LMFDB - Embedded newform 3420.2.bb.d.37.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3420,2,Mod(37,3420)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3420, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("3420.37");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3420.bb (of order $4$, degree $2$, 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:	$27.3088374913$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} + 28 x^{10} - 64 x^{9} + 236 x^{8} - 420 x^{7} + 946 x^{6} - 1216 x^{5} + 1896 x^{4} + \cdots + 1370 $ x^12 - 4x^11 + 28x^10 - 64x^9 + 236x^8 - 420x^7 + 946x^6 - 1216x^5 + 1896x^4 - 1564x^3 + 2284x^2 - 1088*x + 1370
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 380)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			37.1
Root			$-0.585043 - 2.22350i$ of defining polynomial
Character	$\chi$	$=$	3420.37
Dual form			3420.2.bb.d.2773.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.17009 - 0.539189i) q^{5} +(2.17009 + 2.17009i) q^{7} +O(q^{10})$	$q+(-2.17009 - 0.539189i) q^{5} +(2.17009 + 2.17009i) q^{7} +1.70928 q^{11} +(-4.43904 - 4.43904i) q^{13} +(1.63090 + 1.63090i) q^{17} +(-2.80058 - 3.34017i) q^{19} +(-3.24846 + 3.24846i) q^{23} +(4.41855 + 2.34017i) q^{25} +3.27692 q^{29} +7.11120i q^{31} +(-3.53919 - 5.87936i) q^{35} +(0.128419 - 0.128419i) q^{37} -3.83428i q^{41} +(-5.24846 + 5.24846i) q^{43} +(-0.908291 - 0.908291i) q^{47} +2.41855i q^{49} +(5.47274 + 5.47274i) q^{53} +(-3.70928 - 0.921622i) q^{55} -13.9219 q^{59} +2.63090 q^{61} +(7.23962 + 12.0266i) q^{65} +(7.71596 - 7.71596i) q^{67} +1.51004i q^{71} +(-8.70928 + 8.70928i) q^{73} +(3.70928 + 3.70928i) q^{77} -4.78696 q^{79} +(-6.46081 + 6.46081i) q^{83} +(-2.65983 - 4.41855i) q^{85} -5.34432 q^{89} -19.2662i q^{91} +(4.27652 + 8.75851i) q^{95} +(-10.2597 + 10.2597i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 4 q^{5} + 4 q^{7}+O(q^{10}) $ 12 * q - 4 * q^5 + 4 * q^7	$ 12 q - 4 q^{5} + 4 q^{7} - 8 q^{11} + 4 q^{17} - 4 q^{23} - 4 q^{25} - 36 q^{35} - 28 q^{43} - 20 q^{47} - 16 q^{55} + 16 q^{61} - 76 q^{73} + 16 q^{77} - 84 q^{83} - 76 q^{85} + 16 q^{95}+O(q^{100}) $ 12 * q - 4 * q^5 + 4 * q^7 - 8 * q^11 + 4 * q^17 - 4 * q^23 - 4 * q^25 - 36 * q^35 - 28 * q^43 - 20 * q^47 - 16 * q^55 + 16 * q^61 - 76 * q^73 + 16 * q^77 - 84 * q^83 - 76 * q^85 + 16 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3420\mathbb{Z}\right)^\times$.

$n$	$1711$	$1901$	$2737$	$3061$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{4}\right)$	$-1$

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			0
$3$		0			0
$4$		0			0
$5$	−2.17009	−	0.539189i	−0.970492	−	0.241133i
$5$	−2.17009	−	0.539189i	−0.970492	−	0.241133i
$6$		0			0
$7$	2.17009	+	2.17009i	0.820216	+	0.820216i	0.986139	−	0.165923i	$-0.0530604\pi$
$7$	2.17009	+	2.17009i	0.820216	+	0.820216i	−0.165923	+	0.986139i	$0.553060\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.70928			0.515366			0.257683	−	0.966230i	$-0.417041\pi$
$11$	1.70928			0.515366			0.257683	+	0.966230i	$0.417041\pi$
$12$		0			0
$13$	−4.43904	−	4.43904i	−1.23117	−	1.23117i	−0.963516	−	0.267652i	$-0.913752\pi$
$13$	−4.43904	−	4.43904i	−1.23117	−	1.23117i	−0.267652	−	0.963516i	$-0.586248\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.63090	+	1.63090i	0.395551	+	0.395551i	0.876660	−	0.481110i	$-0.159766\pi$
$17$	1.63090	+	1.63090i	0.395551	+	0.395551i	−0.481110	+	0.876660i	$0.659766\pi$
$18$		0			0
$19$	−2.80058	−	3.34017i	−0.642497	−	0.766288i
$19$	−2.80058	−	3.34017i	−0.642497	−	0.766288i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.24846	+	3.24846i	−0.677352	+	0.677352i	−0.959400	−	0.282049i	$-0.908986\pi$
$23$	−3.24846	+	3.24846i	−0.677352	+	0.677352i	0.282049	+	0.959400i	$0.408986\pi$
$24$		0			0
$25$	4.41855	+	2.34017i	0.883710	+	0.468035i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	3.27692			0.608509			0.304254	−	0.952591i	$-0.401593\pi$
$29$	3.27692			0.608509			0.304254	+	0.952591i	$0.401593\pi$
$30$		0			0
$31$			7.11120i			1.27721i	0.769535	+	0.638605i	$0.220488\pi$
$31$			7.11120i			1.27721i	−0.769535	+	0.638605i	$0.779512\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−3.53919	−	5.87936i	−0.598232	−	0.993794i
$36$		0			0
$37$	0.128419	−	0.128419i	0.0211119	−	0.0211119i	−0.696472	−	0.717584i	$-0.745248\pi$
$37$	0.128419	−	0.128419i	0.0211119	−	0.0211119i	0.717584	+	0.696472i	$0.245248\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	3.83428i		−	0.598814i	−0.954126	−	0.299407i	$-0.903211\pi$
$41$		−	3.83428i		−	0.598814i	0.954126	−	0.299407i	$-0.0967888\pi$
$42$		0			0
$43$	−5.24846	+	5.24846i	−0.800383	+	0.800383i	−0.983155	−	0.182772i	$-0.941493\pi$
$43$	−5.24846	+	5.24846i	−0.800383	+	0.800383i	0.182772	+	0.983155i	$0.441493\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−0.908291	−	0.908291i	−0.132488	−	0.132488i	0.637753	−	0.770241i	$-0.279864\pi$
$47$	−0.908291	−	0.908291i	−0.132488	−	0.132488i	−0.770241	+	0.637753i	$0.779864\pi$
$48$		0			0
$49$			2.41855i			0.345507i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	5.47274	+	5.47274i	0.751739	+	0.751739i	0.974804	−	0.223065i	$-0.0716062\pi$
$53$	5.47274	+	5.47274i	0.751739	+	0.751739i	−0.223065	+	0.974804i	$0.571606\pi$
$54$		0			0
$55$	−3.70928	−	0.921622i	−0.500159	−	0.124272i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−13.9219			−1.81247			−0.906237	−	0.422770i	$-0.861058\pi$
$59$	−13.9219			−1.81247			−0.906237	+	0.422770i	$0.861058\pi$
$60$		0			0
$61$	2.63090			0.336852			0.168426	−	0.985714i	$-0.446132\pi$
$61$	2.63090			0.336852			0.168426	+	0.985714i	$0.446132\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	7.23962	+	12.0266i	0.897964	+	1.49171i
$66$		0			0
$67$	7.71596	−	7.71596i	0.942654	−	0.942654i	−0.0557882	−	0.998443i	$-0.517767\pi$
$67$	7.71596	−	7.71596i	0.942654	−	0.942654i	0.998443	+	0.0557882i	$0.0177671\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			1.51004i			0.179209i	0.995977	+	0.0896045i	$0.0285603\pi$
$71$			1.51004i			0.179209i	−0.995977	+	0.0896045i	$0.971440\pi$
$72$		0			0
$73$	−8.70928	+	8.70928i	−1.01934	+	1.01934i	−0.0195344	+	0.999809i	$0.506218\pi$
$73$	−8.70928	+	8.70928i	−1.01934	+	1.01934i	−0.999809	+	0.0195344i	$0.993782\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	3.70928	+	3.70928i	0.422711	+	0.422711i
$78$		0			0
$79$	−4.78696			−0.538575			−0.269288	−	0.963060i	$-0.586788\pi$
$79$	−4.78696			−0.538575			−0.269288	+	0.963060i	$0.586788\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−6.46081	+	6.46081i	−0.709166	+	0.709166i	−0.966360	−	0.257194i	$-0.917202\pi$
$83$	−6.46081	+	6.46081i	−0.709166	+	0.709166i	0.257194	+	0.966360i	$0.417202\pi$
$84$		0			0
$85$	−2.65983	−	4.41855i	−0.288499	−	0.479259i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−5.34432			−0.566497			−0.283248	−	0.959047i	$-0.591412\pi$
$89$	−5.34432			−0.566497			−0.283248	+	0.959047i	$0.591412\pi$
$90$		0			0
$91$		−	19.2662i		−	2.01965i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	4.27652	+	8.75851i	0.438761	+	0.898604i
$96$		0			0
$97$	−10.2597	+	10.2597i	−1.04171	+	1.04171i	−0.0426236	+	0.999091i	$0.513572\pi$
$97$	−10.2597	+	10.2597i	−1.04171	+	1.04171i	−0.999091	+	0.0426236i	$0.986428\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	15.3112			1.52353			0.761763	−	0.647856i	$-0.224334\pi$
$101$	15.3112			1.52353			0.761763	+	0.647856i	$0.224334\pi$
$102$		0			0
$103$	−13.3171	−	13.3171i	−1.31217	−	1.31217i	−0.919810	−	0.392365i	$-0.871657\pi$
$103$	−13.3171	−	13.3171i	−1.31217	−	1.31217i	−0.392365	−	0.919810i	$-0.628343\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	2.67216	−	2.67216i	0.258328	−	0.258328i	−0.566046	−	0.824374i	$-0.691528\pi$
$107$	2.67216	−	2.67216i	0.258328	−	0.258328i	0.824374	+	0.566046i	$0.191528\pi$
$108$		0			0
$109$	6.15852			0.589879			0.294940	−	0.955516i	$-0.404700\pi$
$109$	6.15852			0.589879			0.294940	+	0.955516i	$0.404700\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−11.5502	−	11.5502i	−1.08656	−	1.08656i	−0.995881	−	0.0906745i	$-0.971098\pi$
$113$	−11.5502	−	11.5502i	−1.08656	−	1.08656i	−0.0906745	−	0.995881i	$-0.528902\pi$
$114$		0			0
$115$	8.80098	−	5.29791i	0.820696	−	0.494033i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			7.07838i			0.648874i
$120$		0			0
$121$	−8.07838			−0.734398
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−8.32684	−	7.46081i	−0.744775	−	0.667315i
$126$		0			0
$127$	−3.40534	+	3.40534i	−0.302175	+	0.302175i	−0.841864	−	0.539689i	$-0.818542\pi$
$127$	−3.40534	+	3.40534i	−0.302175	+	0.302175i	0.539689	+	0.841864i	$0.318542\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−15.5174			−1.35577			−0.677883	−	0.735170i	$-0.737102\pi$
$131$	−15.5174			−1.35577			−0.677883	+	0.735170i	$0.737102\pi$
$132$		0			0
$133$	1.17096	−	13.3260i	0.101536	−	1.15551i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	14.1773	+	14.1773i	1.21125	+	1.21125i	0.970617	+	0.240629i	$0.0773538\pi$
$137$	14.1773	+	14.1773i	1.21125	+	1.21125i	0.240629	+	0.970617i	$0.422646\pi$
$138$		0			0
$139$			10.9444i			0.928293i	0.885759	+	0.464146i	$0.153639\pi$
$139$			10.9444i			0.928293i	−0.885759	+	0.464146i	$0.846361\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−7.58754	−	7.58754i	−0.634502	−	0.634502i
$144$		0			0
$145$	−7.11120	−	1.76688i	−0.590553	−	0.146731i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			0.532001i			0.0435832i	0.999763	+	0.0217916i	$0.00693703\pi$
$149$			0.532001i			0.0435832i	−0.999763	+	0.0217916i	$0.993063\pi$
$150$		0			0
$151$			11.8982i			0.968259i	0.874996	+	0.484129i	$0.160864\pi$
$151$			11.8982i			0.968259i	−0.874996	+	0.484129i	$0.839136\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	3.83428	−	15.4319i	0.307977	−	1.23952i
$156$		0			0
$157$	−5.94441	−	5.94441i	−0.474415	−	0.474415i	0.428925	−	0.903340i	$-0.358893\pi$
$157$	−5.94441	−	5.94441i	−0.474415	−	0.474415i	−0.903340	+	0.428925i	$0.858893\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−14.0989			−1.11115
$162$		0			0
$163$	−12.0361	+	12.0361i	−0.942741	+	0.942741i	−0.998447	−	0.0557058i	$-0.982259\pi$
$163$	−12.0361	+	12.0361i	−0.942741	+	0.942741i	0.0557058	+	0.998447i	$0.482259\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−8.74966	+	8.74966i	−0.677069	+	0.677069i	−0.959336	−	0.282267i	$-0.908914\pi$
$167$	−8.74966	+	8.74966i	−0.677069	+	0.677069i	0.282267	+	0.959336i	$0.408914\pi$
$168$		0			0
$169$			26.4101i			2.03155i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1.16212	−	1.16212i	−0.0883543	−	0.0883543i	0.661548	−	0.749903i	$-0.269900\pi$
$173$	−1.16212	−	1.16212i	−0.0883543	−	0.0883543i	−0.749903	+	0.661548i	$0.769900\pi$
$174$		0			0
$175$	4.51026	+	14.6670i	0.340944	+	1.10872i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−25.5632			−1.91068			−0.955342	−	0.295503i	$-0.904513\pi$
$179$	−25.5632			−1.91068			−0.955342	+	0.295503i	$0.904513\pi$
$180$		0			0
$181$		−	13.1077i		−	0.974286i	−0.873322	−	0.487143i	$-0.838039\pi$
$181$		−	13.1077i		−	0.974286i	0.873322	−	0.487143i	$-0.161961\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−0.347922	+	0.209438i	−0.0255797	+	0.0153982i
$186$		0			0
$187$	2.78765	+	2.78765i	0.203853	+	0.203853i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	2.65368			0.192014			0.0960069	−	0.995381i	$-0.469393\pi$
$191$	2.65368			0.192014			0.0960069	+	0.995381i	$0.469393\pi$
$192$		0			0
$193$	8.74966	+	8.74966i	0.629814	+	0.629814i	0.948021	−	0.318207i	$-0.103081\pi$
$193$	8.74966	+	8.74966i	0.629814	+	0.629814i	−0.318207	+	0.948021i	$0.603081\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	4.46800	+	4.46800i	0.318332	+	0.318332i	0.848126	−	0.529794i	$-0.177731\pi$
$197$	4.46800	+	4.46800i	0.318332	+	0.318332i	−0.529794	+	0.848126i	$0.677731\pi$
$198$		0			0
$199$			15.9421i			1.13011i	0.825054	+	0.565054i	$0.191145\pi$
$199$			15.9421i			1.13011i	−0.825054	+	0.565054i	$0.808855\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	7.11120	+	7.11120i	0.499108	+	0.499108i
$204$		0			0
$205$	−2.06740	+	8.32072i	−0.144394	+	0.581144i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−4.78696	−	5.70928i	−0.331121	−	0.394919i
$210$		0			0
$211$			16.3846i			1.12796i	0.825788	+	0.563981i	$0.190731\pi$
$211$			16.3846i			1.12796i	−0.825788	+	0.563981i	$0.809269\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	14.2195	−	8.55971i	0.969764	−	0.583767i
$216$		0			0
$217$	−15.4319	+	15.4319i	−1.04759	+	1.04759i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	14.4792i		−	0.973979i
$222$		0			0
$223$	−2.19582	−	2.19582i	−0.147043	−	0.147043i	0.629753	−	0.776796i	$-0.283156\pi$
$223$	−2.19582	−	2.19582i	−0.147043	−	0.147043i	−0.776796	+	0.629753i	$0.783156\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−1.16212	+	1.16212i	−0.0771326	+	0.0771326i	−0.744621	−	0.667488i	$-0.767370\pi$
$227$	−1.16212	+	1.16212i	−0.0771326	+	0.0771326i	0.667488	+	0.744621i	$0.267370\pi$
$228$		0			0
$229$		−	9.89269i		−	0.653728i	−0.945071	−	0.326864i	$-0.894008\pi$
$229$		−	9.89269i		−	0.653728i	0.945071	−	0.326864i	$-0.105992\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−20.8576	+	20.8576i	−1.36643	+	1.36643i	−0.500955	+	0.865474i	$0.667018\pi$
$233$	−20.8576	+	20.8576i	−1.36643	+	1.36643i	−0.865474	+	0.500955i	$0.832982\pi$
$234$		0			0
$235$	1.48133	+	2.46081i	0.0966313	+	0.160526i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			8.65368i			0.559760i	0.960035	+	0.279880i	$0.0902947\pi$
$239$			8.65368i			0.559760i	−0.960035	+	0.279880i	$0.909705\pi$
$240$		0			0
$241$			1.25320i			0.0807259i	0.999185	+	0.0403630i	$0.0128514\pi$
$241$			1.25320i			0.0807259i	−0.999185	+	0.0403630i	$0.987149\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1.30406	−	5.24846i	0.0833131	−	0.335312i
$246$		0			0
$247$	−2.39528	+	27.2590i	−0.152408	+	1.73445i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−1.65983			−0.104767			−0.0523837	−	0.998627i	$-0.516682\pi$
$251$	−1.65983			−0.104767			−0.0523837	+	0.998627i	$0.516682\pi$
$252$		0			0
$253$	−5.55252	+	5.55252i	−0.349084	+	0.349084i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−4.91538	+	4.91538i	−0.306613	+	0.306613i	−0.843594	−	0.536981i	$-0.819565\pi$
$257$	−4.91538	+	4.91538i	−0.306613	+	0.306613i	0.536981	+	0.843594i	$0.319565\pi$
$258$		0			0
$259$	0.557360			0.0346327
$260$		0			0
$261$		0			0
$262$		0			0
$263$	12.8010	−	12.8010i	0.789342	−	0.789342i	−0.192044	−	0.981386i	$-0.561512\pi$
$263$	12.8010	−	12.8010i	0.789342	−	0.789342i	0.981386	+	0.192044i	$0.0615116\pi$
$264$		0			0
$265$	−8.92548	−	14.8272i	−0.548288	−	0.910825i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	17.1988			1.04863			0.524315	−	0.851525i	$-0.324322\pi$
$269$	17.1988			1.04863			0.524315	+	0.851525i	$0.324322\pi$
$270$		0			0
$271$	−11.2846			−0.685490			−0.342745	−	0.939429i	$-0.611357\pi$
$271$	−11.2846			−0.685490			−0.342745	+	0.939429i	$0.611357\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	7.55252	+	4.00000i	0.455434	+	0.241209i
$276$		0			0
$277$	−7.29072	−	7.29072i	−0.438057	−	0.438057i	0.453301	−	0.891358i	$-0.350246\pi$
$277$	−7.29072	−	7.29072i	−0.438057	−	0.438057i	−0.891358	+	0.453301i	$0.850246\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	25.1242i		−	1.49878i	−0.662127	−	0.749392i	$-0.730346\pi$
$281$		−	25.1242i		−	1.49878i	0.662127	−	0.749392i	$-0.269654\pi$
$282$		0			0
$283$	−4.01333	+	4.01333i	−0.238568	+	0.238568i	−0.816257	−	0.577689i	$-0.803955\pi$
$283$	−4.01333	+	4.01333i	−0.238568	+	0.238568i	0.577689	+	0.816257i	$0.303955\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	8.32072	−	8.32072i	0.491157	−	0.491157i
$288$		0			0
$289$		−	11.6803i		−	0.687079i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	0.824263	+	0.824263i	0.0481539	+	0.0481539i	0.730774	−	0.682620i	$-0.239159\pi$
$293$	0.824263	+	0.824263i	0.0481539	+	0.0481539i	−0.682620	+	0.730774i	$0.739159\pi$
$294$		0			0
$295$	30.2117	+	7.50652i	1.75899	+	0.437047i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	28.8401			1.66787
$300$		0			0
$301$	−22.7792			−1.31297
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−5.70928	−	1.41855i	−0.326912	−	0.0812260i
$306$		0			0
$307$	−7.32064	+	7.32064i	−0.417811	+	0.417811i	−0.884449	−	0.466638i	$-0.845465\pi$
$307$	−7.32064	+	7.32064i	−0.417811	+	0.417811i	0.466638	+	0.884449i	$0.345465\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	11.3835			0.645498			0.322749	−	0.946485i	$-0.395393\pi$
$311$	11.3835			0.645498			0.322749	+	0.946485i	$0.395393\pi$
$312$		0			0
$313$	−5.09890	+	5.09890i	−0.288207	+	0.288207i	−0.836371	−	0.548164i	$-0.815327\pi$
$313$	−5.09890	+	5.09890i	−0.288207	+	0.288207i	0.548164	+	0.836371i	$0.315327\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−17.1514	+	17.1514i	−0.963318	+	0.963318i	−0.999351	−	0.0360322i	$-0.988528\pi$
$317$	−17.1514	+	17.1514i	−0.963318	+	0.963318i	0.0360322	+	0.999351i	$0.488528\pi$
$318$		0			0
$319$	5.60116			0.313605
$320$		0			0
$321$		0			0
$322$		0			0
$323$	0.880022	−	10.0149i	0.0489657	−	0.557246i
$324$		0			0
$325$	−9.22600	−	30.0022i	−0.511766	−	1.66422i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	3.94214i		−	0.217337i
$330$		0			0
$331$			8.36440i			0.459749i	0.973220	+	0.229875i	$0.0738316\pi$
$331$			8.36440i			0.459749i	−0.973220	+	0.229875i	$0.926168\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−20.9047	+	12.5839i	−1.14214	+	0.687534i
$336$		0			0
$337$	24.6579	−	24.6579i	1.34320	−	1.34320i	0.450351	−	0.892852i	$-0.351299\pi$
$337$	24.6579	−	24.6579i	1.34320	−	1.34320i	0.892852	−	0.450351i	$-0.148701\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			12.1550i			0.658230i
$342$		0			0
$343$	9.94214	−	9.94214i	0.536825	−	0.536825i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	9.68753	+	9.68753i	0.520054	+	0.520054i	0.917588	−	0.397534i	$-0.130134\pi$
$347$	9.68753	+	9.68753i	0.520054	+	0.520054i	−0.397534	+	0.917588i	$0.630134\pi$
$348$		0			0
$349$		−	0.130094i		−	0.00696375i	−0.999994	−	0.00348187i	$-0.998892\pi$
$349$		−	0.130094i		−	0.00696375i	0.999994	−	0.00348187i	$-0.00110832\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−17.8865	+	17.8865i	−0.952005	+	0.952005i	−0.998900	−	0.0468948i	$-0.985067\pi$
$353$	−17.8865	+	17.8865i	−0.952005	+	0.952005i	0.0468948	+	0.998900i	$0.485067\pi$
$354$		0			0
$355$	0.814197	−	3.27692i	0.0432131	−	0.173921i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	15.2579i		−	0.805282i	−0.915358	−	0.402641i	$-0.868092\pi$
$359$		−	15.2579i		−	0.805282i	0.915358	−	0.402641i	$-0.131908\pi$
$360$		0			0
$361$	−3.31351	+	18.7088i	−0.174395	+	0.984676i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	23.5958	−	14.2039i	1.23506	−	0.743468i
$366$		0			0
$367$	23.2401	+	23.2401i	1.21312	+	1.21312i	0.969995	+	0.243126i	$0.0781729\pi$
$367$	23.2401	+	23.2401i	1.21312	+	1.21312i	0.243126	+	0.969995i	$0.421827\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			23.7526i			1.23318i
$372$		0			0
$373$	−3.70586	−	3.70586i	−0.191882	−	0.191882i	0.604627	−	0.796509i	$-0.293322\pi$
$373$	−3.70586	−	3.70586i	−0.191882	−	0.191882i	−0.796509	+	0.604627i	$0.793322\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−14.5464	−	14.5464i	−0.749177	−	0.749177i
$378$		0			0
$379$	9.57392			0.491779			0.245890	−	0.969298i	$-0.420920\pi$
$379$	9.57392			0.491779			0.245890	+	0.969298i	$0.420920\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	8.01648	+	8.01648i	0.409623	+	0.409623i	0.881607	−	0.471984i	$-0.156462\pi$
$383$	8.01648	+	8.01648i	0.409623	+	0.409623i	−0.471984	+	0.881607i	$0.656462\pi$
$384$		0			0
$385$	−6.04945	−	10.0494i	−0.308308	−	0.512167i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	30.5692i		−	1.54992i	−0.632011	−	0.774959i	$-0.717770\pi$
$389$		−	30.5692i		−	1.54992i	0.632011	−	0.774959i	$-0.282230\pi$
$390$		0			0
$391$	−10.5958			−0.535854
$392$		0			0
$393$		0			0
$394$		0			0
$395$	10.3881	+	2.58108i	0.522683	+	0.129868i
$396$		0			0
$397$	−15.0494	−	15.0494i	−0.755310	−	0.755310i	0.220155	−	0.975465i	$-0.429344\pi$
$397$	−15.0494	−	15.0494i	−0.755310	−	0.755310i	−0.975465	+	0.220155i	$0.929344\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			2.06740i			0.103241i	0.998667	+	0.0516205i	$0.0164386\pi$
$401$			2.06740i			0.103241i	−0.998667	+	0.0516205i	$0.983561\pi$
$402$		0			0
$403$	31.5669	−	31.5669i	1.57246	−	1.57246i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	0.219503	−	0.219503i	0.0108804	−	0.0108804i
$408$		0			0
$409$	−14.6177			−0.722800			−0.361400	−	0.932411i	$-0.617701\pi$
$409$	−14.6177			−0.722800			−0.361400	+	0.932411i	$0.617701\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−30.2117	−	30.2117i	−1.48662	−	1.48662i
$414$		0			0
$415$	17.5041	−	10.5369i	0.859243	−	0.517237i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			3.29299i			0.160873i	0.996760	+	0.0804366i	$0.0256315\pi$
$419$			3.29299i			0.160873i	−0.996760	+	0.0804366i	$0.974369\pi$
$420$		0			0
$421$		−	27.6306i		−	1.34663i	−0.739354	−	0.673317i	$-0.764869\pi$
$421$		−	27.6306i		−	1.34663i	0.739354	−	0.673317i	$-0.235131\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	3.38962	+	11.0228i	0.164421	+	0.534684i
$426$		0			0
$427$	5.70928	+	5.70928i	0.276291	+	0.276291i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	34.3028i		−	1.65231i	−0.563445	−	0.826154i	$-0.690524\pi$
$431$		−	34.3028i		−	1.65231i	0.563445	−	0.826154i	$-0.309476\pi$
$432$		0			0
$433$	4.69588	+	4.69588i	0.225669	+	0.225669i	0.810881	−	0.585211i	$-0.198988\pi$
$433$	4.69588	+	4.69588i	0.225669	+	0.225669i	−0.585211	+	0.810881i	$0.698988\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	19.9480	+	1.75285i	0.954243	+	0.0838502i
$438$		0			0
$439$	−12.0165			−0.573517			−0.286758	−	0.958003i	$-0.592578\pi$
$439$	−12.0165			−0.573517			−0.286758	+	0.958003i	$0.592578\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	16.8999	−	16.8999i	0.802938	−	0.802938i	−0.180616	−	0.983554i	$-0.557809\pi$
$443$	16.8999	−	16.8999i	0.802938	−	0.802938i	0.983554	+	0.180616i	$0.0578092\pi$
$444$		0			0
$445$	11.5976	+	2.88160i	0.549781	+	0.136601i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	39.6034			1.86900			0.934501	−	0.355961i	$-0.115846\pi$
$449$	39.6034			1.86900			0.934501	+	0.355961i	$0.115846\pi$
$450$		0			0
$451$		−	6.55384i		−	0.308608i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−10.3881	+	41.8093i	−0.487003	+	1.96005i
$456$		0			0
$457$	−10.5174	−	10.5174i	−0.491985	−	0.491985i	0.416946	−	0.908931i	$-0.363100\pi$
$457$	−10.5174	−	10.5174i	−0.491985	−	0.491985i	−0.908931	+	0.416946i	$0.863100\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−25.6163			−1.19307			−0.596536	−	0.802586i	$-0.703457\pi$
$461$	−25.6163			−1.19307			−0.596536	+	0.802586i	$0.703457\pi$
$462$		0			0
$463$	9.58864	−	9.58864i	0.445622	−	0.445622i	−0.448274	−	0.893896i	$-0.647961\pi$
$463$	9.58864	−	9.58864i	0.445622	−	0.445622i	0.893896	+	0.448274i	$0.147961\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	9.27513	+	9.27513i	0.429202	+	0.429202i	0.888356	−	0.459155i	$-0.151848\pi$
$467$	9.27513	+	9.27513i	0.429202	+	0.429202i	−0.459155	+	0.888356i	$0.651848\pi$
$468$		0			0
$469$	33.4886			1.54636
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−8.97107	+	8.97107i	−0.412490	+	0.412490i
$474$		0			0
$475$	−4.55792	−	21.3126i	−0.209132	−	0.977887i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	32.6719i		−	1.49282i	−0.665487	−	0.746409i	$-0.731776\pi$
$479$		−	32.6719i		−	1.49282i	0.665487	−	0.746409i	$-0.268224\pi$
$480$		0			0
$481$	−1.14011			−0.0519846
$482$		0			0
$483$		0			0
$484$		0			0
$485$	27.7964	−	16.7325i	1.26217	−	0.759785i
$486$		0			0
$487$	0.347922	−	0.347922i	0.0157658	−	0.0157658i	−0.699180	−	0.714946i	$-0.746451\pi$
$487$	0.347922	−	0.347922i	0.0157658	−	0.0157658i	0.714946	+	0.699180i	$0.246451\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−1.17727			−0.0531297			−0.0265648	−	0.999647i	$-0.508457\pi$
$491$	−1.17727			−0.0531297			−0.0265648	+	0.999647i	$0.508457\pi$
$492$		0			0
$493$	5.34432	+	5.34432i	0.240696	+	0.240696i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−3.27692	+	3.27692i	−0.146990	+	0.146990i
$498$		0			0
$499$		−	18.4163i		−	0.824426i	−0.911087	−	0.412213i	$-0.864756\pi$
$499$		−	18.4163i		−	0.824426i	0.911087	−	0.412213i	$-0.135244\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−10.9493	+	10.9493i	−0.488206	+	0.488206i	−0.907740	−	0.419533i	$-0.862194\pi$
$503$	−10.9493	+	10.9493i	−0.488206	+	0.488206i	0.419533	+	0.907740i	$0.362194\pi$
$504$		0			0
$505$	−33.2267	−	8.25565i	−1.47857	−	0.367372i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	29.1406			1.29164			0.645818	−	0.763491i	$-0.276516\pi$
$509$	29.1406			1.29164			0.645818	+	0.763491i	$0.276516\pi$
$510$		0			0
$511$	−37.7998			−1.67216
$512$		0			0
$513$		0			0
$514$		0			0
$515$	21.7189	+	36.0797i	0.957047	+	1.58986i
$516$		0			0
$517$	−1.55252	−	1.55252i	−0.0682797	−	0.0682797i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	36.9040i		−	1.61679i	−0.588638	−	0.808397i	$-0.700336\pi$
$521$		−	36.9040i		−	1.61679i	0.588638	−	0.808397i	$-0.299664\pi$
$522$		0			0
$523$	6.72594	+	6.72594i	0.294105	+	0.294105i	0.838699	−	0.544595i	$-0.183316\pi$
$523$	6.72594	+	6.72594i	0.294105	+	0.294105i	−0.544595	+	0.838699i	$0.683316\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−11.5976	+	11.5976i	−0.505201	+	0.505201i
$528$		0			0
$529$			1.89496i			0.0823896i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−17.0205	+	17.0205i	−0.737241	+	0.737241i
$534$		0			0
$535$	−7.23962	+	4.35802i	−0.312996	+	0.188414i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			4.13397i			0.178063i
$540$		0			0
$541$	32.5197			1.39813			0.699066	−	0.715057i	$-0.253599\pi$
$541$	32.5197			1.39813			0.699066	+	0.715057i	$0.253599\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−13.3645	−	3.32060i	−0.572473	−	0.142239i
$546$		0			0
$547$	−23.2289	+	23.2289i	−0.993196	+	0.993196i	−0.999977	−	0.00678109i	$-0.997841\pi$
$547$	−23.2289	+	23.2289i	−0.993196	+	0.993196i	0.00678109	+	0.999977i	$0.497841\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−9.17727	−	10.9455i	−0.390965	−	0.466293i
$552$		0			0
$553$	−10.3881	−	10.3881i	−0.441748	−	0.441748i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	5.14834	+	5.14834i	0.218142	+	0.218142i	0.807715	−	0.589573i	$-0.200704\pi$
$557$	5.14834	+	5.14834i	0.218142	+	0.218142i	−0.589573	+	0.807715i	$0.700704\pi$
$558$		0			0
$559$	46.5963			1.97081
$560$		0			0
$561$		0			0
$562$		0			0
$563$	11.4692	+	11.4692i	0.483370	+	0.483370i	0.906206	−	0.422836i	$-0.138965\pi$
$563$	11.4692	+	11.4692i	0.483370	+	0.483370i	−0.422836	+	0.906206i	$0.638965\pi$
$564$		0			0
$565$	18.8373	+	31.2928i	0.792489	+	1.31650i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−31.8602			−1.33565			−0.667825	−	0.744319i	$-0.732774\pi$
$569$	−31.8602			−1.33565			−0.667825	+	0.744319i	$0.732774\pi$
$570$		0			0
$571$	23.5981			0.987549			0.493775	−	0.869590i	$-0.335617\pi$
$571$	23.5981			0.987549			0.493775	+	0.869590i	$0.335617\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−21.9555	+	6.75154i	−0.915606	+	0.281559i
$576$		0			0
$577$	24.2762	+	24.2762i	1.01063	+	1.01063i	0.999943	+	0.0106873i	$0.00340195\pi$
$577$	24.2762	+	24.2762i	1.01063	+	1.01063i	0.0106873	+	0.999943i	$0.496598\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−28.0410			−1.16334
$582$		0			0
$583$	9.35442	+	9.35442i	0.387420	+	0.387420i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−18.7165	−	18.7165i	−0.772511	−	0.772511i	0.206034	−	0.978545i	$-0.433944\pi$
$587$	−18.7165	−	18.7165i	−0.772511	−	0.772511i	−0.978545	+	0.206034i	$0.933944\pi$
$588$		0			0
$589$	23.7526	−	19.9155i	0.978710	−	0.820603i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−1.26794	+	1.26794i	−0.0520680	+	0.0520680i	−0.732661	−	0.680593i	$-0.761722\pi$
$593$	−1.26794	+	1.26794i	−0.0520680	+	0.0520680i	0.680593	+	0.732661i	$0.261722\pi$
$594$		0			0
$595$	3.81658	−	15.3607i	0.156465	−	0.629727i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−10.5502			−0.431068			−0.215534	−	0.976496i	$-0.569149\pi$
$599$	−10.5502			−0.431068			−0.215534	+	0.976496i	$0.569149\pi$
$600$		0			0
$601$		−	38.8329i		−	1.58403i	−0.610503	−	0.792014i	$-0.709033\pi$
$601$		−	38.8329i		−	1.58403i	0.610503	−	0.792014i	$-0.290967\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	17.5308	+	4.35577i	0.712727	+	0.177087i
$606$		0			0
$607$	−7.97280	+	7.97280i	−0.323606	+	0.323606i	−0.850149	−	0.526543i	$-0.823488\pi$
$607$	−7.97280	+	7.97280i	−0.323606	+	0.323606i	0.526543	+	0.850149i	$0.323488\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			8.06388i			0.326230i
$612$		0			0
$613$	−18.1194	+	18.1194i	−0.731836	+	0.731836i	−0.970983	−	0.239147i	$-0.923132\pi$
$613$	−18.1194	+	18.1194i	−0.731836	+	0.731836i	0.239147	+	0.970983i	$0.423132\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	16.2123	+	16.2123i	0.652685	+	0.652685i	0.953639	−	0.300954i	$-0.0973051\pi$
$617$	16.2123	+	16.2123i	0.652685	+	0.652685i	−0.300954	+	0.953639i	$0.597305\pi$
$618$		0			0
$619$			8.73433i			0.351062i	0.984474	+	0.175531i	$0.0561643\pi$
$619$			8.73433i			0.351062i	−0.984474	+	0.175531i	$0.943836\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−11.5976	−	11.5976i	−0.464650	−	0.464650i
$624$		0			0
$625$	14.0472	+	20.6803i	0.561887	+	0.827214i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	0.418876			0.0167017
$630$		0			0
$631$	29.7093			1.18271			0.591354	−	0.806412i	$-0.298594\pi$
$631$	29.7093			1.18271			0.591354	+	0.806412i	$0.298594\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	9.22600	−	5.55376i	0.366123	−	0.220394i
$636$		0			0
$637$	10.7360	−	10.7360i	0.425377	−	0.425377i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			40.2993i			1.59173i	0.605477	+	0.795863i	$0.292982\pi$
$641$			40.2993i			1.59173i	−0.605477	+	0.795863i	$0.707018\pi$
$642$		0			0
$643$	1.57426	−	1.57426i	0.0620828	−	0.0620828i	−0.675384	−	0.737467i	$-0.736022\pi$
$643$	1.57426	−	1.57426i	0.0620828	−	0.0620828i	0.737467	+	0.675384i	$0.236022\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	5.79484	+	5.79484i	0.227819	+	0.227819i	0.811781	−	0.583962i	$-0.198498\pi$
$647$	5.79484	+	5.79484i	0.227819	+	0.227819i	−0.583962	+	0.811781i	$0.698498\pi$
$648$		0			0
$649$	−23.7963			−0.934087
$650$		0			0
$651$		0			0
$652$		0			0
$653$	21.6537	−	21.6537i	0.847374	−	0.847374i	−0.142431	−	0.989805i	$-0.545492\pi$
$653$	21.6537	−	21.6537i	0.847374	−	0.847374i	0.989805	+	0.142431i	$0.0454918\pi$
$654$		0			0
$655$	33.6742	+	8.36683i	1.31576	+	0.326919i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−16.9420			−0.659965			−0.329983	−	0.943987i	$-0.607043\pi$
$659$	−16.9420			−0.659965			−0.329983	+	0.943987i	$0.607043\pi$
$660$		0			0
$661$			27.1916i			1.05763i	0.848737	+	0.528815i	$0.177364\pi$
$661$			27.1916i			1.05763i	−0.848737	+	0.528815i	$0.822636\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−9.72631	+	28.2871i	−0.377170	+	1.09693i
$666$		0			0
$667$	−10.6450	+	10.6450i	−0.412174	+	0.412174i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	4.49693			0.173602
$672$		0			0
$673$	1.97632	+	1.97632i	0.0761814	+	0.0761814i	0.744171	−	0.667989i	$-0.232845\pi$
$673$	1.97632	+	1.97632i	0.0761814	+	0.0761814i	−0.667989	+	0.744171i	$0.732845\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−21.7189	+	21.7189i	−0.834723	+	0.834723i	−0.988159	−	0.153435i	$-0.950966\pi$
$677$	−21.7189	+	21.7189i	−0.834723	+	0.834723i	0.153435	+	0.988159i	$0.450966\pi$
$678$		0			0
$679$	−44.5289			−1.70886
$680$		0			0
$681$		0			0
$682$		0			0
$683$	22.3710	+	22.3710i	0.856003	+	0.856003i	0.990864	−	0.134861i	$-0.0430589\pi$
$683$	22.3710	+	22.3710i	0.856003	+	0.856003i	−0.134861	+	0.990864i	$0.543059\pi$
$684$		0			0
$685$	−23.1217	−	38.4101i	−0.883434	−	1.46758i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	48.5874i		−	1.85103i
$690$		0			0
$691$	13.7093			0.521525			0.260763	−	0.965403i	$-0.416026\pi$
$691$	13.7093			0.521525			0.260763	+	0.965403i	$0.416026\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	5.90110	−	23.7503i	0.223842	−	0.900901i
$696$		0			0
$697$	6.25332	−	6.25332i	0.236861	−	0.236861i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−47.8225			−1.80623			−0.903116	−	0.429396i	$-0.858727\pi$
$701$	−47.8225			−1.80623			−0.903116	+	0.429396i	$0.858727\pi$
$702$		0			0
$703$	−0.788588	−	0.0692940i	−0.0297422	−	0.00261347i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	33.2267	+	33.2267i	1.24962	+	1.24962i
$708$		0			0
$709$			0.0143758i			0.000539896i	1.00000		0.000269948i	$8.59272e-5\pi$
$709$			0.0143758i			0.000539896i	−1.00000		0.000269948i	$0.999914\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−23.1005	−	23.1005i	−0.865120	−	0.865120i
$714$		0			0
$715$	12.3745	+	20.5567i	0.462780	+	0.768778i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			11.7237i			0.437218i	0.975812	+	0.218609i	$0.0701520\pi$
$719$			11.7237i			0.437218i	−0.975812	+	0.218609i	$0.929848\pi$
$720$		0			0
$721$		−	57.7986i		−	2.15253i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	14.4792	+	7.66856i	0.537745	+	0.284803i
$726$		0			0
$727$	−3.87936	−	3.87936i	−0.143878	−	0.143878i	0.631499	−	0.775377i	$-0.282440\pi$
$727$	−3.87936	−	3.87936i	−0.143878	−	0.143878i	−0.775377	+	0.631499i	$0.782440\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−17.1194			−0.633184
$732$		0			0
$733$	3.97948	−	3.97948i	0.146985	−	0.146985i	−0.629784	−	0.776770i	$-0.716857\pi$
$733$	3.97948	−	3.97948i	0.146985	−	0.146985i	0.776770	+	0.629784i	$0.216857\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	13.1887	−	13.1887i	0.485812	−	0.485812i
$738$		0			0
$739$		−	30.6270i		−	1.12663i	−0.826241	−	0.563317i	$-0.809525\pi$
$739$		−	30.6270i		−	1.12663i	0.826241	−	0.563317i	$-0.190475\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	7.93546	+	7.93546i	0.291124	+	0.291124i	0.837524	−	0.546400i	$-0.184002\pi$
$743$	7.93546	+	7.93546i	0.291124	+	0.291124i	−0.546400	+	0.837524i	$0.684002\pi$
$744$		0			0
$745$	0.286849	−	1.15449i	0.0105093	−	0.0422972i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	11.5976			0.423769
$750$		0			0
$751$			7.92540i			0.289202i	0.989490	+	0.144601i	$0.0461898\pi$
$751$			7.92540i			0.289202i	−0.989490	+	0.144601i	$0.953810\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	6.41536	−	25.8200i	0.233479	−	0.939687i
$756$		0			0
$757$	2.58372	+	2.58372i	0.0939068	+	0.0939068i	0.752500	−	0.658593i	$-0.228848\pi$
$757$	2.58372	+	2.58372i	0.0939068	+	0.0939068i	−0.658593	+	0.752500i	$0.728848\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	33.4680			1.21321			0.606607	−	0.795002i	$-0.292530\pi$
$761$	33.4680			1.21321			0.606607	+	0.795002i	$0.292530\pi$
$762$		0			0
$763$	13.3645	+	13.3645i	0.483828	+	0.483828i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	61.7998	+	61.7998i	2.23146	+	2.23146i
$768$		0			0
$769$		−	49.4557i		−	1.78342i	−0.452609	−	0.891709i	$-0.649506\pi$
$769$		−	49.4557i		−	1.78342i	0.452609	−	0.891709i	$-0.350494\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	37.0698	+	37.0698i	1.33331	+	1.33331i	0.902394	+	0.430913i	$0.141808\pi$
$773$	37.0698	+	37.0698i	1.33331	+	1.33331i	0.430913	+	0.902394i	$0.358192\pi$
$774$		0			0
$775$	−16.6414	+	31.4212i	−0.597778	+	1.12868i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−12.8072	+	10.7382i	−0.458864	+	0.384736i
$780$		0			0
$781$			2.58108i			0.0923582i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	9.69472	+	16.1050i	0.346019	+	0.574814i
$786$		0			0
$787$	13.3981	−	13.3981i	0.477592	−	0.477592i	−0.426769	−	0.904361i	$-0.640348\pi$
$787$	13.3981	−	13.3981i	0.477592	−	0.477592i	0.904361	+	0.426769i	$0.140348\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	50.1300i		−	1.78242i
$792$		0			0
$793$	−11.6787	−	11.6787i	−0.414721	−	0.414721i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	5.61122	−	5.61122i	0.198760	−	0.198760i	−0.600708	−	0.799468i	$-0.705115\pi$
$797$	5.61122	−	5.61122i	0.198760	−	0.198760i	0.799468	+	0.600708i	$0.205115\pi$
$798$		0			0
$799$		−	2.96266i		−	0.104811i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−14.8865	+	14.8865i	−0.525335	+	0.525335i
$804$		0			0
$805$	30.5958	+	7.60197i	1.07836	+	0.267934i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	44.3812i		−	1.56036i	−0.625555	−	0.780180i	$-0.715127\pi$
$809$		−	44.3812i		−	1.56036i	0.625555	−	0.780180i	$-0.284873\pi$
$810$		0			0
$811$		−	19.2662i		−	0.676528i	−0.941051	−	0.338264i	$-0.890160\pi$
$811$		−	19.2662i		−	0.676528i	0.941051	−	0.338264i	$-0.109840\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	32.6092	−	19.6297i	1.14225	−	0.687598i
$816$		0			0
$817$	32.2295	+	2.83204i	1.12757	+	0.0990805i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	5.05172			0.176306			0.0881530	−	0.996107i	$-0.471904\pi$
$821$	5.05172			0.176306			0.0881530	+	0.996107i	$0.471904\pi$
$822$		0			0
$823$	6.53305	−	6.53305i	0.227728	−	0.227728i	−0.584015	−	0.811743i	$-0.698519\pi$
$823$	6.53305	−	6.53305i	0.227728	−	0.227728i	0.811743	+	0.584015i	$0.198519\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−35.9413	+	35.9413i	−1.24980	+	1.24980i	−0.293992	+	0.955808i	$0.594984\pi$
$827$	−35.9413	+	35.9413i	−1.24980	+	1.24980i	−0.955808	+	0.293992i	$0.905016\pi$
$828$		0			0
$829$	−3.57744			−0.124250			−0.0621249	−	0.998068i	$-0.519788\pi$
$829$	−3.57744			−0.124250			−0.0621249	+	0.998068i	$0.519788\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−3.94441	+	3.94441i	−0.136666	+	0.136666i
$834$		0			0
$835$	23.7052	−	14.2698i	0.820354	−	0.493827i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	21.0768			0.727651			0.363825	−	0.931467i	$-0.381471\pi$
$839$	21.0768			0.727651			0.363825	+	0.931467i	$0.381471\pi$
$840$		0			0
$841$	−18.2618			−0.629717
$842$		0			0
$843$		0			0
$844$		0			0
$845$	14.2401	−	57.3123i	0.489873	−	1.97160i
$846$		0			0
$847$	−17.5308	−	17.5308i	−0.602365	−	0.602365i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			0.834328i			0.0286004i
$852$		0			0
$853$	10.1773	−	10.1773i	0.348463	−	0.348463i	−0.511074	−	0.859537i	$-0.670752\pi$
$853$	10.1773	−	10.1773i	0.348463	−	0.348463i	0.859537	+	0.511074i	$0.170752\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	29.2691	−	29.2691i	0.999812	−	0.999812i	−0.000187872	−	1.00000i	$-0.500060\pi$
$857$	29.2691	−	29.2691i	0.999812	−	0.999812i	1.00000		0.000187872i	$5.98016e-5\pi$
$858$		0			0
$859$		−	21.5318i		−	0.734656i	−0.930091	−	0.367328i	$-0.880273\pi$
$859$		−	21.5318i		−	0.734656i	0.930091	−	0.367328i	$-0.119727\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	0.942616	+	0.942616i	0.0320870	+	0.0320870i	0.722968	−	0.690881i	$-0.242777\pi$
$863$	0.942616	+	0.942616i	0.0320870	+	0.0320870i	−0.690881	+	0.722968i	$0.742777\pi$
$864$		0			0
$865$	1.89530	+	3.14850i	0.0644421	+	0.107052i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−8.18223			−0.277563
$870$		0			0
$871$	−68.5029			−2.32113
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1.87936	−	34.2606i	−0.0635340	−	1.15822i
$876$		0			0
$877$	10.8171	−	10.8171i	0.365266	−	0.365266i	−0.500481	−	0.865747i	$-0.666844\pi$
$877$	10.8171	−	10.8171i	0.365266	−	0.365266i	0.865747	+	0.500481i	$0.166844\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−14.6309			−0.492927			−0.246464	−	0.969152i	$-0.579269\pi$
$881$	−14.6309			−0.492927			−0.246464	+	0.969152i	$0.579269\pi$
$882$		0			0
$883$	9.87936	−	9.87936i	0.332467	−	0.332467i	−0.521056	−	0.853523i	$-0.674462\pi$
$883$	9.87936	−	9.87936i	0.332467	−	0.332467i	0.853523	+	0.521056i	$0.174462\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−30.4211	+	30.4211i	−1.02144	+	1.02144i	−0.0216761	+	0.999765i	$0.506900\pi$
$887$	−30.4211	+	30.4211i	−1.02144	+	1.02144i	−0.999765	+	0.0216761i	$0.993100\pi$
$888$		0			0
$889$	−14.7798			−0.495697
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−0.490108	+	5.57759i	−0.0164008	+	0.186647i
$894$		0			0
$895$	55.4744	+	13.7834i	1.85430	+	0.460728i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			23.3028i			0.777193i
$900$		0			0
$901$			17.8510i			0.594702i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−7.06752	+	28.4448i	−0.234932	+	0.945537i
$906$		0			0
$907$	−1.41896	+	1.41896i	−0.0471157	+	0.0471157i	−0.730272	−	0.683156i	$-0.760607\pi$
$907$	−1.41896	+	1.41896i	−0.0471157	+	0.0471157i	0.683156	+	0.730272i	$0.260607\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	49.4779i		−	1.63928i	−0.572882	−	0.819638i	$-0.694175\pi$
$911$		−	49.4779i		−	1.63928i	0.572882	−	0.819638i	$-0.305825\pi$
$912$		0			0
$913$	−11.0433	+	11.0433i	−0.365480	+	0.365480i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−33.6742	−	33.6742i	−1.11202	−	1.11202i
$918$		0			0
$919$			7.44521i			0.245595i	0.992432	+	0.122797i	$0.0391866\pi$
$919$			7.44521i			0.245595i	−0.992432	+	0.122797i	$0.960813\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	6.70313	−	6.70313i	0.220636	−	0.220636i
$924$		0			0
$925$	0.867947	−	0.266903i	0.0285379	−	0.00877571i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			24.1133i			0.791131i	0.918438	+	0.395565i	$0.129451\pi$
$929$			24.1133i			0.791131i	−0.918438	+	0.395565i	$0.870549\pi$
$930$		0			0
$931$	8.07838	−	6.77334i	0.264758	−	0.221987i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−4.54638	−	7.55252i	−0.148682	−	0.246994i
$936$		0			0
$937$	4.23901	+	4.23901i	0.138482	+	0.138482i	0.772950	−	0.634467i	$-0.218780\pi$
$937$	4.23901	+	4.23901i	0.138482	+	0.138482i	−0.634467	+	0.772950i	$0.718780\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			49.5727i			1.61602i	0.589167	+	0.808012i	$0.299456\pi$
$941$			49.5727i			1.61602i	−0.589167	+	0.808012i	$0.700544\pi$
$942$		0			0
$943$	12.4555	+	12.4555i	0.405608	+	0.405608i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−18.8276	−	18.8276i	−0.611816	−	0.611816i	0.331603	−	0.943419i	$-0.392411\pi$
$947$	−18.8276	−	18.8276i	−0.611816	−	0.611816i	−0.943419	+	0.331603i	$0.892411\pi$
$948$		0			0
$949$	77.3216			2.50997
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−24.4384	−	24.4384i	−0.791638	−	0.791638i	0.190122	−	0.981760i	$-0.439112\pi$
$953$	−24.4384	−	24.4384i	−0.791638	−	0.791638i	−0.981760	+	0.190122i	$0.939112\pi$
$954$		0			0
$955$	−5.75872	−	1.43084i	−0.186348	−	0.0463008i
$956$		0			0
$957$		0			0
$958$		0			0
$959$			61.5318i			1.98697i
$960$		0			0
$961$	−19.5692			−0.631263
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−14.2698	−	23.7052i	−0.459361	−	0.763099i
$966$		0			0
$967$	21.3207	+	21.3207i	0.685627	+	0.685627i	0.961262	−	0.275635i	$-0.0888881\pi$
$967$	21.3207	+	21.3207i	0.685627	+	0.685627i	−0.275635	+	0.961262i	$0.588888\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	35.3940i		−	1.13585i	−0.823082	−	0.567923i	$-0.807747\pi$
$971$		−	35.3940i		−	1.13585i	0.823082	−	0.567923i	$-0.192253\pi$
$972$		0			0
$973$	−23.7503	+	23.7503i	−0.761400	+	0.761400i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−34.3939	+	34.3939i	−1.10036	+	1.10036i	−0.105991	+	0.994367i	$0.533801\pi$
$977$	−34.3939	+	34.3939i	−1.10036	+	1.10036i	−0.994367	+	0.105991i	$0.966199\pi$
$978$		0			0
$979$	−9.13492			−0.291953
$980$		0			0
$981$		0			0
$982$		0			0
$983$	18.1041	+	18.1041i	0.577430	+	0.577430i	0.934195	−	0.356764i	$-0.116120\pi$
$983$	18.1041	+	18.1041i	0.577430	+	0.577430i	−0.356764	+	0.934195i	$0.616120\pi$
$984$		0			0
$985$	−7.28685	−	12.1050i	−0.232178	−	0.385699i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	34.0989i		−	1.08428i
$990$		0			0
$991$			30.6070i			0.972263i	0.873886	+	0.486132i	$0.161592\pi$
$991$			30.6070i			0.972263i	−0.873886	+	0.486132i	$0.838408\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	8.59583	−	34.5958i	0.272506	−	1.09676i
$996$		0			0
$997$	−34.3074	−	34.3074i	−1.08653	−	1.08653i	−0.995884	−	0.0906417i	$-0.971108\pi$
$997$	−34.3074	−	34.3074i	−1.08653	−	1.08653i	−0.0906417	−	0.995884i	$-0.528892\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3420.2.bb.d.37.1		12
3.2	odd	2		380.2.l.b.37.5	yes	12
5.3	odd	4	inner	3420.2.bb.d.2773.2		12
15.2	even	4		1900.2.l.b.493.5		12
15.8	even	4		380.2.l.b.113.2	yes	12
15.14	odd	2		1900.2.l.b.1557.2		12
19.18	odd	2	inner	3420.2.bb.d.37.2		12
57.56	even	2		380.2.l.b.37.2	✓	12
95.18	even	4	inner	3420.2.bb.d.2773.1		12
285.113	odd	4		380.2.l.b.113.5	yes	12
285.227	odd	4		1900.2.l.b.493.2		12
285.284	even	2		1900.2.l.b.1557.5		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.2.l.b.37.2	✓	12	57.56	even	2
380.2.l.b.37.5	yes	12	3.2	odd	2
380.2.l.b.113.2	yes	12	15.8	even	4
380.2.l.b.113.5	yes	12	285.113	odd	4
1900.2.l.b.493.2		12	285.227	odd	4
1900.2.l.b.493.5		12	15.2	even	4
1900.2.l.b.1557.2		12	15.14	odd	2
1900.2.l.b.1557.5		12	285.284	even	2
3420.2.bb.d.37.1		12	1.1	even	1	trivial
3420.2.bb.d.37.2		12	19.18	odd	2	inner
3420.2.bb.d.2773.1		12	95.18	even	4	inner
3420.2.bb.d.2773.2		12	5.3	odd	4	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3420.bb (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.17009 - 0.539189i) q^{5} +(2.17009 + 2.17009i) q^{7} +O(q^{10})\)	\(q+(-2.17009 - 0.539189i) q^{5} +(2.17009 + 2.17009i) q^{7} +1.70928 q^{11} +(-4.43904 - 4.43904i) q^{13} +(1.63090 + 1.63090i) q^{17} +(-2.80058 - 3.34017i) q^{19} +(-3.24846 + 3.24846i) q^{23} +(4.41855 + 2.34017i) q^{25} +3.27692 q^{29} +7.11120i q^{31} +(-3.53919 - 5.87936i) q^{35} +(0.128419 - 0.128419i) q^{37} -3.83428i q^{41} +(-5.24846 + 5.24846i) q^{43} +(-0.908291 - 0.908291i) q^{47} +2.41855i q^{49} +(5.47274 + 5.47274i) q^{53} +(-3.70928 - 0.921622i) q^{55} -13.9219 q^{59} +2.63090 q^{61} +(7.23962 + 12.0266i) q^{65} +(7.71596 - 7.71596i) q^{67} +1.51004i q^{71} +(-8.70928 + 8.70928i) q^{73} +(3.70928 + 3.70928i) q^{77} -4.78696 q^{79} +(-6.46081 + 6.46081i) q^{83} +(-2.65983 - 4.41855i) q^{85} -5.34432 q^{89} -19.2662i q^{91} +(4.27652 + 8.75851i) q^{95} +(-10.2597 + 10.2597i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{5} + 4 q^{7}+O(q^{10}) \) 12 * q - 4 * q^5 + 4 * q^7	\( 12 q - 4 q^{5} + 4 q^{7} - 8 q^{11} + 4 q^{17} - 4 q^{23} - 4 q^{25} - 36 q^{35} - 28 q^{43} - 20 q^{47} - 16 q^{55} + 16 q^{61} - 76 q^{73} + 16 q^{77} - 84 q^{83} - 76 q^{85} + 16 q^{95}+O(q^{100}) \) 12 * q - 4 * q^5 + 4 * q^7 - 8 * q^11 + 4 * q^17 - 4 * q^23 - 4 * q^25 - 36 * q^35 - 28 * q^43 - 20 * q^47 - 16 * q^55 + 16 * q^61 - 76 * q^73 + 16 * q^77 - 84 * q^83 - 76 * q^85 + 16 * q^95

Introduction

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