LMFDB - Embedded newform 675.3.d.k.674.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [675,3,Mod(674,675)] 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("675.674"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(675, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	675.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,6,0,14,0,0,0,42,0,0,0,0,0,0,0,46,84,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$18.3924178443$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.60217600.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 16x^{4} + 64x^{2} + 4 $ x^6 + 16x^4 + 64x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2\cdot 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			674.2
Root			$2.94600i$ of defining polynomial
Character	$\chi$	$=$	675.674
Dual form			675.3.d.k.674.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.94600 q^{2} -0.213103 q^{4} +6.41178i q^{7} +8.19868 q^{8} +2.74732i q^{11} -20.3009i q^{13} -12.4773i q^{14} -15.1022 q^{16} +21.0907 q^{17} -13.6615 q^{19} -5.34627i q^{22} -1.87465 q^{23} +39.5054i q^{26} -1.36637i q^{28} +6.98266i q^{29} -26.8602 q^{31} -3.40595 q^{32} -41.0423 q^{34} +19.1899i q^{37} +26.5853 q^{38} +67.2073i q^{41} -67.0135i q^{43} -0.585461i q^{44} +3.64806 q^{46} +70.6846 q^{47} +7.88907 q^{49} +4.32617i q^{52} +81.2159 q^{53} +52.5681i q^{56} -13.5882i q^{58} -106.356i q^{59} +13.4563 q^{61} +52.2699 q^{62} +67.0367 q^{64} +63.0790i q^{67} -4.49448 q^{68} -32.1660i q^{71} +123.844i q^{73} -37.3435i q^{74} +2.91131 q^{76} -17.6152 q^{77} +19.0868 q^{79} -130.785i q^{82} +119.462 q^{83} +130.408i q^{86} +22.5244i q^{88} +152.844i q^{89} +130.165 q^{91} +0.399493 q^{92} -137.552 q^{94} +16.4385i q^{97} -15.3521 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{2} + 14 q^{4} + 42 q^{8} + 46 q^{16} + 84 q^{17} + 16 q^{19} + 102 q^{23} - 56 q^{31} + 174 q^{32} + 80 q^{34} + 96 q^{38} + 234 q^{46} + 138 q^{47} - 74 q^{49} + 120 q^{53} - 46 q^{61} - 36 q^{62}+ \cdots - 318 q^{98}+O(q^{100}) $ 6 * q + 6 * q^2 + 14 * q^4 + 42 * q^8 + 46 * q^16 + 84 * q^17 + 16 * q^19 + 102 * q^23 - 56 * q^31 + 174 * q^32 + 80 * q^34 + 96 * q^38 + 234 * q^46 + 138 * q^47 - 74 * q^49 + 120 * q^53 - 46 * q^61 - 36 * q^62 + 342 * q^64 + 432 * q^68 - 156 * q^76 + 492 * q^77 - 316 * q^79 + 324 * q^83 + 44 * q^91 + 210 * q^92 - 526 * q^94 - 318 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/675\mathbb{Z}\right)^\times$.

$n$	$326$	$352$
$\chi(n)$	$-1$	$-1$

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 $

$2$	−1.94600	−0.972998	−0.486499	−	0.873681i	$-0.661726\pi$
$2$	−1.94600	−0.972998	−0.486499	+	0.873681i	$0.661726\pi$
$3$	0	0
$3$	0	0
$4$	−0.213103	−0.0532757
$5$	0	0
$5$	0	0
$6$	0	0
$7$	6.41178i	0.915969i	0.888960	+	0.457984i	$0.151428\pi$
$7$	6.41178i	0.915969i	−0.888960	+	0.457984i	$0.848572\pi$
$8$	8.19868	1.02483
$9$	0	0
$10$	0	0
$11$	2.74732i	0.249756i	0.992172	+	0.124878i	$0.0398540\pi$
$11$	2.74732i	0.249756i	−0.992172	+	0.124878i	$0.960146\pi$
$12$	0	0
$13$	− 20.3009i	− 1.56160i	−0.624778	−	0.780802i	$-0.714811\pi$
$13$	− 20.3009i	− 1.56160i	0.624778	−	0.780802i	$-0.285189\pi$
$14$	− 12.4773i	− 0.891235i
$15$	0	0
$16$	−15.1022	−0.943886
$17$	21.0907	1.24063	0.620314	−	0.784354i	$-0.287005\pi$
$17$	21.0907	1.24063	0.620314	+	0.784354i	$0.287005\pi$
$18$	0	0
$19$	−13.6615	−0.719029	−0.359514	−	0.933140i	$-0.617058\pi$
$19$	−13.6615	−0.719029	−0.359514	+	0.933140i	$0.617058\pi$
$20$	0	0
$21$	0	0
$22$	− 5.34627i	− 0.243012i
$23$	−1.87465	−0.0815065	−0.0407532	−	0.999169i	$-0.512976\pi$
$23$	−1.87465	−0.0815065	−0.0407532	+	0.999169i	$0.512976\pi$
$24$	0	0
$25$	0	0
$26$	39.5054i	1.51944i
$27$	0	0
$28$	− 1.36637i	− 0.0487988i
$29$	6.98266i	0.240781i	0.992727	+	0.120391i	$0.0384147\pi$
$29$	6.98266i	0.240781i	−0.992727	+	0.120391i	$0.961585\pi$
$30$	0	0
$31$	−26.8602	−0.866459	−0.433229	−	0.901284i	$-0.642626\pi$
$31$	−26.8602	−0.866459	−0.433229	+	0.901284i	$0.642626\pi$
$32$	−3.40595	−0.106436
$33$	0	0
$34$	−41.0423	−1.20713
$35$	0	0
$36$	0	0
$37$	19.1899i	0.518647i	0.965791	+	0.259323i	$0.0834995\pi$
$37$	19.1899i	0.518647i	−0.965791	+	0.259323i	$0.916500\pi$
$38$	26.5853	0.699613
$39$	0	0
$40$	0	0
$41$	67.2073i	1.63920i	0.572935	+	0.819601i	$0.305805\pi$
$41$	67.2073i	1.63920i	−0.572935	+	0.819601i	$0.694195\pi$
$42$	0	0
$43$	− 67.0135i	− 1.55845i	−0.626742	−	0.779227i	$-0.715612\pi$
$43$	− 67.0135i	− 1.55845i	0.626742	−	0.779227i	$-0.284388\pi$
$44$	− 0.585461i	− 0.0133059i
$45$	0	0
$46$	3.64806	0.0793056
$47$	70.6846	1.50393	0.751963	−	0.659205i	$-0.229107\pi$
$47$	70.6846	1.50393	0.751963	+	0.659205i	$0.229107\pi$
$48$	0	0
$49$	7.88907	0.161002
$50$	0	0
$51$	0	0
$52$	4.32617i	0.0831955i
$53$	81.2159	1.53237	0.766187	−	0.642617i	$-0.222151\pi$
$53$	81.2159	1.53237	0.766187	+	0.642617i	$0.222151\pi$
$54$	0	0
$55$	0	0
$56$	52.5681i	0.938716i
$57$	0	0
$58$	− 13.5882i	− 0.234280i
$59$	− 106.356i	− 1.80264i	−0.433151	−	0.901321i	$-0.642598\pi$
$59$	− 106.356i	− 1.80264i	0.433151	−	0.901321i	$-0.357402\pi$
$60$	0	0
$61$	13.4563	0.220594	0.110297	−	0.993899i	$-0.464820\pi$
$61$	13.4563	0.220594	0.110297	+	0.993899i	$0.464820\pi$
$62$	52.2699	0.843062
$63$	0	0
$64$	67.0367	1.04745
$65$	0	0
$66$	0	0
$67$	63.0790i	0.941478i	0.882273	+	0.470739i	$0.156013\pi$
$67$	63.0790i	0.941478i	−0.882273	+	0.470739i	$0.843987\pi$
$68$	−4.49448	−0.0660953
$69$	0	0
$70$	0	0
$71$	− 32.1660i	− 0.453042i	−0.974006	−	0.226521i	$-0.927265\pi$
$71$	− 32.1660i	− 0.453042i	0.974006	−	0.226521i	$-0.0727352\pi$
$72$	0	0
$73$	123.844i	1.69649i	0.529605	+	0.848244i	$0.322340\pi$
$73$	123.844i	1.69649i	−0.529605	+	0.848244i	$0.677660\pi$
$74$	− 37.3435i	− 0.504642i
$75$	0	0
$76$	2.91131	0.0383067
$77$	−17.6152	−0.228769
$78$	0	0
$79$	19.0868	0.241605	0.120803	−	0.992677i	$-0.461453\pi$
$79$	19.0868	0.241605	0.120803	+	0.992677i	$0.461453\pi$
$80$	0	0
$81$	0	0
$82$	− 130.785i	− 1.59494i
$83$	119.462	1.43930	0.719650	−	0.694337i	$-0.244302\pi$
$83$	119.462	1.43930	0.719650	+	0.694337i	$0.244302\pi$
$84$	0	0
$85$	0	0
$86$	130.408i	1.51637i
$87$	0	0
$88$	22.5244i	0.255959i
$89$	152.844i	1.71735i	0.512523	+	0.858674i	$0.328711\pi$
$89$	152.844i	1.71735i	−0.512523	+	0.858674i	$0.671289\pi$
$90$	0	0
$91$	130.165	1.43038
$92$	0.399493	0.00434231
$93$	0	0
$94$	−137.552	−1.46332
$95$	0	0
$96$	0	0
$97$	16.4385i	0.169469i	0.996404	+	0.0847345i	$0.0270042\pi$
$97$	16.4385i	0.169469i	−0.996404	+	0.0847345i	$0.972996\pi$
$98$	−15.3521	−0.156654
$99$	0	0
$100$	0	0
$101$	107.615i	1.06550i	0.846274	+	0.532748i	$0.178841\pi$
$101$	107.615i	1.06550i	−0.846274	+	0.532748i	$0.821159\pi$
$102$	0	0
$103$	28.6540i	0.278194i	0.990279	+	0.139097i	$0.0444200\pi$
$103$	28.6540i	0.278194i	−0.990279	+	0.139097i	$0.955580\pi$
$104$	− 166.440i	− 1.60039i
$105$	0	0
$106$	−158.046	−1.49100
$107$	77.6672	0.725862	0.362931	−	0.931816i	$-0.381776\pi$
$107$	77.6672	0.725862	0.362931	+	0.931816i	$0.381776\pi$
$108$	0	0
$109$	205.842	1.88846	0.944228	−	0.329291i	$-0.106810\pi$
$109$	205.842	1.88846	0.944228	+	0.329291i	$0.106810\pi$
$110$	0	0
$111$	0	0
$112$	− 96.8318i	− 0.864570i
$113$	−88.5285	−0.783438	−0.391719	−	0.920085i	$-0.628119\pi$
$113$	−88.5285	−0.783438	−0.391719	+	0.920085i	$0.628119\pi$
$114$	0	0
$115$	0	0
$116$	− 1.48802i	− 0.0128278i
$117$	0	0
$118$	206.968i	1.75397i
$119$	135.229i	1.13638i
$120$	0	0
$121$	113.452	0.937622
$122$	−26.1858	−0.214638
$123$	0	0
$124$	5.72399	0.0461612
$125$	0	0
$126$	0	0
$127$	− 51.9477i	− 0.409037i	−0.978863	−	0.204519i	$-0.934437\pi$
$127$	− 51.9477i	− 0.409037i	0.978863	−	0.204519i	$-0.0655629\pi$
$128$	−116.829	−0.912728
$129$	0	0
$130$	0	0
$131$	− 98.2420i	− 0.749939i	−0.927037	−	0.374969i	$-0.877653\pi$
$131$	− 98.2420i	− 0.749939i	0.927037	−	0.374969i	$-0.122347\pi$
$132$	0	0
$133$	− 87.5948i	− 0.658608i
$134$	− 122.751i	− 0.916055i
$135$	0	0
$136$	172.916	1.27144
$137$	81.1445	0.592296	0.296148	−	0.955142i	$-0.404298\pi$
$137$	81.1445	0.592296	0.296148	+	0.955142i	$0.404298\pi$
$138$	0	0
$139$	120.844	0.869381	0.434690	−	0.900580i	$-0.356858\pi$
$139$	120.844	0.869381	0.434690	+	0.900580i	$0.356858\pi$
$140$	0	0
$141$	0	0
$142$	62.5948i	0.440809i
$143$	55.7729	0.390020
$144$	0	0
$145$	0	0
$146$	− 240.999i	− 1.65068i
$147$	0	0
$148$	− 4.08942i	− 0.0276312i
$149$	112.793i	0.756998i	0.925602	+	0.378499i	$0.123560\pi$
$149$	112.793i	0.756998i	−0.925602	+	0.378499i	$0.876440\pi$
$150$	0	0
$151$	62.6221	0.414716	0.207358	−	0.978265i	$-0.433514\pi$
$151$	62.6221	0.414716	0.207358	+	0.978265i	$0.433514\pi$
$152$	−112.007	−0.736886
$153$	0	0
$154$	34.2791	0.222591
$155$	0	0
$156$	0	0
$157$	181.896i	1.15857i	0.815124	+	0.579286i	$0.196669\pi$
$157$	181.896i	1.15857i	−0.815124	+	0.579286i	$0.803331\pi$
$158$	−37.1429	−0.235081
$159$	0	0
$160$	0	0
$161$	− 12.0198i	− 0.0746574i
$162$	0	0
$163$	179.817i	1.10317i	0.834119	+	0.551585i	$0.185977\pi$
$163$	179.817i	1.10317i	−0.834119	+	0.551585i	$0.814023\pi$
$164$	− 14.3220i	− 0.0873296i
$165$	0	0
$166$	−232.472	−1.40044
$167$	−218.835	−1.31039	−0.655195	−	0.755460i	$-0.727414\pi$
$167$	−218.835	−1.31039	−0.655195	+	0.755460i	$0.727414\pi$
$168$	0	0
$169$	−243.125	−1.43861
$170$	0	0
$171$	0	0
$172$	14.2808i	0.0830276i
$173$	52.0991	0.301151	0.150575	−	0.988599i	$-0.451887\pi$
$173$	52.0991	0.301151	0.150575	+	0.988599i	$0.451887\pi$
$174$	0	0
$175$	0	0
$176$	− 41.4905i	− 0.235741i
$177$	0	0
$178$	− 297.434i	− 1.67097i
$179$	170.130i	0.950445i	0.879866	+	0.475223i	$0.157632\pi$
$179$	170.130i	0.950445i	−0.879866	+	0.475223i	$0.842368\pi$
$180$	0	0
$181$	−24.5286	−0.135517	−0.0677587	−	0.997702i	$-0.521585\pi$
$181$	−24.5286	−0.135517	−0.0677587	+	0.997702i	$0.521585\pi$
$182$	−253.300	−1.39176
$183$	0	0
$184$	−15.3696	−0.0835307
$185$	0	0
$186$	0	0
$187$	57.9428i	0.309854i
$188$	−15.0631	−0.0801227
$189$	0	0
$190$	0	0
$191$	− 231.080i	− 1.20984i	−0.796285	−	0.604922i	$-0.793204\pi$
$191$	− 231.080i	− 1.20984i	0.796285	−	0.604922i	$-0.206796\pi$
$192$	0	0
$193$	− 290.185i	− 1.50355i	−0.659420	−	0.751775i	$-0.729198\pi$
$193$	− 290.185i	− 1.50355i	0.659420	−	0.751775i	$-0.270802\pi$
$194$	− 31.9892i	− 0.164893i
$195$	0	0
$196$	−1.68118	−0.00857746
$197$	82.9589	0.421111	0.210555	−	0.977582i	$-0.432473\pi$
$197$	82.9589	0.421111	0.210555	+	0.977582i	$0.432473\pi$
$198$	0	0
$199$	37.8701	0.190302	0.0951510	−	0.995463i	$-0.469667\pi$
$199$	37.8701	0.190302	0.0951510	+	0.995463i	$0.469667\pi$
$200$	0	0
$201$	0	0
$202$	− 209.419i	− 1.03673i
$203$	−44.7713	−0.220548
$204$	0	0
$205$	0	0
$206$	− 55.7605i	− 0.270682i
$207$	0	0
$208$	306.587i	1.47398i
$209$	− 37.5326i	− 0.179582i
$210$	0	0
$211$	−223.286	−1.05823	−0.529114	−	0.848551i	$-0.677476\pi$
$211$	−223.286	−1.05823	−0.529114	+	0.848551i	$0.677476\pi$
$212$	−17.3073	−0.0816383
$213$	0	0
$214$	−151.140	−0.706262
$215$	0	0
$216$	0	0
$217$	− 172.222i	− 0.793649i
$218$	−400.567	−1.83746
$219$	0	0
$220$	0	0
$221$	− 428.159i	− 1.93737i
$222$	0	0
$223$	− 239.237i	− 1.07281i	−0.843960	−	0.536406i	$-0.819782\pi$
$223$	− 239.237i	− 1.07281i	0.843960	−	0.536406i	$-0.180218\pi$
$224$	− 21.8382i	− 0.0974919i
$225$	0	0
$226$	172.276	0.762283
$227$	−10.1604	−0.0447594	−0.0223797	−	0.999750i	$-0.507124\pi$
$227$	−10.1604	−0.0447594	−0.0223797	+	0.999750i	$0.507124\pi$
$228$	0	0
$229$	−140.208	−0.612263	−0.306131	−	0.951989i	$-0.599035\pi$
$229$	−140.208	−0.612263	−0.306131	+	0.951989i	$0.599035\pi$
$230$	0	0
$231$	0	0
$232$	57.2486i	0.246761i
$233$	−224.074	−0.961692	−0.480846	−	0.876805i	$-0.659670\pi$
$233$	−224.074	−0.961692	−0.480846	+	0.876805i	$0.659670\pi$
$234$	0	0
$235$	0	0
$236$	22.6647i	0.0960370i
$237$	0	0
$238$	− 263.154i	− 1.10569i
$239$	446.856i	1.86969i	0.355054	+	0.934846i	$0.384463\pi$
$239$	446.856i	1.86969i	−0.355054	+	0.934846i	$0.615537\pi$
$240$	0	0
$241$	−420.097	−1.74314	−0.871572	−	0.490268i	$-0.836899\pi$
$241$	−420.097	−1.74314	−0.871572	+	0.490268i	$0.836899\pi$
$242$	−220.778	−0.912304
$243$	0	0
$244$	−2.86756	−0.0117523
$245$	0	0
$246$	0	0
$247$	277.341i	1.12284i
$248$	−220.218	−0.887977
$249$	0	0
$250$	0	0
$251$	− 393.654i	− 1.56834i	−0.620544	−	0.784171i	$-0.713088\pi$
$251$	− 393.654i	− 1.56834i	0.620544	−	0.784171i	$-0.286912\pi$
$252$	0	0
$253$	− 5.15026i	− 0.0203567i
$254$	101.090i	0.397992i
$255$	0	0
$256$	−40.7975	−0.159365
$257$	350.416	1.36349	0.681743	−	0.731592i	$-0.261222\pi$
$257$	350.416	1.36349	0.681743	+	0.731592i	$0.261222\pi$
$258$	0	0
$259$	−123.042	−0.475064
$260$	0	0
$261$	0	0
$262$	191.178i	0.729688i
$263$	168.686	0.641393	0.320696	−	0.947182i	$-0.396083\pi$
$263$	168.686	0.641393	0.320696	+	0.947182i	$0.396083\pi$
$264$	0	0
$265$	0	0
$266$	170.459i	0.640824i
$267$	0	0
$268$	− 13.4423i	− 0.0501579i
$269$	− 138.025i	− 0.513103i	−0.966531	−	0.256552i	$-0.917414\pi$
$269$	− 138.025i	− 0.513103i	0.966531	−	0.256552i	$-0.0825864\pi$
$270$	0	0
$271$	147.030	0.542547	0.271274	−	0.962502i	$-0.412555\pi$
$271$	147.030	0.542547	0.271274	+	0.962502i	$0.412555\pi$
$272$	−318.515	−1.17101
$273$	0	0
$274$	−157.907	−0.576302
$275$	0	0
$276$	0	0
$277$	244.575i	0.882943i	0.897275	+	0.441472i	$0.145543\pi$
$277$	244.575i	0.882943i	−0.897275	+	0.441472i	$0.854457\pi$
$278$	−235.162	−0.845905
$279$	0	0
$280$	0	0
$281$	274.284i	0.976100i	0.872816	+	0.488050i	$0.162292\pi$
$281$	274.284i	0.976100i	−0.872816	+	0.488050i	$0.837708\pi$
$282$	0	0
$283$	− 166.960i	− 0.589965i	−0.955503	−	0.294982i	$-0.904686\pi$
$283$	− 166.960i	− 0.589965i	0.955503	−	0.294982i	$-0.0953138\pi$
$284$	6.85466i	0.0241361i
$285$	0	0
$286$	−108.534	−0.379489
$287$	−430.918	−1.50146
$288$	0	0
$289$	155.816	0.539157
$290$	0	0
$291$	0	0
$292$	− 26.3914i	− 0.0903816i
$293$	459.087	1.56685	0.783426	−	0.621486i	$-0.213471\pi$
$293$	459.087	1.56685	0.783426	+	0.621486i	$0.213471\pi$
$294$	0	0
$295$	0	0
$296$	157.332i	0.531527i
$297$	0	0
$298$	− 219.494i	− 0.736557i
$299$	38.0570i	0.127281i
$300$	0	0
$301$	429.676	1.42749
$302$	−121.862	−0.403517
$303$	0	0
$304$	206.319	0.678681
$305$	0	0
$306$	0	0
$307$	− 257.545i	− 0.838908i	−0.907777	−	0.419454i	$-0.862222\pi$
$307$	− 257.545i	− 0.838908i	0.907777	−	0.419454i	$-0.137778\pi$
$308$	3.75385	0.0121878
$309$	0	0
$310$	0	0
$311$	− 314.146i	− 1.01012i	−0.863085	−	0.505058i	$-0.831471\pi$
$311$	− 314.146i	− 1.01012i	0.863085	−	0.505058i	$-0.168529\pi$
$312$	0	0
$313$	119.111i	0.380548i	0.981731	+	0.190274i	$0.0609376\pi$
$313$	119.111i	0.380548i	−0.981731	+	0.190274i	$0.939062\pi$
$314$	− 353.969i	− 1.12729i
$315$	0	0
$316$	−4.06745	−0.0128717
$317$	344.122	1.08556	0.542780	−	0.839875i	$-0.317372\pi$
$317$	344.122	1.08556	0.542780	+	0.839875i	$0.317372\pi$
$318$	0	0
$319$	−19.1836	−0.0601366
$320$	0	0
$321$	0	0
$322$	23.3905i	0.0726414i
$323$	−288.131	−0.892047
$324$	0	0
$325$	0	0
$326$	− 349.922i	− 1.07338i
$327$	0	0
$328$	551.011i	1.67991i
$329$	453.214i	1.37755i
$330$	0	0
$331$	−41.0069	−0.123888	−0.0619439	−	0.998080i	$-0.519730\pi$
$331$	−41.0069	−0.123888	−0.0619439	+	0.998080i	$0.519730\pi$
$332$	−25.4577	−0.0766797
$333$	0	0
$334$	425.852	1.27501
$335$	0	0
$336$	0	0
$337$	− 225.925i	− 0.670399i	−0.942147	−	0.335200i	$-0.891196\pi$
$337$	− 225.925i	− 0.670399i	0.942147	−	0.335200i	$-0.108804\pi$
$338$	473.119	1.39976
$339$	0	0
$340$	0	0
$341$	− 73.7936i	− 0.216403i
$342$	0	0
$343$	364.760i	1.06344i
$344$	− 549.422i	− 1.59716i
$345$	0	0
$346$	−101.385	−0.293019
$347$	407.547	1.17449	0.587244	−	0.809410i	$-0.300213\pi$
$347$	407.547	1.17449	0.587244	+	0.809410i	$0.300213\pi$
$348$	0	0
$349$	39.1150	0.112077	0.0560387	−	0.998429i	$-0.482153\pi$
$349$	39.1150	0.112077	0.0560387	+	0.998429i	$0.482153\pi$
$350$	0	0
$351$	0	0
$352$	− 9.35722i	− 0.0265830i
$353$	43.5078	0.123252	0.0616258	−	0.998099i	$-0.480371\pi$
$353$	43.5078	0.123252	0.0616258	+	0.998099i	$0.480371\pi$
$354$	0	0
$355$	0	0
$356$	− 32.5714i	− 0.0914928i
$357$	0	0
$358$	− 331.072i	− 0.924781i
$359$	30.7176i	0.0855643i	0.999084	+	0.0427821i	$0.0136221\pi$
$359$	30.7176i	0.0855643i	−0.999084	+	0.0427821i	$0.986378\pi$
$360$	0	0
$361$	−174.362	−0.482998
$362$	47.7326	0.131858
$363$	0	0
$364$	−27.7384	−0.0762045
$365$	0	0
$366$	0	0
$367$	− 576.528i	− 1.57092i	−0.618912	−	0.785460i	$-0.712426\pi$
$367$	− 576.528i	− 1.57092i	0.618912	−	0.785460i	$-0.287574\pi$
$368$	28.3113	0.0769328
$369$	0	0
$370$	0	0
$371$	520.738i	1.40361i
$372$	0	0
$373$	353.505i	0.947735i	0.880596	+	0.473867i	$0.157142\pi$
$373$	353.505i	0.947735i	−0.880596	+	0.473867i	$0.842858\pi$
$374$	− 112.756i	− 0.301488i
$375$	0	0
$376$	579.520	1.54128
$377$	141.754	0.376005
$378$	0	0
$379$	−545.663	−1.43974	−0.719871	−	0.694107i	$-0.755799\pi$
$379$	−545.663	−1.43974	−0.719871	+	0.694107i	$0.755799\pi$
$380$	0	0
$381$	0	0
$382$	449.681i	1.17717i
$383$	−62.8586	−0.164122	−0.0820609	−	0.996627i	$-0.526150\pi$
$383$	−62.8586	−0.164122	−0.0820609	+	0.996627i	$0.526150\pi$
$384$	0	0
$385$	0	0
$386$	564.699i	1.46295i
$387$	0	0
$388$	− 3.50309i	− 0.00902857i
$389$	− 194.282i	− 0.499441i	−0.968318	−	0.249720i	$-0.919661\pi$
$389$	− 194.282i	− 0.499441i	0.968318	−	0.249720i	$-0.0803387\pi$
$390$	0	0
$391$	−39.5376	−0.101119
$392$	64.6800	0.165000
$393$	0	0
$394$	−161.438	−0.409740
$395$	0	0
$396$	0	0
$397$	− 490.342i	− 1.23512i	−0.786525	−	0.617559i	$-0.788122\pi$
$397$	− 490.342i	− 1.23512i	0.786525	−	0.617559i	$-0.211878\pi$
$398$	−73.6951	−0.185163
$399$	0	0
$400$	0	0
$401$	496.793i	1.23888i	0.785042	+	0.619442i	$0.212641\pi$
$401$	496.793i	1.23888i	−0.785042	+	0.619442i	$0.787359\pi$
$402$	0	0
$403$	545.285i	1.35307i
$404$	− 22.9331i	− 0.0567651i
$405$	0	0
$406$	87.1247	0.214593
$407$	−52.7208	−0.129535
$408$	0	0
$409$	266.707	0.652095	0.326047	−	0.945353i	$-0.394283\pi$
$409$	266.707	0.652095	0.326047	+	0.945353i	$0.394283\pi$
$410$	0	0
$411$	0	0
$412$	− 6.10624i	− 0.0148210i
$413$	681.931	1.65116
$414$	0	0
$415$	0	0
$416$	69.1436i	0.166211i
$417$	0	0
$418$	73.0383i	0.174733i
$419$	329.828i	0.787180i	0.919286	+	0.393590i	$0.128767\pi$
$419$	329.828i	0.787180i	−0.919286	+	0.393590i	$0.871233\pi$
$420$	0	0
$421$	184.257	0.437665	0.218832	−	0.975762i	$-0.429775\pi$
$421$	184.257	0.437665	0.218832	+	0.975762i	$0.429775\pi$
$422$	434.514	1.02965
$423$	0	0
$424$	665.863	1.57043
$425$	0	0
$426$	0	0
$427$	86.2786i	0.202058i
$428$	−16.5511	−0.0386708
$429$	0	0
$430$	0	0
$431$	590.656i	1.37043i	0.728340	+	0.685216i	$0.240292\pi$
$431$	590.656i	1.37043i	−0.728340	+	0.685216i	$0.759708\pi$
$432$	0	0
$433$	308.140i	0.711640i	0.934555	+	0.355820i	$0.115798\pi$
$433$	308.140i	0.711640i	−0.934555	+	0.355820i	$0.884202\pi$
$434$	335.143i	0.772219i
$435$	0	0
$436$	−43.8654	−0.100609
$437$	25.6106	0.0586055
$438$	0	0
$439$	679.266	1.54730	0.773651	−	0.633612i	$-0.218428\pi$
$439$	679.266	1.54730	0.773651	+	0.633612i	$0.218428\pi$
$440$	0	0
$441$	0	0
$442$	833.195i	1.88506i
$443$	−147.846	−0.333738	−0.166869	−	0.985979i	$-0.553366\pi$
$443$	−147.846	−0.333738	−0.166869	+	0.985979i	$0.553366\pi$
$444$	0	0
$445$	0	0
$446$	465.554i	1.04384i
$447$	0	0
$448$	429.824i	0.959429i
$449$	− 315.942i	− 0.703657i	−0.936064	−	0.351829i	$-0.885560\pi$
$449$	− 315.942i	− 0.703657i	0.936064	−	0.351829i	$-0.114440\pi$
$450$	0	0
$451$	−184.640	−0.409401
$452$	18.8657	0.0417382
$453$	0	0
$454$	19.7721	0.0435508
$455$	0	0
$456$	0	0
$457$	− 314.688i	− 0.688596i	−0.938860	−	0.344298i	$-0.888117\pi$
$457$	− 314.688i	− 0.688596i	0.938860	−	0.344298i	$-0.111883\pi$
$458$	272.844	0.595730
$459$	0	0
$460$	0	0
$461$	343.949i	0.746093i	0.927813	+	0.373046i	$0.121687\pi$
$461$	343.949i	0.746093i	−0.927813	+	0.373046i	$0.878313\pi$
$462$	0	0
$463$	− 318.673i	− 0.688278i	−0.938919	−	0.344139i	$-0.888171\pi$
$463$	− 318.673i	− 0.688278i	0.938919	−	0.344139i	$-0.111829\pi$
$464$	− 105.453i	− 0.227270i
$465$	0	0
$466$	436.048	0.935724
$467$	33.4248	0.0715733	0.0357867	−	0.999359i	$-0.488606\pi$
$467$	33.4248	0.0715733	0.0357867	+	0.999359i	$0.488606\pi$
$468$	0	0
$469$	−404.449	−0.862364
$470$	0	0
$471$	0	0
$472$	− 871.978i	− 1.84741i
$473$	184.107	0.389233
$474$	0	0
$475$	0	0
$476$	− 28.8176i	− 0.0605412i
$477$	0	0
$478$	− 869.580i	− 1.81921i
$479$	− 104.499i	− 0.218160i	−0.994033	−	0.109080i	$-0.965209\pi$
$479$	− 104.499i	− 0.218160i	0.994033	−	0.109080i	$-0.0347905\pi$
$480$	0	0
$481$	389.572	0.809921
$482$	817.508	1.69607
$483$	0	0
$484$	−24.1770	−0.0499524
$485$	0	0
$486$	0	0
$487$	220.335i	0.452434i	0.974077	+	0.226217i	$0.0726358\pi$
$487$	220.335i	0.452434i	−0.974077	+	0.226217i	$0.927364\pi$
$488$	110.324	0.226073
$489$	0	0
$490$	0	0
$491$	− 670.746i	− 1.36608i	−0.730380	−	0.683041i	$-0.760657\pi$
$491$	− 670.746i	− 1.36608i	0.730380	−	0.683041i	$-0.239343\pi$
$492$	0	0
$493$	147.269i	0.298720i
$494$	− 539.704i	− 1.09252i
$495$	0	0
$496$	405.648	0.817838
$497$	206.241	0.414972
$498$	0	0
$499$	522.246	1.04658	0.523292	−	0.852153i	$-0.324704\pi$
$499$	522.246	1.04658	0.523292	+	0.852153i	$0.324704\pi$
$500$	0	0
$501$	0	0
$502$	766.049i	1.52599i
$503$	−197.944	−0.393527	−0.196763	−	0.980451i	$-0.563043\pi$
$503$	−197.944	−0.393527	−0.196763	+	0.980451i	$0.563043\pi$
$504$	0	0
$505$	0	0
$506$	10.0224i	0.0198071i
$507$	0	0
$508$	11.0702i	0.0217917i
$509$	− 503.620i	− 0.989431i	−0.869055	−	0.494715i	$-0.835272\pi$
$509$	− 503.620i	− 0.989431i	0.869055	−	0.494715i	$-0.164728\pi$
$510$	0	0
$511$	−794.058	−1.55393
$512$	546.709	1.06779
$513$	0	0
$514$	−681.908	−1.32667
$515$	0	0
$516$	0	0
$517$	194.193i	0.375615i
$518$	239.438	0.462236
$519$	0	0
$520$	0	0
$521$	474.180i	0.910134i	0.890457	+	0.455067i	$0.150385\pi$
$521$	474.180i	0.910134i	−0.890457	+	0.455067i	$0.849615\pi$
$522$	0	0
$523$	− 710.931i	− 1.35933i	−0.733521	−	0.679666i	$-0.762125\pi$
$523$	− 710.931i	− 1.35933i	0.733521	−	0.679666i	$-0.237875\pi$
$524$	20.9356i	0.0399535i
$525$	0	0
$526$	−328.263	−0.624073
$527$	−566.500	−1.07495
$528$	0	0
$529$	−525.486	−0.993357
$530$	0	0
$531$	0	0
$532$	18.6667i	0.0350878i
$533$	1364.36	2.55978
$534$	0	0
$535$	0	0
$536$	517.164i	0.964859i
$537$	0	0
$538$	268.596i	0.499248i
$539$	21.6738i	0.0402111i
$540$	0	0
$541$	−829.835	−1.53389	−0.766946	−	0.641712i	$-0.778224\pi$
$541$	−829.835	−1.53389	−0.766946	+	0.641712i	$0.778224\pi$
$542$	−286.120	−0.527897
$543$	0	0
$544$	−71.8337	−0.132047
$545$	0	0
$546$	0	0
$547$	− 71.3316i	− 0.130405i	−0.997872	−	0.0652026i	$-0.979231\pi$
$547$	− 71.3316i	− 0.130405i	0.997872	−	0.0652026i	$-0.0207694\pi$
$548$	−17.2921	−0.0315550
$549$	0	0
$550$	0	0
$551$	− 95.3939i	− 0.173129i
$552$	0	0
$553$	122.380i	0.221303i
$554$	− 475.942i	− 0.859102i
$555$	0	0
$556$	−25.7522	−0.0463168
$557$	−52.0760	−0.0934937	−0.0467468	−	0.998907i	$-0.514885\pi$
$557$	−52.0760	−0.0934937	−0.0467468	+	0.998907i	$0.514885\pi$
$558$	0	0
$559$	−1360.43	−2.43369
$560$	0	0
$561$	0	0
$562$	− 533.756i	− 0.949743i
$563$	−297.145	−0.527789	−0.263895	−	0.964552i	$-0.585007\pi$
$563$	−297.145	−0.527789	−0.263895	+	0.964552i	$0.585007\pi$
$564$	0	0
$565$	0	0
$566$	324.903i	0.574034i
$567$	0	0
$568$	− 263.718i	− 0.464293i
$569$	− 117.372i	− 0.206278i	−0.994667	−	0.103139i	$-0.967111\pi$
$569$	− 117.372i	− 0.206278i	0.994667	−	0.103139i	$-0.0328887\pi$
$570$	0	0
$571$	595.259	1.04248	0.521242	−	0.853409i	$-0.325469\pi$
$571$	595.259	1.04248	0.521242	+	0.853409i	$0.325469\pi$
$572$	−11.8854	−0.0207786
$573$	0	0
$574$	838.565	1.46091
$575$	0	0
$576$	0	0
$577$	− 12.9488i	− 0.0224415i	−0.999937	−	0.0112208i	$-0.996428\pi$
$577$	− 12.9488i	− 0.0224415i	0.999937	−	0.0112208i	$-0.00357176\pi$
$578$	−303.218	−0.524598
$579$	0	0
$580$	0	0
$581$	765.964i	1.31835i
$582$	0	0
$583$	223.126i	0.382720i
$584$	1015.35i	1.73862i
$585$	0	0
$586$	−893.382	−1.52454
$587$	742.994	1.26575	0.632874	−	0.774255i	$-0.281875\pi$
$587$	742.994	1.26575	0.632874	+	0.774255i	$0.281875\pi$
$588$	0	0
$589$	366.952	0.623009
$590$	0	0
$591$	0	0
$592$	− 289.810i	− 0.489543i
$593$	470.348	0.793167	0.396584	−	0.917999i	$-0.370196\pi$
$593$	470.348	0.793167	0.396584	+	0.917999i	$0.370196\pi$
$594$	0	0
$595$	0	0
$596$	− 24.0364i	− 0.0403296i
$597$	0	0
$598$	− 74.0587i	− 0.123844i
$599$	− 144.766i	− 0.241680i	−0.992672	−	0.120840i	$-0.961441\pi$
$599$	− 144.766i	− 0.241680i	0.992672	−	0.120840i	$-0.0385588\pi$
$600$	0	0
$601$	256.583	0.426926	0.213463	−	0.976951i	$-0.431526\pi$
$601$	256.583	0.426926	0.213463	+	0.976951i	$0.431526\pi$
$602$	−836.147	−1.38895
$603$	0	0
$604$	−13.3449	−0.0220943
$605$	0	0
$606$	0	0
$607$	356.366i	0.587094i	0.955945	+	0.293547i	$0.0948357\pi$
$607$	356.366i	0.587094i	−0.955945	+	0.293547i	$0.905164\pi$
$608$	46.5305	0.0765304
$609$	0	0
$610$	0	0
$611$	− 1434.96i	− 2.34854i
$612$	0	0
$613$	− 1145.01i	− 1.86789i	−0.357422	−	0.933943i	$-0.616344\pi$
$613$	− 1145.01i	− 1.86789i	0.357422	−	0.933943i	$-0.383656\pi$
$614$	501.181i	0.816255i
$615$	0	0
$616$	−144.421	−0.234450
$617$	−811.543	−1.31531	−0.657653	−	0.753321i	$-0.728450\pi$
$617$	−811.543	−1.31531	−0.657653	+	0.753321i	$0.728450\pi$
$618$	0	0
$619$	−595.797	−0.962515	−0.481258	−	0.876579i	$-0.659820\pi$
$619$	−595.797	−0.962515	−0.481258	+	0.876579i	$0.659820\pi$
$620$	0	0
$621$	0	0
$622$	611.327i	0.982841i
$623$	−980.002	−1.57304
$624$	0	0
$625$	0	0
$626$	− 231.790i	− 0.370272i
$627$	0	0
$628$	− 38.7625i	− 0.0617237i
$629$	404.728i	0.643447i
$630$	0	0
$631$	−584.585	−0.926442	−0.463221	−	0.886243i	$-0.653306\pi$
$631$	−584.585	−0.926442	−0.463221	+	0.886243i	$0.653306\pi$
$632$	156.487	0.247605
$633$	0	0
$634$	−669.660	−1.05625
$635$	0	0
$636$	0	0
$637$	− 160.155i	− 0.251421i
$638$	37.3312	0.0585128
$639$	0	0
$640$	0	0
$641$	803.853i	1.25406i	0.778995	+	0.627030i	$0.215730\pi$
$641$	803.853i	1.25406i	−0.778995	+	0.627030i	$0.784270\pi$
$642$	0	0
$643$	− 426.906i	− 0.663928i	−0.943292	−	0.331964i	$-0.892289\pi$
$643$	− 426.906i	− 0.663928i	0.943292	−	0.331964i	$-0.107711\pi$
$644$	2.56146i	0.00397742i
$645$	0	0
$646$	560.702	0.867959
$647$	803.477	1.24185	0.620925	−	0.783870i	$-0.286757\pi$
$647$	803.477	1.24185	0.620925	+	0.783870i	$0.286757\pi$
$648$	0	0
$649$	292.193	0.450221
$650$	0	0
$651$	0	0
$652$	− 38.3194i	− 0.0587721i
$653$	−1002.19	−1.53475	−0.767374	−	0.641200i	$-0.778437\pi$
$653$	−1002.19	−1.53475	−0.767374	+	0.641200i	$0.778437\pi$
$654$	0	0
$655$	0	0
$656$	− 1014.98i	− 1.54722i
$657$	0	0
$658$	− 881.952i	− 1.34035i
$659$	− 381.903i	− 0.579518i	−0.957100	−	0.289759i	$-0.906425\pi$
$659$	− 381.903i	− 0.579518i	0.957100	−	0.289759i	$-0.0935752\pi$
$660$	0	0
$661$	−850.360	−1.28647	−0.643237	−	0.765667i	$-0.722409\pi$
$661$	−850.360	−1.28647	−0.643237	+	0.765667i	$0.722409\pi$
$662$	79.7992	0.120543
$663$	0	0
$664$	979.430	1.47504
$665$	0	0
$666$	0	0
$667$	− 13.0900i	− 0.0196252i
$668$	46.6344	0.0698119
$669$	0	0
$670$	0	0
$671$	36.9686i	0.0550948i
$672$	0	0
$673$	− 200.948i	− 0.298585i	−0.988793	−	0.149292i	$-0.952300\pi$
$673$	− 200.948i	− 0.298585i	0.988793	−	0.149292i	$-0.0476996\pi$
$674$	439.648i	0.652297i
$675$	0	0
$676$	51.8105	0.0766428
$677$	89.2667	0.131856	0.0659281	−	0.997824i	$-0.478999\pi$
$677$	89.2667	0.131856	0.0659281	+	0.997824i	$0.478999\pi$
$678$	0	0
$679$	−105.400	−0.155228
$680$	0	0
$681$	0	0
$682$	143.602i	0.210560i
$683$	−995.726	−1.45787	−0.728936	−	0.684582i	$-0.759985\pi$
$683$	−995.726	−1.45787	−0.728936	+	0.684582i	$0.759985\pi$
$684$	0	0
$685$	0	0
$686$	− 709.822i	− 1.03473i
$687$	0	0
$688$	1012.05i	1.47100i
$689$	− 1648.75i	− 2.39296i
$690$	0	0
$691$	948.518	1.37267	0.686337	−	0.727284i	$-0.259218\pi$
$691$	948.518	1.37267	0.686337	+	0.727284i	$0.259218\pi$
$692$	−11.1025	−0.0160440
$693$	0	0
$694$	−793.085	−1.14277
$695$	0	0
$696$	0	0
$697$	1417.45i	2.03364i
$698$	−76.1176	−0.109051
$699$	0	0
$700$	0	0
$701$	618.434i	0.882217i	0.897454	+	0.441109i	$0.145415\pi$
$701$	618.434i	0.882217i	−0.897454	+	0.441109i	$0.854585\pi$
$702$	0	0
$703$	− 262.164i	− 0.372922i
$704$	184.171i	0.261607i
$705$	0	0
$706$	−84.6660	−0.119924
$707$	−690.005	−0.975962
$708$	0	0
$709$	−208.535	−0.294126	−0.147063	−	0.989127i	$-0.546982\pi$
$709$	−208.535	−0.294126	−0.147063	+	0.989127i	$0.546982\pi$
$710$	0	0
$711$	0	0
$712$	1253.12i	1.76000i
$713$	50.3535	0.0706220
$714$	0	0
$715$	0	0
$716$	− 36.2551i	− 0.0506356i
$717$	0	0
$718$	− 59.7763i	− 0.0832539i
$719$	1077.05i	1.49799i	0.662576	+	0.748995i	$0.269463\pi$
$719$	1077.05i	1.49799i	−0.662576	+	0.748995i	$0.730537\pi$
$720$	0	0
$721$	−183.723	−0.254817
$722$	339.308	0.469956
$723$	0	0
$724$	5.22712	0.00721978
$725$	0	0
$726$	0	0
$727$	− 1352.36i	− 1.86019i	−0.367321	−	0.930094i	$-0.619725\pi$
$727$	− 1352.36i	− 1.86019i	0.367321	−	0.930094i	$-0.380275\pi$
$728$	1067.18	1.46590
$729$	0	0
$730$	0	0
$731$	− 1413.36i	− 1.93346i
$732$	0	0
$733$	394.787i	0.538591i	0.963058	+	0.269295i	$0.0867908\pi$
$733$	394.787i	0.538591i	−0.963058	+	0.269295i	$0.913209\pi$
$734$	1121.92i	1.52850i
$735$	0	0
$736$	6.38495	0.00867521
$737$	−173.298	−0.235140
$738$	0	0
$739$	947.519	1.28216	0.641082	−	0.767473i	$-0.278486\pi$
$739$	947.519	1.28216	0.641082	+	0.767473i	$0.278486\pi$
$740$	0	0
$741$	0	0
$742$	− 1013.35i	− 1.36571i
$743$	−917.875	−1.23536	−0.617682	−	0.786428i	$-0.711928\pi$
$743$	−917.875	−1.23536	−0.617682	+	0.786428i	$0.711928\pi$
$744$	0	0
$745$	0	0
$746$	− 687.919i	− 0.922144i
$747$	0	0
$748$	− 12.3478i	− 0.0165077i
$749$	497.985i	0.664867i
$750$	0	0
$751$	1373.39	1.82875	0.914374	−	0.404871i	$-0.132684\pi$
$751$	1373.39	1.82875	0.914374	+	0.404871i	$0.132684\pi$
$752$	−1067.49	−1.41954
$753$	0	0
$754$	−275.852	−0.365852
$755$	0	0
$756$	0	0
$757$	− 907.831i	− 1.19925i	−0.800282	−	0.599624i	$-0.795317\pi$
$757$	− 907.831i	− 1.19925i	0.800282	−	0.599624i	$-0.204683\pi$
$758$	1061.86	1.40087
$759$	0	0
$760$	0	0
$761$	− 532.362i	− 0.699555i	−0.936833	−	0.349778i	$-0.886257\pi$
$761$	− 532.362i	− 0.699555i	0.936833	−	0.349778i	$-0.113743\pi$
$762$	0	0
$763$	1319.81i	1.72977i
$764$	49.2438i	0.0644552i
$765$	0	0
$766$	122.323	0.159690
$767$	−2159.12	−2.81501
$768$	0	0
$769$	101.705	0.132257	0.0661284	−	0.997811i	$-0.478935\pi$
$769$	101.705	0.132257	0.0661284	+	0.997811i	$0.478935\pi$
$770$	0	0
$771$	0	0
$772$	61.8392i	0.0801026i
$773$	−760.965	−0.984431	−0.492215	−	0.870473i	$-0.663813\pi$
$773$	−760.965	−0.984431	−0.492215	+	0.870473i	$0.663813\pi$
$774$	0	0
$775$	0	0
$776$	134.774i	0.173678i
$777$	0	0
$778$	378.073i	0.485955i
$779$	− 918.155i	− 1.17863i
$780$	0	0
$781$	88.3701	0.113150
$782$	76.9400	0.0983887
$783$	0	0
$784$	−119.142	−0.151967
$785$	0	0
$786$	0	0
$787$	311.448i	0.395741i	0.980228	+	0.197870i	$0.0634025\pi$
$787$	311.448i	0.395741i	−0.980228	+	0.197870i	$0.936598\pi$
$788$	−17.6788	−0.0224350
$789$	0	0
$790$	0	0
$791$	− 567.625i	− 0.717605i
$792$	0	0
$793$	− 273.174i	− 0.344481i
$794$	954.202i	1.20177i
$795$	0	0
$796$	−8.07022	−0.0101385
$797$	658.366	0.826055	0.413027	−	0.910719i	$-0.364471\pi$
$797$	658.366	0.826055	0.413027	+	0.910719i	$0.364471\pi$
$798$	0	0
$799$	1490.78	1.86581
$800$	0	0
$801$	0	0
$802$	− 966.756i	− 1.20543i
$803$	−340.238	−0.423708
$804$	0	0
$805$	0	0
$806$	− 1061.12i	− 1.31653i
$807$	0	0
$808$	882.302i	1.09196i
$809$	− 1026.08i	− 1.26833i	−0.773197	−	0.634166i	$-0.781343\pi$
$809$	− 1026.08i	− 1.26833i	0.773197	−	0.634166i	$-0.218657\pi$
$810$	0	0
$811$	−1489.54	−1.83668	−0.918338	−	0.395798i	$-0.870468\pi$
$811$	−1489.54	−1.83668	−0.918338	+	0.395798i	$0.870468\pi$
$812$	9.54088	0.0117498
$813$	0	0
$814$	102.594	0.126037
$815$	0	0
$816$	0	0
$817$	915.508i	1.12057i
$818$	−519.010	−0.634486
$819$	0	0
$820$	0	0
$821$	− 594.254i	− 0.723818i	−0.932213	−	0.361909i	$-0.882125\pi$
$821$	− 594.254i	− 0.723818i	0.932213	−	0.361909i	$-0.117875\pi$
$822$	0	0
$823$	543.853i	0.660818i	0.943838	+	0.330409i	$0.107187\pi$
$823$	543.853i	0.660818i	−0.943838	+	0.330409i	$0.892813\pi$
$824$	234.925i	0.285103i
$825$	0	0
$826$	−1327.03	−1.60658
$827$	−1566.71	−1.89445	−0.947225	−	0.320570i	$-0.896125\pi$
$827$	−1566.71	−1.89445	−0.947225	+	0.320570i	$0.896125\pi$
$828$	0	0
$829$	−896.771	−1.08175	−0.540875	−	0.841103i	$-0.681907\pi$
$829$	−896.771	−1.08175	−0.540875	+	0.841103i	$0.681907\pi$
$830$	0	0
$831$	0	0
$832$	− 1360.90i	− 1.63570i
$833$	166.386	0.199743
$834$	0	0
$835$	0	0
$836$	7.99830i	0.00956734i
$837$	0	0
$838$	− 641.844i	− 0.765924i
$839$	476.848i	0.568353i	0.958772	+	0.284176i	$0.0917202\pi$
$839$	476.848i	0.568353i	−0.958772	+	0.284176i	$0.908280\pi$
$840$	0	0
$841$	792.242	0.942024
$842$	−358.563	−0.425847
$843$	0	0
$844$	47.5829	0.0563778
$845$	0	0
$846$	0	0
$847$	727.431i	0.858832i
$848$	−1226.54	−1.44639
$849$	0	0
$850$	0	0
$851$	− 35.9744i	− 0.0422731i
$852$	0	0
$853$	195.809i	0.229553i	0.993391	+	0.114777i	$0.0366152\pi$
$853$	195.809i	0.229553i	−0.993391	+	0.114777i	$0.963385\pi$
$854$	− 167.898i	− 0.196601i
$855$	0	0
$856$	636.769	0.743888
$857$	633.759	0.739509	0.369754	−	0.929129i	$-0.379442\pi$
$857$	633.759	0.739509	0.369754	+	0.929129i	$0.379442\pi$
$858$	0	0
$859$	676.916	0.788027	0.394014	−	0.919105i	$-0.371086\pi$
$859$	676.916	0.788027	0.394014	+	0.919105i	$0.371086\pi$
$860$	0	0
$861$	0	0
$862$	− 1149.41i	− 1.33343i
$863$	1373.71	1.59179	0.795895	−	0.605435i	$-0.207001\pi$
$863$	1373.71	1.59179	0.795895	+	0.605435i	$0.207001\pi$
$864$	0	0
$865$	0	0
$866$	− 599.639i	− 0.692424i
$867$	0	0
$868$	36.7009i	0.0422822i
$869$	52.4375i	0.0603424i
$870$	0	0
$871$	1280.56	1.47022
$872$	1687.63	1.93536
$873$	0	0
$874$	−49.8381	−0.0570230
$875$	0	0
$876$	0	0
$877$	338.203i	0.385636i	0.981235	+	0.192818i	$0.0617627\pi$
$877$	338.203i	0.385636i	−0.981235	+	0.192818i	$0.938237\pi$
$878$	−1321.85	−1.50552
$879$	0	0
$880$	0	0
$881$	117.833i	0.133749i	0.997761	+	0.0668747i	$0.0213028\pi$
$881$	117.833i	0.133749i	−0.997761	+	0.0668747i	$0.978697\pi$
$882$	0	0
$883$	741.709i	0.839988i	0.907527	+	0.419994i	$0.137968\pi$
$883$	741.709i	0.839988i	−0.907527	+	0.419994i	$0.862032\pi$
$884$	91.2417i	0.103215i
$885$	0	0
$886$	287.707	0.324726
$887$	−467.896	−0.527504	−0.263752	−	0.964590i	$-0.584960\pi$
$887$	−467.896	−0.527504	−0.263752	+	0.964590i	$0.584960\pi$
$888$	0	0
$889$	333.077	0.374665
$890$	0	0
$891$	0	0
$892$	50.9820i	0.0571548i
$893$	−965.660	−1.08137
$894$	0	0
$895$	0	0
$896$	− 749.083i	− 0.836031i
$897$	0	0
$898$	614.822i	0.684657i
$899$	− 187.556i	− 0.208627i
$900$	0	0
$901$	1712.90	1.90111
$902$	359.308	0.398346
$903$	0	0
$904$	−725.817	−0.802894
$905$	0	0
$906$	0	0
$907$	822.407i	0.906733i	0.891324	+	0.453366i	$0.149777\pi$
$907$	822.407i	0.906733i	−0.891324	+	0.453366i	$0.850223\pi$
$908$	2.16521	0.00238459
$909$	0	0
$910$	0	0
$911$	− 55.3187i	− 0.0607230i	−0.999539	−	0.0303615i	$-0.990334\pi$
$911$	− 55.3187i	− 0.0607230i	0.999539	−	0.0303615i	$-0.00966586\pi$
$912$	0	0
$913$	328.200i	0.359474i
$914$	612.382i	0.670002i
$915$	0	0
$916$	29.8787	0.0326187
$917$	629.906	0.686920
$918$	0	0
$919$	52.0847	0.0566755	0.0283377	−	0.999598i	$-0.490979\pi$
$919$	52.0847	0.0566755	0.0283377	+	0.999598i	$0.490979\pi$
$920$	0	0
$921$	0	0
$922$	− 669.323i	− 0.725947i
$923$	−652.997	−0.707472
$924$	0	0
$925$	0	0
$926$	620.136i	0.669693i
$927$	0	0
$928$	− 23.7826i	− 0.0256278i
$929$	− 932.074i	− 1.00331i	−0.865068	−	0.501655i	$-0.832725\pi$
$929$	− 932.074i	− 1.00331i	0.865068	−	0.501655i	$-0.167275\pi$
$930$	0	0
$931$	−107.777	−0.115765
$932$	47.7508	0.0512348
$933$	0	0
$934$	−65.0444	−0.0696407
$935$	0	0
$936$	0	0
$937$	− 283.275i	− 0.302322i	−0.988509	−	0.151161i	$-0.951699\pi$
$937$	− 283.275i	− 0.302322i	0.988509	−	0.151161i	$-0.0483011\pi$
$938$	787.055	0.839078
$939$	0	0
$940$	0	0
$941$	1579.56i	1.67860i	0.543672	+	0.839298i	$0.317034\pi$
$941$	1579.56i	1.67860i	−0.543672	+	0.839298i	$0.682966\pi$
$942$	0	0
$943$	− 125.990i	− 0.133606i
$944$	1606.21i	1.70149i
$945$	0	0
$946$	−358.272	−0.378723
$947$	−1108.91	−1.17097	−0.585484	−	0.810684i	$-0.699096\pi$
$947$	−1108.91	−1.17097	−0.585484	+	0.810684i	$0.699096\pi$
$948$	0	0
$949$	2514.13	2.64924
$950$	0	0
$951$	0	0
$952$	1108.70i	1.16460i
$953$	−352.378	−0.369757	−0.184878	−	0.982761i	$-0.559189\pi$
$953$	−352.378	−0.369757	−0.184878	+	0.982761i	$0.559189\pi$
$954$	0	0
$955$	0	0
$956$	− 95.2263i	− 0.0996091i
$957$	0	0
$958$	203.354i	0.212269i
$959$	520.281i	0.542524i
$960$	0	0
$961$	−239.528	−0.249249
$962$	−758.105	−0.788051
$963$	0	0
$964$	89.5239	0.0928671
$965$	0	0
$966$	0	0
$967$	639.884i	0.661721i	0.943680	+	0.330861i	$0.107339\pi$
$967$	639.884i	0.661721i	−0.943680	+	0.330861i	$0.892661\pi$
$968$	930.158	0.960907
$969$	0	0
$970$	0	0
$971$	26.1957i	0.0269781i	0.999909	+	0.0134890i	$0.00429383\pi$
$971$	26.1957i	0.0269781i	−0.999909	+	0.0134890i	$0.995706\pi$
$972$	0	0
$973$	774.825i	0.796325i
$974$	− 428.771i	− 0.440217i
$975$	0	0
$976$	−203.219	−0.208216
$977$	−523.893	−0.536226	−0.268113	−	0.963387i	$-0.586400\pi$
$977$	−523.893	−0.536226	−0.268113	+	0.963387i	$0.586400\pi$
$978$	0	0
$979$	−419.911	−0.428918
$980$	0	0
$981$	0	0
$982$	1305.27i	1.32919i
$983$	−161.009	−0.163793	−0.0818966	−	0.996641i	$-0.526098\pi$
$983$	−161.009	−0.163793	−0.0818966	+	0.996641i	$0.526098\pi$
$984$	0	0
$985$	0	0
$986$	− 286.585i	− 0.290654i
$987$	0	0
$988$	− 59.1021i	− 0.0598200i
$989$	125.627i	0.127024i
$990$	0	0
$991$	−345.150	−0.348285	−0.174142	−	0.984720i	$-0.555715\pi$
$991$	−345.150	−0.348285	−0.174142	+	0.984720i	$0.555715\pi$
$992$	91.4845	0.0922223
$993$	0	0
$994$	−401.344	−0.403767
$995$	0	0
$996$	0	0
$997$	− 1311.95i	− 1.31590i	−0.753061	−	0.657951i	$-0.771423\pi$
$997$	− 1311.95i	− 1.31590i	0.753061	−	0.657951i	$-0.228577\pi$
$998$	−1016.29	−1.01832
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	675.3.d.k.674.2		6
3.2	odd	2		675.3.d.j.674.6		6
5.2	odd	4		675.3.c.r.26.2	✓	6
5.3	odd	4		675.3.c.s.26.5	yes	6
5.4	even	2		675.3.d.j.674.5		6
15.2	even	4		675.3.c.r.26.5	yes	6
15.8	even	4		675.3.c.s.26.2	yes	6
15.14	odd	2	inner	675.3.d.k.674.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
675.3.c.r.26.2	✓	6	5.2	odd	4
675.3.c.r.26.5	yes	6	15.2	even	4
675.3.c.s.26.2	yes	6	15.8	even	4
675.3.c.s.26.5	yes	6	5.3	odd	4
675.3.d.j.674.5		6	5.4	even	2
675.3.d.j.674.6		6	3.2	odd	2
675.3.d.k.674.1		6	15.14	odd	2	inner
675.3.d.k.674.2		6	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:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	675.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.94600 q^{2} -0.213103 q^{4} +6.41178i q^{7} +8.19868 q^{8} +2.74732i q^{11} -20.3009i q^{13} -12.4773i q^{14} -15.1022 q^{16} +21.0907 q^{17} -13.6615 q^{19} -5.34627i q^{22} -1.87465 q^{23} +39.5054i q^{26} -1.36637i q^{28} +6.98266i q^{29} -26.8602 q^{31} -3.40595 q^{32} -41.0423 q^{34} +19.1899i q^{37} +26.5853 q^{38} +67.2073i q^{41} -67.0135i q^{43} -0.585461i q^{44} +3.64806 q^{46} +70.6846 q^{47} +7.88907 q^{49} +4.32617i q^{52} +81.2159 q^{53} +52.5681i q^{56} -13.5882i q^{58} -106.356i q^{59} +13.4563 q^{61} +52.2699 q^{62} +67.0367 q^{64} +63.0790i q^{67} -4.49448 q^{68} -32.1660i q^{71} +123.844i q^{73} -37.3435i q^{74} +2.91131 q^{76} -17.6152 q^{77} +19.0868 q^{79} -130.785i q^{82} +119.462 q^{83} +130.408i q^{86} +22.5244i q^{88} +152.844i q^{89} +130.165 q^{91} +0.399493 q^{92} -137.552 q^{94} +16.4385i q^{97} -15.3521 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{2} + 14 q^{4} + 42 q^{8} + 46 q^{16} + 84 q^{17} + 16 q^{19} + 102 q^{23} - 56 q^{31} + 174 q^{32} + 80 q^{34} + 96 q^{38} + 234 q^{46} + 138 q^{47} - 74 q^{49} + 120 q^{53} - 46 q^{61} - 36 q^{62}+ \cdots - 318 q^{98}+O(q^{100}) \) 6 * q + 6 * q^2 + 14 * q^4 + 42 * q^8 + 46 * q^16 + 84 * q^17 + 16 * q^19 + 102 * q^23 - 56 * q^31 + 174 * q^32 + 80 * q^34 + 96 * q^38 + 234 * q^46 + 138 * q^47 - 74 * q^49 + 120 * q^53 - 46 * q^61 - 36 * q^62 + 342 * q^64 + 432 * q^68 - 156 * q^76 + 492 * q^77 - 316 * q^79 + 324 * q^83 + 44 * q^91 + 210 * q^92 - 526 * q^94 - 318 * q^98

Introduction

L-functions

Modular forms

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Database

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