LMFDB - Embedded newform 525.2.g.b.524.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [525,2,Mod(524,525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(525, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("525.524");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 525 = 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	525.g (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$4.19214610612$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 105)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			524.1
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	525.524
Dual form			525.2.g.b.524.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205 q^{2} +1.73205 q^{3} +1.00000 q^{4} -3.00000 q^{6} +(1.73205 - 2.00000i) q^{7} +1.73205 q^{8} +3.00000 q^{9} +O(q^{10})$	\(q-1.73205 q^{2} +1.73205 q^{3} +1.00000 q^{4} -3.00000 q^{6} +(1.73205 - 2.00000i) q^{7} +1.73205 q^{8} +3.00000 q^{9} +3.46410i q^{11} +1.73205 q^{12} +(-3.00000 + 3.46410i) q^{14} -5.00000 q^{16} -6.00000i q^{17} -5.19615 q^{18} +3.46410i q^{19} +(3.00000 - 3.46410i) q^{21} -6.00000i q^{22} +3.46410 q^{23} +3.00000 q^{24} +5.19615 q^{27} +(1.73205 - 2.00000i) q^{28} -6.92820i q^{29} +3.46410i q^{31} +5.19615 q^{32} +6.00000i q^{33} +10.3923i q^{34} +3.00000 q^{36} +2.00000i q^{37} -6.00000i q^{38} +6.00000 q^{41} +(-5.19615 + 6.00000i) q^{42} -8.00000i q^{43} +3.46410i q^{44} -6.00000 q^{46} +12.0000i q^{47} -8.66025 q^{48} +(-1.00000 - 6.92820i) q^{49} -10.3923i q^{51} -9.00000 q^{54} +(3.00000 - 3.46410i) q^{56} +6.00000i q^{57} +12.0000i q^{58} +12.0000 q^{59} +6.92820i q^{61} -6.00000i q^{62} +(5.19615 - 6.00000i) q^{63} +1.00000 q^{64} -10.3923i q^{66} -8.00000i q^{67} -6.00000i q^{68} +6.00000 q^{69} -3.46410i q^{71} +5.19615 q^{72} +6.92820 q^{73} -3.46410i q^{74} +3.46410i q^{76} +(6.92820 + 6.00000i) q^{77} -8.00000 q^{79} +9.00000 q^{81} -10.3923 q^{82} +(3.00000 - 3.46410i) q^{84} +13.8564i q^{86} -12.0000i q^{87} +6.00000i q^{88} -6.00000 q^{89} +3.46410 q^{92} +6.00000i q^{93} -20.7846i q^{94} +9.00000 q^{96} -6.92820 q^{97} +(1.73205 + 12.0000i) q^{98} +10.3923i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{4} - 12 q^{6} + 12 q^{9}+O(q^{10}) $ 4 * q + 4 * q^4 - 12 * q^6 + 12 * q^9	$ 4 q + 4 q^{4} - 12 q^{6} + 12 q^{9} - 12 q^{14} - 20 q^{16} + 12 q^{21} + 12 q^{24} + 12 q^{36} + 24 q^{41} - 24 q^{46} - 4 q^{49} - 36 q^{54} + 12 q^{56} + 48 q^{59} + 4 q^{64} + 24 q^{69} - 32 q^{79} + 36 q^{81} + 12 q^{84} - 24 q^{89} + 36 q^{96}+O(q^{100}) $ 4 * q + 4 * q^4 - 12 * q^6 + 12 * q^9 - 12 * q^14 - 20 * q^16 + 12 * q^21 + 12 * q^24 + 12 * q^36 + 24 * q^41 - 24 * q^46 - 4 * q^49 - 36 * q^54 + 12 * q^56 + 48 * q^59 + 4 * q^64 + 24 * q^69 - 32 * q^79 + 36 * q^81 + 12 * q^84 - 24 * q^89 + 36 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/525\mathbb{Z}\right)^\times$.

$n$	$127$	$176$	$451$
$\chi(n)$	$-1$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−1.73205			−1.22474			−0.612372	−	0.790569i	$-0.709785\pi$
$2$	−1.73205			−1.22474			−0.612372	+	0.790569i	$0.709785\pi$
$3$	1.73205			1.00000
$3$	1.73205			1.00000
$4$	1.00000			0.500000
$5$		0			0
$5$		0			0
$6$	−3.00000			−1.22474
$7$	1.73205	−	2.00000i	0.654654	−	0.755929i
$7$	1.73205	−	2.00000i	0.654654	−	0.755929i
$8$	1.73205			0.612372
$9$	3.00000			1.00000
$10$		0			0
$11$			3.46410i			1.04447i	0.852803	+	0.522233i	$0.174901\pi$
$11$			3.46410i			1.04447i	−0.852803	+	0.522233i	$0.825099\pi$
$12$	1.73205			0.500000
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$	−3.00000	+	3.46410i	−0.801784	+	0.925820i
$15$		0			0
$16$	−5.00000			−1.25000
$17$		−	6.00000i		−	1.45521i	−0.685994	−	0.727607i	$-0.740633\pi$
$17$		−	6.00000i		−	1.45521i	0.685994	−	0.727607i	$-0.259367\pi$
$18$	−5.19615			−1.22474
$19$			3.46410i			0.794719i	0.917663	+	0.397360i	$0.130073\pi$
$19$			3.46410i			0.794719i	−0.917663	+	0.397360i	$0.869927\pi$
$20$		0			0
$21$	3.00000	−	3.46410i	0.654654	−	0.755929i
$22$		−	6.00000i		−	1.27920i
$23$	3.46410			0.722315			0.361158	−	0.932505i	$-0.382382\pi$
$23$	3.46410			0.722315			0.361158	+	0.932505i	$0.382382\pi$
$24$	3.00000			0.612372
$25$		0			0
$26$		0			0
$27$	5.19615			1.00000
$28$	1.73205	−	2.00000i	0.327327	−	0.377964i
$29$		−	6.92820i		−	1.28654i	−0.765641	−	0.643268i	$-0.777578\pi$
$29$		−	6.92820i		−	1.28654i	0.765641	−	0.643268i	$-0.222422\pi$
$30$		0			0
$31$			3.46410i			0.622171i	0.950382	+	0.311086i	$0.100693\pi$
$31$			3.46410i			0.622171i	−0.950382	+	0.311086i	$0.899307\pi$
$32$	5.19615			0.918559
$33$			6.00000i			1.04447i
$34$			10.3923i			1.78227i
$35$		0			0
$36$	3.00000			0.500000
$37$			2.00000i			0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$			2.00000i			0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$		−	6.00000i		−	0.973329i
$39$		0			0
$40$		0			0
$41$	6.00000			0.937043			0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000			0.937043			0.468521	+	0.883452i	$0.344787\pi$
$42$	−5.19615	+	6.00000i	−0.801784	+	0.925820i
$43$		−	8.00000i		−	1.21999i	−0.792406	−	0.609994i	$-0.791172\pi$
$43$		−	8.00000i		−	1.21999i	0.792406	−	0.609994i	$-0.208828\pi$
$44$			3.46410i			0.522233i
$45$		0			0
$46$	−6.00000			−0.884652
$47$			12.0000i			1.75038i	0.483779	+	0.875190i	$0.339264\pi$
$47$			12.0000i			1.75038i	−0.483779	+	0.875190i	$0.660736\pi$
$48$	−8.66025			−1.25000
$49$	−1.00000	−	6.92820i	−0.142857	−	0.989743i
$50$		0			0
$51$		−	10.3923i		−	1.45521i
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$	−9.00000			−1.22474
$55$		0			0
$56$	3.00000	−	3.46410i	0.400892	−	0.462910i
$57$			6.00000i			0.794719i
$58$			12.0000i			1.57568i
$59$	12.0000			1.56227			0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000			1.56227			0.781133	+	0.624364i	$0.214642\pi$
$60$		0			0
$61$			6.92820i			0.887066i	0.896258	+	0.443533i	$0.146275\pi$
$61$			6.92820i			0.887066i	−0.896258	+	0.443533i	$0.853725\pi$
$62$		−	6.00000i		−	0.762001i
$63$	5.19615	−	6.00000i	0.654654	−	0.755929i
$64$	1.00000			0.125000
$65$		0			0
$66$		−	10.3923i		−	1.27920i
$67$		−	8.00000i		−	0.977356i	−0.872464	−	0.488678i	$-0.837479\pi$
$67$		−	8.00000i		−	0.977356i	0.872464	−	0.488678i	$-0.162521\pi$
$68$		−	6.00000i		−	0.727607i
$69$	6.00000			0.722315
$70$		0			0
$71$		−	3.46410i		−	0.411113i	−0.978645	−	0.205557i	$-0.934100\pi$
$71$		−	3.46410i		−	0.411113i	0.978645	−	0.205557i	$-0.0659005\pi$
$72$	5.19615			0.612372
$73$	6.92820			0.810885			0.405442	−	0.914121i	$-0.367117\pi$
$73$	6.92820			0.810885			0.405442	+	0.914121i	$0.367117\pi$
$74$		−	3.46410i		−	0.402694i
$75$		0			0
$76$			3.46410i			0.397360i
$77$	6.92820	+	6.00000i	0.789542	+	0.683763i
$78$		0			0
$79$	−8.00000			−0.900070			−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000			−0.900070			−0.450035	+	0.893011i	$0.648589\pi$
$80$		0			0
$81$	9.00000			1.00000
$82$	−10.3923			−1.14764
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$	3.00000	−	3.46410i	0.327327	−	0.377964i
$85$		0			0
$86$			13.8564i			1.49417i
$87$		−	12.0000i		−	1.28654i
$88$			6.00000i			0.639602i
$89$	−6.00000			−0.635999			−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000			−0.635999			−0.317999	+	0.948091i	$0.603011\pi$
$90$		0			0
$91$		0			0
$92$	3.46410			0.361158
$93$			6.00000i			0.622171i
$94$		−	20.7846i		−	2.14377i
$95$		0			0
$96$	9.00000			0.918559
$97$	−6.92820			−0.703452			−0.351726	−	0.936103i	$-0.614405\pi$
$97$	−6.92820			−0.703452			−0.351726	+	0.936103i	$0.614405\pi$
$98$	1.73205	+	12.0000i	0.174964	+	1.21218i
$99$			10.3923i			1.04447i
$100$		0			0
$101$	−6.00000			−0.597022			−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000			−0.597022			−0.298511	+	0.954406i	$0.596490\pi$
$102$			18.0000i			1.78227i
$103$	−3.46410			−0.341328			−0.170664	−	0.985329i	$-0.554591\pi$
$103$	−3.46410			−0.341328			−0.170664	+	0.985329i	$0.554591\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−10.3923			−1.00466			−0.502331	−	0.864675i	$-0.667524\pi$
$107$	−10.3923			−1.00466			−0.502331	+	0.864675i	$0.667524\pi$
$108$	5.19615			0.500000
$109$	2.00000			0.191565			0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000			0.191565			0.0957826	+	0.995402i	$0.469465\pi$
$110$		0			0
$111$			3.46410i			0.328798i
$112$	−8.66025	+	10.0000i	−0.818317	+	0.944911i
$113$	−6.92820			−0.651751			−0.325875	−	0.945413i	$-0.605659\pi$
$113$	−6.92820			−0.651751			−0.325875	+	0.945413i	$0.605659\pi$
$114$		−	10.3923i		−	0.973329i
$115$		0			0
$116$		−	6.92820i		−	0.643268i
$117$		0			0
$118$	−20.7846			−1.91338
$119$	−12.0000	−	10.3923i	−1.10004	−	0.952661i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		−	12.0000i		−	1.08643i
$123$	10.3923			0.937043
$124$			3.46410i			0.311086i
$125$		0			0
$126$	−9.00000	+	10.3923i	−0.801784	+	0.925820i
$127$			4.00000i			0.354943i	0.984126	+	0.177471i	$0.0567917\pi$
$127$			4.00000i			0.354943i	−0.984126	+	0.177471i	$0.943208\pi$
$128$	−12.1244			−1.07165
$129$		−	13.8564i		−	1.21999i
$130$		0			0
$131$	−12.0000			−1.04844			−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000			−1.04844			−0.524222	+	0.851581i	$0.675644\pi$
$132$			6.00000i			0.522233i
$133$	6.92820	+	6.00000i	0.600751	+	0.520266i
$134$			13.8564i			1.19701i
$135$		0			0
$136$		−	10.3923i		−	0.891133i
$137$	−20.7846			−1.77575			−0.887875	−	0.460086i	$-0.847819\pi$
$137$	−20.7846			−1.77575			−0.887875	+	0.460086i	$0.847819\pi$
$138$	−10.3923			−0.884652
$139$			17.3205i			1.46911i	0.678551	+	0.734553i	$0.262608\pi$
$139$			17.3205i			1.46911i	−0.678551	+	0.734553i	$0.737392\pi$
$140$		0			0
$141$			20.7846i			1.75038i
$142$			6.00000i			0.503509i
$143$		0			0
$144$	−15.0000			−1.25000
$145$		0			0
$146$	−12.0000			−0.993127
$147$	−1.73205	−	12.0000i	−0.142857	−	0.989743i
$148$			2.00000i			0.164399i
$149$			6.92820i			0.567581i	0.958886	+	0.283790i	$0.0915919\pi$
$149$			6.92820i			0.567581i	−0.958886	+	0.283790i	$0.908408\pi$
$150$		0			0
$151$	−8.00000			−0.651031			−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000			−0.651031			−0.325515	+	0.945537i	$0.605538\pi$
$152$			6.00000i			0.486664i
$153$		−	18.0000i		−	1.45521i
$154$	−12.0000	−	10.3923i	−0.966988	−	0.837436i
$155$		0			0
$156$		0			0
$157$	−13.8564			−1.10586			−0.552931	−	0.833227i	$-0.686491\pi$
$157$	−13.8564			−1.10586			−0.552931	+	0.833227i	$0.686491\pi$
$158$	13.8564			1.10236
$159$		0			0
$160$		0			0
$161$	6.00000	−	6.92820i	0.472866	−	0.546019i
$162$	−15.5885			−1.22474
$163$		−	16.0000i		−	1.25322i	−0.779334	−	0.626608i	$-0.784443\pi$
$163$		−	16.0000i		−	1.25322i	0.779334	−	0.626608i	$-0.215557\pi$
$164$	6.00000			0.468521
$165$		0			0
$166$		0			0
$167$		−	12.0000i		−	0.928588i	−0.885681	−	0.464294i	$-0.846308\pi$
$167$		−	12.0000i		−	0.928588i	0.885681	−	0.464294i	$-0.153692\pi$
$168$	5.19615	−	6.00000i	0.400892	−	0.462910i
$169$	−13.0000			−1.00000
$170$		0			0
$171$			10.3923i			0.794719i
$172$		−	8.00000i		−	0.609994i
$173$			18.0000i			1.36851i	0.729241	+	0.684257i	$0.239873\pi$
$173$			18.0000i			1.36851i	−0.729241	+	0.684257i	$0.760127\pi$
$174$			20.7846i			1.57568i
$175$		0			0
$176$		−	17.3205i		−	1.30558i
$177$	20.7846			1.56227
$178$	10.3923			0.778936
$179$			10.3923i			0.776757i	0.921500	+	0.388379i	$0.126965\pi$
$179$			10.3923i			0.776757i	−0.921500	+	0.388379i	$0.873035\pi$
$180$		0			0
$181$			20.7846i			1.54491i	0.635071	+	0.772454i	$0.280971\pi$
$181$			20.7846i			1.54491i	−0.635071	+	0.772454i	$0.719029\pi$
$182$		0			0
$183$			12.0000i			0.887066i
$184$	6.00000			0.442326
$185$		0			0
$186$		−	10.3923i		−	0.762001i
$187$	20.7846			1.51992
$188$			12.0000i			0.875190i
$189$	9.00000	−	10.3923i	0.654654	−	0.755929i
$190$		0			0
$191$			10.3923i			0.751961i	0.926628	+	0.375980i	$0.122694\pi$
$191$			10.3923i			0.751961i	−0.926628	+	0.375980i	$0.877306\pi$
$192$	1.73205			0.125000
$193$		−	14.0000i		−	1.00774i	−0.863779	−	0.503871i	$-0.831909\pi$
$193$		−	14.0000i		−	1.00774i	0.863779	−	0.503871i	$-0.168091\pi$
$194$	12.0000			0.861550
$195$		0			0
$196$	−1.00000	−	6.92820i	−0.0714286	−	0.494872i
$197$	13.8564			0.987228			0.493614	−	0.869681i	$-0.335676\pi$
$197$	13.8564			0.987228			0.493614	+	0.869681i	$0.335676\pi$
$198$		−	18.0000i		−	1.27920i
$199$			10.3923i			0.736691i	0.929689	+	0.368345i	$0.120076\pi$
$199$			10.3923i			0.736691i	−0.929689	+	0.368345i	$0.879924\pi$
$200$		0			0
$201$		−	13.8564i		−	0.977356i
$202$	10.3923			0.731200
$203$	−13.8564	−	12.0000i	−0.972529	−	0.842235i
$204$		−	10.3923i		−	0.727607i
$205$		0			0
$206$	6.00000			0.418040
$207$	10.3923			0.722315
$208$		0			0
$209$	−12.0000			−0.830057
$210$		0			0
$211$	4.00000			0.275371			0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000			0.275371			0.137686	+	0.990476i	$0.456034\pi$
$212$		0			0
$213$		−	6.00000i		−	0.411113i
$214$	18.0000			1.23045
$215$		0			0
$216$	9.00000			0.612372
$217$	6.92820	+	6.00000i	0.470317	+	0.407307i
$218$	−3.46410			−0.234619
$219$	12.0000			0.810885
$220$		0			0
$221$		0			0
$222$		−	6.00000i		−	0.402694i
$223$	−17.3205			−1.15987			−0.579934	−	0.814664i	$-0.696921\pi$
$223$	−17.3205			−1.15987			−0.579934	+	0.814664i	$0.696921\pi$
$224$	9.00000	−	10.3923i	0.601338	−	0.694365i
$225$		0			0
$226$	12.0000			0.798228
$227$			24.0000i			1.59294i	0.604681	+	0.796468i	$0.293301\pi$
$227$			24.0000i			1.59294i	−0.604681	+	0.796468i	$0.706699\pi$
$228$			6.00000i			0.397360i
$229$		−	6.92820i		−	0.457829i	−0.973447	−	0.228914i	$-0.926482\pi$
$229$		−	6.92820i		−	0.457829i	0.973447	−	0.228914i	$-0.0735176\pi$
$230$		0			0
$231$	12.0000	+	10.3923i	0.789542	+	0.683763i
$232$		−	12.0000i		−	0.787839i
$233$	6.92820			0.453882			0.226941	−	0.973909i	$-0.427128\pi$
$233$	6.92820			0.453882			0.226941	+	0.973909i	$0.427128\pi$
$234$		0			0
$235$		0			0
$236$	12.0000			0.781133
$237$	−13.8564			−0.900070
$238$	20.7846	+	18.0000i	1.34727	+	1.16677i
$239$		−	10.3923i		−	0.672222i	−0.941822	−	0.336111i	$-0.890888\pi$
$239$		−	10.3923i		−	0.672222i	0.941822	−	0.336111i	$-0.109112\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	1.73205			0.111340
$243$	15.5885			1.00000
$244$			6.92820i			0.443533i
$245$		0			0
$246$	−18.0000			−1.14764
$247$		0			0
$248$			6.00000i			0.381000i
$249$		0			0
$250$		0			0
$251$	12.0000			0.757433			0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000			0.757433			0.378717	+	0.925513i	$0.376365\pi$
$252$	5.19615	−	6.00000i	0.327327	−	0.377964i
$253$			12.0000i			0.754434i
$254$		−	6.92820i		−	0.434714i
$255$		0			0
$256$	19.0000			1.18750
$257$		−	6.00000i		−	0.374270i	−0.982334	−	0.187135i	$-0.940080\pi$
$257$		−	6.00000i		−	0.374270i	0.982334	−	0.187135i	$-0.0599201\pi$
$258$			24.0000i			1.49417i
$259$	4.00000	+	3.46410i	0.248548	+	0.215249i
$260$		0			0
$261$		−	20.7846i		−	1.28654i
$262$	20.7846			1.28408
$263$	−24.2487			−1.49524			−0.747620	−	0.664127i	$-0.768803\pi$
$263$	−24.2487			−1.49524			−0.747620	+	0.664127i	$0.768803\pi$
$264$			10.3923i			0.639602i
$265$		0			0
$266$	−12.0000	−	10.3923i	−0.735767	−	0.637193i
$267$	−10.3923			−0.635999
$268$		−	8.00000i		−	0.488678i
$269$	−18.0000			−1.09748			−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000			−1.09748			−0.548740	+	0.835993i	$0.684892\pi$
$270$		0			0
$271$		−	24.2487i		−	1.47300i	−0.676435	−	0.736502i	$-0.736476\pi$
$271$		−	24.2487i		−	1.47300i	0.676435	−	0.736502i	$-0.263524\pi$
$272$			30.0000i			1.81902i
$273$		0			0
$274$	36.0000			2.17484
$275$		0			0
$276$	6.00000			0.361158
$277$		−	14.0000i		−	0.841178i	−0.907251	−	0.420589i	$-0.861823\pi$
$277$		−	14.0000i		−	0.841178i	0.907251	−	0.420589i	$-0.138177\pi$
$278$		−	30.0000i		−	1.79928i
$279$			10.3923i			0.622171i
$280$		0			0
$281$		−	13.8564i		−	0.826604i	−0.910594	−	0.413302i	$-0.864375\pi$
$281$		−	13.8564i		−	0.826604i	0.910594	−	0.413302i	$-0.135625\pi$
$282$		−	36.0000i		−	2.14377i
$283$	17.3205			1.02960			0.514799	−	0.857311i	$-0.327867\pi$
$283$	17.3205			1.02960			0.514799	+	0.857311i	$0.327867\pi$
$284$		−	3.46410i		−	0.205557i
$285$		0			0
$286$		0			0
$287$	10.3923	−	12.0000i	0.613438	−	0.708338i
$288$	15.5885			0.918559
$289$	−19.0000			−1.11765
$290$		0			0
$291$	−12.0000			−0.703452
$292$	6.92820			0.405442
$293$		−	6.00000i		−	0.350524i	−0.984522	−	0.175262i	$-0.943923\pi$
$293$		−	6.00000i		−	0.350524i	0.984522	−	0.175262i	$-0.0560772\pi$
$294$	3.00000	+	20.7846i	0.174964	+	1.21218i
$295$		0			0
$296$			3.46410i			0.201347i
$297$			18.0000i			1.04447i
$298$		−	12.0000i		−	0.695141i
$299$		0			0
$300$		0			0
$301$	−16.0000	−	13.8564i	−0.922225	−	0.798670i
$302$	13.8564			0.797347
$303$	−10.3923			−0.597022
$304$		−	17.3205i		−	0.993399i
$305$		0			0
$306$			31.1769i			1.78227i
$307$	24.2487			1.38395			0.691974	−	0.721923i	$-0.256741\pi$
$307$	24.2487			1.38395			0.691974	+	0.721923i	$0.256741\pi$
$308$	6.92820	+	6.00000i	0.394771	+	0.341882i
$309$	−6.00000			−0.341328
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$	−20.7846			−1.17482			−0.587408	−	0.809291i	$-0.699852\pi$
$313$	−20.7846			−1.17482			−0.587408	+	0.809291i	$0.699852\pi$
$314$	24.0000			1.35440
$315$		0			0
$316$	−8.00000			−0.450035
$317$	−27.7128			−1.55651			−0.778253	−	0.627950i	$-0.783894\pi$
$317$	−27.7128			−1.55651			−0.778253	+	0.627950i	$0.783894\pi$
$318$		0			0
$319$	24.0000			1.34374
$320$		0			0
$321$	−18.0000			−1.00466
$322$	−10.3923	+	12.0000i	−0.579141	+	0.668734i
$323$	20.7846			1.15649
$324$	9.00000			0.500000
$325$		0			0
$326$			27.7128i			1.53487i
$327$	3.46410			0.191565
$328$	10.3923			0.573819
$329$	24.0000	+	20.7846i	1.32316	+	1.14589i
$330$		0			0
$331$	28.0000			1.53902			0.769510	−	0.638635i	$-0.220501\pi$
$331$	28.0000			1.53902			0.769510	+	0.638635i	$0.220501\pi$
$332$		0			0
$333$			6.00000i			0.328798i
$334$			20.7846i			1.13728i
$335$		0			0
$336$	−15.0000	+	17.3205i	−0.818317	+	0.944911i
$337$			14.0000i			0.762629i	0.924445	+	0.381314i	$0.124528\pi$
$337$			14.0000i			0.762629i	−0.924445	+	0.381314i	$0.875472\pi$
$338$	22.5167			1.22474
$339$	−12.0000			−0.651751
$340$		0			0
$341$	−12.0000			−0.649836
$342$		−	18.0000i		−	0.973329i
$343$	−15.5885	−	10.0000i	−0.841698	−	0.539949i
$344$		−	13.8564i		−	0.747087i
$345$		0			0
$346$		−	31.1769i		−	1.67608i
$347$	17.3205			0.929814			0.464907	−	0.885360i	$-0.346088\pi$
$347$	17.3205			0.929814			0.464907	+	0.885360i	$0.346088\pi$
$348$		−	12.0000i		−	0.643268i
$349$			6.92820i			0.370858i	0.982658	+	0.185429i	$0.0593675\pi$
$349$			6.92820i			0.370858i	−0.982658	+	0.185429i	$0.940632\pi$
$350$		0			0
$351$		0			0
$352$			18.0000i			0.959403i
$353$			6.00000i			0.319348i	0.987170	+	0.159674i	$0.0510443\pi$
$353$			6.00000i			0.319348i	−0.987170	+	0.159674i	$0.948956\pi$
$354$	−36.0000			−1.91338
$355$		0			0
$356$	−6.00000			−0.317999
$357$	−20.7846	−	18.0000i	−1.10004	−	0.952661i
$358$		−	18.0000i		−	0.951330i
$359$			3.46410i			0.182828i	0.995813	+	0.0914141i	$0.0291387\pi$
$359$			3.46410i			0.182828i	−0.995813	+	0.0914141i	$0.970861\pi$
$360$		0			0
$361$	7.00000			0.368421
$362$		−	36.0000i		−	1.89212i
$363$	−1.73205			−0.0909091
$364$		0			0
$365$		0			0
$366$		−	20.7846i		−	1.08643i
$367$	−10.3923			−0.542474			−0.271237	−	0.962513i	$-0.587433\pi$
$367$	−10.3923			−0.542474			−0.271237	+	0.962513i	$0.587433\pi$
$368$	−17.3205			−0.902894
$369$	18.0000			0.937043
$370$		0			0
$371$		0			0
$372$			6.00000i			0.311086i
$373$			14.0000i			0.724893i	0.932005	+	0.362446i	$0.118058\pi$
$373$			14.0000i			0.724893i	−0.932005	+	0.362446i	$0.881942\pi$
$374$	−36.0000			−1.86152
$375$		0			0
$376$			20.7846i			1.07188i
$377$		0			0
$378$	−15.5885	+	18.0000i	−0.801784	+	0.925820i
$379$	−4.00000			−0.205466			−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000			−0.205466			−0.102733	+	0.994709i	$0.532759\pi$
$380$		0			0
$381$			6.92820i			0.354943i
$382$		−	18.0000i		−	0.920960i
$383$			12.0000i			0.613171i	0.951843	+	0.306586i	$0.0991866\pi$
$383$			12.0000i			0.613171i	−0.951843	+	0.306586i	$0.900813\pi$
$384$	−21.0000			−1.07165
$385$		0			0
$386$			24.2487i			1.23423i
$387$		−	24.0000i		−	1.21999i
$388$	−6.92820			−0.351726
$389$			6.92820i			0.351274i	0.984455	+	0.175637i	$0.0561985\pi$
$389$			6.92820i			0.351274i	−0.984455	+	0.175637i	$0.943802\pi$
$390$		0			0
$391$		−	20.7846i		−	1.05112i
$392$	−1.73205	−	12.0000i	−0.0874818	−	0.606092i
$393$	−20.7846			−1.04844
$394$	−24.0000			−1.20910
$395$		0			0
$396$			10.3923i			0.522233i
$397$	−13.8564			−0.695433			−0.347717	−	0.937600i	$-0.613043\pi$
$397$	−13.8564			−0.695433			−0.347717	+	0.937600i	$0.613043\pi$
$398$		−	18.0000i		−	0.902258i
$399$	12.0000	+	10.3923i	0.600751	+	0.520266i
$400$		0			0
$401$			27.7128i			1.38391i	0.721940	+	0.691956i	$0.243251\pi$
$401$			27.7128i			1.38391i	−0.721940	+	0.691956i	$0.756749\pi$
$402$			24.0000i			1.19701i
$403$		0			0
$404$	−6.00000			−0.298511
$405$		0			0
$406$	24.0000	+	20.7846i	1.19110	+	1.03152i
$407$	−6.92820			−0.343418
$408$		−	18.0000i		−	0.891133i
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$	−36.0000			−1.77575
$412$	−3.46410			−0.170664
$413$	20.7846	−	24.0000i	1.02274	−	1.18096i
$414$	−18.0000			−0.884652
$415$		0			0
$416$		0			0
$417$			30.0000i			1.46911i
$418$	20.7846			1.01661
$419$	−12.0000			−0.586238			−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000			−0.586238			−0.293119	+	0.956076i	$0.594693\pi$
$420$		0			0
$421$	−10.0000			−0.487370			−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000			−0.487370			−0.243685	+	0.969854i	$0.578356\pi$
$422$	−6.92820			−0.337260
$423$			36.0000i			1.75038i
$424$		0			0
$425$		0			0
$426$			10.3923i			0.503509i
$427$	13.8564	+	12.0000i	0.670559	+	0.580721i
$428$	−10.3923			−0.502331
$429$		0			0
$430$		0			0
$431$			10.3923i			0.500580i	0.968171	+	0.250290i	$0.0805259\pi$
$431$			10.3923i			0.500580i	−0.968171	+	0.250290i	$0.919474\pi$
$432$	−25.9808			−1.25000
$433$	−6.92820			−0.332948			−0.166474	−	0.986046i	$-0.553238\pi$
$433$	−6.92820			−0.332948			−0.166474	+	0.986046i	$0.553238\pi$
$434$	−12.0000	−	10.3923i	−0.576018	−	0.498847i
$435$		0			0
$436$	2.00000			0.0957826
$437$			12.0000i			0.574038i
$438$	−20.7846			−0.993127
$439$		−	17.3205i		−	0.826663i	−0.910581	−	0.413331i	$-0.864365\pi$
$439$		−	17.3205i		−	0.826663i	0.910581	−	0.413331i	$-0.135635\pi$
$440$		0			0
$441$	−3.00000	−	20.7846i	−0.142857	−	0.989743i
$442$		0			0
$443$	10.3923			0.493753			0.246877	−	0.969047i	$-0.420596\pi$
$443$	10.3923			0.493753			0.246877	+	0.969047i	$0.420596\pi$
$444$			3.46410i			0.164399i
$445$		0			0
$446$	30.0000			1.42054
$447$			12.0000i			0.567581i
$448$	1.73205	−	2.00000i	0.0818317	−	0.0944911i
$449$		−	13.8564i		−	0.653924i	−0.945037	−	0.326962i	$-0.893975\pi$
$449$		−	13.8564i		−	0.653924i	0.945037	−	0.326962i	$-0.106025\pi$
$450$		0			0
$451$			20.7846i			0.978709i
$452$	−6.92820			−0.325875
$453$	−13.8564			−0.651031
$454$		−	41.5692i		−	1.95094i
$455$		0			0
$456$			10.3923i			0.486664i
$457$			38.0000i			1.77757i	0.458329	+	0.888783i	$0.348448\pi$
$457$			38.0000i			1.77757i	−0.458329	+	0.888783i	$0.651552\pi$
$458$			12.0000i			0.560723i
$459$		−	31.1769i		−	1.45521i
$460$		0			0
$461$	18.0000			0.838344			0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000			0.838344			0.419172	+	0.907907i	$0.362320\pi$
$462$	−20.7846	−	18.0000i	−0.966988	−	0.837436i
$463$			20.0000i			0.929479i	0.885448	+	0.464739i	$0.153852\pi$
$463$			20.0000i			0.929479i	−0.885448	+	0.464739i	$0.846148\pi$
$464$			34.6410i			1.60817i
$465$		0			0
$466$	−12.0000			−0.555889
$467$		−	24.0000i		−	1.11059i	−0.831654	−	0.555294i	$-0.812606\pi$
$467$		−	24.0000i		−	1.11059i	0.831654	−	0.555294i	$-0.187394\pi$
$468$		0			0
$469$	−16.0000	−	13.8564i	−0.738811	−	0.639829i
$470$		0			0
$471$	−24.0000			−1.10586
$472$	20.7846			0.956689
$473$	27.7128			1.27424
$474$	24.0000			1.10236
$475$		0			0
$476$	−12.0000	−	10.3923i	−0.550019	−	0.476331i
$477$		0			0
$478$			18.0000i			0.823301i
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$	10.3923	−	12.0000i	0.472866	−	0.546019i
$484$	−1.00000			−0.0454545
$485$		0			0
$486$	−27.0000			−1.22474
$487$		−	4.00000i		−	0.181257i	−0.995885	−	0.0906287i	$-0.971112\pi$
$487$		−	4.00000i		−	0.181257i	0.995885	−	0.0906287i	$-0.0288876\pi$
$488$			12.0000i			0.543214i
$489$		−	27.7128i		−	1.25322i
$490$		0			0
$491$		−	24.2487i		−	1.09433i	−0.837025	−	0.547165i	$-0.815707\pi$
$491$		−	24.2487i		−	1.09433i	0.837025	−	0.547165i	$-0.184293\pi$
$492$	10.3923			0.468521
$493$	−41.5692			−1.87218
$494$		0			0
$495$		0			0
$496$		−	17.3205i		−	0.777714i
$497$	−6.92820	−	6.00000i	−0.310772	−	0.269137i
$498$		0			0
$499$	20.0000			0.895323			0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000			0.895323			0.447661	+	0.894203i	$0.352257\pi$
$500$		0			0
$501$		−	20.7846i		−	0.928588i
$502$	−20.7846			−0.927663
$503$		−	12.0000i		−	0.535054i	−0.963550	−	0.267527i	$-0.913794\pi$
$503$		−	12.0000i		−	0.535054i	0.963550	−	0.267527i	$-0.0862064\pi$
$504$	9.00000	−	10.3923i	0.400892	−	0.462910i
$505$		0			0
$506$		−	20.7846i		−	0.923989i
$507$	−22.5167			−1.00000
$508$			4.00000i			0.177471i
$509$	30.0000			1.32973			0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000			1.32973			0.664863	+	0.746965i	$0.268490\pi$
$510$		0			0
$511$	12.0000	−	13.8564i	0.530849	−	0.612971i
$512$	−8.66025			−0.382733
$513$			18.0000i			0.794719i
$514$			10.3923i			0.458385i
$515$		0			0
$516$		−	13.8564i		−	0.609994i
$517$	−41.5692			−1.82821
$518$	−6.92820	−	6.00000i	−0.304408	−	0.263625i
$519$			31.1769i			1.36851i
$520$		0			0
$521$	6.00000			0.262865			0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000			0.262865			0.131432	+	0.991325i	$0.458042\pi$
$522$			36.0000i			1.57568i
$523$	17.3205			0.757373			0.378686	−	0.925525i	$-0.376376\pi$
$523$	17.3205			0.757373			0.378686	+	0.925525i	$0.376376\pi$
$524$	−12.0000			−0.524222
$525$		0			0
$526$	42.0000			1.83129
$527$	20.7846			0.905392
$528$		−	30.0000i		−	1.30558i
$529$	−11.0000			−0.478261
$530$		0			0
$531$	36.0000			1.56227
$532$	6.92820	+	6.00000i	0.300376	+	0.260133i
$533$		0			0
$534$	18.0000			0.778936
$535$		0			0
$536$		−	13.8564i		−	0.598506i
$537$			18.0000i			0.776757i
$538$	31.1769			1.34413
$539$	24.0000	−	3.46410i	1.03375	−	0.149209i
$540$		0			0
$541$	−34.0000			−1.46177			−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000			−1.46177			−0.730887	+	0.682498i	$0.760893\pi$
$542$			42.0000i			1.80405i
$543$			36.0000i			1.54491i
$544$		−	31.1769i		−	1.33670i
$545$		0			0
$546$		0			0
$547$		−	8.00000i		−	0.342055i	−0.985266	−	0.171028i	$-0.945291\pi$
$547$		−	8.00000i		−	0.342055i	0.985266	−	0.171028i	$-0.0547087\pi$
$548$	−20.7846			−0.887875
$549$			20.7846i			0.887066i
$550$		0			0
$551$	24.0000			1.02243
$552$	10.3923			0.442326
$553$	−13.8564	+	16.0000i	−0.589234	+	0.680389i
$554$			24.2487i			1.03023i
$555$		0			0
$556$			17.3205i			0.734553i
$557$	27.7128			1.17423			0.587115	−	0.809504i	$-0.300264\pi$
$557$	27.7128			1.17423			0.587115	+	0.809504i	$0.300264\pi$
$558$		−	18.0000i		−	0.762001i
$559$		0			0
$560$		0			0
$561$	36.0000			1.51992
$562$			24.0000i			1.01238i
$563$		0			0		1.00000			$0$
$563$		0			0		−1.00000			$\pi$
$564$			20.7846i			0.875190i
$565$		0			0
$566$	−30.0000			−1.26099
$567$	15.5885	−	18.0000i	0.654654	−	0.755929i
$568$		−	6.00000i		−	0.251754i
$569$		−	27.7128i		−	1.16178i	−0.813982	−	0.580891i	$-0.802704\pi$
$569$		−	27.7128i		−	1.16178i	0.813982	−	0.580891i	$-0.197296\pi$
$570$		0			0
$571$	−4.00000			−0.167395			−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000			−0.167395			−0.0836974	+	0.996491i	$0.526673\pi$
$572$		0			0
$573$			18.0000i			0.751961i
$574$	−18.0000	+	20.7846i	−0.751305	+	0.867533i
$575$		0			0
$576$	3.00000			0.125000
$577$	−34.6410			−1.44212			−0.721062	−	0.692870i	$-0.756346\pi$
$577$	−34.6410			−1.44212			−0.721062	+	0.692870i	$0.756346\pi$
$578$	32.9090			1.36883
$579$		−	24.2487i		−	1.00774i
$580$		0			0
$581$		0			0
$582$	20.7846			0.861550
$583$		0			0
$584$	12.0000			0.496564
$585$		0			0
$586$			10.3923i			0.429302i
$587$		0			0		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$	−1.73205	−	12.0000i	−0.0714286	−	0.494872i
$589$	−12.0000			−0.494451
$590$		0			0
$591$	24.0000			0.987228
$592$		−	10.0000i		−	0.410997i
$593$			6.00000i			0.246390i	0.992382	+	0.123195i	$0.0393141\pi$
$593$			6.00000i			0.246390i	−0.992382	+	0.123195i	$0.960686\pi$
$594$		−	31.1769i		−	1.27920i
$595$		0			0
$596$			6.92820i			0.283790i
$597$			18.0000i			0.736691i
$598$		0			0
$599$		−	24.2487i		−	0.990775i	−0.868672	−	0.495388i	$-0.835026\pi$
$599$		−	24.2487i		−	0.990775i	0.868672	−	0.495388i	$-0.164974\pi$
$600$		0			0
$601$		−	13.8564i		−	0.565215i	−0.959236	−	0.282607i	$-0.908801\pi$
$601$		−	13.8564i		−	0.565215i	0.959236	−	0.282607i	$-0.0911993\pi$
$602$	27.7128	+	24.0000i	1.12949	+	0.978167i
$603$		−	24.0000i		−	0.977356i
$604$	−8.00000			−0.325515
$605$		0			0
$606$	18.0000			0.731200
$607$	17.3205			0.703018			0.351509	−	0.936185i	$-0.385669\pi$
$607$	17.3205			0.703018			0.351509	+	0.936185i	$0.385669\pi$
$608$			18.0000i			0.729996i
$609$	−24.0000	−	20.7846i	−0.972529	−	0.842235i
$610$		0			0
$611$		0			0
$612$		−	18.0000i		−	0.727607i
$613$		−	34.0000i		−	1.37325i	−0.727013	−	0.686624i	$-0.759092\pi$
$613$		−	34.0000i		−	1.37325i	0.727013	−	0.686624i	$-0.240908\pi$
$614$	−42.0000			−1.69498
$615$		0			0
$616$	12.0000	+	10.3923i	0.483494	+	0.418718i
$617$	6.92820			0.278919			0.139459	−	0.990228i	$-0.455464\pi$
$617$	6.92820			0.278919			0.139459	+	0.990228i	$0.455464\pi$
$618$	10.3923			0.418040
$619$			17.3205i			0.696170i	0.937463	+	0.348085i	$0.113168\pi$
$619$			17.3205i			0.696170i	−0.937463	+	0.348085i	$0.886832\pi$
$620$		0			0
$621$	18.0000			0.722315
$622$		0			0
$623$	−10.3923	+	12.0000i	−0.416359	+	0.480770i
$624$		0			0
$625$		0			0
$626$	36.0000			1.43885
$627$	−20.7846			−0.830057
$628$	−13.8564			−0.552931
$629$	12.0000			0.478471
$630$		0			0
$631$	−40.0000			−1.59237			−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000			−1.59237			−0.796187	+	0.605050i	$0.793153\pi$
$632$	−13.8564			−0.551178
$633$	6.92820			0.275371
$634$	48.0000			1.90632
$635$		0			0
$636$		0			0
$637$		0			0
$638$	−41.5692			−1.64574
$639$		−	10.3923i		−	0.411113i
$640$		0			0
$641$		−	27.7128i		−	1.09459i	−0.836940	−	0.547295i	$-0.815658\pi$
$641$		−	27.7128i		−	1.09459i	0.836940	−	0.547295i	$-0.184342\pi$
$642$	31.1769			1.23045
$643$	31.1769			1.22950			0.614749	−	0.788723i	$-0.289257\pi$
$643$	31.1769			1.22950			0.614749	+	0.788723i	$0.289257\pi$
$644$	6.00000	−	6.92820i	0.236433	−	0.273009i
$645$		0			0
$646$	−36.0000			−1.41640
$647$		−	12.0000i		−	0.471769i	−0.971781	−	0.235884i	$-0.924201\pi$
$647$		−	12.0000i		−	0.471769i	0.971781	−	0.235884i	$-0.0757987\pi$
$648$	15.5885			0.612372
$649$			41.5692i			1.63173i
$650$		0			0
$651$	12.0000	+	10.3923i	0.470317	+	0.407307i
$652$		−	16.0000i		−	0.626608i
$653$	−41.5692			−1.62673			−0.813365	−	0.581754i	$-0.802367\pi$
$653$	−41.5692			−1.62673			−0.813365	+	0.581754i	$0.802367\pi$
$654$	−6.00000			−0.234619
$655$		0			0
$656$	−30.0000			−1.17130
$657$	20.7846			0.810885
$658$	−41.5692	−	36.0000i	−1.62054	−	1.40343i
$659$			10.3923i			0.404827i	0.979300	+	0.202413i	$0.0648785\pi$
$659$			10.3923i			0.404827i	−0.979300	+	0.202413i	$0.935122\pi$
$660$		0			0
$661$			48.4974i			1.88633i	0.332323	+	0.943166i	$0.392168\pi$
$661$			48.4974i			1.88633i	−0.332323	+	0.943166i	$0.607832\pi$
$662$	−48.4974			−1.88491
$663$		0			0
$664$		0			0
$665$		0			0
$666$		−	10.3923i		−	0.402694i
$667$		−	24.0000i		−	0.929284i
$668$		−	12.0000i		−	0.464294i
$669$	−30.0000			−1.15987
$670$		0			0
$671$	−24.0000			−0.926510
$672$	15.5885	−	18.0000i	0.601338	−	0.694365i
$673$		−	14.0000i		−	0.539660i	−0.962908	−	0.269830i	$-0.913032\pi$
$673$		−	14.0000i		−	0.539660i	0.962908	−	0.269830i	$-0.0869676\pi$
$674$		−	24.2487i		−	0.934025i
$675$		0			0
$676$	−13.0000			−0.500000
$677$			6.00000i			0.230599i	0.993331	+	0.115299i	$0.0367827\pi$
$677$			6.00000i			0.230599i	−0.993331	+	0.115299i	$0.963217\pi$
$678$	20.7846			0.798228
$679$	−12.0000	+	13.8564i	−0.460518	+	0.531760i
$680$		0			0
$681$			41.5692i			1.59294i
$682$	20.7846			0.795884
$683$	−17.3205			−0.662751			−0.331375	−	0.943499i	$-0.607513\pi$
$683$	−17.3205			−0.662751			−0.331375	+	0.943499i	$0.607513\pi$
$684$			10.3923i			0.397360i
$685$		0			0
$686$	27.0000	+	17.3205i	1.03086	+	0.661300i
$687$		−	12.0000i		−	0.457829i
$688$			40.0000i			1.52499i
$689$		0			0
$690$		0			0
$691$		−	31.1769i		−	1.18603i	−0.805193	−	0.593013i	$-0.797938\pi$
$691$		−	31.1769i		−	1.18603i	0.805193	−	0.593013i	$-0.202062\pi$
$692$			18.0000i			0.684257i
$693$	20.7846	+	18.0000i	0.789542	+	0.683763i
$694$	−30.0000			−1.13878
$695$		0			0
$696$		−	20.7846i		−	0.787839i
$697$		−	36.0000i		−	1.36360i
$698$		−	12.0000i		−	0.454207i
$699$	12.0000			0.453882
$700$		0			0
$701$			20.7846i			0.785024i	0.919747	+	0.392512i	$0.128394\pi$
$701$			20.7846i			0.785024i	−0.919747	+	0.392512i	$0.871606\pi$
$702$		0			0
$703$	−6.92820			−0.261302
$704$			3.46410i			0.130558i
$705$		0			0
$706$		−	10.3923i		−	0.391120i
$707$	−10.3923	+	12.0000i	−0.390843	+	0.451306i
$708$	20.7846			0.781133
$709$	−22.0000			−0.826227			−0.413114	−	0.910679i	$-0.635559\pi$
$709$	−22.0000			−0.826227			−0.413114	+	0.910679i	$0.635559\pi$
$710$		0			0
$711$	−24.0000			−0.900070
$712$	−10.3923			−0.389468
$713$			12.0000i			0.449404i
$714$	36.0000	+	31.1769i	1.34727	+	1.16677i
$715$		0			0
$716$			10.3923i			0.388379i
$717$		−	18.0000i		−	0.672222i
$718$		−	6.00000i		−	0.223918i
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$	−6.00000	+	6.92820i	−0.223452	+	0.258020i
$722$	−12.1244			−0.451222
$723$		0			0
$724$			20.7846i			0.772454i
$725$		0			0
$726$	3.00000			0.111340
$727$	3.46410			0.128476			0.0642382	−	0.997935i	$-0.479538\pi$
$727$	3.46410			0.128476			0.0642382	+	0.997935i	$0.479538\pi$
$728$		0			0
$729$	27.0000			1.00000
$730$		0			0
$731$	−48.0000			−1.77534
$732$			12.0000i			0.443533i
$733$		0			0			−	1.00000i	$-0.5\pi$
$733$		0			0				1.00000i	$0.5\pi$
$734$	18.0000			0.664392
$735$		0			0
$736$	18.0000			0.663489
$737$	27.7128			1.02081
$738$	−31.1769			−1.14764
$739$	4.00000			0.147142			0.0735712	−	0.997290i	$-0.476560\pi$
$739$	4.00000			0.147142			0.0735712	+	0.997290i	$0.476560\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	31.1769			1.14377			0.571885	−	0.820334i	$-0.306212\pi$
$743$	31.1769			1.14377			0.571885	+	0.820334i	$0.306212\pi$
$744$			10.3923i			0.381000i
$745$		0			0
$746$		−	24.2487i		−	0.887808i
$747$		0			0
$748$	20.7846			0.759961
$749$	−18.0000	+	20.7846i	−0.657706	+	0.759453i
$750$		0			0
$751$	16.0000			0.583848			0.291924	−	0.956441i	$-0.405705\pi$
$751$	16.0000			0.583848			0.291924	+	0.956441i	$0.405705\pi$
$752$		−	60.0000i		−	2.18797i
$753$	20.7846			0.757433
$754$		0			0
$755$		0			0
$756$	9.00000	−	10.3923i	0.327327	−	0.377964i
$757$			2.00000i			0.0726912i	0.999339	+	0.0363456i	$0.0115717\pi$
$757$			2.00000i			0.0726912i	−0.999339	+	0.0363456i	$0.988428\pi$
$758$	6.92820			0.251644
$759$			20.7846i			0.754434i
$760$		0			0
$761$	−42.0000			−1.52250			−0.761249	−	0.648459i	$-0.775414\pi$
$761$	−42.0000			−1.52250			−0.761249	+	0.648459i	$0.775414\pi$
$762$		−	12.0000i		−	0.434714i
$763$	3.46410	−	4.00000i	0.125409	−	0.144810i
$764$			10.3923i			0.375980i
$765$		0			0
$766$		−	20.7846i		−	0.750978i
$767$		0			0
$768$	32.9090			1.18750
$769$		−	41.5692i		−	1.49902i	−0.661991	−	0.749512i	$-0.730288\pi$
$769$		−	41.5692i		−	1.49902i	0.661991	−	0.749512i	$-0.269712\pi$
$770$		0			0
$771$		−	10.3923i		−	0.374270i
$772$		−	14.0000i		−	0.503871i
$773$		−	6.00000i		−	0.215805i	−0.994161	−	0.107903i	$-0.965587\pi$
$773$		−	6.00000i		−	0.215805i	0.994161	−	0.107903i	$-0.0344134\pi$
$774$			41.5692i			1.49417i
$775$		0			0
$776$	−12.0000			−0.430775
$777$	6.92820	+	6.00000i	0.248548	+	0.215249i
$778$		−	12.0000i		−	0.430221i
$779$			20.7846i			0.744686i
$780$		0			0
$781$	12.0000			0.429394
$782$			36.0000i			1.28736i
$783$		−	36.0000i		−	1.28654i
$784$	5.00000	+	34.6410i	0.178571	+	1.23718i
$785$		0			0
$786$	36.0000			1.28408
$787$	24.2487			0.864373			0.432187	−	0.901784i	$-0.357742\pi$
$787$	24.2487			0.864373			0.432187	+	0.901784i	$0.357742\pi$
$788$	13.8564			0.493614
$789$	−42.0000			−1.49524
$790$		0			0
$791$	−12.0000	+	13.8564i	−0.426671	+	0.492677i
$792$			18.0000i			0.639602i
$793$		0			0
$794$	24.0000			0.851728
$795$		0			0
$796$			10.3923i			0.368345i
$797$		−	18.0000i		−	0.637593i	−0.947823	−	0.318796i	$-0.896721\pi$
$797$		−	18.0000i		−	0.637593i	0.947823	−	0.318796i	$-0.103279\pi$
$798$	−20.7846	−	18.0000i	−0.735767	−	0.637193i
$799$	72.0000			2.54718
$800$		0			0
$801$	−18.0000			−0.635999
$802$		−	48.0000i		−	1.69494i
$803$			24.0000i			0.846942i
$804$		−	13.8564i		−	0.488678i
$805$		0			0
$806$		0			0
$807$	−31.1769			−1.09748
$808$	−10.3923			−0.365600
$809$			55.4256i			1.94866i	0.225122	+	0.974331i	$0.427722\pi$
$809$			55.4256i			1.94866i	−0.225122	+	0.974331i	$0.572278\pi$
$810$		0			0
$811$			38.1051i			1.33805i	0.743239	+	0.669026i	$0.233288\pi$
$811$			38.1051i			1.33805i	−0.743239	+	0.669026i	$0.766712\pi$
$812$	−13.8564	−	12.0000i	−0.486265	−	0.421117i
$813$		−	42.0000i		−	1.47300i
$814$	12.0000			0.420600
$815$		0			0
$816$			51.9615i			1.81902i
$817$	27.7128			0.969549
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	20.7846i		−	0.725388i	−0.931908	−	0.362694i	$-0.881857\pi$
$821$		−	20.7846i		−	0.725388i	0.931908	−	0.362694i	$-0.118143\pi$
$822$	62.3538			2.17484
$823$			44.0000i			1.53374i	0.641800	+	0.766872i	$0.278188\pi$
$823$			44.0000i			1.53374i	−0.641800	+	0.766872i	$0.721812\pi$
$824$	−6.00000			−0.209020
$825$		0			0
$826$	−36.0000	+	41.5692i	−1.25260	+	1.44638i
$827$	−38.1051			−1.32504			−0.662522	−	0.749042i	$-0.730514\pi$
$827$	−38.1051			−1.32504			−0.662522	+	0.749042i	$0.730514\pi$
$828$	10.3923			0.361158
$829$			34.6410i			1.20313i	0.798823	+	0.601566i	$0.205456\pi$
$829$			34.6410i			1.20313i	−0.798823	+	0.601566i	$0.794544\pi$
$830$		0			0
$831$		−	24.2487i		−	0.841178i
$832$		0			0
$833$	−41.5692	+	6.00000i	−1.44029	+	0.207888i
$834$		−	51.9615i		−	1.79928i
$835$		0			0
$836$	−12.0000			−0.415029
$837$			18.0000i			0.622171i
$838$	20.7846			0.717992
$839$	24.0000			0.828572			0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000			0.828572			0.414286	+	0.910147i	$0.364031\pi$
$840$		0			0
$841$	−19.0000			−0.655172
$842$	17.3205			0.596904
$843$		−	24.0000i		−	0.826604i
$844$	4.00000			0.137686
$845$		0			0
$846$		−	62.3538i		−	2.14377i
$847$	−1.73205	+	2.00000i	−0.0595140	+	0.0687208i
$848$		0			0
$849$	30.0000			1.02960
$850$		0			0
$851$			6.92820i			0.237496i
$852$		−	6.00000i		−	0.205557i
$853$	41.5692			1.42330			0.711651	−	0.702533i	$-0.247948\pi$
$853$	41.5692			1.42330			0.711651	+	0.702533i	$0.247948\pi$
$854$	−24.0000	−	20.7846i	−0.821263	−	0.711235i
$855$		0			0
$856$	−18.0000			−0.615227
$857$		−	30.0000i		−	1.02478i	−0.858753	−	0.512390i	$-0.828760\pi$
$857$		−	30.0000i		−	1.02478i	0.858753	−	0.512390i	$-0.171240\pi$
$858$		0			0
$859$		−	38.1051i		−	1.30013i	−0.759879	−	0.650065i	$-0.774742\pi$
$859$		−	38.1051i		−	1.30013i	0.759879	−	0.650065i	$-0.225258\pi$
$860$		0			0
$861$	18.0000	−	20.7846i	0.613438	−	0.708338i
$862$		−	18.0000i		−	0.613082i
$863$	17.3205			0.589597			0.294798	−	0.955559i	$-0.404747\pi$
$863$	17.3205			0.589597			0.294798	+	0.955559i	$0.404747\pi$
$864$	27.0000			0.918559
$865$		0			0
$866$	12.0000			0.407777
$867$	−32.9090			−1.11765
$868$	6.92820	+	6.00000i	0.235159	+	0.203653i
$869$		−	27.7128i		−	0.940093i
$870$		0			0
$871$		0			0
$872$	3.46410			0.117309
$873$	−20.7846			−0.703452
$874$		−	20.7846i		−	0.703050i
$875$		0			0
$876$	12.0000			0.405442
$877$		−	38.0000i		−	1.28317i	−0.767052	−	0.641584i	$-0.778277\pi$
$877$		−	38.0000i		−	1.28317i	0.767052	−	0.641584i	$-0.221723\pi$
$878$			30.0000i			1.01245i
$879$		−	10.3923i		−	0.350524i
$880$		0			0
$881$	30.0000			1.01073			0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000			1.01073			0.505363	+	0.862907i	$0.331359\pi$
$882$	5.19615	+	36.0000i	0.174964	+	1.21218i
$883$			16.0000i			0.538443i	0.963078	+	0.269221i	$0.0867663\pi$
$883$			16.0000i			0.538443i	−0.963078	+	0.269221i	$0.913234\pi$
$884$		0			0
$885$		0			0
$886$	−18.0000			−0.604722
$887$			36.0000i			1.20876i	0.796696	+	0.604381i	$0.206579\pi$
$887$			36.0000i			1.20876i	−0.796696	+	0.604381i	$0.793421\pi$
$888$			6.00000i			0.201347i
$889$	8.00000	+	6.92820i	0.268311	+	0.232364i
$890$		0			0
$891$			31.1769i			1.04447i
$892$	−17.3205			−0.579934
$893$	−41.5692			−1.39106
$894$		−	20.7846i		−	0.695141i
$895$		0			0
$896$	−21.0000	+	24.2487i	−0.701561	+	0.810093i
$897$		0			0
$898$			24.0000i			0.800890i
$899$	24.0000			0.800445
$900$		0			0
$901$		0			0
$902$		−	36.0000i		−	1.19867i
$903$	−27.7128	−	24.0000i	−0.922225	−	0.798670i
$904$	−12.0000			−0.399114
$905$		0			0
$906$	24.0000			0.797347
$907$			32.0000i			1.06254i	0.847202	+	0.531271i	$0.178286\pi$
$907$			32.0000i			1.06254i	−0.847202	+	0.531271i	$0.821714\pi$
$908$			24.0000i			0.796468i
$909$	−18.0000			−0.597022
$910$		0			0
$911$		−	17.3205i		−	0.573854i	−0.957952	−	0.286927i	$-0.907366\pi$
$911$		−	17.3205i		−	0.573854i	0.957952	−	0.286927i	$-0.0926337\pi$
$912$		−	30.0000i		−	0.993399i
$913$		0			0
$914$		−	65.8179i		−	2.17706i
$915$		0			0
$916$		−	6.92820i		−	0.228914i
$917$	−20.7846	+	24.0000i	−0.686368	+	0.792550i
$918$			54.0000i			1.78227i
$919$	−16.0000			−0.527791			−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000			−0.527791			−0.263896	+	0.964551i	$0.585007\pi$
$920$		0			0
$921$	42.0000			1.38395
$922$	−31.1769			−1.02676
$923$		0			0
$924$	12.0000	+	10.3923i	0.394771	+	0.341882i
$925$		0			0
$926$		−	34.6410i		−	1.13837i
$927$	−10.3923			−0.341328
$928$		−	36.0000i		−	1.18176i
$929$	−30.0000			−0.984268			−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000			−0.984268			−0.492134	+	0.870519i	$0.663783\pi$
$930$		0			0
$931$	24.0000	−	3.46410i	0.786568	−	0.113531i
$932$	6.92820			0.226941
$933$		0			0
$934$			41.5692i			1.36019i
$935$		0			0
$936$		0			0
$937$	6.92820			0.226335			0.113167	−	0.993576i	$-0.463900\pi$
$937$	6.92820			0.226335			0.113167	+	0.993576i	$0.463900\pi$
$938$	27.7128	+	24.0000i	0.904855	+	0.783628i
$939$	−36.0000			−1.17482
$940$		0			0
$941$	18.0000			0.586783			0.293392	−	0.955992i	$-0.405216\pi$
$941$	18.0000			0.586783			0.293392	+	0.955992i	$0.405216\pi$
$942$	41.5692			1.35440
$943$	20.7846			0.676840
$944$	−60.0000			−1.95283
$945$		0			0
$946$	−48.0000			−1.56061
$947$	31.1769			1.01311			0.506557	−	0.862207i	$-0.330918\pi$
$947$	31.1769			1.01311			0.506557	+	0.862207i	$0.330918\pi$
$948$	−13.8564			−0.450035
$949$		0			0
$950$		0			0
$951$	−48.0000			−1.55651
$952$	−20.7846	−	18.0000i	−0.673633	−	0.583383i
$953$	34.6410			1.12213			0.561066	−	0.827771i	$-0.310391\pi$
$953$	34.6410			1.12213			0.561066	+	0.827771i	$0.310391\pi$
$954$		0			0
$955$		0			0
$956$		−	10.3923i		−	0.336111i
$957$	41.5692			1.34374
$958$		0			0
$959$	−36.0000	+	41.5692i	−1.16250	+	1.34234i
$960$		0			0
$961$	19.0000			0.612903
$962$		0			0
$963$	−31.1769			−1.00466
$964$		0			0
$965$		0			0
$966$	−18.0000	+	20.7846i	−0.579141	+	0.668734i
$967$		−	20.0000i		−	0.643157i	−0.946883	−	0.321578i	$-0.895787\pi$
$967$		−	20.0000i		−	0.643157i	0.946883	−	0.321578i	$-0.104213\pi$
$968$	−1.73205			−0.0556702
$969$	36.0000			1.15649
$970$		0			0
$971$	60.0000			1.92549			0.962746	−	0.270408i	$-0.0871586\pi$
$971$	60.0000			1.92549			0.962746	+	0.270408i	$0.0871586\pi$
$972$	15.5885			0.500000
$973$	34.6410	+	30.0000i	1.11054	+	0.961756i
$974$			6.92820i			0.221994i
$975$		0			0
$976$		−	34.6410i		−	1.10883i
$977$	−6.92820			−0.221653			−0.110826	−	0.993840i	$-0.535350\pi$
$977$	−6.92820			−0.221653			−0.110826	+	0.993840i	$0.535350\pi$
$978$			48.0000i			1.53487i
$979$		−	20.7846i		−	0.664279i
$980$		0			0
$981$	6.00000			0.191565
$982$			42.0000i			1.34027i
$983$		−	60.0000i		−	1.91370i	−0.290578	−	0.956851i	$-0.593847\pi$
$983$		−	60.0000i		−	1.91370i	0.290578	−	0.956851i	$-0.406153\pi$
$984$	18.0000			0.573819
$985$		0			0
$986$	72.0000			2.29295
$987$	41.5692	+	36.0000i	1.32316	+	1.14589i
$988$		0			0
$989$		−	27.7128i		−	0.881216i
$990$		0			0
$991$	16.0000			0.508257			0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000			0.508257			0.254128	+	0.967170i	$0.418211\pi$
$992$			18.0000i			0.571501i
$993$	48.4974			1.53902
$994$	12.0000	+	10.3923i	0.380617	+	0.329624i
$995$		0			0
$996$		0			0
$997$		0			0			−	1.00000i	$-0.5\pi$
$997$		0			0				1.00000i	$0.5\pi$
$998$	−34.6410			−1.09654
$999$			10.3923i			0.328798i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.2.g.b.524.1		4
3.2	odd	2		525.2.g.c.524.3		4
5.2	odd	4		105.2.b.a.41.1	✓	2
5.3	odd	4		525.2.b.a.251.2		2
5.4	even	2	inner	525.2.g.b.524.4		4
7.6	odd	2		525.2.g.c.524.1		4
15.2	even	4		105.2.b.b.41.2	yes	2
15.8	even	4		525.2.b.b.251.1		2
15.14	odd	2		525.2.g.c.524.2		4
20.7	even	4		1680.2.f.b.881.2		2
21.20	even	2	inner	525.2.g.b.524.3		4
35.2	odd	12		735.2.s.b.521.1		2
35.12	even	12		735.2.s.a.521.1		2
35.13	even	4		525.2.b.b.251.2		2
35.17	even	12		735.2.s.f.656.1		2
35.27	even	4		105.2.b.b.41.1	yes	2
35.32	odd	12		735.2.s.d.656.1		2
35.34	odd	2		525.2.g.c.524.4		4
60.47	odd	4		1680.2.f.c.881.2		2
105.2	even	12		735.2.s.f.521.1		2
105.17	odd	12		735.2.s.b.656.1		2
105.32	even	12		735.2.s.a.656.1		2
105.47	odd	12		735.2.s.d.521.1		2
105.62	odd	4		105.2.b.a.41.2	yes	2
105.83	odd	4		525.2.b.a.251.1		2
105.104	even	2	inner	525.2.g.b.524.2		4
140.27	odd	4		1680.2.f.c.881.1		2
420.167	even	4		1680.2.f.b.881.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.b.a.41.1	✓	2	5.2	odd	4
105.2.b.a.41.2	yes	2	105.62	odd	4
105.2.b.b.41.1	yes	2	35.27	even	4
105.2.b.b.41.2	yes	2	15.2	even	4
525.2.b.a.251.1		2	105.83	odd	4
525.2.b.a.251.2		2	5.3	odd	4
525.2.b.b.251.1		2	15.8	even	4
525.2.b.b.251.2		2	35.13	even	4
525.2.g.b.524.1		4	1.1	even	1	trivial
525.2.g.b.524.2		4	105.104	even	2	inner
525.2.g.b.524.3		4	21.20	even	2	inner
525.2.g.b.524.4		4	5.4	even	2	inner
525.2.g.c.524.1		4	7.6	odd	2
525.2.g.c.524.2		4	15.14	odd	2
525.2.g.c.524.3		4	3.2	odd	2
525.2.g.c.524.4		4	35.34	odd	2
735.2.s.a.521.1		2	35.12	even	12
735.2.s.a.656.1		2	105.32	even	12
735.2.s.b.521.1		2	35.2	odd	12
735.2.s.b.656.1		2	105.17	odd	12
735.2.s.d.521.1		2	105.47	odd	12
735.2.s.d.656.1		2	35.32	odd	12
735.2.s.f.521.1		2	105.2	even	12
735.2.s.f.656.1		2	35.17	even	12
1680.2.f.b.881.1		2	420.167	even	4
1680.2.f.b.881.2		2	20.7	even	4
1680.2.f.c.881.1		2	140.27	odd	4
1680.2.f.c.881.2		2	60.47	odd	4

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:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	525.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.73205 q^{2} +1.73205 q^{3} +1.00000 q^{4} -3.00000 q^{6} +(1.73205 - 2.00000i) q^{7} +1.73205 q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q-1.73205 q^{2} +1.73205 q^{3} +1.00000 q^{4} -3.00000 q^{6} +(1.73205 - 2.00000i) q^{7} +1.73205 q^{8} +3.00000 q^{9} +3.46410i q^{11} +1.73205 q^{12} +(-3.00000 + 3.46410i) q^{14} -5.00000 q^{16} -6.00000i q^{17} -5.19615 q^{18} +3.46410i q^{19} +(3.00000 - 3.46410i) q^{21} -6.00000i q^{22} +3.46410 q^{23} +3.00000 q^{24} +5.19615 q^{27} +(1.73205 - 2.00000i) q^{28} -6.92820i q^{29} +3.46410i q^{31} +5.19615 q^{32} +6.00000i q^{33} +10.3923i q^{34} +3.00000 q^{36} +2.00000i q^{37} -6.00000i q^{38} +6.00000 q^{41} +(-5.19615 + 6.00000i) q^{42} -8.00000i q^{43} +3.46410i q^{44} -6.00000 q^{46} +12.0000i q^{47} -8.66025 q^{48} +(-1.00000 - 6.92820i) q^{49} -10.3923i q^{51} -9.00000 q^{54} +(3.00000 - 3.46410i) q^{56} +6.00000i q^{57} +12.0000i q^{58} +12.0000 q^{59} +6.92820i q^{61} -6.00000i q^{62} +(5.19615 - 6.00000i) q^{63} +1.00000 q^{64} -10.3923i q^{66} -8.00000i q^{67} -6.00000i q^{68} +6.00000 q^{69} -3.46410i q^{71} +5.19615 q^{72} +6.92820 q^{73} -3.46410i q^{74} +3.46410i q^{76} +(6.92820 + 6.00000i) q^{77} -8.00000 q^{79} +9.00000 q^{81} -10.3923 q^{82} +(3.00000 - 3.46410i) q^{84} +13.8564i q^{86} -12.0000i q^{87} +6.00000i q^{88} -6.00000 q^{89} +3.46410 q^{92} +6.00000i q^{93} -20.7846i q^{94} +9.00000 q^{96} -6.92820 q^{97} +(1.73205 + 12.0000i) q^{98} +10.3923i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{4} - 12 q^{6} + 12 q^{9}+O(q^{10}) \) 4 * q + 4 * q^4 - 12 * q^6 + 12 * q^9	\( 4 q + 4 q^{4} - 12 q^{6} + 12 q^{9} - 12 q^{14} - 20 q^{16} + 12 q^{21} + 12 q^{24} + 12 q^{36} + 24 q^{41} - 24 q^{46} - 4 q^{49} - 36 q^{54} + 12 q^{56} + 48 q^{59} + 4 q^{64} + 24 q^{69} - 32 q^{79} + 36 q^{81} + 12 q^{84} - 24 q^{89} + 36 q^{96}+O(q^{100}) \) 4 * q + 4 * q^4 - 12 * q^6 + 12 * q^9 - 12 * q^14 - 20 * q^16 + 12 * q^21 + 12 * q^24 + 12 * q^36 + 24 * q^41 - 24 * q^46 - 4 * q^49 - 36 * q^54 + 12 * q^56 + 48 * q^59 + 4 * q^64 + 24 * q^69 - 32 * q^79 + 36 * q^81 + 12 * q^84 - 24 * q^89 + 36 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more