LMFDB - Embedded newform 1350.4.c.v.649.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1350,4,Mod(649,1350)]

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([0, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1350.649");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1350 = 2 \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1350.c (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:	$79.6525785077$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{21})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 11x^{2} + 25 $ x^4 + 11*x^2 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.1
Root			$-1.79129i$ of defining polynomial
Character	$\chi$	$=$	1350.649
Dual form			1350.4.c.v.649.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{2} -4.00000 q^{4} -8.74773i q^{7} +8.00000i q^{8} +O(q^{10})$	$q-2.00000i q^{2} -4.00000 q^{4} -8.74773i q^{7} +8.00000i q^{8} -69.9909 q^{11} -6.25227i q^{13} -17.4955 q^{14} +16.0000 q^{16} -83.7386i q^{17} +45.7386 q^{19} +139.982i q^{22} +56.7386i q^{23} -12.5045 q^{26} +34.9909i q^{28} -51.2341 q^{29} -284.955 q^{31} -32.0000i q^{32} -167.477 q^{34} -21.2432i q^{37} -91.4773i q^{38} +14.9727 q^{41} -373.739i q^{43} +279.964 q^{44} +113.477 q^{46} +380.739i q^{47} +266.477 q^{49} +25.0091i q^{52} +54.2614i q^{53} +69.9818 q^{56} +102.468i q^{58} +402.441 q^{59} +686.648 q^{61} +569.909i q^{62} -64.0000 q^{64} +818.730i q^{67} +334.955i q^{68} -596.216 q^{71} -454.973i q^{73} -42.4864 q^{74} -182.955 q^{76} +612.261i q^{77} +16.7841 q^{79} -29.9455i q^{82} +816.523i q^{83} -747.477 q^{86} -559.927i q^{88} -663.593 q^{89} -54.6932 q^{91} -226.955i q^{92} +761.477 q^{94} +817.559i q^{97} -532.955i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 16 q^{4}+O(q^{10}) $ 4 * q - 16 * q^4	$ 4 q - 16 q^{4} - 60 q^{11} + 40 q^{14} + 64 q^{16} - 92 q^{19} - 160 q^{26} + 180 q^{29} - 40 q^{31} - 120 q^{34} - 600 q^{41} + 240 q^{44} - 96 q^{46} + 516 q^{49} - 160 q^{56} + 180 q^{59} + 272 q^{61} - 256 q^{64} - 1560 q^{71} + 160 q^{74} + 368 q^{76} + 892 q^{79} - 2440 q^{86} + 1140 q^{89} + 1156 q^{91} + 2496 q^{94}+O(q^{100}) $ 4 * q - 16 * q^4 - 60 * q^11 + 40 * q^14 + 64 * q^16 - 92 * q^19 - 160 * q^26 + 180 * q^29 - 40 * q^31 - 120 * q^34 - 600 * q^41 + 240 * q^44 - 96 * q^46 + 516 * q^49 - 160 * q^56 + 180 * q^59 + 272 * q^61 - 256 * q^64 - 1560 * q^71 + 160 * q^74 + 368 * q^76 + 892 * q^79 - 2440 * q^86 + 1140 * q^89 + 1156 * q^91 + 2496 * q^94

Display 100 coefficients 10 coefficients

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 $

$2$	− 2.00000i	− 0.707107i
$2$	− 2.00000i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−4.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 8.74773i	− 0.472333i	−0.971713	−	0.236166i	$-0.924109\pi$
$7$	− 8.74773i	− 0.472333i	0.971713	−	0.236166i	$-0.0758911\pi$
$8$	8.00000i	0.353553i
$9$	0	0
$10$	0	0
$11$	−69.9909	−1.91846	−0.959230	−	0.282628i	$-0.908794\pi$
$11$	−69.9909	−1.91846	−0.959230	+	0.282628i	$0.908794\pi$
$12$	0	0
$13$	− 6.25227i	− 0.133390i	−0.997773	−	0.0666949i	$-0.978755\pi$
$13$	− 6.25227i	− 0.133390i	0.997773	−	0.0666949i	$-0.0212454\pi$
$14$	−17.4955	−0.333990
$15$	0	0
$16$	16.0000	0.250000
$17$	− 83.7386i	− 1.19468i	−0.801987	−	0.597341i	$-0.796224\pi$
$17$	− 83.7386i	− 1.19468i	0.801987	−	0.597341i	$-0.203776\pi$
$18$	0	0
$19$	45.7386	0.552272	0.276136	−	0.961119i	$-0.410946\pi$
$19$	45.7386	0.552272	0.276136	+	0.961119i	$0.410946\pi$
$20$	0	0
$21$	0	0
$22$	139.982i	1.35656i
$23$	56.7386i	0.514384i	0.966360	+	0.257192i	$0.0827972\pi$
$23$	56.7386i	0.514384i	−0.966360	+	0.257192i	$0.917203\pi$
$24$	0	0
$25$	0	0
$26$	−12.5045	−0.0943209
$27$	0	0
$28$	34.9909i	0.236166i
$29$	−51.2341	−0.328067	−0.164033	−	0.986455i	$-0.552450\pi$
$29$	−51.2341	−0.328067	−0.164033	+	0.986455i	$0.552450\pi$
$30$	0	0
$31$	−284.955	−1.65095	−0.825473	−	0.564441i	$-0.809092\pi$
$31$	−284.955	−1.65095	−0.825473	+	0.564441i	$0.809092\pi$
$32$	− 32.0000i	− 0.176777i
$33$	0	0
$34$	−167.477	−0.844768
$35$	0	0
$36$	0	0
$37$	− 21.2432i	− 0.0943880i	−0.998886	−	0.0471940i	$-0.984972\pi$
$37$	− 21.2432i	− 0.0943880i	0.998886	−	0.0471940i	$-0.0150279\pi$
$38$	− 91.4773i	− 0.390515i
$39$	0	0
$40$	0	0
$41$	14.9727	0.0570328	0.0285164	−	0.999593i	$-0.490922\pi$
$41$	14.9727	0.0570328	0.0285164	+	0.999593i	$0.490922\pi$
$42$	0	0
$43$	− 373.739i	− 1.32546i	−0.748860	−	0.662728i	$-0.769399\pi$
$43$	− 373.739i	− 1.32546i	0.748860	−	0.662728i	$-0.230601\pi$
$44$	279.964	0.959230
$45$	0	0
$46$	113.477	0.363724
$47$	380.739i	1.18163i	0.806808	+	0.590813i	$0.201193\pi$
$47$	380.739i	1.18163i	−0.806808	+	0.590813i	$0.798807\pi$
$48$	0	0
$49$	266.477	0.776902
$50$	0	0
$51$	0	0
$52$	25.0091i	0.0666949i
$53$	54.2614i	0.140630i	0.997525	+	0.0703148i	$0.0224004\pi$
$53$	54.2614i	0.140630i	−0.997525	+	0.0703148i	$0.977600\pi$
$54$	0	0
$55$	0	0
$56$	69.9818	0.166995
$57$	0	0
$58$	102.468i	0.231978i
$59$	402.441	0.888023	0.444011	−	0.896021i	$-0.353555\pi$
$59$	402.441	0.888023	0.444011	+	0.896021i	$0.353555\pi$
$60$	0	0
$61$	686.648	1.44125	0.720625	−	0.693325i	$-0.243855\pi$
$61$	686.648	1.44125	0.720625	+	0.693325i	$0.243855\pi$
$62$	569.909i	1.16740i
$63$	0	0
$64$	−64.0000	−0.125000
$65$	0	0
$66$	0	0
$67$	818.730i	1.49289i	0.665446	+	0.746446i	$0.268241\pi$
$67$	818.730i	1.49289i	−0.665446	+	0.746446i	$0.731759\pi$
$68$	334.955i	0.597341i
$69$	0	0
$70$	0	0
$71$	−596.216	−0.996589	−0.498294	−	0.867008i	$-0.666040\pi$
$71$	−596.216	−0.996589	−0.498294	+	0.867008i	$0.666040\pi$
$72$	0	0
$73$	− 454.973i	− 0.729459i	−0.931113	−	0.364730i	$-0.881161\pi$
$73$	− 454.973i	− 0.729459i	0.931113	−	0.364730i	$-0.118839\pi$
$74$	−42.4864	−0.0667424
$75$	0	0
$76$	−182.955	−0.276136
$77$	612.261i	0.906151i
$78$	0	0
$79$	16.7841	0.0239033	0.0119516	−	0.999929i	$-0.496196\pi$
$79$	16.7841	0.0239033	0.0119516	+	0.999929i	$0.496196\pi$
$80$	0	0
$81$	0	0
$82$	− 29.9455i	− 0.0403283i
$83$	816.523i	1.07982i	0.841723	+	0.539910i	$0.181542\pi$
$83$	816.523i	1.07982i	−0.841723	+	0.539910i	$0.818458\pi$
$84$	0	0
$85$	0	0
$86$	−747.477	−0.937239
$87$	0	0
$88$	− 559.927i	− 0.678278i
$89$	−663.593	−0.790345	−0.395173	−	0.918607i	$-0.629315\pi$
$89$	−663.593	−0.790345	−0.395173	+	0.918607i	$0.629315\pi$
$90$	0	0
$91$	−54.6932	−0.0630044
$92$	− 226.955i	− 0.257192i
$93$	0	0
$94$	761.477	0.835536
$95$	0	0
$96$	0	0
$97$	817.559i	0.855779i	0.903831	+	0.427889i	$0.140743\pi$
$97$	817.559i	0.855779i	−0.903831	+	0.427889i	$0.859257\pi$
$98$	− 532.955i	− 0.549352i
$99$	0	0
$100$	0	0
$101$	−988.639	−0.973992	−0.486996	−	0.873404i	$-0.661907\pi$
$101$	−988.639	−0.973992	−0.486996	+	0.873404i	$0.661907\pi$
$102$	0	0
$103$	602.486i	0.576357i	0.957577	+	0.288178i	$0.0930496\pi$
$103$	602.486i	0.576357i	−0.957577	+	0.288178i	$0.906950\pi$
$104$	50.0182	0.0471604
$105$	0	0
$106$	108.523	0.0994402
$107$	− 1276.82i	− 1.15360i	−0.816887	−	0.576798i	$-0.804302\pi$
$107$	− 1276.82i	− 1.15360i	0.816887	−	0.576798i	$-0.195698\pi$
$108$	0	0
$109$	−155.386	−0.136544	−0.0682721	−	0.997667i	$-0.521749\pi$
$109$	−155.386	−0.136544	−0.0682721	+	0.997667i	$0.521749\pi$
$110$	0	0
$111$	0	0
$112$	− 139.964i	− 0.118083i
$113$	1012.95i	0.843281i	0.906763	+	0.421640i	$0.138546\pi$
$113$	1012.95i	0.843281i	−0.906763	+	0.421640i	$0.861454\pi$
$114$	0	0
$115$	0	0
$116$	204.936	0.164033
$117$	0	0
$118$	− 804.882i	− 0.627927i
$119$	−732.523	−0.564288
$120$	0	0
$121$	3567.73	2.68049
$122$	− 1373.30i	− 1.01912i
$123$	0	0
$124$	1139.82	0.825473
$125$	0	0
$126$	0	0
$127$	2352.32i	1.64358i	0.569789	+	0.821791i	$0.307025\pi$
$127$	2352.32i	1.64358i	−0.569789	+	0.821791i	$0.692975\pi$
$128$	128.000i	0.0883883i
$129$	0	0
$130$	0	0
$131$	−642.659	−0.428621	−0.214311	−	0.976766i	$-0.568750\pi$
$131$	−642.659	−0.428621	−0.214311	+	0.976766i	$0.568750\pi$
$132$	0	0
$133$	− 400.109i	− 0.260856i
$134$	1637.46	1.05563
$135$	0	0
$136$	669.909	0.422384
$137$	2078.08i	1.29593i	0.761670	+	0.647965i	$0.224380\pi$
$137$	2078.08i	1.29593i	−0.761670	+	0.647965i	$0.775620\pi$
$138$	0	0
$139$	1602.35	0.977768	0.488884	−	0.872349i	$-0.337404\pi$
$139$	1602.35	0.977768	0.488884	+	0.872349i	$0.337404\pi$
$140$	0	0
$141$	0	0
$142$	1192.43i	0.704695i
$143$	437.602i	0.255903i
$144$	0	0
$145$	0	0
$146$	−909.945	−0.515806
$147$	0	0
$148$	84.9727i	0.0471940i
$149$	1282.38	0.705077	0.352538	−	0.935797i	$-0.385319\pi$
$149$	1282.38	0.705077	0.352538	+	0.935797i	$0.385319\pi$
$150$	0	0
$151$	−1703.81	−0.918237	−0.459119	−	0.888375i	$-0.651835\pi$
$151$	−1703.81	−0.918237	−0.459119	+	0.888375i	$0.651835\pi$
$152$	365.909i	0.195258i
$153$	0	0
$154$	1224.52	0.640746
$155$	0	0
$156$	0	0
$157$	− 312.286i	− 0.158746i	−0.996845	−	0.0793731i	$-0.974708\pi$
$157$	− 312.286i	− 0.158746i	0.996845	−	0.0793731i	$-0.0252919\pi$
$158$	− 33.5682i	− 0.0169022i
$159$	0	0
$160$	0	0
$161$	496.334	0.242960
$162$	0	0
$163$	1094.68i	0.526025i	0.964792	+	0.263013i	$0.0847161\pi$
$163$	1094.68i	0.526025i	−0.964792	+	0.263013i	$0.915284\pi$
$164$	−59.8909	−0.0285164
$165$	0	0
$166$	1633.05	0.763548
$167$	3743.85i	1.73478i	0.497630	+	0.867389i	$0.334204\pi$
$167$	3743.85i	1.73478i	−0.497630	+	0.867389i	$0.665796\pi$
$168$	0	0
$169$	2157.91	0.982207
$170$	0	0
$171$	0	0
$172$	1494.95i	0.662728i
$173$	− 473.466i	− 0.208075i	−0.994573	−	0.104037i	$-0.966824\pi$
$173$	− 473.466i	− 0.208075i	0.994573	−	0.104037i	$-0.0331762\pi$
$174$	0	0
$175$	0	0
$176$	−1119.85	−0.479615
$177$	0	0
$178$	1327.19i	0.558859i
$179$	1257.14	0.524934	0.262467	−	0.964941i	$-0.415464\pi$
$179$	1257.14	0.524934	0.262467	+	0.964941i	$0.415464\pi$
$180$	0	0
$181$	1467.97	0.602835	0.301417	−	0.953492i	$-0.402540\pi$
$181$	1467.97	0.602835	0.301417	+	0.953492i	$0.402540\pi$
$182$	109.386i	0.0445509i
$183$	0	0
$184$	−453.909	−0.181862
$185$	0	0
$186$	0	0
$187$	5860.94i	2.29195i
$188$	− 1522.95i	− 0.590813i
$189$	0	0
$190$	0	0
$191$	3212.01	1.21682	0.608411	−	0.793622i	$-0.291807\pi$
$191$	3212.01	1.21682	0.608411	+	0.793622i	$0.291807\pi$
$192$	0	0
$193$	− 2089.45i	− 0.779286i	−0.920966	−	0.389643i	$-0.872598\pi$
$193$	− 2089.45i	− 0.779286i	0.920966	−	0.389643i	$-0.127402\pi$
$194$	1635.12	0.605127
$195$	0	0
$196$	−1065.91	−0.388451
$197$	− 52.1818i	− 0.0188721i	−0.999955	−	0.00943604i	$-0.996996\pi$
$197$	− 52.1818i	− 0.0188721i	0.999955	−	0.00943604i	$-0.00300363\pi$
$198$	0	0
$199$	5324.93	1.89686	0.948428	−	0.316992i	$-0.102673\pi$
$199$	5324.93	1.89686	0.948428	+	0.316992i	$0.102673\pi$
$200$	0	0
$201$	0	0
$202$	1977.28i	0.688717i
$203$	448.182i	0.154957i
$204$	0	0
$205$	0	0
$206$	1204.97	0.407546
$207$	0	0
$208$	− 100.036i	− 0.0333475i
$209$	−3201.29	−1.05951
$210$	0	0
$211$	3605.39	1.17633	0.588164	−	0.808742i	$-0.299851\pi$
$211$	3605.39	1.17633	0.588164	+	0.808742i	$0.299851\pi$
$212$	− 217.045i	− 0.0703148i
$213$	0	0
$214$	−2553.64	−0.815715
$215$	0	0
$216$	0	0
$217$	2492.70i	0.779796i
$218$	310.773i	0.0965513i
$219$	0	0
$220$	0	0
$221$	−523.557	−0.159359
$222$	0	0
$223$	− 5560.96i	− 1.66991i	−0.550320	−	0.834954i	$-0.685494\pi$
$223$	− 5560.96i	− 1.66991i	0.550320	−	0.834954i	$-0.314506\pi$
$224$	−279.927	−0.0834974
$225$	0	0
$226$	2025.91	0.596290
$227$	− 1358.45i	− 0.397197i	−0.980081	−	0.198599i	$-0.936361\pi$
$227$	− 1358.45i	− 0.397197i	0.980081	−	0.198599i	$-0.0636390\pi$
$228$	0	0
$229$	2870.31	0.828276	0.414138	−	0.910214i	$-0.364083\pi$
$229$	2870.31	0.828276	0.414138	+	0.910214i	$0.364083\pi$
$230$	0	0
$231$	0	0
$232$	− 409.873i	− 0.115989i
$233$	− 2922.51i	− 0.821717i	−0.911699	−	0.410859i	$-0.865229\pi$
$233$	− 2922.51i	− 0.821717i	0.911699	−	0.410859i	$-0.134771\pi$
$234$	0	0
$235$	0	0
$236$	−1609.76	−0.444011
$237$	0	0
$238$	1465.05i	0.399012i
$239$	−6390.62	−1.72960	−0.864802	−	0.502114i	$-0.832556\pi$
$239$	−6390.62	−1.72960	−0.864802	+	0.502114i	$0.832556\pi$
$240$	0	0
$241$	2729.50	0.729554	0.364777	−	0.931095i	$-0.381145\pi$
$241$	2729.50	0.729554	0.364777	+	0.931095i	$0.381145\pi$
$242$	− 7135.45i	− 1.89539i
$243$	0	0
$244$	−2746.59	−0.720625
$245$	0	0
$246$	0	0
$247$	− 285.970i	− 0.0736675i
$248$	− 2279.64i	− 0.583698i
$249$	0	0
$250$	0	0
$251$	5577.52	1.40259	0.701295	−	0.712871i	$-0.252606\pi$
$251$	5577.52	1.40259	0.701295	+	0.712871i	$0.252606\pi$
$252$	0	0
$253$	− 3971.19i	− 0.986824i
$254$	4704.65	1.16219
$255$	0	0
$256$	256.000	0.0625000
$257$	− 3321.62i	− 0.806215i	−0.915153	−	0.403108i	$-0.867930\pi$
$257$	− 3321.62i	− 0.806215i	0.915153	−	0.403108i	$-0.132070\pi$
$258$	0	0
$259$	−185.830	−0.0445826
$260$	0	0
$261$	0	0
$262$	1285.32i	0.303081i
$263$	7182.60i	1.68402i	0.539459	+	0.842012i	$0.318629\pi$
$263$	7182.60i	1.68402i	−0.539459	+	0.842012i	$0.681371\pi$
$264$	0	0
$265$	0	0
$266$	−800.218	−0.184453
$267$	0	0
$268$	− 3274.92i	− 0.746446i
$269$	−8517.06	−1.93046	−0.965231	−	0.261399i	$-0.915816\pi$
$269$	−8517.06	−1.93046	−0.965231	+	0.261399i	$0.915816\pi$
$270$	0	0
$271$	949.727	0.212885	0.106442	−	0.994319i	$-0.466054\pi$
$271$	949.727	0.212885	0.106442	+	0.994319i	$0.466054\pi$
$272$	− 1339.82i	− 0.298671i
$273$	0	0
$274$	4156.16	0.916360
$275$	0	0
$276$	0	0
$277$	4125.21i	0.894801i	0.894334	+	0.447400i	$0.147650\pi$
$277$	4125.21i	0.894801i	−0.894334	+	0.447400i	$0.852350\pi$
$278$	− 3204.70i	− 0.691386i
$279$	0	0
$280$	0	0
$281$	1764.12	0.374515	0.187257	−	0.982311i	$-0.440040\pi$
$281$	1764.12	0.374515	0.187257	+	0.982311i	$0.440040\pi$
$282$	0	0
$283$	7510.43i	1.57756i	0.614677	+	0.788779i	$0.289286\pi$
$283$	7510.43i	1.57756i	−0.614677	+	0.788779i	$0.710714\pi$
$284$	2384.86	0.498294
$285$	0	0
$286$	875.205	0.180951
$287$	− 130.977i	− 0.0269385i
$288$	0	0
$289$	−2099.16	−0.427266
$290$	0	0
$291$	0	0
$292$	1819.89i	0.364730i
$293$	3771.92i	0.752075i	0.926604	+	0.376037i	$0.122714\pi$
$293$	3771.92i	0.752075i	−0.926604	+	0.376037i	$0.877286\pi$
$294$	0	0
$295$	0	0
$296$	169.945	0.0333712
$297$	0	0
$298$	− 2564.75i	− 0.498565i
$299$	354.745	0.0686136
$300$	0	0
$301$	−3269.36	−0.626056
$302$	3407.61i	0.649292i
$303$	0	0
$304$	731.818	0.138068
$305$	0	0
$306$	0	0
$307$	2119.95i	0.394110i	0.980392	+	0.197055i	$0.0631377\pi$
$307$	2119.95i	0.394110i	−0.980392	+	0.197055i	$0.936862\pi$
$308$	− 2449.05i	− 0.453076i
$309$	0	0
$310$	0	0
$311$	−7980.24	−1.45504	−0.727521	−	0.686085i	$-0.759328\pi$
$311$	−7980.24	−1.45504	−0.727521	+	0.686085i	$0.759328\pi$
$312$	0	0
$313$	− 7649.85i	− 1.38145i	−0.723115	−	0.690727i	$-0.757290\pi$
$313$	− 7649.85i	− 1.38145i	0.723115	−	0.690727i	$-0.242710\pi$
$314$	−624.573	−0.112251
$315$	0	0
$316$	−67.1364	−0.0119516
$317$	1105.03i	0.195788i	0.995197	+	0.0978942i	$0.0312107\pi$
$317$	1105.03i	0.195788i	−0.995197	+	0.0978942i	$0.968789\pi$
$318$	0	0
$319$	3585.92	0.629382
$320$	0	0
$321$	0	0
$322$	− 992.668i	− 0.171799i
$323$	− 3830.09i	− 0.659789i
$324$	0	0
$325$	0	0
$326$	2189.36	0.371956
$327$	0	0
$328$	119.782i	0.0201642i
$329$	3330.60	0.558121
$330$	0	0
$331$	5816.75	0.965914	0.482957	−	0.875644i	$-0.339563\pi$
$331$	5816.75	0.965914	0.482957	+	0.875644i	$0.339563\pi$
$332$	− 3266.09i	− 0.539910i
$333$	0	0
$334$	7487.70	1.22667
$335$	0	0
$336$	0	0
$337$	7918.71i	1.28000i	0.768375	+	0.639999i	$0.221065\pi$
$337$	7918.71i	1.28000i	−0.768375	+	0.639999i	$0.778935\pi$
$338$	− 4315.82i	− 0.694525i
$339$	0	0
$340$	0	0
$341$	19944.2	3.16727
$342$	0	0
$343$	− 5331.54i	− 0.839289i
$344$	2989.91	0.468619
$345$	0	0
$346$	−946.932	−0.147131
$347$	6246.27i	0.966333i	0.875529	+	0.483166i	$0.160513\pi$
$347$	6246.27i	0.966333i	−0.875529	+	0.483166i	$0.839487\pi$
$348$	0	0
$349$	−9446.48	−1.44888	−0.724439	−	0.689339i	$-0.757901\pi$
$349$	−9446.48	−1.44888	−0.724439	+	0.689339i	$0.757901\pi$
$350$	0	0
$351$	0	0
$352$	2239.71i	0.339139i
$353$	− 8105.69i	− 1.22216i	−0.791569	−	0.611080i	$-0.790735\pi$
$353$	− 8105.69i	− 1.22216i	0.791569	−	0.611080i	$-0.209265\pi$
$354$	0	0
$355$	0	0
$356$	2654.37	0.395173
$357$	0	0
$358$	− 2514.28i	− 0.371184i
$359$	−1007.18	−0.148069	−0.0740347	−	0.997256i	$-0.523588\pi$
$359$	−1007.18	−0.148069	−0.0740347	+	0.997256i	$0.523588\pi$
$360$	0	0
$361$	−4766.98	−0.694996
$362$	− 2935.93i	− 0.426268i
$363$	0	0
$364$	218.773	0.0315022
$365$	0	0
$366$	0	0
$367$	− 5326.10i	− 0.757547i	−0.925489	−	0.378774i	$-0.876346\pi$
$367$	− 5326.10i	− 0.757547i	0.925489	−	0.378774i	$-0.123654\pi$
$368$	907.818i	0.128596i
$369$	0	0
$370$	0	0
$371$	474.664	0.0664240
$372$	0	0
$373$	11496.7i	1.59591i	0.602715	+	0.797956i	$0.294085\pi$
$373$	11496.7i	1.59591i	−0.602715	+	0.797956i	$0.705915\pi$
$374$	11721.9	1.62065
$375$	0	0
$376$	−3045.91	−0.417768
$377$	320.330i	0.0437608i
$378$	0	0
$379$	−13701.2	−1.85695	−0.928477	−	0.371389i	$-0.878882\pi$
$379$	−13701.2	−1.85695	−0.928477	+	0.371389i	$0.878882\pi$
$380$	0	0
$381$	0	0
$382$	− 6424.03i	− 0.860424i
$383$	4798.58i	0.640198i	0.947384	+	0.320099i	$0.103716\pi$
$383$	4798.58i	0.640198i	−0.947384	+	0.320099i	$0.896284\pi$
$384$	0	0
$385$	0	0
$386$	−4178.91	−0.551039
$387$	0	0
$388$	− 3270.24i	− 0.427889i
$389$	−8825.06	−1.15025	−0.575126	−	0.818065i	$-0.695047\pi$
$389$	−8825.06	−1.15025	−0.575126	+	0.818065i	$0.695047\pi$
$390$	0	0
$391$	4751.22	0.614525
$392$	2131.82i	0.274676i
$393$	0	0
$394$	−104.364	−0.0133446
$395$	0	0
$396$	0	0
$397$	− 4196.85i	− 0.530564i	−0.964171	−	0.265282i	$-0.914535\pi$
$397$	− 4196.85i	− 0.530564i	0.964171	−	0.265282i	$-0.0854651\pi$
$398$	− 10649.9i	− 1.34128i
$399$	0	0
$400$	0	0
$401$	−1527.12	−0.190176	−0.0950879	−	0.995469i	$-0.530313\pi$
$401$	−1527.12	−0.190176	−0.0950879	+	0.995469i	$0.530313\pi$
$402$	0	0
$403$	1781.61i	0.220220i
$404$	3954.55	0.486996
$405$	0	0
$406$	896.364	0.109571
$407$	1486.83i	0.181080i
$408$	0	0
$409$	10576.3	1.27864	0.639322	−	0.768939i	$-0.279215\pi$
$409$	10576.3	1.27864	0.639322	+	0.768939i	$0.279215\pi$
$410$	0	0
$411$	0	0
$412$	− 2409.95i	− 0.288178i
$413$	− 3520.44i	− 0.419442i
$414$	0	0
$415$	0	0
$416$	−200.073	−0.0235802
$417$	0	0
$418$	6402.58i	0.749187i
$419$	−12675.8	−1.47793	−0.738965	−	0.673744i	$-0.764685\pi$
$419$	−12675.8	−1.47793	−0.738965	+	0.673744i	$0.764685\pi$
$420$	0	0
$421$	−10292.0	−1.19145	−0.595725	−	0.803188i	$-0.703135\pi$
$421$	−10292.0	−1.19145	−0.595725	+	0.803188i	$0.703135\pi$
$422$	− 7210.77i	− 0.831789i
$423$	0	0
$424$	−434.091	−0.0497201
$425$	0	0
$426$	0	0
$427$	− 6006.61i	− 0.680750i
$428$	5107.27i	0.576798i
$429$	0	0
$430$	0	0
$431$	14373.5	1.60637	0.803185	−	0.595730i	$-0.203137\pi$
$431$	14373.5	1.60637	0.803185	+	0.595730i	$0.203137\pi$
$432$	0	0
$433$	14568.6i	1.61691i	0.588556	+	0.808457i	$0.299697\pi$
$433$	14568.6i	1.61691i	−0.588556	+	0.808457i	$0.700303\pi$
$434$	4985.41	0.551399
$435$	0	0
$436$	621.545	0.0682721
$437$	2595.15i	0.284080i
$438$	0	0
$439$	10210.4	1.11006	0.555032	−	0.831829i	$-0.312706\pi$
$439$	10210.4	1.11006	0.555032	+	0.831829i	$0.312706\pi$
$440$	0	0
$441$	0	0
$442$	1047.11i	0.112684i
$443$	11305.4i	1.21250i	0.795275	+	0.606249i	$0.207326\pi$
$443$	11305.4i	1.21250i	−0.795275	+	0.606249i	$0.792674\pi$
$444$	0	0
$445$	0	0
$446$	−11121.9	−1.18080
$447$	0	0
$448$	559.855i	0.0590416i
$449$	−15372.5	−1.61575	−0.807875	−	0.589354i	$-0.799382\pi$
$449$	−15372.5	−1.61575	−0.807875	+	0.589354i	$0.799382\pi$
$450$	0	0
$451$	−1047.95	−0.109415
$452$	− 4051.82i	− 0.421640i
$453$	0	0
$454$	−2716.91	−0.280861
$455$	0	0
$456$	0	0
$457$	11365.5i	1.16336i	0.813419	+	0.581679i	$0.197604\pi$
$457$	11365.5i	1.16336i	−0.813419	+	0.581679i	$0.802396\pi$
$458$	− 5740.61i	− 0.585680i
$459$	0	0
$460$	0	0
$461$	−17225.5	−1.74029	−0.870145	−	0.492795i	$-0.835975\pi$
$461$	−17225.5	−1.74029	−0.870145	+	0.492795i	$0.835975\pi$
$462$	0	0
$463$	− 9224.33i	− 0.925899i	−0.886385	−	0.462949i	$-0.846791\pi$
$463$	− 9224.33i	− 0.925899i	0.886385	−	0.462949i	$-0.153209\pi$
$464$	−819.745	−0.0820167
$465$	0	0
$466$	−5845.02	−0.581042
$467$	− 7807.09i	− 0.773595i	−0.922165	−	0.386798i	$-0.873581\pi$
$467$	− 7807.09i	− 0.773595i	0.922165	−	0.386798i	$-0.126419\pi$
$468$	0	0
$469$	7162.02	0.705142
$470$	0	0
$471$	0	0
$472$	3219.53i	0.313963i
$473$	26158.3i	2.54283i
$474$	0	0
$475$	0	0
$476$	2930.09	0.282144
$477$	0	0
$478$	12781.2i	1.22301i
$479$	11748.3	1.12066	0.560329	−	0.828270i	$-0.310675\pi$
$479$	11748.3	1.12066	0.560329	+	0.828270i	$0.310675\pi$
$480$	0	0
$481$	−132.818	−0.0125904
$482$	− 5459.00i	− 0.515873i
$483$	0	0
$484$	−14270.9	−1.34024
$485$	0	0
$486$	0	0
$487$	− 19545.2i	− 1.81864i	−0.416095	−	0.909321i	$-0.636602\pi$
$487$	− 19545.2i	− 1.81864i	0.416095	−	0.909321i	$-0.363398\pi$
$488$	5493.18i	0.509559i
$489$	0	0
$490$	0	0
$491$	−16172.8	−1.48649	−0.743246	−	0.669019i	$-0.766715\pi$
$491$	−16172.8	−1.48649	−0.743246	+	0.669019i	$0.766715\pi$
$492$	0	0
$493$	4290.27i	0.391935i
$494$	−571.941	−0.0520908
$495$	0	0
$496$	−4559.27	−0.412737
$497$	5215.53i	0.470722i
$498$	0	0
$499$	−20895.7	−1.87459	−0.937295	−	0.348538i	$-0.886678\pi$
$499$	−20895.7	−1.87459	−0.937295	+	0.348538i	$0.886678\pi$
$500$	0	0
$501$	0	0
$502$	− 11155.0i	− 0.991781i
$503$	− 4296.98i	− 0.380900i	−0.981697	−	0.190450i	$-0.939005\pi$
$503$	− 4296.98i	− 0.380900i	0.981697	−	0.190450i	$-0.0609947\pi$
$504$	0	0
$505$	0	0
$506$	−7942.38	−0.697790
$507$	0	0
$508$	− 9409.29i	− 0.821791i
$509$	8315.66	0.724136	0.362068	−	0.932152i	$-0.382071\pi$
$509$	8315.66	0.724136	0.362068	+	0.932152i	$0.382071\pi$
$510$	0	0
$511$	−3979.98	−0.344548
$512$	− 512.000i	− 0.0441942i
$513$	0	0
$514$	−6643.25	−0.570080
$515$	0	0
$516$	0	0
$517$	− 26648.2i	− 2.26690i
$518$	371.659i	0.0315246i
$519$	0	0
$520$	0	0
$521$	5884.90	0.494860	0.247430	−	0.968906i	$-0.420414\pi$
$521$	5884.90	0.494860	0.247430	+	0.968906i	$0.420414\pi$
$522$	0	0
$523$	− 1578.69i	− 0.131991i	−0.997820	−	0.0659955i	$-0.978978\pi$
$523$	− 1578.69i	− 0.131991i	0.997820	−	0.0659955i	$-0.0210223\pi$
$524$	2570.64	0.214311
$525$	0	0
$526$	14365.2	1.19078
$527$	23861.7i	1.97236i
$528$	0	0
$529$	8947.73	0.735409
$530$	0	0
$531$	0	0
$532$	1600.44i	0.130428i
$533$	− 93.6136i	− 0.00760761i
$534$	0	0
$535$	0	0
$536$	−6549.84	−0.527817
$537$	0	0
$538$	17034.1i	1.36504i
$539$	−18651.0	−1.49045
$540$	0	0
$541$	−5702.35	−0.453167	−0.226583	−	0.973992i	$-0.572756\pi$
$541$	−5702.35	−0.453167	−0.226583	+	0.973992i	$0.572756\pi$
$542$	− 1899.45i	− 0.150532i
$543$	0	0
$544$	−2679.64	−0.211192
$545$	0	0
$546$	0	0
$547$	− 4083.09i	− 0.319159i	−0.987185	−	0.159580i	$-0.948986\pi$
$547$	− 4083.09i	− 0.319159i	0.987185	−	0.159580i	$-0.0510139\pi$
$548$	− 8312.32i	− 0.647965i
$549$	0	0
$550$	0	0
$551$	−2343.38	−0.181182
$552$	0	0
$553$	− 146.823i	− 0.0112903i
$554$	8250.42	0.632720
$555$	0	0
$556$	−6409.41	−0.488884
$557$	− 3498.58i	− 0.266139i	−0.991107	−	0.133070i	$-0.957517\pi$
$557$	− 3498.58i	− 0.266139i	0.991107	−	0.133070i	$-0.0424834\pi$
$558$	0	0
$559$	−2336.72	−0.176802
$560$	0	0
$561$	0	0
$562$	− 3528.24i	− 0.264822i
$563$	− 3079.07i	− 0.230492i	−0.993337	−	0.115246i	$-0.963234\pi$
$563$	− 3079.07i	− 0.230492i	0.993337	−	0.115246i	$-0.0367657\pi$
$564$	0	0
$565$	0	0
$566$	15020.9	1.11550
$567$	0	0
$568$	− 4769.73i	− 0.352347i
$569$	−694.464	−0.0511660	−0.0255830	−	0.999673i	$-0.508144\pi$
$569$	−694.464	−0.0511660	−0.0255830	+	0.999673i	$0.508144\pi$
$570$	0	0
$571$	−2115.27	−0.155029	−0.0775144	−	0.996991i	$-0.524698\pi$
$571$	−2115.27	−0.155029	−0.0775144	+	0.996991i	$0.524698\pi$
$572$	− 1750.41i	− 0.127952i
$573$	0	0
$574$	−261.955	−0.0190484
$575$	0	0
$576$	0	0
$577$	− 25561.0i	− 1.84423i	−0.386917	−	0.922114i	$-0.626460\pi$
$577$	− 25561.0i	− 1.84423i	0.386917	−	0.922114i	$-0.373540\pi$
$578$	4198.32i	0.302123i
$579$	0	0
$580$	0	0
$581$	7142.72	0.510034
$582$	0	0
$583$	− 3797.80i	− 0.269792i
$584$	3639.78	0.257903
$585$	0	0
$586$	7543.84	0.531797
$587$	17377.5i	1.22188i	0.791675	+	0.610942i	$0.209209\pi$
$587$	17377.5i	1.22188i	−0.791675	+	0.610942i	$0.790791\pi$
$588$	0	0
$589$	−13033.4	−0.911771
$590$	0	0
$591$	0	0
$592$	− 339.891i	− 0.0235970i
$593$	− 7559.50i	− 0.523493i	−0.965137	−	0.261747i	$-0.915702\pi$
$593$	− 7559.50i	− 0.523493i	0.965137	−	0.261747i	$-0.0842985\pi$
$594$	0	0
$595$	0	0
$596$	−5129.51	−0.352538
$597$	0	0
$598$	− 709.491i	− 0.0485171i
$599$	24162.9	1.64820	0.824098	−	0.566447i	$-0.191683\pi$
$599$	24162.9	1.64820	0.824098	+	0.566447i	$0.191683\pi$
$600$	0	0
$601$	−8330.05	−0.565374	−0.282687	−	0.959212i	$-0.591226\pi$
$601$	−8330.05	−0.565374	−0.282687	+	0.959212i	$0.591226\pi$
$602$	6538.73i	0.442689i
$603$	0	0
$604$	6815.23	0.459119
$605$	0	0
$606$	0	0
$607$	− 9280.71i	− 0.620581i	−0.950642	−	0.310290i	$-0.899574\pi$
$607$	− 9280.71i	− 0.620581i	0.950642	−	0.310290i	$-0.100426\pi$
$608$	− 1463.64i	− 0.0976288i
$609$	0	0
$610$	0	0
$611$	2380.48	0.157617
$612$	0	0
$613$	8818.70i	0.581051i	0.956867	+	0.290525i	$0.0938300\pi$
$613$	8818.70i	0.581051i	−0.956867	+	0.290525i	$0.906170\pi$
$614$	4239.89	0.278678
$615$	0	0
$616$	−4898.09	−0.320373
$617$	24177.1i	1.57752i	0.614699	+	0.788762i	$0.289278\pi$
$617$	24177.1i	1.57752i	−0.614699	+	0.788762i	$0.710722\pi$
$618$	0	0
$619$	−17113.9	−1.11126	−0.555628	−	0.831431i	$-0.687522\pi$
$619$	−17113.9	−1.11126	−0.555628	+	0.831431i	$0.687522\pi$
$620$	0	0
$621$	0	0
$622$	15960.5i	1.02887i
$623$	5804.93i	0.373306i
$624$	0	0
$625$	0	0
$626$	−15299.7	−0.976836
$627$	0	0
$628$	1249.15i	0.0793731i
$629$	−1778.88	−0.112764
$630$	0	0
$631$	−20590.2	−1.29902	−0.649512	−	0.760352i	$-0.725027\pi$
$631$	−20590.2	−1.29902	−0.649512	+	0.760352i	$0.725027\pi$
$632$	134.273i	0.00845108i
$633$	0	0
$634$	2210.07	0.138443
$635$	0	0
$636$	0	0
$637$	− 1666.09i	− 0.103631i
$638$	− 7171.84i	− 0.445041i
$639$	0	0
$640$	0	0
$641$	18360.4	1.13135	0.565673	−	0.824629i	$-0.308616\pi$
$641$	18360.4	1.13135	0.565673	+	0.824629i	$0.308616\pi$
$642$	0	0
$643$	18547.6i	1.13755i	0.822492	+	0.568776i	$0.192583\pi$
$643$	18547.6i	1.13755i	−0.822492	+	0.568776i	$0.807417\pi$
$644$	−1985.34	−0.121480
$645$	0	0
$646$	−7660.18	−0.466542
$647$	− 2335.51i	− 0.141914i	−0.997479	−	0.0709571i	$-0.977395\pi$
$647$	− 2335.51i	− 0.141914i	0.997479	−	0.0709571i	$-0.0226053\pi$
$648$	0	0
$649$	−28167.2	−1.70364
$650$	0	0
$651$	0	0
$652$	− 4378.73i	− 0.263013i
$653$	19720.9i	1.18184i	0.806731	+	0.590918i	$0.201234\pi$
$653$	19720.9i	1.18184i	−0.806731	+	0.590918i	$0.798766\pi$
$654$	0	0
$655$	0	0
$656$	239.564	0.0142582
$657$	0	0
$658$	− 6661.20i	− 0.394651i
$659$	−368.709	−0.0217949	−0.0108975	−	0.999941i	$-0.503469\pi$
$659$	−368.709	−0.0217949	−0.0108975	+	0.999941i	$0.503469\pi$
$660$	0	0
$661$	−17508.4	−1.03026	−0.515128	−	0.857114i	$-0.672255\pi$
$661$	−17508.4	−1.03026	−0.515128	+	0.857114i	$0.672255\pi$
$662$	− 11633.5i	− 0.683004i
$663$	0	0
$664$	−6532.18	−0.381774
$665$	0	0
$666$	0	0
$667$	− 2906.95i	− 0.168752i
$668$	− 14975.4i	− 0.867389i
$669$	0	0
$670$	0	0
$671$	−48059.1	−2.76498
$672$	0	0
$673$	12382.6i	0.709233i	0.935012	+	0.354617i	$0.115389\pi$
$673$	12382.6i	0.709233i	−0.935012	+	0.354617i	$0.884611\pi$
$674$	15837.4	0.905096
$675$	0	0
$676$	−8631.64	−0.491104
$677$	− 6547.95i	− 0.371726i	−0.982576	−	0.185863i	$-0.940492\pi$
$677$	− 6547.95i	− 0.371726i	0.982576	−	0.185863i	$-0.0595080\pi$
$678$	0	0
$679$	7151.78	0.404212
$680$	0	0
$681$	0	0
$682$	− 39888.5i	− 2.23960i
$683$	19849.5i	1.11204i	0.831170	+	0.556019i	$0.187672\pi$
$683$	19849.5i	1.11204i	−0.831170	+	0.556019i	$0.812328\pi$
$684$	0	0
$685$	0	0
$686$	−10663.1	−0.593467
$687$	0	0
$688$	− 5979.82i	− 0.331364i
$689$	339.257	0.0187586
$690$	0	0
$691$	9554.55	0.526009	0.263004	−	0.964795i	$-0.415287\pi$
$691$	9554.55	0.526009	0.263004	+	0.964795i	$0.415287\pi$
$692$	1893.86i	0.104037i
$693$	0	0
$694$	12492.5	0.683300
$695$	0	0
$696$	0	0
$697$	− 1253.80i	− 0.0681361i
$698$	18893.0i	1.02451i
$699$	0	0
$700$	0	0
$701$	15940.1	0.858842	0.429421	−	0.903104i	$-0.358718\pi$
$701$	15940.1	0.858842	0.429421	+	0.903104i	$0.358718\pi$
$702$	0	0
$703$	− 971.634i	− 0.0521278i
$704$	4479.42	0.239807
$705$	0	0
$706$	−16211.4	−0.864198
$707$	8648.34i	0.460049i
$708$	0	0
$709$	−10439.1	−0.552961	−0.276481	−	0.961019i	$-0.589168\pi$
$709$	−10439.1	−0.552961	−0.276481	+	0.961019i	$0.589168\pi$
$710$	0	0
$711$	0	0
$712$	− 5308.75i	− 0.279429i
$713$	− 16167.9i	− 0.849220i
$714$	0	0
$715$	0	0
$716$	−5028.56	−0.262467
$717$	0	0
$718$	2014.36i	0.104701i
$719$	−10720.6	−0.556067	−0.278034	−	0.960571i	$-0.589683\pi$
$719$	−10720.6	−0.556067	−0.278034	+	0.960571i	$0.589683\pi$
$720$	0	0
$721$	5270.39	0.272232
$722$	9533.95i	0.491436i
$723$	0	0
$724$	−5871.86	−0.301417
$725$	0	0
$726$	0	0
$727$	− 3931.94i	− 0.200588i	−0.994958	−	0.100294i	$-0.968022\pi$
$727$	− 3931.94i	− 0.200588i	0.994958	−	0.100294i	$-0.0319784\pi$
$728$	− 437.545i	− 0.0222754i
$729$	0	0
$730$	0	0
$731$	−31296.4	−1.58350
$732$	0	0
$733$	− 16142.0i	− 0.813395i	−0.913563	−	0.406698i	$-0.866680\pi$
$733$	− 16142.0i	− 0.813395i	0.913563	−	0.406698i	$-0.133320\pi$
$734$	−10652.2	−0.535667
$735$	0	0
$736$	1815.64	0.0909310
$737$	− 57303.6i	− 2.86405i
$738$	0	0
$739$	−17856.4	−0.888847	−0.444424	−	0.895817i	$-0.646592\pi$
$739$	−17856.4	−0.888847	−0.444424	+	0.895817i	$0.646592\pi$
$740$	0	0
$741$	0	0
$742$	− 949.327i	− 0.0469689i
$743$	5436.99i	0.268457i	0.990950	+	0.134229i	$0.0428557\pi$
$743$	5436.99i	0.268457i	−0.990950	+	0.134229i	$0.957144\pi$
$744$	0	0
$745$	0	0
$746$	22993.4	1.12848
$747$	0	0
$748$	− 23443.8i	− 1.14597i
$749$	−11169.3	−0.544881
$750$	0	0
$751$	−36670.2	−1.78178	−0.890889	−	0.454222i	$-0.849917\pi$
$751$	−36670.2	−1.78178	−0.890889	+	0.454222i	$0.849917\pi$
$752$	6091.82i	0.295407i
$753$	0	0
$754$	640.659	0.0309435
$755$	0	0
$756$	0	0
$757$	5075.21i	0.243675i	0.992550	+	0.121837i	$0.0388786\pi$
$757$	5075.21i	0.243675i	−0.992550	+	0.121837i	$0.961121\pi$
$758$	27402.5i	1.31307i
$759$	0	0
$760$	0	0
$761$	−2690.15	−0.128144	−0.0640721	−	0.997945i	$-0.520409\pi$
$761$	−2690.15	−0.128144	−0.0640721	+	0.997945i	$0.520409\pi$
$762$	0	0
$763$	1359.28i	0.0644943i
$764$	−12848.1	−0.608411
$765$	0	0
$766$	9597.16	0.452689
$767$	− 2516.17i	− 0.118453i
$768$	0	0
$769$	−20842.3	−0.977363	−0.488682	−	0.872462i	$-0.662522\pi$
$769$	−20842.3	−0.977363	−0.488682	+	0.872462i	$0.662522\pi$
$770$	0	0
$771$	0	0
$772$	8357.82i	0.389643i
$773$	− 28879.3i	− 1.34375i	−0.740666	−	0.671873i	$-0.765490\pi$
$773$	− 28879.3i	− 1.34375i	0.740666	−	0.671873i	$-0.234510\pi$
$774$	0	0
$775$	0	0
$776$	−6540.47	−0.302564
$777$	0	0
$778$	17650.1i	0.813352i
$779$	684.832	0.0314976
$780$	0	0
$781$	41729.7	1.91192
$782$	− 9502.43i	− 0.434535i
$783$	0	0
$784$	4263.64	0.194225
$785$	0	0
$786$	0	0
$787$	35384.4i	1.60269i	0.598203	+	0.801345i	$0.295882\pi$
$787$	35384.4i	1.60269i	−0.598203	+	0.801345i	$0.704118\pi$
$788$	208.727i	0.00943604i
$789$	0	0
$790$	0	0
$791$	8861.05	0.398309
$792$	0	0
$793$	− 4293.11i	− 0.192248i
$794$	−8393.70	−0.375166
$795$	0	0
$796$	−21299.7	−0.948428
$797$	− 5072.31i	− 0.225433i	−0.993627	−	0.112717i	$-0.964045\pi$
$797$	− 5072.31i	− 0.225433i	0.993627	−	0.112717i	$-0.0359552\pi$
$798$	0	0
$799$	31882.5	1.41167
$800$	0	0
$801$	0	0
$802$	3054.23i	0.134475i
$803$	31844.0i	1.39944i
$804$	0	0
$805$	0	0
$806$	3563.23	0.155719
$807$	0	0
$808$	− 7909.11i	− 0.344358i
$809$	5154.39	0.224003	0.112002	−	0.993708i	$-0.464274\pi$
$809$	5154.39	0.224003	0.112002	+	0.993708i	$0.464274\pi$
$810$	0	0
$811$	−12612.5	−0.546099	−0.273049	−	0.962000i	$-0.588032\pi$
$811$	−12612.5	−0.546099	−0.273049	+	0.962000i	$0.588032\pi$
$812$	− 1792.73i	− 0.0774783i
$813$	0	0
$814$	2973.66	0.128043
$815$	0	0
$816$	0	0
$817$	− 17094.3i	− 0.732012i
$818$	− 21152.6i	− 0.904138i
$819$	0	0
$820$	0	0
$821$	26726.7	1.13614	0.568069	−	0.822981i	$-0.307691\pi$
$821$	26726.7	1.13614	0.568069	+	0.822981i	$0.307691\pi$
$822$	0	0
$823$	13722.7i	0.581217i	0.956842	+	0.290609i	$0.0938578\pi$
$823$	13722.7i	0.581217i	−0.956842	+	0.290609i	$0.906142\pi$
$824$	−4819.89	−0.203773
$825$	0	0
$826$	−7040.89	−0.296591
$827$	− 895.727i	− 0.0376632i	−0.999823	−	0.0188316i	$-0.994005\pi$
$827$	− 895.727i	− 0.0376632i	0.999823	−	0.0188316i	$-0.00599464\pi$
$828$	0	0
$829$	4788.76	0.200628	0.100314	−	0.994956i	$-0.468015\pi$
$829$	4788.76	0.200628	0.100314	+	0.994956i	$0.468015\pi$
$830$	0	0
$831$	0	0
$832$	400.145i	0.0166737i
$833$	− 22314.4i	− 0.928151i
$834$	0	0
$835$	0	0
$836$	12805.2	0.529755
$837$	0	0
$838$	25351.6i	1.04505i
$839$	−10345.8	−0.425719	−0.212859	−	0.977083i	$-0.568278\pi$
$839$	−10345.8	−0.425719	−0.212859	+	0.977083i	$0.568278\pi$
$840$	0	0
$841$	−21764.1	−0.892372
$842$	20584.0i	0.842483i
$843$	0	0
$844$	−14421.5	−0.588164
$845$	0	0
$846$	0	0
$847$	− 31209.5i	− 1.26608i
$848$	868.182i	0.0351574i
$849$	0	0
$850$	0	0
$851$	1205.31	0.0485517
$852$	0	0
$853$	− 10604.3i	− 0.425656i	−0.977090	−	0.212828i	$-0.931733\pi$
$853$	− 10604.3i	− 0.425656i	0.977090	−	0.212828i	$-0.0682674\pi$
$854$	−12013.2	−0.481363
$855$	0	0
$856$	10214.5	0.407857
$857$	7158.11i	0.285317i	0.989772	+	0.142658i	$0.0455650\pi$
$857$	7158.11i	0.285317i	−0.989772	+	0.142658i	$0.954435\pi$
$858$	0	0
$859$	−7522.01	−0.298775	−0.149387	−	0.988779i	$-0.547730\pi$
$859$	−7522.01	−0.298775	−0.149387	+	0.988779i	$0.547730\pi$
$860$	0	0
$861$	0	0
$862$	− 28746.9i	− 1.13587i
$863$	− 37744.6i	− 1.48881i	−0.667730	−	0.744404i	$-0.732734\pi$
$863$	− 37744.6i	− 1.48881i	0.667730	−	0.744404i	$-0.267266\pi$
$864$	0	0
$865$	0	0
$866$	29137.3	1.14333
$867$	0	0
$868$	− 9970.82i	− 0.389898i
$869$	−1174.73	−0.0458574
$870$	0	0
$871$	5118.92	0.199137
$872$	− 1243.09i	− 0.0482757i
$873$	0	0
$874$	5190.30	0.200875
$875$	0	0
$876$	0	0
$877$	− 7362.69i	− 0.283490i	−0.989903	−	0.141745i	$-0.954729\pi$
$877$	− 7362.69i	− 0.283490i	0.989903	−	0.141745i	$-0.0452712\pi$
$878$	− 20420.9i	− 0.784933i
$879$	0	0
$880$	0	0
$881$	29498.8	1.12808	0.564040	−	0.825747i	$-0.309246\pi$
$881$	29498.8	1.12808	0.564040	+	0.825747i	$0.309246\pi$
$882$	0	0
$883$	42118.4i	1.60521i	0.596514	+	0.802603i	$0.296552\pi$
$883$	42118.4i	1.60521i	−0.596514	+	0.802603i	$0.703448\pi$
$884$	2094.23	0.0796793
$885$	0	0
$886$	22610.8	0.857365
$887$	15665.2i	0.592996i	0.955034	+	0.296498i	$0.0958188\pi$
$887$	15665.2i	0.592996i	−0.955034	+	0.296498i	$0.904181\pi$
$888$	0	0
$889$	20577.5	0.776318
$890$	0	0
$891$	0	0
$892$	22243.8i	0.834954i
$893$	17414.5i	0.652579i
$894$	0	0
$895$	0	0
$896$	1119.71	0.0417487
$897$	0	0
$898$	30744.9i	1.14251i
$899$	14599.4	0.541620
$900$	0	0
$901$	4543.77	0.168008
$902$	2095.91i	0.0773682i
$903$	0	0
$904$	−8103.64	−0.298145
$905$	0	0
$906$	0	0
$907$	38776.7i	1.41958i	0.704414	+	0.709790i	$0.251210\pi$
$907$	38776.7i	1.41958i	−0.704414	+	0.709790i	$0.748790\pi$
$908$	5433.82i	0.198599i
$909$	0	0
$910$	0	0
$911$	12966.3	0.471563	0.235781	−	0.971806i	$-0.424235\pi$
$911$	12966.3	0.471563	0.235781	+	0.971806i	$0.424235\pi$
$912$	0	0
$913$	− 57149.2i	− 2.07159i
$914$	22730.9	0.822618
$915$	0	0
$916$	−11481.2	−0.414138
$917$	5621.81i	0.202452i
$918$	0	0
$919$	21454.1	0.770082	0.385041	−	0.922899i	$-0.374187\pi$
$919$	21454.1	0.770082	0.385041	+	0.922899i	$0.374187\pi$
$920$	0	0
$921$	0	0
$922$	34451.1i	1.23057i
$923$	3727.70i	0.132935i
$924$	0	0
$925$	0	0
$926$	−18448.7	−0.654709
$927$	0	0
$928$	1639.49i	0.0579945i
$929$	19943.2	0.704323	0.352162	−	0.935939i	$-0.385447\pi$
$929$	19943.2	0.704323	0.352162	+	0.935939i	$0.385447\pi$
$930$	0	0
$931$	12188.3	0.429061
$932$	11690.0i	0.410859i
$933$	0	0
$934$	−15614.2	−0.547015
$935$	0	0
$936$	0	0
$937$	− 23471.1i	− 0.818320i	−0.912463	−	0.409160i	$-0.865822\pi$
$937$	− 23471.1i	− 0.818320i	0.912463	−	0.409160i	$-0.134178\pi$
$938$	− 14324.0i	− 0.498611i
$939$	0	0
$940$	0	0
$941$	−16961.5	−0.587597	−0.293798	−	0.955867i	$-0.594919\pi$
$941$	−16961.5	−0.587597	−0.293798	+	0.955867i	$0.594919\pi$
$942$	0	0
$943$	849.532i	0.0293368i
$944$	6439.05	0.222006
$945$	0	0
$946$	52316.6	1.79805
$947$	3160.75i	0.108459i	0.998529	+	0.0542294i	$0.0172702\pi$
$947$	3160.75i	0.108459i	−0.998529	+	0.0542294i	$0.982730\pi$
$948$	0	0
$949$	−2844.61	−0.0973025
$950$	0	0
$951$	0	0
$952$	− 5860.18i	− 0.199506i
$953$	− 24337.6i	− 0.827253i	−0.910447	−	0.413626i	$-0.864262\pi$
$953$	− 24337.6i	− 0.827253i	0.910447	−	0.413626i	$-0.135738\pi$
$954$	0	0
$955$	0	0
$956$	25562.5	0.864802
$957$	0	0
$958$	− 23496.7i	− 0.792425i
$959$	18178.5	0.612110
$960$	0	0
$961$	51408.1	1.72562
$962$	265.636i	0.00890276i
$963$	0	0
$964$	−10918.0	−0.364777
$965$	0	0
$966$	0	0
$967$	− 21718.2i	− 0.722244i	−0.932519	−	0.361122i	$-0.882394\pi$
$967$	− 21718.2i	− 0.722244i	0.932519	−	0.361122i	$-0.117606\pi$
$968$	28541.8i	0.947695i
$969$	0	0
$970$	0	0
$971$	7315.13	0.241765	0.120882	−	0.992667i	$-0.461428\pi$
$971$	7315.13	0.241765	0.120882	+	0.992667i	$0.461428\pi$
$972$	0	0
$973$	− 14016.9i	− 0.461832i
$974$	−39090.4	−1.28597
$975$	0	0
$976$	10986.4	0.360312
$977$	− 41775.1i	− 1.36797i	−0.729498	−	0.683983i	$-0.760246\pi$
$977$	− 41775.1i	− 1.36797i	0.729498	−	0.683983i	$-0.239754\pi$
$978$	0	0
$979$	46445.5	1.51625
$980$	0	0
$981$	0	0
$982$	32345.6i	1.05111i
$983$	− 34768.4i	− 1.12812i	−0.825734	−	0.564059i	$-0.809239\pi$
$983$	− 34768.4i	− 1.12812i	0.825734	−	0.564059i	$-0.190761\pi$
$984$	0	0
$985$	0	0
$986$	8580.55	0.277140
$987$	0	0
$988$	1143.88i	0.0368337i
$989$	21205.4	0.681793
$990$	0	0
$991$	−8620.41	−0.276323	−0.138162	−	0.990410i	$-0.544119\pi$
$991$	−8620.41	−0.276323	−0.138162	+	0.990410i	$0.544119\pi$
$992$	9118.55i	0.291849i
$993$	0	0
$994$	10431.1	0.332850
$995$	0	0
$996$	0	0
$997$	31582.0i	1.00322i	0.865094	+	0.501611i	$0.167259\pi$
$997$	31582.0i	1.00322i	−0.865094	+	0.501611i	$0.832741\pi$
$998$	41791.4i	1.32553i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1350.4.c.v.649.1		4
3.2	odd	2		1350.4.c.ba.649.3		4
5.2	odd	4		1350.4.a.bl.1.2	yes	2
5.3	odd	4		1350.4.a.bg.1.1	yes	2
5.4	even	2	inner	1350.4.c.v.649.4		4
15.2	even	4		1350.4.a.be.1.2	✓	2
15.8	even	4		1350.4.a.bn.1.1	yes	2
15.14	odd	2		1350.4.c.ba.649.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1350.4.a.be.1.2	✓	2	15.2	even	4
1350.4.a.bg.1.1	yes	2	5.3	odd	4
1350.4.a.bl.1.2	yes	2	5.2	odd	4
1350.4.a.bn.1.1	yes	2	15.8	even	4
1350.4.c.v.649.1		4	1.1	even	1	trivial
1350.4.c.v.649.4		4	5.4	even	2	inner
1350.4.c.ba.649.2		4	15.14	odd	2
1350.4.c.ba.649.3		4	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1350.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{2} -4.00000 q^{4} -8.74773i q^{7} +8.00000i q^{8} +O(q^{10})\)	\(q-2.00000i q^{2} -4.00000 q^{4} -8.74773i q^{7} +8.00000i q^{8} -69.9909 q^{11} -6.25227i q^{13} -17.4955 q^{14} +16.0000 q^{16} -83.7386i q^{17} +45.7386 q^{19} +139.982i q^{22} +56.7386i q^{23} -12.5045 q^{26} +34.9909i q^{28} -51.2341 q^{29} -284.955 q^{31} -32.0000i q^{32} -167.477 q^{34} -21.2432i q^{37} -91.4773i q^{38} +14.9727 q^{41} -373.739i q^{43} +279.964 q^{44} +113.477 q^{46} +380.739i q^{47} +266.477 q^{49} +25.0091i q^{52} +54.2614i q^{53} +69.9818 q^{56} +102.468i q^{58} +402.441 q^{59} +686.648 q^{61} +569.909i q^{62} -64.0000 q^{64} +818.730i q^{67} +334.955i q^{68} -596.216 q^{71} -454.973i q^{73} -42.4864 q^{74} -182.955 q^{76} +612.261i q^{77} +16.7841 q^{79} -29.9455i q^{82} +816.523i q^{83} -747.477 q^{86} -559.927i q^{88} -663.593 q^{89} -54.6932 q^{91} -226.955i q^{92} +761.477 q^{94} +817.559i q^{97} -532.955i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 16 q^{4}+O(q^{10}) \) 4 * q - 16 * q^4	\( 4 q - 16 q^{4} - 60 q^{11} + 40 q^{14} + 64 q^{16} - 92 q^{19} - 160 q^{26} + 180 q^{29} - 40 q^{31} - 120 q^{34} - 600 q^{41} + 240 q^{44} - 96 q^{46} + 516 q^{49} - 160 q^{56} + 180 q^{59} + 272 q^{61} - 256 q^{64} - 1560 q^{71} + 160 q^{74} + 368 q^{76} + 892 q^{79} - 2440 q^{86} + 1140 q^{89} + 1156 q^{91} + 2496 q^{94}+O(q^{100}) \) 4 * q - 16 * q^4 - 60 * q^11 + 40 * q^14 + 64 * q^16 - 92 * q^19 - 160 * q^26 + 180 * q^29 - 40 * q^31 - 120 * q^34 - 600 * q^41 + 240 * q^44 - 96 * q^46 + 516 * q^49 - 160 * q^56 + 180 * q^59 + 272 * q^61 - 256 * q^64 - 1560 * q^71 + 160 * q^74 + 368 * q^76 + 892 * q^79 - 2440 * q^86 + 1140 * q^89 + 1156 * q^91 + 2496 * q^94

Introduction

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