LMFDB - Embedded newform 348.3.j.a.133.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [348,3,Mod(133,348)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(348, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 3]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("348.133");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 348 = 2^{2} \cdot 3 \cdot 29 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	348.j (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:	$9.48231319974$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 4 x^{19} + 8 x^{18} + 480 x^{17} + 8664 x^{16} - 4140 x^{15} + 62448 x^{14} + 2397840 x^{13} + \cdots + 1236544 $ x^20 - 4x^19 + 8x^18 + 480x^17 + 8664x^16 - 4140x^15 + 62448x^14 + 2397840x^13 + 25572348x^12 + 75112832x^11 + 113453416x^10 + 34892812x^9 + 32828793x^8 + 79741068x^7 + 115779336x^6 + 36690744x^5 + 11691156x^4 + 17421120x^3 + 25804928x^2 + 7988608*x + 1236544
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			133.7
Root			$0.531346 + 0.531346i$ of defining polynomial
Character	$\chi$	$=$	348.133
Dual form			348.3.j.a.157.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.22474 - 1.22474i) q^{3} -6.22487i q^{5} -8.38632 q^{7} -3.00000i q^{9} +O(q^{10})$	\(q+(1.22474 - 1.22474i) q^{3} -6.22487i q^{5} -8.38632 q^{7} -3.00000i q^{9} +(-11.2596 + 11.2596i) q^{11} -12.5547i q^{13} +(-7.62388 - 7.62388i) q^{15} +(-2.97621 + 2.97621i) q^{17} +(-19.0514 + 19.0514i) q^{19} +(-10.2711 + 10.2711i) q^{21} -0.990288 q^{23} -13.7491 q^{25} +(-3.67423 - 3.67423i) q^{27} +(23.7707 + 16.6118i) q^{29} +(9.11485 - 9.11485i) q^{31} +27.5803i q^{33} +52.2038i q^{35} +(-41.7531 - 41.7531i) q^{37} +(-15.3764 - 15.3764i) q^{39} +(-18.0867 - 18.0867i) q^{41} +(50.0423 - 50.0423i) q^{43} -18.6746 q^{45} +(-53.3333 - 53.3333i) q^{47} +21.3304 q^{49} +7.29020i q^{51} -42.5439 q^{53} +(70.0895 + 70.0895i) q^{55} +46.6662i q^{57} -61.9343 q^{59} +(36.0094 - 36.0094i) q^{61} +25.1590i q^{63} -78.1517 q^{65} -66.8704i q^{67} +(-1.21285 + 1.21285i) q^{69} +89.1659i q^{71} +(16.0513 + 16.0513i) q^{73} +(-16.8391 + 16.8391i) q^{75} +(94.4266 - 94.4266i) q^{77} +(74.3011 - 74.3011i) q^{79} -9.00000 q^{81} -80.2278 q^{83} +(18.5265 + 18.5265i) q^{85} +(49.4583 - 8.76782i) q^{87} +(3.67823 - 3.67823i) q^{89} +105.288i q^{91} -22.3267i q^{93} +(118.592 + 118.592i) q^{95} +(-101.725 - 101.725i) q^{97} +(33.7788 + 33.7788i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q+O(q^{10}) $ 20 * q	$ 20 q - 36 q^{15} + 8 q^{17} + 8 q^{19} - 72 q^{23} - 124 q^{25} + 36 q^{29} + 44 q^{31} + 32 q^{37} - 48 q^{39} - 56 q^{41} + 64 q^{43} + 48 q^{45} - 56 q^{47} + 44 q^{49} + 176 q^{53} + 168 q^{55} - 80 q^{59} - 64 q^{61} - 296 q^{65} - 48 q^{69} - 172 q^{73} + 48 q^{75} + 336 q^{77} - 244 q^{79} - 180 q^{81} - 368 q^{83} + 592 q^{85} + 12 q^{87} + 176 q^{89} - 184 q^{95} + 188 q^{97}+O(q^{100}) $ 20 * q - 36 * q^15 + 8 * q^17 + 8 * q^19 - 72 * q^23 - 124 * q^25 + 36 * q^29 + 44 * q^31 + 32 * q^37 - 48 * q^39 - 56 * q^41 + 64 * q^43 + 48 * q^45 - 56 * q^47 + 44 * q^49 + 176 * q^53 + 168 * q^55 - 80 * q^59 - 64 * q^61 - 296 * q^65 - 48 * q^69 - 172 * q^73 + 48 * q^75 + 336 * q^77 - 244 * q^79 - 180 * q^81 - 368 * q^83 + 592 * q^85 + 12 * q^87 + 176 * q^89 - 184 * q^95 + 188 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/348\mathbb{Z}\right)^\times$.

$n$	$175$	$205$	$233$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$	$1$

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
$3$	1.22474	−	1.22474i	0.408248	−	0.408248i
$3$	1.22474	−	1.22474i	0.408248	−	0.408248i
$4$		0			0
$5$		−	6.22487i		−	1.24497i	−0.782630	−	0.622487i	$-0.786122\pi$
$5$		−	6.22487i		−	1.24497i	0.782630	−	0.622487i	$-0.213878\pi$
$6$		0			0
$7$	−8.38632			−1.19805			−0.599023	−	0.800732i	$-0.704444\pi$
$7$	−8.38632			−1.19805			−0.599023	+	0.800732i	$0.704444\pi$
$8$		0			0
$9$		−	3.00000i		−	0.333333i
$10$		0			0
$11$	−11.2596	+	11.2596i	−1.02360	+	1.02360i	−0.0238844	+	0.999715i	$0.507603\pi$
$11$	−11.2596	+	11.2596i	−1.02360	+	1.02360i	−0.999715	+	0.0238844i	$0.992397\pi$
$12$		0			0
$13$		−	12.5547i		−	0.965750i	−0.875689	−	0.482875i	$-0.839593\pi$
$13$		−	12.5547i		−	0.965750i	0.875689	−	0.482875i	$-0.160407\pi$
$14$		0			0
$15$	−7.62388	−	7.62388i	−0.508259	−	0.508259i
$16$		0			0
$17$	−2.97621	+	2.97621i	−0.175071	+	0.175071i	−0.789203	−	0.614132i	$-0.789506\pi$
$17$	−2.97621	+	2.97621i	−0.175071	+	0.175071i	0.614132	+	0.789203i	$0.289506\pi$
$18$		0			0
$19$	−19.0514	+	19.0514i	−1.00270	+	1.00270i	−0.00270834	+	0.999996i	$0.500862\pi$
$19$	−19.0514	+	19.0514i	−1.00270	+	1.00270i	−0.999996	+	0.00270834i	$0.999138\pi$
$20$		0			0
$21$	−10.2711	+	10.2711i	−0.489100	+	0.489100i
$22$		0			0
$23$	−0.990288			−0.0430560			−0.0215280	−	0.999768i	$-0.506853\pi$
$23$	−0.990288			−0.0430560			−0.0215280	+	0.999768i	$0.506853\pi$
$24$		0			0
$25$	−13.7491			−0.549962
$26$		0			0
$27$	−3.67423	−	3.67423i	−0.136083	−	0.136083i
$28$		0			0
$29$	23.7707	+	16.6118i	0.819680	+	0.572822i
$29$	23.7707	+	16.6118i	0.819680	+	0.572822i
$30$		0			0
$31$	9.11485	−	9.11485i	0.294027	−	0.294027i	−0.544642	−	0.838669i	$-0.683334\pi$
$31$	9.11485	−	9.11485i	0.294027	−	0.294027i	0.838669	+	0.544642i	$0.183334\pi$
$32$		0			0
$33$			27.5803i			0.835765i
$34$		0			0
$35$			52.2038i			1.49154i
$36$		0			0
$37$	−41.7531	−	41.7531i	−1.12846	−	1.12846i	−0.990427	−	0.138035i	$-0.955921\pi$
$37$	−41.7531	−	41.7531i	−1.12846	−	1.12846i	−0.138035	−	0.990427i	$-0.544079\pi$
$38$		0			0
$39$	−15.3764	−	15.3764i	−0.394266	−	0.394266i
$40$		0			0
$41$	−18.0867	−	18.0867i	−0.441138	−	0.441138i	0.451256	−	0.892394i	$-0.350976\pi$
$41$	−18.0867	−	18.0867i	−0.441138	−	0.441138i	−0.892394	+	0.451256i	$0.850976\pi$
$42$		0			0
$43$	50.0423	−	50.0423i	1.16378	−	1.16378i	0.180133	−	0.983642i	$-0.442347\pi$
$43$	50.0423	−	50.0423i	1.16378	−	1.16378i	0.983642	−	0.180133i	$-0.0576529\pi$
$44$		0			0
$45$	−18.6746			−0.414992
$46$		0			0
$47$	−53.3333	−	53.3333i	−1.13475	−	1.13475i	−0.989377	−	0.145375i	$-0.953561\pi$
$47$	−53.3333	−	53.3333i	−1.13475	−	1.13475i	−0.145375	−	0.989377i	$-0.546439\pi$
$48$		0			0
$49$	21.3304			0.435315
$50$		0			0
$51$			7.29020i			0.142945i
$52$		0			0
$53$	−42.5439			−0.802715			−0.401358	−	0.915921i	$-0.631462\pi$
$53$	−42.5439			−0.802715			−0.401358	+	0.915921i	$0.631462\pi$
$54$		0			0
$55$	70.0895	+	70.0895i	1.27436	+	1.27436i
$56$		0			0
$57$			46.6662i			0.818705i
$58$		0			0
$59$	−61.9343			−1.04973			−0.524867	−	0.851184i	$-0.675885\pi$
$59$	−61.9343			−1.04973			−0.524867	+	0.851184i	$0.675885\pi$
$60$		0			0
$61$	36.0094	−	36.0094i	0.590318	−	0.590318i	−0.347399	−	0.937717i	$-0.612935\pi$
$61$	36.0094	−	36.0094i	0.590318	−	0.590318i	0.937717	+	0.347399i	$0.112935\pi$
$62$		0			0
$63$			25.1590i			0.399349i
$64$		0			0
$65$	−78.1517			−1.20233
$66$		0			0
$67$		−	66.8704i		−	0.998066i	−0.866583	−	0.499033i	$-0.833689\pi$
$67$		−	66.8704i		−	0.998066i	0.866583	−	0.499033i	$-0.166311\pi$
$68$		0			0
$69$	−1.21285	+	1.21285i	−0.0175775	+	0.0175775i
$70$		0			0
$71$			89.1659i			1.25586i	0.778271	+	0.627929i	$0.216097\pi$
$71$			89.1659i			1.25586i	−0.778271	+	0.627929i	$0.783903\pi$
$72$		0			0
$73$	16.0513	+	16.0513i	0.219881	+	0.219881i	0.808448	−	0.588567i	$-0.200308\pi$
$73$	16.0513	+	16.0513i	0.219881	+	0.219881i	−0.588567	+	0.808448i	$0.700308\pi$
$74$		0			0
$75$	−16.8391	+	16.8391i	−0.224521	+	0.224521i
$76$		0			0
$77$	94.4266	−	94.4266i	1.22632	−	1.22632i
$78$		0			0
$79$	74.3011	−	74.3011i	0.940521	−	0.940521i	−0.0578072	−	0.998328i	$-0.518411\pi$
$79$	74.3011	−	74.3011i	0.940521	−	0.940521i	0.998328	+	0.0578072i	$0.0184109\pi$
$80$		0			0
$81$	−9.00000			−0.111111
$82$		0			0
$83$	−80.2278			−0.966601			−0.483300	−	0.875455i	$-0.660562\pi$
$83$	−80.2278			−0.966601			−0.483300	+	0.875455i	$0.660562\pi$
$84$		0			0
$85$	18.5265	+	18.5265i	0.217959	+	0.217959i
$86$		0			0
$87$	49.4583	−	8.76782i	0.568486	−	0.100779i
$88$		0			0
$89$	3.67823	−	3.67823i	0.0413285	−	0.0413285i	−0.686141	−	0.727469i	$-0.740697\pi$
$89$	3.67823	−	3.67823i	0.0413285	−	0.0413285i	0.727469	+	0.686141i	$0.240697\pi$
$90$		0			0
$91$			105.288i			1.15701i
$92$		0			0
$93$		−	22.3267i		−	0.240072i
$94$		0			0
$95$	118.592	+	118.592i	1.24834	+	1.24834i
$96$		0			0
$97$	−101.725	−	101.725i	−1.04871	−	1.04871i	−0.998751	−	0.0499585i	$-0.984091\pi$
$97$	−101.725	−	101.725i	−1.04871	−	1.04871i	−0.0499585	−	0.998751i	$-0.515909\pi$
$98$		0			0
$99$	33.7788	+	33.7788i	0.341200	+	0.341200i
$100$		0			0
$101$	78.5057	−	78.5057i	0.777284	−	0.777284i	−0.202084	−	0.979368i	$-0.564772\pi$
$101$	78.5057	−	78.5057i	0.777284	−	0.777284i	0.979368	+	0.202084i	$0.0647715\pi$
$102$		0			0
$103$	167.789			1.62902			0.814510	−	0.580149i	$-0.197006\pi$
$103$	167.789			1.62902			0.814510	+	0.580149i	$0.197006\pi$
$104$		0			0
$105$	63.9363	+	63.9363i	0.608918	+	0.608918i
$106$		0			0
$107$	41.6570			0.389317			0.194659	−	0.980871i	$-0.437640\pi$
$107$	41.6570			0.389317			0.194659	+	0.980871i	$0.437640\pi$
$108$		0			0
$109$		−	1.95402i		−	0.0179267i	−0.999960	−	0.00896337i	$-0.997147\pi$
$109$		−	1.95402i		−	0.0179267i	0.999960	−	0.00896337i	$-0.00285317\pi$
$110$		0			0
$111$	−102.274			−0.921386
$112$		0			0
$113$	128.633	+	128.633i	1.13835	+	1.13835i	0.988746	+	0.149603i	$0.0477996\pi$
$113$	128.633	+	128.633i	1.13835	+	1.13835i	0.149603	+	0.988746i	$0.452200\pi$
$114$		0			0
$115$			6.16442i			0.0536036i
$116$		0			0
$117$	−37.6642			−0.321917
$118$		0			0
$119$	24.9595	−	24.9595i	0.209744	−	0.209744i
$120$		0			0
$121$		−	132.557i		−	1.09551i
$122$		0			0
$123$	−44.3031			−0.360188
$124$		0			0
$125$		−	70.0357i		−	0.560286i
$126$		0			0
$127$	−61.7993	+	61.7993i	−0.486608	+	0.486608i	−0.907234	−	0.420626i	$-0.861811\pi$
$127$	−61.7993	+	61.7993i	−0.486608	+	0.486608i	0.420626	+	0.907234i	$0.361811\pi$
$128$		0			0
$129$		−	122.578i		−	0.950219i
$130$		0			0
$131$	−61.6928	−	61.6928i	−0.470937	−	0.470937i	0.431280	−	0.902218i	$-0.358062\pi$
$131$	−61.6928	−	61.6928i	−0.470937	−	0.470937i	−0.902218	+	0.431280i	$0.858062\pi$
$132$		0			0
$133$	159.771	−	159.771i	1.20129	−	1.20129i
$134$		0			0
$135$	−22.8716	+	22.8716i	−0.169420	+	0.169420i
$136$		0			0
$137$	−43.1336	+	43.1336i	−0.314844	+	0.314844i	−0.846783	−	0.531939i	$-0.821464\pi$
$137$	−43.1336	+	43.1336i	−0.314844	+	0.314844i	0.531939	+	0.846783i	$0.321464\pi$
$138$		0			0
$139$	122.071			0.878208			0.439104	−	0.898436i	$-0.355296\pi$
$139$	122.071			0.878208			0.439104	+	0.898436i	$0.355296\pi$
$140$		0			0
$141$	−130.639			−0.926521
$142$		0			0
$143$	141.361	+	141.361i	0.988540	+	0.988540i
$144$		0			0
$145$	103.407	−	147.970i	0.713149	−	1.02048i
$146$		0			0
$147$	26.1243	−	26.1243i	0.177717	−	0.177717i
$148$		0			0
$149$			91.8081i			0.616161i	0.951360	+	0.308081i	$0.0996867\pi$
$149$			91.8081i			0.616161i	−0.951360	+	0.308081i	$0.900313\pi$
$150$		0			0
$151$			189.682i			1.25617i	0.778144	+	0.628087i	$0.216162\pi$
$151$			189.682i			1.25617i	−0.778144	+	0.628087i	$0.783838\pi$
$152$		0			0
$153$	8.92864	+	8.92864i	0.0583571	+	0.0583571i
$154$		0			0
$155$	−56.7388	−	56.7388i	−0.366057	−	0.366057i
$156$		0			0
$157$	−26.2750	−	26.2750i	−0.167357	−	0.167357i	0.618460	−	0.785816i	$-0.287757\pi$
$157$	−26.2750	−	26.2750i	−0.167357	−	0.167357i	−0.785816	+	0.618460i	$0.787757\pi$
$158$		0			0
$159$	−52.1054	+	52.1054i	−0.327707	+	0.327707i
$160$		0			0
$161$	8.30488			0.0515831
$162$		0			0
$163$	69.9956	+	69.9956i	0.429421	+	0.429421i	0.888431	−	0.459010i	$-0.151796\pi$
$163$	69.9956	+	69.9956i	0.429421	+	0.429421i	−0.459010	+	0.888431i	$0.651796\pi$
$164$		0			0
$165$	171.684			1.04051
$166$		0			0
$167$			165.548i			0.991306i	0.868521	+	0.495653i	$0.165071\pi$
$167$			165.548i			0.991306i	−0.868521	+	0.495653i	$0.834929\pi$
$168$		0			0
$169$	11.3784			0.0673277
$170$		0			0
$171$	57.1542	+	57.1542i	0.334235	+	0.334235i
$172$		0			0
$173$		−	77.5776i		−	0.448426i	−0.974540	−	0.224213i	$-0.928019\pi$
$173$		−	77.5776i		−	0.448426i	0.974540	−	0.224213i	$-0.0719811\pi$
$174$		0			0
$175$	115.304			0.658880
$176$		0			0
$177$	−75.8538	+	75.8538i	−0.428552	+	0.428552i
$178$		0			0
$179$			50.7547i			0.283546i	0.989899	+	0.141773i	$0.0452803\pi$
$179$			50.7547i			0.283546i	−0.989899	+	0.141773i	$0.954720\pi$
$180$		0			0
$181$	−83.8733			−0.463388			−0.231694	−	0.972789i	$-0.574427\pi$
$181$	−83.8733			−0.463388			−0.231694	+	0.972789i	$0.574427\pi$
$182$		0			0
$183$		−	88.2047i		−	0.481993i
$184$		0			0
$185$	−259.908	+	259.908i	−1.40491	+	1.40491i
$186$		0			0
$187$		−	67.0219i		−	0.358406i
$188$		0			0
$189$	30.8133	+	30.8133i	0.163033	+	0.163033i
$190$		0			0
$191$	−132.202	+	132.202i	−0.692157	+	0.692157i	−0.962706	−	0.270549i	$-0.912795\pi$
$191$	−132.202	+	132.202i	−0.692157	+	0.692157i	0.270549	+	0.962706i	$0.412795\pi$
$192$		0			0
$193$	150.453	−	150.453i	0.779552	−	0.779552i	−0.200203	−	0.979754i	$-0.564160\pi$
$193$	150.453	−	150.453i	0.779552	−	0.779552i	0.979754	+	0.200203i	$0.0641601\pi$
$194$		0			0
$195$	−95.7159	+	95.7159i	−0.490851	+	0.490851i
$196$		0			0
$197$	360.216			1.82851			0.914253	−	0.405143i	$-0.132778\pi$
$197$	360.216			1.82851			0.914253	+	0.405143i	$0.132778\pi$
$198$		0			0
$199$	−338.298			−1.69999			−0.849995	−	0.526791i	$-0.823395\pi$
$199$	−338.298			−1.69999			−0.849995	+	0.526791i	$0.823395\pi$
$200$		0			0
$201$	−81.8992	−	81.8992i	−0.407459	−	0.407459i
$202$		0			0
$203$	−199.349	−	139.312i	−0.982015	−	0.686267i
$204$		0			0
$205$	−112.587	+	112.587i	−0.549206	+	0.549206i
$206$		0			0
$207$			2.97086i			0.0143520i
$208$		0			0
$209$		−	429.022i		−	2.05274i
$210$		0			0
$211$	−256.472	−	256.472i	−1.21551	−	1.21551i	−0.969189	−	0.246319i	$-0.920779\pi$
$211$	−256.472	−	256.472i	−1.21551	−	1.21551i	−0.246319	−	0.969189i	$-0.579221\pi$
$212$		0			0
$213$	109.205	+	109.205i	0.512702	+	0.512702i
$214$		0			0
$215$	−311.507	−	311.507i	−1.44887	−	1.44887i
$216$		0			0
$217$	−76.4401	+	76.4401i	−0.352258	+	0.352258i
$218$		0			0
$219$	39.3176			0.179532
$220$		0			0
$221$	37.3656	+	37.3656i	0.169075	+	0.169075i
$222$		0			0
$223$	−154.108			−0.691067			−0.345533	−	0.938406i	$-0.612302\pi$
$223$	−154.108			−0.691067			−0.345533	+	0.938406i	$0.612302\pi$
$224$		0			0
$225$			41.2472i			0.183321i
$226$		0			0
$227$	−136.424			−0.600987			−0.300493	−	0.953784i	$-0.597151\pi$
$227$	−136.424			−0.600987			−0.300493	+	0.953784i	$0.597151\pi$
$228$		0			0
$229$	−125.468	−	125.468i	−0.547896	−	0.547896i	0.377936	−	0.925832i	$-0.376634\pi$
$229$	−125.468	−	125.468i	−0.547896	−	0.547896i	−0.925832	+	0.377936i	$0.876634\pi$
$230$		0			0
$231$		−	231.297i		−	1.00129i
$232$		0			0
$233$	−448.242			−1.92378			−0.961892	−	0.273430i	$-0.911842\pi$
$233$	−448.242			−1.92378			−0.961892	+	0.273430i	$0.911842\pi$
$234$		0			0
$235$	−331.993	+	331.993i	−1.41274	+	1.41274i
$236$		0			0
$237$		−	182.000i		−	0.767932i
$238$		0			0
$239$	289.674			1.21202			0.606012	−	0.795456i	$-0.292768\pi$
$239$	289.674			1.21202			0.606012	+	0.795456i	$0.292768\pi$
$240$		0			0
$241$			42.2795i			0.175434i	0.996145	+	0.0877169i	$0.0279571\pi$
$241$			42.2795i			0.175434i	−0.996145	+	0.0877169i	$0.972043\pi$
$242$		0			0
$243$	−11.0227	+	11.0227i	−0.0453609	+	0.0453609i
$244$		0			0
$245$		−	132.779i		−	0.541956i
$246$		0			0
$247$	239.185	+	239.185i	0.968362	+	0.968362i
$248$		0			0
$249$	−98.2586	+	98.2586i	−0.394613	+	0.394613i
$250$		0			0
$251$	154.680	−	154.680i	0.616256	−	0.616256i	−0.328313	−	0.944569i	$-0.606480\pi$
$251$	154.680	−	154.680i	0.616256	−	0.616256i	0.944569	+	0.328313i	$0.106480\pi$
$252$		0			0
$253$	11.1502	−	11.1502i	0.0440721	−	0.0440721i
$254$		0			0
$255$	45.3806			0.177963
$256$		0			0
$257$	16.2375			0.0631809			0.0315905	−	0.999501i	$-0.489943\pi$
$257$	16.2375			0.0631809			0.0315905	+	0.999501i	$0.489943\pi$
$258$		0			0
$259$	350.155	+	350.155i	1.35195	+	1.35195i
$260$		0			0
$261$	49.8355	−	71.3122i	0.190941	−	0.273227i
$262$		0			0
$263$	53.6604	−	53.6604i	0.204032	−	0.204032i	−0.597693	−	0.801725i	$-0.703916\pi$
$263$	53.6604	−	53.6604i	0.204032	−	0.204032i	0.801725	+	0.597693i	$0.203916\pi$
$264$		0			0
$265$			264.830i			0.999360i
$266$		0			0
$267$		−	9.00979i		−	0.0337446i
$268$		0			0
$269$	183.535	+	183.535i	0.682288	+	0.682288i	0.960515	−	0.278227i	$-0.0897469\pi$
$269$	183.535	+	183.535i	0.682288	+	0.682288i	−0.278227	+	0.960515i	$0.589747\pi$
$270$		0			0
$271$	−351.496	−	351.496i	−1.29703	−	1.29703i	−0.930342	−	0.366693i	$-0.880490\pi$
$271$	−351.496	−	351.496i	−1.29703	−	1.29703i	−0.366693	−	0.930342i	$-0.619510\pi$
$272$		0			0
$273$	128.951	+	128.951i	0.472348	+	0.472348i
$274$		0			0
$275$	154.809	−	154.809i	0.562941	−	0.562941i
$276$		0			0
$277$	−185.886			−0.671069			−0.335534	−	0.942028i	$-0.608917\pi$
$277$	−185.886			−0.671069			−0.335534	+	0.942028i	$0.608917\pi$
$278$		0			0
$279$	−27.3445	−	27.3445i	−0.0980091	−	0.0980091i
$280$		0			0
$281$	−357.621			−1.27267			−0.636336	−	0.771412i	$-0.719551\pi$
$281$	−357.621			−1.27267			−0.636336	+	0.771412i	$0.719551\pi$
$282$		0			0
$283$			12.9326i			0.0456983i	0.999739	+	0.0228492i	$0.00727374\pi$
$283$			12.9326i			0.0456983i	−0.999739	+	0.0228492i	$0.992726\pi$
$284$		0			0
$285$	290.491			1.01927
$286$		0			0
$287$	151.681	+	151.681i	0.528504	+	0.528504i
$288$		0			0
$289$			271.284i			0.938700i
$290$		0			0
$291$	−249.174			−0.856268
$292$		0			0
$293$	−92.7504	+	92.7504i	−0.316554	+	0.316554i	−0.847442	−	0.530888i	$-0.821859\pi$
$293$	−92.7504	+	92.7504i	−0.316554	+	0.316554i	0.530888	+	0.847442i	$0.321859\pi$
$294$		0			0
$295$			385.533i			1.30689i
$296$		0			0
$297$	82.7408			0.278588
$298$		0			0
$299$			12.4328i			0.0415813i
$300$		0			0
$301$	−419.671	+	419.671i	−1.39426	+	1.39426i
$302$		0			0
$303$		−	192.299i		−	0.634650i
$304$		0			0
$305$	−224.154	−	224.154i	−0.734931	−	0.734931i
$306$		0			0
$307$	417.035	−	417.035i	1.35842	−	1.35842i	0.482552	−	0.875867i	$-0.339710\pi$
$307$	417.035	−	417.035i	1.35842	−	1.35842i	0.875867	−	0.482552i	$-0.160290\pi$
$308$		0			0
$309$	205.499	−	205.499i	0.665045	−	0.665045i
$310$		0			0
$311$	297.022	−	297.022i	0.955055	−	0.955055i	−0.0439779	−	0.999033i	$-0.514003\pi$
$311$	297.022	−	297.022i	0.955055	−	0.955055i	0.999033	+	0.0439779i	$0.0140031\pi$
$312$		0			0
$313$	−395.782			−1.26448			−0.632239	−	0.774773i	$-0.717864\pi$
$313$	−395.782			−1.26448			−0.632239	+	0.774773i	$0.717864\pi$
$314$		0			0
$315$	156.611			0.497179
$316$		0			0
$317$	−35.4067	−	35.4067i	−0.111693	−	0.111693i	0.649052	−	0.760744i	$-0.275166\pi$
$317$	−35.4067	−	35.4067i	−0.111693	−	0.111693i	−0.760744	+	0.649052i	$0.775166\pi$
$318$		0			0
$319$	−454.691	+	80.6062i	−1.42536	+	0.252684i
$320$		0			0
$321$	51.0192	−	51.0192i	0.158938	−	0.158938i
$322$		0			0
$323$		−	113.402i		−	0.351090i
$324$		0			0
$325$			172.616i			0.531126i
$326$		0			0
$327$	−2.39317	−	2.39317i	−0.00731856	−	0.00731856i
$328$		0			0
$329$	447.271	+	447.271i	1.35949	+	1.35949i
$330$		0			0
$331$	337.771	+	337.771i	1.02046	+	1.02046i	0.999786	+	0.0206709i	$0.00658023\pi$
$331$	337.771	+	337.771i	1.02046	+	1.02046i	0.0206709	+	0.999786i	$0.493420\pi$
$332$		0			0
$333$	−125.259	+	125.259i	−0.376154	+	0.376154i
$334$		0			0
$335$	−416.260			−1.24257
$336$		0			0
$337$	−367.977	−	367.977i	−1.09192	−	1.09192i	−0.995324	−	0.0965967i	$-0.969204\pi$
$337$	−367.977	−	367.977i	−1.09192	−	1.09192i	−0.0965967	−	0.995324i	$-0.530796\pi$
$338$		0			0
$339$	315.086			0.929458
$340$		0			0
$341$			205.259i			0.601932i
$342$		0			0
$343$	232.046			0.676519
$344$		0			0
$345$	7.54984	+	7.54984i	0.0218836	+	0.0218836i
$346$		0			0
$347$			191.746i			0.552581i	0.961074	+	0.276290i	$0.0891052\pi$
$347$			191.746i			0.552581i	−0.961074	+	0.276290i	$0.910895\pi$
$348$		0			0
$349$	−94.2770			−0.270135			−0.135067	−	0.990836i	$-0.543125\pi$
$349$	−94.2770			−0.270135			−0.135067	+	0.990836i	$0.543125\pi$
$350$		0			0
$351$	−46.1291	+	46.1291i	−0.131422	+	0.131422i
$352$		0			0
$353$			420.448i			1.19107i	0.803329	+	0.595536i	$0.203060\pi$
$353$			420.448i			1.19107i	−0.803329	+	0.595536i	$0.796940\pi$
$354$		0			0
$355$	555.046			1.56351
$356$		0			0
$357$		−	61.1380i		−	0.171255i
$358$		0			0
$359$	186.902	−	186.902i	0.520619	−	0.520619i	−0.397139	−	0.917758i	$-0.629997\pi$
$359$	186.902	−	186.902i	0.520619	−	0.520619i	0.917758	+	0.397139i	$0.129997\pi$
$360$		0			0
$361$		−	364.911i		−	1.01083i
$362$		0			0
$363$	−162.348	−	162.348i	−0.447240	−	0.447240i
$364$		0			0
$365$	99.9176	−	99.9176i	0.273747	−	0.273747i
$366$		0			0
$367$	148.125	−	148.125i	0.403610	−	0.403610i	−0.475893	−	0.879503i	$-0.657875\pi$
$367$	148.125	−	148.125i	0.403610	−	0.403610i	0.879503	+	0.475893i	$0.157875\pi$
$368$		0			0
$369$	−54.2600	+	54.2600i	−0.147046	+	0.147046i
$370$		0			0
$371$	356.787			0.961690
$372$		0			0
$373$	372.127			0.997659			0.498830	−	0.866700i	$-0.333763\pi$
$373$	372.127			0.997659			0.498830	+	0.866700i	$0.333763\pi$
$374$		0			0
$375$	−85.7759	−	85.7759i	−0.228736	−	0.228736i
$376$		0			0
$377$	208.557	−	298.435i	0.553202	−	0.791606i
$378$		0			0
$379$	−94.3069	+	94.3069i	−0.248831	+	0.248831i	−0.820491	−	0.571660i	$-0.806300\pi$
$379$	−94.3069	+	94.3069i	−0.248831	+	0.248831i	0.571660	+	0.820491i	$0.306300\pi$
$380$		0			0
$381$			151.377i			0.397314i
$382$		0			0
$383$		−	305.251i		−	0.797000i	−0.917168	−	0.398500i	$-0.869531\pi$
$383$		−	305.251i		−	0.797000i	0.917168	−	0.398500i	$-0.130469\pi$
$384$		0			0
$385$	−587.794	−	587.794i	−1.52674	−	1.52674i
$386$		0			0
$387$	−150.127	−	150.127i	−0.387925	−	0.387925i
$388$		0			0
$389$	499.186	+	499.186i	1.28325	+	1.28325i	0.938805	+	0.344450i	$0.111935\pi$
$389$	499.186	+	499.186i	1.28325	+	1.28325i	0.344450	+	0.938805i	$0.388065\pi$
$390$		0			0
$391$	2.94731	−	2.94731i	0.00753787	−	0.00753787i
$392$		0			0
$393$	−151.116			−0.384519
$394$		0			0
$395$	−462.515	−	462.515i	−1.17092	−	1.17092i
$396$		0			0
$397$	140.405			0.353666			0.176833	−	0.984241i	$-0.443415\pi$
$397$	140.405			0.353666			0.176833	+	0.984241i	$0.443415\pi$
$398$		0			0
$399$		−	391.358i		−	0.980846i
$400$		0			0
$401$	178.555			0.445274			0.222637	−	0.974901i	$-0.428534\pi$
$401$	178.555			0.445274			0.222637	+	0.974901i	$0.428534\pi$
$402$		0			0
$403$	−114.435	−	114.435i	−0.283957	−	0.283957i
$404$		0			0
$405$			56.0239i			0.138331i
$406$		0			0
$407$	940.246			2.31019
$408$		0			0
$409$	265.947	−	265.947i	0.650237	−	0.650237i	−0.302813	−	0.953050i	$-0.597926\pi$
$409$	265.947	−	265.947i	0.650237	−	0.650237i	0.953050	+	0.302813i	$0.0979258\pi$
$410$		0			0
$411$			105.655i			0.257069i
$412$		0			0
$413$	519.402			1.25763
$414$		0			0
$415$			499.408i			1.20339i
$416$		0			0
$417$	149.506	−	149.506i	0.358527	−	0.358527i
$418$		0			0
$419$		−	152.135i		−	0.363090i	−0.983383	−	0.181545i	$-0.941890\pi$
$419$		−	152.135i		−	0.363090i	0.983383	−	0.181545i	$-0.0581098\pi$
$420$		0			0
$421$	98.1758	+	98.1758i	0.233197	+	0.233197i	0.814026	−	0.580829i	$-0.197272\pi$
$421$	98.1758	+	98.1758i	0.233197	+	0.233197i	−0.580829	+	0.814026i	$0.697272\pi$
$422$		0			0
$423$	−160.000	+	160.000i	−0.378251	+	0.378251i
$424$		0			0
$425$	40.9201	−	40.9201i	0.0962826	−	0.0962826i
$426$		0			0
$427$	−301.987	+	301.987i	−0.707228	+	0.707228i
$428$		0			0
$429$	346.263			0.807140
$430$		0			0
$431$	−727.779			−1.68858			−0.844291	−	0.535885i	$-0.819978\pi$
$431$	−727.779			−1.68858			−0.844291	+	0.535885i	$0.819978\pi$
$432$		0			0
$433$	−112.939	−	112.939i	−0.260829	−	0.260829i	0.564562	−	0.825391i	$-0.309045\pi$
$433$	−112.939	−	112.939i	−0.260829	−	0.260829i	−0.825391	+	0.564562i	$0.809045\pi$
$434$		0			0
$435$	−54.5785	−	307.872i	−0.125468	−	0.707751i
$436$		0			0
$437$	18.8664	−	18.8664i	0.0431725	−	0.0431725i
$438$		0			0
$439$			405.711i			0.924170i	0.886836	+	0.462085i	$0.152899\pi$
$439$			405.711i			0.924170i	−0.886836	+	0.462085i	$0.847101\pi$
$440$		0			0
$441$		−	63.9913i		−	0.145105i
$442$		0			0
$443$	−129.892	−	129.892i	−0.293210	−	0.293210i	0.545137	−	0.838347i	$-0.316478\pi$
$443$	−129.892	−	129.892i	−0.293210	−	0.293210i	−0.838347	+	0.545137i	$0.816478\pi$
$444$		0			0
$445$	−22.8965	−	22.8965i	−0.0514529	−	0.0514529i
$446$		0			0
$447$	112.441	+	112.441i	0.251547	+	0.251547i
$448$		0			0
$449$	435.307	−	435.307i	0.969503	−	0.969503i	−0.0300457	−	0.999549i	$-0.509565\pi$
$449$	435.307	−	435.307i	0.969503	−	0.969503i	0.999549	+	0.0300457i	$0.00956527\pi$
$450$		0			0
$451$	407.297			0.903097
$452$		0			0
$453$	232.312	+	232.312i	0.512830	+	0.512830i
$454$		0			0
$455$	655.406			1.44045
$456$		0			0
$457$			632.050i			1.38304i	0.722357	+	0.691521i	$0.243059\pi$
$457$			632.050i			1.38304i	−0.722357	+	0.691521i	$0.756941\pi$
$458$		0			0
$459$	21.8706			0.0476484
$460$		0			0
$461$	271.189	+	271.189i	0.588263	+	0.588263i	0.937161	−	0.348898i	$-0.113444\pi$
$461$	271.189	+	271.189i	0.588263	+	0.588263i	−0.348898	+	0.937161i	$0.613444\pi$
$462$		0			0
$463$		−	439.080i		−	0.948336i	−0.880434	−	0.474168i	$-0.842749\pi$
$463$		−	439.080i		−	0.948336i	0.880434	−	0.474168i	$-0.157251\pi$
$464$		0			0
$465$	−138.981			−0.298884
$466$		0			0
$467$	−356.789	+	356.789i	−0.764003	+	0.764003i	−0.977043	−	0.213041i	$-0.931663\pi$
$467$	−356.789	+	356.789i	−0.764003	+	0.764003i	0.213041	+	0.977043i	$0.431663\pi$
$468$		0			0
$469$			560.797i			1.19573i
$470$		0			0
$471$	−64.3604			−0.136646
$472$		0			0
$473$			1126.91i			2.38248i
$474$		0			0
$475$	261.939	−	261.939i	0.551450	−	0.551450i
$476$		0			0
$477$			127.632i			0.267572i
$478$		0			0
$479$	−228.554	−	228.554i	−0.477149	−	0.477149i	0.427070	−	0.904219i	$-0.359546\pi$
$479$	−228.554	−	228.554i	−0.477149	−	0.477149i	−0.904219	+	0.427070i	$0.859546\pi$
$480$		0			0
$481$	−524.200	+	524.200i	−1.08981	+	1.08981i
$482$		0			0
$483$	10.1714	−	10.1714i	0.0210587	−	0.0210587i
$484$		0			0
$485$	−633.224	+	633.224i	−1.30562	+	1.30562i
$486$		0			0
$487$	228.487			0.469172			0.234586	−	0.972095i	$-0.424627\pi$
$487$	228.487			0.469172			0.234586	+	0.972095i	$0.424627\pi$
$488$		0			0
$489$	171.453			0.350621
$490$		0			0
$491$	−330.119	−	330.119i	−0.672340	−	0.672340i	0.285915	−	0.958255i	$-0.407702\pi$
$491$	−330.119	−	330.119i	−0.672340	−	0.672340i	−0.958255	+	0.285915i	$0.907702\pi$
$492$		0			0
$493$	−120.187	+	21.3064i	−0.243787	+	0.0432178i
$494$		0			0
$495$	210.269	−	210.269i	0.424785	−	0.424785i
$496$		0			0
$497$		−	747.774i		−	1.50458i
$498$		0			0
$499$			739.860i			1.48268i	0.671127	+	0.741342i	$0.265811\pi$
$499$			739.860i			1.48268i	−0.671127	+	0.741342i	$0.734189\pi$
$500$		0			0
$501$	202.754	+	202.754i	0.404699	+	0.404699i
$502$		0			0
$503$	−201.129	−	201.129i	−0.399859	−	0.399859i	0.478324	−	0.878183i	$-0.341244\pi$
$503$	−201.129	−	201.129i	−0.399859	−	0.399859i	−0.878183	+	0.478324i	$0.841244\pi$
$504$		0			0
$505$	−488.688	−	488.688i	−0.967699	−	0.967699i
$506$		0			0
$507$	13.9356	−	13.9356i	0.0274864	−	0.0274864i
$508$		0			0
$509$	−10.0139			−0.0196736			−0.00983682	−	0.999952i	$-0.503131\pi$
$509$	−10.0139			−0.0196736			−0.00983682	+	0.999952i	$0.503131\pi$
$510$		0			0
$511$	−134.612	−	134.612i	−0.263428	−	0.263428i
$512$		0			0
$513$	139.999			0.272902
$514$		0			0
$515$		−	1044.47i		−	2.02809i
$516$		0			0
$517$	1201.02			2.32306
$518$		0			0
$519$	−95.0128	−	95.0128i	−0.183069	−	0.183069i
$520$		0			0
$521$			199.135i			0.382216i	0.981569	+	0.191108i	$0.0612081\pi$
$521$			199.135i			0.382216i	−0.981569	+	0.191108i	$0.938792\pi$
$522$		0			0
$523$	277.854			0.531269			0.265635	−	0.964074i	$-0.414419\pi$
$523$	277.854			0.531269			0.265635	+	0.964074i	$0.414419\pi$
$524$		0			0
$525$	141.218	−	141.218i	0.268987	−	0.268987i
$526$		0			0
$527$			54.2555i			0.102952i
$528$		0			0
$529$	−528.019			−0.998146
$530$		0			0
$531$			185.803i			0.349912i
$532$		0			0
$533$	−227.074	+	227.074i	−0.426029	+	0.426029i
$534$		0			0
$535$		−	259.309i		−	0.484690i
$536$		0			0
$537$	62.1615	+	62.1615i	0.115757	+	0.115757i
$538$		0			0
$539$	−240.172	+	240.172i	−0.445588	+	0.445588i
$540$		0			0
$541$	327.619	−	327.619i	0.605580	−	0.605580i	−0.336208	−	0.941788i	$-0.609144\pi$
$541$	327.619	−	327.619i	0.605580	−	0.605580i	0.941788	+	0.336208i	$0.109144\pi$
$542$		0			0
$543$	−102.723	+	102.723i	−0.189178	+	0.189178i
$544$		0			0
$545$	−12.1635			−0.0223183
$546$		0			0
$547$	536.401			0.980623			0.490311	−	0.871547i	$-0.336883\pi$
$547$	536.401			0.980623			0.490311	+	0.871547i	$0.336883\pi$
$548$		0			0
$549$	−108.028	−	108.028i	−0.196773	−	0.196773i
$550$		0			0
$551$	−769.344	+	136.387i	−1.39627	+	0.247526i
$552$		0			0
$553$	−623.113	+	623.113i	−1.12679	+	1.12679i
$554$		0			0
$555$			636.642i			1.14710i
$556$		0			0
$557$		−	1108.07i		−	1.98934i	−0.103087	−	0.994672i	$-0.532872\pi$
$557$		−	1108.07i		−	1.98934i	0.103087	−	0.994672i	$-0.467128\pi$
$558$		0			0
$559$	−628.269	−	628.269i	−1.12392	−	1.12392i
$560$		0			0
$561$	−82.0847	−	82.0847i	−0.146319	−	0.146319i
$562$		0			0
$563$	323.646	+	323.646i	0.574860	+	0.574860i	0.933483	−	0.358622i	$-0.116753\pi$
$563$	323.646	+	323.646i	0.574860	+	0.574860i	−0.358622	+	0.933483i	$0.616753\pi$
$564$		0			0
$565$	800.727	−	800.727i	1.41722	−	1.41722i
$566$		0			0
$567$	75.4769			0.133116
$568$		0			0
$569$	−534.525	−	534.525i	−0.939412	−	0.939412i	0.0588546	−	0.998267i	$-0.481255\pi$
$569$	−534.525	−	534.525i	−0.939412	−	0.939412i	−0.998267	+	0.0588546i	$0.981255\pi$
$570$		0			0
$571$	−707.131			−1.23841			−0.619204	−	0.785230i	$-0.712545\pi$
$571$	−707.131			−1.23841			−0.619204	+	0.785230i	$0.712545\pi$
$572$		0			0
$573$			323.827i			0.565144i
$574$		0			0
$575$	13.6155			0.0236792
$576$		0			0
$577$	569.533	+	569.533i	0.987059	+	0.987059i	0.999917	−	0.0128581i	$-0.00409298\pi$
$577$	569.533	+	569.533i	0.987059	+	0.987059i	−0.0128581	+	0.999917i	$0.504093\pi$
$578$		0			0
$579$		−	368.534i		−	0.636501i
$580$		0			0
$581$	672.817			1.15803
$582$		0			0
$583$	479.027	−	479.027i	0.821659	−	0.821659i
$584$		0			0
$585$			234.455i			0.400778i
$586$		0			0
$587$	−453.981			−0.773392			−0.386696	−	0.922207i	$-0.626384\pi$
$587$	−453.981			−0.773392			−0.386696	+	0.922207i	$0.626384\pi$
$588$		0			0
$589$			347.301i			0.589645i
$590$		0			0
$591$	441.172	−	441.172i	0.746485	−	0.746485i
$592$		0			0
$593$			118.602i			0.200004i	0.994987	+	0.100002i	$0.0318849\pi$
$593$			118.602i			0.200004i	−0.994987	+	0.100002i	$0.968115\pi$
$594$		0			0
$595$	−155.370	−	155.370i	−0.261125	−	0.261125i
$596$		0			0
$597$	−414.329	+	414.329i	−0.694018	+	0.694018i
$598$		0			0
$599$	−169.686	+	169.686i	−0.283282	+	0.283282i	−0.834416	−	0.551134i	$-0.814195\pi$
$599$	−169.686	+	169.686i	−0.283282	+	0.283282i	0.551134	+	0.834416i	$0.314195\pi$
$600$		0			0
$601$	−118.429	+	118.429i	−0.197052	+	0.197052i	−0.798735	−	0.601683i	$-0.794497\pi$
$601$	−118.429	+	118.429i	−0.197052	+	0.197052i	0.601683	+	0.798735i	$0.294497\pi$
$602$		0			0
$603$	−200.611			−0.332689
$604$		0			0
$605$	−825.149			−1.36388
$606$		0			0
$607$	−479.221	−	479.221i	−0.789491	−	0.789491i	0.191920	−	0.981411i	$-0.438529\pi$
$607$	−479.221	−	479.221i	−0.789491	−	0.789491i	−0.981411	+	0.191920i	$0.938529\pi$
$608$		0			0
$609$	−414.774	+	73.5297i	−0.681073	+	0.120739i
$610$		0			0
$611$	−669.586	+	669.586i	−1.09589	+	1.09589i
$612$		0			0
$613$		−	419.425i		−	0.684217i	−0.939660	−	0.342109i	$-0.888859\pi$
$613$		−	419.425i		−	0.684217i	0.939660	−	0.342109i	$-0.111141\pi$
$614$		0			0
$615$			275.781i			0.448425i
$616$		0			0
$617$	707.633	+	707.633i	1.14689	+	1.14689i	0.987160	+	0.159733i	$0.0510633\pi$
$617$	707.633	+	707.633i	1.14689	+	1.14689i	0.159733	+	0.987160i	$0.448937\pi$
$618$		0			0
$619$	−579.890	−	579.890i	−0.936818	−	0.936818i	0.0613011	−	0.998119i	$-0.480475\pi$
$619$	−579.890	−	579.890i	−0.936818	−	0.936818i	−0.998119	+	0.0613011i	$0.980475\pi$
$620$		0			0
$621$	3.63855	+	3.63855i	0.00585918	+	0.00585918i
$622$		0			0
$623$	−30.8469	+	30.8469i	−0.0495134	+	0.0495134i
$624$		0			0
$625$	−779.690			−1.24750
$626$		0			0
$627$	−525.442	−	525.442i	−0.838026	−	0.838026i
$628$		0			0
$629$	248.532			0.395123
$630$		0			0
$631$			600.538i			0.951725i	0.879520	+	0.475863i	$0.157864\pi$
$631$			600.538i			0.951725i	−0.879520	+	0.475863i	$0.842136\pi$
$632$		0			0
$633$	−628.226			−0.992458
$634$		0			0
$635$	384.693	+	384.693i	0.605815	+	0.605815i
$636$		0			0
$637$		−	267.798i		−	0.420405i
$638$		0			0
$639$	267.498			0.418619
$640$		0			0
$641$	152.928	−	152.928i	0.238578	−	0.238578i	−0.577683	−	0.816261i	$-0.696043\pi$
$641$	152.928	−	152.928i	0.238578	−	0.238578i	0.816261	+	0.577683i	$0.196043\pi$
$642$		0			0
$643$		−	472.606i		−	0.735001i	−0.930023	−	0.367500i	$-0.880214\pi$
$643$		−	472.606i		−	0.735001i	0.930023	−	0.367500i	$-0.119786\pi$
$644$		0			0
$645$	−763.034			−1.18300
$646$		0			0
$647$			436.764i			0.675060i	0.941315	+	0.337530i	$0.109591\pi$
$647$			436.764i			0.675060i	−0.941315	+	0.337530i	$0.890409\pi$
$648$		0			0
$649$	697.355	−	697.355i	1.07451	−	1.07451i
$650$		0			0
$651$			187.239i			0.287618i
$652$		0			0
$653$	40.4261	+	40.4261i	0.0619083	+	0.0619083i	0.737383	−	0.675475i	$-0.236061\pi$
$653$	40.4261	+	40.4261i	0.0619083	+	0.0619083i	−0.675475	+	0.737383i	$0.736061\pi$
$654$		0			0
$655$	−384.030	+	384.030i	−0.586305	+	0.586305i
$656$		0			0
$657$	48.1540	−	48.1540i	0.0732938	−	0.0732938i
$658$		0			0
$659$	−282.795	+	282.795i	−0.429127	+	0.429127i	−0.888331	−	0.459204i	$-0.848135\pi$
$659$	−282.795	+	282.795i	−0.429127	+	0.429127i	0.459204	+	0.888331i	$0.348135\pi$
$660$		0			0
$661$	−896.310			−1.35599			−0.677996	−	0.735066i	$-0.737151\pi$
$661$	−896.310			−1.35599			−0.677996	+	0.735066i	$0.737151\pi$
$662$		0			0
$663$	91.5266			0.138049
$664$		0			0
$665$	−994.555	−	994.555i	−1.49557	−	1.49557i
$666$		0			0
$667$	−23.5399	−	16.4505i	−0.0352921	−	0.0246634i
$668$		0			0
$669$	−188.743	+	188.743i	−0.282127	+	0.282127i
$670$		0			0
$671$			810.902i			1.20850i
$672$		0			0
$673$		−	499.060i		−	0.741545i	−0.928724	−	0.370772i	$-0.879093\pi$
$673$		−	499.060i		−	0.741545i	0.928724	−	0.370772i	$-0.120907\pi$
$674$		0			0
$675$	50.5172	+	50.5172i	0.0748404	+	0.0748404i
$676$		0			0
$677$	−630.278	−	630.278i	−0.930987	−	0.930987i	0.0667803	−	0.997768i	$-0.478727\pi$
$677$	−630.278	−	630.278i	−0.930987	−	0.930987i	−0.997768	+	0.0667803i	$0.978727\pi$
$678$		0			0
$679$	853.098	+	853.098i	1.25640	+	1.25640i
$680$		0			0
$681$	−167.085	+	167.085i	−0.245352	+	0.245352i
$682$		0			0
$683$	−561.416			−0.821985			−0.410993	−	0.911639i	$-0.634818\pi$
$683$	−561.416			−0.821985			−0.410993	+	0.911639i	$0.634818\pi$
$684$		0			0
$685$	268.501	+	268.501i	0.391973	+	0.391973i
$686$		0			0
$687$	−307.333			−0.447355
$688$		0			0
$689$			534.128i			0.775222i
$690$		0			0
$691$	360.414			0.521583			0.260792	−	0.965395i	$-0.416016\pi$
$691$	360.414			0.521583			0.260792	+	0.965395i	$0.416016\pi$
$692$		0			0
$693$	−283.280	−	283.280i	−0.408773	−	0.408773i
$694$		0			0
$695$		−	759.876i		−	1.09335i
$696$		0			0
$697$	107.660			0.154461
$698$		0			0
$699$	−548.982	+	548.982i	−0.785382	+	0.785382i
$700$		0			0
$701$			741.677i			1.05803i	0.848613	+	0.529013i	$0.177438\pi$
$701$			741.677i			1.05803i	−0.848613	+	0.529013i	$0.822562\pi$
$702$		0			0
$703$	1590.91			2.26303
$704$		0			0
$705$			813.214i			1.15350i
$706$		0			0
$707$	−658.374	+	658.374i	−0.931222	+	0.931222i
$708$		0			0
$709$		−	1207.44i		−	1.70301i	−0.524344	−	0.851506i	$-0.675690\pi$
$709$		−	1207.44i		−	1.70301i	0.524344	−	0.851506i	$-0.324310\pi$
$710$		0			0
$711$	−222.903	−	222.903i	−0.313507	−	0.313507i
$712$		0			0
$713$	−9.02633	+	9.02633i	−0.0126596	+	0.0126596i
$714$		0			0
$715$	879.956	−	879.956i	1.23071	−	1.23071i
$716$		0			0
$717$	354.776	−	354.776i	0.494806	−	0.494806i
$718$		0			0
$719$	1289.31			1.79320			0.896602	−	0.442838i	$-0.146028\pi$
$719$	1289.31			1.79320			0.896602	+	0.442838i	$0.146028\pi$
$720$		0			0
$721$	−1407.13			−1.95164
$722$		0			0
$723$	51.7816	+	51.7816i	0.0716205	+	0.0716205i
$724$		0			0
$725$	−326.825	−	228.397i	−0.450793	−	0.315030i
$726$		0			0
$727$	−492.316	+	492.316i	−0.677188	+	0.677188i	−0.959363	−	0.282175i	$-0.908944\pi$
$727$	−492.316	+	492.316i	−0.677188	+	0.677188i	0.282175	+	0.959363i	$0.408944\pi$
$728$		0			0
$729$			27.0000i			0.0370370i
$730$		0			0
$731$			297.873i			0.407487i
$732$		0			0
$733$	713.708	+	713.708i	0.973680	+	0.973680i	0.999662	−	0.0259822i	$-0.00827132\pi$
$733$	713.708	+	713.708i	0.973680	+	0.973680i	−0.0259822	+	0.999662i	$0.508271\pi$
$734$		0			0
$735$	−162.621	−	162.621i	−0.221253	−	0.221253i
$736$		0			0
$737$	752.934	+	752.934i	1.02162	+	1.02162i
$738$		0			0
$739$	686.867	−	686.867i	0.929455	−	0.929455i	−0.0682158	−	0.997671i	$-0.521731\pi$
$739$	686.867	−	686.867i	0.929455	−	0.929455i	0.997671	+	0.0682158i	$0.0217306\pi$
$740$		0			0
$741$	585.882			0.790664
$742$		0			0
$743$	799.060	+	799.060i	1.07545	+	1.07545i	0.996911	+	0.0785399i	$0.0250258\pi$
$743$	799.060	+	799.060i	1.07545	+	1.07545i	0.0785399	+	0.996911i	$0.474974\pi$
$744$		0			0
$745$	571.494			0.767105
$746$		0			0
$747$			240.684i			0.322200i
$748$		0			0
$749$	−349.349			−0.466420
$750$		0			0
$751$	−892.272	−	892.272i	−1.18811	−	1.18811i	−0.977589	−	0.210522i	$-0.932484\pi$
$751$	−892.272	−	892.272i	−1.18811	−	1.18811i	−0.210522	−	0.977589i	$-0.567516\pi$
$752$		0			0
$753$		−	378.887i		−	0.503171i
$754$		0			0
$755$	1180.75			1.56390
$756$		0			0
$757$	748.084	−	748.084i	0.988222	−	0.988222i	−0.0117098	−	0.999931i	$-0.503727\pi$
$757$	748.084	−	748.084i	0.988222	−	0.988222i	0.999931	+	0.0117098i	$0.00372742\pi$
$758$		0			0
$759$		−	27.3124i		−	0.0359847i
$760$		0			0
$761$	−910.116			−1.19595			−0.597974	−	0.801516i	$-0.704027\pi$
$761$	−910.116			−1.19595			−0.597974	+	0.801516i	$0.704027\pi$
$762$		0			0
$763$			16.3870i			0.0214771i
$764$		0			0
$765$	55.5796	−	55.5796i	0.0726531	−	0.0726531i
$766$		0			0
$767$			777.570i			1.01378i
$768$		0			0
$769$	791.847	+	791.847i	1.02971	+	1.02971i	0.999545	+	0.0301653i	$0.00960338\pi$
$769$	791.847	+	791.847i	1.02971	+	1.02971i	0.0301653	+	0.999545i	$0.490397\pi$
$770$		0			0
$771$	19.8868	−	19.8868i	0.0257935	−	0.0257935i
$772$		0			0
$773$	361.424	−	361.424i	0.467561	−	0.467561i	−0.433563	−	0.901123i	$-0.642744\pi$
$773$	361.424	−	361.424i	0.467561	−	0.467561i	0.901123	+	0.433563i	$0.142744\pi$
$774$		0			0
$775$	−125.321	+	125.321i	−0.161704	+	0.161704i
$776$		0			0
$777$	857.702			1.10386
$778$		0			0
$779$	689.152			0.884663
$780$		0			0
$781$	−1003.97	−	1003.97i	−1.28550	−	1.28550i
$782$		0			0
$783$	−26.3034	−	148.375i	−0.0335932	−	0.189495i
$784$		0			0
$785$	−163.559	+	163.559i	−0.208355	+	0.208355i
$786$		0			0
$787$		−	868.958i		−	1.10414i	−0.833798	−	0.552070i	$-0.813838\pi$
$787$		−	868.958i		−	1.10414i	0.833798	−	0.552070i	$-0.186162\pi$
$788$		0			0
$789$		−	131.441i		−	0.166591i
$790$		0			0
$791$	−1078.76	−	1078.76i	−1.36380	−	1.36380i
$792$		0			0
$793$	−452.089	−	452.089i	−0.570099	−	0.570099i
$794$		0			0
$795$	324.350	+	324.350i	0.407987	+	0.407987i
$796$		0			0
$797$	702.869	−	702.869i	0.881893	−	0.881893i	−0.111834	−	0.993727i	$-0.535672\pi$
$797$	702.869	−	702.869i	0.881893	−	0.881893i	0.993727	+	0.111834i	$0.0356724\pi$
$798$		0			0
$799$	317.463			0.397325
$800$		0			0
$801$	−11.0347	−	11.0347i	−0.0137762	−	0.0137762i
$802$		0			0
$803$	−361.463			−0.450141
$804$		0			0
$805$		−	51.6968i		−	0.0642196i
$806$		0			0
$807$	449.568			0.557086
$808$		0			0
$809$	−475.902	−	475.902i	−0.588259	−	0.588259i	0.348901	−	0.937160i	$-0.386555\pi$
$809$	−475.902	−	475.902i	−0.588259	−	0.588259i	−0.937160	+	0.348901i	$0.886555\pi$
$810$		0			0
$811$			160.098i			0.197408i	0.995117	+	0.0987040i	$0.0314697\pi$
$811$			160.098i			0.197408i	−0.995117	+	0.0987040i	$0.968530\pi$
$812$		0			0
$813$	−860.987			−1.05902
$814$		0			0
$815$	435.714	−	435.714i	0.534618	−	0.534618i
$816$		0			0
$817$			1906.75i			2.33385i
$818$		0			0
$819$	315.864			0.385671
$820$		0			0
$821$		−	417.838i		−	0.508937i	−0.967081	−	0.254469i	$-0.918099\pi$
$821$		−	417.838i		−	0.508937i	0.967081	−	0.254469i	$-0.0819006\pi$
$822$		0			0
$823$	−881.681	+	881.681i	−1.07130	+	1.07130i	−0.0740470	+	0.997255i	$0.523591\pi$
$823$	−881.681	+	881.681i	−1.07130	+	1.07130i	−0.997255	+	0.0740470i	$0.976409\pi$
$824$		0			0
$825$		−	379.202i		−	0.459639i
$826$		0			0
$827$	980.281	+	980.281i	1.18535	+	1.18535i	0.978340	+	0.207006i	$0.0663719\pi$
$827$	980.281	+	980.281i	1.18535	+	1.18535i	0.207006	+	0.978340i	$0.433628\pi$
$828$		0			0
$829$	712.861	−	712.861i	0.859905	−	0.859905i	−0.131422	−	0.991327i	$-0.541954\pi$
$829$	712.861	−	712.861i	0.859905	−	0.859905i	0.991327	+	0.131422i	$0.0419542\pi$
$830$		0			0
$831$	−227.663	+	227.663i	−0.273963	+	0.273963i
$832$		0			0
$833$	−63.4839	+	63.4839i	−0.0762112	+	0.0762112i
$834$		0			0
$835$	1030.52			1.23415
$836$		0			0
$837$	−66.9802			−0.0800241
$838$		0			0
$839$	−981.031	−	981.031i	−1.16929	−	1.16929i	−0.982377	−	0.186909i	$-0.940153\pi$
$839$	−981.031	−	981.031i	−1.16929	−	1.16929i	−0.186909	−	0.982377i	$-0.559847\pi$
$840$		0			0
$841$	289.094	+	789.750i	0.343751	+	0.939061i
$842$		0			0
$843$	−437.994	+	437.994i	−0.519566	+	0.519566i
$844$		0			0
$845$		−	70.8290i		−	0.0838213i
$846$		0			0
$847$			1111.66i			1.31247i
$848$		0			0
$849$	15.8392	+	15.8392i	0.0186563	+	0.0186563i
$850$		0			0
$851$	41.3476	+	41.3476i	0.0485871	+	0.0485871i
$852$		0			0
$853$	289.400	+	289.400i	0.339273	+	0.339273i	0.856094	−	0.516821i	$-0.172885\pi$
$853$	289.400	+	289.400i	0.339273	+	0.339273i	−0.516821	+	0.856094i	$0.672885\pi$
$854$		0			0
$855$	355.777	−	355.777i	0.416114	−	0.416114i
$856$		0			0
$857$	74.3861			0.0867982			0.0433991	−	0.999058i	$-0.486181\pi$
$857$	74.3861			0.0867982			0.0433991	+	0.999058i	$0.486181\pi$
$858$		0			0
$859$	−224.397	−	224.397i	−0.261230	−	0.261230i	0.564324	−	0.825554i	$-0.309137\pi$
$859$	−224.397	−	224.397i	−0.261230	−	0.261230i	−0.825554	+	0.564324i	$0.809137\pi$
$860$		0			0
$861$	371.540			0.431522
$862$		0			0
$863$		−	1041.69i		−	1.20705i	−0.797343	−	0.603526i	$-0.793762\pi$
$863$		−	1041.69i		−	1.20705i	0.797343	−	0.603526i	$-0.206238\pi$
$864$		0			0
$865$	−482.911			−0.558279
$866$		0			0
$867$	332.254	+	332.254i	0.383223	+	0.383223i
$868$		0			0
$869$			1673.20i			1.92543i
$870$		0			0
$871$	−839.541			−0.963882
$872$		0			0
$873$	−305.175	+	305.175i	−0.349570	+	0.349570i
$874$		0			0
$875$			587.342i			0.671248i
$876$		0			0
$877$	1346.71			1.53559			0.767796	−	0.640695i	$-0.221354\pi$
$877$	1346.71			1.53559			0.767796	+	0.640695i	$0.221354\pi$
$878$		0			0
$879$			227.191i			0.258466i
$880$		0			0
$881$	−450.862	+	450.862i	−0.511762	+	0.511762i	−0.915066	−	0.403304i	$-0.867862\pi$
$881$	−450.862	+	450.862i	−0.511762	+	0.511762i	0.403304	+	0.915066i	$0.367862\pi$
$882$		0			0
$883$		−	1155.28i		−	1.30836i	−0.756339	−	0.654180i	$-0.773014\pi$
$883$		−	1155.28i		−	1.30836i	0.756339	−	0.654180i	$-0.226986\pi$
$884$		0			0
$885$	472.180	+	472.180i	0.533537	+	0.533537i
$886$		0			0
$887$	154.986	−	154.986i	0.174731	−	0.174731i	−0.614323	−	0.789054i	$-0.710571\pi$
$887$	154.986	−	154.986i	0.174731	−	0.174731i	0.789054	+	0.614323i	$0.210571\pi$
$888$		0			0
$889$	518.269	−	518.269i	0.582980	−	0.582980i
$890$		0			0
$891$	101.336	−	101.336i	0.113733	−	0.113733i
$892$		0			0
$893$	2032.15			2.27564
$894$		0			0
$895$	315.942			0.353007
$896$		0			0
$897$	15.2270	+	15.2270i	0.0169755	+	0.0169755i
$898$		0			0
$899$	368.081	−	65.2522i	0.409434	−	0.0725831i
$900$		0			0
$901$	126.620	−	126.620i	0.140532	−	0.140532i
$902$		0			0
$903$			1027.98i			1.13841i
$904$		0			0
$905$			522.101i			0.576907i
$906$		0			0
$907$	543.696	+	543.696i	0.599444	+	0.599444i	0.940165	−	0.340721i	$-0.110671\pi$
$907$	543.696	+	543.696i	0.599444	+	0.599444i	−0.340721	+	0.940165i	$0.610671\pi$
$908$		0			0
$909$	−235.517	−	235.517i	−0.259095	−	0.259095i
$910$		0			0
$911$	−1196.69	−	1196.69i	−1.31360	−	1.31360i	−0.918741	−	0.394861i	$-0.870793\pi$
$911$	−1196.69	−	1196.69i	−1.31360	−	1.31360i	−0.394861	−	0.918741i	$-0.629207\pi$
$912$		0			0
$913$	903.333	−	903.333i	0.989412	−	0.989412i
$914$		0			0
$915$	−549.063			−0.600069
$916$		0			0
$917$	517.376	+	517.376i	0.564205	+	0.564205i
$918$		0			0
$919$	446.342			0.485682			0.242841	−	0.970066i	$-0.421921\pi$
$919$	446.342			0.485682			0.242841	+	0.970066i	$0.421921\pi$
$920$		0			0
$921$		−	1021.52i		−	1.10914i
$922$		0			0
$923$	1119.46			1.21284
$924$		0			0
$925$	574.066	+	574.066i	0.620612	+	0.620612i
$926$		0			0
$927$		−	503.367i		−	0.543007i
$928$		0			0
$929$	−1017.18			−1.09492			−0.547458	−	0.836833i	$-0.684405\pi$
$929$	−1017.18			−1.09492			−0.547458	+	0.836833i	$0.684405\pi$
$930$		0			0
$931$	−406.375	+	406.375i	−0.436492	+	0.436492i
$932$		0			0
$933$		−	727.552i		−	0.779799i
$934$		0			0
$935$	−417.203			−0.446206
$936$		0			0
$937$			504.667i			0.538598i	0.963057	+	0.269299i	$0.0867920\pi$
$937$			504.667i			0.538598i	−0.963057	+	0.269299i	$0.913208\pi$
$938$		0			0
$939$	−484.732	+	484.732i	−0.516221	+	0.516221i
$940$		0			0
$941$		−	1144.29i		−	1.21604i	−0.793921	−	0.608021i	$-0.791964\pi$
$941$		−	1144.29i		−	1.21604i	0.793921	−	0.608021i	$-0.208036\pi$
$942$		0			0
$943$	17.9110	+	17.9110i	0.0189936	+	0.0189936i
$944$		0			0
$945$	191.809	−	191.809i	0.202973	−	0.202973i
$946$		0			0
$947$	458.125	−	458.125i	0.483764	−	0.483764i	−0.422567	−	0.906332i	$-0.638871\pi$
$947$	458.125	−	458.125i	0.483764	−	0.483764i	0.906332	+	0.422567i	$0.138871\pi$
$948$		0			0
$949$	201.521	−	201.521i	0.212350	−	0.212350i
$950$		0			0
$951$	−86.7283			−0.0911969
$952$		0			0
$953$	1072.75			1.12565			0.562827	−	0.826575i	$-0.309714\pi$
$953$	1072.75			1.12565			0.562827	+	0.826575i	$0.309714\pi$
$954$		0			0
$955$	822.940	+	822.940i	0.861718	+	0.861718i
$956$		0			0
$957$	−458.158	+	655.602i	−0.478744	+	0.685060i
$958$		0			0
$959$	361.733	−	361.733i	0.377198	−	0.377198i
$960$		0			0
$961$			794.839i			0.827096i
$962$		0			0
$963$		−	124.971i		−	0.129772i
$964$		0			0
$965$	−936.554	−	936.554i	−0.970522	−	0.970522i
$966$		0			0
$967$	108.375	+	108.375i	0.112074	+	0.112074i	0.760920	−	0.648846i	$-0.224748\pi$
$967$	108.375	+	108.375i	0.112074	+	0.112074i	−0.648846	+	0.760920i	$0.724748\pi$
$968$		0			0
$969$	−138.888	−	138.888i	−0.143332	−	0.143332i
$970$		0			0
$971$	−602.558	+	602.558i	−0.620555	+	0.620555i	−0.945673	−	0.325119i	$-0.894596\pi$
$971$	−602.558	+	602.558i	−0.620555	+	0.620555i	0.325119	+	0.945673i	$0.394596\pi$
$972$		0			0
$973$	−1023.73			−1.05213
$974$		0			0
$975$	211.410	+	211.410i	0.216831	+	0.216831i
$976$		0			0
$977$	−272.492			−0.278907			−0.139453	−	0.990229i	$-0.544535\pi$
$977$	−272.492			−0.278907			−0.139453	+	0.990229i	$0.544535\pi$
$978$		0			0
$979$			82.8308i			0.0846076i
$980$		0			0
$981$	−5.86205			−0.00597558
$982$		0			0
$983$	79.4702	+	79.4702i	0.0808445	+	0.0808445i	0.746373	−	0.665528i	$-0.231794\pi$
$983$	79.4702	+	79.4702i	0.0808445	+	0.0808445i	−0.665528	+	0.746373i	$0.731794\pi$
$984$		0			0
$985$		−	2242.30i		−	2.27644i
$986$		0			0
$987$	1095.58			1.11002
$988$		0			0
$989$	−49.5563	+	49.5563i	−0.0501075	+	0.0501075i
$990$		0			0
$991$		−	267.139i		−	0.269565i	−0.990875	−	0.134782i	$-0.956966\pi$
$991$		−	267.139i		−	0.269565i	0.990875	−	0.134782i	$-0.0430335\pi$
$992$		0			0
$993$	827.367			0.833200
$994$		0			0
$995$			2105.86i			2.11644i
$996$		0			0
$997$	164.885	−	164.885i	0.165381	−	0.165381i	−0.619565	−	0.784946i	$-0.712691\pi$
$997$	164.885	−	164.885i	0.165381	−	0.165381i	0.784946	+	0.619565i	$0.212691\pi$
$998$		0			0
$999$			306.821i			0.307129i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	348.3.j.a.133.7	✓	20
3.2	odd	2		1044.3.k.b.829.9		20
29.12	odd	4	inner	348.3.j.a.157.9	yes	20
87.41	even	4		1044.3.k.b.505.2		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
348.3.j.a.133.7	✓	20	1.1	even	1	trivial
348.3.j.a.157.9	yes	20	29.12	odd	4	inner
1044.3.k.b.505.2		20	87.41	even	4
1044.3.k.b.829.9		20	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 348 = 2^{2} \cdot 3 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	348.j (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.22474 - 1.22474i) q^{3} -6.22487i q^{5} -8.38632 q^{7} -3.00000i q^{9} +O(q^{10})\)	\(q+(1.22474 - 1.22474i) q^{3} -6.22487i q^{5} -8.38632 q^{7} -3.00000i q^{9} +(-11.2596 + 11.2596i) q^{11} -12.5547i q^{13} +(-7.62388 - 7.62388i) q^{15} +(-2.97621 + 2.97621i) q^{17} +(-19.0514 + 19.0514i) q^{19} +(-10.2711 + 10.2711i) q^{21} -0.990288 q^{23} -13.7491 q^{25} +(-3.67423 - 3.67423i) q^{27} +(23.7707 + 16.6118i) q^{29} +(9.11485 - 9.11485i) q^{31} +27.5803i q^{33} +52.2038i q^{35} +(-41.7531 - 41.7531i) q^{37} +(-15.3764 - 15.3764i) q^{39} +(-18.0867 - 18.0867i) q^{41} +(50.0423 - 50.0423i) q^{43} -18.6746 q^{45} +(-53.3333 - 53.3333i) q^{47} +21.3304 q^{49} +7.29020i q^{51} -42.5439 q^{53} +(70.0895 + 70.0895i) q^{55} +46.6662i q^{57} -61.9343 q^{59} +(36.0094 - 36.0094i) q^{61} +25.1590i q^{63} -78.1517 q^{65} -66.8704i q^{67} +(-1.21285 + 1.21285i) q^{69} +89.1659i q^{71} +(16.0513 + 16.0513i) q^{73} +(-16.8391 + 16.8391i) q^{75} +(94.4266 - 94.4266i) q^{77} +(74.3011 - 74.3011i) q^{79} -9.00000 q^{81} -80.2278 q^{83} +(18.5265 + 18.5265i) q^{85} +(49.4583 - 8.76782i) q^{87} +(3.67823 - 3.67823i) q^{89} +105.288i q^{91} -22.3267i q^{93} +(118.592 + 118.592i) q^{95} +(-101.725 - 101.725i) q^{97} +(33.7788 + 33.7788i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q+O(q^{10}) \) 20 * q	\( 20 q - 36 q^{15} + 8 q^{17} + 8 q^{19} - 72 q^{23} - 124 q^{25} + 36 q^{29} + 44 q^{31} + 32 q^{37} - 48 q^{39} - 56 q^{41} + 64 q^{43} + 48 q^{45} - 56 q^{47} + 44 q^{49} + 176 q^{53} + 168 q^{55} - 80 q^{59} - 64 q^{61} - 296 q^{65} - 48 q^{69} - 172 q^{73} + 48 q^{75} + 336 q^{77} - 244 q^{79} - 180 q^{81} - 368 q^{83} + 592 q^{85} + 12 q^{87} + 176 q^{89} - 184 q^{95} + 188 q^{97}+O(q^{100}) \) 20 * q - 36 * q^15 + 8 * q^17 + 8 * q^19 - 72 * q^23 - 124 * q^25 + 36 * q^29 + 44 * q^31 + 32 * q^37 - 48 * q^39 - 56 * q^41 + 64 * q^43 + 48 * q^45 - 56 * q^47 + 44 * q^49 + 176 * q^53 + 168 * q^55 - 80 * q^59 - 64 * q^61 - 296 * q^65 - 48 * q^69 - 172 * q^73 + 48 * q^75 + 336 * q^77 - 244 * q^79 - 180 * q^81 - 368 * q^83 + 592 * q^85 + 12 * q^87 + 176 * q^89 - 184 * q^95 + 188 * q^97

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