LMFDB - Embedded newform 3060.2.w.f.2177.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3060,2,Mod(953,3060)] 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("3060.953"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3060, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 2, 3, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3060 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3060.w (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [28,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$24.4342230185$
Analytic rank:	$0$
Dimension:	$28$
Relative dimension:	$14$ over $\Q(i)$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			2177.8
Character	$\chi$	$=$	3060.2177
Dual form			3060.2.w.f.953.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.0301846 + 2.23586i) q^{5} +(-0.0761394 - 0.0761394i) q^{7} -3.96321i q^{11} +(0.813455 - 0.813455i) q^{13} +(0.707107 - 0.707107i) q^{17} -3.30260i q^{19} +(2.91151 + 2.91151i) q^{23} +(-4.99818 + 0.134977i) q^{25} +4.02358 q^{29} -3.06509 q^{31} +(0.167939 - 0.172536i) q^{35} +(-4.09970 - 4.09970i) q^{37} +1.86567i q^{41} +(7.28362 - 7.28362i) q^{43} +(-0.576219 + 0.576219i) q^{47} -6.98841i q^{49} +(-8.09941 - 8.09941i) q^{53} +(8.86119 - 0.119628i) q^{55} -1.40560 q^{59} -11.3956 q^{61} +(1.84333 + 1.79422i) q^{65} +(1.53597 + 1.53597i) q^{67} -8.56989i q^{71} +(-5.01252 + 5.01252i) q^{73} +(-0.301756 + 0.301756i) q^{77} -14.7881i q^{79} +(10.0852 + 10.0852i) q^{83} +(1.60234 + 1.55965i) q^{85} +17.8344 q^{89} -0.123872 q^{91} +(7.38416 - 0.0996875i) q^{95} +(2.06743 + 2.06743i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 8 q^{13} + 16 q^{25} - 24 q^{31} - 4 q^{37} + 36 q^{43} + 40 q^{55} - 40 q^{61} + 64 q^{67} + 28 q^{73} - 4 q^{85} - 120 q^{91} + 44 q^{97}+O(q^{100}) $ 28 * q + 8 * q^13 + 16 * q^25 - 24 * q^31 - 4 * q^37 + 36 * q^43 + 40 * q^55 - 40 * q^61 + 64 * q^67 + 28 * q^73 - 4 * q^85 - 120 * q^91 + 44 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3060\mathbb{Z}\right)^\times$.

$n$	$1261$	$1361$	$1531$	$1837$
$\chi(n)$	$1$	$-1$	$1$	$e\left(\frac{1}{4}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.0301846	+	2.23586i	0.0134990	+	0.999909i
$5$	0.0301846	+	2.23586i	0.0134990	+	0.999909i
$6$		0			0
$7$	−0.0761394	−	0.0761394i	−0.0287780	−	0.0287780i	0.692571	−	0.721349i	$-0.256478\pi$
$7$	−0.0761394	−	0.0761394i	−0.0287780	−	0.0287780i	−0.721349	+	0.692571i	$0.756478\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	3.96321i		−	1.19495i	−0.801887	−	0.597476i	$-0.796170\pi$
$11$		−	3.96321i		−	1.19495i	0.801887	−	0.597476i	$-0.203830\pi$
$12$		0			0
$13$	0.813455	−	0.813455i	0.225612	−	0.225612i	−0.585245	−	0.810857i	$-0.699002\pi$
$13$	0.813455	−	0.813455i	0.225612	−	0.225612i	0.810857	+	0.585245i	$0.199002\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	0.707107	−	0.707107i	0.171499	−	0.171499i
$17$	0.707107	−	0.707107i	0.171499	−	0.171499i
$18$		0			0
$19$		−	3.30260i		−	0.757668i	−0.925465	−	0.378834i	$-0.876325\pi$
$19$		−	3.30260i		−	0.757668i	0.925465	−	0.378834i	$-0.123675\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.91151	+	2.91151i	0.607091	+	0.607091i	0.942185	−	0.335094i	$-0.108768\pi$
$23$	2.91151	+	2.91151i	0.607091	+	0.607091i	−0.335094	+	0.942185i	$0.608768\pi$
$24$		0			0
$25$	−4.99818	+	0.134977i	−0.999636	+	0.0269955i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	4.02358			0.747159			0.373580	−	0.927598i	$-0.378130\pi$
$29$	4.02358			0.747159			0.373580	+	0.927598i	$0.378130\pi$
$30$		0			0
$31$	−3.06509			−0.550507			−0.275254	−	0.961372i	$-0.588762\pi$
$31$	−3.06509			−0.550507			−0.275254	+	0.961372i	$0.588762\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	0.167939	−	0.172536i	0.0283869	−	0.0291638i
$36$		0			0
$37$	−4.09970	−	4.09970i	−0.673987	−	0.673987i	0.284646	−	0.958633i	$-0.408124\pi$
$37$	−4.09970	−	4.09970i	−0.673987	−	0.673987i	−0.958633	+	0.284646i	$0.908124\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			1.86567i			0.291368i	0.989331	+	0.145684i	$0.0465383\pi$
$41$			1.86567i			0.291368i	−0.989331	+	0.145684i	$0.953462\pi$
$42$		0			0
$43$	7.28362	−	7.28362i	1.11074	−	1.11074i	0.117692	−	0.993050i	$-0.462450\pi$
$43$	7.28362	−	7.28362i	1.11074	−	1.11074i	0.993050	−	0.117692i	$-0.0375496\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−0.576219	+	0.576219i	−0.0840501	+	0.0840501i	−0.747882	−	0.663832i	$-0.768929\pi$
$47$	−0.576219	+	0.576219i	−0.0840501	+	0.0840501i	0.663832	+	0.747882i	$0.268929\pi$
$48$		0			0
$49$		−	6.98841i		−	0.998344i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−8.09941	−	8.09941i	−1.11254	−	1.11254i	−0.992806	−	0.119734i	$-0.961796\pi$
$53$	−8.09941	−	8.09941i	−1.11254	−	1.11254i	−0.119734	−	0.992806i	$-0.538204\pi$
$54$		0			0
$55$	8.86119	−	0.119628i	1.19484	−	0.0161306i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−1.40560			−0.182994			−0.0914970	−	0.995805i	$-0.529165\pi$
$59$	−1.40560			−0.182994			−0.0914970	+	0.995805i	$0.529165\pi$
$60$		0			0
$61$	−11.3956			−1.45905			−0.729527	−	0.683952i	$-0.760260\pi$
$61$	−11.3956			−1.45905			−0.729527	+	0.683952i	$0.760260\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.84333	+	1.79422i	0.228637	+	0.222546i
$66$		0			0
$67$	1.53597	+	1.53597i	0.187649	+	0.187649i	0.794679	−	0.607030i	$-0.207639\pi$
$67$	1.53597	+	1.53597i	0.187649	+	0.187649i	−0.607030	+	0.794679i	$0.707639\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	8.56989i		−	1.01706i	−0.861045	−	0.508529i	$-0.830189\pi$
$71$		−	8.56989i		−	1.01706i	0.861045	−	0.508529i	$-0.169811\pi$
$72$		0			0
$73$	−5.01252	+	5.01252i	−0.586671	+	0.586671i	−0.936728	−	0.350057i	$-0.886162\pi$
$73$	−5.01252	+	5.01252i	−0.586671	+	0.586671i	0.350057	+	0.936728i	$0.386162\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−0.301756	+	0.301756i	−0.0343883	+	0.0343883i
$78$		0			0
$79$		−	14.7881i		−	1.66379i	−0.554935	−	0.831893i	$-0.687257\pi$
$79$		−	14.7881i		−	1.66379i	0.554935	−	0.831893i	$-0.312743\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	10.0852	+	10.0852i	1.10700	+	1.10700i	0.993543	+	0.113457i	$0.0361923\pi$
$83$	10.0852	+	10.0852i	1.10700	+	1.10700i	0.113457	+	0.993543i	$0.463808\pi$
$84$		0			0
$85$	1.60234	+	1.55965i	0.173798	+	0.169168i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	17.8344			1.89045			0.945223	−	0.326424i	$-0.105844\pi$
$89$	17.8344			1.89045			0.945223	+	0.326424i	$0.105844\pi$
$90$		0			0
$91$	−0.123872			−0.0129853
$92$		0			0
$93$		0			0
$94$		0			0
$95$	7.38416	−	0.0996875i	0.757599	−	0.0102277i
$96$		0			0
$97$	2.06743	+	2.06743i	0.209916	+	0.209916i	0.804232	−	0.594316i	$-0.202577\pi$
$97$	2.06743	+	2.06743i	0.209916	+	0.209916i	−0.594316	+	0.804232i	$0.702577\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			0.983719i			0.0978837i	0.998802	+	0.0489418i	$0.0155849\pi$
$101$			0.983719i			0.0978837i	−0.998802	+	0.0489418i	$0.984415\pi$
$102$		0			0
$103$	7.27911	−	7.27911i	0.717232	−	0.717232i	−0.250805	−	0.968038i	$-0.580695\pi$
$103$	7.27911	−	7.27911i	0.717232	−	0.717232i	0.968038	+	0.250805i	$0.0806954\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	11.3722	−	11.3722i	1.09939	−	1.09939i	0.104907	−	0.994482i	$-0.466545\pi$
$107$	11.3722	−	11.3722i	1.09939	−	1.09939i	0.994482	−	0.104907i	$-0.0334546\pi$
$108$		0			0
$109$		−	7.53792i		−	0.722002i	−0.932566	−	0.361001i	$-0.882435\pi$
$109$		−	7.53792i		−	0.722002i	0.932566	−	0.361001i	$-0.117565\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	13.6371	+	13.6371i	1.28287	+	1.28287i	0.939025	+	0.343849i	$0.111731\pi$
$113$	13.6371	+	13.6371i	1.28287	+	1.28287i	0.343849	+	0.939025i	$0.388269\pi$
$114$		0			0
$115$	−6.42185	+	6.59762i	−0.598841	+	0.615231i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−0.107677			−0.00987077
$120$		0			0
$121$	−4.70701			−0.427910
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−0.452659	−	11.1712i	−0.0404870	−	0.999180i
$126$		0			0
$127$	6.44506	+	6.44506i	0.571906	+	0.571906i	0.932661	−	0.360755i	$-0.117481\pi$
$127$	6.44506	+	6.44506i	0.571906	+	0.571906i	−0.360755	+	0.932661i	$0.617481\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			8.57897i			0.749548i	0.927116	+	0.374774i	$0.122280\pi$
$131$			8.57897i			0.749548i	−0.927116	+	0.374774i	$0.877720\pi$
$132$		0			0
$133$	−0.251458	+	0.251458i	−0.0218042	+	0.0218042i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	11.6965	−	11.6965i	0.999296	−	0.999296i	−0.000703436	−	1.00000i	$-0.500224\pi$
$137$	11.6965	−	11.6965i	0.999296	−	0.999296i	1.00000		0.000703436i	$0.000223911\pi$
$138$		0			0
$139$			18.5729i			1.57533i	0.616104	+	0.787665i	$0.288710\pi$
$139$			18.5729i			1.57533i	−0.616104	+	0.787665i	$0.711290\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−3.22389	−	3.22389i	−0.269595	−	0.269595i
$144$		0			0
$145$	0.121450	+	8.99617i	0.0100859	+	0.747091i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	12.6194			1.03382			0.516909	−	0.856040i	$-0.327082\pi$
$149$	12.6194			1.03382			0.516909	+	0.856040i	$0.327082\pi$
$150$		0			0
$151$	0.455155			0.0370400			0.0185200	−	0.999828i	$-0.494105\pi$
$151$	0.455155			0.0370400			0.0185200	+	0.999828i	$0.494105\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−0.0925186	−	6.85314i	−0.00743127	−	0.550457i
$156$		0			0
$157$	−13.3139	−	13.3139i	−1.06257	−	1.06257i	−0.997907	−	0.0646603i	$-0.979404\pi$
$157$	−13.3139	−	13.3139i	−1.06257	−	1.06257i	−0.0646603	−	0.997907i	$-0.520596\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	0.443361i		−	0.0349417i
$162$		0			0
$163$	6.39362	−	6.39362i	0.500787	−	0.500787i	−0.410896	−	0.911682i	$-0.634784\pi$
$163$	6.39362	−	6.39362i	0.500787	−	0.500787i	0.911682	+	0.410896i	$0.134784\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	5.46063	−	5.46063i	0.422556	−	0.422556i	−0.463527	−	0.886083i	$-0.653416\pi$
$167$	5.46063	−	5.46063i	0.422556	−	0.422556i	0.886083	+	0.463527i	$0.153416\pi$
$168$		0			0
$169$			11.6766i			0.898199i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−6.07524	−	6.07524i	−0.461892	−	0.461892i	0.437383	−	0.899275i	$-0.355905\pi$
$173$	−6.07524	−	6.07524i	−0.461892	−	0.461892i	−0.899275	+	0.437383i	$0.855905\pi$
$174$		0			0
$175$	0.390835	+	0.370281i	0.0295444	+	0.0279906i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	11.8877			0.888530			0.444265	−	0.895895i	$-0.353465\pi$
$179$	11.8877			0.888530			0.444265	+	0.895895i	$0.353465\pi$
$180$		0			0
$181$	1.33303			0.0990836			0.0495418	−	0.998772i	$-0.484224\pi$
$181$	1.33303			0.0990836			0.0495418	+	0.998772i	$0.484224\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	9.04263	−	9.29013i	0.664828	−	0.683024i
$186$		0			0
$187$	−2.80241	−	2.80241i	−0.204933	−	0.204933i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			20.6539i			1.49446i	0.664565	+	0.747230i	$0.268617\pi$
$191$			20.6539i			1.49446i	−0.664565	+	0.747230i	$0.731383\pi$
$192$		0			0
$193$	5.42963	−	5.42963i	0.390833	−	0.390833i	−0.484151	−	0.874984i	$-0.660871\pi$
$193$	5.42963	−	5.42963i	0.390833	−	0.390833i	0.874984	+	0.484151i	$0.160871\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−4.74669	+	4.74669i	−0.338188	+	0.338188i	−0.855685	−	0.517497i	$-0.826864\pi$
$197$	−4.74669	+	4.74669i	−0.338188	+	0.338188i	0.517497	+	0.855685i	$0.326864\pi$
$198$		0			0
$199$		−	22.2154i		−	1.57481i	−0.616437	−	0.787405i	$-0.711424\pi$
$199$		−	22.2154i		−	1.57481i	0.616437	−	0.787405i	$-0.288576\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−0.306353	−	0.306353i	−0.0215017	−	0.0215017i
$204$		0			0
$205$	−4.17137	+	0.0563143i	−0.291341	+	0.00393316i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−13.0889			−0.905376
$210$		0			0
$211$	7.08612			0.487828			0.243914	−	0.969797i	$-0.421568\pi$
$211$	7.08612			0.487828			0.243914	+	0.969797i	$0.421568\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	16.5050	+	16.0653i	1.12564	+	1.09565i
$216$		0			0
$217$	0.233375	+	0.233375i	0.0158425	+	0.0158425i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	1.15040i		−	0.0773842i
$222$		0			0
$223$	3.37593	−	3.37593i	0.226069	−	0.226069i	−0.584979	−	0.811048i	$-0.698897\pi$
$223$	3.37593	−	3.37593i	0.226069	−	0.226069i	0.811048	+	0.584979i	$0.198897\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	8.60725	−	8.60725i	0.571283	−	0.571283i	−0.361204	−	0.932487i	$-0.617634\pi$
$227$	8.60725	−	8.60725i	0.571283	−	0.571283i	0.932487	+	0.361204i	$0.117634\pi$
$228$		0			0
$229$		−	9.06035i		−	0.598725i	−0.954140	−	0.299362i	$-0.903226\pi$
$229$		−	9.06035i		−	0.598725i	0.954140	−	0.299362i	$-0.0967739\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−15.6054	−	15.6054i	−1.02234	−	1.02234i	−0.999745	−	0.0225975i	$-0.992806\pi$
$233$	−15.6054	−	15.6054i	−1.02234	−	1.02234i	−0.0225975	−	0.999745i	$-0.507194\pi$
$234$		0			0
$235$	−1.30574	−	1.27095i	−0.0851771	−	0.0829079i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−15.5313			−1.00464			−0.502320	−	0.864682i	$-0.667520\pi$
$239$	−15.5313			−1.00464			−0.502320	+	0.864682i	$0.667520\pi$
$240$		0			0
$241$	−24.2407			−1.56148			−0.780741	−	0.624854i	$-0.785158\pi$
$241$	−24.2407			−1.56148			−0.780741	+	0.624854i	$0.785158\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	15.6251	−	0.210942i	0.998253	−	0.0134766i
$246$		0			0
$247$	−2.68651	−	2.68651i	−0.170939	−	0.170939i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			4.21845i			0.266266i	0.991098	+	0.133133i	$0.0425038\pi$
$251$			4.21845i			0.266266i	−0.991098	+	0.133133i	$0.957496\pi$
$252$		0			0
$253$	11.5389	−	11.5389i	0.725445	−	0.725445i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−6.18145	+	6.18145i	−0.385588	+	0.385588i	−0.873110	−	0.487522i	$-0.837901\pi$
$257$	−6.18145	+	6.18145i	−0.385588	+	0.385588i	0.487522	+	0.873110i	$0.337901\pi$
$258$		0			0
$259$			0.624298i			0.0387920i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−2.87813	−	2.87813i	−0.177473	−	0.177473i	0.612780	−	0.790253i	$-0.290051\pi$
$263$	−2.87813	−	2.87813i	−0.177473	−	0.177473i	−0.790253	+	0.612780i	$0.790051\pi$
$264$		0			0
$265$	17.8647	−	18.3537i	1.09742	−	1.12746i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−15.5001			−0.945060			−0.472530	−	0.881315i	$-0.656659\pi$
$269$	−15.5001			−0.945060			−0.472530	+	0.881315i	$0.656659\pi$
$270$		0			0
$271$	−26.0269			−1.58102			−0.790512	−	0.612446i	$-0.790186\pi$
$271$	−26.0269			−1.58102			−0.790512	+	0.612446i	$0.790186\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	0.534943	+	19.8088i	0.0322583	+	1.19452i
$276$		0			0
$277$	5.79861	+	5.79861i	0.348405	+	0.348405i	0.859515	−	0.511111i	$-0.170766\pi$
$277$	5.79861	+	5.79861i	0.348405	+	0.348405i	−0.511111	+	0.859515i	$0.670766\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	1.23601i		−	0.0737341i	−0.999320	−	0.0368671i	$-0.988262\pi$
$281$		−	1.23601i		−	0.0737341i	0.999320	−	0.0368671i	$-0.0117378\pi$
$282$		0			0
$283$	4.15805	−	4.15805i	0.247171	−	0.247171i	−0.572638	−	0.819808i	$-0.694080\pi$
$283$	4.15805	−	4.15805i	0.247171	−	0.247171i	0.819808	+	0.572638i	$0.194080\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	0.142051	−	0.142051i	0.00838499	−	0.00838499i
$288$		0			0
$289$		−	1.00000i		−	0.0588235i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	6.58019	+	6.58019i	0.384419	+	0.384419i	0.872691	−	0.488272i	$-0.162373\pi$
$293$	6.58019	+	6.58019i	0.384419	+	0.384419i	−0.488272	+	0.872691i	$0.662373\pi$
$294$		0			0
$295$	−0.0424276	−	3.14274i	−0.00247023	−	0.182977i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	4.73676			0.273934
$300$		0			0
$301$	−1.10914			−0.0639299
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−0.343971	−	25.4790i	−0.0196957	−	1.45892i
$306$		0			0
$307$	0.440704	+	0.440704i	0.0251523	+	0.0251523i	0.719571	−	0.694419i	$-0.244338\pi$
$307$	0.440704	+	0.440704i	0.0251523	+	0.0251523i	−0.694419	+	0.719571i	$0.744338\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			10.5637i			0.599013i	0.954094	+	0.299507i	$0.0968221\pi$
$311$			10.5637i			0.599013i	−0.954094	+	0.299507i	$0.903178\pi$
$312$		0			0
$313$	−10.2206	+	10.2206i	−0.577704	+	0.577704i	−0.934270	−	0.356566i	$-0.883948\pi$
$313$	−10.2206	+	10.2206i	−0.577704	+	0.577704i	0.356566	+	0.934270i	$0.383948\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	9.50766	−	9.50766i	0.534003	−	0.534003i	−0.387758	−	0.921761i	$-0.626750\pi$
$317$	9.50766	−	9.50766i	0.534003	−	0.534003i	0.921761	+	0.387758i	$0.126750\pi$
$318$		0			0
$319$		−	15.9463i		−	0.892819i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−2.33529	−	2.33529i	−0.129939	−	0.129939i
$324$		0			0
$325$	−3.95600	+	4.17559i	−0.219439	+	0.231620i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	0.0877459			0.00483759
$330$		0			0
$331$	−11.7397			−0.645270			−0.322635	−	0.946523i	$-0.604569\pi$
$331$	−11.7397			−0.645270			−0.322635	+	0.946523i	$0.604569\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−3.38786	+	3.48059i	−0.185099	+	0.190165i
$336$		0			0
$337$	7.31088	+	7.31088i	0.398249	+	0.398249i	0.877615	−	0.479366i	$-0.159133\pi$
$337$	7.31088	+	7.31088i	0.398249	+	0.398249i	−0.479366	+	0.877615i	$0.659133\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			12.1476i			0.657830i
$342$		0			0
$343$	−1.06507	+	1.06507i	−0.0575083	+	0.0575083i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−6.20743	+	6.20743i	−0.333232	+	0.333232i	−0.853813	−	0.520580i	$-0.825716\pi$
$347$	−6.20743	+	6.20743i	−0.333232	+	0.333232i	0.520580	+	0.853813i	$0.325716\pi$
$348$		0			0
$349$		−	23.1200i		−	1.23759i	−0.785553	−	0.618794i	$-0.787622\pi$
$349$		−	23.1200i		−	1.23759i	0.785553	−	0.618794i	$-0.212378\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−15.3258	−	15.3258i	−0.815711	−	0.815711i	0.169773	−	0.985483i	$-0.445697\pi$
$353$	−15.3258	−	15.3258i	−0.815711	−	0.815711i	−0.985483	+	0.169773i	$0.945697\pi$
$354$		0			0
$355$	19.1611	−	0.258678i	1.01697	−	0.0137292i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−30.9673			−1.63439			−0.817195	−	0.576361i	$-0.804472\pi$
$359$	−30.9673			−1.63439			−0.817195	+	0.576361i	$0.804472\pi$
$360$		0			0
$361$	8.09285			0.425940
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−11.3586	−	11.0560i	−0.594537	−	0.578698i
$366$		0			0
$367$	4.15037	+	4.15037i	0.216648	+	0.216648i	0.807084	−	0.590436i	$-0.201044\pi$
$367$	4.15037	+	4.15037i	0.216648	+	0.216648i	−0.590436	+	0.807084i	$0.701044\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			1.23337i			0.0640333i
$372$		0			0
$373$	−9.91982	+	9.91982i	−0.513629	+	0.513629i	−0.915636	−	0.402008i	$-0.868313\pi$
$373$	−9.91982	+	9.91982i	−0.513629	+	0.513629i	0.402008	+	0.915636i	$0.368313\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	3.27300	−	3.27300i	0.168568	−	0.168568i
$378$		0			0
$379$		−	16.1016i		−	0.827084i	−0.910485	−	0.413542i	$-0.864292\pi$
$379$		−	16.1016i		−	0.827084i	0.910485	−	0.413542i	$-0.135708\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−2.08442	−	2.08442i	−0.106509	−	0.106509i	0.651844	−	0.758353i	$-0.273996\pi$
$383$	−2.08442	−	2.08442i	−0.106509	−	0.106509i	−0.758353	+	0.651844i	$0.773996\pi$
$384$		0			0
$385$	−0.683794	−	0.665578i	−0.0348494	−	0.0339210i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−13.2415			−0.671372			−0.335686	−	0.941974i	$-0.608968\pi$
$389$	−13.2415			−0.671372			−0.335686	+	0.941974i	$0.608968\pi$
$390$		0			0
$391$	4.11749			0.208231
$392$		0			0
$393$		0			0
$394$		0			0
$395$	33.0641	−	0.446371i	1.66364	−	0.0224594i
$396$		0			0
$397$	8.44355	+	8.44355i	0.423770	+	0.423770i	0.886499	−	0.462730i	$-0.153130\pi$
$397$	8.44355	+	8.44355i	0.423770	+	0.423770i	−0.462730	+	0.886499i	$0.653130\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			5.72761i			0.286023i	0.989721	+	0.143012i	$0.0456786\pi$
$401$			5.72761i			0.286023i	−0.989721	+	0.143012i	$0.954321\pi$
$402$		0			0
$403$	−2.49332	+	2.49332i	−0.124201	+	0.124201i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−16.2480	+	16.2480i	−0.805382	+	0.805382i
$408$		0			0
$409$			4.99079i			0.246779i	0.992358	+	0.123389i	$0.0393764\pi$
$409$			4.99079i			0.246779i	−0.992358	+	0.123389i	$0.960624\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	0.107022	+	0.107022i	0.00526620	+	0.00526620i
$414$		0			0
$415$	−22.2448	+	22.8537i	−1.09196	+	1.12184i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	16.2594			0.794324			0.397162	−	0.917748i	$-0.369995\pi$
$419$	16.2594			0.794324			0.397162	+	0.917748i	$0.369995\pi$
$420$		0			0
$421$	1.77155			0.0863401			0.0431700	−	0.999068i	$-0.486254\pi$
$421$	1.77155			0.0863401			0.0431700	+	0.999068i	$0.486254\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−3.43880	+	3.62969i	−0.166806	+	0.176066i
$426$		0			0
$427$	0.867652	+	0.867652i	0.0419886	+	0.0419886i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			38.8433i			1.87102i	0.353305	+	0.935508i	$0.385058\pi$
$431$			38.8433i			1.87102i	−0.353305	+	0.935508i	$0.614942\pi$
$432$		0			0
$433$	−7.72967	+	7.72967i	−0.371464	+	0.371464i	−0.868010	−	0.496546i	$-0.834601\pi$
$433$	−7.72967	+	7.72967i	−0.371464	+	0.371464i	0.496546	+	0.868010i	$0.334601\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	9.61553	−	9.61553i	0.459973	−	0.459973i
$438$		0			0
$439$			26.0752i			1.24450i	0.782817	+	0.622252i	$0.213782\pi$
$439$			26.0752i			1.24450i	−0.782817	+	0.622252i	$0.786218\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	8.38643	+	8.38643i	0.398452	+	0.398452i	0.877687	−	0.479235i	$-0.159086\pi$
$443$	8.38643	+	8.38643i	0.398452	+	0.398452i	−0.479235	+	0.877687i	$0.659086\pi$
$444$		0			0
$445$	0.538325	+	39.8754i	0.0255191	+	1.89027i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−2.91069			−0.137364			−0.0686819	−	0.997639i	$-0.521879\pi$
$449$	−2.91069			−0.137364			−0.0686819	+	0.997639i	$0.521879\pi$
$450$		0			0
$451$	7.39402			0.348171
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−0.00373902	−	0.276961i	−0.000175288	−	0.0129841i
$456$		0			0
$457$	−8.81377	−	8.81377i	−0.412291	−	0.412291i	0.470245	−	0.882536i	$-0.344166\pi$
$457$	−8.81377	−	8.81377i	−0.412291	−	0.412291i	−0.882536	+	0.470245i	$0.844166\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	28.8380i		−	1.34312i	−0.740951	−	0.671559i	$-0.765625\pi$
$461$		−	28.8380i		−	1.34312i	0.740951	−	0.671559i	$-0.234375\pi$
$462$		0			0
$463$	22.8686	−	22.8686i	1.06279	−	1.06279i	0.0649029	−	0.997892i	$-0.479326\pi$
$463$	22.8686	−	22.8686i	1.06279	−	1.06279i	0.997892	−	0.0649029i	$-0.0206738\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	12.7684	−	12.7684i	0.590849	−	0.590849i	−0.347012	−	0.937861i	$-0.612804\pi$
$467$	12.7684	−	12.7684i	0.590849	−	0.590849i	0.937861	+	0.347012i	$0.112804\pi$
$468$		0			0
$469$		−	0.233896i		−	0.0108003i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−28.8665	−	28.8665i	−1.32728	−	1.32728i
$474$		0			0
$475$	0.445775	+	16.5070i	0.0204536	+	0.757392i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−25.3325			−1.15747			−0.578737	−	0.815514i	$-0.696454\pi$
$479$	−25.3325			−1.15747			−0.578737	+	0.815514i	$0.696454\pi$
$480$		0			0
$481$	−6.66985			−0.304119
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−4.56008	+	4.68489i	−0.207063	+	0.212730i
$486$		0			0
$487$	21.8762	+	21.8762i	0.991305	+	0.991305i	0.999963	−	0.00865751i	$-0.00275581\pi$
$487$	21.8762	+	21.8762i	0.991305	+	0.991305i	−0.00865751	+	0.999963i	$0.502756\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			15.0767i			0.680402i	0.940353	+	0.340201i	$0.110495\pi$
$491$			15.0767i			0.680402i	−0.940353	+	0.340201i	$0.889505\pi$
$492$		0			0
$493$	2.84510	−	2.84510i	0.128137	−	0.128137i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−0.652506	+	0.652506i	−0.0292689	+	0.0292689i
$498$		0			0
$499$			7.69432i			0.344445i	0.985058	+	0.172222i	$0.0550948\pi$
$499$			7.69432i			0.344445i	−0.985058	+	0.172222i	$0.944905\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−3.92845	−	3.92845i	−0.175161	−	0.175161i	0.614082	−	0.789243i	$-0.289527\pi$
$503$	−3.92845	−	3.92845i	−0.175161	−	0.175161i	−0.789243	+	0.614082i	$0.789527\pi$
$504$		0			0
$505$	−2.19946	+	0.0296931i	−0.0978748	+	0.00132133i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−2.31527			−0.102623			−0.0513113	−	0.998683i	$-0.516340\pi$
$509$	−2.31527			−0.102623			−0.0513113	+	0.998683i	$0.516340\pi$
$510$		0			0
$511$	0.763301			0.0337664
$512$		0			0
$513$		0			0
$514$		0			0
$515$	16.4948	+	16.0554i	0.726849	+	0.707485i
$516$		0			0
$517$	2.28367	+	2.28367i	0.100436	+	0.100436i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			29.8493i			1.30772i	0.756616	+	0.653860i	$0.226851\pi$
$521$			29.8493i			1.30772i	−0.756616	+	0.653860i	$0.773149\pi$
$522$		0			0
$523$	24.5256	−	24.5256i	1.07243	−	1.07243i	0.0752669	−	0.997163i	$-0.476019\pi$
$523$	24.5256	−	24.5256i	1.07243	−	1.07243i	0.997163	−	0.0752669i	$-0.0239809\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−2.16735	+	2.16735i	−0.0944112	+	0.0944112i
$528$		0			0
$529$		−	6.04625i		−	0.262880i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	1.51764	+	1.51764i	0.0657361	+	0.0657361i
$534$		0			0
$535$	25.7699	+	25.0834i	1.11413	+	1.08445i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−27.6965			−1.19297
$540$		0			0
$541$	−33.6913			−1.44850			−0.724251	−	0.689537i	$-0.757814\pi$
$541$	−33.6913			−1.44850			−0.724251	+	0.689537i	$0.757814\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	16.8538	−	0.227529i	0.721936	−	0.00974627i
$546$		0			0
$547$	8.39757	+	8.39757i	0.359054	+	0.359054i	0.863464	−	0.504410i	$-0.168290\pi$
$547$	8.39757	+	8.39757i	0.359054	+	0.359054i	−0.504410	+	0.863464i	$0.668290\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	13.2882i		−	0.566098i
$552$		0			0
$553$	−1.12595	+	1.12595i	−0.0478804	+	0.0478804i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−12.7597	+	12.7597i	−0.540644	+	0.540644i	−0.923718	−	0.383074i	$-0.874866\pi$
$557$	−12.7597	+	12.7597i	−0.540644	+	0.540644i	0.383074	+	0.923718i	$0.374866\pi$
$558$		0			0
$559$		−	11.8498i		−	0.501193i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−4.71370	−	4.71370i	−0.198659	−	0.198659i	0.600766	−	0.799425i	$-0.294862\pi$
$563$	−4.71370	−	4.71370i	−0.198659	−	0.198659i	−0.799425	+	0.600766i	$0.794862\pi$
$564$		0			0
$565$	−30.0792	+	30.9024i	−1.26544	+	1.30007i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	3.17937			0.133286			0.0666432	−	0.997777i	$-0.478771\pi$
$569$	3.17937			0.133286			0.0666432	+	0.997777i	$0.478771\pi$
$570$		0			0
$571$	0.0596794			0.00249750			0.00124875	−	0.999999i	$-0.499603\pi$
$571$	0.0596794			0.00249750			0.00124875	+	0.999999i	$0.499603\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−14.9452	−	14.1592i	−0.623259	−	0.590481i
$576$		0			0
$577$	20.4269	+	20.4269i	0.850385	+	0.850385i	0.990180	−	0.139796i	$-0.0446446\pi$
$577$	20.4269	+	20.4269i	0.850385	+	0.850385i	−0.139796	+	0.990180i	$0.544645\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	1.53577i		−	0.0637145i
$582$		0			0
$583$	−32.0996	+	32.0996i	−1.32943	+	1.32943i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−3.80824	+	3.80824i	−0.157183	+	0.157183i	−0.781317	−	0.624134i	$-0.785452\pi$
$587$	−3.80824	+	3.80824i	−0.157183	+	0.157183i	0.624134	+	0.781317i	$0.285452\pi$
$588$		0			0
$589$			10.1228i			0.417102i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−13.2206	−	13.2206i	−0.542906	−	0.542906i	0.381474	−	0.924380i	$-0.375417\pi$
$593$	−13.2206	−	13.2206i	−0.542906	−	0.542906i	−0.924380	+	0.381474i	$0.875417\pi$
$594$		0			0
$595$	−0.00325020	−	0.240752i	−0.000133245	−	0.00986987i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−11.3034			−0.461844			−0.230922	−	0.972972i	$-0.574174\pi$
$599$	−11.3034			−0.461844			−0.230922	+	0.972972i	$0.574174\pi$
$600$		0			0
$601$	−45.2400			−1.84538			−0.922688	−	0.385547i	$-0.874013\pi$
$601$	−45.2400			−1.84538			−0.922688	+	0.385547i	$0.874013\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−0.142079	−	10.5242i	−0.00577634	−	0.427871i
$606$		0			0
$607$	13.9256	+	13.9256i	0.565224	+	0.565224i	0.930787	−	0.365562i	$-0.119123\pi$
$607$	13.9256	+	13.9256i	0.565224	+	0.565224i	−0.365562	+	0.930787i	$0.619123\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			0.937456i			0.0379254i
$612$		0			0
$613$	26.8856	−	26.8856i	1.08590	−	1.08590i	0.0899532	−	0.995946i	$-0.471328\pi$
$613$	26.8856	−	26.8856i	1.08590	−	1.08590i	0.995946	−	0.0899532i	$-0.0286718\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−7.13493	+	7.13493i	−0.287242	+	0.287242i	−0.835989	−	0.548747i	$-0.815105\pi$
$617$	−7.13493	+	7.13493i	−0.287242	+	0.287242i	0.548747	+	0.835989i	$0.315105\pi$
$618$		0			0
$619$			7.03028i			0.282571i	0.989969	+	0.141285i	$0.0451235\pi$
$619$			7.03028i			0.282571i	−0.989969	+	0.141285i	$0.954876\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−1.35790	−	1.35790i	−0.0544033	−	0.0544033i
$624$		0			0
$625$	24.9636	−	1.34928i	0.998542	−	0.0539712i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−5.79786			−0.231176
$630$		0			0
$631$	40.3490			1.60627			0.803134	−	0.595798i	$-0.203164\pi$
$631$	40.3490			1.60627			0.803134	+	0.595798i	$0.203164\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−14.2157	+	14.6048i	−0.564134	+	0.579574i
$636$		0			0
$637$	−5.68475	−	5.68475i	−0.225238	−	0.225238i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	20.9516i		−	0.827537i	−0.910382	−	0.413769i	$-0.864212\pi$
$641$		−	20.9516i		−	0.827537i	0.910382	−	0.413769i	$-0.135788\pi$
$642$		0			0
$643$	−5.86016	+	5.86016i	−0.231102	+	0.231102i	−0.813153	−	0.582050i	$-0.802251\pi$
$643$	−5.86016	+	5.86016i	−0.231102	+	0.231102i	0.582050	+	0.813153i	$0.302251\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−27.9506	+	27.9506i	−1.09885	+	1.09885i	−0.104308	+	0.994545i	$0.533263\pi$
$647$	−27.9506	+	27.9506i	−1.09885	+	1.09885i	−0.994545	+	0.104308i	$0.966737\pi$
$648$		0			0
$649$			5.57070i			0.218669i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−23.4270	−	23.4270i	−0.916768	−	0.916768i	0.0800244	−	0.996793i	$-0.474500\pi$
$653$	−23.4270	−	23.4270i	−0.916768	−	0.916768i	−0.996793	+	0.0800244i	$0.974500\pi$
$654$		0			0
$655$	−19.1814	+	0.258953i	−0.749480	+	0.0101181i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	0.224465			0.00874390			0.00437195	−	0.999990i	$-0.498608\pi$
$659$	0.224465			0.00874390			0.00437195	+	0.999990i	$0.498608\pi$
$660$		0			0
$661$	−26.7542			−1.04062			−0.520309	−	0.853978i	$-0.674183\pi$
$661$	−26.7542			−1.04062			−0.520309	+	0.853978i	$0.674183\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−0.569816	−	0.554635i	−0.0220965	−	0.0215078i
$666$		0			0
$667$	11.7147	+	11.7147i	0.453594	+	0.453594i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			45.1630i			1.74350i
$672$		0			0
$673$	20.4784	−	20.4784i	0.789384	−	0.789384i	−0.192009	−	0.981393i	$-0.561500\pi$
$673$	20.4784	−	20.4784i	0.789384	−	0.789384i	0.981393	+	0.192009i	$0.0615004\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−3.41567	+	3.41567i	−0.131275	+	0.131275i	−0.769691	−	0.638416i	$-0.779590\pi$
$677$	−3.41567	+	3.41567i	−0.131275	+	0.131275i	0.638416	+	0.769691i	$0.279590\pi$
$678$		0			0
$679$		−	0.314826i		−	0.0120819i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	22.9464	+	22.9464i	0.878018	+	0.878018i	0.993329	−	0.115311i	$-0.0367866\pi$
$683$	22.9464	+	22.9464i	0.878018	+	0.878018i	−0.115311	+	0.993329i	$0.536787\pi$
$684$		0			0
$685$	26.5048	+	25.7987i	1.01269	+	0.985716i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−13.1770			−0.502004
$690$		0			0
$691$	−37.5965			−1.43024			−0.715120	−	0.699002i	$-0.753628\pi$
$691$	−37.5965			−1.43024			−0.715120	+	0.699002i	$0.753628\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−41.5264	+	0.560614i	−1.57519	+	0.0212653i
$696$		0			0
$697$	1.31922	+	1.31922i	0.0499692	+	0.0499692i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	8.99719i		−	0.339819i	−0.985460	−	0.169910i	$-0.945652\pi$
$701$		−	8.99719i		−	0.339819i	0.985460	−	0.169910i	$-0.0543476\pi$
$702$		0			0
$703$	−13.5397	+	13.5397i	−0.510658	+	0.510658i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	0.0748998	−	0.0748998i	0.00281690	−	0.00281690i
$708$		0			0
$709$		−	3.01276i		−	0.113147i	−0.998398	−	0.0565734i	$-0.981983\pi$
$709$		−	3.01276i		−	0.113147i	0.998398	−	0.0565734i	$-0.0180175\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−8.92405	−	8.92405i	−0.334208	−	0.334208i
$714$		0			0
$715$	7.11087	−	7.30550i	0.265932	−	0.273210i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	36.1265			1.34729			0.673646	−	0.739054i	$-0.264727\pi$
$719$	36.1265			1.34729			0.673646	+	0.739054i	$0.264727\pi$
$720$		0			0
$721$	−1.10845			−0.0412810
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−20.1105	+	0.543091i	−0.746887	+	0.0201699i
$726$		0			0
$727$	−19.0586	−	19.0586i	−0.706845	−	0.706845i	0.259025	−	0.965871i	$-0.416599\pi$
$727$	−19.0586	−	19.0586i	−0.706845	−	0.706845i	−0.965871	+	0.259025i	$0.916599\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	10.3006i		−	0.380981i
$732$		0			0
$733$	32.0899	−	32.0899i	1.18527	−	1.18527i	0.206908	−	0.978360i	$-0.433660\pi$
$733$	32.0899	−	32.0899i	1.18527	−	1.18527i	0.978360	−	0.206908i	$-0.0663402\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	6.08737	−	6.08737i	0.224231	−	0.224231i
$738$		0			0
$739$		−	6.47723i		−	0.238269i	−0.992878	−	0.119134i	$-0.961988\pi$
$739$		−	6.47723i		−	0.238269i	0.992878	−	0.119134i	$-0.0380119\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−15.2112	−	15.2112i	−0.558043	−	0.558043i	0.370707	−	0.928750i	$-0.379116\pi$
$743$	−15.2112	−	15.2112i	−0.558043	−	0.558043i	−0.928750	+	0.370707i	$0.879116\pi$
$744$		0			0
$745$	0.380910	+	28.2152i	0.0139555	+	1.03372i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−1.73174			−0.0632764
$750$		0			0
$751$	7.67420			0.280036			0.140018	−	0.990149i	$-0.455284\pi$
$751$	7.67420			0.280036			0.140018	+	0.990149i	$0.455284\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	0.0137387	+	1.01766i	0.000500001	+	0.0370366i
$756$		0			0
$757$	27.4806	+	27.4806i	0.998801	+	0.998801i	0.999999	−	0.00119835i	$-0.000381446\pi$
$757$	27.4806	+	27.4806i	0.998801	+	0.998801i	−0.00119835	+	0.999999i	$0.500381\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			9.27086i			0.336068i	0.985781	+	0.168034i	$0.0537419\pi$
$761$			9.27086i			0.336068i	−0.985781	+	0.168034i	$0.946258\pi$
$762$		0			0
$763$	−0.573933	+	0.573933i	−0.0207778	+	0.0207778i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1.14340	+	1.14340i	−0.0412856	+	0.0412856i
$768$		0			0
$769$			31.2228i			1.12592i	0.826483	+	0.562961i	$0.190338\pi$
$769$			31.2228i			1.12592i	−0.826483	+	0.562961i	$0.809662\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	23.6922	+	23.6922i	0.852150	+	0.852150i	0.990398	−	0.138247i	$-0.0441469\pi$
$773$	23.6922	+	23.6922i	0.852150	+	0.852150i	−0.138247	+	0.990398i	$0.544147\pi$
$774$		0			0
$775$	15.3199	−	0.413718i	0.550307	−	0.0148612i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	6.16154			0.220760
$780$		0			0
$781$	−33.9642			−1.21534
$782$		0			0
$783$		0			0
$784$		0			0
$785$	29.3663	−	30.1700i	1.04813	−	1.07681i
$786$		0			0
$787$	−4.26096	−	4.26096i	−0.151887	−	0.151887i	0.627073	−	0.778960i	$-0.284253\pi$
$787$	−4.26096	−	4.26096i	−0.151887	−	0.151887i	−0.778960	+	0.627073i	$0.784253\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	2.07665i		−	0.0738371i
$792$		0			0
$793$	−9.26979	+	9.26979i	−0.329180	+	0.329180i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	17.9022	−	17.9022i	0.634127	−	0.634127i	−0.314973	−	0.949101i	$-0.601996\pi$
$797$	17.9022	−	17.9022i	0.634127	−	0.634127i	0.949101	+	0.314973i	$0.101996\pi$
$798$		0			0
$799$			0.814896i			0.0288290i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	19.8657	+	19.8657i	0.701044	+	0.701044i
$804$		0			0
$805$	0.991295	−	0.0133827i	0.0349386	−	0.000471677i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−15.8032			−0.555610			−0.277805	−	0.960637i	$-0.589607\pi$
$809$	−15.8032			−0.555610			−0.277805	+	0.960637i	$0.589607\pi$
$810$		0			0
$811$	18.0904			0.635241			0.317621	−	0.948218i	$-0.397116\pi$
$811$	18.0904			0.635241			0.317621	+	0.948218i	$0.397116\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	14.4883	+	14.1023i	0.507501	+	0.493981i
$816$		0			0
$817$	−24.0549	−	24.0549i	−0.841574	−	0.841574i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			39.2029i			1.36819i	0.729392	+	0.684096i	$0.239803\pi$
$821$			39.2029i			1.36819i	−0.729392	+	0.684096i	$0.760197\pi$
$822$		0			0
$823$	23.3634	−	23.3634i	0.814396	−	0.814396i	−0.170894	−	0.985289i	$-0.554665\pi$
$823$	23.3634	−	23.3634i	0.814396	−	0.814396i	0.985289	+	0.170894i	$0.0546655\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	7.92686	−	7.92686i	0.275644	−	0.275644i	−0.555723	−	0.831367i	$-0.687559\pi$
$827$	7.92686	−	7.92686i	0.275644	−	0.275644i	0.831367	+	0.555723i	$0.187559\pi$
$828$		0			0
$829$			47.8230i			1.66096i	0.557047	+	0.830481i	$0.311934\pi$
$829$			47.8230i			1.66096i	−0.557047	+	0.830481i	$0.688066\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−4.94155	−	4.94155i	−0.171215	−	0.171215i
$834$		0			0
$835$	12.3741	+	12.0444i	0.428222	+	0.416814i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−49.9531			−1.72457			−0.862287	−	0.506420i	$-0.830969\pi$
$839$	−49.9531			−1.72457			−0.862287	+	0.506420i	$0.830969\pi$
$840$		0			0
$841$	−12.8108			−0.441753
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−26.1073	+	0.352453i	−0.898117	+	0.0121247i
$846$		0			0
$847$	0.358389	+	0.358389i	0.0123144	+	0.0123144i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	23.8726i		−	0.818343i
$852$		0			0
$853$	−19.7094	+	19.7094i	−0.674837	+	0.674837i	−0.958827	−	0.283990i	$-0.908342\pi$
$853$	−19.7094	+	19.7094i	−0.674837	+	0.674837i	0.283990	+	0.958827i	$0.408342\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−28.0114	+	28.0114i	−0.956852	+	0.956852i	−0.999107	−	0.0422549i	$-0.986546\pi$
$857$	−28.0114	+	28.0114i	−0.956852	+	0.956852i	0.0422549	+	0.999107i	$0.486546\pi$
$858$		0			0
$859$		−	24.1559i		−	0.824189i	−0.911141	−	0.412094i	$-0.864797\pi$
$859$		−	24.1559i		−	0.824189i	0.911141	−	0.412094i	$-0.135203\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−5.40501	−	5.40501i	−0.183989	−	0.183989i	0.609103	−	0.793091i	$-0.291530\pi$
$863$	−5.40501	−	5.40501i	−0.183989	−	0.183989i	−0.793091	+	0.609103i	$0.791530\pi$
$864$		0			0
$865$	13.4000	−	13.7668i	0.455615	−	0.468085i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−58.6081			−1.98815
$870$		0			0
$871$	2.49889			0.0846716
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−0.816101	+	0.885032i	−0.0275893	+	0.0299195i
$876$		0			0
$877$	19.3090	+	19.3090i	0.652020	+	0.652020i	0.953479	−	0.301459i	$-0.0974736\pi$
$877$	19.3090	+	19.3090i	0.652020	+	0.652020i	−0.301459	+	0.953479i	$0.597474\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			35.1337i			1.18368i	0.806054	+	0.591842i	$0.201599\pi$
$881$			35.1337i			1.18368i	−0.806054	+	0.591842i	$0.798401\pi$
$882$		0			0
$883$	−25.9751	+	25.9751i	−0.874131	+	0.874131i	−0.992920	−	0.118788i	$-0.962099\pi$
$883$	−25.9751	+	25.9751i	−0.874131	+	0.874131i	0.118788	+	0.992920i	$0.462099\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−8.62683	+	8.62683i	−0.289661	+	0.289661i	−0.836946	−	0.547285i	$-0.815661\pi$
$887$	−8.62683	+	8.62683i	−0.289661	+	0.289661i	0.547285	+	0.836946i	$0.315661\pi$
$888$		0			0
$889$		−	0.981446i		−	0.0329166i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	1.90302	+	1.90302i	0.0636821	+	0.0636821i
$894$		0			0
$895$	0.358826	+	26.5793i	0.0119942	+	0.888449i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−12.3326			−0.411317
$900$		0			0
$901$	−11.4543			−0.381598
$902$		0			0
$903$		0			0
$904$		0			0
$905$	0.0402370	+	2.98048i	0.00133752	+	0.0990745i
$906$		0			0
$907$	36.3576	+	36.3576i	1.20723	+	1.20723i	0.971919	+	0.235314i	$0.0756117\pi$
$907$	36.3576	+	36.3576i	1.20723	+	1.20723i	0.235314	+	0.971919i	$0.424388\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			2.32266i			0.0769530i	0.999260	+	0.0384765i	$0.0122505\pi$
$911$			2.32266i			0.0769530i	−0.999260	+	0.0384765i	$0.987750\pi$
$912$		0			0
$913$	39.9699	−	39.9699i	1.32281	−	1.32281i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	0.653198	−	0.653198i	0.0215705	−	0.0215705i
$918$		0			0
$919$		−	48.8677i		−	1.61200i	−0.591918	−	0.805998i	$-0.701629\pi$
$919$		−	48.8677i		−	1.61200i	0.591918	−	0.805998i	$-0.298371\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−6.97122	−	6.97122i	−0.229460	−	0.229460i
$924$		0			0
$925$	21.0444	+	19.9377i	0.691936	+	0.655547i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−9.28766			−0.304718			−0.152359	−	0.988325i	$-0.548687\pi$
$929$	−9.28766			−0.304718			−0.152359	+	0.988325i	$0.548687\pi$
$930$		0			0
$931$	−23.0799			−0.756413
$932$		0			0
$933$		0			0
$934$		0			0
$935$	6.18122	−	6.35040i	0.202148	−	0.207680i
$936$		0			0
$937$	−33.4773	−	33.4773i	−1.09366	−	1.09366i	−0.995135	−	0.0985213i	$-0.968589\pi$
$937$	−33.4773	−	33.4773i	−1.09366	−	1.09366i	−0.0985213	−	0.995135i	$-0.531411\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	50.6878i		−	1.65237i	−0.563396	−	0.826187i	$-0.690505\pi$
$941$		−	50.6878i		−	1.65237i	0.563396	−	0.826187i	$-0.309495\pi$
$942$		0			0
$943$	−5.43190	+	5.43190i	−0.176887	+	0.176887i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−0.0477502	+	0.0477502i	−0.00155167	+	0.00155167i	−0.707882	−	0.706331i	$-0.750349\pi$
$947$	−0.0477502	+	0.0477502i	−0.00155167	+	0.00155167i	0.706331	+	0.707882i	$0.250349\pi$
$948$		0			0
$949$			8.15492i			0.264720i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−35.5661	−	35.5661i	−1.15210	−	1.15210i	−0.986131	−	0.165966i	$-0.946926\pi$
$953$	−35.5661	−	35.5661i	−1.15210	−	1.15210i	−0.165966	−	0.986131i	$-0.553074\pi$
$954$		0			0
$955$	−46.1792	+	0.623428i	−1.49432	+	0.0201737i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−1.78112			−0.0575155
$960$		0			0
$961$	−21.6052			−0.696942
$962$		0			0
$963$		0			0
$964$		0			0
$965$	12.3038	+	11.9760i	0.396074	+	0.385522i
$966$		0			0
$967$	30.6024	+	30.6024i	0.984108	+	0.984108i	0.999876	−	0.0157674i	$-0.00501913\pi$
$967$	30.6024	+	30.6024i	0.984108	+	0.984108i	−0.0157674	+	0.999876i	$0.505019\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			6.46324i			0.207415i	0.994608	+	0.103708i	$0.0330706\pi$
$971$			6.46324i			0.207415i	−0.994608	+	0.103708i	$0.966929\pi$
$972$		0			0
$973$	1.41413	−	1.41413i	0.0453348	−	0.0453348i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	16.3015	−	16.3015i	0.521531	−	0.521531i	−0.396503	−	0.918034i	$-0.629776\pi$
$977$	16.3015	−	16.3015i	0.521531	−	0.521531i	0.918034	+	0.396503i	$0.129776\pi$
$978$		0			0
$979$		−	70.6816i		−	2.25899i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−8.78487	−	8.78487i	−0.280194	−	0.280194i	0.552992	−	0.833186i	$-0.313486\pi$
$983$	−8.78487	−	8.78487i	−0.280194	−	0.280194i	−0.833186	+	0.552992i	$0.813486\pi$
$984$		0			0
$985$	−10.7562	−	10.4697i	−0.342722	−	0.333592i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	42.4127			1.34864
$990$		0			0
$991$	−32.2420			−1.02420			−0.512101	−	0.858925i	$-0.671133\pi$
$991$	−32.2420			−1.02420			−0.512101	+	0.858925i	$0.671133\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	49.6707	−	0.670563i	1.57467	−	0.0212583i
$996$		0			0
$997$	3.17926	+	3.17926i	0.100688	+	0.100688i	0.755656	−	0.654968i	$-0.227318\pi$
$997$	3.17926	+	3.17926i	0.100688	+	0.100688i	−0.654968	+	0.755656i	$0.727318\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3060.2.w.f.2177.8	yes	28
3.2	odd	2	inner	3060.2.w.f.2177.7	yes	28
5.3	odd	4	inner	3060.2.w.f.953.7	✓	28
15.8	even	4	inner	3060.2.w.f.953.8	yes	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3060.2.w.f.953.7	✓	28	5.3	odd	4	inner
3060.2.w.f.953.8	yes	28	15.8	even	4	inner
3060.2.w.f.2177.7	yes	28	3.2	odd	2	inner
3060.2.w.f.2177.8	yes	28	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3060 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3060.w (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.0301846 + 2.23586i) q^{5} +(-0.0761394 - 0.0761394i) q^{7} -3.96321i q^{11} +(0.813455 - 0.813455i) q^{13} +(0.707107 - 0.707107i) q^{17} -3.30260i q^{19} +(2.91151 + 2.91151i) q^{23} +(-4.99818 + 0.134977i) q^{25} +4.02358 q^{29} -3.06509 q^{31} +(0.167939 - 0.172536i) q^{35} +(-4.09970 - 4.09970i) q^{37} +1.86567i q^{41} +(7.28362 - 7.28362i) q^{43} +(-0.576219 + 0.576219i) q^{47} -6.98841i q^{49} +(-8.09941 - 8.09941i) q^{53} +(8.86119 - 0.119628i) q^{55} -1.40560 q^{59} -11.3956 q^{61} +(1.84333 + 1.79422i) q^{65} +(1.53597 + 1.53597i) q^{67} -8.56989i q^{71} +(-5.01252 + 5.01252i) q^{73} +(-0.301756 + 0.301756i) q^{77} -14.7881i q^{79} +(10.0852 + 10.0852i) q^{83} +(1.60234 + 1.55965i) q^{85} +17.8344 q^{89} -0.123872 q^{91} +(7.38416 - 0.0996875i) q^{95} +(2.06743 + 2.06743i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 8 q^{13} + 16 q^{25} - 24 q^{31} - 4 q^{37} + 36 q^{43} + 40 q^{55} - 40 q^{61} + 64 q^{67} + 28 q^{73} - 4 q^{85} - 120 q^{91} + 44 q^{97}+O(q^{100}) \) 28 * q + 8 * q^13 + 16 * q^25 - 24 * q^31 - 4 * q^37 + 36 * q^43 + 40 * q^55 - 40 * q^61 + 64 * q^67 + 28 * q^73 - 4 * q^85 - 120 * q^91 + 44 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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