LMFDB - Embedded newform 1904.2.c.g.1121.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1904,2,Mod(1121,1904)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1904.1121"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1904, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1904 = 2^{4} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1904.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$15.2035165449$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.6179217664.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 13x^{6} + 40x^{4} + 13x^{2} + 1 $ x^8 + 13x^6 + 40x^4 + 13*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 952)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1121.1
Root			$0.344151i$ of defining polynomial
Character	$\chi$	$=$	1904.1121
Dual form			1904.2.c.g.1121.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.04948i q^{3} +1.48793i q^{5} -1.00000i q^{7} -6.29934 q^{9} +3.81141i q^{11} -5.12311 q^{13} +4.53741 q^{15} +(-3.24985 - 2.53741i) q^{17} +2.09896 q^{19} -3.04948 q^{21} +2.97586i q^{23} +2.78607 q^{25} +10.0613i q^{27} +6.78726i q^{29} -3.92273i q^{31} +11.6228 q^{33} +1.48793 q^{35} +0.688301i q^{37} +15.6228i q^{39} +11.8850i q^{41} +9.83674 q^{43} -9.37296i q^{45} -10.0990 q^{47} -1.00000 q^{49} +(-7.73778 + 9.91037i) q^{51} -4.32348 q^{53} -5.67110 q^{55} -6.40075i q^{57} -1.02414 q^{59} +4.22087i q^{61} +6.29934i q^{63} -7.62281i q^{65} +9.15209 q^{67} +9.07482 q^{69} +3.42489i q^{71} +13.2616i q^{73} -8.49607i q^{75} +3.81141 q^{77} +10.4514i q^{79} +11.7836 q^{81} +3.69822 q^{83} +(3.77548 - 4.83555i) q^{85} +20.6976 q^{87} -11.3693 q^{89} +5.12311i q^{91} -11.9623 q^{93} +3.12311i q^{95} -14.3235i q^{97} -24.0093i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 6 q^{9} - 8 q^{13} + 16 q^{15} - 20 q^{19} - 6 q^{21} + 2 q^{25} + 8 q^{33} + 10 q^{35} + 14 q^{43} - 44 q^{47} - 8 q^{49} - 34 q^{51} + 6 q^{53} + 32 q^{55} - 12 q^{59} + 58 q^{67} + 32 q^{69} - 12 q^{77}+ \cdots - 64 q^{93}+O(q^{100}) $ 8 * q - 6 * q^9 - 8 * q^13 + 16 * q^15 - 20 * q^19 - 6 * q^21 + 2 * q^25 + 8 * q^33 + 10 * q^35 + 14 * q^43 - 44 * q^47 - 8 * q^49 - 34 * q^51 + 6 * q^53 + 32 * q^55 - 12 * q^59 + 58 * q^67 + 32 * q^69 - 12 * q^77 + 8 * q^81 + 8 * q^83 + 34 * q^85 + 40 * q^87 + 8 * q^89 - 64 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1904\mathbb{Z}\right)^\times$.

$n$	$239$	$785$	$1361$	$1429$
$\chi(n)$	$1$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		−	3.04948i		−	1.76062i	−0.474400	−	0.880309i	$-0.657335\pi$
$3$		−	3.04948i		−	1.76062i	0.474400	−	0.880309i	$-0.342665\pi$
$4$		0			0
$5$			1.48793i			0.665422i	0.943029	+	0.332711i	$0.107963\pi$
$5$			1.48793i			0.665422i	−0.943029	+	0.332711i	$0.892037\pi$
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	−6.29934			−2.09978
$10$		0			0
$11$			3.81141i			1.14918i	0.818441	+	0.574591i	$0.194839\pi$
$11$			3.81141i			1.14918i	−0.818441	+	0.574591i	$0.805161\pi$
$12$		0			0
$13$	−5.12311			−1.42089			−0.710447	−	0.703751i	$-0.751507\pi$
$13$	−5.12311			−1.42089			−0.710447	+	0.703751i	$0.751507\pi$
$14$		0			0
$15$	4.53741			1.17155
$16$		0			0
$17$	−3.24985	−	2.53741i	−0.788205	−	0.615412i
$17$	−3.24985	−	2.53741i	−0.788205	−	0.615412i
$18$		0			0
$19$	2.09896			0.481535			0.240767	−	0.970583i	$-0.422601\pi$
$19$	2.09896			0.481535			0.240767	+	0.970583i	$0.422601\pi$
$20$		0			0
$21$	−3.04948			−0.665451
$22$		0			0
$23$			2.97586i			0.620509i	0.950654	+	0.310255i	$0.100414\pi$
$23$			2.97586i			0.620509i	−0.950654	+	0.310255i	$0.899586\pi$
$24$		0			0
$25$	2.78607			0.557214
$26$		0			0
$27$			10.0613i			1.93629i
$28$		0			0
$29$			6.78726i			1.26036i	0.776448	+	0.630182i	$0.217020\pi$
$29$			6.78726i			1.26036i	−0.776448	+	0.630182i	$0.782980\pi$
$30$		0			0
$31$		−	3.92273i		−	0.704544i	−0.935898	−	0.352272i	$-0.885409\pi$
$31$		−	3.92273i		−	0.704544i	0.935898	−	0.352272i	$-0.114591\pi$
$32$		0			0
$33$	11.6228			2.02327
$34$		0			0
$35$	1.48793			0.251506
$36$		0			0
$37$			0.688301i			0.113156i	0.998398	+	0.0565780i	$0.0180190\pi$
$37$			0.688301i			0.113156i	−0.998398	+	0.0565780i	$0.981981\pi$
$38$		0			0
$39$			15.6228i			2.50165i
$40$		0			0
$41$			11.8850i			1.85613i	0.372418	+	0.928065i	$0.378529\pi$
$41$			11.8850i			1.85613i	−0.372418	+	0.928065i	$0.621471\pi$
$42$		0			0
$43$	9.83674			1.50009			0.750045	−	0.661387i	$-0.230032\pi$
$43$	9.83674			1.50009			0.750045	+	0.661387i	$0.230032\pi$
$44$		0			0
$45$		−	9.37296i		−	1.39724i
$46$		0			0
$47$	−10.0990			−1.47309			−0.736543	−	0.676391i	$-0.763543\pi$
$47$	−10.0990			−1.47309			−0.736543	+	0.676391i	$0.763543\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−7.73778	+	9.91037i	−1.08351	+	1.38773i
$52$		0			0
$53$	−4.32348			−0.593875			−0.296938	−	0.954897i	$-0.595965\pi$
$53$	−4.32348			−0.593875			−0.296938	+	0.954897i	$0.595965\pi$
$54$		0			0
$55$	−5.67110			−0.764691
$56$		0			0
$57$		−	6.40075i		−	0.847800i
$58$		0			0
$59$	−1.02414			−0.133332			−0.0666660	−	0.997775i	$-0.521236\pi$
$59$	−1.02414			−0.133332			−0.0666660	+	0.997775i	$0.521236\pi$
$60$		0			0
$61$			4.22087i			0.540427i	0.962800	+	0.270214i	$0.0870944\pi$
$61$			4.22087i			0.540427i	−0.962800	+	0.270214i	$0.912906\pi$
$62$		0			0
$63$			6.29934i			0.793642i
$64$		0			0
$65$		−	7.62281i		−	0.945494i
$66$		0			0
$67$	9.15209			1.11811			0.559053	−	0.829132i	$-0.311165\pi$
$67$	9.15209			1.11811			0.559053	+	0.829132i	$0.311165\pi$
$68$		0			0
$69$	9.07482			1.09248
$70$		0			0
$71$			3.42489i			0.406460i	0.979131	+	0.203230i	$0.0651438\pi$
$71$			3.42489i			0.406460i	−0.979131	+	0.203230i	$0.934856\pi$
$72$		0			0
$73$			13.2616i			1.55216i	0.630636	+	0.776078i	$0.282794\pi$
$73$			13.2616i			1.55216i	−0.630636	+	0.776078i	$0.717206\pi$
$74$		0			0
$75$		−	8.49607i		−	0.981041i
$76$		0			0
$77$	3.81141			0.434350
$78$		0			0
$79$			10.4514i			1.17588i	0.808906	+	0.587938i	$0.200060\pi$
$79$			10.4514i			1.17588i	−0.808906	+	0.587938i	$0.799940\pi$
$80$		0			0
$81$	11.7836			1.30929
$82$		0			0
$83$	3.69822			0.405932			0.202966	−	0.979186i	$-0.434942\pi$
$83$	3.69822			0.405932			0.202966	+	0.979186i	$0.434942\pi$
$84$		0			0
$85$	3.77548	−	4.83555i	0.409509	−	0.524489i
$86$		0			0
$87$	20.6976			2.21902
$88$		0			0
$89$	−11.3693			−1.20515			−0.602573	−	0.798064i	$-0.705858\pi$
$89$	−11.3693			−1.20515			−0.602573	+	0.798064i	$0.705858\pi$
$90$		0			0
$91$			5.12311i			0.537047i
$92$		0			0
$93$	−11.9623			−1.24043
$94$		0			0
$95$			3.12311i			0.320424i
$96$		0			0
$97$		−	14.3235i		−	1.45433i	−0.686463	−	0.727164i	$-0.740838\pi$
$97$		−	14.3235i		−	1.45433i	0.686463	−	0.727164i	$-0.259162\pi$
$98$		0			0
$99$		−	24.0093i		−	2.41303i
$100$		0			0
$101$	0.674073			0.0670728			0.0335364	−	0.999437i	$-0.489323\pi$
$101$	0.674073			0.0670728			0.0335364	+	0.999437i	$0.489323\pi$
$102$		0			0
$103$	−16.9994			−1.67500			−0.837501	−	0.546436i	$-0.815984\pi$
$103$	−16.9994			−1.67500			−0.837501	+	0.546436i	$0.815984\pi$
$104$		0			0
$105$		−	4.53741i		−	0.442806i
$106$		0			0
$107$		−	5.71244i		−	0.552243i	−0.961123	−	0.276121i	$-0.910951\pi$
$107$		−	5.71244i		−	0.552243i	0.961123	−	0.276121i	$-0.0890492\pi$
$108$		0			0
$109$		−	3.95866i		−	0.379170i	−0.981864	−	0.189585i	$-0.939286\pi$
$109$		−	3.95866i		−	0.379170i	0.981864	−	0.189585i	$-0.0607143\pi$
$110$		0			0
$111$	2.09896			0.199225
$112$		0			0
$113$			8.55038i			0.804353i	0.915562	+	0.402176i	$0.131746\pi$
$113$			8.55038i			0.804353i	−0.915562	+	0.402176i	$0.868254\pi$
$114$		0			0
$115$	−4.42786			−0.412900
$116$		0			0
$117$	32.2722			2.98356
$118$		0			0
$119$	−2.53741	+	3.24985i	−0.232604	+	0.297914i
$120$		0			0
$121$	−3.52682			−0.320620
$122$		0			0
$123$	36.2432			3.26794
$124$		0			0
$125$			11.5851i			1.03620i
$126$		0			0
$127$	−1.37474			−0.121988			−0.0609941	−	0.998138i	$-0.519427\pi$
$127$	−1.37474			−0.121988			−0.0609941	+	0.998138i	$0.519427\pi$
$128$		0			0
$129$		−	29.9970i		−	2.64109i
$130$		0			0
$131$		−	15.4342i		−	1.34849i	−0.738506	−	0.674247i	$-0.764468\pi$
$131$		−	15.4342i		−	1.34849i	0.738506	−	0.674247i	$-0.235532\pi$
$132$		0			0
$133$		−	2.09896i		−	0.182003i
$134$		0			0
$135$	−14.9704			−1.28845
$136$		0			0
$137$	−14.0323			−1.19886			−0.599429	−	0.800428i	$-0.704606\pi$
$137$	−14.0323			−1.19886			−0.599429	+	0.800428i	$0.704606\pi$
$138$		0			0
$139$			14.3328i			1.21569i	0.794055	+	0.607847i	$0.207966\pi$
$139$			14.3328i			1.21569i	−0.794055	+	0.607847i	$0.792034\pi$
$140$		0			0
$141$			30.7966i			2.59354i
$142$		0			0
$143$		−	19.5262i		−	1.63287i
$144$		0			0
$145$	−10.0990			−0.838673
$146$		0			0
$147$			3.04948i			0.251517i
$148$		0			0
$149$	16.0347			1.31361			0.656806	−	0.754060i	$-0.271907\pi$
$149$	16.0347			1.31361			0.656806	+	0.754060i	$0.271907\pi$
$150$		0			0
$151$	11.2133			0.912529			0.456265	−	0.889844i	$-0.349187\pi$
$151$	11.2133			0.912529			0.456265	+	0.889844i	$0.349187\pi$
$152$		0			0
$153$	20.4719	+	15.9840i	1.65506	+	1.29223i
$154$		0			0
$155$	5.83674			0.468819
$156$		0			0
$157$	−17.6735			−1.41050			−0.705249	−	0.708960i	$-0.749165\pi$
$157$	−17.6735			−1.41050			−0.705249	+	0.708960i	$0.749165\pi$
$158$		0			0
$159$			13.1844i			1.04559i
$160$		0			0
$161$	2.97586			0.234530
$162$		0			0
$163$		−	19.2870i		−	1.51067i	−0.655338	−	0.755336i	$-0.727474\pi$
$163$		−	19.2870i		−	1.51067i	0.655338	−	0.755336i	$-0.272526\pi$
$164$		0			0
$165$			17.2939i			1.34633i
$166$		0			0
$167$		−	2.08785i		−	0.161563i	−0.996732	−	0.0807815i	$-0.974258\pi$
$167$		−	2.08785i		−	0.161563i	0.996732	−	0.0807815i	$-0.0257416\pi$
$168$		0			0
$169$	13.2462			1.01894
$170$		0			0
$171$	−13.2221			−1.01112
$172$		0			0
$173$			22.4496i			1.70681i	0.521244	+	0.853407i	$0.325468\pi$
$173$			22.4496i			1.70681i	−0.521244	+	0.853407i	$0.674532\pi$
$174$		0			0
$175$		−	2.78607i		−	0.210607i
$176$		0			0
$177$			3.12311i			0.234747i
$178$		0			0
$179$	10.3612			0.774431			0.387215	−	0.921989i	$-0.373437\pi$
$179$	10.3612			0.774431			0.387215	+	0.921989i	$0.373437\pi$
$180$		0			0
$181$		−	5.26341i		−	0.391226i	−0.980681	−	0.195613i	$-0.937330\pi$
$181$		−	5.26341i		−	0.391226i	0.980681	−	0.195613i	$-0.0626697\pi$
$182$		0			0
$183$	12.8715			0.951487
$184$		0			0
$185$	−1.02414			−0.0752965
$186$		0			0
$187$	9.67110	−	12.3865i	0.707221	−	0.905792i
$188$		0			0
$189$	10.0613			0.731849
$190$		0			0
$191$	−19.9487			−1.44344			−0.721718	−	0.692187i	$-0.756647\pi$
$191$	−19.9487			−1.44344			−0.721718	+	0.692187i	$0.756647\pi$
$192$		0			0
$193$			6.09896i			0.439013i	0.975611	+	0.219506i	$0.0704447\pi$
$193$			6.09896i			0.439013i	−0.975611	+	0.219506i	$0.929555\pi$
$194$		0			0
$195$	−23.2456			−1.66465
$196$		0			0
$197$			6.26283i			0.446208i	0.974795	+	0.223104i	$0.0716190\pi$
$197$			6.26283i			0.446208i	−0.974795	+	0.223104i	$0.928381\pi$
$198$		0			0
$199$			7.18740i			0.509501i	0.967007	+	0.254751i	$0.0819934\pi$
$199$			7.18740i			0.509501i	−0.967007	+	0.254751i	$0.918007\pi$
$200$		0			0
$201$		−	27.9091i		−	1.96856i
$202$		0			0
$203$	6.78726			0.476373
$204$		0			0
$205$	−17.6841			−1.23511
$206$		0			0
$207$		−	18.7459i		−	1.30293i
$208$		0			0
$209$			8.00000i			0.553372i
$210$		0			0
$211$			5.93510i			0.408589i	0.978909	+	0.204294i	$0.0654900\pi$
$211$			5.93510i			0.408589i	−0.978909	+	0.204294i	$0.934510\pi$
$212$		0			0
$213$	10.4441			0.715620
$214$		0			0
$215$			14.6364i			0.998192i
$216$		0			0
$217$	−3.92273			−0.266292
$218$		0			0
$219$	40.4411			2.73276
$220$		0			0
$221$	16.6493	+	12.9994i	1.11996	+	0.874435i
$222$		0			0
$223$	21.5166			1.44086			0.720428	−	0.693530i	$-0.243945\pi$
$223$	21.5166			1.44086			0.720428	+	0.693530i	$0.243945\pi$
$224$		0			0
$225$	−17.5504			−1.17003
$226$		0			0
$227$		−	6.03831i		−	0.400777i	−0.979717	−	0.200388i	$-0.935780\pi$
$227$		−	6.03831i		−	0.400777i	0.979717	−	0.200388i	$-0.0642204\pi$
$228$		0			0
$229$	17.0477			1.12654			0.563272	−	0.826272i	$-0.309542\pi$
$229$	17.0477			1.12654			0.563272	+	0.826272i	$0.309542\pi$
$230$		0			0
$231$		−	11.6228i		−	0.764725i
$232$		0			0
$233$			2.82861i			0.185308i	0.995698	+	0.0926541i	$0.0295351\pi$
$233$			2.82861i			0.185308i	−0.995698	+	0.0926541i	$0.970465\pi$
$234$		0			0
$235$		−	15.0265i		−	0.980223i
$236$		0			0
$237$	31.8714			2.07027
$238$		0			0
$239$	−3.08354			−0.199458			−0.0997288	−	0.995015i	$-0.531798\pi$
$239$	−3.08354			−0.199458			−0.0997288	+	0.995015i	$0.531798\pi$
$240$		0			0
$241$		−	15.3229i		−	0.987034i	−0.869736	−	0.493517i	$-0.835711\pi$
$241$		−	15.3229i		−	0.987034i	0.869736	−	0.493517i	$-0.164289\pi$
$242$		0			0
$243$		−	5.75015i		−	0.368872i
$244$		0			0
$245$		−	1.48793i		−	0.0950603i
$246$		0			0
$247$	−10.7532			−0.684210
$248$		0			0
$249$		−	11.2776i		−	0.714691i
$250$		0			0
$251$	−22.8473			−1.44211			−0.721053	−	0.692879i	$-0.756342\pi$
$251$	−22.8473			−1.44211			−0.721053	+	0.692879i	$0.756342\pi$
$252$		0			0
$253$	−11.3422			−0.713078
$254$		0			0
$255$	−14.7459	−	11.5133i	−0.923425	−	0.720989i
$256$		0			0
$257$	−25.5262			−1.59228			−0.796142	−	0.605110i	$-0.793129\pi$
$257$	−25.5262			−1.59228			−0.796142	+	0.605110i	$0.793129\pi$
$258$		0			0
$259$	0.688301			0.0427690
$260$		0			0
$261$		−	42.7553i		−	2.64648i
$262$		0			0
$263$	−27.0192			−1.66608			−0.833039	−	0.553214i	$-0.813401\pi$
$263$	−27.0192			−1.66608			−0.833039	+	0.553214i	$0.813401\pi$
$264$		0			0
$265$		−	6.43303i		−	0.395178i
$266$		0			0
$267$			34.6705i			2.12180i
$268$		0			0
$269$			14.6400i			0.892617i	0.894879	+	0.446309i	$0.147262\pi$
$269$			14.6400i			0.892617i	−0.894879	+	0.446309i	$0.852738\pi$
$270$		0			0
$271$	−22.8473			−1.38787			−0.693936	−	0.720036i	$-0.744125\pi$
$271$	−22.8473			−1.38787			−0.693936	+	0.720036i	$0.744125\pi$
$272$		0			0
$273$	15.6228			0.945536
$274$		0			0
$275$			10.6188i			0.640340i
$276$		0			0
$277$		−	9.23869i		−	0.555099i	−0.960711	−	0.277549i	$-0.910478\pi$
$277$		−	9.23869i		−	0.555099i	0.960711	−	0.277549i	$-0.0895222\pi$
$278$		0			0
$279$			24.7106i			1.47939i
$280$		0			0
$281$	−10.0525			−0.599684			−0.299842	−	0.953989i	$-0.596934\pi$
$281$	−10.0525			−0.599684			−0.299842	+	0.953989i	$0.596934\pi$
$282$		0			0
$283$		−	19.3570i		−	1.15065i	−0.817924	−	0.575326i	$-0.804875\pi$
$283$		−	19.3570i		−	1.15065i	0.817924	−	0.575326i	$-0.195125\pi$
$284$		0			0
$285$	9.52385			0.564144
$286$		0			0
$287$	11.8850			0.701551
$288$		0			0
$289$	4.12311	+	16.4924i	0.242536	+	0.970143i
$290$		0			0
$291$	−43.6792			−2.56052
$292$		0			0
$293$	6.35246			0.371115			0.185557	−	0.982633i	$-0.440591\pi$
$293$	6.35246			0.371115			0.185557	+	0.982633i	$0.440591\pi$
$294$		0			0
$295$		−	1.52385i		−	0.0887221i
$296$		0			0
$297$	−38.3476			−2.22515
$298$		0			0
$299$		−	15.2456i		−	0.881677i
$300$		0			0
$301$		−	9.83674i		−	0.566981i
$302$		0			0
$303$		−	2.05557i		−	0.118090i
$304$		0			0
$305$	−6.28036			−0.359612
$306$		0			0
$307$	−1.64993			−0.0941665			−0.0470832	−	0.998891i	$-0.514993\pi$
$307$	−1.64993			−0.0941665			−0.0470832	+	0.998891i	$0.514993\pi$
$308$		0			0
$309$			51.8394i			2.94904i
$310$		0			0
$311$			2.47801i			0.140515i	0.997529	+	0.0702576i	$0.0223821\pi$
$311$			2.47801i			0.140515i	−0.997529	+	0.0702576i	$0.977618\pi$
$312$		0			0
$313$		−	13.1201i		−	0.741590i	−0.928715	−	0.370795i	$-0.879085\pi$
$313$		−	13.1201i		−	0.741590i	0.928715	−	0.370795i	$-0.120915\pi$
$314$		0			0
$315$	−9.37296			−0.528106
$316$		0			0
$317$		−	15.0099i		−	0.843041i	−0.906819	−	0.421520i	$-0.861497\pi$
$317$		−	15.0099i		−	0.843041i	0.906819	−	0.421520i	$-0.138503\pi$
$318$		0			0
$319$	−25.8690			−1.44839
$320$		0			0
$321$	−17.4200			−0.972289
$322$		0			0
$323$	−6.82132	−	5.32593i	−0.379548	−	0.296343i
$324$		0			0
$325$	−14.2733			−0.791742
$326$		0			0
$327$	−12.0718			−0.667575
$328$		0			0
$329$			10.0990i			0.556774i
$330$		0			0
$331$	−22.1949			−1.21994			−0.609971	−	0.792424i	$-0.708819\pi$
$331$	−22.1949			−1.21994			−0.609971	+	0.792424i	$0.708819\pi$
$332$		0			0
$333$		−	4.33584i		−	0.237603i
$334$		0			0
$335$			13.6176i			0.744012i
$336$		0			0
$337$			16.4713i			0.897246i	0.893721	+	0.448623i	$0.148085\pi$
$337$			16.4713i			0.897246i	−0.893721	+	0.448623i	$0.851915\pi$
$338$		0			0
$339$	26.0742			1.41616
$340$		0			0
$341$	14.9511			0.809649
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$			13.5027i			0.726960i
$346$		0			0
$347$			8.31112i			0.446164i	0.974800	+	0.223082i	$0.0716118\pi$
$347$			8.31112i			0.446164i	−0.974800	+	0.223082i	$0.928388\pi$
$348$		0			0
$349$	−22.1014			−1.18306			−0.591529	−	0.806284i	$-0.701476\pi$
$349$	−22.1014			−1.18306			−0.591529	+	0.806284i	$0.701476\pi$
$350$		0			0
$351$		−	51.5449i		−	2.75126i
$352$		0			0
$353$	35.0501			1.86553			0.932764	−	0.360487i	$-0.117390\pi$
$353$	35.0501			1.86553			0.932764	+	0.360487i	$0.117390\pi$
$354$		0			0
$355$	−5.09599			−0.270467
$356$		0			0
$357$	9.91037	+	7.73778i	0.524512	+	0.409527i
$358$		0			0
$359$	−14.6005			−0.770587			−0.385293	−	0.922794i	$-0.625900\pi$
$359$	−14.6005			−0.770587			−0.385293	+	0.922794i	$0.625900\pi$
$360$		0			0
$361$	−14.5944			−0.768124
$362$		0			0
$363$			10.7550i			0.564490i
$364$		0			0
$365$	−19.7324			−1.03284
$366$		0			0
$367$		−	7.28875i		−	0.380470i	−0.981739	−	0.190235i	$-0.939075\pi$
$367$		−	7.28875i		−	0.380470i	0.981739	−	0.190235i	$-0.0609249\pi$
$368$		0			0
$369$		−	74.8678i		−	3.89746i
$370$		0			0
$371$			4.32348i			0.224464i
$372$		0			0
$373$	−14.8844			−0.770688			−0.385344	−	0.922773i	$-0.625917\pi$
$373$	−14.8844			−0.770688			−0.385344	+	0.922773i	$0.625917\pi$
$374$		0			0
$375$	35.3286			1.82436
$376$		0			0
$377$		−	34.7719i		−	1.79084i
$378$		0			0
$379$			36.2791i			1.86353i	0.363059	+	0.931766i	$0.381732\pi$
$379$			36.2791i			1.86353i	−0.363059	+	0.931766i	$0.618268\pi$
$380$		0			0
$381$			4.19224i			0.214775i
$382$		0			0
$383$	29.1973			1.49191			0.745957	−	0.665994i	$-0.231992\pi$
$383$	29.1973			1.49191			0.745957	+	0.665994i	$0.231992\pi$
$384$		0			0
$385$			5.67110i			0.289026i
$386$		0			0
$387$	−61.9650			−3.14986
$388$		0			0
$389$	12.4634			0.631922			0.315961	−	0.948772i	$-0.397673\pi$
$389$	12.4634			0.631922			0.315961	+	0.948772i	$0.397673\pi$
$390$		0			0
$391$	7.55097	−	9.67110i	0.381869	−	0.489089i
$392$		0			0
$393$	−47.0664			−2.37418
$394$		0			0
$395$	−15.5510			−0.782454
$396$		0			0
$397$			4.38110i			0.219881i	0.993938	+	0.109940i	$0.0350660\pi$
$397$			4.38110i			0.219881i	−0.993938	+	0.109940i	$0.964934\pi$
$398$		0			0
$399$	−6.40075			−0.320438
$400$		0			0
$401$			17.3234i			0.865090i	0.901612	+	0.432545i	$0.142384\pi$
$401$			17.3234i			0.865090i	−0.901612	+	0.432545i	$0.857616\pi$
$402$		0			0
$403$			20.0966i			1.00108i
$404$		0			0
$405$			17.5332i			0.871231i
$406$		0			0
$407$	−2.62340			−0.130037
$408$		0			0
$409$	−28.3228			−1.40047			−0.700237	−	0.713910i	$-0.746922\pi$
$409$	−28.3228			−1.40047			−0.700237	+	0.713910i	$0.746922\pi$
$410$		0			0
$411$			42.7912i			2.11073i
$412$		0			0
$413$			1.02414i			0.0503948i
$414$		0			0
$415$			5.50268i			0.270116i
$416$		0			0
$417$	43.7076			2.14037
$418$		0			0
$419$			12.9562i			0.632952i	0.948600	+	0.316476i	$0.102500\pi$
$419$			12.9562i			0.632952i	−0.948600	+	0.316476i	$0.897500\pi$
$420$		0			0
$421$	−38.6913			−1.88570			−0.942849	−	0.333219i	$-0.891865\pi$
$421$	−38.6913			−1.88570			−0.942849	+	0.333219i	$0.891865\pi$
$422$		0			0
$423$	63.6168			3.09315
$424$		0			0
$425$	−9.05432	−	7.06940i	−0.439199	−	0.342916i
$426$		0			0
$427$	4.22087			0.204262
$428$		0			0
$429$	−59.5449			−2.87486
$430$		0			0
$431$			13.5486i			0.652612i	0.945264	+	0.326306i	$0.105804\pi$
$431$			13.5486i			0.652612i	−0.945264	+	0.326306i	$0.894196\pi$
$432$		0			0
$433$	−15.4273			−0.741388			−0.370694	−	0.928755i	$-0.620880\pi$
$433$	−15.4273			−0.741388			−0.370694	+	0.928755i	$0.620880\pi$
$434$		0			0
$435$			30.7966i			1.47658i
$436$		0			0
$437$			6.24621i			0.298797i
$438$		0			0
$439$		−	2.98947i		−	0.142680i	−0.997452	−	0.0713399i	$-0.977273\pi$
$439$		−	2.98947i		−	0.142680i	0.997452	−	0.0713399i	$-0.0227275\pi$
$440$		0			0
$441$	6.29934			0.299968
$442$		0			0
$443$	−11.8492			−0.562972			−0.281486	−	0.959565i	$-0.590827\pi$
$443$	−11.8492			−0.562972			−0.281486	+	0.959565i	$0.590827\pi$
$444$		0			0
$445$		−	16.9167i		−	0.801930i
$446$		0			0
$447$		−	48.8974i		−	2.31277i
$448$		0			0
$449$			10.2709i			0.484716i	0.970187	+	0.242358i	$0.0779208\pi$
$449$			10.2709i			0.484716i	−0.970187	+	0.242358i	$0.922079\pi$
$450$		0			0
$451$	−45.2987			−2.13303
$452$		0			0
$453$		−	34.1949i		−	1.60662i
$454$		0			0
$455$	−7.62281			−0.357363
$456$		0			0
$457$	15.3241			0.716829			0.358415	−	0.933562i	$-0.383317\pi$
$457$	15.3241			0.716829			0.358415	+	0.933562i	$0.383317\pi$
$458$		0			0
$459$	25.5295	−	32.6976i	1.19162	−	1.52619i
$460$		0			0
$461$	9.55097			0.444833			0.222416	−	0.974952i	$-0.428606\pi$
$461$	9.55097			0.444833			0.222416	+	0.974952i	$0.428606\pi$
$462$		0			0
$463$	22.7855			1.05893			0.529466	−	0.848331i	$-0.322392\pi$
$463$	22.7855			1.05893			0.529466	+	0.848331i	$0.322392\pi$
$464$		0			0
$465$		−	17.7990i		−	0.825411i
$466$		0			0
$467$	21.5957			0.999330			0.499665	−	0.866219i	$-0.333456\pi$
$467$	21.5957			0.999330			0.499665	+	0.866219i	$0.333456\pi$
$468$		0			0
$469$		−	9.15209i		−	0.422604i
$470$		0			0
$471$			53.8950i			2.48335i
$472$		0			0
$473$			37.4918i			1.72388i
$474$		0			0
$475$	5.84785			0.268318
$476$		0			0
$477$	27.2350			1.24701
$478$		0			0
$479$		−	38.6357i		−	1.76531i	−0.470020	−	0.882656i	$-0.655753\pi$
$479$		−	38.6357i		−	1.76531i	0.470020	−	0.882656i	$-0.344247\pi$
$480$		0			0
$481$		−	3.52624i		−	0.160783i
$482$		0			0
$483$		−	9.07482i		−	0.412919i
$484$		0			0
$485$	21.3123			0.967742
$486$		0			0
$487$		−	19.5262i		−	0.884818i	−0.896813	−	0.442409i	$-0.854124\pi$
$487$		−	19.5262i		−	0.884818i	0.896813	−	0.442409i	$-0.145876\pi$
$488$		0			0
$489$	−58.8153			−2.65972
$490$		0			0
$491$	1.75192			0.0790632			0.0395316	−	0.999218i	$-0.487413\pi$
$491$	1.75192			0.0790632			0.0395316	+	0.999218i	$0.487413\pi$
$492$		0			0
$493$	17.2221	−	22.0576i	0.775643	−	0.993425i
$494$		0			0
$495$	35.7242			1.60568
$496$		0			0
$497$	3.42489			0.153627
$498$		0			0
$499$		−	18.0503i		−	0.808044i	−0.914749	−	0.404022i	$-0.867612\pi$
$499$		−	18.0503i		−	0.808044i	0.914749	−	0.404022i	$-0.132388\pi$
$500$		0			0
$501$	−6.36687			−0.284451
$502$		0			0
$503$		−	18.7007i		−	0.833821i	−0.908947	−	0.416911i	$-0.863113\pi$
$503$		−	18.7007i		−	0.833821i	0.908947	−	0.416911i	$-0.136887\pi$
$504$		0			0
$505$			1.00297i			0.0446317i
$506$		0			0
$507$		−	40.3941i		−	1.79396i
$508$		0			0
$509$	39.1219			1.73405			0.867025	−	0.498265i	$-0.166029\pi$
$509$	39.1219			1.73405			0.867025	+	0.498265i	$0.166029\pi$
$510$		0			0
$511$	13.2616			0.586660
$512$		0			0
$513$			21.1182i			0.932392i
$514$		0			0
$515$		−	25.2939i		−	1.11458i
$516$		0			0
$517$		−	38.4913i		−	1.69284i
$518$		0			0
$519$	68.4598			3.00505
$520$		0			0
$521$			31.9544i			1.39995i	0.714169	+	0.699973i	$0.246805\pi$
$521$			31.9544i			1.39995i	−0.714169	+	0.699973i	$0.753195\pi$
$522$		0			0
$523$	21.8516			0.955503			0.477752	−	0.878495i	$-0.341452\pi$
$523$	21.8516			0.955503			0.477752	+	0.878495i	$0.341452\pi$
$524$		0			0
$525$	−8.49607			−0.370799
$526$		0			0
$527$	−9.95358	+	12.7483i	−0.433585	+	0.555325i
$528$		0			0
$529$	14.1443			0.614969
$530$		0			0
$531$	6.45142			0.279968
$532$		0			0
$533$		−	60.8883i		−	2.63736i
$534$		0			0
$535$	8.49971			0.367474
$536$		0			0
$537$		−	31.5962i		−	1.36348i
$538$		0			0
$539$		−	3.81141i		−	0.164169i
$540$		0			0
$541$			32.0847i			1.37943i	0.724081	+	0.689715i	$0.242264\pi$
$541$			32.0847i			1.37943i	−0.724081	+	0.689715i	$0.757736\pi$
$542$		0			0
$543$	−16.0507			−0.688800
$544$		0			0
$545$	5.89020			0.252308
$546$		0			0
$547$		−	34.9098i		−	1.49263i	−0.665590	−	0.746317i	$-0.731820\pi$
$547$		−	34.9098i		−	1.49263i	0.665590	−	0.746317i	$-0.268180\pi$
$548$		0			0
$549$		−	26.5887i		−	1.13478i
$550$		0			0
$551$			14.2462i			0.606909i
$552$		0			0
$553$	10.4514			0.444440
$554$		0			0
$555$			3.12311i			0.132568i
$556$		0			0
$557$	−28.9511			−1.22670			−0.613349	−	0.789812i	$-0.710178\pi$
$557$	−28.9511			−1.22670			−0.613349	+	0.789812i	$0.710178\pi$
$558$		0			0
$559$	−50.3947			−2.13147
$560$		0			0
$561$	−37.7725	−	29.4918i	−1.59475	−	1.24515i
$562$		0			0
$563$	24.2698			1.02285			0.511424	−	0.859328i	$-0.329118\pi$
$563$	24.2698			1.02285			0.511424	+	0.859328i	$0.329118\pi$
$564$		0			0
$565$	−12.7224			−0.535234
$566$		0			0
$567$		−	11.7836i		−	0.494866i
$568$		0			0
$569$	25.3606			1.06317			0.531586	−	0.847004i	$-0.321596\pi$
$569$	25.3606			1.06317			0.531586	+	0.847004i	$0.321596\pi$
$570$		0			0
$571$			8.79694i			0.368140i	0.982913	+	0.184070i	$0.0589274\pi$
$571$			8.79694i			0.368140i	−0.982913	+	0.184070i	$0.941073\pi$
$572$		0			0
$573$			60.8331i			2.54134i
$574$		0			0
$575$			8.29094i			0.345756i
$576$		0			0
$577$	12.2215			0.508787			0.254394	−	0.967101i	$-0.418124\pi$
$577$	12.2215			0.508787			0.254394	+	0.967101i	$0.418124\pi$
$578$		0			0
$579$	18.5987			0.772934
$580$		0			0
$581$		−	3.69822i		−	0.153428i
$582$		0			0
$583$		−	16.4785i		−	0.682471i
$584$		0			0
$585$			48.0187i			1.98533i
$586$		0			0
$587$	−15.2704			−0.630275			−0.315137	−	0.949046i	$-0.602051\pi$
$587$	−15.2704			−0.630275			−0.315137	+	0.949046i	$0.602051\pi$
$588$		0			0
$589$		−	8.23367i		−	0.339262i
$590$		0			0
$591$	19.0984			0.785602
$592$		0			0
$593$	42.7163			1.75415			0.877074	−	0.480355i	$-0.159492\pi$
$593$	42.7163			1.75415			0.877074	+	0.480355i	$0.159492\pi$
$594$		0			0
$595$	−4.83555	−	3.77548i	−0.198238	−	0.154780i
$596$		0			0
$597$	21.9178			0.897037
$598$		0			0
$599$	1.06371			0.0434620			0.0217310	−	0.999764i	$-0.493082\pi$
$599$	1.06371			0.0434620			0.0217310	+	0.999764i	$0.493082\pi$
$600$		0			0
$601$		−	14.3501i		−	0.585352i	−0.956212	−	0.292676i	$-0.905454\pi$
$601$		−	14.3501i		−	0.585352i	0.956212	−	0.292676i	$-0.0945457\pi$
$602$		0			0
$603$	−57.6521			−2.34777
$604$		0			0
$605$		−	5.24766i		−	0.213348i
$606$		0			0
$607$			34.1689i			1.38687i	0.720517	+	0.693437i	$0.243905\pi$
$607$			34.1689i			1.38687i	−0.720517	+	0.693437i	$0.756095\pi$
$608$		0			0
$609$		−	20.6976i		−	0.838710i
$610$		0			0
$611$	51.7381			2.09310
$612$		0			0
$613$	−29.9227			−1.20857			−0.604284	−	0.796769i	$-0.706541\pi$
$613$	−29.9227			−1.20857			−0.604284	+	0.796769i	$0.706541\pi$
$614$		0			0
$615$			53.9273i			2.17456i
$616$		0			0
$617$		−	21.4127i		−	0.862043i	−0.902342	−	0.431022i	$-0.858153\pi$
$617$		−	21.4127i		−	0.862043i	0.902342	−	0.431022i	$-0.141847\pi$
$618$		0			0
$619$			26.1874i			1.05256i	0.850311	+	0.526281i	$0.176414\pi$
$619$			26.1874i			1.05256i	−0.850311	+	0.526281i	$0.823586\pi$
$620$		0			0
$621$	−29.9409			−1.20149
$622$		0			0
$623$			11.3693i			0.455502i
$624$		0			0
$625$	−3.30747			−0.132299
$626$		0			0
$627$	24.3958			0.974276
$628$		0			0
$629$	1.74650	−	2.23688i	0.0696376	−	0.0891902i
$630$		0			0
$631$	−22.8913			−0.911288			−0.455644	−	0.890162i	$-0.650591\pi$
$631$	−22.8913			−0.911288			−0.455644	+	0.890162i	$0.650591\pi$
$632$		0			0
$633$	18.0990			0.719369
$634$		0			0
$635$		−	2.04551i		−	0.0811736i
$636$		0			0
$637$	5.12311			0.202985
$638$		0			0
$639$		−	21.5745i		−	0.853475i
$640$		0			0
$641$		−	34.4514i		−	1.36075i	−0.732864	−	0.680375i	$-0.761817\pi$
$641$		−	34.4514i		−	1.36075i	0.732864	−	0.680375i	$-0.238183\pi$
$642$		0			0
$643$			6.57572i			0.259321i	0.991558	+	0.129661i	$0.0413888\pi$
$643$			6.57572i			0.259321i	−0.991558	+	0.129661i	$0.958611\pi$
$644$		0			0
$645$	44.6333			1.75744
$646$		0			0
$647$	−18.4689			−0.726086			−0.363043	−	0.931772i	$-0.618262\pi$
$647$	−18.4689			−0.726086			−0.363043	+	0.931772i	$0.618262\pi$
$648$		0			0
$649$		−	3.90343i		−	0.153223i
$650$		0			0
$651$			11.9623i			0.468839i
$652$		0			0
$653$			41.7523i			1.63389i	0.576713	+	0.816946i	$0.304335\pi$
$653$			41.7523i			1.63389i	−0.576713	+	0.816946i	$0.695665\pi$
$654$		0			0
$655$	22.9650			0.897317
$656$		0			0
$657$		−	83.5395i		−	3.25919i
$658$		0			0
$659$	−0.552250			−0.0215126			−0.0107563	−	0.999942i	$-0.503424\pi$
$659$	−0.552250			−0.0215126			−0.0107563	+	0.999942i	$0.503424\pi$
$660$		0			0
$661$	11.2294			0.436771			0.218386	−	0.975863i	$-0.429921\pi$
$661$	11.2294			0.436771			0.218386	+	0.975863i	$0.429921\pi$
$662$		0			0
$663$	39.6415	−	50.7719i	1.53955	−	1.97182i
$664$		0			0
$665$	3.12311			0.121109
$666$		0			0
$667$	−20.1979			−0.782067
$668$		0			0
$669$		−	65.6144i		−	2.53680i
$670$		0			0
$671$	−16.0875			−0.621050
$672$		0			0
$673$		−	27.4471i		−	1.05801i	−0.848619	−	0.529004i	$-0.822566\pi$
$673$		−	27.4471i		−	1.05801i	0.848619	−	0.529004i	$-0.177434\pi$
$674$		0			0
$675$			28.0314i			1.07893i
$676$		0			0
$677$			15.6062i			0.599795i	0.953971	+	0.299897i	$0.0969525\pi$
$677$			15.6062i			0.599795i	−0.953971	+	0.299897i	$0.903048\pi$
$678$		0			0
$679$	−14.3235			−0.549685
$680$		0			0
$681$	−18.4137			−0.705615
$682$		0			0
$683$			6.68591i			0.255829i	0.991785	+	0.127915i	$0.0408284\pi$
$683$			6.68591i			0.255829i	−0.991785	+	0.127915i	$0.959172\pi$
$684$		0			0
$685$		−	20.8790i		−	0.797747i
$686$		0			0
$687$		−	51.9867i		−	1.98341i
$688$		0			0
$689$	22.1496			0.843834
$690$		0			0
$691$			26.4459i			1.00605i	0.864272	+	0.503025i	$0.167780\pi$
$691$			26.4459i			1.00605i	−0.864272	+	0.503025i	$0.832220\pi$
$692$		0			0
$693$	−24.0093			−0.912039
$694$		0			0
$695$	−21.3262			−0.808949
$696$		0			0
$697$	30.1572	−	38.6246i	1.14229	−	1.46301i
$698$		0			0
$699$	8.62579			0.326257
$700$		0			0
$701$	37.4436			1.41422			0.707112	−	0.707102i	$-0.249998\pi$
$701$	37.4436			1.41422			0.707112	+	0.707102i	$0.249998\pi$
$702$		0			0
$703$			1.44472i			0.0544886i
$704$		0			0
$705$	−45.8231			−1.72580
$706$		0			0
$707$		−	0.674073i		−	0.0253511i
$708$		0			0
$709$			11.6110i			0.436059i	0.975942	+	0.218030i	$0.0699630\pi$
$709$			11.6110i			0.436059i	−0.975942	+	0.218030i	$0.930037\pi$
$710$		0			0
$711$		−	65.8370i		−	2.46908i
$712$		0			0
$713$	11.6735			0.437176
$714$		0			0
$715$	29.0536			1.08654
$716$		0			0
$717$			9.40319i			0.351169i
$718$		0			0
$719$			9.04254i			0.337230i	0.985682	+	0.168615i	$0.0539294\pi$
$719$			9.04254i			0.337230i	−0.985682	+	0.168615i	$0.946071\pi$
$720$		0			0
$721$			16.9994i			0.633091i
$722$		0			0
$723$	−46.7269			−1.73779
$724$		0			0
$725$			18.9098i			0.702292i
$726$		0			0
$727$	8.12608			0.301380			0.150690	−	0.988581i	$-0.451851\pi$
$727$	8.12608			0.301380			0.150690	+	0.988581i	$0.451851\pi$
$728$		0			0
$729$	17.8159			0.659848
$730$		0			0
$731$	−31.9680	−	24.9599i	−1.18238	−	0.923173i
$732$		0			0
$733$	9.25647			0.341896			0.170948	−	0.985280i	$-0.445317\pi$
$733$	9.25647			0.341896			0.170948	+	0.985280i	$0.445317\pi$
$734$		0			0
$735$	−4.53741			−0.167365
$736$		0			0
$737$			34.8823i			1.28491i
$738$		0			0
$739$	−19.9864			−0.735211			−0.367605	−	0.929982i	$-0.619822\pi$
$739$	−19.9864			−0.735211			−0.367605	+	0.929982i	$0.619822\pi$
$740$		0			0
$741$			32.7917i			1.20463i
$742$		0			0
$743$		−	20.9674i		−	0.769219i	−0.923079	−	0.384610i	$-0.874336\pi$
$743$		−	20.9674i		−	0.769219i	0.923079	−	0.384610i	$-0.125664\pi$
$744$		0			0
$745$			23.8584i			0.874106i
$746$		0			0
$747$	−23.2963			−0.852367
$748$		0			0
$749$	−5.71244			−0.208728
$750$		0			0
$751$			28.8918i			1.05428i	0.849779	+	0.527139i	$0.176735\pi$
$751$			28.8918i			1.05428i	−0.849779	+	0.527139i	$0.823265\pi$
$752$		0			0
$753$			69.6723i			2.53900i
$754$		0			0
$755$			16.6847i			0.607217i
$756$		0			0
$757$	48.8216			1.77445			0.887225	−	0.461336i	$-0.152630\pi$
$757$	48.8216			1.77445			0.887225	+	0.461336i	$0.152630\pi$
$758$		0			0
$759$			34.5878i			1.25546i
$760$		0			0
$761$	−18.0507			−0.654336			−0.327168	−	0.944966i	$-0.606094\pi$
$761$	−18.0507			−0.654336			−0.327168	+	0.944966i	$0.606094\pi$
$762$		0			0
$763$	−3.95866			−0.143313
$764$		0			0
$765$	−23.7830	+	30.4608i	−0.859878	+	1.10131i
$766$		0			0
$767$	5.24679			0.189451
$768$		0			0
$769$	−9.57942			−0.345443			−0.172721	−	0.984971i	$-0.555256\pi$
$769$	−9.57942			−0.345443			−0.172721	+	0.984971i	$0.555256\pi$
$770$		0			0
$771$			77.8418i			2.80340i
$772$		0			0
$773$	14.3572			0.516394			0.258197	−	0.966092i	$-0.416872\pi$
$773$	14.3572			0.516394			0.258197	+	0.966092i	$0.416872\pi$
$774$		0			0
$775$		−	10.9290i		−	0.392581i
$776$		0			0
$777$		−	2.09896i		−	0.0752999i
$778$		0			0
$779$			24.9462i			0.893792i
$780$		0			0
$781$	−13.0536			−0.467096
$782$		0			0
$783$	−68.2884			−2.44043
$784$		0			0
$785$		−	26.2969i		−	0.938576i
$786$		0			0
$787$			10.4063i			0.370946i	0.982649	+	0.185473i	$0.0593818\pi$
$787$			10.4063i			0.370946i	−0.982649	+	0.185473i	$0.940618\pi$
$788$		0			0
$789$			82.3947i			2.93333i
$790$		0			0
$791$	8.55038			0.304017
$792$		0			0
$793$		−	21.6240i		−	0.767890i
$794$		0			0
$795$	−19.6174			−0.695757
$796$		0			0
$797$	−22.8981			−0.811091			−0.405546	−	0.914075i	$-0.632918\pi$
$797$	−22.8981			−0.811091			−0.405546	+	0.914075i	$0.632918\pi$
$798$		0			0
$799$	32.8202	+	25.6252i	1.16109	+	0.906555i
$800$		0			0
$801$	71.6191			2.53054
$802$		0			0
$803$	−50.5455			−1.78371
$804$		0			0
$805$			4.42786i			0.156062i
$806$		0			0
$807$	44.6444			1.57156
$808$		0			0
$809$		−	43.3609i		−	1.52449i	−0.647290	−	0.762243i	$-0.724098\pi$
$809$		−	43.3609i		−	1.52449i	0.647290	−	0.762243i	$-0.275902\pi$
$810$		0			0
$811$		−	30.9680i		−	1.08743i	−0.839269	−	0.543717i	$-0.817016\pi$
$811$		−	30.9680i		−	1.08743i	0.839269	−	0.543717i	$-0.182984\pi$
$812$		0			0
$813$			69.6723i			2.44352i
$814$		0			0
$815$	28.6976			1.00523
$816$		0			0
$817$	20.6470			0.722346
$818$		0			0
$819$		−	32.2722i		−	1.12768i
$820$		0			0
$821$		−	4.35451i		−	0.151973i	−0.997109	−	0.0759866i	$-0.975789\pi$
$821$		−	4.35451i		−	0.151973i	0.997109	−	0.0759866i	$-0.0242106\pi$
$822$		0			0
$823$			20.7942i			0.724840i	0.932015	+	0.362420i	$0.118049\pi$
$823$			20.7942i			0.724840i	−0.932015	+	0.362420i	$0.881951\pi$
$824$		0			0
$825$	32.3820			1.12740
$826$		0			0
$827$		−	44.7244i		−	1.55522i	−0.628747	−	0.777610i	$-0.716432\pi$
$827$		−	44.7244i		−	1.55522i	0.628747	−	0.777610i	$-0.283568\pi$
$828$		0			0
$829$	54.0471			1.87713			0.938567	−	0.345097i	$-0.112154\pi$
$829$	54.0471			1.87713			0.938567	+	0.345097i	$0.112154\pi$
$830$		0			0
$831$	−28.1732			−0.977317
$832$		0			0
$833$	3.24985	+	2.53741i	0.112601	+	0.0879160i
$834$		0			0
$835$	3.10658			0.107507
$836$		0			0
$837$	39.4676			1.36420
$838$		0			0
$839$			8.41042i			0.290360i	0.989405	+	0.145180i	$0.0463761\pi$
$839$			8.41042i			0.290360i	−0.989405	+	0.145180i	$0.953624\pi$
$840$		0			0
$841$	−17.0670			−0.588516
$842$		0			0
$843$			30.6550i			1.05582i
$844$		0			0
$845$			19.7094i			0.678024i
$846$		0			0
$847$			3.52682i			0.121183i
$848$		0			0
$849$	−59.0287			−2.02586
$850$		0			0
$851$	−2.04829			−0.0702144
$852$		0			0
$853$			19.7360i			0.675748i	0.941191	+	0.337874i	$0.109708\pi$
$853$			19.7360i			0.675748i	−0.941191	+	0.337874i	$0.890292\pi$
$854$		0			0
$855$		−	19.6735i		−	0.672819i
$856$		0			0
$857$			51.3663i			1.75464i	0.479907	+	0.877319i	$0.340670\pi$
$857$			51.3663i			1.75464i	−0.479907	+	0.877319i	$0.659330\pi$
$858$		0			0
$859$	32.0012			1.09187			0.545934	−	0.837828i	$-0.316175\pi$
$859$	32.0012			1.09187			0.545934	+	0.837828i	$0.316175\pi$
$860$		0			0
$861$		−	36.2432i		−	1.23516i
$862$		0			0
$863$	16.6204			0.565764			0.282882	−	0.959155i	$-0.408710\pi$
$863$	16.6204			0.565764			0.282882	+	0.959155i	$0.408710\pi$
$864$		0			0
$865$	−33.4035			−1.13575
$866$		0			0
$867$	50.2933	−	12.5733i	1.70805	−	0.427013i
$868$		0			0
$869$	−39.8346			−1.35130
$870$		0			0
$871$	−46.8871			−1.58871
$872$		0			0
$873$			90.2284i			3.05377i
$874$		0			0
$875$	11.5851			0.391648
$876$		0			0
$877$			15.3672i			0.518912i	0.965755	+	0.259456i	$0.0835432\pi$
$877$			15.3672i			0.518912i	−0.965755	+	0.259456i	$0.916457\pi$
$878$		0			0
$879$		−	19.3717i		−	0.653391i
$880$		0			0
$881$			20.2071i			0.680794i	0.940282	+	0.340397i	$0.110561\pi$
$881$			20.2071i			0.680794i	−0.940282	+	0.340397i	$0.889439\pi$
$882$		0			0
$883$	8.88163			0.298891			0.149445	−	0.988770i	$-0.452251\pi$
$883$	8.88163			0.298891			0.149445	+	0.988770i	$0.452251\pi$
$884$		0			0
$885$	−4.64696			−0.156206
$886$		0			0
$887$		−	18.8406i		−	0.632606i	−0.948658	−	0.316303i	$-0.897558\pi$
$887$		−	18.8406i		−	0.632606i	0.948658	−	0.316303i	$-0.102442\pi$
$888$		0			0
$889$			1.37474i			0.0461072i
$890$		0			0
$891$			44.9122i			1.50461i
$892$		0			0
$893$	−21.1973			−0.709342
$894$		0			0
$895$			15.4167i			0.515323i
$896$		0			0
$897$	−46.4913			−1.55230
$898$		0			0
$899$	26.6246			0.887981
$900$		0			0
$901$	14.0507	+	10.9704i	0.468096	+	0.365478i
$902$		0			0
$903$	−29.9970			−0.998237
$904$		0			0
$905$	7.83158			0.260331
$906$		0			0
$907$		−	28.5882i		−	0.949255i	−0.880187	−	0.474627i	$-0.842583\pi$
$907$		−	28.5882i		−	0.949255i	0.880187	−	0.474627i	$-0.157417\pi$
$908$		0			0
$909$	−4.24621			−0.140838
$910$		0			0
$911$			27.3644i			0.906624i	0.891352	+	0.453312i	$0.149758\pi$
$911$			27.3644i			0.906624i	−0.891352	+	0.453312i	$0.850242\pi$
$912$		0			0
$913$			14.0954i			0.466490i
$914$		0			0
$915$			19.1518i			0.633140i
$916$		0			0
$917$	−15.4342			−0.509683
$918$		0			0
$919$	18.9327			0.624533			0.312267	−	0.949994i	$-0.398912\pi$
$919$	18.9327			0.624533			0.312267	+	0.949994i	$0.398912\pi$
$920$		0			0
$921$			5.03143i			0.165791i
$922$		0			0
$923$		−	17.5461i		−	0.577536i
$924$		0			0
$925$			1.91766i			0.0630521i
$926$		0			0
$927$	107.085			3.51713
$928$		0			0
$929$		−	44.5697i		−	1.46228i	−0.682225	−	0.731142i	$-0.738988\pi$
$929$		−	44.5697i		−	1.46228i	0.682225	−	0.731142i	$-0.261012\pi$
$930$		0			0
$931$	−2.09896			−0.0687907
$932$		0			0
$933$	7.55666			0.247394
$934$		0			0
$935$	18.4303	+	14.3899i	0.602734	+	0.470600i
$936$		0			0
$937$	−19.4026			−0.633854			−0.316927	−	0.948450i	$-0.602651\pi$
$937$	−19.4026			−0.633854			−0.316927	+	0.948450i	$0.602651\pi$
$938$		0			0
$939$	−40.0094			−1.30566
$940$		0			0
$941$		−	34.6143i		−	1.12840i	−0.825640	−	0.564198i	$-0.809186\pi$
$941$		−	34.6143i		−	1.12840i	0.825640	−	0.564198i	$-0.190814\pi$
$942$		0			0
$943$	−35.3682			−1.15175
$944$		0			0
$945$			14.9704i			0.486988i
$946$		0			0
$947$			54.1656i			1.76014i	0.474840	+	0.880072i	$0.342506\pi$
$947$			54.1656i			1.76014i	−0.474840	+	0.880072i	$0.657494\pi$
$948$		0			0
$949$		−	67.9408i		−	2.20545i
$950$		0			0
$951$	−45.7725			−1.48427
$952$		0			0
$953$	15.3760			0.498076			0.249038	−	0.968494i	$-0.419886\pi$
$953$	15.3760			0.498076			0.249038	+	0.968494i	$0.419886\pi$
$954$		0			0
$955$		−	29.6822i		−	0.960494i
$956$		0			0
$957$			78.8871i			2.55006i
$958$		0			0
$959$			14.0323i			0.453126i
$960$		0			0
$961$	15.6122			0.503618
$962$		0			0
$963$			35.9846i			1.15959i
$964$		0			0
$965$	−9.07482			−0.292129
$966$		0			0
$967$	−52.6593			−1.69341			−0.846704	−	0.532064i	$-0.821417\pi$
$967$	−52.6593			−1.69341			−0.846704	+	0.532064i	$0.821417\pi$
$968$		0			0
$969$	−16.2413	+	20.8015i	−0.521746	+	0.668240i
$970$		0			0
$971$	5.75023			0.184534			0.0922669	−	0.995734i	$-0.470589\pi$
$971$	5.75023			0.184534			0.0922669	+	0.995734i	$0.470589\pi$
$972$		0			0
$973$	14.3328			0.459489
$974$		0			0
$975$			43.5262i			1.39396i
$976$		0			0
$977$	−10.9221			−0.349431			−0.174715	−	0.984619i	$-0.555900\pi$
$977$	−10.9221			−0.349431			−0.174715	+	0.984619i	$0.555900\pi$
$978$		0			0
$979$		−	43.3331i		−	1.38493i
$980$		0			0
$981$			24.9369i			0.796174i
$982$		0			0
$983$			13.4539i			0.429112i	0.976712	+	0.214556i	$0.0688304\pi$
$983$			13.4539i			0.429112i	−0.976712	+	0.214556i	$0.931170\pi$
$984$		0			0
$985$	−9.31864			−0.296917
$986$		0			0
$987$	30.7966			0.980266
$988$		0			0
$989$			29.2727i			0.930819i
$990$		0			0
$991$		−	34.1461i		−	1.08469i	−0.840157	−	0.542343i	$-0.817537\pi$
$991$		−	34.1461i		−	1.08469i	0.840157	−	0.542343i	$-0.182463\pi$
$992$		0			0
$993$			67.6829i			2.14785i
$994$		0			0
$995$	−10.6943			−0.339033
$996$		0			0
$997$			38.7664i			1.22774i	0.789406	+	0.613871i	$0.210389\pi$
$997$			38.7664i			1.22774i	−0.789406	+	0.613871i	$0.789611\pi$
$998$		0			0
$999$	−6.92518			−0.219103

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1904.2.c.g.1121.1		8
4.3	odd	2		952.2.c.d.169.8	yes	8
17.16	even	2	inner	1904.2.c.g.1121.8		8
68.67	odd	2		952.2.c.d.169.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
952.2.c.d.169.1	✓	8	68.67	odd	2
952.2.c.d.169.8	yes	8	4.3	odd	2
1904.2.c.g.1121.1		8	1.1	even	1	trivial
1904.2.c.g.1121.8		8	17.16	even	2	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1904 = 2^{4} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1904.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.04948i q^{3} +1.48793i q^{5} -1.00000i q^{7} -6.29934 q^{9} +3.81141i q^{11} -5.12311 q^{13} +4.53741 q^{15} +(-3.24985 - 2.53741i) q^{17} +2.09896 q^{19} -3.04948 q^{21} +2.97586i q^{23} +2.78607 q^{25} +10.0613i q^{27} +6.78726i q^{29} -3.92273i q^{31} +11.6228 q^{33} +1.48793 q^{35} +0.688301i q^{37} +15.6228i q^{39} +11.8850i q^{41} +9.83674 q^{43} -9.37296i q^{45} -10.0990 q^{47} -1.00000 q^{49} +(-7.73778 + 9.91037i) q^{51} -4.32348 q^{53} -5.67110 q^{55} -6.40075i q^{57} -1.02414 q^{59} +4.22087i q^{61} +6.29934i q^{63} -7.62281i q^{65} +9.15209 q^{67} +9.07482 q^{69} +3.42489i q^{71} +13.2616i q^{73} -8.49607i q^{75} +3.81141 q^{77} +10.4514i q^{79} +11.7836 q^{81} +3.69822 q^{83} +(3.77548 - 4.83555i) q^{85} +20.6976 q^{87} -11.3693 q^{89} +5.12311i q^{91} -11.9623 q^{93} +3.12311i q^{95} -14.3235i q^{97} -24.0093i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 6 q^{9} - 8 q^{13} + 16 q^{15} - 20 q^{19} - 6 q^{21} + 2 q^{25} + 8 q^{33} + 10 q^{35} + 14 q^{43} - 44 q^{47} - 8 q^{49} - 34 q^{51} + 6 q^{53} + 32 q^{55} - 12 q^{59} + 58 q^{67} + 32 q^{69} - 12 q^{77}+ \cdots - 64 q^{93}+O(q^{100}) \) 8 * q - 6 * q^9 - 8 * q^13 + 16 * q^15 - 20 * q^19 - 6 * q^21 + 2 * q^25 + 8 * q^33 + 10 * q^35 + 14 * q^43 - 44 * q^47 - 8 * q^49 - 34 * q^51 + 6 * q^53 + 32 * q^55 - 12 * q^59 + 58 * q^67 + 32 * q^69 - 12 * q^77 + 8 * q^81 + 8 * q^83 + 34 * q^85 + 40 * q^87 + 8 * q^89 - 64 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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