LMFDB - Embedded newform 1200.2.v.b.593.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,2,Mod(257,1200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1200, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 2, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1200.257");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1200 = 2^{4} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1200.v (of order $4$, degree $2$, 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:	$9.58204824255$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 30)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			593.2
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1200.593
Dual form			1200.2.v.b.257.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.292893 - 1.70711i) q^{3} +(-1.00000 + 1.00000i) q^{7} +(-2.82843 + 1.00000i) q^{9} +O(q^{10})$	$q+(-0.292893 - 1.70711i) q^{3} +(-1.00000 + 1.00000i) q^{7} +(-2.82843 + 1.00000i) q^{9} +1.41421i q^{11} +(1.41421 + 1.41421i) q^{17} +4.00000i q^{19} +(2.00000 + 1.41421i) q^{21} +(-2.82843 + 2.82843i) q^{23} +(2.53553 + 4.53553i) q^{27} +7.07107 q^{29} +2.00000 q^{31} +(2.41421 - 0.414214i) q^{33} +(-6.00000 + 6.00000i) q^{37} -5.65685i q^{41} +(6.00000 + 6.00000i) q^{43} +5.00000i q^{49} +(2.00000 - 2.82843i) q^{51} +(-2.82843 + 2.82843i) q^{53} +(6.82843 - 1.17157i) q^{57} +9.89949 q^{59} -6.00000 q^{61} +(1.82843 - 3.82843i) q^{63} +(-4.00000 + 4.00000i) q^{67} +(5.65685 + 4.00000i) q^{69} -14.1421i q^{71} +(5.00000 + 5.00000i) q^{73} +(-1.41421 - 1.41421i) q^{77} +6.00000i q^{79} +(7.00000 - 5.65685i) q^{81} +(8.48528 - 8.48528i) q^{83} +(-2.07107 - 12.0711i) q^{87} -2.82843 q^{89} +(-0.585786 - 3.41421i) q^{93} +(-3.00000 + 3.00000i) q^{97} +(-1.41421 - 4.00000i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} - 4 q^{7}+O(q^{10}) $ 4 * q - 4 * q^3 - 4 * q^7	$ 4 q - 4 q^{3} - 4 q^{7} + 8 q^{21} - 4 q^{27} + 8 q^{31} + 4 q^{33} - 24 q^{37} + 24 q^{43} + 8 q^{51} + 16 q^{57} - 24 q^{61} - 4 q^{63} - 16 q^{67} + 20 q^{73} + 28 q^{81} + 20 q^{87} - 8 q^{93} - 12 q^{97}+O(q^{100}) $ 4 * q - 4 * q^3 - 4 * q^7 + 8 * q^21 - 4 * q^27 + 8 * q^31 + 4 * q^33 - 24 * q^37 + 24 * q^43 + 8 * q^51 + 16 * q^57 - 24 * q^61 - 4 * q^63 - 16 * q^67 + 20 * q^73 + 28 * q^81 + 20 * q^87 - 8 * q^93 - 12 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$.

$n$	$401$	$577$	$751$	$901$
$\chi(n)$	$-1$	$e\left(\frac{3}{4}\right)$	$1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	−0.292893	−	1.70711i	−0.169102	−	0.985599i
$3$	−0.292893	−	1.70711i	−0.169102	−	0.985599i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−1.00000	+	1.00000i	−0.377964	+	0.377964i	−0.870367	−	0.492403i	$-0.836119\pi$
$7$	−1.00000	+	1.00000i	−0.377964	+	0.377964i	0.492403	+	0.870367i	$0.336119\pi$
$8$		0			0
$9$	−2.82843	+	1.00000i	−0.942809	+	0.333333i
$10$		0			0
$11$			1.41421i			0.426401i	0.977008	+	0.213201i	$0.0683888\pi$
$11$			1.41421i			0.426401i	−0.977008	+	0.213201i	$0.931611\pi$
$12$		0			0
$13$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$13$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.41421	+	1.41421i	0.342997	+	0.342997i	0.857493	−	0.514496i	$-0.172021\pi$
$17$	1.41421	+	1.41421i	0.342997	+	0.342997i	−0.514496	+	0.857493i	$0.672021\pi$
$18$		0			0
$19$			4.00000i			0.917663i	0.888523	+	0.458831i	$0.151732\pi$
$19$			4.00000i			0.917663i	−0.888523	+	0.458831i	$0.848268\pi$
$20$		0			0
$21$	2.00000	+	1.41421i	0.436436	+	0.308607i
$22$		0			0
$23$	−2.82843	+	2.82843i	−0.589768	+	0.589768i	−0.937568	−	0.347801i	$-0.886929\pi$
$23$	−2.82843	+	2.82843i	−0.589768	+	0.589768i	0.347801	+	0.937568i	$0.386929\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	2.53553	+	4.53553i	0.487964	+	0.872864i
$28$		0			0
$29$	7.07107			1.31306			0.656532	−	0.754298i	$-0.272023\pi$
$29$	7.07107			1.31306			0.656532	+	0.754298i	$0.272023\pi$
$30$		0			0
$31$	2.00000			0.359211			0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000			0.359211			0.179605	+	0.983739i	$0.442518\pi$
$32$		0			0
$33$	2.41421	−	0.414214i	0.420261	−	0.0721053i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−6.00000	+	6.00000i	−0.986394	+	0.986394i	−0.999909	−	0.0135147i	$-0.995698\pi$
$37$	−6.00000	+	6.00000i	−0.986394	+	0.986394i	0.0135147	+	0.999909i	$0.495698\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	5.65685i		−	0.883452i	−0.897150	−	0.441726i	$-0.854366\pi$
$41$		−	5.65685i		−	0.883452i	0.897150	−	0.441726i	$-0.145634\pi$
$42$		0			0
$43$	6.00000	+	6.00000i	0.914991	+	0.914991i	0.996660	−	0.0816682i	$-0.0260248\pi$
$43$	6.00000	+	6.00000i	0.914991	+	0.914991i	−0.0816682	+	0.996660i	$0.526025\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$47$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$48$		0			0
$49$			5.00000i			0.714286i
$50$		0			0
$51$	2.00000	−	2.82843i	0.280056	−	0.396059i
$52$		0			0
$53$	−2.82843	+	2.82843i	−0.388514	+	0.388514i	−0.874157	−	0.485643i	$-0.838586\pi$
$53$	−2.82843	+	2.82843i	−0.388514	+	0.388514i	0.485643	+	0.874157i	$0.338586\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	6.82843	−	1.17157i	0.904447	−	0.155179i
$58$		0			0
$59$	9.89949			1.28880			0.644402	−	0.764687i	$-0.277106\pi$
$59$	9.89949			1.28880			0.644402	+	0.764687i	$0.277106\pi$
$60$		0			0
$61$	−6.00000			−0.768221			−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000			−0.768221			−0.384111	+	0.923287i	$0.625492\pi$
$62$		0			0
$63$	1.82843	−	3.82843i	0.230360	−	0.482336i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−4.00000	+	4.00000i	−0.488678	+	0.488678i	−0.907889	−	0.419211i	$-0.862307\pi$
$67$	−4.00000	+	4.00000i	−0.488678	+	0.488678i	0.419211	+	0.907889i	$0.362307\pi$
$68$		0			0
$69$	5.65685	+	4.00000i	0.681005	+	0.481543i
$70$		0			0
$71$		−	14.1421i		−	1.67836i	−0.543852	−	0.839181i	$-0.683035\pi$
$71$		−	14.1421i		−	1.67836i	0.543852	−	0.839181i	$-0.316965\pi$
$72$		0			0
$73$	5.00000	+	5.00000i	0.585206	+	0.585206i	0.936329	−	0.351123i	$-0.114200\pi$
$73$	5.00000	+	5.00000i	0.585206	+	0.585206i	−0.351123	+	0.936329i	$0.614200\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−1.41421	−	1.41421i	−0.161165	−	0.161165i
$78$		0			0
$79$			6.00000i			0.675053i	0.941316	+	0.337526i	$0.109590\pi$
$79$			6.00000i			0.675053i	−0.941316	+	0.337526i	$0.890410\pi$
$80$		0			0
$81$	7.00000	−	5.65685i	0.777778	−	0.628539i
$82$		0			0
$83$	8.48528	−	8.48528i	0.931381	−	0.931381i	−0.0664117	−	0.997792i	$-0.521155\pi$
$83$	8.48528	−	8.48528i	0.931381	−	0.931381i	0.997792	+	0.0664117i	$0.0211551\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−2.07107	−	12.0711i	−0.222042	−	1.29415i
$88$		0			0
$89$	−2.82843			−0.299813			−0.149906	−	0.988700i	$-0.547897\pi$
$89$	−2.82843			−0.299813			−0.149906	+	0.988700i	$0.547897\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−0.585786	−	3.41421i	−0.0607432	−	0.354037i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−3.00000	+	3.00000i	−0.304604	+	0.304604i	−0.842812	−	0.538208i	$-0.819101\pi$
$97$	−3.00000	+	3.00000i	−0.304604	+	0.304604i	0.538208	+	0.842812i	$0.319101\pi$
$98$		0			0
$99$	−1.41421	−	4.00000i	−0.142134	−	0.402015i
$100$		0			0
$101$			9.89949i			0.985037i	0.870302	+	0.492518i	$0.163924\pi$
$101$			9.89949i			0.985037i	−0.870302	+	0.492518i	$0.836076\pi$
$102$		0			0
$103$	1.00000	+	1.00000i	0.0985329	+	0.0985329i	0.754655	−	0.656122i	$-0.227804\pi$
$103$	1.00000	+	1.00000i	0.0985329	+	0.0985329i	−0.656122	+	0.754655i	$0.727804\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	2.82843	+	2.82843i	0.273434	+	0.273434i	0.830481	−	0.557047i	$-0.188066\pi$
$107$	2.82843	+	2.82843i	0.273434	+	0.273434i	−0.557047	+	0.830481i	$0.688066\pi$
$108$		0			0
$109$			10.0000i			0.957826i	0.877862	+	0.478913i	$0.158969\pi$
$109$			10.0000i			0.957826i	−0.877862	+	0.478913i	$0.841031\pi$
$110$		0			0
$111$	12.0000	+	8.48528i	1.13899	+	0.805387i
$112$		0			0
$113$	9.89949	−	9.89949i	0.931266	−	0.931266i	−0.0665190	−	0.997785i	$-0.521189\pi$
$113$	9.89949	−	9.89949i	0.931266	−	0.931266i	0.997785	+	0.0665190i	$0.0211893\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−2.82843			−0.259281
$120$		0			0
$121$	9.00000			0.818182
$122$		0			0
$123$	−9.65685	+	1.65685i	−0.870729	+	0.149394i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−7.00000	+	7.00000i	−0.621150	+	0.621150i	−0.945825	−	0.324676i	$-0.894745\pi$
$127$	−7.00000	+	7.00000i	−0.621150	+	0.621150i	0.324676	+	0.945825i	$0.394745\pi$
$128$		0			0
$129$	8.48528	−	12.0000i	0.747087	−	1.05654i
$130$		0			0
$131$			18.3848i			1.60629i	0.595787	+	0.803143i	$0.296840\pi$
$131$			18.3848i			1.60629i	−0.595787	+	0.803143i	$0.703160\pi$
$132$		0			0
$133$	−4.00000	−	4.00000i	−0.346844	−	0.346844i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−4.24264	−	4.24264i	−0.362473	−	0.362473i	0.502249	−	0.864723i	$-0.332506\pi$
$137$	−4.24264	−	4.24264i	−0.362473	−	0.362473i	−0.864723	+	0.502249i	$0.832506\pi$
$138$		0			0
$139$			8.00000i			0.678551i	0.940687	+	0.339276i	$0.110182\pi$
$139$			8.00000i			0.678551i	−0.940687	+	0.339276i	$0.889818\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	8.53553	−	1.46447i	0.703999	−	0.120787i
$148$		0			0
$149$	−12.7279			−1.04271			−0.521356	−	0.853339i	$-0.674574\pi$
$149$	−12.7279			−1.04271			−0.521356	+	0.853339i	$0.674574\pi$
$150$		0			0
$151$	−16.0000			−1.30206			−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000			−1.30206			−0.651031	+	0.759051i	$0.725663\pi$
$152$		0			0
$153$	−5.41421	−	2.58579i	−0.437713	−	0.209048i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$157$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$158$		0			0
$159$	5.65685	+	4.00000i	0.448618	+	0.317221i
$160$		0			0
$161$		−	5.65685i		−	0.445823i
$162$		0			0
$163$	−4.00000	−	4.00000i	−0.313304	−	0.313304i	0.532884	−	0.846188i	$-0.321108\pi$
$163$	−4.00000	−	4.00000i	−0.313304	−	0.313304i	−0.846188	+	0.532884i	$0.821108\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	5.65685	+	5.65685i	0.437741	+	0.437741i	0.891251	−	0.453510i	$-0.149829\pi$
$167$	5.65685	+	5.65685i	0.437741	+	0.437741i	−0.453510	+	0.891251i	$0.649829\pi$
$168$		0			0
$169$		−	13.0000i		−	1.00000i
$170$		0			0
$171$	−4.00000	−	11.3137i	−0.305888	−	0.865181i
$172$		0			0
$173$	−1.41421	+	1.41421i	−0.107521	+	0.107521i	−0.758820	−	0.651300i	$-0.774224\pi$
$173$	−1.41421	+	1.41421i	−0.107521	+	0.107521i	0.651300	+	0.758820i	$0.274224\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−2.89949	−	16.8995i	−0.217939	−	1.27024i
$178$		0			0
$179$	−18.3848			−1.37414			−0.687071	−	0.726590i	$-0.741104\pi$
$179$	−18.3848			−1.37414			−0.687071	+	0.726590i	$0.741104\pi$
$180$		0			0
$181$	−22.0000			−1.63525			−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000			−1.63525			−0.817624	+	0.575753i	$0.804709\pi$
$182$		0			0
$183$	1.75736	+	10.2426i	0.129908	+	0.757158i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−2.00000	+	2.00000i	−0.146254	+	0.146254i
$188$		0			0
$189$	−7.07107	−	2.00000i	−0.514344	−	0.145479i
$190$		0			0
$191$			2.82843i			0.204658i	0.994751	+	0.102329i	$0.0326294\pi$
$191$			2.82843i			0.204658i	−0.994751	+	0.102329i	$0.967371\pi$
$192$		0			0
$193$	15.0000	+	15.0000i	1.07972	+	1.07972i	0.996534	+	0.0831899i	$0.0265108\pi$
$193$	15.0000	+	15.0000i	1.07972	+	1.07972i	0.0831899	+	0.996534i	$0.473489\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	16.9706	+	16.9706i	1.20910	+	1.20910i	0.971318	+	0.237785i	$0.0764212\pi$
$197$	16.9706	+	16.9706i	1.20910	+	1.20910i	0.237785	+	0.971318i	$0.423579\pi$
$198$		0			0
$199$		−	24.0000i		−	1.70131i	−0.525720	−	0.850657i	$-0.676204\pi$
$199$		−	24.0000i		−	1.70131i	0.525720	−	0.850657i	$-0.323796\pi$
$200$		0			0
$201$	8.00000	+	5.65685i	0.564276	+	0.399004i
$202$		0			0
$203$	−7.07107	+	7.07107i	−0.496292	+	0.496292i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	5.17157	−	10.8284i	0.359449	−	0.752628i
$208$		0			0
$209$	−5.65685			−0.391293
$210$		0			0
$211$	−8.00000			−0.550743			−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000			−0.550743			−0.275371	+	0.961338i	$0.588801\pi$
$212$		0			0
$213$	−24.1421	+	4.14214i	−1.65419	+	0.283814i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−2.00000	+	2.00000i	−0.135769	+	0.135769i
$218$		0			0
$219$	7.07107	−	10.0000i	0.477818	−	0.675737i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−9.00000	−	9.00000i	−0.602685	−	0.602685i	0.338340	−	0.941024i	$-0.390135\pi$
$223$	−9.00000	−	9.00000i	−0.602685	−	0.602685i	−0.941024	+	0.338340i	$0.890135\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−15.5563	−	15.5563i	−1.03251	−	1.03251i	−0.999453	−	0.0330577i	$-0.989475\pi$
$227$	−15.5563	−	15.5563i	−1.03251	−	1.03251i	−0.0330577	−	0.999453i	$-0.510525\pi$
$228$		0			0
$229$			6.00000i			0.396491i	0.980152	+	0.198246i	$0.0635244\pi$
$229$			6.00000i			0.396491i	−0.980152	+	0.198246i	$0.936476\pi$
$230$		0			0
$231$	−2.00000	+	2.82843i	−0.131590	+	0.186097i
$232$		0			0
$233$	−12.7279	+	12.7279i	−0.833834	+	0.833834i	−0.988039	−	0.154205i	$-0.950718\pi$
$233$	−12.7279	+	12.7279i	−0.833834	+	0.833834i	0.154205	+	0.988039i	$0.450718\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	10.2426	−	1.75736i	0.665331	−	0.114153i
$238$		0			0
$239$	−8.48528			−0.548867			−0.274434	−	0.961606i	$-0.588490\pi$
$239$	−8.48528			−0.548867			−0.274434	+	0.961606i	$0.588490\pi$
$240$		0			0
$241$	4.00000			0.257663			0.128831	−	0.991667i	$-0.458877\pi$
$241$	4.00000			0.257663			0.128831	+	0.991667i	$0.458877\pi$
$242$		0			0
$243$	−11.7071	−	10.2929i	−0.751011	−	0.660289i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	−16.9706	−	12.0000i	−1.07547	−	0.760469i
$250$		0			0
$251$		−	12.7279i		−	0.803379i	−0.915776	−	0.401690i	$-0.868423\pi$
$251$		−	12.7279i		−	0.803379i	0.915776	−	0.401690i	$-0.131577\pi$
$252$		0			0
$253$	−4.00000	−	4.00000i	−0.251478	−	0.251478i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−9.89949	−	9.89949i	−0.617514	−	0.617514i	0.327379	−	0.944893i	$-0.393834\pi$
$257$	−9.89949	−	9.89949i	−0.617514	−	0.617514i	−0.944893	+	0.327379i	$0.893834\pi$
$258$		0			0
$259$		−	12.0000i		−	0.745644i
$260$		0			0
$261$	−20.0000	+	7.07107i	−1.23797	+	0.437688i
$262$		0			0
$263$	−5.65685	+	5.65685i	−0.348817	+	0.348817i	−0.859669	−	0.510852i	$-0.829330\pi$
$263$	−5.65685	+	5.65685i	−0.348817	+	0.348817i	0.510852	+	0.859669i	$0.329330\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	0.828427	+	4.82843i	0.0506989	+	0.295495i
$268$		0			0
$269$	15.5563			0.948487			0.474244	−	0.880394i	$-0.342722\pi$
$269$	15.5563			0.948487			0.474244	+	0.880394i	$0.342722\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	6.00000	−	6.00000i	0.360505	−	0.360505i	−0.503494	−	0.863999i	$-0.667952\pi$
$277$	6.00000	−	6.00000i	0.360505	−	0.360505i	0.863999	+	0.503494i	$0.167952\pi$
$278$		0			0
$279$	−5.65685	+	2.00000i	−0.338667	+	0.119737i
$280$		0			0
$281$			8.48528i			0.506189i	0.967442	+	0.253095i	$0.0814484\pi$
$281$			8.48528i			0.506189i	−0.967442	+	0.253095i	$0.918552\pi$
$282$		0			0
$283$	20.0000	+	20.0000i	1.18888	+	1.18888i	0.977378	+	0.211498i	$0.0678343\pi$
$283$	20.0000	+	20.0000i	1.18888	+	1.18888i	0.211498	+	0.977378i	$0.432166\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	5.65685	+	5.65685i	0.333914	+	0.333914i
$288$		0			0
$289$		−	13.0000i		−	0.764706i
$290$		0			0
$291$	6.00000	+	4.24264i	0.351726	+	0.248708i
$292$		0			0
$293$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$293$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−6.41421	+	3.58579i	−0.372190	+	0.208068i
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−12.0000			−0.691669
$302$		0			0
$303$	16.8995	−	2.89949i	0.970851	−	0.166572i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	18.0000	−	18.0000i	1.02731	−	1.02731i	0.0276979	−	0.999616i	$-0.491182\pi$
$307$	18.0000	−	18.0000i	1.02731	−	1.02731i	0.999616	−	0.0276979i	$-0.00881765\pi$
$308$		0			0
$309$	1.41421	−	2.00000i	0.0804518	−	0.113776i
$310$		0			0
$311$			19.7990i			1.12270i	0.827579	+	0.561349i	$0.189717\pi$
$311$			19.7990i			1.12270i	−0.827579	+	0.561349i	$0.810283\pi$
$312$		0			0
$313$	−9.00000	−	9.00000i	−0.508710	−	0.508710i	0.405420	−	0.914130i	$-0.367125\pi$
$313$	−9.00000	−	9.00000i	−0.508710	−	0.508710i	−0.914130	+	0.405420i	$0.867125\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−12.7279	−	12.7279i	−0.714871	−	0.714871i	0.252679	−	0.967550i	$-0.418688\pi$
$317$	−12.7279	−	12.7279i	−0.714871	−	0.714871i	−0.967550	+	0.252679i	$0.918688\pi$
$318$		0			0
$319$			10.0000i			0.559893i
$320$		0			0
$321$	4.00000	−	5.65685i	0.223258	−	0.315735i
$322$		0			0
$323$	−5.65685	+	5.65685i	−0.314756	+	0.314756i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	17.0711	−	2.92893i	0.944032	−	0.161970i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	8.00000			0.439720			0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000			0.439720			0.219860	+	0.975531i	$0.429440\pi$
$332$		0			0
$333$	10.9706	−	22.9706i	0.601183	−	1.25878i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	9.00000	−	9.00000i	0.490261	−	0.490261i	−0.418127	−	0.908388i	$-0.637313\pi$
$337$	9.00000	−	9.00000i	0.490261	−	0.490261i	0.908388	+	0.418127i	$0.137313\pi$
$338$		0			0
$339$	−19.7990	−	14.0000i	−1.07533	−	0.760376i
$340$		0			0
$341$			2.82843i			0.153168i
$342$		0			0
$343$	−12.0000	−	12.0000i	−0.647939	−	0.647939i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−9.89949	−	9.89949i	−0.531433	−	0.531433i	0.389566	−	0.920999i	$-0.372625\pi$
$347$	−9.89949	−	9.89949i	−0.531433	−	0.531433i	−0.920999	+	0.389566i	$0.872625\pi$
$348$		0			0
$349$			2.00000i			0.107058i	0.998566	+	0.0535288i	$0.0170469\pi$
$349$			2.00000i			0.107058i	−0.998566	+	0.0535288i	$0.982953\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	12.7279	−	12.7279i	0.677439	−	0.677439i	−0.281981	−	0.959420i	$-0.590992\pi$
$353$	12.7279	−	12.7279i	0.677439	−	0.677439i	0.959420	+	0.281981i	$0.0909915\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	0.828427	+	4.82843i	0.0438450	+	0.255547i
$358$		0			0
$359$	11.3137			0.597115			0.298557	−	0.954392i	$-0.403495\pi$
$359$	11.3137			0.597115			0.298557	+	0.954392i	$0.403495\pi$
$360$		0			0
$361$	3.00000			0.157895
$362$		0			0
$363$	−2.63604	−	15.3640i	−0.138356	−	0.806399i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−19.0000	+	19.0000i	−0.991792	+	0.991792i	−0.999967	−	0.00817466i	$-0.997398\pi$
$367$	−19.0000	+	19.0000i	−0.991792	+	0.991792i	0.00817466	+	0.999967i	$0.497398\pi$
$368$		0			0
$369$	5.65685	+	16.0000i	0.294484	+	0.832927i
$370$		0			0
$371$		−	5.65685i		−	0.293689i
$372$		0			0
$373$	−4.00000	−	4.00000i	−0.207112	−	0.207112i	0.595927	−	0.803039i	$-0.296785\pi$
$373$	−4.00000	−	4.00000i	−0.207112	−	0.207112i	−0.803039	+	0.595927i	$0.796785\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$			16.0000i			0.821865i	0.911666	+	0.410932i	$0.134797\pi$
$379$			16.0000i			0.821865i	−0.911666	+	0.410932i	$0.865203\pi$
$380$		0			0
$381$	14.0000	+	9.89949i	0.717242	+	0.507166i
$382$		0			0
$383$	16.9706	−	16.9706i	0.867155	−	0.867155i	−0.125001	−	0.992157i	$-0.539894\pi$
$383$	16.9706	−	16.9706i	0.867155	−	0.867155i	0.992157	+	0.125001i	$0.0398935\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−22.9706	−	10.9706i	−1.16766	−	0.557665i
$388$		0			0
$389$	−4.24264			−0.215110			−0.107555	−	0.994199i	$-0.534302\pi$
$389$	−4.24264			−0.215110			−0.107555	+	0.994199i	$0.534302\pi$
$390$		0			0
$391$	−8.00000			−0.404577
$392$		0			0
$393$	31.3848	−	5.38478i	1.58315	−	0.271626i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−22.0000	+	22.0000i	−1.10415	+	1.10415i	−0.110244	+	0.993905i	$0.535163\pi$
$397$	−22.0000	+	22.0000i	−1.10415	+	1.10415i	−0.993905	+	0.110244i	$0.964837\pi$
$398$		0			0
$399$	−5.65685	+	8.00000i	−0.283197	+	0.400501i
$400$		0			0
$401$		−	8.48528i		−	0.423735i	−0.977298	−	0.211867i	$-0.932046\pi$
$401$		−	8.48528i		−	0.423735i	0.977298	−	0.211867i	$-0.0679545\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−8.48528	−	8.48528i	−0.420600	−	0.420600i
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$	−6.00000	+	8.48528i	−0.295958	+	0.418548i
$412$		0			0
$413$	−9.89949	+	9.89949i	−0.487122	+	0.487122i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	13.6569	−	2.34315i	0.668779	−	0.114744i
$418$		0			0
$419$	21.2132			1.03633			0.518166	−	0.855280i	$-0.326615\pi$
$419$	21.2132			1.03633			0.518166	+	0.855280i	$0.326615\pi$
$420$		0			0
$421$	14.0000			0.682318			0.341159	−	0.940006i	$-0.389181\pi$
$421$	14.0000			0.682318			0.341159	+	0.940006i	$0.389181\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	6.00000	−	6.00000i	0.290360	−	0.290360i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	11.3137i		−	0.544962i	−0.962161	−	0.272481i	$-0.912156\pi$
$431$		−	11.3137i		−	0.544962i	0.962161	−	0.272481i	$-0.0878442\pi$
$432$		0			0
$433$	1.00000	+	1.00000i	0.0480569	+	0.0480569i	0.730727	−	0.682670i	$-0.239181\pi$
$433$	1.00000	+	1.00000i	0.0480569	+	0.0480569i	−0.682670	+	0.730727i	$0.739181\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−11.3137	−	11.3137i	−0.541208	−	0.541208i
$438$		0			0
$439$		−	16.0000i		−	0.763638i	−0.924237	−	0.381819i	$-0.875298\pi$
$439$		−	16.0000i		−	0.763638i	0.924237	−	0.381819i	$-0.124702\pi$
$440$		0			0
$441$	−5.00000	−	14.1421i	−0.238095	−	0.673435i
$442$		0			0
$443$	4.24264	−	4.24264i	0.201574	−	0.201574i	−0.599100	−	0.800674i	$-0.704475\pi$
$443$	4.24264	−	4.24264i	0.201574	−	0.201574i	0.800674	+	0.599100i	$0.204475\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	3.72792	+	21.7279i	0.176325	+	1.02770i
$448$		0			0
$449$	−14.1421			−0.667409			−0.333704	−	0.942678i	$-0.608299\pi$
$449$	−14.1421			−0.667409			−0.333704	+	0.942678i	$0.608299\pi$
$450$		0			0
$451$	8.00000			0.376705
$452$		0			0
$453$	4.68629	+	27.3137i	0.220181	+	1.28331i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	15.0000	−	15.0000i	0.701670	−	0.701670i	−0.263099	−	0.964769i	$-0.584744\pi$
$457$	15.0000	−	15.0000i	0.701670	−	0.701670i	0.964769	+	0.263099i	$0.0847444\pi$
$458$		0			0
$459$	−2.82843	+	10.0000i	−0.132020	+	0.466760i
$460$		0			0
$461$			7.07107i			0.329332i	0.986349	+	0.164666i	$0.0526547\pi$
$461$			7.07107i			0.329332i	−0.986349	+	0.164666i	$0.947345\pi$
$462$		0			0
$463$	5.00000	+	5.00000i	0.232370	+	0.232370i	0.813681	−	0.581311i	$-0.197460\pi$
$463$	5.00000	+	5.00000i	0.232370	+	0.232370i	−0.581311	+	0.813681i	$0.697460\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	19.7990	+	19.7990i	0.916188	+	0.916188i	0.996750	−	0.0805616i	$-0.0256714\pi$
$467$	19.7990	+	19.7990i	0.916188	+	0.916188i	−0.0805616	+	0.996750i	$0.525671\pi$
$468$		0			0
$469$		−	8.00000i		−	0.369406i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−8.48528	+	8.48528i	−0.390154	+	0.390154i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	5.17157	−	10.8284i	0.236790	−	0.495800i
$478$		0			0
$479$	31.1127			1.42158			0.710788	−	0.703407i	$-0.248339\pi$
$479$	31.1127			1.42158			0.710788	+	0.703407i	$0.248339\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$	−9.65685	+	1.65685i	−0.439402	+	0.0753895i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	9.00000	−	9.00000i	0.407829	−	0.407829i	−0.473152	−	0.880981i	$-0.656884\pi$
$487$	9.00000	−	9.00000i	0.407829	−	0.407829i	0.880981	+	0.473152i	$0.156884\pi$
$488$		0			0
$489$	−5.65685	+	8.00000i	−0.255812	+	0.361773i
$490$		0			0
$491$		−	26.8701i		−	1.21263i	−0.795225	−	0.606314i	$-0.792647\pi$
$491$		−	26.8701i		−	1.21263i	0.795225	−	0.606314i	$-0.207353\pi$
$492$		0			0
$493$	10.0000	+	10.0000i	0.450377	+	0.450377i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	14.1421	+	14.1421i	0.634361	+	0.634361i
$498$		0			0
$499$			20.0000i			0.895323i	0.894203	+	0.447661i	$0.147743\pi$
$499$			20.0000i			0.895323i	−0.894203	+	0.447661i	$0.852257\pi$
$500$		0			0
$501$	8.00000	−	11.3137i	0.357414	−	0.505459i
$502$		0			0
$503$	−8.48528	+	8.48528i	−0.378340	+	0.378340i	−0.870503	−	0.492163i	$-0.836206\pi$
$503$	−8.48528	+	8.48528i	−0.378340	+	0.378340i	0.492163	+	0.870503i	$0.336206\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−22.1924	+	3.80761i	−0.985599	+	0.169102i
$508$		0			0
$509$	24.0416			1.06563			0.532813	−	0.846233i	$-0.321135\pi$
$509$	24.0416			1.06563			0.532813	+	0.846233i	$0.321135\pi$
$510$		0			0
$511$	−10.0000			−0.442374
$512$		0			0
$513$	−18.1421	+	10.1421i	−0.800995	+	0.447786i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	2.82843	+	2.00000i	0.124154	+	0.0877903i
$520$		0			0
$521$		−	25.4558i		−	1.11524i	−0.830096	−	0.557620i	$-0.811714\pi$
$521$		−	25.4558i		−	1.11524i	0.830096	−	0.557620i	$-0.188286\pi$
$522$		0			0
$523$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$523$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	2.82843	+	2.82843i	0.123208	+	0.123208i
$528$		0			0
$529$			7.00000i			0.304348i
$530$		0			0
$531$	−28.0000	+	9.89949i	−1.21510	+	0.429601i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	5.38478	+	31.3848i	0.232370	+	1.35435i
$538$		0			0
$539$	−7.07107			−0.304572
$540$		0			0
$541$	−2.00000			−0.0859867			−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000			−0.0859867			−0.0429934	+	0.999075i	$0.513689\pi$
$542$		0			0
$543$	6.44365	+	37.5563i	0.276524	+	1.61170i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	6.00000	−	6.00000i	0.256541	−	0.256541i	−0.567104	−	0.823646i	$-0.691936\pi$
$547$	6.00000	−	6.00000i	0.256541	−	0.256541i	0.823646	+	0.567104i	$0.191936\pi$
$548$		0			0
$549$	16.9706	−	6.00000i	0.724286	−	0.256074i
$550$		0			0
$551$			28.2843i			1.20495i
$552$		0			0
$553$	−6.00000	−	6.00000i	−0.255146	−	0.255146i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$557$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	4.00000	+	2.82843i	0.168880	+	0.119416i
$562$		0			0
$563$	−21.2132	+	21.2132i	−0.894030	+	0.894030i	−0.994900	−	0.100870i	$-0.967837\pi$
$563$	−21.2132	+	21.2132i	−0.894030	+	0.894030i	0.100870	+	0.994900i	$0.467837\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−1.34315	+	12.6569i	−0.0564068	+	0.531538i
$568$		0			0
$569$	14.1421			0.592869			0.296435	−	0.955053i	$-0.404202\pi$
$569$	14.1421			0.592869			0.296435	+	0.955053i	$0.404202\pi$
$570$		0			0
$571$	40.0000			1.67395			0.836974	−	0.547243i	$-0.184323\pi$
$571$	40.0000			1.67395			0.836974	+	0.547243i	$0.184323\pi$
$572$		0			0
$573$	4.82843	−	0.828427i	0.201710	−	0.0346080i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−13.0000	+	13.0000i	−0.541197	+	0.541197i	−0.923880	−	0.382683i	$-0.875000\pi$
$577$	−13.0000	+	13.0000i	−0.541197	+	0.541197i	0.382683	+	0.923880i	$0.375000\pi$
$578$		0			0
$579$	21.2132	−	30.0000i	0.881591	−	1.24676i
$580$		0			0
$581$			16.9706i			0.704058i
$582$		0			0
$583$	−4.00000	−	4.00000i	−0.165663	−	0.165663i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	25.4558	+	25.4558i	1.05068	+	1.05068i	0.998646	+	0.0520296i	$0.0165690\pi$
$587$	25.4558	+	25.4558i	1.05068	+	1.05068i	0.0520296	+	0.998646i	$0.483431\pi$
$588$		0			0
$589$			8.00000i			0.329634i
$590$		0			0
$591$	24.0000	−	33.9411i	0.987228	−	1.39615i
$592$		0			0
$593$	15.5563	−	15.5563i	0.638823	−	0.638823i	−0.311442	−	0.950265i	$-0.600812\pi$
$593$	15.5563	−	15.5563i	0.638823	−	0.638823i	0.950265	+	0.311442i	$0.100812\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−40.9706	+	7.02944i	−1.67681	+	0.287696i
$598$		0			0
$599$	−45.2548			−1.84906			−0.924531	−	0.381106i	$-0.875543\pi$
$599$	−45.2548			−1.84906			−0.924531	+	0.381106i	$0.875543\pi$
$600$		0			0
$601$	−10.0000			−0.407909			−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000			−0.407909			−0.203954	+	0.978980i	$0.565379\pi$
$602$		0			0
$603$	7.31371	−	15.3137i	0.297837	−	0.623622i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	3.00000	−	3.00000i	0.121766	−	0.121766i	−0.643598	−	0.765364i	$-0.722559\pi$
$607$	3.00000	−	3.00000i	0.121766	−	0.121766i	0.765364	+	0.643598i	$0.222559\pi$
$608$		0			0
$609$	14.1421	+	10.0000i	0.573068	+	0.405220i
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−18.0000	−	18.0000i	−0.727013	−	0.727013i	0.243011	−	0.970024i	$-0.421865\pi$
$613$	−18.0000	−	18.0000i	−0.727013	−	0.727013i	−0.970024	+	0.243011i	$0.921865\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	21.2132	+	21.2132i	0.854011	+	0.854011i	0.990624	−	0.136613i	$-0.0436217\pi$
$617$	21.2132	+	21.2132i	0.854011	+	0.854011i	−0.136613	+	0.990624i	$0.543622\pi$
$618$		0			0
$619$			24.0000i			0.964641i	0.875995	+	0.482321i	$0.160206\pi$
$619$			24.0000i			0.964641i	−0.875995	+	0.482321i	$0.839794\pi$
$620$		0			0
$621$	−20.0000	−	5.65685i	−0.802572	−	0.227002i
$622$		0			0
$623$	2.82843	−	2.82843i	0.113319	−	0.113319i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	1.65685	+	9.65685i	0.0661684	+	0.385658i
$628$		0			0
$629$	−16.9706			−0.676661
$630$		0			0
$631$	14.0000			0.557331			0.278666	−	0.960388i	$-0.410108\pi$
$631$	14.0000			0.557331			0.278666	+	0.960388i	$0.410108\pi$
$632$		0			0
$633$	2.34315	+	13.6569i	0.0931317	+	0.542811i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$	14.1421	+	40.0000i	0.559454	+	1.58238i
$640$		0			0
$641$			5.65685i			0.223432i	0.993740	+	0.111716i	$0.0356347\pi$
$641$			5.65685i			0.223432i	−0.993740	+	0.111716i	$0.964365\pi$
$642$		0			0
$643$	24.0000	+	24.0000i	0.946468	+	0.946468i	0.998638	−	0.0521706i	$-0.0166140\pi$
$643$	24.0000	+	24.0000i	0.946468	+	0.946468i	−0.0521706	+	0.998638i	$0.516614\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−19.7990	−	19.7990i	−0.778379	−	0.778379i	0.201176	−	0.979555i	$-0.435524\pi$
$647$	−19.7990	−	19.7990i	−0.778379	−	0.778379i	−0.979555	+	0.201176i	$0.935524\pi$
$648$		0			0
$649$			14.0000i			0.549548i
$650$		0			0
$651$	4.00000	+	2.82843i	0.156772	+	0.110855i
$652$		0			0
$653$	4.24264	−	4.24264i	0.166027	−	0.166027i	−0.619203	−	0.785231i	$-0.712544\pi$
$653$	4.24264	−	4.24264i	0.166027	−	0.166027i	0.785231	+	0.619203i	$0.212544\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−19.1421	−	9.14214i	−0.746806	−	0.356669i
$658$		0			0
$659$	−35.3553			−1.37725			−0.688624	−	0.725118i	$-0.741785\pi$
$659$	−35.3553			−1.37725			−0.688624	+	0.725118i	$0.741785\pi$
$660$		0			0
$661$	50.0000			1.94477			0.972387	−	0.233373i	$-0.0749763\pi$
$661$	50.0000			1.94477			0.972387	+	0.233373i	$0.0749763\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−20.0000	+	20.0000i	−0.774403	+	0.774403i
$668$		0			0
$669$	−12.7279	+	18.0000i	−0.492090	+	0.695920i
$670$		0			0
$671$		−	8.48528i		−	0.327571i
$672$		0			0
$673$	−13.0000	−	13.0000i	−0.501113	−	0.501113i	0.410671	−	0.911784i	$-0.365295\pi$
$673$	−13.0000	−	13.0000i	−0.501113	−	0.501113i	−0.911784	+	0.410671i	$0.865295\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	18.3848	+	18.3848i	0.706584	+	0.706584i	0.965815	−	0.259231i	$-0.0834691\pi$
$677$	18.3848	+	18.3848i	0.706584	+	0.706584i	−0.259231	+	0.965815i	$0.583469\pi$
$678$		0			0
$679$		−	6.00000i		−	0.230259i
$680$		0			0
$681$	−22.0000	+	31.1127i	−0.843042	+	1.19224i
$682$		0			0
$683$	25.4558	−	25.4558i	0.974041	−	0.974041i	−0.0256307	−	0.999671i	$-0.508159\pi$
$683$	25.4558	−	25.4558i	0.974041	−	0.974041i	0.999671	+	0.0256307i	$0.00815939\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	10.2426	−	1.75736i	0.390781	−	0.0670474i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−12.0000			−0.456502			−0.228251	−	0.973602i	$-0.573301\pi$
$691$	−12.0000			−0.456502			−0.228251	+	0.973602i	$0.573301\pi$
$692$		0			0
$693$	5.41421	+	2.58579i	0.205669	+	0.0982259i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	8.00000	−	8.00000i	0.303022	−	0.303022i
$698$		0			0
$699$	25.4558	+	18.0000i	0.962828	+	0.680823i
$700$		0			0
$701$		−	26.8701i		−	1.01487i	−0.861691	−	0.507434i	$-0.830594\pi$
$701$		−	26.8701i		−	1.01487i	0.861691	−	0.507434i	$-0.169406\pi$
$702$		0			0
$703$	−24.0000	−	24.0000i	−0.905177	−	0.905177i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−9.89949	−	9.89949i	−0.372309	−	0.372309i
$708$		0			0
$709$		−	10.0000i		−	0.375558i	−0.982211	−	0.187779i	$-0.939871\pi$
$709$		−	10.0000i		−	0.375558i	0.982211	−	0.187779i	$-0.0601289\pi$
$710$		0			0
$711$	−6.00000	−	16.9706i	−0.225018	−	0.636446i
$712$		0			0
$713$	−5.65685	+	5.65685i	−0.211851	+	0.211851i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	2.48528	+	14.4853i	0.0928145	+	0.540963i
$718$		0			0
$719$	22.6274			0.843860			0.421930	−	0.906628i	$-0.361353\pi$
$719$	22.6274			0.843860			0.421930	+	0.906628i	$0.361353\pi$
$720$		0			0
$721$	−2.00000			−0.0744839
$722$		0			0
$723$	−1.17157	−	6.82843i	−0.0435713	−	0.253952i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−3.00000	+	3.00000i	−0.111264	+	0.111264i	−0.760547	−	0.649283i	$-0.775069\pi$
$727$	−3.00000	+	3.00000i	−0.111264	+	0.111264i	0.649283	+	0.760547i	$0.275069\pi$
$728$		0			0
$729$	−14.1421	+	23.0000i	−0.523783	+	0.851852i
$730$		0			0
$731$			16.9706i			0.627679i
$732$		0			0
$733$	26.0000	+	26.0000i	0.960332	+	0.960332i	0.999243	−	0.0389108i	$-0.0123888\pi$
$733$	26.0000	+	26.0000i	0.960332	+	0.960332i	−0.0389108	+	0.999243i	$0.512389\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−5.65685	−	5.65685i	−0.208373	−	0.208373i
$738$		0			0
$739$		−	40.0000i		−	1.47142i	−0.677295	−	0.735712i	$-0.736848\pi$
$739$		−	40.0000i		−	1.47142i	0.677295	−	0.735712i	$-0.263152\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	36.7696	−	36.7696i	1.34894	−	1.34894i	0.462134	−	0.886810i	$-0.347084\pi$
$743$	36.7696	−	36.7696i	1.34894	−	1.34894i	0.886810	−	0.462134i	$-0.152916\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−15.5147	+	32.4853i	−0.567654	+	1.18857i
$748$		0			0
$749$	−5.65685			−0.206697
$750$		0			0
$751$	30.0000			1.09472			0.547358	−	0.836899i	$-0.315634\pi$
$751$	30.0000			1.09472			0.547358	+	0.836899i	$0.315634\pi$
$752$		0			0
$753$	−21.7279	+	3.72792i	−0.791809	+	0.135853i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	30.0000	−	30.0000i	1.09037	−	1.09037i	0.0948798	−	0.995489i	$-0.469753\pi$
$757$	30.0000	−	30.0000i	1.09037	−	1.09037i	0.995489	−	0.0948798i	$-0.0302467\pi$
$758$		0			0
$759$	−5.65685	+	8.00000i	−0.205331	+	0.290382i
$760$		0			0
$761$			36.7696i			1.33290i	0.745552	+	0.666448i	$0.232186\pi$
$761$			36.7696i			1.33290i	−0.745552	+	0.666448i	$0.767814\pi$
$762$		0			0
$763$	−10.0000	−	10.0000i	−0.362024	−	0.362024i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$			22.0000i			0.793340i	0.917961	+	0.396670i	$0.129834\pi$
$769$			22.0000i			0.793340i	−0.917961	+	0.396670i	$0.870166\pi$
$770$		0			0
$771$	−14.0000	+	19.7990i	−0.504198	+	0.713043i
$772$		0			0
$773$	29.6985	−	29.6985i	1.06818	−	1.06818i	0.0706813	−	0.997499i	$-0.477483\pi$
$773$	29.6985	−	29.6985i	1.06818	−	1.06818i	0.997499	−	0.0706813i	$-0.0225173\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−20.4853	+	3.51472i	−0.734905	+	0.126090i
$778$		0			0
$779$	22.6274			0.810711
$780$		0			0
$781$	20.0000			0.715656
$782$		0			0
$783$	17.9289	+	32.0711i	0.640728	+	1.14613i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	28.0000	−	28.0000i	0.998092	−	0.998092i	−0.00190598	−	0.999998i	$-0.500607\pi$
$787$	28.0000	−	28.0000i	0.998092	−	0.998092i	0.999998	+	0.00190598i	$0.000606691\pi$
$788$		0			0
$789$	11.3137	+	8.00000i	0.402779	+	0.284808i
$790$		0			0
$791$			19.7990i			0.703971i
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	12.7279	+	12.7279i	0.450846	+	0.450846i	0.895635	−	0.444789i	$-0.146721\pi$
$797$	12.7279	+	12.7279i	0.450846	+	0.450846i	−0.444789	+	0.895635i	$0.646721\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	8.00000	−	2.82843i	0.282666	−	0.0999376i
$802$		0			0
$803$	−7.07107	+	7.07107i	−0.249533	+	0.249533i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−4.55635	−	26.5563i	−0.160391	−	0.934828i
$808$		0			0
$809$	22.6274			0.795538			0.397769	−	0.917486i	$-0.369785\pi$
$809$	22.6274			0.795538			0.397769	+	0.917486i	$0.369785\pi$
$810$		0			0
$811$	4.00000			0.140459			0.0702295	−	0.997531i	$-0.477627\pi$
$811$	4.00000			0.140459			0.0702295	+	0.997531i	$0.477627\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−24.0000	+	24.0000i	−0.839654	+	0.839654i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	43.8406i		−	1.53005i	−0.644002	−	0.765024i	$-0.722727\pi$
$821$		−	43.8406i		−	1.53005i	0.644002	−	0.765024i	$-0.277273\pi$
$822$		0			0
$823$	−25.0000	−	25.0000i	−0.871445	−	0.871445i	0.121185	−	0.992630i	$-0.461331\pi$
$823$	−25.0000	−	25.0000i	−0.871445	−	0.871445i	−0.992630	+	0.121185i	$0.961331\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	12.7279	+	12.7279i	0.442593	+	0.442593i	0.892883	−	0.450289i	$-0.148679\pi$
$827$	12.7279	+	12.7279i	0.442593	+	0.442593i	−0.450289	+	0.892883i	$0.648679\pi$
$828$		0			0
$829$		−	14.0000i		−	0.486240i	−0.969996	−	0.243120i	$-0.921829\pi$
$829$		−	14.0000i		−	0.486240i	0.969996	−	0.243120i	$-0.0781709\pi$
$830$		0			0
$831$	−12.0000	−	8.48528i	−0.416275	−	0.294351i
$832$		0			0
$833$	−7.07107	+	7.07107i	−0.244998	+	0.244998i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	5.07107	+	9.07107i	0.175282	+	0.313542i
$838$		0			0
$839$	−5.65685			−0.195296			−0.0976481	−	0.995221i	$-0.531132\pi$
$839$	−5.65685			−0.195296			−0.0976481	+	0.995221i	$0.531132\pi$
$840$		0			0
$841$	21.0000			0.724138
$842$		0			0
$843$	14.4853	−	2.48528i	0.498900	−	0.0855976i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−9.00000	+	9.00000i	−0.309244	+	0.309244i
$848$		0			0
$849$	28.2843	−	40.0000i	0.970714	−	1.37280i
$850$		0			0
$851$		−	33.9411i		−	1.16349i
$852$		0			0
$853$	2.00000	+	2.00000i	0.0684787	+	0.0684787i	0.740517	−	0.672038i	$-0.234581\pi$
$853$	2.00000	+	2.00000i	0.0684787	+	0.0684787i	−0.672038	+	0.740517i	$0.734581\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−38.1838	−	38.1838i	−1.30433	−	1.30433i	−0.925441	−	0.378892i	$-0.876305\pi$
$857$	−38.1838	−	38.1838i	−1.30433	−	1.30433i	−0.378892	−	0.925441i	$-0.623695\pi$
$858$		0			0
$859$			32.0000i			1.09183i	0.837842	+	0.545913i	$0.183817\pi$
$859$			32.0000i			1.09183i	−0.837842	+	0.545913i	$0.816183\pi$
$860$		0			0
$861$	8.00000	−	11.3137i	0.272639	−	0.385570i
$862$		0			0
$863$	−36.7696	+	36.7696i	−1.25165	+	1.25165i	−0.296670	+	0.954980i	$0.595876\pi$
$863$	−36.7696	+	36.7696i	−1.25165	+	1.25165i	−0.954980	+	0.296670i	$0.904124\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−22.1924	+	3.80761i	−0.753693	+	0.129313i
$868$		0			0
$869$	−8.48528			−0.287843
$870$		0			0
$871$		0			0
$872$		0			0
$873$	5.48528	−	11.4853i	0.185649	−	0.388718i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	36.0000	−	36.0000i	1.21563	−	1.21563i	0.246488	−	0.969146i	$-0.420724\pi$
$877$	36.0000	−	36.0000i	1.21563	−	1.21563i	0.969146	−	0.246488i	$-0.0792765\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			19.7990i			0.667045i	0.942742	+	0.333522i	$0.108237\pi$
$881$			19.7990i			0.667045i	−0.942742	+	0.333522i	$0.891763\pi$
$882$		0			0
$883$	4.00000	+	4.00000i	0.134611	+	0.134611i	0.771202	−	0.636591i	$-0.219656\pi$
$883$	4.00000	+	4.00000i	0.134611	+	0.134611i	−0.636591	+	0.771202i	$0.719656\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−8.48528	−	8.48528i	−0.284908	−	0.284908i	0.550155	−	0.835063i	$-0.314569\pi$
$887$	−8.48528	−	8.48528i	−0.284908	−	0.284908i	−0.835063	+	0.550155i	$0.814569\pi$
$888$		0			0
$889$		−	14.0000i		−	0.469545i
$890$		0			0
$891$	8.00000	+	9.89949i	0.268010	+	0.331646i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	14.1421			0.471667
$900$		0			0
$901$	−8.00000			−0.266519
$902$		0			0
$903$	3.51472	+	20.4853i	0.116963	+	0.681707i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	22.0000	−	22.0000i	0.730498	−	0.730498i	−0.240220	−	0.970718i	$-0.577220\pi$
$907$	22.0000	−	22.0000i	0.730498	−	0.730498i	0.970718	+	0.240220i	$0.0772197\pi$
$908$		0			0
$909$	−9.89949	−	28.0000i	−0.328346	−	0.928701i
$910$		0			0
$911$		−	39.5980i		−	1.31194i	−0.754787	−	0.655970i	$-0.772260\pi$
$911$		−	39.5980i		−	1.31194i	0.754787	−	0.655970i	$-0.227740\pi$
$912$		0			0
$913$	12.0000	+	12.0000i	0.397142	+	0.397142i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−18.3848	−	18.3848i	−0.607119	−	0.607119i
$918$		0			0
$919$		−	34.0000i		−	1.12156i	−0.827966	−	0.560778i	$-0.810502\pi$
$919$		−	34.0000i		−	1.12156i	0.827966	−	0.560778i	$-0.189498\pi$
$920$		0			0
$921$	−36.0000	−	25.4558i	−1.18624	−	0.838799i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−3.82843	−	1.82843i	−0.125742	−	0.0600534i
$928$		0			0
$929$	2.82843			0.0927977			0.0463988	−	0.998923i	$-0.485225\pi$
$929$	2.82843			0.0927977			0.0463988	+	0.998923i	$0.485225\pi$
$930$		0			0
$931$	−20.0000			−0.655474
$932$		0			0
$933$	33.7990	−	5.79899i	1.10653	−	0.189850i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	5.00000	−	5.00000i	0.163343	−	0.163343i	−0.620703	−	0.784046i	$-0.713153\pi$
$937$	5.00000	−	5.00000i	0.163343	−	0.163343i	0.784046	+	0.620703i	$0.213153\pi$
$938$		0			0
$939$	−12.7279	+	18.0000i	−0.415360	+	0.587408i
$940$		0			0
$941$		−	12.7279i		−	0.414918i	−0.978244	−	0.207459i	$-0.933481\pi$
$941$		−	12.7279i		−	0.414918i	0.978244	−	0.207459i	$-0.0665194\pi$
$942$		0			0
$943$	16.0000	+	16.0000i	0.521032	+	0.521032i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	18.3848	+	18.3848i	0.597425	+	0.597425i	0.939627	−	0.342202i	$-0.111173\pi$
$947$	18.3848	+	18.3848i	0.597425	+	0.597425i	−0.342202	+	0.939627i	$0.611173\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	−18.0000	+	25.4558i	−0.583690	+	0.825462i
$952$		0			0
$953$	−4.24264	+	4.24264i	−0.137433	+	0.137433i	−0.772476	−	0.635044i	$-0.780982\pi$
$953$	−4.24264	+	4.24264i	−0.137433	+	0.137433i	0.635044	+	0.772476i	$0.280982\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	17.0711	−	2.92893i	0.551829	−	0.0946789i
$958$		0			0
$959$	8.48528			0.274004
$960$		0			0
$961$	−27.0000			−0.870968
$962$		0			0
$963$	−10.8284	−	5.17157i	−0.348941	−	0.166652i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	19.0000	−	19.0000i	0.610999	−	0.610999i	−0.332208	−	0.943206i	$-0.607793\pi$
$967$	19.0000	−	19.0000i	0.610999	−	0.610999i	0.943206	+	0.332208i	$0.107793\pi$
$968$		0			0
$969$	11.3137	+	8.00000i	0.363449	+	0.256997i
$970$		0			0
$971$			41.0122i			1.31614i	0.752955	+	0.658072i	$0.228628\pi$
$971$			41.0122i			1.31614i	−0.752955	+	0.658072i	$0.771372\pi$
$972$		0			0
$973$	−8.00000	−	8.00000i	−0.256468	−	0.256468i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	4.24264	+	4.24264i	0.135734	+	0.135734i	0.771709	−	0.635975i	$-0.219402\pi$
$977$	4.24264	+	4.24264i	0.135734	+	0.135734i	−0.635975	+	0.771709i	$0.719402\pi$
$978$		0			0
$979$		−	4.00000i		−	0.127841i
$980$		0			0
$981$	−10.0000	−	28.2843i	−0.319275	−	0.903047i
$982$		0			0
$983$	−14.1421	+	14.1421i	−0.451064	+	0.451064i	−0.895708	−	0.444644i	$-0.853330\pi$
$983$	−14.1421	+	14.1421i	−0.451064	+	0.451064i	0.444644	+	0.895708i	$0.353330\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−33.9411			−1.07927
$990$		0			0
$991$	−40.0000			−1.27064			−0.635321	−	0.772248i	$-0.719132\pi$
$991$	−40.0000			−1.27064			−0.635321	+	0.772248i	$0.719132\pi$
$992$		0			0
$993$	−2.34315	−	13.6569i	−0.0743575	−	0.433387i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	8.00000	−	8.00000i	0.253363	−	0.253363i	−0.568985	−	0.822348i	$-0.692664\pi$
$997$	8.00000	−	8.00000i	0.253363	−	0.253363i	0.822348	+	0.568985i	$0.192664\pi$
$998$		0			0
$999$	−42.4264	−	12.0000i	−1.34231	−	0.379663i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.2.v.b.593.2		4
3.2	odd	2	inner	1200.2.v.b.593.1		4
4.3	odd	2		150.2.e.a.143.2		4
5.2	odd	4	inner	1200.2.v.b.257.1		4
5.3	odd	4		240.2.v.e.17.2		4
5.4	even	2		240.2.v.e.113.1		4
12.11	even	2		150.2.e.a.143.1		4
15.2	even	4	inner	1200.2.v.b.257.2		4
15.8	even	4		240.2.v.e.17.1		4
15.14	odd	2		240.2.v.e.113.2		4
20.3	even	4		30.2.e.a.17.2	yes	4
20.7	even	4		150.2.e.a.107.1		4
20.19	odd	2		30.2.e.a.23.1	yes	4
40.3	even	4		960.2.v.k.257.2		4
40.13	odd	4		960.2.v.c.257.1		4
40.19	odd	2		960.2.v.k.833.1		4
40.29	even	2		960.2.v.c.833.2		4
60.23	odd	4		30.2.e.a.17.1	✓	4
60.47	odd	4		150.2.e.a.107.2		4
60.59	even	2		30.2.e.a.23.2	yes	4
120.29	odd	2		960.2.v.c.833.1		4
120.53	even	4		960.2.v.c.257.2		4
120.59	even	2		960.2.v.k.833.2		4
120.83	odd	4		960.2.v.k.257.1		4
180.23	odd	12		810.2.m.f.107.1		8
180.43	even	12		810.2.m.f.377.1		8
180.59	even	6		810.2.m.f.593.1		8
180.79	odd	6		810.2.m.f.53.1		8
180.83	odd	12		810.2.m.f.377.2		8
180.103	even	12		810.2.m.f.107.2		8
180.119	even	6		810.2.m.f.53.2		8
180.139	odd	6		810.2.m.f.593.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.2.e.a.17.1	✓	4	60.23	odd	4
30.2.e.a.17.2	yes	4	20.3	even	4
30.2.e.a.23.1	yes	4	20.19	odd	2
30.2.e.a.23.2	yes	4	60.59	even	2
150.2.e.a.107.1		4	20.7	even	4
150.2.e.a.107.2		4	60.47	odd	4
150.2.e.a.143.1		4	12.11	even	2
150.2.e.a.143.2		4	4.3	odd	2
240.2.v.e.17.1		4	15.8	even	4
240.2.v.e.17.2		4	5.3	odd	4
240.2.v.e.113.1		4	5.4	even	2
240.2.v.e.113.2		4	15.14	odd	2
810.2.m.f.53.1		8	180.79	odd	6
810.2.m.f.53.2		8	180.119	even	6
810.2.m.f.107.1		8	180.23	odd	12
810.2.m.f.107.2		8	180.103	even	12
810.2.m.f.377.1		8	180.43	even	12
810.2.m.f.377.2		8	180.83	odd	12
810.2.m.f.593.1		8	180.59	even	6
810.2.m.f.593.2		8	180.139	odd	6
960.2.v.c.257.1		4	40.13	odd	4
960.2.v.c.257.2		4	120.53	even	4
960.2.v.c.833.1		4	120.29	odd	2
960.2.v.c.833.2		4	40.29	even	2
960.2.v.k.257.1		4	120.83	odd	4
960.2.v.k.257.2		4	40.3	even	4
960.2.v.k.833.1		4	40.19	odd	2
960.2.v.k.833.2		4	120.59	even	2
1200.2.v.b.257.1		4	5.2	odd	4	inner
1200.2.v.b.257.2		4	15.2	even	4	inner
1200.2.v.b.593.1		4	3.2	odd	2	inner
1200.2.v.b.593.2		4	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1200.v (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.292893 - 1.70711i) q^{3} +(-1.00000 + 1.00000i) q^{7} +(-2.82843 + 1.00000i) q^{9} +O(q^{10})\)	\(q+(-0.292893 - 1.70711i) q^{3} +(-1.00000 + 1.00000i) q^{7} +(-2.82843 + 1.00000i) q^{9} +1.41421i q^{11} +(1.41421 + 1.41421i) q^{17} +4.00000i q^{19} +(2.00000 + 1.41421i) q^{21} +(-2.82843 + 2.82843i) q^{23} +(2.53553 + 4.53553i) q^{27} +7.07107 q^{29} +2.00000 q^{31} +(2.41421 - 0.414214i) q^{33} +(-6.00000 + 6.00000i) q^{37} -5.65685i q^{41} +(6.00000 + 6.00000i) q^{43} +5.00000i q^{49} +(2.00000 - 2.82843i) q^{51} +(-2.82843 + 2.82843i) q^{53} +(6.82843 - 1.17157i) q^{57} +9.89949 q^{59} -6.00000 q^{61} +(1.82843 - 3.82843i) q^{63} +(-4.00000 + 4.00000i) q^{67} +(5.65685 + 4.00000i) q^{69} -14.1421i q^{71} +(5.00000 + 5.00000i) q^{73} +(-1.41421 - 1.41421i) q^{77} +6.00000i q^{79} +(7.00000 - 5.65685i) q^{81} +(8.48528 - 8.48528i) q^{83} +(-2.07107 - 12.0711i) q^{87} -2.82843 q^{89} +(-0.585786 - 3.41421i) q^{93} +(-3.00000 + 3.00000i) q^{97} +(-1.41421 - 4.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} - 4 q^{7}+O(q^{10}) \) 4 * q - 4 * q^3 - 4 * q^7	\( 4 q - 4 q^{3} - 4 q^{7} + 8 q^{21} - 4 q^{27} + 8 q^{31} + 4 q^{33} - 24 q^{37} + 24 q^{43} + 8 q^{51} + 16 q^{57} - 24 q^{61} - 4 q^{63} - 16 q^{67} + 20 q^{73} + 28 q^{81} + 20 q^{87} - 8 q^{93} - 12 q^{97}+O(q^{100}) \) 4 * q - 4 * q^3 - 4 * q^7 + 8 * q^21 - 4 * q^27 + 8 * q^31 + 4 * q^33 - 24 * q^37 + 24 * q^43 + 8 * q^51 + 16 * q^57 - 24 * q^61 - 4 * q^63 - 16 * q^67 + 20 * q^73 + 28 * q^81 + 20 * q^87 - 8 * q^93 - 12 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more