LMFDB - Embedded newform 324.5.k.a.89.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [324,5,Mod(17,324)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(324, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 11]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("324.17");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 324 = 2^{2} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	324.k (of order $18$, degree $6$, 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:	$33.4918680392$
Analytic rank:	$0$
Dimension:	$72$
Relative dimension:	$12$ over $\Q(\zeta_{18})$
Twist minimal:	no (minimal twist has level 108)
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			89.3
Character	$\chi$	$=$	324.89
Dual form			324.5.k.a.233.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-18.1168 - 21.5908i) q^{5} +(22.9581 - 8.35608i) q^{7} +O(q^{10})$	$q+(-18.1168 - 21.5908i) q^{5} +(22.9581 - 8.35608i) q^{7} +(73.4456 - 87.5290i) q^{11} +(-0.427402 - 2.42392i) q^{13} +(208.105 - 120.150i) q^{17} +(-246.932 + 427.698i) q^{19} +(179.414 - 492.937i) q^{23} +(-29.4131 + 166.810i) q^{25} +(95.9782 + 16.9236i) q^{29} +(-1499.65 - 545.827i) q^{31} +(-596.344 - 344.299i) q^{35} +(414.251 + 717.504i) q^{37} +(-2401.19 + 423.394i) q^{41} +(-1294.95 - 1086.59i) q^{43} +(379.164 + 1041.74i) q^{47} +(-1382.02 + 1159.65i) q^{49} -3880.06i q^{53} -3220.42 q^{55} +(-2429.52 - 2895.39i) q^{59} +(1823.41 - 663.667i) q^{61} +(-44.5912 + 53.1417i) q^{65} +(-145.480 - 825.059i) q^{67} +(-1966.62 + 1135.43i) q^{71} +(2940.78 - 5093.57i) q^{73} +(954.774 - 2623.22i) q^{77} +(-988.613 + 5606.70i) q^{79} +(-7283.42 - 1284.26i) q^{83} +(-6364.34 - 2316.43i) q^{85} +(-10189.8 - 5883.10i) q^{89} +(-30.0668 - 52.0773i) q^{91} +(13708.0 - 2417.09i) q^{95} +(2197.40 + 1843.84i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 72 q - 9 q^{5}+O(q^{10}) $ 72 * q - 9 * q^5	$ 72 q - 9 q^{5} - 18 q^{11} + 1278 q^{23} + 441 q^{25} - 1854 q^{29} - 1665 q^{31} + 2673 q^{35} + 5472 q^{41} + 1260 q^{43} - 5103 q^{47} - 5904 q^{49} + 10944 q^{59} + 8352 q^{61} - 8757 q^{65} + 378 q^{67} + 19764 q^{71} + 6111 q^{73} + 5679 q^{77} - 5652 q^{79} + 20061 q^{83} + 26100 q^{85} - 15633 q^{89} - 6039 q^{91} - 48024 q^{95} - 37530 q^{97}+O(q^{100}) $ 72 * q - 9 * q^5 - 18 * q^11 + 1278 * q^23 + 441 * q^25 - 1854 * q^29 - 1665 * q^31 + 2673 * q^35 + 5472 * q^41 + 1260 * q^43 - 5103 * q^47 - 5904 * q^49 + 10944 * q^59 + 8352 * q^61 - 8757 * q^65 + 378 * q^67 + 19764 * q^71 + 6111 * q^73 + 5679 * q^77 - 5652 * q^79 + 20061 * q^83 + 26100 * q^85 - 15633 * q^89 - 6039 * q^91 - 48024 * q^95 - 37530 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/324\mathbb{Z}\right)^\times$.

$n$	$163$	$245$
$\chi(n)$	$1$	$e\left(\frac{1}{18}\right)$

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 $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−18.1168	−	21.5908i	−0.724674	−	0.863632i	0.270402	−	0.962747i	$-0.412843\pi$
$5$	−18.1168	−	21.5908i	−0.724674	−	0.863632i	−0.995076	+	0.0991149i	$0.968399\pi$
$6$		0			0
$7$	22.9581	−	8.35608i	0.468534	−	0.170532i	−0.0969544	−	0.995289i	$-0.530910\pi$
$7$	22.9581	−	8.35608i	0.468534	−	0.170532i	0.565488	+	0.824757i	$0.308688\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	73.4456	−	87.5290i	0.606988	−	0.723380i	−0.371787	−	0.928318i	$-0.621255\pi$
$11$	73.4456	−	87.5290i	0.606988	−	0.723380i	0.978775	+	0.204938i	$0.0656992\pi$
$12$		0			0
$13$	−0.427402	−	2.42392i	−0.00252901	−	0.0143427i	0.983517	−	0.180815i	$-0.0578735\pi$
$13$	−0.427402	−	2.42392i	−0.00252901	−	0.0143427i	−0.986046	+	0.166472i	$0.946762\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	208.105	−	120.150i	0.720088	−	0.415743i	−0.0946973	−	0.995506i	$-0.530188\pi$
$17$	208.105	−	120.150i	0.720088	−	0.415743i	0.814785	+	0.579763i	$0.196855\pi$
$18$		0			0
$19$	−246.932	+	427.698i	−0.684021	+	1.18476i	0.289722	+	0.957111i	$0.406437\pi$
$19$	−246.932	+	427.698i	−0.684021	+	1.18476i	−0.973743	+	0.227649i	$0.926896\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	179.414	−	492.937i	0.339158	−	0.931828i	−0.646477	−	0.762934i	$-0.723758\pi$
$23$	179.414	−	492.937i	0.339158	−	0.931828i	0.985634	−	0.168894i	$-0.0540196\pi$
$24$		0			0
$25$	−29.4131	+	166.810i	−0.0470609	+	0.266896i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	95.9782	+	16.9236i	0.114124	+	0.0201231i	0.230418	−	0.973092i	$-0.425991\pi$
$29$	95.9782	+	16.9236i	0.114124	+	0.0201231i	−0.116294	+	0.993215i	$0.537102\pi$
$30$		0			0
$31$	−1499.65	−	545.827i	−1.56051	−	0.567978i	−0.589654	−	0.807656i	$-0.700736\pi$
$31$	−1499.65	−	545.827i	−1.56051	−	0.567978i	−0.970852	+	0.239679i	$0.922958\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−596.344	−	344.299i	−0.486811	−	0.281061i
$36$		0			0
$37$	414.251	+	717.504i	0.302594	+	0.524108i	0.976723	−	0.214506i	$-0.0688141\pi$
$37$	414.251	+	717.504i	0.302594	+	0.524108i	−0.674129	+	0.738614i	$0.735481\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−2401.19	+	423.394i	−1.42843	+	0.251870i	−0.833772	−	0.552110i	$-0.813823\pi$
$41$	−2401.19	+	423.394i	−1.42843	+	0.251870i	−0.594656	+	0.803980i	$0.702712\pi$
$42$		0			0
$43$	−1294.95	−	1086.59i	−0.700351	−	0.587664i	0.221523	−	0.975155i	$-0.428897\pi$
$43$	−1294.95	−	1086.59i	−0.700351	−	0.587664i	−0.921873	+	0.387491i	$0.873342\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	379.164	+	1041.74i	0.171645	+	0.471591i	0.995450	−	0.0952820i	$-0.0303753\pi$
$47$	379.164	+	1041.74i	0.171645	+	0.471591i	−0.823805	+	0.566873i	$0.808153\pi$
$48$		0			0
$49$	−1382.02	+	1159.65i	−0.575602	+	0.482987i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	3880.06i		−	1.38130i	−0.723191	−	0.690648i	$-0.757325\pi$
$53$		−	3880.06i		−	1.38130i	0.723191	−	0.690648i	$-0.242675\pi$
$54$		0			0
$55$	−3220.42			−1.06460
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−2429.52	−	2895.39i	−0.697939	−	0.831771i	0.294353	−	0.955697i	$-0.404896\pi$
$59$	−2429.52	−	2895.39i	−0.697939	−	0.831771i	−0.992291	+	0.123926i	$0.960451\pi$
$60$		0			0
$61$	1823.41	−	663.667i	0.490032	−	0.178357i	−0.0851732	−	0.996366i	$-0.527144\pi$
$61$	1823.41	−	663.667i	0.490032	−	0.178357i	0.575205	+	0.818009i	$0.304922\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−44.5912	+	53.1417i	−0.0105541	+	0.0125779i
$66$		0			0
$67$	−145.480	−	825.059i	−0.0324082	−	0.183796i	0.964307	−	0.264788i	$-0.0853021\pi$
$67$	−145.480	−	825.059i	−0.0324082	−	0.183796i	−0.996715	+	0.0809926i	$0.974191\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−1966.62	+	1135.43i	−0.390126	+	0.225239i	−0.682215	−	0.731152i	$-0.738983\pi$
$71$	−1966.62	+	1135.43i	−0.390126	+	0.225239i	0.292089	+	0.956391i	$0.405650\pi$
$72$		0			0
$73$	2940.78	−	5093.57i	0.551844	−	0.955822i	−0.446298	−	0.894885i	$-0.647258\pi$
$73$	2940.78	−	5093.57i	0.551844	−	0.955822i	0.998142	−	0.0609373i	$-0.0194090\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	954.774	−	2623.22i	0.161035	−	0.442439i
$78$		0			0
$79$	−988.613	+	5606.70i	−0.158406	+	0.898366i	0.797199	+	0.603717i	$0.206314\pi$
$79$	−988.613	+	5606.70i	−0.158406	+	0.898366i	−0.955605	+	0.294650i	$0.904797\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−7283.42	−	1284.26i	−1.05725	−	0.186422i	−0.382116	−	0.924114i	$-0.624804\pi$
$83$	−7283.42	−	1284.26i	−1.05725	−	0.186422i	−0.675137	+	0.737692i	$0.735916\pi$
$84$		0			0
$85$	−6364.34	−	2316.43i	−0.880877	−	0.320613i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−10189.8	−	5883.10i	−1.28643	−	0.742722i	−0.308416	−	0.951252i	$-0.599799\pi$
$89$	−10189.8	−	5883.10i	−1.28643	−	0.742722i	−0.978016	+	0.208530i	$0.933132\pi$
$90$		0			0
$91$	−30.0668	−	52.0773i	−0.00363082	−	0.00628877i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	13708.0	−	2417.09i	1.51889	−	0.267821i
$96$		0			0
$97$	2197.40	+	1843.84i	0.233542	+	0.195965i	0.752047	−	0.659110i	$-0.229067\pi$
$97$	2197.40	+	1843.84i	0.233542	+	0.195965i	−0.518505	+	0.855075i	$0.673511\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−573.994	−	1577.03i	−0.0562684	−	0.154596i	0.908374	−	0.418159i	$-0.137325\pi$
$101$	−573.994	−	1577.03i	−0.0562684	−	0.154596i	−0.964642	+	0.263563i	$0.915102\pi$
$102$		0			0
$103$	−11083.9	+	9300.48i	−1.04476	+	0.876659i	−0.992533	−	0.121976i	$-0.961077\pi$
$103$	−11083.9	+	9300.48i	−1.04476	+	0.876659i	−0.0522289	+	0.998635i	$0.516633\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			6102.13i			0.532983i	0.963837	+	0.266492i	$0.0858645\pi$
$107$			6102.13i			0.532983i	−0.963837	+	0.266492i	$0.914136\pi$
$108$		0			0
$109$	−19211.9			−1.61703			−0.808513	−	0.588479i	$-0.799727\pi$
$109$	−19211.9			−1.61703			−0.808513	+	0.588479i	$0.799727\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−176.048	−	209.806i	−0.0137871	−	0.0164309i	0.759107	−	0.650966i	$-0.225636\pi$
$113$	−176.048	−	209.806i	−0.0137871	−	0.0164309i	−0.772894	+	0.634535i	$0.781192\pi$
$114$		0			0
$115$	−13893.3	+	5056.76i	−1.05054	+	0.382364i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	3773.73	−	4497.36i	0.266488	−	0.317588i
$120$		0			0
$121$	275.306	+	1561.34i	0.0188037	+	0.106641i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−11121.0	+	6420.74i	−0.711747	+	0.410927i
$126$		0			0
$127$	14641.1	−	25359.1i	0.907749	−	1.57227i	0.0905660	−	0.995890i	$-0.471132\pi$
$127$	14641.1	−	25359.1i	0.907749	−	1.57227i	0.817183	−	0.576378i	$-0.195534\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	8692.53	−	23882.5i	0.506528	−	1.39167i	−0.378268	−	0.925696i	$-0.623480\pi$
$131$	8692.53	−	23882.5i	0.506528	−	1.39167i	0.884796	−	0.465979i	$-0.154298\pi$
$132$		0			0
$133$	−2095.21	+	11882.5i	−0.118447	+	0.671747i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	12879.6	+	2271.01i	0.686214	+	0.120998i	0.505875	−	0.862607i	$-0.331170\pi$
$137$	12879.6	+	2271.01i	0.686214	+	0.120998i	0.180339	+	0.983605i	$0.442281\pi$
$138$		0			0
$139$	8423.60	+	3065.94i	0.435982	+	0.158684i	0.550681	−	0.834716i	$-0.314368\pi$
$139$	8423.60	+	3065.94i	0.435982	+	0.158684i	−0.114699	+	0.993400i	$0.536590\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−243.554	−	140.616i	−0.0119103	−	0.00687642i
$144$		0			0
$145$	−1373.43	−	2378.85i	−0.0653236	−	0.113144i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−2532.71	+	446.586i	−0.114081	+	0.0201156i	−0.230397	−	0.973097i	$-0.574003\pi$
$149$	−2532.71	+	446.586i	−0.114081	+	0.0201156i	0.116316	+	0.993212i	$0.462891\pi$
$150$		0			0
$151$	27267.3	+	22880.0i	1.19588	+	1.00347i	0.999738	+	0.0228908i	$0.00728701\pi$
$151$	27267.3	+	22880.0i	1.19588	+	1.00347i	0.196146	+	0.980575i	$0.437157\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	15384.0	+	42267.2i	0.640334	+	1.75930i
$156$		0			0
$157$	6637.42	−	5569.46i	0.269277	−	0.225951i	−0.498143	−	0.867095i	$-0.665984\pi$
$157$	6637.42	−	5569.46i	0.269277	−	0.225951i	0.767420	+	0.641144i	$0.221540\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	12816.1i		−	0.494430i
$162$		0			0
$163$	44505.1			1.67508			0.837538	−	0.546380i	$-0.183994\pi$
$163$	44505.1			1.67508			0.837538	+	0.546380i	$0.183994\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	20696.9	+	24665.6i	0.742116	+	0.884420i	0.996578	−	0.0826633i	$-0.0263426\pi$
$167$	20696.9	+	24665.6i	0.742116	+	0.884420i	−0.254461	+	0.967083i	$0.581898\pi$
$168$		0			0
$169$	26832.9	−	9766.37i	0.939493	−	0.341948i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−19826.3	+	23628.1i	−0.662445	+	0.789472i	−0.987734	−	0.156143i	$-0.950094\pi$
$173$	−19826.3	+	23628.1i	−0.662445	+	0.789472i	0.325289	+	0.945615i	$0.394538\pi$
$174$		0			0
$175$	718.607	+	4075.42i	0.0234647	+	0.133075i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−26159.6	+	15103.2i	−0.816441	+	0.471372i	−0.849187	−	0.528091i	$-0.822908\pi$
$179$	−26159.6	+	15103.2i	−0.816441	+	0.471372i	0.0327469	+	0.999464i	$0.489574\pi$
$180$		0			0
$181$	26379.0	−	45689.8i	0.805196	−	1.39464i	−0.110963	−	0.993824i	$-0.535394\pi$
$181$	26379.0	−	45689.8i	0.805196	−	1.39464i	0.916159	−	0.400815i	$-0.131273\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	7986.57	−	21942.9i	0.233355	−	0.641137i
$186$		0			0
$187$	4767.83	−	27039.7i	0.136344	−	0.773248i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	43117.1	+	7602.71i	1.18191	+	0.208402i	0.729862	−	0.683594i	$-0.239584\pi$
$191$	43117.1	+	7602.71i	1.18191	+	0.208402i	0.452043	+	0.891996i	$0.350695\pi$
$192$		0			0
$193$	−65952.9	−	24004.9i	−1.77060	−	0.644444i	−0.999974	−	0.00715076i	$-0.997724\pi$
$193$	−65952.9	−	24004.9i	−1.77060	−	0.644444i	−0.770621	−	0.637293i	$-0.780054\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−24057.2	−	13889.5i	−0.619888	−	0.357893i	0.156937	−	0.987609i	$-0.449838\pi$
$197$	−24057.2	−	13889.5i	−0.619888	−	0.357893i	−0.776825	+	0.629716i	$0.783171\pi$
$198$		0			0
$199$	15512.8	+	26868.9i	0.391727	+	0.678491i	0.992677	−	0.120795i	$-0.0385445\pi$
$199$	15512.8	+	26868.9i	0.391727	+	0.678491i	−0.600951	+	0.799286i	$0.705211\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	2344.90	−	413.469i	0.0569025	−	0.0100335i
$204$		0			0
$205$	52643.4	+	44173.0i	1.25267	+	1.05111i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	19300.0	+	53026.2i	0.441839	+	1.21394i
$210$		0			0
$211$	−19010.7	+	15951.9i	−0.427005	+	0.358300i	−0.830820	−	0.556541i	$-0.812128\pi$
$211$	−19010.7	+	15951.9i	−0.427005	+	0.358300i	0.403815	+	0.914841i	$0.367684\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			47644.6i			1.03071i
$216$		0			0
$217$	−38990.1			−0.828008
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−380.178	−	453.078i	−0.00778399	−	0.00927659i
$222$		0			0
$223$	74226.5	−	27016.2i	1.49262	−	0.543269i	0.538482	−	0.842637i	$-0.318998\pi$
$223$	74226.5	−	27016.2i	1.49262	−	0.543269i	0.954138	+	0.299368i	$0.0967760\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−48220.6	+	57467.1i	−0.935795	+	1.11524i	0.0573506	+	0.998354i	$0.481735\pi$
$227$	−48220.6	+	57467.1i	−0.935795	+	1.11524i	−0.993146	+	0.116883i	$0.962710\pi$
$228$		0			0
$229$	−2505.13	−	14207.3i	−0.0477705	−	0.270920i	0.951562	−	0.307458i	$-0.0994782\pi$
$229$	−2505.13	−	14207.3i	−0.0477705	−	0.270920i	−0.999332	+	0.0365377i	$0.988367\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	21793.0	−	12582.2i	0.401425	−	0.231763i	−0.285674	−	0.958327i	$-0.592217\pi$
$233$	21793.0	−	12582.2i	0.401425	−	0.231763i	0.687099	+	0.726564i	$0.258884\pi$
$234$		0			0
$235$	15622.9	−	27059.6i	0.282895	−	0.489988i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	12305.3	−	33808.6i	0.215425	−	0.591876i	−0.784163	−	0.620554i	$-0.786908\pi$
$239$	12305.3	−	33808.6i	0.215425	−	0.591876i	0.999589	+	0.0286783i	$0.00912983\pi$
$240$		0			0
$241$	−11393.0	+	64612.9i	−0.196157	+	1.11246i	0.714605	+	0.699529i	$0.246607\pi$
$241$	−11393.0	+	64612.9i	−0.196157	+	1.11246i	−0.910762	+	0.412933i	$0.864504\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	50075.7	+	8829.70i	0.834247	+	0.147100i
$246$		0			0
$247$	1142.24	+	415.743i	0.0187226	+	0.00681445i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−35938.0	−	20748.8i	−0.570435	−	0.329341i	0.186888	−	0.982381i	$-0.440160\pi$
$251$	−35938.0	−	20748.8i	−0.570435	−	0.329341i	−0.757323	+	0.653040i	$0.773493\pi$
$252$		0			0
$253$	−29969.1	−	51908.0i	−0.468201	−	0.810948i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	67142.0	−	11838.9i	1.01655	−	0.179245i	0.359541	−	0.933129i	$-0.382933\pi$
$257$	67142.0	−	11838.9i	1.01655	−	0.179245i	0.657007	+	0.753884i	$0.271822\pi$
$258$		0			0
$259$	15506.0	+	13011.0i	0.231153	+	0.193960i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	10711.9	+	29430.8i	0.154866	+	0.425492i	0.992726	−	0.120394i	$-0.0384157\pi$
$263$	10711.9	+	29430.8i	0.154866	+	0.425492i	−0.837860	+	0.545885i	$0.816193\pi$
$264$		0			0
$265$	−83773.7	+	70294.5i	−1.19293	+	1.00099i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	136677.i		−	1.88882i	−0.328766	−	0.944411i	$-0.606633\pi$
$269$		−	136677.i		−	1.88882i	0.328766	−	0.944411i	$-0.393367\pi$
$270$		0			0
$271$	122527.			1.66837			0.834187	−	0.551481i	$-0.185937\pi$
$271$	122527.			1.66837			0.834187	+	0.551481i	$0.185937\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	12440.4	+	14825.9i	0.164502	+	0.196046i
$276$		0			0
$277$	−39521.0	+	14384.5i	−0.515073	+	0.187471i	−0.586461	−	0.809978i	$-0.699479\pi$
$277$	−39521.0	+	14384.5i	−0.515073	+	0.187471i	0.0713884	+	0.997449i	$0.477257\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	36512.2	−	43513.5i	0.462408	−	0.551076i	−0.483571	−	0.875305i	$-0.660660\pi$
$281$	36512.2	−	43513.5i	0.462408	−	0.551076i	0.945979	+	0.324229i	$0.105105\pi$
$282$		0			0
$283$	1441.88	+	8177.30i	0.0180035	+	0.102103i	0.992485	−	0.122363i	$-0.0390473\pi$
$283$	1441.88	+	8177.30i	0.0180035	+	0.102103i	−0.974482	+	0.224466i	$0.927936\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−51588.9	+	29784.9i	−0.626314	+	0.361603i
$288$		0			0
$289$	−12888.6	+	22323.7i	−0.154316	+	0.267283i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	51954.9	−	142745.i	0.605189	−	1.66274i	−0.135404	−	0.990791i	$-0.543233\pi$
$293$	51954.9	−	142745.i	0.605189	−	1.66274i	0.740593	−	0.671954i	$-0.234545\pi$
$294$		0			0
$295$	−18498.6	+	104911.i	−0.212567	+	1.20553i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1271.52	−	224.203i	−0.0142227	−	0.00250784i
$300$		0			0
$301$	−38809.2	−	14125.4i	−0.428353	−	0.155908i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−47363.5	−	27345.3i	−0.509148	−	0.293957i
$306$		0			0
$307$	−90294.2	−	156394.i	−0.958039	−	1.65937i	−0.727255	−	0.686367i	$-0.759204\pi$
$307$	−90294.2	−	156394.i	−0.958039	−	1.65937i	−0.230783	−	0.973005i	$-0.574129\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−71897.1	+	12677.4i	−0.743345	+	0.131072i	−0.532479	−	0.846443i	$-0.678740\pi$
$311$	−71897.1	+	12677.4i	−0.743345	+	0.131072i	−0.210866	+	0.977515i	$0.567628\pi$
$312$		0			0
$313$	−74410.2	−	62437.5i	−0.759528	−	0.637319i	0.178476	−	0.983944i	$-0.442883\pi$
$313$	−74410.2	−	62437.5i	−0.759528	−	0.637319i	−0.938004	+	0.346625i	$0.887328\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−27314.7	−	75046.5i	−0.271818	−	0.746813i	−0.998225	−	0.0595488i	$-0.981034\pi$
$317$	−27314.7	−	75046.5i	−0.271818	−	0.746813i	0.726408	−	0.687264i	$-0.241188\pi$
$318$		0			0
$319$	8530.48	−	7157.92i	0.0838286	−	0.0703405i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			118675.i			1.13751i
$324$		0			0
$325$	416.905			0.00394703
$326$		0			0
$327$		0			0
$328$		0			0
$329$	17409.8	+	20748.2i	0.160843	+	0.191685i
$330$		0			0
$331$	−51094.6	+	18596.9i	−0.466357	+	0.169740i	−0.564502	−	0.825432i	$-0.690932\pi$
$331$	−51094.6	+	18596.9i	−0.466357	+	0.169740i	0.0981442	+	0.995172i	$0.468709\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−15178.1	+	18088.5i	−0.135247	+	0.161181i
$336$		0			0
$337$	3454.56	+	19591.8i	0.0304181	+	0.172510i	0.996232	−	0.0867260i	$-0.0276405\pi$
$337$	3454.56	+	19591.8i	0.0304181	+	0.172510i	−0.965814	+	0.259236i	$0.916529\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−157918.	+	91174.0i	−1.35807	+	0.784083i
$342$		0			0
$343$	−51368.5	+	88972.9i	−0.436625	+	0.756257i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	19743.2	−	54243.9i	0.163967	−	0.450497i	−0.830313	−	0.557297i	$-0.811838\pi$
$347$	19743.2	−	54243.9i	0.163967	−	0.450497i	0.994280	+	0.106801i	$0.0340606\pi$
$348$		0			0
$349$	4802.56	−	27236.6i	0.0394295	−	0.223616i	−0.958725	−	0.284333i	$-0.908228\pi$
$349$	4802.56	−	27236.6i	0.0394295	−	0.223616i	0.998155	+	0.0607175i	$0.0193389\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−163139.	−	28765.8i	−1.30921	−	0.230848i	−0.524869	−	0.851183i	$-0.675886\pi$
$353$	−163139.	−	28765.8i	−1.30921	−	0.230848i	−0.784337	+	0.620334i	$0.786997\pi$
$354$		0			0
$355$	60143.9	+	21890.6i	0.477238	+	0.173700i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−20895.8	−	12064.2i	−0.162133	−	0.0936074i	0.416738	−	0.909027i	$-0.363173\pi$
$359$	−20895.8	−	12064.2i	−0.162133	−	0.0936074i	−0.578871	+	0.815419i	$0.696507\pi$
$360$		0			0
$361$	−56789.9	−	98363.1i	−0.435770	−	0.754775i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−163252.	+	28785.7i	−1.22539	+	0.216069i
$366$		0			0
$367$	7464.93	+	6263.82i	0.0554235	+	0.0465058i	0.670079	−	0.742290i	$-0.266260\pi$
$367$	7464.93	+	6263.82i	0.0554235	+	0.0465058i	−0.614655	+	0.788796i	$0.710705\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−32422.1	−	89079.0i	−0.235556	−	0.647184i
$372$		0			0
$373$	19173.2	−	16088.2i	0.137809	−	0.115635i	−0.571278	−	0.820757i	$-0.693552\pi$
$373$	19173.2	−	16088.2i	0.137809	−	0.115635i	0.709086	+	0.705122i	$0.249108\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	239.877i		−	0.00168774i
$378$		0			0
$379$	−57436.8			−0.399863			−0.199932	−	0.979810i	$-0.564072\pi$
$379$	−57436.8			−0.399863			−0.199932	+	0.979810i	$0.564072\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	82991.0	+	98904.8i	0.565762	+	0.674249i	0.970755	−	0.240071i	$-0.0771707\pi$
$383$	82991.0	+	98904.8i	0.565762	+	0.674249i	−0.404994	+	0.914319i	$0.632726\pi$
$384$		0			0
$385$	−73935.0	+	26910.1i	−0.498802	+	0.181549i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−24505.0	+	29203.9i	−0.161941	+	0.192993i	−0.840913	−	0.541171i	$-0.817981\pi$
$389$	−24505.0	+	29203.9i	−0.161941	+	0.192993i	0.678972	+	0.734164i	$0.262426\pi$
$390$		0			0
$391$	−21889.1	−	124139.i	−0.143177	−	0.812000i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	138964.	−	80230.8i	0.890651	−	0.514218i
$396$		0			0
$397$	5111.47	−	8853.33i	0.0324313	−	0.0561727i	−0.849354	−	0.527823i	$-0.823008\pi$
$397$	5111.47	−	8853.33i	0.0324313	−	0.0561727i	0.881786	+	0.471651i	$0.156342\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	92985.8	−	255477.i	0.578267	−	1.58877i	−0.212834	−	0.977088i	$-0.568269\pi$
$401$	92985.8	−	255477.i	0.578267	−	1.58877i	0.791101	−	0.611686i	$-0.209508\pi$
$402$		0			0
$403$	−682.087	+	3868.31i	−0.00419981	+	0.0238183i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	93227.3	+	16438.5i	0.562800	+	0.0992368i
$408$		0			0
$409$	−236318.	−	86012.7i	−1.41270	−	0.514181i	−0.480779	−	0.876842i	$-0.659646\pi$
$409$	−236318.	−	86012.7i	−1.41270	−	0.514181i	−0.931921	+	0.362661i	$0.881868\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−79971.5	−	46171.6i	−0.468851	−	0.270692i
$414$		0			0
$415$	104224.	+	180522.i	0.605163	+	1.04817i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−226077.	+	39863.4i	−1.28774	+	0.227063i	−0.775263	−	0.631638i	$-0.782383\pi$
$419$	−226077.	+	39863.4i	−1.28774	+	0.227063i	−0.512476	+	0.858702i	$0.671272\pi$
$420$		0			0
$421$	34300.2	+	28781.3i	0.193523	+	0.162385i	0.734401	−	0.678716i	$-0.237463\pi$
$421$	34300.2	+	28781.3i	0.193523	+	0.162385i	−0.540878	+	0.841101i	$0.681908\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	13921.1	+	38248.0i	0.0770720	+	0.211754i
$426$		0			0
$427$	36316.4	−	30473.1i	0.199181	−	0.167133i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	128951.i		−	0.694176i	−0.937833	−	0.347088i	$-0.887170\pi$
$431$		−	128951.i		−	0.694176i	0.937833	−	0.347088i	$-0.112830\pi$
$432$		0			0
$433$	68399.9			0.364821			0.182410	−	0.983222i	$-0.441610\pi$
$433$	68399.9			0.364821			0.182410	+	0.983222i	$0.441610\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	166525.	+	198457.i	0.872001	+	1.03921i
$438$		0			0
$439$	6048.33	−	2201.41i	0.0313839	−	0.0114228i	−0.326281	−	0.945273i	$-0.605795\pi$
$439$	6048.33	−	2201.41i	0.0313839	−	0.0114228i	0.357664	+	0.933850i	$0.383573\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	181179.	−	215921.i	0.923212	−	1.10024i	−0.0714898	−	0.997441i	$-0.522775\pi$
$443$	181179.	−	215921.i	0.923212	−	1.10024i	0.994702	−	0.102800i	$-0.0327802\pi$
$444$		0			0
$445$	57586.6	+	326590.i	0.290805	+	1.64924i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	250048.	−	144365.i	1.24031	−	0.716094i	0.271154	−	0.962536i	$-0.412595\pi$
$449$	250048.	−	144365.i	1.24031	−	0.716094i	0.969158	+	0.246442i	$0.0792614\pi$
$450$		0			0
$451$	−139297.	+	241270.i	−0.684841	+	1.18618i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−579.675	+	1592.64i	−0.00280002	+	0.00769300i
$456$		0			0
$457$	56283.2	−	319198.i	0.269492	−	1.52837i	−0.486438	−	0.873715i	$-0.661704\pi$
$457$	56283.2	−	319198.i	0.269492	−	1.52837i	0.755930	−	0.654652i	$-0.227185\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−254561.	−	44886.0i	−1.19782	−	0.211207i	−0.461062	−	0.887368i	$-0.652531\pi$
$461$	−254561.	−	44886.0i	−1.19782	−	0.211207i	−0.736754	+	0.676161i	$0.763642\pi$
$462$		0			0
$463$	249634.	+	90859.5i	1.16451	+	0.423846i	0.850706	−	0.525642i	$-0.176175\pi$
$463$	249634.	+	90859.5i	1.16451	+	0.423846i	0.313802	+	0.949488i	$0.398397\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	234369.	+	135313.i	1.07465	+	0.620448i	0.929448	−	0.368954i	$-0.120284\pi$
$467$	234369.	+	135313.i	1.07465	+	0.620448i	0.145200	+	0.989402i	$0.453617\pi$
$468$		0			0
$469$	−10234.2	−	17726.2i	−0.0465274	−	0.0805878i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−190216.	+	33540.3i	−0.850209	+	0.149915i
$474$		0			0
$475$	−64081.2	−	53770.5i	−0.284017	−	0.238318i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	43361.6	+	119135.i	0.188988	+	0.519240i	0.997611	−	0.0690874i	$-0.0220087\pi$
$479$	43361.6	+	119135.i	0.188988	+	0.519240i	−0.808623	+	0.588328i	$0.799787\pi$
$480$		0			0
$481$	1562.12	−	1310.77i	0.00675187	−	0.00566549i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	80848.2i		−	0.343706i
$486$		0			0
$487$	−99411.6			−0.419159			−0.209580	−	0.977792i	$-0.567210\pi$
$487$	−99411.6			−0.419159			−0.209580	+	0.977792i	$0.567210\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−59612.1	−	71043.0i	−0.247270	−	0.294685i	0.628106	−	0.778128i	$-0.283831\pi$
$491$	−59612.1	−	71043.0i	−0.247270	−	0.294685i	−0.875376	+	0.483443i	$0.839386\pi$
$492$		0			0
$493$	22006.9	−	8009.87i	0.0905453	−	0.0329558i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−35662.3	+	42500.7i	−0.144376	+	0.172061i
$498$		0			0
$499$	−58901.1	−	334045.i	−0.236550	−	1.34154i	−0.839326	−	0.543629i	$-0.817050\pi$
$499$	−58901.1	−	334045.i	−0.236550	−	1.34154i	0.602776	−	0.797910i	$-0.294061\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	22048.1	−	12729.5i	0.0871435	−	0.0503123i	−0.455795	−	0.890085i	$-0.650645\pi$
$503$	22048.1	−	12729.5i	0.0871435	−	0.0503123i	0.542939	+	0.839772i	$0.317312\pi$
$504$		0			0
$505$	−23650.5	+	40963.9i	−0.0927380	+	0.160627i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−149571.	+	410944.i	−0.577315	+	1.58616i	0.215373	+	0.976532i	$0.430903\pi$
$509$	−149571.	+	410944.i	−0.577315	+	1.58616i	−0.792688	+	0.609628i	$0.791319\pi$
$510$		0			0
$511$	24952.5	−	141512.i	0.0955590	−	0.541942i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	401610.	+	70814.6i	1.51422	+	0.266998i
$516$		0			0
$517$	119031.	+	43323.7i	0.445326	+	0.162085i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	389228.	+	224721.i	1.43393	+	0.827882i	0.997418	−	0.0718152i	$-0.0228792\pi$
$521$	389228.	+	224721.i	1.43393	+	0.827882i	0.436515	+	0.899697i	$0.356213\pi$
$522$		0			0
$523$	179231.	+	310437.i	0.655254	+	1.13493i	0.981830	+	0.189762i	$0.0607717\pi$
$523$	179231.	+	310437.i	0.655254	+	1.13493i	−0.326576	+	0.945171i	$0.605895\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−377665.	+	66592.6i	−1.35983	+	0.239775i
$528$		0			0
$529$	3573.37	+	2998.41i	0.0127693	+	0.0107147i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	2052.55	+	5639.32i	0.00722501	+	0.0198506i
$534$		0			0
$535$	131750.	−	110551.i	0.460302	−	0.386239i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			206138.i			0.709547i
$540$		0			0
$541$	116585.			0.398333			0.199167	−	0.979966i	$-0.436177\pi$
$541$	116585.			0.398333			0.199167	+	0.979966i	$0.436177\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	348059.	+	414800.i	1.17182	+	1.39652i
$546$		0			0
$547$	−51478.9	+	18736.8i	−0.172050	+	0.0626210i	−0.426609	−	0.904436i	$-0.640292\pi$
$547$	−51478.9	+	18736.8i	−0.172050	+	0.0626210i	0.254559	+	0.967057i	$0.418070\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−30938.2	+	36870.8i	−0.101904	+	0.121445i
$552$		0			0
$553$	24153.4	+	136980.i	0.0789818	+	0.447928i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	30508.0	−	17613.8i	0.0983339	−	0.0567731i	−0.450027	−	0.893015i	$-0.648586\pi$
$557$	30508.0	−	17613.8i	0.0983339	−	0.0567731i	0.548360	+	0.836242i	$0.315252\pi$
$558$		0			0
$559$	−2080.34	+	3603.26i	−0.00665750	+	0.0115311i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	61925.1	−	170138.i	0.195366	−	0.536765i	−0.802868	−	0.596156i	$-0.796694\pi$
$563$	61925.1	−	170138.i	0.195366	−	0.536765i	0.998235	+	0.0593917i	$0.0189161\pi$
$564$		0			0
$565$	−1340.44	+	7602.04i	−0.00419906	+	0.0238140i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−368377.	−	64954.7i	−1.13780	−	0.200626i	−0.427159	−	0.904176i	$-0.640486\pi$
$569$	−368377.	−	64954.7i	−1.13780	−	0.200626i	−0.710645	+	0.703551i	$0.751597\pi$
$570$		0			0
$571$	459788.	+	167349.i	1.41022	+	0.513277i	0.931193	−	0.364526i	$-0.118769\pi$
$571$	459788.	+	167349.i	1.41022	+	0.513277i	0.479023	+	0.877803i	$0.340991\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	76949.6	+	44426.9i	0.232740	+	0.134372i
$576$		0			0
$577$	113291.	+	196225.i	0.340285	+	0.589390i	0.984485	−	0.175466i	$-0.0561433\pi$
$577$	113291.	+	196225.i	0.340285	+	0.589390i	−0.644201	+	0.764856i	$0.722810\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−177945.	+	31376.5i	−0.527149	+	0.0929507i
$582$		0			0
$583$	−339618.	−	284973.i	−0.999203	−	0.838431i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−136938.	−	376234.i	−0.397418	−	1.09190i	−0.963538	−	0.267573i	$-0.913778\pi$
$587$	−136938.	−	376234.i	−0.397418	−	1.09190i	0.566120	−	0.824323i	$-0.308444\pi$
$588$		0			0
$589$	603759.	−	506614.i	1.74034	−	1.46032i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			332164.i			0.944590i	0.881441	+	0.472295i	$0.156574\pi$
$593$			332164.i			0.944590i	−0.881441	+	0.472295i	$0.843426\pi$
$594$		0			0
$595$	−165470.			−0.467396
$596$		0			0
$597$		0			0
$598$		0			0
$599$	403032.	+	480315.i	1.12327	+	1.33867i	0.934219	+	0.356700i	$0.116098\pi$
$599$	403032.	+	480315.i	1.12327	+	1.33867i	0.189056	+	0.981966i	$0.439457\pi$
$600$		0			0
$601$	565600.	−	205862.i	1.56589	−	0.569936i	0.593811	−	0.804604i	$-0.297623\pi$
$601$	565600.	−	205862.i	1.56589	−	0.569936i	0.972076	+	0.234668i	$0.0754003\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	28722.8	−	34230.6i	0.0784724	−	0.0935197i
$606$		0			0
$607$	−72553.0	−	411469.i	−0.196915	−	1.11676i	−0.909665	−	0.415342i	$-0.863662\pi$
$607$	−72553.0	−	411469.i	−0.196915	−	1.11676i	0.712751	−	0.701417i	$-0.247449\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	2363.05	−	1364.31i	0.00632980	−	0.00365451i
$612$		0			0
$613$	165625.	−	286871.i	0.440764	−	0.763425i	−0.556983	−	0.830524i	$-0.688041\pi$
$613$	165625.	−	286871.i	0.440764	−	0.763425i	0.997746	+	0.0670991i	$0.0213744\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	182752.	−	502108.i	0.480057	−	1.31895i	−0.429387	−	0.903120i	$-0.641270\pi$
$617$	182752.	−	502108.i	0.480057	−	1.31895i	0.909444	−	0.415826i	$-0.136507\pi$
$618$		0			0
$619$	18382.4	−	104252.i	0.0479756	−	0.272083i	−0.951378	−	0.308025i	$-0.900332\pi$
$619$	18382.4	−	104252.i	0.0479756	−	0.272083i	0.999354	+	0.0359418i	$0.0114431\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−283099.	−	49918.0i	−0.729395	−	0.128612i
$624$		0			0
$625$	439587.	+	159997.i	1.12534	+	0.409591i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	172416.	+	99544.2i	0.435788	+	0.251602i
$630$		0			0
$631$	288702.	+	500047.i	0.725089	+	1.25589i	0.958937	+	0.283618i	$0.0915348\pi$
$631$	288702.	+	500047.i	0.725089	+	1.25589i	−0.233848	+	0.972273i	$0.575132\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−812774.	+	143314.i	−2.01568	+	0.355419i
$636$		0			0
$637$	3401.58	+	2854.27i	0.00838305	+	0.00703422i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−181206.	−	497859.i	−0.441018	−	1.21169i	−0.938824	−	0.344397i	$-0.888083\pi$
$641$	−181206.	−	497859.i	−0.441018	−	1.21169i	0.497806	−	0.867288i	$-0.334139\pi$
$642$		0			0
$643$	68330.7	−	57336.3i	0.165270	−	0.138678i	−0.556402	−	0.830913i	$-0.687818\pi$
$643$	68330.7	−	57336.3i	0.165270	−	0.138678i	0.721672	+	0.692235i	$0.243374\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			690108.i			1.64857i	0.566172	+	0.824287i	$0.308424\pi$
$647$			690108.i			1.64857i	−0.566172	+	0.824287i	$0.691576\pi$
$648$		0			0
$649$	−431869.			−1.02533
$650$		0			0
$651$		0			0
$652$		0			0
$653$	360793.	+	429977.i	0.846121	+	1.00837i	0.999795	+	0.0202458i	$0.00644489\pi$
$653$	360793.	+	429977.i	0.846121	+	1.00837i	−0.153674	+	0.988122i	$0.549111\pi$
$654$		0			0
$655$	−673125.	+	244997.i	−1.56896	+	0.571056i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	470549.	−	560778.i	1.08351	−	1.29128i	0.129479	−	0.991582i	$-0.458670\pi$
$659$	470549.	−	560778.i	1.08351	−	1.29128i	0.954034	−	0.299698i	$-0.0968859\pi$
$660$		0			0
$661$	118352.	+	671209.i	0.270878	+	1.53623i	0.751758	+	0.659439i	$0.229206\pi$
$661$	118352.	+	671209.i	0.270878	+	1.53623i	−0.480880	+	0.876786i	$0.659683\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	294512.	−	170037.i	0.665978	−	0.384503i
$666$		0			0
$667$	25562.1	−	44274.9i	0.0574573	−	0.0995190i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	75831.2	−	208345.i	0.168424	−	0.462740i
$672$		0			0
$673$	−50789.8	+	288043.i	−0.112136	+	0.635957i	0.875992	+	0.482326i	$0.160208\pi$
$673$	−50789.8	+	288043.i	−0.112136	+	0.635957i	−0.988128	+	0.153631i	$0.950903\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	547224.	+	96490.4i	1.19396	+	0.210527i	0.735084	−	0.677976i	$-0.237143\pi$
$677$	547224.	+	96490.4i	1.19396	+	0.210527i	0.458871	+	0.888503i	$0.348254\pi$
$678$		0			0
$679$	65855.5	+	23969.4i	0.142841	+	0.0519898i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	8476.38	+	4893.84i	0.0181706	+	0.0104908i	0.509058	−	0.860732i	$-0.329994\pi$
$683$	8476.38	+	4893.84i	0.0181706	+	0.0104908i	−0.490887	+	0.871223i	$0.663327\pi$
$684$		0			0
$685$	−184304.	−	319224.i	−0.392783	−	0.680321i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−9404.96	+	1658.35i	−0.0198115	+	0.00349331i
$690$		0			0
$691$	−604854.	−	507533.i	−1.26676	−	1.06294i	−0.994928	−	0.100594i	$-0.967926\pi$
$691$	−604854.	−	507533.i	−1.26676	−	1.06294i	−0.271833	−	0.962344i	$-0.587630\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−86412.9	−	237418.i	−0.178900	−	0.491522i
$696$		0			0
$697$	−448829.	+	376612.i	−0.923880	+	0.775227i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	765720.i		−	1.55824i	−0.626876	−	0.779119i	$-0.715667\pi$
$701$		−	765720.i		−	1.55824i	0.626876	−	0.779119i	$-0.284333\pi$
$702$		0			0
$703$	−409167.			−0.827922
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−26355.6	−	31409.4i	−0.0527272	−	0.0628379i
$708$		0			0
$709$	−132595.	+	48260.7i	−0.263776	+	0.0960066i	−0.470523	−	0.882388i	$-0.655935\pi$
$709$	−132595.	+	48260.7i	−0.263776	+	0.0960066i	0.206747	+	0.978394i	$0.433712\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−538116.	+	641302.i	−1.05851	+	1.26149i
$714$		0			0
$715$	1376.42	+	7806.05i	0.00269239	+	0.0152693i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−76955.1	+	44430.1i	−0.148861	+	0.0859447i	−0.572580	−	0.819849i	$-0.694058\pi$
$719$	−76955.1	+	44430.1i	−0.148861	+	0.0859447i	0.423720	+	0.905793i	$0.360724\pi$
$720$		0			0
$721$	−176750.	+	306140.i	−0.340007	+	0.588910i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−5646.03	+	15512.3i	−0.0107416	+	0.0295122i
$726$		0			0
$727$	−2153.14	+	12211.1i	−0.00407384	+	0.0231039i	−0.986777	−	0.162084i	$-0.948178\pi$
$727$	−2153.14	+	12211.1i	−0.00407384	+	0.0231039i	0.982703	+	0.185188i	$0.0592895\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−400039.	−	70537.7i	−0.748631	−	0.132004i
$732$		0			0
$733$	−405994.	−	147770.i	−0.755634	−	0.275028i	−0.0646595	−	0.997907i	$-0.520596\pi$
$733$	−405994.	−	147770.i	−0.755634	−	0.275028i	−0.690975	+	0.722879i	$0.742818\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−82901.5	−	47863.2i	−0.152626	−	0.0881184i
$738$		0			0
$739$	95170.2	+	164840.i	0.174266	+	0.301837i	0.939907	−	0.341431i	$-0.110911\pi$
$739$	95170.2	+	164840.i	0.174266	+	0.301837i	−0.765641	+	0.643268i	$0.777578\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−920487.	+	162307.i	−1.66740	+	0.294008i	−0.926132	−	0.377200i	$-0.876887\pi$
$743$	−920487.	+	162307.i	−1.66740	+	0.294008i	−0.741269	+	0.671208i	$0.765776\pi$
$744$		0			0
$745$	55526.9	+	46592.6i	0.100044	+	0.0839469i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	50989.9	+	140093.i	0.0908908	+	0.249721i
$750$		0			0
$751$	193215.	−	162126.i	0.342578	−	0.287457i	−0.455223	−	0.890377i	$-0.650441\pi$
$751$	193215.	−	162126.i	0.342578	−	0.287457i	0.797802	+	0.602920i	$0.205996\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	1.00324e6i		−	1.75999i
$756$		0			0
$757$	−449009.			−0.783544			−0.391772	−	0.920062i	$-0.628138\pi$
$757$	−449009.			−0.783544			−0.391772	+	0.920062i	$0.628138\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	380749.	+	453759.i	0.657460	+	0.783530i	0.987019	−	0.160605i	$-0.0513444\pi$
$761$	380749.	+	453759.i	0.657460	+	0.783530i	−0.329559	+	0.944135i	$0.606900\pi$
$762$		0			0
$763$	−441069.	+	160536.i	−0.757631	+	0.275755i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−5979.82	+	7126.47i	−0.0101648	+	0.0121139i
$768$		0			0
$769$	−35286.0	−	200117.i	−0.0596692	−	0.338401i	0.940329	−	0.340267i	$-0.110517\pi$
$769$	−35286.0	−	200117.i	−0.0596692	−	0.338401i	−0.999998	+	0.00186555i	$0.999406\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	93598.2	−	54039.0i	0.156642	−	0.0904374i	−0.419630	−	0.907695i	$-0.637840\pi$
$773$	93598.2	−	54039.0i	0.156642	−	0.0904374i	0.576272	+	0.817258i	$0.304507\pi$
$774$		0			0
$775$	135158.	−	234101.i	0.225030	−	0.389763i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	411844.	−	1.13153e6i	0.678669	−	1.86463i
$780$		0			0
$781$	−45056.7	+	255529.i	−0.0738681	+	0.418927i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−240498.	−	42406.3i	−0.390277	−	0.0688163i
$786$		0			0
$787$	−819692.	−	298343.i	−1.32343	−	0.481689i	−0.418874	−	0.908044i	$-0.637575\pi$
$787$	−819692.	−	298343.i	−1.32343	−	0.481689i	−0.904556	+	0.426355i	$0.859797\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−5794.89	−	3345.68i	−0.00926173	−	0.00534726i
$792$		0			0
$793$	−2388.00	−	4136.14i	−0.00379742	−	0.00657732i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−105519.	+	18605.8i	−0.166116	+	0.0292908i	−0.256088	−	0.966654i	$-0.582434\pi$
$797$	−105519.	+	18605.8i	−0.166116	+	0.0292908i	0.0899713	+	0.995944i	$0.471322\pi$
$798$		0			0
$799$	204071.	+	171236.i	0.319660	+	0.268227i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−229849.	−	631504.i	−0.356460	−	0.979366i
$804$		0			0
$805$	−276710.	+	232188.i	−0.427006	+	0.358300i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			15837.0i			0.0241977i	0.999927	+	0.0120989i	$0.00385128\pi$
$809$			15837.0i			0.0241977i	−0.999927	+	0.0120989i	$0.996149\pi$
$810$		0			0
$811$	77890.8			0.118425			0.0592127	−	0.998245i	$-0.481141\pi$
$811$	77890.8			0.118425			0.0592127	+	0.998245i	$0.481141\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−806291.	−	960901.i	−1.21388	−	1.44665i
$816$		0			0
$817$	784496.	−	285533.i	1.17529	−	0.427772i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	281744.	−	335769.i	0.417992	−	0.498143i	−0.515426	−	0.856934i	$-0.672366\pi$
$821$	281744.	−	335769.i	0.417992	−	0.498143i	0.933418	+	0.358791i	$0.116811\pi$
$822$		0			0
$823$	43692.6	+	247793.i	0.0645072	+	0.365838i	0.999924	+	0.0122928i	$0.00391301\pi$
$823$	43692.6	+	247793.i	0.0645072	+	0.365838i	−0.935417	+	0.353546i	$0.884976\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−317420.	+	183262.i	−0.464112	+	0.267955i	−0.713772	−	0.700379i	$-0.753015\pi$
$827$	−317420.	+	183262.i	−0.464112	+	0.267955i	0.249660	+	0.968334i	$0.419681\pi$
$828$		0			0
$829$	−260535.	+	451260.i	−0.379103	+	0.656625i	−0.990932	−	0.134365i	$-0.957101\pi$
$829$	−260535.	+	451260.i	−0.379103	+	0.656625i	0.611829	+	0.790990i	$0.290434\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−148274.	+	407379.i	−0.213685	+	0.587096i
$834$		0			0
$835$	157588.	−	893725.i	0.226021	−	1.28183i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	488823.	+	86192.7i	0.694429	+	0.122447i	0.509712	−	0.860345i	$-0.329752\pi$
$839$	488823.	+	86192.7i	0.694429	+	0.122447i	0.184717	+	0.982792i	$0.440863\pi$
$840$		0			0
$841$	−655701.	−	238656.i	−0.927073	−	0.337427i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−696991.	−	402408.i	−0.976143	−	0.563577i
$846$		0			0
$847$	19367.2	+	33544.9i	0.0269960	+	0.0467584i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	428007.	−	75469.1i	0.591005	−	0.104210i
$852$		0			0
$853$	740679.	+	621504.i	1.01796	+	0.854172i	0.989370	−	0.145419i	$-0.0464529\pi$
$853$	740679.	+	621504.i	1.01796	+	0.854172i	0.0285928	+	0.999591i	$0.490897\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	396643.	+	1.08977e6i	0.540055	+	1.48379i	0.846755	+	0.531982i	$0.178553\pi$
$857$	396643.	+	1.08977e6i	0.540055	+	1.48379i	−0.306701	+	0.951806i	$0.599225\pi$
$858$		0			0
$859$	−138587.	+	116288.i	−0.187817	+	0.157597i	−0.731848	−	0.681468i	$-0.761342\pi$
$859$	−138587.	+	116288.i	−0.187817	+	0.157597i	0.544031	+	0.839065i	$0.316897\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			941281.i			1.26386i	0.775027	+	0.631928i	$0.217736\pi$
$863$			941281.i			1.26386i	−0.775027	+	0.631928i	$0.782264\pi$
$864$		0			0
$865$	869340.			1.16187
$866$		0			0
$867$		0			0
$868$		0			0
$869$	418140.	+	498320.i	0.553710	+	0.659886i
$870$		0			0
$871$	−1937.70	+	705.264i	−0.00255417	+	0.000929642i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−201666.	+	240336.i	−0.263401	+	0.313909i
$876$		0			0
$877$	−123251.	−	698994.i	−0.160248	−	0.908812i	−0.953830	−	0.300348i	$-0.902897\pi$
$877$	−123251.	−	698994.i	−0.160248	−	0.908812i	0.793582	−	0.608464i	$-0.208214\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	479040.	−	276574.i	0.617191	−	0.356336i	−0.158583	−	0.987346i	$-0.550693\pi$
$881$	479040.	−	276574.i	0.617191	−	0.356336i	0.775775	+	0.631010i	$0.217359\pi$
$882$		0			0
$883$	−567258.	+	982519.i	−0.727543	+	1.26014i	0.230375	+	0.973102i	$0.426005\pi$
$883$	−567258.	+	982519.i	−0.727543	+	1.26014i	−0.957919	+	0.287040i	$0.907329\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	201922.	−	554777.i	0.256648	−	0.705133i	−0.742721	−	0.669601i	$-0.766465\pi$
$887$	201922.	−	554777.i	0.256648	−	0.705133i	0.999369	−	0.0355323i	$-0.0113127\pi$
$888$		0			0
$889$	124229.	−	704540.i	0.157189	−	0.891461i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−539180.	−	95071.9i	−0.676131	−	0.119220i
$894$		0			0
$895$	800020.	+	291183.i	0.998745	+	0.363514i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−134696.	−	77766.8i	−0.166662	−	0.0962221i
$900$		0			0
$901$	−466188.	−	807462.i	−0.574264	−	0.994655i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−1.46438e6	+	258210.i	−1.78796	+	0.315266i
$906$		0			0
$907$	−452398.	−	379607.i	−0.549928	−	0.461444i	0.324989	−	0.945718i	$-0.394639\pi$
$907$	−452398.	−	379607.i	−0.549928	−	0.461444i	−0.874917	+	0.484274i	$0.839084\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−56684.2	−	155738.i	−0.0683007	−	0.187655i	0.900846	−	0.434138i	$-0.142947\pi$
$911$	−56684.2	−	155738.i	−0.0683007	−	0.187655i	−0.969147	+	0.246484i	$0.920725\pi$
$912$		0			0
$913$	−647345.	+	543187.i	−0.776594	+	0.651640i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	620934.i		−	0.738426i
$918$		0			0
$919$	1.06675e6			1.26309			0.631543	−	0.775341i	$-0.282422\pi$
$919$	1.06675e6			1.26309			0.631543	+	0.775341i	$0.282422\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	3592.73	+	4281.65i	0.00421717	+	0.00502583i
$924$		0			0
$925$	−131871.	+	47997.2i	−0.154123	+	0.0560960i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−389468.	+	464150.i	−0.451274	+	0.537807i	−0.942934	−	0.332980i	$-0.891946\pi$
$929$	−389468.	+	464150.i	−0.451274	+	0.537807i	0.491660	+	0.870787i	$0.336390\pi$
$930$		0			0
$931$	−154717.	−	877443.i	−0.178500	−	1.01232i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−670187.	+	386933.i	−0.766607	+	0.442601i
$936$		0			0
$937$	650792.	−	1.12720e6i	0.741247	−	1.28388i	−0.210681	−	0.977555i	$-0.567568\pi$
$937$	650792.	−	1.12720e6i	0.741247	−	1.28388i	0.951928	−	0.306323i	$-0.0990987\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	285013.	−	783067.i	0.321874	−	0.884341i	−0.668224	−	0.743960i	$-0.732945\pi$
$941$	285013.	−	783067.i	0.321874	−	0.884341i	0.990098	−	0.140381i	$-0.0448326\pi$
$942$		0			0
$943$	−222101.	+	1.25960e6i	−0.249762	+	1.41647i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1.02383e6	+	180529.i	1.14164	+	0.201302i	0.712323	−	0.701852i	$-0.247643\pi$
$947$	1.02383e6	+	180529.i	1.14164	+	0.201302i	0.429317	+	0.903154i	$0.358754\pi$
$948$		0			0
$949$	−13603.3	−	4951.20i	−0.0151047	−	0.00549766i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1.26089e6	−	727976.i	−1.38833	−	0.801551i	−0.395200	−	0.918595i	$-0.629325\pi$
$953$	−1.26089e6	−	727976.i	−1.38833	−	0.801551i	−0.993127	+	0.117044i	$0.962658\pi$
$954$		0			0
$955$	−616997.	−	1.06867e6i	−0.676513	−	1.17176i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	314667.	−	55484.3i	0.342148	−	0.0603300i
$960$		0			0
$961$	1.24355e6	+	1.04347e6i	1.34654	+	1.12988i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	676573.	+	1.85887e6i	0.726541	+	1.99616i
$966$		0			0
$967$	−215918.	+	181176.i	−0.230906	+	0.193753i	−0.750898	−	0.660418i	$-0.770379\pi$
$967$	−215918.	+	181176.i	−0.230906	+	0.193753i	0.519992	+	0.854171i	$0.325935\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			450143.i			0.477433i	0.971089	+	0.238716i	$0.0767267\pi$
$971$			450143.i			0.477433i	−0.971089	+	0.238716i	$0.923273\pi$
$972$		0			0
$973$	219010.			0.231333
$974$		0			0
$975$		0			0
$976$		0			0
$977$	601049.	+	716302.i	0.629681	+	0.750424i	0.982703	−	0.185191i	$-0.0592903\pi$
$977$	601049.	+	716302.i	0.629681	+	0.750424i	−0.353022	+	0.935615i	$0.614846\pi$
$978$		0			0
$979$	−1.26334e6	+	459818.i	−1.31812	+	0.479756i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	32996.0	−	39323.2i	0.0341472	−	0.0406950i	−0.748701	−	0.662908i	$-0.769322\pi$
$983$	32996.0	−	39323.2i	0.0341472	−	0.0406950i	0.782848	+	0.622213i	$0.213766\pi$
$984$		0			0
$985$	135957.	+	771048.i	0.140129	+	0.794711i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−767953.	+	443378.i	−0.785131	+	0.453295i
$990$		0			0
$991$	124272.	−	215246.i	0.126540	−	0.219173i	−0.795794	−	0.605567i	$-0.792946\pi$
$991$	124272.	−	215246.i	0.126540	−	0.219173i	0.922334	+	0.386394i	$0.126280\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	299079.	−	821713.i	0.302093	−	0.829992i
$996$		0			0
$997$	−209863.	+	1.19019e6i	−0.211128	+	1.19737i	0.676373	+	0.736560i	$0.263551\pi$
$997$	−209863.	+	1.19019e6i	−0.211128	+	1.19737i	−0.887501	+	0.460807i	$0.847560\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	324.5.k.a.89.3		72
3.2	odd	2		108.5.k.a.29.2	✓	72
27.13	even	9		108.5.k.a.41.2	yes	72
27.14	odd	18	inner	324.5.k.a.233.3		72

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.5.k.a.29.2	✓	72	3.2	odd	2
108.5.k.a.41.2	yes	72	27.13	even	9
324.5.k.a.89.3		72	1.1	even	1	trivial
324.5.k.a.233.3		72	27.14	odd	18	inner

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:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	324.k (of order \(18\), degree \(6\), not minimal)

\(f(q)\)	\(=\)	\(q+(-18.1168 - 21.5908i) q^{5} +(22.9581 - 8.35608i) q^{7} +O(q^{10})\)	\(q+(-18.1168 - 21.5908i) q^{5} +(22.9581 - 8.35608i) q^{7} +(73.4456 - 87.5290i) q^{11} +(-0.427402 - 2.42392i) q^{13} +(208.105 - 120.150i) q^{17} +(-246.932 + 427.698i) q^{19} +(179.414 - 492.937i) q^{23} +(-29.4131 + 166.810i) q^{25} +(95.9782 + 16.9236i) q^{29} +(-1499.65 - 545.827i) q^{31} +(-596.344 - 344.299i) q^{35} +(414.251 + 717.504i) q^{37} +(-2401.19 + 423.394i) q^{41} +(-1294.95 - 1086.59i) q^{43} +(379.164 + 1041.74i) q^{47} +(-1382.02 + 1159.65i) q^{49} -3880.06i q^{53} -3220.42 q^{55} +(-2429.52 - 2895.39i) q^{59} +(1823.41 - 663.667i) q^{61} +(-44.5912 + 53.1417i) q^{65} +(-145.480 - 825.059i) q^{67} +(-1966.62 + 1135.43i) q^{71} +(2940.78 - 5093.57i) q^{73} +(954.774 - 2623.22i) q^{77} +(-988.613 + 5606.70i) q^{79} +(-7283.42 - 1284.26i) q^{83} +(-6364.34 - 2316.43i) q^{85} +(-10189.8 - 5883.10i) q^{89} +(-30.0668 - 52.0773i) q^{91} +(13708.0 - 2417.09i) q^{95} +(2197.40 + 1843.84i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 9 q^{5}+O(q^{10}) \) 72 * q - 9 * q^5	\( 72 q - 9 q^{5} - 18 q^{11} + 1278 q^{23} + 441 q^{25} - 1854 q^{29} - 1665 q^{31} + 2673 q^{35} + 5472 q^{41} + 1260 q^{43} - 5103 q^{47} - 5904 q^{49} + 10944 q^{59} + 8352 q^{61} - 8757 q^{65} + 378 q^{67} + 19764 q^{71} + 6111 q^{73} + 5679 q^{77} - 5652 q^{79} + 20061 q^{83} + 26100 q^{85} - 15633 q^{89} - 6039 q^{91} - 48024 q^{95} - 37530 q^{97}+O(q^{100}) \) 72 * q - 9 * q^5 - 18 * q^11 + 1278 * q^23 + 441 * q^25 - 1854 * q^29 - 1665 * q^31 + 2673 * q^35 + 5472 * q^41 + 1260 * q^43 - 5103 * q^47 - 5904 * q^49 + 10944 * q^59 + 8352 * q^61 - 8757 * q^65 + 378 * q^67 + 19764 * q^71 + 6111 * q^73 + 5679 * q^77 - 5652 * q^79 + 20061 * q^83 + 26100 * q^85 - 15633 * q^89 - 6039 * q^91 - 48024 * q^95 - 37530 * q^97

Introduction

L-functions

Modular forms

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Database

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