LMFDB - Embedded newform 270.3.d.a.161.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [270,3,Mod(161,270)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(270, 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("270.161");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 270 = 2 \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	270.d (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:	$7.35696713773$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 9 $ x^4 - 4*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.1
Root			$-1.58114 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	270.161
Dual form			270.3.d.a.161.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.41421i q^{2} -2.00000 q^{4} -2.23607i q^{5} -1.00000 q^{7} +2.82843i q^{8} -3.16228 q^{10} -13.4164i q^{11} -13.9737 q^{13} +1.41421i q^{14} +4.00000 q^{16} -8.48528i q^{17} -25.9737 q^{19} +4.47214i q^{20} -18.9737 q^{22} +3.55415i q^{23} -5.00000 q^{25} +19.7617i q^{26} +2.00000 q^{28} -4.93113i q^{29} -35.9473 q^{31} -5.65685i q^{32} -12.0000 q^{34} +2.23607i q^{35} -13.9737 q^{37} +36.7323i q^{38} +6.32456 q^{40} -29.0100i q^{41} +75.9473 q^{43} +26.8328i q^{44} +5.02633 q^{46} -69.2592i q^{47} -48.0000 q^{49} +7.07107i q^{50} +27.9473 q^{52} +89.7839i q^{53} -30.0000 q^{55} -2.82843i q^{56} -6.97367 q^{58} +8.48528i q^{59} +50.8947 q^{61} +50.8372i q^{62} -8.00000 q^{64} +31.2461i q^{65} +71.0000 q^{67} +16.9706i q^{68} +3.16228 q^{70} -126.479i q^{71} +22.0263 q^{73} +19.7617i q^{74} +51.9473 q^{76} +13.4164i q^{77} +75.8683 q^{79} -8.94427i q^{80} -41.0263 q^{82} -34.5179i q^{83} -18.9737 q^{85} -107.406i q^{86} +37.9473 q^{88} -147.004i q^{89} +13.9737 q^{91} -7.10831i q^{92} -97.9473 q^{94} +58.0789i q^{95} +99.8683 q^{97} +67.8823i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{4} - 4 q^{7} + 20 q^{13} + 16 q^{16} - 28 q^{19} - 20 q^{25} + 8 q^{28} + 8 q^{31} - 48 q^{34} + 20 q^{37} + 152 q^{43} + 96 q^{46} - 192 q^{49} - 40 q^{52} - 120 q^{55} + 48 q^{58} - 100 q^{61}+ \cdots + 20 q^{97}+O(q^{100}) $ 4 * q - 8 * q^4 - 4 * q^7 + 20 * q^13 + 16 * q^16 - 28 * q^19 - 20 * q^25 + 8 * q^28 + 8 * q^31 - 48 * q^34 + 20 * q^37 + 152 * q^43 + 96 * q^46 - 192 * q^49 - 40 * q^52 - 120 * q^55 + 48 * q^58 - 100 * q^61 - 32 * q^64 + 284 * q^67 + 164 * q^73 + 56 * q^76 - 76 * q^79 - 240 * q^82 - 20 * q^91 - 240 * q^94 + 20 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/270\mathbb{Z}\right)^\times$.

$n$	$191$	$217$
$\chi(n)$	$-1$	$1$

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
$5$	− 2.23607i	− 0.447214i
$5$	− 2.23607i	− 0.447214i
$6$	0	0
$7$	−1.00000	−0.142857	−0.0714286	−	0.997446i	$-0.522756\pi$
$7$	−1.00000	−0.142857	−0.0714286	+	0.997446i	$0.522756\pi$
$8$	2.82843i	0.353553i
$9$	0	0
$10$	−3.16228	−0.316228
$11$	− 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
$13$	−13.9737	−1.07490	−0.537449	−	0.843296i	$-0.680612\pi$
$13$	−13.9737	−1.07490	−0.537449	+	0.843296i	$0.680612\pi$
$14$	1.41421i	0.101015i
$15$	0	0
$16$	4.00000	0.250000
$17$	− 8.48528i	− 0.499134i	−0.968358	−	0.249567i	$-0.919712\pi$
$17$	− 8.48528i	− 0.499134i	0.968358	−	0.249567i	$-0.0802883\pi$
$18$	0	0
$19$	−25.9737	−1.36704	−0.683518	−	0.729934i	$-0.739551\pi$
$19$	−25.9737	−1.36704	−0.683518	+	0.729934i	$0.739551\pi$
$20$	4.47214i	0.223607i
$21$	0	0
$22$	−18.9737	−0.862439
$23$	3.55415i	0.154528i	0.997011	+	0.0772642i	$0.0246185\pi$
$23$	3.55415i	0.154528i	−0.997011	+	0.0772642i	$0.975381\pi$
$24$	0	0
$25$	−5.00000	−0.200000
$26$	19.7617i	0.760067i
$27$	0	0
$28$	2.00000	0.0714286
$29$	− 4.93113i	− 0.170039i	−0.996379	−	0.0850194i	$-0.972905\pi$
$29$	− 4.93113i	− 0.170039i	0.996379	−	0.0850194i	$-0.0270952\pi$
$30$	0	0
$31$	−35.9473	−1.15959	−0.579796	−	0.814762i	$-0.696868\pi$
$31$	−35.9473	−1.15959	−0.579796	+	0.814762i	$0.696868\pi$
$32$	− 5.65685i	− 0.176777i
$33$	0	0
$34$	−12.0000	−0.352941
$35$	2.23607i	0.0638877i
$36$	0	0
$37$	−13.9737	−0.377667	−0.188833	−	0.982009i	$-0.560471\pi$
$37$	−13.9737	−0.377667	−0.188833	+	0.982009i	$0.560471\pi$
$38$	36.7323i	0.966640i
$39$	0	0
$40$	6.32456	0.158114
$41$	− 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
$43$	75.9473	1.76622	0.883109	−	0.469169i	$-0.155446\pi$
$43$	75.9473	1.76622	0.883109	+	0.469169i	$0.155446\pi$
$44$	26.8328i	0.609837i
$45$	0	0
$46$	5.02633	0.109268
$47$	− 69.2592i	− 1.47360i	−0.676110	−	0.736800i	$-0.736336\pi$
$47$	− 69.2592i	− 1.47360i	0.676110	−	0.736800i	$-0.263664\pi$
$48$	0	0
$49$	−48.0000	−0.979592
$50$	7.07107i	0.141421i
$51$	0	0
$52$	27.9473	0.537449
$53$	89.7839i	1.69404i	0.531564	+	0.847018i	$0.321605\pi$
$53$	89.7839i	1.69404i	−0.531564	+	0.847018i	$0.678395\pi$
$54$	0	0
$55$	−30.0000	−0.545455
$56$	− 2.82843i	− 0.0505076i
$57$	0	0
$58$	−6.97367	−0.120236
$59$	8.48528i	0.143818i	0.997411	+	0.0719092i	$0.0229092\pi$
$59$	8.48528i	0.143818i	−0.997411	+	0.0719092i	$0.977091\pi$
$60$	0	0
$61$	50.8947	0.834339	0.417169	−	0.908829i	$-0.363022\pi$
$61$	50.8947	0.834339	0.417169	+	0.908829i	$0.363022\pi$
$62$	50.8372i	0.819955i
$63$	0	0
$64$	−8.00000	−0.125000
$65$	31.2461i	0.480709i
$66$	0	0
$67$	71.0000	1.05970	0.529851	−	0.848091i	$-0.322248\pi$
$67$	71.0000	1.05970	0.529851	+	0.848091i	$0.322248\pi$
$68$	16.9706i	0.249567i
$69$	0	0
$70$	3.16228	0.0451754
$71$	− 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
$73$	22.0263	0.301731	0.150865	−	0.988554i	$-0.451794\pi$
$73$	22.0263	0.301731	0.150865	+	0.988554i	$0.451794\pi$
$74$	19.7617i	0.267051i
$75$	0	0
$76$	51.9473	0.683518
$77$	13.4164i	0.174239i
$78$	0	0
$79$	75.8683	0.960359	0.480179	−	0.877170i	$-0.340572\pi$
$79$	75.8683	0.960359	0.480179	+	0.877170i	$0.340572\pi$
$80$	− 8.94427i	− 0.111803i
$81$	0	0
$82$	−41.0263	−0.500321
$83$	− 34.5179i	− 0.415878i	−0.978142	−	0.207939i	$-0.933324\pi$
$83$	− 34.5179i	− 0.415878i	0.978142	−	0.207939i	$-0.0666756\pi$
$84$	0	0
$85$	−18.9737	−0.223220
$86$	− 107.406i	− 1.24890i
$87$	0	0
$88$	37.9473	0.431220
$89$	− 147.004i	− 1.65173i	−0.563870	−	0.825864i	$-0.690688\pi$
$89$	− 147.004i	− 1.65173i	0.563870	−	0.825864i	$-0.309312\pi$
$90$	0	0
$91$	13.9737	0.153557
$92$	− 7.10831i	− 0.0772642i
$93$	0	0
$94$	−97.9473	−1.04199
$95$	58.0789i	0.611357i
$96$	0	0
$97$	99.8683	1.02957	0.514785	−	0.857319i	$-0.327872\pi$
$97$	99.8683	1.02957	0.514785	+	0.857319i	$0.327872\pi$
$98$	67.8823i	0.692676i
$99$	0	0
$100$	10.0000	0.100000
$101$	− 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
$103$	12.9473	0.125702	0.0628511	−	0.998023i	$-0.479981\pi$
$103$	12.9473	0.125702	0.0628511	+	0.998023i	$0.479981\pi$
$104$	− 39.5235i	− 0.380034i
$105$	0	0
$106$	126.974	1.19786
$107$	− 103.777i	− 0.969880i	−0.874548	−	0.484940i	$-0.838842\pi$
$107$	− 103.777i	− 0.969880i	0.874548	−	0.484940i	$-0.161158\pi$
$108$	0	0
$109$	99.9473	0.916948	0.458474	−	0.888708i	$-0.348396\pi$
$109$	99.9473	0.916948	0.458474	+	0.888708i	$0.348396\pi$
$110$	42.4264i	0.385695i
$111$	0	0
$112$	−4.00000	−0.0357143
$113$	205.600i	1.81947i	0.415186	+	0.909737i	$0.363717\pi$
$113$	205.600i	1.81947i	−0.415186	+	0.909737i	$0.636283\pi$
$114$	0	0
$115$	7.94733	0.0691072
$116$	9.86225i	0.0850194i
$117$	0	0
$118$	12.0000	0.101695
$119$	8.48528i	0.0713049i
$120$	0	0
$121$	−59.0000	−0.487603
$122$	− 71.9759i	− 0.589967i
$123$	0	0
$124$	71.8947	0.579796
$125$	11.1803i	0.0894427i
$126$	0	0
$127$	−34.0000	−0.267717	−0.133858	−	0.991000i	$-0.542737\pi$
$127$	−34.0000	−0.267717	−0.133858	+	0.991000i	$0.542737\pi$
$128$	11.3137i	0.0883883i
$129$	0	0
$130$	44.1886	0.339912
$131$	− 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
$133$	25.9737	0.195291
$134$	− 100.409i	− 0.749322i
$135$	0	0
$136$	24.0000	0.176471
$137$	52.8654i	0.385879i	0.981211	+	0.192939i	$0.0618021\pi$
$137$	52.8654i	0.385879i	−0.981211	+	0.192939i	$0.938198\pi$
$138$	0	0
$139$	−175.816	−1.26486	−0.632430	−	0.774617i	$-0.717943\pi$
$139$	−175.816	−1.26486	−0.632430	+	0.774617i	$0.717943\pi$
$140$	− 4.47214i	− 0.0319438i
$141$	0	0
$142$	−178.868	−1.25964
$143$	187.476i	1.31102i
$144$	0	0
$145$	−11.0263	−0.0760437
$146$	− 31.1499i	− 0.213356i
$147$	0	0
$148$	27.9473	0.188833
$149$	238.741i	1.60229i	0.598469	+	0.801146i	$0.295776\pi$
$149$	238.741i	1.60229i	−0.598469	+	0.801146i	$0.704224\pi$
$150$	0	0
$151$	−209.868	−1.38986	−0.694928	−	0.719079i	$-0.744564\pi$
$151$	−209.868	−1.38986	−0.694928	+	0.719079i	$0.744564\pi$
$152$	− 73.4646i	− 0.483320i
$153$	0	0
$154$	18.9737	0.123206
$155$	80.3807i	0.518585i
$156$	0	0
$157$	154.105	0.981563	0.490781	−	0.871283i	$-0.336711\pi$
$157$	154.105	0.981563	0.490781	+	0.871283i	$0.336711\pi$
$158$	− 107.294i	− 0.679076i
$159$	0	0
$160$	−12.6491	−0.0790569
$161$	− 3.55415i	− 0.0220755i
$162$	0	0
$163$	−14.9473	−0.0917014	−0.0458507	−	0.998948i	$-0.514600\pi$
$163$	−14.9473	−0.0917014	−0.0458507	+	0.998948i	$0.514600\pi$
$164$	58.0200i	0.353780i
$165$	0	0
$166$	−48.8157	−0.294070
$167$	138.295i	0.828114i	0.910251	+	0.414057i	$0.135889\pi$
$167$	138.295i	0.828114i	−0.910251	+	0.414057i	$0.864111\pi$
$168$	0	0
$169$	26.2633	0.155404
$170$	26.8328i	0.157840i
$171$	0	0
$172$	−151.895	−0.883109
$173$	197.692i	1.14273i	0.820697	+	0.571364i	$0.193586\pi$
$173$	197.692i	1.14273i	−0.820697	+	0.571364i	$0.806414\pi$
$174$	0	0
$175$	5.00000	0.0285714
$176$	− 53.6656i	− 0.304918i
$177$	0	0
$178$	−207.895	−1.16795
$179$	− 242.519i	− 1.35485i	−0.735590	−	0.677427i	$-0.763095\pi$
$179$	− 242.519i	− 1.35485i	0.735590	−	0.677427i	$-0.236905\pi$
$180$	0	0
$181$	−308.789	−1.70602	−0.853009	−	0.521896i	$-0.825225\pi$
$181$	−308.789	−1.70602	−0.853009	+	0.521896i	$0.825225\pi$
$182$	− 19.7617i	− 0.108581i
$183$	0	0
$184$	−10.0527	−0.0546341
$185$	31.2461i	0.168898i
$186$	0	0
$187$	−113.842	−0.608781
$188$	138.518i	0.736800i
$189$	0	0
$190$	82.1359	0.432294
$191$	− 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
$193$	−341.868	−1.77134	−0.885669	−	0.464317i	$-0.846300\pi$
$193$	−341.868	−1.77134	−0.885669	+	0.464317i	$0.846300\pi$
$194$	− 141.235i	− 0.728016i
$195$	0	0
$196$	96.0000	0.489796
$197$	128.433i	0.651943i	0.945380	+	0.325971i	$0.105691\pi$
$197$	128.433i	0.651943i	−0.945380	+	0.325971i	$0.894309\pi$
$198$	0	0
$199$	178.026	0.894605	0.447302	−	0.894383i	$-0.352385\pi$
$199$	178.026	0.894605	0.447302	+	0.894383i	$0.352385\pi$
$200$	− 14.1421i	− 0.0707107i
$201$	0	0
$202$	−6.15800	−0.0304852
$203$	4.93113i	0.0242913i
$204$	0	0
$205$	−64.8683	−0.316431
$206$	− 18.3103i	− 0.0888849i
$207$	0	0
$208$	−55.8947	−0.268724
$209$	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	$0.447176\pi$
$212$	− 179.568i	− 0.847018i
$213$	0	0
$214$	−146.763	−0.685808
$215$	− 169.823i	− 0.789876i
$216$	0	0
$217$	35.9473	0.165656
$218$	− 141.347i	− 0.648380i
$219$	0	0
$220$	60.0000	0.272727
$221$	118.570i	0.536518i
$222$	0	0
$223$	−131.631	−0.590275	−0.295137	−	0.955455i	$-0.595365\pi$
$223$	−131.631	−0.590275	−0.295137	+	0.955455i	$0.595365\pi$
$224$	5.65685i	0.0252538i
$225$	0	0
$226$	290.763	1.28656
$227$	− 343.895i	− 1.51496i	−0.652859	−	0.757479i	$-0.726431\pi$
$227$	− 343.895i	− 1.51496i	0.652859	−	0.757479i	$-0.273569\pi$
$228$	0	0
$229$	413.579	1.80602	0.903010	−	0.429619i	$-0.141352\pi$
$229$	413.579	1.80602	0.903010	+	0.429619i	$0.141352\pi$
$230$	− 11.2392i	− 0.0488662i
$231$	0	0
$232$	13.9473	0.0601178
$233$	154.335i	0.662384i	0.943563	+	0.331192i	$0.107451\pi$
$233$	154.335i	0.662384i	−0.943563	+	0.331192i	$0.892549\pi$
$234$	0	0
$235$	−154.868	−0.659014
$236$	− 16.9706i	− 0.0719092i
$237$	0	0
$238$	12.0000	0.0504202
$239$	47.5810i	0.199084i	0.995033	+	0.0995418i	$0.0317377\pi$
$239$	47.5810i	0.199084i	−0.995033	+	0.0995418i	$0.968262\pi$
$240$	0	0
$241$	−440.789	−1.82900	−0.914501	−	0.404584i	$-0.867416\pi$
$241$	−440.789	−1.82900	−0.914501	+	0.404584i	$0.867416\pi$
$242$	83.4386i	0.344788i
$243$	0	0
$244$	−101.789	−0.417169
$245$	107.331i	0.438087i
$246$	0	0
$247$	362.947	1.46942
$248$	− 101.674i	− 0.409977i
$249$	0	0
$250$	15.8114	0.0632456
$251$	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
$253$	47.6840	0.188474
$254$	48.0833i	0.189304i
$255$	0	0
$256$	16.0000	0.0625000
$257$	− 309.378i	− 1.20380i	−0.798570	−	0.601902i	$-0.794410\pi$
$257$	− 309.378i	− 1.20380i	0.798570	−	0.601902i	$-0.205590\pi$
$258$	0	0
$259$	13.9737	0.0539524
$260$	− 62.4921i	− 0.240354i
$261$	0	0
$262$	−140.105	−0.534753
$263$	− 359.136i	− 1.36554i	−0.730636	−	0.682768i	$-0.760776\pi$
$263$	− 359.136i	− 1.36554i	0.730636	−	0.682768i	$-0.239224\pi$
$264$	0	0
$265$	200.763	0.757596
$266$	− 36.7323i	− 0.138091i
$267$	0	0
$268$	−142.000	−0.529851
$269$	− 257.536i	− 0.957382i	−0.877983	−	0.478691i	$-0.841111\pi$
$269$	− 257.536i	− 0.957382i	0.877983	−	0.478691i	$-0.158889\pi$
$270$	0	0
$271$	177.710	0.655758	0.327879	−	0.944720i	$-0.393666\pi$
$271$	177.710	0.655758	0.327879	+	0.944720i	$0.393666\pi$
$272$	− 33.9411i	− 0.124784i
$273$	0	0
$274$	74.7630	0.272858
$275$	67.0820i	0.243935i
$276$	0	0
$277$	−311.631	−1.12502	−0.562511	−	0.826790i	$-0.690165\pi$
$277$	−311.631	−1.12502	−0.562511	+	0.826790i	$0.690165\pi$
$278$	248.641i	0.894392i
$279$	0	0
$280$	−6.32456	−0.0225877
$281$	− 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
$283$	155.737	0.550306	0.275153	−	0.961400i	$-0.411271\pi$
$283$	155.737	0.550306	0.275153	+	0.961400i	$0.411271\pi$
$284$	252.958i	0.890697i
$285$	0	0
$286$	265.132	0.927034
$287$	29.0100i	0.101080i
$288$	0	0
$289$	217.000	0.750865
$290$	15.5936i	0.0537710i
$291$	0	0
$292$	−44.0527	−0.150865
$293$	− 338.611i	− 1.15567i	−0.816154	−	0.577835i	$-0.803898\pi$
$293$	− 338.611i	− 1.15567i	0.816154	−	0.577835i	$-0.196102\pi$
$294$	0	0
$295$	18.9737	0.0643175
$296$	− 39.5235i	− 0.133525i
$297$	0	0
$298$	337.631	1.13299
$299$	− 49.6646i	− 0.166102i
$300$	0	0
$301$	−75.9473	−0.252317
$302$	296.799i	0.982777i
$303$	0	0
$304$	−103.895	−0.341759
$305$	− 113.804i	− 0.373128i
$306$	0	0
$307$	229.684	0.748156	0.374078	−	0.927397i	$-0.377959\pi$
$307$	229.684	0.748156	0.374078	+	0.927397i	$0.377959\pi$
$308$	− 26.8328i	− 0.0871195i
$309$	0	0
$310$	113.675	0.366695
$311$	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
$313$	335.658	1.07239	0.536194	−	0.844095i	$-0.319861\pi$
$313$	335.658	1.07239	0.536194	+	0.844095i	$0.319861\pi$
$314$	− 217.938i	− 0.694070i
$315$	0	0
$316$	−151.737	−0.480179
$317$	430.925i	1.35939i	0.733497	+	0.679693i	$0.237887\pi$
$317$	430.925i	1.35939i	−0.733497	+	0.679693i	$0.762113\pi$
$318$	0	0
$319$	−66.1580	−0.207392
$320$	17.8885i	0.0559017i
$321$	0	0
$322$	−5.02633	−0.0156097
$323$	220.394i	0.682334i
$324$	0	0
$325$	69.8683	0.214979
$326$	21.1387i	0.0648427i
$327$	0	0
$328$	82.0527	0.250161
$329$	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	$0.588640\pi$
$332$	69.0358i	0.207939i
$333$	0	0
$334$	195.579	0.585565
$335$	− 158.761i	− 0.473913i
$336$	0	0
$337$	353.500	1.04896	0.524480	−	0.851423i	$-0.324260\pi$
$337$	353.500	1.04896	0.524480	+	0.851423i	$0.324260\pi$
$338$	− 37.1420i	− 0.109887i
$339$	0	0
$340$	37.9473	0.111610
$341$	482.284i	1.41432i
$342$	0	0
$343$	97.0000	0.282799
$344$	214.811i	0.624452i
$345$	0	0
$346$	279.579	0.808031
$347$	123.725i	0.356556i	0.983980	+	0.178278i	$0.0570527\pi$
$347$	123.725i	0.356556i	−0.983980	+	0.178278i	$0.942947\pi$
$348$	0	0
$349$	67.1053	0.192279	0.0961395	−	0.995368i	$-0.469351\pi$
$349$	67.1053	0.192279	0.0961395	+	0.995368i	$0.469351\pi$
$350$	− 7.07107i	− 0.0202031i
$351$	0	0
$352$	−75.8947	−0.215610
$353$	− 665.759i	− 1.88600i	−0.332787	−	0.943002i	$-0.607989\pi$
$353$	− 665.759i	− 1.88600i	0.332787	−	0.943002i	$-0.392011\pi$
$354$	0	0
$355$	−282.816	−0.796664
$356$	294.007i	0.825864i
$357$	0	0
$358$	−342.974	−0.958027
$359$	− 434.927i	− 1.21149i	−0.795657	−	0.605747i	$-0.792874\pi$
$359$	− 434.927i	− 1.21149i	0.795657	−	0.605747i	$-0.207126\pi$
$360$	0	0
$361$	313.631	0.868785
$362$	436.694i	1.20634i
$363$	0	0
$364$	−27.9473	−0.0767784
$365$	− 49.2524i	− 0.134938i
$366$	0	0
$367$	286.684	0.781155	0.390578	−	0.920570i	$-0.372275\pi$
$367$	286.684	0.781155	0.390578	+	0.920570i	$0.372275\pi$
$368$	14.2166i	0.0386321i
$369$	0	0
$370$	44.1886	0.119429
$371$	− 89.7839i	− 0.242005i
$372$	0	0
$373$	3.86833	0.0103709	0.00518543	−	0.999987i	$-0.498349\pi$
$373$	3.86833	0.0103709	0.00518543	+	0.999987i	$0.498349\pi$
$374$	160.997i	0.430473i
$375$	0	0
$376$	195.895	0.520996
$377$	68.9059i	0.182774i
$378$	0	0
$379$	−114.184	−0.301278	−0.150639	−	0.988589i	$-0.548133\pi$
$379$	−114.184	−0.301278	−0.150639	+	0.988589i	$0.548133\pi$
$380$	− 116.158i	− 0.305678i
$381$	0	0
$382$	−435.579	−1.14026
$383$	− 333.903i	− 0.871810i	−0.899993	−	0.435905i	$-0.856428\pi$
$383$	− 333.903i	− 0.871810i	0.899993	−	0.435905i	$-0.143572\pi$
$384$	0	0
$385$	30.0000	0.0779221
$386$	483.475i	1.25253i
$387$	0	0
$388$	−199.737	−0.514785
$389$	682.507i	1.75452i	0.480020	+	0.877258i	$0.340630\pi$
$389$	682.507i	1.75452i	−0.480020	+	0.877258i	$0.659370\pi$
$390$	0	0
$391$	30.1580	0.0771304
$392$	− 135.765i	− 0.346338i
$393$	0	0
$394$	181.631	0.460993
$395$	− 169.647i	− 0.429485i
$396$	0	0
$397$	258.211	0.650405	0.325202	−	0.945644i	$-0.394568\pi$
$397$	258.211	0.650405	0.325202	+	0.945644i	$0.394568\pi$
$398$	− 251.767i	− 0.632581i
$399$	0	0
$400$	−20.0000	−0.0500000
$401$	− 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
$403$	502.316	1.24644
$404$	8.70873i	0.0215563i
$405$	0	0
$406$	6.97367	0.0171765
$407$	187.476i	0.460630i
$408$	0	0
$409$	−302.315	−0.739157	−0.369579	−	0.929199i	$-0.620498\pi$
$409$	−302.315	−0.739157	−0.369579	+	0.929199i	$0.620498\pi$
$410$	91.7377i	0.223750i
$411$	0	0
$412$	−25.8947	−0.0628511
$413$	− 8.48528i	− 0.0205455i
$414$	0	0
$415$	−77.1843	−0.185986
$416$	79.0470i	0.190017i
$417$	0	0
$418$	492.816	1.17898
$419$	− 264.197i	− 0.630542i	−0.949002	−	0.315271i	$-0.897905\pi$
$419$	− 264.197i	− 0.630542i	0.949002	−	0.315271i	$-0.102095\pi$
$420$	0	0
$421$	−131.369	−0.312040	−0.156020	−	0.987754i	$-0.549866\pi$
$421$	−131.369	−0.312040	−0.156020	+	0.987754i	$0.549866\pi$
$422$	− 98.5853i	− 0.233614i
$423$	0	0
$424$	−253.947	−0.598932
$425$	42.4264i	0.0998268i
$426$	0	0
$427$	−50.8947	−0.119191
$428$	207.554i	0.484940i
$429$	0	0
$430$	−240.167	−0.558527
$431$	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
$433$	−793.263	−1.83202	−0.916008	−	0.401161i	$-0.868607\pi$
$433$	−793.263	−1.83202	−0.916008	+	0.401161i	$0.868607\pi$
$434$	− 50.8372i	− 0.117136i
$435$	0	0
$436$	−199.895	−0.458474
$437$	− 92.3144i	− 0.211246i
$438$	0	0
$439$	−598.000	−1.36219	−0.681093	−	0.732197i	$-0.738495\pi$
$439$	−598.000	−1.36219	−0.681093	+	0.732197i	$0.738495\pi$
$440$	− 84.8528i	− 0.192847i
$441$	0	0
$442$	167.684	0.379376
$443$	− 740.750i	− 1.67212i	−0.548637	−	0.836061i	$-0.684853\pi$
$443$	− 740.750i	− 1.67212i	0.548637	−	0.836061i	$-0.315147\pi$
$444$	0	0
$445$	−328.710	−0.738675
$446$	186.155i	0.417387i
$447$	0	0
$448$	8.00000	0.0178571
$449$	− 881.799i	− 1.96392i	−0.189096	−	0.981959i	$-0.560556\pi$
$449$	− 881.799i	− 1.96392i	0.189096	−	0.981959i	$-0.439444\pi$
$450$	0	0
$451$	−389.210	−0.862993
$452$	− 411.201i	− 0.909737i
$453$	0	0
$454$	−486.342	−1.07124
$455$	− 31.2461i	− 0.0686727i
$456$	0	0
$457$	−238.000	−0.520788	−0.260394	−	0.965502i	$-0.583852\pi$
$457$	−238.000	−0.520788	−0.260394	+	0.965502i	$0.583852\pi$
$458$	− 584.889i	− 1.27705i
$459$	0	0
$460$	−15.8947	−0.0345536
$461$	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
$463$	−178.737	−0.386040	−0.193020	−	0.981195i	$-0.561828\pi$
$463$	−178.737	−0.386040	−0.193020	+	0.981195i	$0.561828\pi$
$464$	− 19.7245i	− 0.0425097i
$465$	0	0
$466$	218.263	0.468376
$467$	147.004i	0.314783i	0.987536	+	0.157392i	$0.0503085\pi$
$467$	147.004i	0.314783i	−0.987536	+	0.157392i	$0.949692\pi$
$468$	0	0
$469$	−71.0000	−0.151386
$470$	219.017i	0.465993i
$471$	0	0
$472$	−24.0000	−0.0508475
$473$	− 1018.94i	− 2.15421i
$474$	0	0
$475$	129.868	0.273407
$476$	− 16.9706i	− 0.0356524i
$477$	0	0
$478$	67.2897	0.140773
$479$	472.999i	0.987471i	0.869612	+	0.493735i	$0.164369\pi$
$479$	472.999i	0.987471i	−0.869612	+	0.493735i	$0.835631\pi$
$480$	0	0
$481$	195.263	0.405953
$482$	623.370i	1.29330i
$483$	0	0
$484$	118.000	0.243802
$485$	− 223.312i	− 0.460438i
$486$	0	0
$487$	−502.105	−1.03102	−0.515508	−	0.856885i	$-0.672397\pi$
$487$	−502.105	−1.03102	−0.515508	+	0.856885i	$0.672397\pi$
$488$	143.952i	0.294983i
$489$	0	0
$490$	151.789	0.309774
$491$	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
$493$	−41.8420	−0.0848722
$494$	− 513.285i	− 1.03904i
$495$	0	0
$496$	−143.789	−0.289898
$497$	126.479i	0.254485i
$498$	0	0
$499$	−226.000	−0.452906	−0.226453	−	0.974022i	$-0.572713\pi$
$499$	−226.000	−0.452906	−0.226453	+	0.974022i	$0.572713\pi$
$500$	− 22.3607i	− 0.0447214i
$501$	0	0
$502$	484.710	0.965558
$503$	361.089i	0.717872i	0.933362	+	0.358936i	$0.116860\pi$
$503$	361.089i	0.717872i	−0.933362	+	0.358936i	$0.883140\pi$
$504$	0	0
$505$	−9.73666	−0.0192805
$506$	− 67.4353i	− 0.133271i
$507$	0	0
$508$	68.0000	0.133858
$509$	103.070i	0.202496i	0.994861	+	0.101248i	$0.0322836\pi$
$509$	103.070i	0.202496i	−0.994861	+	0.101248i	$0.967716\pi$
$510$	0	0
$511$	−22.0263	−0.0431044
$512$	− 22.6274i	− 0.0441942i
$513$	0	0
$514$	−437.526	−0.851218
$515$	− 28.9511i	− 0.0562158i
$516$	0	0
$517$	−929.210	−1.79731
$518$	− 19.7617i	− 0.0381501i
$519$	0	0
$520$	−88.3772	−0.169956
$521$	− 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
$523$	668.421	1.27805	0.639025	−	0.769186i	$-0.279338\pi$
$523$	668.421	1.27805	0.639025	+	0.769186i	$0.279338\pi$
$524$	198.139i	0.378128i
$525$	0	0
$526$	−507.895	−0.965579
$527$	305.023i	0.578792i
$528$	0	0
$529$	516.368	0.976121
$530$	− 283.922i	− 0.535701i
$531$	0	0
$532$	−51.9473	−0.0976454
$533$	405.376i	0.760555i
$534$	0	0
$535$	−232.053	−0.433743
$536$	200.818i	0.374661i
$537$	0	0
$538$	−364.211	−0.676972
$539$	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	$0.546351\pi$
$542$	− 251.320i	− 0.463691i
$543$	0	0
$544$	−48.0000	−0.0882353
$545$	− 223.489i	− 0.410072i
$546$	0	0
$547$	396.947	0.725681	0.362840	−	0.931851i	$-0.381807\pi$
$547$	396.947	0.725681	0.362840	+	0.931851i	$0.381807\pi$
$548$	− 105.731i	− 0.192939i
$549$	0	0
$550$	94.8683	0.172488
$551$	128.079i	0.232449i
$552$	0	0
$553$	−75.8683	−0.137194
$554$	440.713i	0.795511i
$555$	0	0
$556$	351.631	0.632430
$557$	235.764i	0.423275i	0.977348	+	0.211637i	$0.0678796\pi$
$557$	235.764i	0.423275i	−0.977348	+	0.211637i	$0.932120\pi$
$558$	0	0
$559$	−1061.26	−1.89850
$560$	8.94427i	0.0159719i
$561$	0	0
$562$	−78.1580	−0.139071
$563$	− 202.046i	− 0.358874i	−0.983769	−	0.179437i	$-0.942572\pi$
$563$	− 202.046i	− 0.358874i	0.983769	−	0.179437i	$-0.0574277\pi$
$564$	0	0
$565$	459.737	0.813693
$566$	− 220.245i	− 0.389125i
$567$	0	0
$568$	357.737	0.629818
$569$	− 192.408i	− 0.338150i	−0.985603	−	0.169075i	$-0.945922\pi$
$569$	− 192.408i	− 0.338150i	0.985603	−	0.169075i	$-0.0540781\pi$
$570$	0	0
$571$	206.553	0.361739	0.180870	−	0.983507i	$-0.442109\pi$
$571$	206.553	0.361739	0.180870	+	0.983507i	$0.442109\pi$
$572$	− 374.953i	− 0.655512i
$573$	0	0
$574$	41.0263	0.0714744
$575$	− 17.7708i	− 0.0309057i
$576$	0	0
$577$	263.658	0.456946	0.228473	−	0.973550i	$-0.426627\pi$
$577$	263.658	0.456946	0.228473	+	0.973550i	$0.426627\pi$
$578$	− 306.884i	− 0.530942i
$579$	0	0
$580$	22.0527	0.0380218
$581$	34.5179i	0.0594112i
$582$	0	0
$583$	1204.58	2.06617
$584$	62.2999i	0.106678i
$585$	0	0
$586$	−478.868	−0.817181
$587$	− 293.654i	− 0.500263i	−0.968212	−	0.250131i	$-0.919526\pi$
$587$	− 293.654i	− 0.500263i	0.968212	−	0.250131i	$-0.0804738\pi$
$588$	0	0
$589$	933.684	1.58520
$590$	− 26.8328i	− 0.0454793i
$591$	0	0
$592$	−55.8947	−0.0944167
$593$	139.095i	0.234562i	0.993099	+	0.117281i	$0.0374178\pi$
$593$	139.095i	0.234562i	−0.993099	+	0.117281i	$0.962582\pi$
$594$	0	0
$595$	18.9737	0.0318885
$596$	− 477.483i	− 0.801146i
$597$	0	0
$598$	−70.2363	−0.117452
$599$	1069.98i	1.78628i	0.449778	+	0.893140i	$0.351503\pi$
$599$	1069.98i	1.78628i	−0.449778	+	0.893140i	$0.648497\pi$
$600$	0	0
$601$	75.9473	0.126368	0.0631841	−	0.998002i	$-0.479874\pi$
$601$	75.9473	0.126368	0.0631841	+	0.998002i	$0.479874\pi$
$602$	107.406i	0.178415i
$603$	0	0
$604$	419.737	0.694928
$605$	131.928i	0.218063i
$606$	0	0
$607$	−383.053	−0.631059	−0.315529	−	0.948916i	$-0.602182\pi$
$607$	−383.053	−0.631059	−0.315529	+	0.948916i	$0.602182\pi$
$608$	146.929i	0.241660i
$609$	0	0
$610$	−160.943	−0.263841
$611$	967.805i	1.58397i
$612$	0	0
$613$	−817.658	−1.33386	−0.666931	−	0.745119i	$-0.732393\pi$
$613$	−817.658	−1.33386	−0.666931	+	0.745119i	$0.732393\pi$
$614$	− 324.822i	− 0.529026i
$615$	0	0
$616$	−37.9473	−0.0616028
$617$	− 525.994i	− 0.852502i	−0.904605	−	0.426251i	$-0.859834\pi$
$617$	− 525.994i	− 0.852502i	0.904605	−	0.426251i	$-0.140166\pi$
$618$	0	0
$619$	−600.026	−0.969348	−0.484674	−	0.874695i	$-0.661062\pi$
$619$	−600.026	−0.969348	−0.484674	+	0.874695i	$0.661062\pi$
$620$	− 160.761i	− 0.259293i
$621$	0	0
$622$	144.316	0.232019
$623$	147.004i	0.235961i
$624$	0	0
$625$	25.0000	0.0400000
$626$	− 474.692i	− 0.758293i
$627$	0	0
$628$	−308.211	−0.490781
$629$	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	$0.200702\pi$
$632$	214.588i	0.339538i
$633$	0	0
$634$	609.421	0.961231
$635$	76.0263i	0.119726i
$636$	0	0
$637$	670.736	1.05296
$638$	93.5615i	0.146648i
$639$	0	0
$640$	25.2982	0.0395285
$641$	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
$643$	−609.368	−0.947695	−0.473848	−	0.880607i	$-0.657135\pi$
$643$	−609.368	−0.947695	−0.473848	+	0.880607i	$0.657135\pi$
$644$	7.10831i	0.0110377i
$645$	0	0
$646$	311.684	0.482483
$647$	545.942i	0.843805i	0.906641	+	0.421902i	$0.138638\pi$
$647$	545.942i	0.843805i	−0.906641	+	0.421902i	$0.861362\pi$
$648$	0	0
$649$	113.842	0.175411
$650$	− 98.8087i	− 0.152013i
$651$	0	0
$652$	29.8947	0.0458507
$653$	239.188i	0.366291i	0.983086	+	0.183146i	$0.0586280\pi$
$653$	239.188i	0.366291i	−0.983086	+	0.183146i	$0.941372\pi$
$654$	0	0
$655$	−221.526	−0.338208
$656$	− 116.040i	− 0.176890i
$657$	0	0
$658$	97.9473	0.148856
$659$	11.4627i	0.0173940i	0.999962	+	0.00869702i	$0.00276838\pi$
$659$	11.4627i	0.0173940i	−0.999962	+	0.00869702i	$0.997232\pi$
$660$	0	0
$661$	210.473	0.318417	0.159208	−	0.987245i	$-0.449106\pi$
$661$	210.473	0.318417	0.159208	+	0.987245i	$0.449106\pi$
$662$	257.350i	0.388746i
$663$	0	0
$664$	97.6313	0.147035
$665$	− 58.0789i	− 0.0873367i
$666$	0	0
$667$	17.5260	0.0262758
$668$	− 276.590i	− 0.414057i
$669$	0	0
$670$	−224.522	−0.335107
$671$	− 682.824i	− 1.01762i
$672$	0	0
$673$	316.184	0.469813	0.234907	−	0.972018i	$-0.424522\pi$
$673$	316.184	0.469813	0.234907	+	0.972018i	$0.424522\pi$
$674$	− 499.924i	− 0.741727i
$675$	0	0
$676$	−52.5267	−0.0777022
$677$	− 77.9680i	− 0.115167i	−0.998341	−	0.0575834i	$-0.981660\pi$
$677$	− 77.9680i	− 0.115167i	0.998341	−	0.0575834i	$-0.0183395\pi$
$678$	0	0
$679$	−99.8683	−0.147081
$680$	− 53.6656i	− 0.0789200i
$681$	0	0
$682$	682.053	1.00008
$683$	718.401i	1.05183i	0.850536	+	0.525916i	$0.176277\pi$
$683$	718.401i	1.05183i	−0.850536	+	0.525916i	$0.823723\pi$
$684$	0	0
$685$	118.211	0.172570
$686$	− 137.179i	− 0.199969i
$687$	0	0
$688$	303.789	0.441554
$689$	− 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	$0.264852\pi$
$692$	− 395.384i	− 0.571364i
$693$	0	0
$694$	174.974	0.252123
$695$	393.136i	0.565663i
$696$	0	0
$697$	−246.158	−0.353168
$698$	− 94.9013i	− 0.135962i
$699$	0	0
$700$	−10.0000	−0.0142857
$701$	− 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
$703$	362.947	0.516284
$704$	107.331i	0.152459i
$705$	0	0
$706$	−941.526	−1.33361
$707$	4.35437i	0.00615893i
$708$	0	0
$709$	930.789	1.31282	0.656410	−	0.754404i	$-0.272074\pi$
$709$	930.789	1.31282	0.656410	+	0.754404i	$0.272074\pi$
$710$	399.962i	0.563326i
$711$	0	0
$712$	415.789	0.583974
$713$	− 127.762i	− 0.179190i
$714$	0	0
$715$	419.210	0.586308
$716$	485.038i	0.677427i
$717$	0	0
$718$	−615.079	−0.856656
$719$	− 403.963i	− 0.561840i	−0.959731	−	0.280920i	$-0.909360\pi$
$719$	− 403.963i	− 0.561840i	0.959731	−	0.280920i	$-0.0906395\pi$
$720$	0	0
$721$	−12.9473	−0.0179575
$722$	− 443.542i	− 0.614324i
$723$	0	0
$724$	617.579	0.853009
$725$	24.6556i	0.0340078i
$726$	0	0
$727$	−490.000	−0.674003	−0.337001	−	0.941504i	$-0.609413\pi$
$727$	−490.000	−0.674003	−0.337001	+	0.941504i	$0.609413\pi$
$728$	39.5235i	0.0542905i
$729$	0	0
$730$	−69.6534	−0.0954156
$731$	− 644.434i	− 0.881579i
$732$	0	0
$733$	−1012.16	−1.38084	−0.690422	−	0.723407i	$-0.742575\pi$
$733$	−1012.16	−1.38084	−0.690422	+	0.723407i	$0.742575\pi$
$734$	− 405.432i	− 0.552360i
$735$	0	0
$736$	20.1053	0.0273170
$737$	− 952.565i	− 1.29249i
$738$	0	0
$739$	499.210	0.675521	0.337760	−	0.941232i	$-0.390331\pi$
$739$	499.210	0.675521	0.337760	+	0.941232i	$0.390331\pi$
$740$	− 62.4921i	− 0.0844488i
$741$	0	0
$742$	−126.974	−0.171124
$743$	748.788i	1.00779i	0.863765	+	0.503895i	$0.168100\pi$
$743$	748.788i	1.00779i	−0.863765	+	0.503895i	$0.831900\pi$
$744$	0	0
$745$	533.842	0.716566
$746$	− 5.47064i	− 0.00733330i
$747$	0	0
$748$	227.684	0.304390
$749$	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	$0.326757\pi$
$752$	− 277.037i	− 0.368400i
$753$	0	0
$754$	97.4477	0.129241
$755$	469.280i	0.621563i
$756$	0	0
$757$	1145.18	1.51279	0.756396	−	0.654114i	$-0.226958\pi$
$757$	1145.18	1.51279	0.756396	+	0.654114i	$0.226958\pi$
$758$	161.481i	0.213036i
$759$	0	0
$760$	−164.272	−0.216147
$761$	− 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
$763$	−99.9473	−0.130993
$764$	616.001i	0.806284i
$765$	0	0
$766$	−472.211	−0.616463
$767$	− 118.570i	− 0.154590i
$768$	0	0
$769$	991.105	1.28882	0.644412	−	0.764679i	$-0.277102\pi$
$769$	991.105	1.28882	0.644412	+	0.764679i	$0.277102\pi$
$770$	− 42.4264i	− 0.0550992i
$771$	0	0
$772$	683.737	0.885669
$773$	389.746i	0.504199i	0.967701	+	0.252100i	$0.0811211\pi$
$773$	389.746i	0.504199i	−0.967701	+	0.252100i	$0.918879\pi$
$774$	0	0
$775$	179.737	0.231918
$776$	282.470i	0.364008i
$777$	0	0
$778$	965.210	1.24063
$779$	753.496i	0.967261i
$780$	0	0
$781$	−1696.89	−2.17272
$782$	− 42.6499i	− 0.0545395i
$783$	0	0
$784$	−192.000	−0.244898
$785$	− 344.590i	− 0.438968i
$786$	0	0
$787$	−206.631	−0.262556	−0.131278	−	0.991346i	$-0.541908\pi$
$787$	−206.631	−0.262556	−0.131278	+	0.991346i	$0.541908\pi$
$788$	− 256.865i	− 0.325971i
$789$	0	0
$790$	−239.917	−0.303692
$791$	− 205.600i	− 0.259925i
$792$	0	0
$793$	−711.185	−0.896829
$794$	− 365.165i	− 0.459906i
$795$	0	0
$796$	−356.053	−0.447302
$797$	207.107i	0.259859i	0.991523	+	0.129929i	$0.0414750\pi$
$797$	207.107i	0.259859i	−0.991523	+	0.129929i	$0.958525\pi$
$798$	0	0
$799$	−587.684	−0.735524
$800$	28.2843i	0.0353553i
$801$	0	0
$802$	−549.920	−0.685686
$803$	− 295.514i	− 0.368013i
$804$	0	0
$805$	−7.94733	−0.00987246
$806$	− 710.382i	− 0.881367i
$807$	0	0
$808$	12.3160	0.0152426
$809$	663.452i	0.820089i	0.912065	+	0.410045i	$0.134487\pi$
$809$	663.452i	0.820089i	−0.912065	+	0.410045i	$0.865513\pi$
$810$	0	0
$811$	−394.683	−0.486663	−0.243331	−	0.969943i	$-0.578240\pi$
$811$	−394.683	−0.486663	−0.243331	+	0.969943i	$0.578240\pi$
$812$	− 9.86225i	− 0.0121456i
$813$	0	0
$814$	265.132	0.325715
$815$	33.4233i	0.0410101i
$816$	0	0
$817$	−1972.63	−2.41448
$818$	427.538i	0.522663i
$819$	0	0
$820$	129.737	0.158215
$821$	− 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
$823$	930.789	1.13097	0.565486	−	0.824758i	$-0.308689\pi$
$823$	930.789	1.13097	0.565486	+	0.824758i	$0.308689\pi$
$824$	36.6206i	0.0444425i
$825$	0	0
$826$	−12.0000	−0.0145278
$827$	1012.32i	1.22408i	0.790826	+	0.612041i	$0.209651\pi$
$827$	1012.32i	1.22408i	−0.790826	+	0.612041i	$0.790349\pi$
$828$	0	0
$829$	−514.105	−0.620150	−0.310075	−	0.950712i	$-0.600354\pi$
$829$	−514.105	−0.620150	−0.310075	+	0.950712i	$0.600354\pi$
$830$	109.155i	0.131512i
$831$	0	0
$832$	111.789	0.134362
$833$	407.294i	0.488948i
$834$	0	0
$835$	309.237	0.370344
$836$	− 696.947i	− 0.833668i
$837$	0	0
$838$	−373.631	−0.445861
$839$	− 549.013i	− 0.654366i	−0.944961	−	0.327183i	$-0.893901\pi$
$839$	− 549.013i	− 0.654366i	0.944961	−	0.327183i	$-0.106099\pi$
$840$	0	0
$841$	816.684	0.971087
$842$	185.783i	0.220645i
$843$	0	0
$844$	−139.421	−0.165190
$845$	− 58.7266i	− 0.0694990i
$846$	0	0
$847$	59.0000	0.0696576
$848$	359.136i	0.423509i
$849$	0	0
$850$	60.0000	0.0705882
$851$	− 49.6646i	− 0.0583603i
$852$	0	0
$853$	−477.131	−0.559356	−0.279678	−	0.960094i	$-0.590228\pi$
$853$	−477.131	−0.559356	−0.279678	+	0.960094i	$0.590228\pi$
$854$	71.9759i	0.0842809i
$855$	0	0
$856$	293.526	0.342904
$857$	− 518.402i	− 0.604904i	−0.953165	−	0.302452i	$-0.902195\pi$
$857$	− 518.402i	− 0.604904i	0.953165	−	0.302452i	$-0.0978051\pi$
$858$	0	0
$859$	−224.448	−0.261289	−0.130645	−	0.991429i	$-0.541705\pi$
$859$	−224.448	−0.261289	−0.130645	+	0.991429i	$0.541705\pi$
$860$	339.647i	0.394938i
$861$	0	0
$862$	261.237	0.303059
$863$	− 765.759i	− 0.887322i	−0.896195	−	0.443661i	$-0.853679\pi$
$863$	− 765.759i	− 0.887322i	0.896195	−	0.443661i	$-0.146321\pi$
$864$	0	0
$865$	442.053	0.511044
$866$	1121.84i	1.29543i
$867$	0	0
$868$	−71.8947	−0.0828280
$869$	− 1017.88i	− 1.17132i
$870$	0	0
$871$	−992.130	−1.13907
$872$	282.694i	0.324190i
$873$	0	0
$874$	−130.552	−0.149373
$875$	− 11.1803i	− 0.0127775i
$876$	0	0
$877$	−844.237	−0.962642	−0.481321	−	0.876544i	$-0.659843\pi$
$877$	−844.237	−0.962642	−0.481321	+	0.876544i	$0.659843\pi$
$878$	845.700i	0.963212i
$879$	0	0
$880$	−120.000	−0.136364
$881$	− 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
$883$	−795.631	−0.901054	−0.450527	−	0.892763i	$-0.648764\pi$
$883$	−795.631	−0.901054	−0.450527	+	0.892763i	$0.648764\pi$
$884$	− 237.141i	− 0.268259i
$885$	0	0
$886$	−1047.58	−1.18237
$887$	249.051i	0.280779i	0.990096	+	0.140389i	$0.0448354\pi$
$887$	249.051i	0.280779i	−0.990096	+	0.140389i	$0.955165\pi$
$888$	0	0
$889$	34.0000	0.0382452
$890$	464.867i	0.522322i
$891$	0	0
$892$	263.263	0.295137
$893$	1798.92i	2.01446i
$894$	0	0
$895$	−542.289	−0.605909
$896$	− 11.3137i	− 0.0126269i
$897$	0	0
$898$	−1247.05	−1.38870
$899$	177.261i	0.197176i
$900$	0	0
$901$	761.842	0.845552
$902$	550.426i	0.610228i
$903$	0	0
$904$	−581.526	−0.643281
$905$	690.474i	0.762955i
$906$	0	0
$907$	−951.631	−1.04921	−0.524603	−	0.851347i	$-0.675786\pi$
$907$	−951.631	−1.04921	−0.524603	+	0.851347i	$0.675786\pi$
$908$	687.791i	0.757479i
$909$	0	0
$910$	−44.1886	−0.0485589
$911$	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
$913$	−463.106	−0.507236
$914$	336.583i	0.368253i
$915$	0	0
$916$	−827.157	−0.903010
$917$	99.0694i	0.108036i
$918$	0	0
$919$	313.368	0.340988	0.170494	−	0.985359i	$-0.445464\pi$
$919$	313.368	0.340988	0.170494	+	0.985359i	$0.445464\pi$
$920$	22.4784i	0.0244331i
$921$	0	0
$922$	655.473	0.710926
$923$	1767.38i	1.91482i
$924$	0	0
$925$	69.8683	0.0755333
$926$	252.772i	0.272972i
$927$	0	0
$928$	−27.8947	−0.0300589
$929$	− 709.693i	− 0.763932i	−0.924176	−	0.381966i	$-0.875247\pi$
$929$	− 709.693i	− 0.763932i	0.924176	−	0.381966i	$-0.124753\pi$
$930$	0	0
$931$	1246.74	1.33914
$932$	− 308.671i	− 0.331192i
$933$	0	0
$934$	207.895	0.222585
$935$	254.558i	0.272255i
$936$	0	0
$937$	−1351.87	−1.44276	−0.721380	−	0.692539i	$-0.756492\pi$
$937$	−1351.87	−1.44276	−0.721380	+	0.692539i	$0.756492\pi$
$938$	100.409i	0.107046i
$939$	0	0
$940$	309.737	0.329507
$941$	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
$943$	103.106	0.109338
$944$	33.9411i	0.0359546i
$945$	0	0
$946$	−1441.00	−1.52326
$947$	− 2.08359i	− 0.00220020i	−0.999999	−	0.00110010i	$-0.999650\pi$
$947$	− 2.08359i	− 0.00220020i	0.999999	−	0.00110010i	$-0.000350173\pi$
$948$	0	0
$949$	−307.789	−0.324329
$950$	− 183.662i	− 0.193328i
$951$	0	0
$952$	−24.0000	−0.0252101
$953$	− 1427.87i	− 1.49829i	−0.662406	−	0.749145i	$-0.730465\pi$
$953$	− 1427.87i	− 1.49829i	0.662406	−	0.749145i	$-0.269535\pi$
$954$	0	0
$955$	−688.710	−0.721163
$956$	− 95.1620i	− 0.0995418i
$957$	0	0
$958$	668.921	0.698247
$959$	− 52.8654i	− 0.0551256i
$960$	0	0
$961$	331.211	0.344652
$962$	− 276.144i	− 0.287052i
$963$	0	0
$964$	881.579	0.914501
$965$	764.441i	0.792167i
$966$	0	0
$967$	1097.84	1.13531	0.567653	−	0.823268i	$-0.307851\pi$
$967$	1097.84	1.13531	0.567653	+	0.823268i	$0.307851\pi$
$968$	− 166.877i	− 0.172394i
$969$	0	0
$970$	−315.811	−0.325579
$971$	− 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
$973$	175.816	0.180694
$974$	710.083i	0.729038i
$975$	0	0
$976$	203.579	0.208585
$977$	− 231.056i	− 0.236496i	−0.992984	−	0.118248i	$-0.962272\pi$
$977$	− 231.056i	− 0.236496i	0.992984	−	0.118248i	$-0.0377277\pi$
$978$	0	0
$979$	−1972.26	−2.01457
$980$	− 214.663i	− 0.219043i
$981$	0	0
$982$	958.236	0.975801
$983$	1362.35i	1.38591i	0.720979	+	0.692957i	$0.243692\pi$
$983$	1362.35i	1.38591i	−0.720979	+	0.692957i	$0.756308\pi$
$984$	0	0
$985$	287.184	0.291558
$986$	59.1735i	0.0600137i
$987$	0	0
$988$	−725.895	−0.734711
$989$	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	$0.299292\pi$
$992$	203.349i	0.204989i
$993$	0	0
$994$	178.868	0.179948
$995$	− 398.079i	− 0.400079i
$996$	0	0
$997$	−291.526	−0.292403	−0.146202	−	0.989255i	$-0.546705\pi$
$997$	−291.526	−0.292403	−0.146202	+	0.989255i	$0.546705\pi$
$998$	319.612i	0.320253i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	270.3.d.a.161.1	✓	4
3.2	odd	2	inner	270.3.d.a.161.4	yes	4
4.3	odd	2		2160.3.l.f.161.1		4
5.2	odd	4		1350.3.b.h.1349.5		8
5.3	odd	4		1350.3.b.h.1349.3		8
5.4	even	2		1350.3.d.m.701.3		4
9.2	odd	6		810.3.h.b.701.3		8
9.4	even	3		810.3.h.b.431.3		8
9.5	odd	6		810.3.h.b.431.2		8
9.7	even	3		810.3.h.b.701.2		8
12.11	even	2		2160.3.l.f.161.3		4
15.2	even	4		1350.3.b.h.1349.2		8
15.8	even	4		1350.3.b.h.1349.8		8
15.14	odd	2		1350.3.d.m.701.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.3.d.a.161.1	✓	4	1.1	even	1	trivial
270.3.d.a.161.4	yes	4	3.2	odd	2	inner
810.3.h.b.431.2		8	9.5	odd	6
810.3.h.b.431.3		8	9.4	even	3
810.3.h.b.701.2		8	9.7	even	3
810.3.h.b.701.3		8	9.2	odd	6
1350.3.b.h.1349.2		8	15.2	even	4
1350.3.b.h.1349.3		8	5.3	odd	4
1350.3.b.h.1349.5		8	5.2	odd	4
1350.3.b.h.1349.8		8	15.8	even	4
1350.3.d.m.701.2		4	15.14	odd	2
1350.3.d.m.701.3		4	5.4	even	2
2160.3.l.f.161.1		4	4.3	odd	2
2160.3.l.f.161.3		4	12.11	even	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:	\( N \)	\(=\)	\( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	270.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} -2.00000 q^{4} -2.23607i q^{5} -1.00000 q^{7} +2.82843i q^{8} -3.16228 q^{10} -13.4164i q^{11} -13.9737 q^{13} +1.41421i q^{14} +4.00000 q^{16} -8.48528i q^{17} -25.9737 q^{19} +4.47214i q^{20} -18.9737 q^{22} +3.55415i q^{23} -5.00000 q^{25} +19.7617i q^{26} +2.00000 q^{28} -4.93113i q^{29} -35.9473 q^{31} -5.65685i q^{32} -12.0000 q^{34} +2.23607i q^{35} -13.9737 q^{37} +36.7323i q^{38} +6.32456 q^{40} -29.0100i q^{41} +75.9473 q^{43} +26.8328i q^{44} +5.02633 q^{46} -69.2592i q^{47} -48.0000 q^{49} +7.07107i q^{50} +27.9473 q^{52} +89.7839i q^{53} -30.0000 q^{55} -2.82843i q^{56} -6.97367 q^{58} +8.48528i q^{59} +50.8947 q^{61} +50.8372i q^{62} -8.00000 q^{64} +31.2461i q^{65} +71.0000 q^{67} +16.9706i q^{68} +3.16228 q^{70} -126.479i q^{71} +22.0263 q^{73} +19.7617i q^{74} +51.9473 q^{76} +13.4164i q^{77} +75.8683 q^{79} -8.94427i q^{80} -41.0263 q^{82} -34.5179i q^{83} -18.9737 q^{85} -107.406i q^{86} +37.9473 q^{88} -147.004i q^{89} +13.9737 q^{91} -7.10831i q^{92} -97.9473 q^{94} +58.0789i q^{95} +99.8683 q^{97} +67.8823i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4} - 4 q^{7} + 20 q^{13} + 16 q^{16} - 28 q^{19} - 20 q^{25} + 8 q^{28} + 8 q^{31} - 48 q^{34} + 20 q^{37} + 152 q^{43} + 96 q^{46} - 192 q^{49} - 40 q^{52} - 120 q^{55} + 48 q^{58} - 100 q^{61}+ \cdots + 20 q^{97}+O(q^{100}) \) 4 * q - 8 * q^4 - 4 * q^7 + 20 * q^13 + 16 * q^16 - 28 * q^19 - 20 * q^25 + 8 * q^28 + 8 * q^31 - 48 * q^34 + 20 * q^37 + 152 * q^43 + 96 * q^46 - 192 * q^49 - 40 * q^52 - 120 * q^55 + 48 * q^58 - 100 * q^61 - 32 * q^64 + 284 * q^67 + 164 * q^73 + 56 * q^76 - 76 * q^79 - 240 * q^82 - 20 * q^91 - 240 * q^94 + 20 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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