	0.981211i	$0.561802\pi$
$138$	0	0
$139$	175.816	1.26486	0.632430	−	0.774617i	$-0.282057\pi$
$139$	175.816	1.26486	0.632430	+	0.774617i	$0.282057\pi$
$140$	0	0
$141$	0	0
$142$	− 178.868i	− 1.25964i
$143$	187.476	1.31102
$144$	0	0
$145$	0	0
$146$	− 31.1499i	− 0.213356i
$147$	0	0
$148$	− 27.9473i	− 0.188833i
$149$	− 238.741i	− 1.60229i	−0.598469	−	0.801146i	$-0.704224\pi$
$149$	− 238.741i	− 1.60229i	0.598469	−	0.801146i	$-0.295776\pi$
$150$	0	0
$151$	−209.868	−1.38986	−0.694928	−	0.719079i	$-0.744564\pi$
$151$	−209.868	−1.38986	−0.694928	+	0.719079i	$0.744564\pi$
$152$	73.4646	0.483320
$153$	0	0
$154$	−18.9737	−0.123206
$155$	0	0
$156$	0	0
$157$	154.105i	0.981563i	0.871283	+	0.490781i	$0.163289\pi$
$157$	154.105i	0.981563i	−0.871283	+	0.490781i	$0.836711\pi$
$158$	−107.294	−0.679076
$159$	0	0
$160$	0	0
$161$	− 3.55415i	− 0.0220755i
$162$	0	0
$163$	14.9473i	0.0917014i	0.998948	+	0.0458507i	$0.0145999\pi$
$163$	14.9473i	0.0917014i	−0.998948	+	0.0458507i	$0.985400\pi$
$164$	− 58.0200i	− 0.353780i
$165$	0	0
$166$	−48.8157	−0.294070
$167$	−138.295	−0.828114	−0.414057	−	0.910251i	$-0.635889\pi$
$167$	−138.295	−0.828114	−0.414057	+	0.910251i	$0.635889\pi$
$168$	0	0
$169$	−26.2633	−0.155404
$170$	0	0
$171$	0	0
$172$	− 151.895i	− 0.883109i
$173$	197.692	1.14273	0.571364	−	0.820697i	$-0.306414\pi$
$173$	197.692	1.14273	0.571364	+	0.820697i	$0.306414\pi$
$174$	0	0
$175$	0	0
$176$	− 53.6656i	− 0.304918i
$177$	0	0
$178$	207.895i	1.16795i
$179$	242.519i	1.35485i	0.735590	+	0.677427i	$0.236905\pi$
$179$	242.519i	1.35485i	−0.735590	+	0.677427i	$0.763095\pi$
$180$	0	0
$181$	−308.789	−1.70602	−0.853009	−	0.521896i	$-0.825225\pi$
$181$	−308.789	−1.70602	−0.853009	+	0.521896i	$0.825225\pi$
$182$	19.7617	0.108581
$183$	0	0
$184$	10.0527	0.0546341
$185$	0	0
$186$	0	0
$187$	− 113.842i	− 0.608781i
$188$	138.518	0.736800
$189$	0	0
$190$	0	0
$191$	− 308.001i	− 1.61257i	−0.591528	−	0.806284i	$-0.701475\pi$
$191$	− 308.001i	− 1.61257i	0.591528	−	0.806284i	$-0.298525\pi$
$192$	0	0
$193$	341.868i	1.77134i	0.464317	+	0.885669i	$0.346300\pi$
$193$	341.868i	1.77134i	−0.464317	+	0.885669i	$0.653700\pi$
$194$	141.235i	0.728016i
$195$	0	0
$196$	96.0000	0.489796
$197$	−128.433	−0.651943	−0.325971	−	0.945380i	$-0.605691\pi$
$197$	−128.433	−0.651943	−0.325971	+	0.945380i	$0.605691\pi$
$198$	0	0
$199$	−178.026	−0.894605	−0.447302	−	0.894383i	$-0.647615\pi$
$199$	−178.026	−0.894605	−0.447302	+	0.894383i	$0.647615\pi$
$200$	0	0
$201$	0	0
$202$	− 6.15800i	− 0.0304852i
$203$	4.93113	0.0242913
$204$	0	0
$205$	0	0
$206$	− 18.3103i	− 0.0888849i
$207$	0	0
$208$	55.8947i	0.268724i
$209$	− 348.473i	− 1.66734i
$210$	0	0
$211$	69.7103	0.330381	0.165190	−	0.986262i	$-0.447176\pi$
$211$	69.7103	0.330381	0.165190	+	0.986262i	$0.447176\pi$
$212$	179.568	0.847018
$213$	0	0
$214$	146.763	0.685808
$215$	0	0
$216$	0	0
$217$	35.9473i	0.165656i
$218$	−141.347	−0.648380
$219$	0	0
$220$	0	0
$221$	118.570i	0.536518i
$222$	0	0
$223$	131.631i	0.590275i	0.955455	+	0.295137i	$0.0953655\pi$
$223$	131.631i	0.590275i	−0.955455	+	0.295137i	$0.904635\pi$
$224$	− 5.65685i	− 0.0252538i
$225$	0	0
$226$	290.763	1.28656
$227$	343.895	1.51496	0.757479	−	0.652859i	$-0.226431\pi$
$227$	343.895	1.51496	0.757479	+	0.652859i	$0.226431\pi$
$228$	0	0
$229$	−413.579	−1.80602	−0.903010	−	0.429619i	$-0.858648\pi$
$229$	−413.579	−1.80602	−0.903010	+	0.429619i	$0.858648\pi$
$230$	0	0
$231$	0	0
$232$	13.9473i	0.0601178i
$233$	154.335	0.662384	0.331192	−	0.943563i	$-0.392549\pi$
$233$	154.335	0.662384	0.331192	+	0.943563i	$0.392549\pi$
$234$	0	0
$235$	0	0
$236$	− 16.9706i	− 0.0719092i
$237$	0	0
$238$	− 12.0000i	− 0.0504202i
$239$	− 47.5810i	− 0.199084i	−0.995033	−	0.0995418i	$-0.968262\pi$
$239$	− 47.5810i	− 0.199084i	0.995033	−	0.0995418i	$-0.0317377\pi$
$240$	0	0
$241$	−440.789	−1.82900	−0.914501	−	0.404584i	$-0.867416\pi$
$241$	−440.789	−1.82900	−0.914501	+	0.404584i	$0.867416\pi$
$242$	−83.4386	−0.344788
$243$	0	0
$244$	101.789	0.417169
$245$	0	0
$246$	0	0
$247$	362.947i	1.46942i
$248$	−101.674	−0.409977
$249$	0	0
$250$	0	0
$251$	342.742i	1.36551i	0.730649	+	0.682753i	$0.239217\pi$
$251$	342.742i	1.36551i	−0.730649	+	0.682753i	$0.760783\pi$
$252$	0	0
$253$	− 47.6840i	− 0.188474i
$254$	− 48.0833i	− 0.189304i
$255$	0	0
$256$	16.0000	0.0625000
$257$	309.378	1.20380	0.601902	−	0.798570i	$-0.294410\pi$
$257$	309.378	1.20380	0.601902	+	0.798570i	$0.294410\pi$
$258$	0	0
$259$	−13.9737	−0.0539524
$260$	0	0
$261$	0	0
$262$	− 140.105i	− 0.534753i
$263$	−359.136	−1.36554	−0.682768	−	0.730636i	$-0.739224\pi$
$263$	−359.136	−1.36554	−0.682768	+	0.730636i	$0.739224\pi$
$264$	0	0
$265$	0	0
$266$	− 36.7323i	− 0.138091i
$267$	0	0
$268$	142.000i	0.529851i
$269$	257.536i	0.957382i	0.877983	+	0.478691i	$0.158889\pi$
$269$	257.536i	0.957382i	−0.877983	+	0.478691i	$0.841111\pi$
$270$	0	0
$271$	177.710	0.655758	0.327879	−	0.944720i	$-0.393666\pi$
$271$	177.710	0.655758	0.327879	+	0.944720i	$0.393666\pi$
$272$	33.9411	0.124784
$273$	0	0
$274$	−74.7630	−0.272858
$275$	0	0
$276$	0	0
$277$	− 311.631i	− 1.12502i	−0.826790	−	0.562511i	$-0.809835\pi$
$277$	− 311.631i	− 1.12502i	0.826790	−	0.562511i	$-0.190165\pi$
$278$	248.641	0.894392
$279$	0	0
$280$	0	0
$281$	− 55.2661i	− 0.196676i	−0.995153	−	0.0983382i	$-0.968647\pi$
$281$	− 55.2661i	− 0.196676i	0.995153	−	0.0983382i	$-0.0313527\pi$
$282$	0	0
$283$	− 155.737i	− 0.550306i	−0.961400	−	0.275153i	$-0.911271\pi$
$283$	− 155.737i	− 0.550306i	0.961400	−	0.275153i	$-0.0887285\pi$
$284$	− 252.958i	− 0.890697i
$285$	0	0
$286$	265.132	0.927034
$287$	−29.0100	−0.101080
$288$	0	0
$289$	−217.000	−0.750865
$290$	0	0
$291$	0	0
$292$	− 44.0527i	− 0.150865i
$293$	−338.611	−1.15567	−0.577835	−	0.816154i	$-0.696102\pi$
$293$	−338.611	−1.15567	−0.577835	+	0.816154i	$0.696102\pi$
$294$	0	0
$295$	0	0
$296$	− 39.5235i	− 0.133525i
$297$	0	0
$298$	− 337.631i	− 1.13299i
$299$	49.6646i	0.166102i
$300$	0	0
$301$	−75.9473	−0.252317
$302$	−296.799	−0.982777
$303$	0	0
$304$	103.895	0.341759
$305$	0	0
$306$	0	0
$307$	229.684i	0.748156i	0.927397	+	0.374078i	$0.122041\pi$
$307$	229.684i	0.748156i	−0.927397	+	0.374078i	$0.877959\pi$
$308$	−26.8328	−0.0871195
$309$	0	0
$310$	0	0
$311$	102.047i	0.328125i	0.986450	+	0.164062i	$0.0524599\pi$
$311$	102.047i	0.328125i	−0.986450	+	0.164062i	$0.947540\pi$
$312$	0	0
$313$	− 335.658i	− 1.07239i	−0.844095	−	0.536194i	$-0.819861\pi$
$313$	− 335.658i	− 1.07239i	0.844095	−	0.536194i	$-0.180139\pi$
$314$	217.938i	0.694070i
$315$	0	0
$316$	−151.737	−0.480179
$317$	−430.925	−1.35939	−0.679693	−	0.733497i	$-0.737887\pi$
$317$	−430.925	−1.35939	−0.679693	+	0.733497i	$0.737887\pi$
$318$	0	0
$319$	66.1580	0.207392
$320$	0	0
$321$	0	0
$322$	− 5.02633i	− 0.0156097i
$323$	220.394	0.682334
$324$	0	0
$325$	0	0
$326$	21.1387i	0.0648427i
$327$	0	0
$328$	− 82.0527i	− 0.250161i
$329$	− 69.2592i	− 0.210514i
$330$	0	0
$331$	−181.974	−0.549769	−0.274885	−	0.961477i	$-0.588640\pi$
$331$	−181.974	−0.549769	−0.274885	+	0.961477i	$0.588640\pi$
$332$	−69.0358	−0.207939
$333$	0	0
$334$	−195.579	−0.585565
$335$	0	0
$336$	0	0
$337$	353.500i	1.04896i	0.851423	+	0.524480i	$0.175740\pi$
$337$	353.500i	1.04896i	−0.851423	+	0.524480i	$0.824260\pi$
$338$	−37.1420	−0.109887
$339$	0	0
$340$	0	0
$341$	482.284i	1.41432i
$342$	0	0
$343$	− 97.0000i	− 0.282799i
$344$	− 214.811i	− 0.624452i
$345$	0	0
$346$	279.579	0.808031
$347$	−123.725	−0.356556	−0.178278	−	0.983980i	$-0.557053\pi$
$347$	−123.725	−0.356556	−0.178278	+	0.983980i	$0.557053\pi$
$348$	0	0
$349$	−67.1053	−0.192279	−0.0961395	−	0.995368i	$-0.530649\pi$
$349$	−67.1053	−0.192279	−0.0961395	+	0.995368i	$0.530649\pi$
$350$	0	0
$351$	0	0
$352$	− 75.8947i	− 0.215610i
$353$	−665.759	−1.88600	−0.943002	−	0.332787i	$-0.892011\pi$
$353$	−665.759	−1.88600	−0.943002	+	0.332787i	$0.892011\pi$
$354$	0	0
$355$	0	0
$356$	294.007i	0.825864i
$357$	0	0
$358$	342.974i	0.958027i
$359$	434.927i	1.21149i	0.795657	+	0.605747i	$0.207126\pi$
$359$	434.927i	1.21149i	−0.795657	+	0.605747i	$0.792874\pi$
$360$	0	0
$361$	313.631	0.868785
$362$	−436.694	−1.20634
$363$	0	0
$364$	27.9473	0.0767784
$365$	0	0
$366$	0	0
$367$	286.684i	0.781155i	0.920570	+	0.390578i	$0.127725\pi$
$367$	286.684i	0.781155i	−0.920570	+	0.390578i	$0.872275\pi$
$368$	14.2166	0.0386321
$369$	0	0
$370$	0	0
$371$	− 89.7839i	− 0.242005i
$372$	0	0
$373$	− 3.86833i	− 0.0103709i	−0.999987	−	0.00518543i	$-0.998349\pi$
$373$	− 3.86833i	− 0.0103709i	0.999987	−	0.00518543i	$-0.00165058\pi$
$374$	− 160.997i	− 0.430473i
$375$	0	0
$376$	195.895	0.520996
$377$	−68.9059	−0.182774
$378$	0	0
$379$	114.184	0.301278	0.150639	−	0.988589i	$-0.451867\pi$
$379$	114.184	0.301278	0.150639	+	0.988589i	$0.451867\pi$
$380$	0	0
$381$	0	0
$382$	− 435.579i	− 1.14026i
$383$	−333.903	−0.871810	−0.435905	−	0.899993i	$-0.643572\pi$
$383$	−333.903	−0.871810	−0.435905	+	0.899993i	$0.643572\pi$
$384$	0	0
$385$	0	0
$386$	483.475i	1.25253i
$387$	0	0
$388$	199.737i	0.514785i
$389$	− 682.507i	− 1.75452i	−0.480020	−	0.877258i	$-0.659370\pi$
$389$	− 682.507i	− 1.75452i	0.480020	−	0.877258i	$-0.340630\pi$
$390$	0	0
$391$	30.1580	0.0771304
$392$	135.765	0.346338
$393$	0	0
$394$	−181.631	−0.460993
$395$	0	0
$396$	0	0
$397$	258.211i	0.650405i	0.945644	+	0.325202i	$0.105432\pi$
$397$	258.211i	0.650405i	−0.945644	+	0.325202i	$0.894568\pi$
$398$	−251.767	−0.632581
$399$	0	0
$400$	0	0
$401$	− 388.852i	− 0.969707i	−0.874595	−	0.484853i	$-0.838873\pi$
$401$	− 388.852i	− 0.969707i	0.874595	−	0.484853i	$-0.161127\pi$
$402$	0	0
$403$	− 502.316i	− 1.24644i
$404$	− 8.70873i	− 0.0215563i
$405$	0	0
$406$	6.97367	0.0171765
$407$	−187.476	−0.460630
$408$	0	0
$409$	302.315	0.739157	0.369579	−	0.929199i	$-0.379502\pi$
$409$	302.315	0.739157	0.369579	+	0.929199i	$0.379502\pi$
$410$	0	0
$411$	0	0
$412$	− 25.8947i	− 0.0628511i
$413$	−8.48528	−0.0205455
$414$	0	0
$415$	0	0
$416$	79.0470i	0.190017i
$417$	0	0
$418$	− 492.816i	− 1.17898i
$419$	264.197i	0.630542i	0.949002	+	0.315271i	$0.102095\pi$
$419$	264.197i	0.630542i	−0.949002	+	0.315271i	$0.897905\pi$
$420$	0	0
$421$	−131.369	−0.312040	−0.156020	−	0.987754i	$-0.549866\pi$
$421$	−131.369	−0.312040	−0.156020	+	0.987754i	$0.549866\pi$
$422$	98.5853	0.233614
$423$	0	0
$424$	253.947	0.598932
$425$	0	0
$426$	0	0
$427$	− 50.8947i	− 0.119191i
$428$	207.554	0.484940
$429$	0	0
$430$	0	0
$431$	184.722i	0.428590i	0.976769	+	0.214295i	$0.0687455\pi$
$431$	184.722i	0.428590i	−0.976769	+	0.214295i	$0.931255\pi$
$432$	0	0
$433$	793.263i	1.83202i	0.401161	+	0.916008i	$0.368607\pi$
$433$	793.263i	1.83202i	−0.401161	+	0.916008i	$0.631393\pi$
$434$	50.8372i	0.117136i
$435$	0	0
$436$	−199.895	−0.458474
$437$	92.3144	0.211246
$438$	0	0
$439$	598.000	1.36219	0.681093	−	0.732197i	$-0.261505\pi$
$439$	598.000	1.36219	0.681093	+	0.732197i	$0.261505\pi$
$440$	0	0
$441$	0	0
$442$	167.684i	0.379376i
$443$	−740.750	−1.67212	−0.836061	−	0.548637i	$-0.815147\pi$
$443$	−740.750	−1.67212	−0.836061	+	0.548637i	$0.815147\pi$
$444$	0	0
$445$	0	0
$446$	186.155i	0.417387i
$447$	0	0
$448$	− 8.00000i	− 0.0178571i
$449$	881.799i	1.96392i	0.189096	+	0.981959i	$0.439444\pi$
$449$	881.799i	1.96392i	−0.189096	+	0.981959i	$0.560556\pi$
$450$	0	0
$451$	−389.210	−0.862993
$452$	411.201	0.909737
$453$	0	0
$454$	486.342	1.07124
$455$	0	0
$456$	0	0
$457$	− 238.000i	− 0.520788i	−0.965502	−	0.260394i	$-0.916148\pi$
$457$	− 238.000i	− 0.520788i	0.965502	−	0.260394i	$-0.0838524\pi$
$458$	−584.889	−1.27705
$459$	0	0
$460$	0	0
$461$	463.490i	1.00540i	0.864461	+	0.502700i	$0.167660\pi$
$461$	463.490i	1.00540i	−0.864461	+	0.502700i	$0.832340\pi$
$462$	0	0
$463$	178.737i	0.386040i	0.981195	+	0.193020i	$0.0618283\pi$
$463$	178.737i	0.386040i	−0.981195	+	0.193020i	$0.938172\pi$
$464$	19.7245i	0.0425097i
$465$	0	0
$466$	218.263	0.468376
$467$	−147.004	−0.314783	−0.157392	−	0.987536i	$-0.550308\pi$
$467$	−147.004	−0.314783	−0.157392	+	0.987536i	$0.550308\pi$
$468$	0	0
$469$	71.0000	0.151386
$470$	0	0
$471$	0	0
$472$	− 24.0000i	− 0.0508475i
$473$	−1018.94	−2.15421
$474$	0	0
$475$	0	0
$476$	− 16.9706i	− 0.0356524i
$477$	0	0
$478$	− 67.2897i	− 0.140773i
$479$	− 472.999i	− 0.987471i	−0.869612	−	0.493735i	$-0.835631\pi$
$479$	− 472.999i	− 0.987471i	0.869612	−	0.493735i	$-0.164369\pi$
$480$	0	0
$481$	195.263	0.405953
$482$	−623.370	−1.29330
$483$	0	0
$484$	−118.000	−0.243802
$485$	0	0
$486$	0	0
$487$	− 502.105i	− 1.03102i	−0.856885	−	0.515508i	$-0.827603\pi$
$487$	− 502.105i	− 1.03102i	0.856885	−	0.515508i	$-0.172397\pi$
$488$	143.952	0.294983
$489$	0	0
$490$	0	0
$491$	677.575i	1.37999i	0.723814	+	0.689995i	$0.242387\pi$
$491$	677.575i	1.37999i	−0.723814	+	0.689995i	$0.757613\pi$
$492$	0	0
$493$	41.8420i	0.0848722i
$494$	513.285i	1.03904i
$495$	0	0
$496$	−143.789	−0.289898
$497$	−126.479	−0.254485
$498$	0	0
$499$	226.000	0.452906	0.226453	−	0.974022i	$-0.427287\pi$
$499$	226.000	0.452906	0.226453	+	0.974022i	$0.427287\pi$
$500$	0	0
$501$	0	0
$502$	484.710i	0.965558i
$503$	361.089	0.717872	0.358936	−	0.933362i	$-0.383140\pi$
$503$	361.089	0.717872	0.358936	+	0.933362i	$0.383140\pi$
$504$	0	0
$505$	0	0
$506$	− 67.4353i	− 0.133271i
$507$	0	0
$508$	− 68.0000i	− 0.133858i
$509$	− 103.070i	− 0.202496i	−0.994861	−	0.101248i	$-0.967716\pi$
$509$	− 103.070i	− 0.202496i	0.994861	−	0.101248i	$-0.0322836\pi$
$510$	0	0
$511$	−22.0263	−0.0431044
$512$	22.6274	0.0441942
$513$	0	0
$514$	437.526	0.851218
$515$	0	0
$516$	0	0
$517$	− 929.210i	− 1.79731i
$518$	−19.7617	−0.0381501
$519$	0	0
$520$	0	0
$521$	− 280.144i	− 0.537705i	−0.963181	−	0.268852i	$-0.913356\pi$
$521$	− 280.144i	− 0.537705i	0.963181	−	0.268852i	$-0.0866444\pi$
$522$	0	0
$523$	− 668.421i	− 1.27805i	−0.769186	−	0.639025i	$-0.779338\pi$
$523$	− 668.421i	− 1.27805i	0.769186	−	0.639025i	$-0.220662\pi$
$524$	− 198.139i	− 0.378128i
$525$	0	0
$526$	−507.895	−0.965579
$527$	−305.023	−0.578792
$528$	0	0
$529$	−516.368	−0.976121
$530$	0	0
$531$	0	0
$532$	− 51.9473i	− 0.0976454i
$533$	405.376	0.760555
$534$	0	0
$535$	0	0
$536$	200.818i	0.374661i
$537$	0	0
$538$	364.211i	0.676972i
$539$	− 643.988i	− 1.19478i
$540$	0	0
$541$	−157.000	−0.290203	−0.145102	−	0.989417i	$-0.546351\pi$
$541$	−157.000	−0.290203	−0.145102	+	0.989417i	$0.546351\pi$
$542$	251.320	0.463691
$543$	0	0
$544$	48.0000	0.0882353
$545$	0	0
$546$	0	0
$547$	396.947i	0.725681i	0.931851	+	0.362840i	$0.118193\pi$
$547$	396.947i	0.725681i	−0.931851	+	0.362840i	$0.881807\pi$
$548$	−105.731	−0.192939
$549$	0	0
$550$	0	0
$551$	128.079i	0.232449i
$552$	0	0
$553$	75.8683i	0.137194i
$554$	− 440.713i	− 0.795511i
$555$	0	0
$556$	351.631	0.632430
$557$	−235.764	−0.423275	−0.211637	−	0.977348i	$-0.567880\pi$
$557$	−235.764	−0.423275	−0.211637	+	0.977348i	$0.567880\pi$
$558$	0	0
$559$	1061.26	1.89850
$560$	0	0
$561$	0	0
$562$	− 78.1580i	− 0.139071i
$563$	−202.046	−0.358874	−0.179437	−	0.983769i	$-0.557428\pi$
$563$	−202.046	−0.358874	−0.179437	+	0.983769i	$0.557428\pi$
$564$	0	0
$565$	0	0
$566$	− 220.245i	− 0.389125i
$567$	0	0
$568$	− 357.737i	− 0.629818i
$569$	192.408i	0.338150i	0.985603	+	0.169075i	$0.0540781\pi$
$569$	192.408i	0.338150i	−0.985603	+	0.169075i	$0.945922\pi$
$570$	0	0
$571$	206.553	0.361739	0.180870	−	0.983507i	$-0.442109\pi$
$571$	206.553	0.361739	0.180870	+	0.983507i	$0.442109\pi$
$572$	374.953	0.655512
$573$	0	0
$574$	−41.0263	−0.0714744
$575$	0	0
$576$	0	0
$577$	263.658i	0.456946i	0.973550	+	0.228473i	$0.0733732\pi$
$577$	263.658i	0.456946i	−0.973550	+	0.228473i	$0.926627\pi$
$578$	−306.884	−0.530942
$579$	0	0
$580$	0	0
$581$	34.5179i	0.0594112i
$582$	0	0
$583$	− 1204.58i	− 2.06617i
$584$	− 62.2999i	− 0.106678i
$585$	0	0
$586$	−478.868	−0.817181
$587$	293.654	0.500263	0.250131	−	0.968212i	$-0.419526\pi$
$587$	293.654	0.500263	0.250131	+	0.968212i	$0.419526\pi$
$588$	0	0
$589$	−933.684	−1.58520
$590$	0	0
$591$	0	0
$592$	− 55.8947i	− 0.0944167i
$593$	139.095	0.234562	0.117281	−	0.993099i	$-0.462582\pi$
$593$	139.095	0.234562	0.117281	+	0.993099i	$0.462582\pi$
$594$	0	0
$595$	0	0
$596$	− 477.483i	− 0.801146i
$597$	0	0
$598$	70.2363i	0.117452i
$599$	− 1069.98i	− 1.78628i	−0.449778	−	0.893140i	$-0.648497\pi$
$599$	− 1069.98i	− 1.78628i	0.449778	−	0.893140i	$-0.351503\pi$
$600$	0	0
$601$	75.9473	0.126368	0.0631841	−	0.998002i	$-0.479874\pi$
$601$	75.9473	0.126368	0.0631841	+	0.998002i	$0.479874\pi$
$602$	−107.406	−0.178415
$603$	0	0
$604$	−419.737	−0.694928
$605$	0	0
$606$	0	0
$607$	− 383.053i	− 0.631059i	−0.948916	−	0.315529i	$-0.897818\pi$
$607$	− 383.053i	− 0.631059i	0.948916	−	0.315529i	$-0.102182\pi$
$608$	146.929	0.241660
$609$	0	0
$610$	0	0
$611$	967.805i	1.58397i
$612$	0	0
$613$	817.658i	1.33386i	0.745119	+	0.666931i	$0.232393\pi$
$613$	817.658i	1.33386i	−0.745119	+	0.666931i	$0.767607\pi$
$614$	324.822i	0.529026i
$615$	0	0
$616$	−37.9473	−0.0616028
$617$	525.994	0.852502	0.426251	−	0.904605i	$-0.359834\pi$
$617$	525.994	0.852502	0.426251	+	0.904605i	$0.359834\pi$
$618$	0	0
$619$	600.026	0.969348	0.484674	−	0.874695i	$-0.338938\pi$
$619$	600.026	0.969348	0.484674	+	0.874695i	$0.338938\pi$
$620$	0	0
$621$	0	0
$622$	144.316i	0.232019i
$623$	147.004	0.235961
$624$	0	0
$625$	0	0
$626$	− 474.692i	− 0.758293i
$627$	0	0
$628$	308.211i	0.490781i
$629$	− 118.570i	− 0.188506i
$630$	0	0
$631$	1019.34	1.61544	0.807719	−	0.589567i	$-0.200702\pi$
$631$	1019.34	1.61544	0.807719	+	0.589567i	$0.200702\pi$
$632$	−214.588	−0.339538
$633$	0	0
$634$	−609.421	−0.961231
$635$	0	0
$636$	0	0
$637$	670.736i	1.05296i
$638$	93.5615	0.146648
$639$	0	0
$640$	0	0
$641$	865.052i	1.34953i	0.738030	+	0.674767i	$0.235756\pi$
$641$	865.052i	1.34953i	−0.738030	+	0.674767i	$0.764244\pi$
$642$	0	0
$643$	609.368i	0.947695i	0.880607	+	0.473848i	$0.157135\pi$
$643$	609.368i	0.947695i	−0.880607	+	0.473848i	$0.842865\pi$
$644$	− 7.10831i	− 0.0110377i
$645$	0	0
$646$	311.684	0.482483
$647$	−545.942	−0.843805	−0.421902	−	0.906641i	$-0.638638\pi$
$647$	−545.942	−0.843805	−0.421902	+	0.906641i	$0.638638\pi$
$648$	0	0
$649$	−113.842	−0.175411
$650$	0	0
$651$	0	0
$652$	29.8947i	0.0458507i
$653$	239.188	0.366291	0.183146	−	0.983086i	$-0.441372\pi$
$653$	239.188	0.366291	0.183146	+	0.983086i	$0.441372\pi$
$654$	0	0
$655$	0	0
$656$	− 116.040i	− 0.176890i
$657$	0	0
$658$	− 97.9473i	− 0.148856i
$659$	− 11.4627i	− 0.0173940i	−0.999962	−	0.00869702i	$-0.997232\pi$
$659$	− 11.4627i	− 0.0173940i	0.999962	−	0.00869702i	$-0.00276838\pi$
$660$	0	0
$661$	210.473	0.318417	0.159208	−	0.987245i	$-0.449106\pi$
$661$	210.473	0.318417	0.159208	+	0.987245i	$0.449106\pi$
$662$	−257.350	−0.388746
$663$	0	0
$664$	−97.6313	−0.147035
$665$	0	0
$666$	0	0
$667$	17.5260i	0.0262758i
$668$	−276.590	−0.414057
$669$	0	0
$670$	0	0
$671$	− 682.824i	− 1.01762i
$672$	0	0
$673$	− 316.184i	− 0.469813i	−0.972018	−	0.234907i	$-0.924522\pi$
$673$	− 316.184i	− 0.469813i	0.972018	−	0.234907i	$-0.0754784\pi$
$674$	499.924i	0.741727i
$675$	0	0
$676$	−52.5267	−0.0777022
$677$	77.9680	0.115167	0.0575834	−	0.998341i	$-0.481660\pi$
$677$	77.9680	0.115167	0.0575834	+	0.998341i	$0.481660\pi$
$678$	0	0
$679$	99.8683	0.147081
$680$	0	0
$681$	0	0
$682$	682.053i	1.00008i
$683$	718.401	1.05183	0.525916	−	0.850536i	$-0.323723\pi$
$683$	718.401	1.05183	0.525916	+	0.850536i	$0.323723\pi$
$684$	0	0
$685$	0	0
$686$	− 137.179i	− 0.199969i
$687$	0	0
$688$	− 303.789i	− 0.441554i
$689$	1254.61i	1.82092i
$690$	0	0
$691$	930.578	1.34671	0.673356	−	0.739318i	$-0.264852\pi$
$691$	930.578	1.34671	0.673356	+	0.739318i	$0.264852\pi$
$692$	395.384	0.571364
$693$	0	0
$694$	−174.974	−0.252123
$695$	0	0
$696$	0	0
$697$	− 246.158i	− 0.353168i
$698$	−94.9013	−0.135962
$699$	0	0
$700$	0	0
$701$	− 24.1725i	− 0.0344828i	−0.999851	−	0.0172414i	$-0.994512\pi$
$701$	− 24.1725i	− 0.0344828i	0.999851	−	0.0172414i	$-0.00548839\pi$
$702$	0	0
$703$	− 362.947i	− 0.516284i
$704$	− 107.331i	− 0.152459i
$705$	0	0
$706$	−941.526	−1.33361
$707$	−4.35437	−0.00615893
$708$	0	0
$709$	−930.789	−1.31282	−0.656410	−	0.754404i	$-0.727926\pi$
$709$	−930.789	−1.31282	−0.656410	+	0.754404i	$0.727926\pi$
$710$	0	0
$711$	0	0
$712$	415.789i	0.583974i
$713$	−127.762	−0.179190
$714$	0	0
$715$	0	0
$716$	485.038i	0.677427i
$717$	0	0
$718$	615.079i	0.856656i
$719$	403.963i	0.561840i	0.959731	+	0.280920i	$0.0906395\pi$
$719$	403.963i	0.561840i	−0.959731	+	0.280920i	$0.909360\pi$
$720$	0	0
$721$	−12.9473	−0.0179575
$722$	443.542	0.614324
$723$	0	0
$724$	−617.579	−0.853009
$725$	0	0
$726$	0	0
$727$	− 490.000i	− 0.674003i	−0.941504	−	0.337001i	$-0.890587\pi$
$727$	− 490.000i	− 0.674003i	0.941504	−	0.337001i	$-0.109413\pi$
$728$	39.5235	0.0542905
$729$	0	0
$730$	0	0
$731$	− 644.434i	− 0.881579i
$732$	0	0
$733$	1012.16i	1.38084i	0.723407	+	0.690422i	$0.242575\pi$
$733$	1012.16i	1.38084i	−0.723407	+	0.690422i	$0.757425\pi$
$734$	405.432i	0.552360i
$735$	0	0
$736$	20.1053	0.0273170
$737$	952.565	1.29249
$738$	0	0
$739$	−499.210	−0.675521	−0.337760	−	0.941232i	$-0.609669\pi$
$739$	−499.210	−0.675521	−0.337760	+	0.941232i	$0.609669\pi$
$740$	0	0
$741$	0	0
$742$	− 126.974i	− 0.171124i
$743$	748.788	1.00779	0.503895	−	0.863765i	$-0.331900\pi$
$743$	748.788	1.00779	0.503895	+	0.863765i	$0.331900\pi$
$744$	0	0
$745$	0	0
$746$	− 5.47064i	− 0.00733330i
$747$	0	0
$748$	− 227.684i	− 0.304390i
$749$	− 103.777i	− 0.138554i
$750$	0	0
$751$	777.710	1.03557	0.517783	−	0.855512i	$-0.326757\pi$
$751$	777.710	1.03557	0.517783	+	0.855512i	$0.326757\pi$
$752$	277.037	0.368400
$753$	0	0
$754$	−97.4477	−0.129241
$755$	0	0
$756$	0	0
$757$	1145.18i	1.51279i	0.654114	+	0.756396i	$0.273042\pi$
$757$	1145.18i	1.51279i	−0.654114	+	0.756396i	$0.726958\pi$
$758$	161.481	0.213036
$759$	0	0
$760$	0	0
$761$	− 1042.93i	− 1.37047i	−0.728323	−	0.685234i	$-0.759700\pi$
$761$	− 1042.93i	− 1.37047i	0.728323	−	0.685234i	$-0.240300\pi$
$762$	0	0
$763$	99.9473i	0.130993i
$764$	− 616.001i	− 0.806284i
$765$	0	0
$766$	−472.211	−0.616463
$767$	118.570	0.154590
$768$	0	0
$769$	−991.105	−1.28882	−0.644412	−	0.764679i	$-0.722898\pi$
$769$	−991.105	−1.28882	−0.644412	+	0.764679i	$0.722898\pi$
$770$	0	0
$771$	0	0
$772$	683.737i	0.885669i
$773$	389.746	0.504199	0.252100	−	0.967701i	$-0.418879\pi$
$773$	389.746	0.504199	0.252100	+	0.967701i	$0.418879\pi$
$774$	0	0
$775$	0	0
$776$	282.470i	0.364008i
$777$	0	0
$778$	− 965.210i	− 1.24063i
$779$	− 753.496i	− 0.967261i
$780$	0	0
$781$	−1696.89	−2.17272
$782$	42.6499	0.0545395
$783$	0	0
$784$	192.000	0.244898
$785$	0	0
$786$	0	0
$787$	− 206.631i	− 0.262556i	−0.991346	−	0.131278i	$-0.958092\pi$
$787$	− 206.631i	− 0.262556i	0.991346	−	0.131278i	$-0.0419080\pi$
$788$	−256.865	−0.325971
$789$	0	0
$790$	0	0
$791$	− 205.600i	− 0.259925i
$792$	0	0
$793$	711.185i	0.896829i
$794$	365.165i	0.459906i
$795$	0	0
$796$	−356.053	−0.447302
$797$	−207.107	−0.259859	−0.129929	−	0.991523i	$-0.541475\pi$
$797$	−207.107	−0.259859	−0.129929	+	0.991523i	$0.541475\pi$
$798$	0	0
$799$	587.684	0.735524
$800$	0	0
$801$	0	0
$802$	− 549.920i	− 0.685686i
$803$	−295.514	−0.368013
$804$	0	0
$805$	0	0
$806$	− 710.382i	− 0.881367i
$807$	0	0
$808$	− 12.3160i	− 0.0152426i
$809$	− 663.452i	− 0.820089i	−0.912065	−	0.410045i	$-0.865513\pi$
$809$	− 663.452i	− 0.820089i	0.912065	−	0.410045i	$-0.134487\pi$
$810$	0	0
$811$	−394.683	−0.486663	−0.243331	−	0.969943i	$-0.578240\pi$
$811$	−394.683	−0.486663	−0.243331	+	0.969943i	$0.578240\pi$
$812$	9.86225	0.0121456
$813$	0	0
$814$	−265.132	−0.325715
$815$	0	0
$816$	0	0
$817$	− 1972.63i	− 2.41448i
$818$	427.538	0.522663
$819$	0	0
$820$	0	0
$821$	− 1156.53i	− 1.40868i	−0.709861	−	0.704342i	$-0.751242\pi$
$821$	− 1156.53i	− 1.40868i	0.709861	−	0.704342i	$-0.248758\pi$
$822$	0	0
$823$	− 930.789i	− 1.13097i	−0.824758	−	0.565486i	$-0.808689\pi$
$823$	− 930.789i	− 1.13097i	0.824758	−	0.565486i	$-0.191311\pi$
$824$	− 36.6206i	− 0.0444425i
$825$	0	0
$826$	−12.0000	−0.0145278
$827$	−1012.32	−1.22408	−0.612041	−	0.790826i	$-0.709651\pi$
$827$	−1012.32	−1.22408	−0.612041	+	0.790826i	$0.709651\pi$
$828$	0	0
$829$	514.105	0.620150	0.310075	−	0.950712i	$-0.399646\pi$
$829$	514.105	0.620150	0.310075	+	0.950712i	$0.399646\pi$
$830$	0	0
$831$	0	0
$832$	111.789i	0.134362i
$833$	407.294	0.488948
$834$	0	0
$835$	0	0
$836$	− 696.947i	− 0.833668i
$837$	0	0
$838$	373.631i	0.445861i
$839$	549.013i	0.654366i	0.944961	+	0.327183i	$0.106099\pi$
$839$	549.013i	0.654366i	−0.944961	+	0.327183i	$0.893901\pi$
$840$	0	0
$841$	816.684	0.971087
$842$	−185.783	−0.220645
$843$	0	0
$844$	139.421	0.165190
$845$	0	0
$846$	0	0
$847$	59.0000i	0.0696576i
$848$	359.136	0.423509
$849$	0	0
$850$	0	0
$851$	− 49.6646i	− 0.0583603i
$852$	0	0
$853$	477.131i	0.559356i	0.960094	+	0.279678i	$0.0902278\pi$
$853$	477.131i	0.559356i	−0.960094	+	0.279678i	$0.909772\pi$
$854$	− 71.9759i	− 0.0842809i
$855$	0	0
$856$	293.526	0.342904
$857$	518.402	0.604904	0.302452	−	0.953165i	$-0.402195\pi$
$857$	518.402	0.604904	0.302452	+	0.953165i	$0.402195\pi$
$858$	0	0
$859$	224.448	0.261289	0.130645	−	0.991429i	$-0.458295\pi$
$859$	224.448	0.261289	0.130645	+	0.991429i	$0.458295\pi$
$860$	0	0
$861$	0	0
$862$	261.237i	0.303059i
$863$	−765.759	−0.887322	−0.443661	−	0.896195i	$-0.646321\pi$
$863$	−765.759	−0.887322	−0.443661	+	0.896195i	$0.646321\pi$
$864$	0	0
$865$	0	0
$866$	1121.84i	1.29543i
$867$	0	0
$868$	71.8947i	0.0828280i
$869$	1017.88i	1.17132i
$870$	0	0
$871$	−992.130	−1.13907
$872$	−282.694	−0.324190
$873$	0	0
$874$	130.552	0.149373
$875$	0	0
$876$	0	0
$877$	− 844.237i	− 0.962642i	−0.876544	−	0.481321i	$-0.840157\pi$
$877$	− 844.237i	− 0.962642i	0.876544	−	0.481321i	$-0.159843\pi$
$878$	845.700	0.963212
$879$	0	0
$880$	0	0
$881$	− 1528.41i	− 1.73486i	−0.497560	−	0.867429i	$-0.665771\pi$
$881$	− 1528.41i	− 1.73486i	0.497560	−	0.867429i	$-0.334229\pi$
$882$	0	0
$883$	795.631i	0.901054i	0.892763	+	0.450527i	$0.148764\pi$
$883$	795.631i	0.901054i	−0.892763	+	0.450527i	$0.851236\pi$
$884$	237.141i	0.268259i
$885$	0	0
$886$	−1047.58	−1.18237
$887$	−249.051	−0.280779	−0.140389	−	0.990096i	$-0.544835\pi$
$887$	−249.051	−0.280779	−0.140389	+	0.990096i	$0.544835\pi$
$888$	0	0
$889$	−34.0000	−0.0382452
$890$	0	0
$891$	0	0
$892$	263.263i	0.295137i
$893$	1798.92	2.01446
$894$	0	0
$895$	0	0
$896$	− 11.3137i	− 0.0126269i
$897$	0	0
$898$	1247.05i	1.38870i
$899$	− 177.261i	− 0.197176i
$900$	0	0
$901$	761.842	0.845552
$902$	−550.426	−0.610228
$903$	0	0
$904$	581.526	0.643281
$905$	0	0
$906$	0	0
$907$	− 951.631i	− 1.04921i	−0.851347	−	0.524603i	$-0.824214\pi$
$907$	− 951.631i	− 1.04921i	0.851347	−	0.524603i	$-0.175786\pi$
$908$	687.791	0.757479
$909$	0	0
$910$	0	0
$911$	357.535i	0.392465i	0.980557	+	0.196232i	$0.0628707\pi$
$911$	357.535i	0.392465i	−0.980557	+	0.196232i	$0.937129\pi$
$912$	0	0
$913$	463.106i	0.507236i
$914$	− 336.583i	− 0.368253i
$915$	0	0
$916$	−827.157	−0.903010
$917$	−99.0694	−0.108036
$918$	0	0
$919$	−313.368	−0.340988	−0.170494	−	0.985359i	$-0.554536\pi$
$919$	−313.368	−0.340988	−0.170494	+	0.985359i	$0.554536\pi$
$920$	0	0
$921$	0	0
$922$	655.473i	0.710926i
$923$	1767.38	1.91482
$924$	0	0
$925$	0	0
$926$	252.772i	0.272972i
$927$	0	0
$928$	27.8947i	0.0300589i
$929$	709.693i	0.763932i	0.924176	+	0.381966i	$0.124753\pi$
$929$	709.693i	0.763932i	−0.924176	+	0.381966i	$0.875247\pi$
$930$	0	0
$931$	1246.74	1.33914
$932$	308.671	0.331192
$933$	0	0
$934$	−207.895	−0.222585
$935$	0	0
$936$	0	0
$937$	− 1351.87i	− 1.44276i	−0.692539	−	0.721380i	$-0.743508\pi$
$937$	− 1351.87i	− 1.44276i	0.692539	−	0.721380i	$-0.256492\pi$
$938$	100.409	0.107046
$939$	0	0
$940$	0	0
$941$	16.8770i	0.0179351i	0.999960	+	0.00896757i	$0.00285450\pi$
$941$	16.8770i	0.0179351i	−0.999960	+	0.00896757i	$0.997145\pi$
$942$	0	0
$943$	− 103.106i	− 0.109338i
$944$	− 33.9411i	− 0.0359546i
$945$	0	0
$946$	−1441.00	−1.52326
$947$	2.08359	0.00220020	0.00110010	−	0.999999i	$-0.499650\pi$
$947$	2.08359	0.00220020	0.00110010	+	0.999999i	$0.499650\pi$
$948$	0	0
$949$	307.789	0.324329
$950$	0	0
$951$	0	0
$952$	− 24.0000i	− 0.0252101i
$953$	−1427.87	−1.49829	−0.749145	−	0.662406i	$-0.769535\pi$
$953$	−1427.87	−1.49829	−0.749145	+	0.662406i	$0.769535\pi$
$954$	0	0
$955$	0	0
$956$	− 95.1620i	− 0.0995418i
$957$	0	0
$958$	− 668.921i	− 0.698247i
$959$	52.8654i	0.0551256i
$960$	0	0
$961$	331.211	0.344652
$962$	276.144	0.287052
$963$	0	0
$964$	−881.579	−0.914501
$965$	0	0
$966$	0	0
$967$	1097.84i	1.13531i	0.823268	+	0.567653i	$0.192149\pi$
$967$	1097.84i	1.13531i	−0.823268	+	0.567653i	$0.807851\pi$
$968$	−166.877	−0.172394
$969$	0	0
$970$	0	0
$971$	− 1709.90i	− 1.76096i	−0.474080	−	0.880482i	$-0.657219\pi$
$971$	− 1709.90i	− 1.76096i	0.474080	−	0.880482i	$-0.342781\pi$
$972$	0	0
$973$	− 175.816i	− 0.180694i
$974$	− 710.083i	− 0.729038i
$975$	0	0
$976$	203.579	0.208585
$977$	231.056	0.236496	0.118248	−	0.992984i	$-0.462272\pi$
$977$	231.056	0.236496	0.118248	+	0.992984i	$0.462272\pi$
$978$	0	0
$979$	1972.26	2.01457
$980$	0	0
$981$	0	0
$982$	958.236i	0.975801i
$983$	1362.35	1.38591	0.692957	−	0.720979i	$-0.256308\pi$
$983$	1362.35	1.38591	0.692957	+	0.720979i	$0.256308\pi$
$984$	0	0
$985$	0	0
$986$	59.1735i	0.0600137i
$987$	0	0
$988$	725.895i	0.734711i
$989$	− 269.929i	− 0.272931i
$990$	0	0
$991$	1168.55	1.17916	0.589582	−	0.807708i	$-0.299292\pi$
$991$	1168.55	1.17916	0.589582	+	0.807708i	$0.299292\pi$
$992$	−203.349	−0.204989
$993$	0	0
$994$	−178.868	−0.179948
$995$	0	0
$996$	0	0
$997$	− 291.526i	− 0.292403i	−0.989255	−	0.146202i	$-0.953295\pi$
$997$	− 291.526i	− 0.292403i	0.989255	−	0.146202i	$-0.0467048\pi$
$998$	319.612	0.320253
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1350.3.b.h.1349.5		8
3.2	odd	2	inner	1350.3.b.h.1349.2		8
5.2	odd	4		1350.3.d.m.701.3		4
5.3	odd	4		270.3.d.a.161.1	✓	4
5.4	even	2	inner	1350.3.b.h.1349.3		8
15.2	even	4		1350.3.d.m.701.2		4
15.8	even	4		270.3.d.a.161.4	yes	4
15.14	odd	2	inner	1350.3.b.h.1349.8		8
20.3	even	4		2160.3.l.f.161.1		4
45.13	odd	12		810.3.h.b.431.3		8
45.23	even	12		810.3.h.b.431.2		8
45.38	even	12		810.3.h.b.701.3		8
45.43	odd	12		810.3.h.b.701.2		8
60.23	odd	4		2160.3.l.f.161.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.3.d.a.161.1	✓	4	5.3	odd	4
270.3.d.a.161.4	yes	4	15.8	even	4
810.3.h.b.431.2		8	45.23	even	12
810.3.h.b.431.3		8	45.13	odd	12
810.3.h.b.701.2		8	45.43	odd	12
810.3.h.b.701.3		8	45.38	even	12
1350.3.b.h.1349.2		8	3.2	odd	2	inner
1350.3.b.h.1349.3		8	5.4	even	2	inner
1350.3.b.h.1349.5		8	1.1	even	1	trivial
1350.3.b.h.1349.8		8	15.14	odd	2	inner
1350.3.d.m.701.2		4	15.2	even	4
1350.3.d.m.701.3		4	5.2	odd	4
2160.3.l.f.161.1		4	20.3	even	4
2160.3.l.f.161.3		4	60.23	odd	4

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:	\( N \)	\(=\)	\( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1350.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} +2.00000 q^{4} -1.00000i q^{7} +2.82843 q^{8} -13.4164i q^{11} +13.9737i q^{13} -1.41421i q^{14} +4.00000 q^{16} +8.48528 q^{17} +25.9737 q^{19} -18.9737i q^{22} +3.55415 q^{23} +19.7617i q^{26} -2.00000i q^{28} +4.93113i q^{29} -35.9473 q^{31} +5.65685 q^{32} +12.0000 q^{34} -13.9737i q^{37} +36.7323 q^{38} -29.0100i q^{41} -75.9473i q^{43} -26.8328i q^{44} +5.02633 q^{46} +69.2592 q^{47} +48.0000 q^{49} +27.9473i q^{52} +89.7839 q^{53} -2.82843i q^{56} +6.97367i q^{58} -8.48528i q^{59} +50.8947 q^{61} -50.8372 q^{62} +8.00000 q^{64} +71.0000i q^{67} +16.9706 q^{68} -126.479i q^{71} -22.0263i q^{73} -19.7617i q^{74} +51.9473 q^{76} -13.4164 q^{77} -75.8683 q^{79} -41.0263i q^{82} -34.5179 q^{83} -107.406i q^{86} -37.9473i q^{88} +147.004i q^{89} +13.9737 q^{91} +7.10831 q^{92} +97.9473 q^{94} +99.8683i q^{97} +67.8823 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{4} + 32 q^{16} + 56 q^{19} + 16 q^{31} + 96 q^{34} + 192 q^{46} + 384 q^{49} - 200 q^{61} + 64 q^{64} + 112 q^{76} + 152 q^{79} - 40 q^{91} + 480 q^{94}+O(q^{100}) \) 8 * q + 16 * q^4 + 32 * q^16 + 56 * q^19 + 16 * q^31 + 96 * q^34 + 192 * q^46 + 384 * q^49 - 200 * q^61 + 64 * q^64 + 112 * q^76 + 152 * q^79 - 40 * q^91 + 480 * q^94

Introduction

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