LMFDB - Embedded newform 4012.2.b.a.237.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4012,2,Mod(237,4012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4012, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4012.237");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4012 = 2^{2} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4012.b (of order $2$, degree $1$, 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:	$32.0359812909$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			237.10
Character	$\chi$	$=$	4012.237
Dual form			4012.2.b.a.237.31

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.86007i q^{3} +2.89135i q^{5} +5.10759i q^{7} -0.459864 q^{9} +O(q^{10})$	$q-1.86007i q^{3} +2.89135i q^{5} +5.10759i q^{7} -0.459864 q^{9} +3.69418i q^{11} +2.99788 q^{13} +5.37811 q^{15} +(1.11873 - 3.96843i) q^{17} +6.95127 q^{19} +9.50047 q^{21} +5.32187i q^{23} -3.35988 q^{25} -4.72483i q^{27} +2.80081i q^{29} +1.47417i q^{31} +6.87143 q^{33} -14.7678 q^{35} -0.220986i q^{37} -5.57627i q^{39} -10.9762i q^{41} +5.46188 q^{43} -1.32963i q^{45} +2.85760 q^{47} -19.0874 q^{49} +(-7.38156 - 2.08092i) q^{51} +13.5562 q^{53} -10.6811 q^{55} -12.9299i q^{57} -1.00000 q^{59} +6.60627i q^{61} -2.34880i q^{63} +8.66791i q^{65} +8.19721 q^{67} +9.89906 q^{69} -7.53883i q^{71} +12.2337i q^{73} +6.24961i q^{75} -18.8683 q^{77} -0.707053i q^{79} -10.1681 q^{81} -2.96347 q^{83} +(11.4741 + 3.23465i) q^{85} +5.20971 q^{87} +4.12214 q^{89} +15.3119i q^{91} +2.74206 q^{93} +20.0985i q^{95} -0.479638i q^{97} -1.69882i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 36 q^{9}+O(q^{10}) $ 40 * q - 36 * q^9	$ 40 q - 36 q^{9} + 8 q^{13} + 20 q^{15} - 8 q^{17} + 20 q^{19} + 6 q^{21} - 24 q^{25} - 14 q^{33} - 52 q^{35} + 22 q^{43} - 10 q^{47} + 8 q^{49} - 6 q^{51} - 2 q^{53} - 12 q^{55} - 40 q^{59} - 24 q^{67} - 36 q^{69} - 38 q^{77} + 16 q^{81} + 32 q^{83} - 22 q^{85} - 18 q^{87} + 40 q^{89} + 22 q^{93}+O(q^{100}) $ 40 * q - 36 * q^9 + 8 * q^13 + 20 * q^15 - 8 * q^17 + 20 * q^19 + 6 * q^21 - 24 * q^25 - 14 * q^33 - 52 * q^35 + 22 * q^43 - 10 * q^47 + 8 * q^49 - 6 * q^51 - 2 * q^53 - 12 * q^55 - 40 * q^59 - 24 * q^67 - 36 * q^69 - 38 * q^77 + 16 * q^81 + 32 * q^83 - 22 * q^85 - 18 * q^87 + 40 * q^89 + 22 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4012\mathbb{Z}\right)^\times$.

$n$	$2007$	$3129$	$3777$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		−	1.86007i		−	1.07391i	−0.843610	−	0.536956i	$-0.819574\pi$
$3$		−	1.86007i		−	1.07391i	0.843610	−	0.536956i	$-0.180426\pi$
$4$		0			0
$5$			2.89135i			1.29305i	0.762893	+	0.646524i	$0.223778\pi$
$5$			2.89135i			1.29305i	−0.762893	+	0.646524i	$0.776222\pi$
$6$		0			0
$7$			5.10759i			1.93049i	0.261353	+	0.965243i	$0.415831\pi$
$7$			5.10759i			1.93049i	−0.261353	+	0.965243i	$0.584169\pi$
$8$		0			0
$9$	−0.459864			−0.153288
$10$		0			0
$11$			3.69418i			1.11384i	0.830567	+	0.556918i	$0.188016\pi$
$11$			3.69418i			1.11384i	−0.830567	+	0.556918i	$0.811984\pi$
$12$		0			0
$13$	2.99788			0.831462			0.415731	−	0.909488i	$-0.363526\pi$
$13$	2.99788			0.831462			0.415731	+	0.909488i	$0.363526\pi$
$14$		0			0
$15$	5.37811			1.38862
$16$		0			0
$17$	1.11873	−	3.96843i	0.271333	−	0.962486i
$17$	1.11873	−	3.96843i	0.271333	−	0.962486i
$18$		0			0
$19$	6.95127			1.59473			0.797366	−	0.603496i	$-0.206226\pi$
$19$	6.95127			1.59473			0.797366	+	0.603496i	$0.206226\pi$
$20$		0			0
$21$	9.50047			2.07317
$22$		0			0
$23$			5.32187i			1.10969i	0.831955	+	0.554843i	$0.187222\pi$
$23$			5.32187i			1.10969i	−0.831955	+	0.554843i	$0.812778\pi$
$24$		0			0
$25$	−3.35988			−0.671976
$26$		0			0
$27$		−	4.72483i		−	0.909295i
$28$		0			0
$29$			2.80081i			0.520098i	0.965595	+	0.260049i	$0.0837386\pi$
$29$			2.80081i			0.520098i	−0.965595	+	0.260049i	$0.916261\pi$
$30$		0			0
$31$			1.47417i			0.264769i	0.991198	+	0.132384i	$0.0422633\pi$
$31$			1.47417i			0.264769i	−0.991198	+	0.132384i	$0.957737\pi$
$32$		0			0
$33$	6.87143			1.19616
$34$		0			0
$35$	−14.7678			−2.49621
$36$		0			0
$37$		−	0.220986i		−	0.0363298i	−0.999835	−	0.0181649i	$-0.994218\pi$
$37$		−	0.220986i		−	0.0363298i	0.999835	−	0.0181649i	$-0.00578239\pi$
$38$		0			0
$39$		−	5.57627i		−	0.892918i
$40$		0			0
$41$		−	10.9762i		−	1.71419i	−0.515160	−	0.857094i	$-0.672267\pi$
$41$		−	10.9762i		−	1.71419i	0.515160	−	0.857094i	$-0.327733\pi$
$42$		0			0
$43$	5.46188			0.832928			0.416464	−	0.909152i	$-0.363269\pi$
$43$	5.46188			0.832928			0.416464	+	0.909152i	$0.363269\pi$
$44$		0			0
$45$		−	1.32963i		−	0.198209i
$46$		0			0
$47$	2.85760			0.416824			0.208412	−	0.978041i	$-0.433170\pi$
$47$	2.85760			0.416824			0.208412	+	0.978041i	$0.433170\pi$
$48$		0			0
$49$	−19.0874			−2.72678
$50$		0			0
$51$	−7.38156	−	2.08092i	−1.03363	−	0.291388i
$52$		0			0
$53$	13.5562			1.86209			0.931043	−	0.364910i	$-0.118900\pi$
$53$	13.5562			1.86209			0.931043	+	0.364910i	$0.118900\pi$
$54$		0			0
$55$	−10.6811			−1.44024
$56$		0			0
$57$		−	12.9299i		−	1.71260i
$58$		0			0
$59$	−1.00000			−0.130189
$59$	−1.00000			−0.130189
$60$		0			0
$61$			6.60627i			0.845846i	0.906166	+	0.422923i	$0.138996\pi$
$61$			6.60627i			0.845846i	−0.906166	+	0.422923i	$0.861004\pi$
$62$		0			0
$63$		−	2.34880i		−	0.295920i
$64$		0			0
$65$			8.66791i			1.07512i
$66$		0			0
$67$	8.19721			1.00145			0.500724	−	0.865607i	$-0.333067\pi$
$67$	8.19721			1.00145			0.500724	+	0.865607i	$0.333067\pi$
$68$		0			0
$69$	9.89906			1.19171
$70$		0			0
$71$		−	7.53883i		−	0.894694i	−0.894360	−	0.447347i	$-0.852369\pi$
$71$		−	7.53883i		−	0.894694i	0.894360	−	0.447347i	$-0.147631\pi$
$72$		0			0
$73$			12.2337i			1.43185i	0.698176	+	0.715926i	$0.253995\pi$
$73$			12.2337i			1.43185i	−0.698176	+	0.715926i	$0.746005\pi$
$74$		0			0
$75$			6.24961i			0.721643i
$76$		0			0
$77$	−18.8683			−2.15025
$78$		0			0
$79$		−	0.707053i		−	0.0795497i	−0.999209	−	0.0397749i	$-0.987336\pi$
$79$		−	0.707053i		−	0.0795497i	0.999209	−	0.0397749i	$-0.0126641\pi$
$80$		0			0
$81$	−10.1681			−1.12979
$82$		0			0
$83$	−2.96347			−0.325283			−0.162641	−	0.986685i	$-0.552001\pi$
$83$	−2.96347			−0.325283			−0.162641	+	0.986685i	$0.552001\pi$
$84$		0			0
$85$	11.4741	+	3.23465i	1.24454	+	0.350847i
$86$		0			0
$87$	5.20971			0.558539
$88$		0			0
$89$	4.12214			0.436946			0.218473	−	0.975843i	$-0.429893\pi$
$89$	4.12214			0.436946			0.218473	+	0.975843i	$0.429893\pi$
$90$		0			0
$91$			15.3119i			1.60513i
$92$		0			0
$93$	2.74206			0.284339
$94$		0			0
$95$			20.0985i			2.06207i
$96$		0			0
$97$		−	0.479638i		−	0.0486998i	−0.999703	−	0.0243499i	$-0.992248\pi$
$97$		−	0.479638i		−	0.0486998i	0.999703	−	0.0243499i	$-0.00775158\pi$
$98$		0			0
$99$		−	1.69882i		−	0.170738i
$100$		0			0
$101$	−16.7520			−1.66689			−0.833443	−	0.552606i	$-0.813634\pi$
$101$	−16.7520			−1.66689			−0.833443	+	0.552606i	$0.813634\pi$
$102$		0			0
$103$	−11.1672			−1.10033			−0.550166	−	0.835055i	$-0.685436\pi$
$103$	−11.1672			−1.10033			−0.550166	+	0.835055i	$0.685436\pi$
$104$		0			0
$105$			27.4692i			2.68071i
$106$		0			0
$107$			7.26880i			0.702701i	0.936244	+	0.351351i	$0.114278\pi$
$107$			7.26880i			0.702701i	−0.936244	+	0.351351i	$0.885722\pi$
$108$		0			0
$109$			10.8396i			1.03825i	0.854699	+	0.519123i	$0.173741\pi$
$109$			10.8396i			1.03825i	−0.854699	+	0.519123i	$0.826259\pi$
$110$		0			0
$111$	−0.411049			−0.0390151
$112$		0			0
$113$		−	8.55169i		−	0.804476i	−0.915535	−	0.402238i	$-0.868233\pi$
$113$		−	8.55169i		−	0.804476i	0.915535	−	0.402238i	$-0.131767\pi$
$114$		0			0
$115$	−15.3874			−1.43488
$116$		0			0
$117$	−1.37862			−0.127453
$118$		0			0
$119$	20.2691	+	5.71403i	1.85807	+	0.523804i
$120$		0			0
$121$	−2.64694			−0.240631
$122$		0			0
$123$	−20.4164			−1.84089
$124$		0			0
$125$			4.74216i			0.424152i
$126$		0			0
$127$	−13.1044			−1.16283			−0.581416	−	0.813607i	$-0.697501\pi$
$127$	−13.1044			−1.16283			−0.581416	+	0.813607i	$0.697501\pi$
$128$		0			0
$129$		−	10.1595i		−	0.894492i
$130$		0			0
$131$		−	22.7174i		−	1.98483i	−0.122952	−	0.992413i	$-0.539236\pi$
$131$		−	22.7174i		−	1.98483i	0.122952	−	0.992413i	$-0.460764\pi$
$132$		0			0
$133$			35.5042i			3.07861i
$134$		0			0
$135$	13.6611			1.17576
$136$		0			0
$137$	−6.80555			−0.581438			−0.290719	−	0.956809i	$-0.593894\pi$
$137$	−6.80555			−0.581438			−0.290719	+	0.956809i	$0.593894\pi$
$138$		0			0
$139$			16.4701i			1.39698i	0.715622	+	0.698488i	$0.246143\pi$
$139$			16.4701i			1.39698i	−0.715622	+	0.698488i	$0.753857\pi$
$140$		0			0
$141$		−	5.31535i		−	0.447633i
$142$		0			0
$143$			11.0747i			0.926113i
$144$		0			0
$145$	−8.09811			−0.672512
$146$		0			0
$147$			35.5040i			2.92832i
$148$		0			0
$149$	7.04758			0.577360			0.288680	−	0.957426i	$-0.406784\pi$
$149$	7.04758			0.577360			0.288680	+	0.957426i	$0.406784\pi$
$150$		0			0
$151$	19.8534			1.61565			0.807825	−	0.589423i	$-0.200645\pi$
$151$	19.8534			1.61565			0.807825	+	0.589423i	$0.200645\pi$
$152$		0			0
$153$	−0.514465	+	1.82494i	−0.0415921	+	0.147537i
$154$		0			0
$155$	−4.26234			−0.342359
$156$		0			0
$157$	−22.1275			−1.76596			−0.882982	−	0.469408i	$-0.844468\pi$
$157$	−22.1275			−1.76596			−0.882982	+	0.469408i	$0.844468\pi$
$158$		0			0
$159$		−	25.2155i		−	1.99972i
$160$		0			0
$161$	−27.1819			−2.14224
$162$		0			0
$163$		−	2.22792i		−	0.174504i	−0.996186	−	0.0872522i	$-0.972191\pi$
$163$		−	2.22792i		−	0.174504i	0.996186	−	0.0872522i	$-0.0278086\pi$
$164$		0			0
$165$			19.8677i			1.54670i
$166$		0			0
$167$			17.1828i			1.32964i	0.747002	+	0.664822i	$0.231493\pi$
$167$			17.1828i			1.32964i	−0.747002	+	0.664822i	$0.768507\pi$
$168$		0			0
$169$	−4.01272			−0.308671
$170$		0			0
$171$	−3.19664			−0.244453
$172$		0			0
$173$		−	11.6921i		−	0.888933i	−0.895795	−	0.444467i	$-0.853393\pi$
$173$		−	11.6921i		−	0.888933i	0.895795	−	0.444467i	$-0.146607\pi$
$174$		0			0
$175$		−	17.1609i		−	1.29724i
$176$		0			0
$177$			1.86007i			0.139811i
$178$		0			0
$179$	−15.0858			−1.12757			−0.563784	−	0.825922i	$-0.690655\pi$
$179$	−15.0858			−1.12757			−0.563784	+	0.825922i	$0.690655\pi$
$180$		0			0
$181$		−	16.5621i		−	1.23105i	−0.788116	−	0.615527i	$-0.788943\pi$
$181$		−	16.5621i		−	1.23105i	0.788116	−	0.615527i	$-0.211057\pi$
$182$		0			0
$183$	12.2881			0.908365
$184$		0			0
$185$	0.638946			0.0469763
$186$		0			0
$187$	14.6601	+	4.13280i	1.07205	+	0.302220i
$188$		0			0
$189$	24.1325			1.75538
$190$		0			0
$191$	16.9895			1.22932			0.614659	−	0.788793i	$-0.289294\pi$
$191$	16.9895			1.22932			0.614659	+	0.788793i	$0.289294\pi$
$192$		0			0
$193$		−	11.3216i		−	0.814944i	−0.913218	−	0.407472i	$-0.866410\pi$
$193$		−	11.3216i		−	0.814944i	0.913218	−	0.407472i	$-0.133590\pi$
$194$		0			0
$195$	16.1229			1.15459
$196$		0			0
$197$		−	10.0831i		−	0.718388i	−0.933263	−	0.359194i	$-0.883052\pi$
$197$		−	10.0831i		−	0.718388i	0.933263	−	0.359194i	$-0.116948\pi$
$198$		0			0
$199$			11.5885i			0.821484i	0.911752	+	0.410742i	$0.134730\pi$
$199$			11.5885i			0.821484i	−0.911752	+	0.410742i	$0.865270\pi$
$200$		0			0
$201$		−	15.2474i		−	1.07547i
$202$		0			0
$203$	−14.3054			−1.00404
$204$		0			0
$205$	31.7359			2.21653
$206$		0			0
$207$		−	2.44734i		−	0.170102i
$208$		0			0
$209$			25.6792i			1.77627i
$210$		0			0
$211$		−	19.6612i		−	1.35354i	−0.736197	−	0.676768i	$-0.763380\pi$
$211$		−	19.6612i		−	1.35354i	0.736197	−	0.676768i	$-0.236620\pi$
$212$		0			0
$213$	−14.0228			−0.960823
$214$		0			0
$215$			15.7922i			1.07702i
$216$		0			0
$217$	−7.52945			−0.511133
$218$		0			0
$219$	22.7556			1.53768
$220$		0			0
$221$	3.35383	−	11.8969i	0.225603	−	0.800270i
$222$		0			0
$223$	8.78624			0.588370			0.294185	−	0.955748i	$-0.404952\pi$
$223$	8.78624			0.588370			0.294185	+	0.955748i	$0.404952\pi$
$224$		0			0
$225$	1.54509			0.103006
$226$		0			0
$227$		−	16.3872i		−	1.08765i	−0.839197	−	0.543827i	$-0.816975\pi$
$227$		−	16.3872i		−	1.08765i	0.839197	−	0.543827i	$-0.183025\pi$
$228$		0			0
$229$	−8.38430			−0.554050			−0.277025	−	0.960863i	$-0.589348\pi$
$229$	−8.38430			−0.554050			−0.277025	+	0.960863i	$0.589348\pi$
$230$		0			0
$231$			35.0964i			2.30918i
$232$		0			0
$233$		−	21.5420i		−	1.41126i	−0.708579	−	0.705632i	$-0.750663\pi$
$233$		−	21.5420i		−	1.41126i	0.708579	−	0.705632i	$-0.249337\pi$
$234$		0			0
$235$			8.26232i			0.538974i
$236$		0			0
$237$	−1.31517			−0.0854294
$238$		0			0
$239$	−11.3729			−0.735652			−0.367826	−	0.929895i	$-0.619898\pi$
$239$	−11.3729			−0.735652			−0.367826	+	0.929895i	$0.619898\pi$
$240$		0			0
$241$		−	8.01204i		−	0.516101i	−0.966131	−	0.258050i	$-0.916920\pi$
$241$		−	8.01204i		−	0.516101i	0.966131	−	0.258050i	$-0.0830801\pi$
$242$		0			0
$243$			4.73892i			0.304002i
$244$		0			0
$245$		−	55.1884i		−	3.52586i
$246$		0			0
$247$	20.8391			1.32596
$248$		0			0
$249$			5.51226i			0.349325i
$250$		0			0
$251$	24.3527			1.53713			0.768564	−	0.639773i	$-0.220972\pi$
$251$	24.3527			1.53713			0.768564	+	0.639773i	$0.220972\pi$
$252$		0			0
$253$	−19.6599			−1.23601
$254$		0			0
$255$	6.01667	−	21.3426i	0.376779	−	1.33653i
$256$		0			0
$257$	−15.4001			−0.960629			−0.480315	−	0.877096i	$-0.659477\pi$
$257$	−15.4001			−0.960629			−0.480315	+	0.877096i	$0.659477\pi$
$258$		0			0
$259$	1.12870			0.0701343
$260$		0			0
$261$		−	1.28799i		−	0.0797247i
$262$		0			0
$263$	11.3456			0.699600			0.349800	−	0.936824i	$-0.386249\pi$
$263$	11.3456			0.699600			0.349800	+	0.936824i	$0.386249\pi$
$264$		0			0
$265$			39.1956i			2.40777i
$266$		0			0
$267$		−	7.66747i		−	0.469241i
$268$		0			0
$269$			24.0600i			1.46696i	0.679709	+	0.733482i	$0.262106\pi$
$269$			24.0600i			1.46696i	−0.679709	+	0.733482i	$0.737894\pi$
$270$		0			0
$271$	−7.61246			−0.462424			−0.231212	−	0.972903i	$-0.574269\pi$
$271$	−7.61246			−0.462424			−0.231212	+	0.972903i	$0.574269\pi$
$272$		0			0
$273$	28.4813			1.72377
$274$		0			0
$275$		−	12.4120i		−	0.748471i
$276$		0			0
$277$		−	10.9397i		−	0.657302i	−0.944451	−	0.328651i	$-0.893406\pi$
$277$		−	10.9397i		−	0.657302i	0.944451	−	0.328651i	$-0.106594\pi$
$278$		0			0
$279$		−	0.677918i		−	0.0405859i
$280$		0			0
$281$	−4.18727			−0.249791			−0.124896	−	0.992170i	$-0.539860\pi$
$281$	−4.18727			−0.249791			−0.124896	+	0.992170i	$0.539860\pi$
$282$		0			0
$283$			15.7253i			0.934775i	0.884053	+	0.467387i	$0.154805\pi$
$283$			15.7253i			0.934775i	−0.884053	+	0.467387i	$0.845195\pi$
$284$		0			0
$285$	37.3847			2.21448
$286$		0			0
$287$	56.0617			3.30922
$288$		0			0
$289$	−14.4969	−	8.87923i	−0.852757	−	0.522308i
$290$		0			0
$291$	−0.892160			−0.0522993
$292$		0			0
$293$	−3.69173			−0.215673			−0.107837	−	0.994169i	$-0.534392\pi$
$293$	−3.69173			−0.215673			−0.107837	+	0.994169i	$0.534392\pi$
$294$		0			0
$295$		−	2.89135i		−	0.168341i
$296$		0			0
$297$	17.4544			1.01280
$298$		0			0
$299$			15.9543i			0.922663i
$300$		0			0
$301$			27.8970i			1.60796i
$302$		0			0
$303$			31.1599i			1.79009i
$304$		0			0
$305$	−19.1010			−1.09372
$306$		0			0
$307$	14.3249			0.817568			0.408784	−	0.912631i	$-0.365953\pi$
$307$	14.3249			0.817568			0.408784	+	0.912631i	$0.365953\pi$
$308$		0			0
$309$			20.7717i			1.18166i
$310$		0			0
$311$			17.8949i			1.01473i	0.861732	+	0.507364i	$0.169380\pi$
$311$			17.8949i			1.01473i	−0.861732	+	0.507364i	$0.830620\pi$
$312$		0			0
$313$			7.16121i			0.404775i	0.979305	+	0.202388i	$0.0648701\pi$
$313$			7.16121i			0.404775i	−0.979305	+	0.202388i	$0.935130\pi$
$314$		0			0
$315$	6.79118			0.382640
$316$		0			0
$317$		−	22.3627i		−	1.25601i	−0.778208	−	0.628007i	$-0.783871\pi$
$317$		−	22.3627i		−	1.25601i	0.778208	−	0.628007i	$-0.216129\pi$
$318$		0			0
$319$	−10.3467			−0.579303
$320$		0			0
$321$	13.5205			0.754640
$322$		0			0
$323$	7.77662	−	27.5856i	0.432703	−	1.53491i
$324$		0			0
$325$	−10.0725			−0.558722
$326$		0			0
$327$	20.1624			1.11499
$328$		0			0
$329$			14.5955i			0.804674i
$330$		0			0
$331$	−7.93069			−0.435910			−0.217955	−	0.975959i	$-0.569939\pi$
$331$	−7.93069			−0.435910			−0.217955	+	0.975959i	$0.569939\pi$
$332$		0			0
$333$			0.101623i			0.00556893i
$334$		0			0
$335$			23.7010i			1.29492i
$336$		0			0
$337$			9.94518i			0.541749i	0.962615	+	0.270874i	$0.0873128\pi$
$337$			9.94518i			0.541749i	−0.962615	+	0.270874i	$0.912687\pi$
$338$		0			0
$339$	−15.9068			−0.863936
$340$		0			0
$341$	−5.44585			−0.294909
$342$		0			0
$343$		−	61.7377i		−	3.33352i
$344$		0			0
$345$			28.6216i			1.54093i
$346$		0			0
$347$			7.21487i			0.387315i	0.981069	+	0.193657i	$0.0620350\pi$
$347$			7.21487i			0.387315i	−0.981069	+	0.193657i	$0.937965\pi$
$348$		0			0
$349$	−28.7116			−1.53690			−0.768448	−	0.639913i	$-0.778970\pi$
$349$	−28.7116			−1.53690			−0.768448	+	0.639913i	$0.778970\pi$
$350$		0			0
$351$		−	14.1645i		−	0.756044i
$352$		0			0
$353$	−16.1753			−0.860923			−0.430462	−	0.902609i	$-0.641649\pi$
$353$	−16.1753			−0.860923			−0.430462	+	0.902609i	$0.641649\pi$
$354$		0			0
$355$	21.7973			1.15688
$356$		0			0
$357$	10.6285	−	37.7020i	0.562520	−	1.99540i
$358$		0			0
$359$	24.2609			1.28044			0.640219	−	0.768192i	$-0.278843\pi$
$359$	24.2609			1.28044			0.640219	+	0.768192i	$0.278843\pi$
$360$		0			0
$361$	29.3202			1.54317
$362$		0			0
$363$			4.92349i			0.258416i
$364$		0			0
$365$	−35.3720			−1.85145
$366$		0			0
$367$		−	24.4869i		−	1.27821i	−0.769120	−	0.639104i	$-0.779305\pi$
$367$		−	24.4869i		−	1.27821i	0.769120	−	0.639104i	$-0.220695\pi$
$368$		0			0
$369$			5.04754i			0.262764i
$370$		0			0
$371$			69.2394i			3.59473i
$372$		0			0
$373$	8.79180			0.455222			0.227611	−	0.973752i	$-0.426908\pi$
$373$	8.79180			0.455222			0.227611	+	0.973752i	$0.426908\pi$
$374$		0			0
$375$	8.82075			0.455502
$376$		0			0
$377$			8.39649i			0.432441i
$378$		0			0
$379$			0.632877i			0.0325087i	0.999868	+	0.0162544i	$0.00517415\pi$
$379$			0.632877i			0.0325087i	−0.999868	+	0.0162544i	$0.994826\pi$
$380$		0			0
$381$			24.3752i			1.24878i
$382$		0			0
$383$	−18.0539			−0.922510			−0.461255	−	0.887268i	$-0.652601\pi$
$383$	−18.0539			−0.922510			−0.461255	+	0.887268i	$0.652601\pi$
$384$		0			0
$385$		−	54.5548i		−	2.78037i
$386$		0			0
$387$	−2.51172			−0.127678
$388$		0			0
$389$	1.06197			0.0538439			0.0269219	−	0.999638i	$-0.491429\pi$
$389$	1.06197			0.0538439			0.0269219	+	0.999638i	$0.491429\pi$
$390$		0			0
$391$	21.1195	+	5.95376i	1.06806	+	0.301094i
$392$		0			0
$393$	−42.2559			−2.13153
$394$		0			0
$395$	2.04434			0.102862
$396$		0			0
$397$		−	38.0078i		−	1.90756i	−0.300510	−	0.953779i	$-0.597157\pi$
$397$		−	38.0078i		−	1.90756i	0.300510	−	0.953779i	$-0.402843\pi$
$398$		0			0
$399$	66.0404			3.30615
$400$		0			0
$401$		−	18.1438i		−	0.906057i	−0.891496	−	0.453028i	$-0.850344\pi$
$401$		−	18.1438i		−	0.906057i	0.891496	−	0.453028i	$-0.149656\pi$
$402$		0			0
$403$			4.41939i			0.220145i
$404$		0			0
$405$		−	29.3995i		−	1.46087i
$406$		0			0
$407$	0.816360			0.0404655
$408$		0			0
$409$	11.3829			0.562847			0.281424	−	0.959584i	$-0.409193\pi$
$409$	11.3829			0.562847			0.281424	+	0.959584i	$0.409193\pi$
$410$		0			0
$411$			12.6588i			0.624413i
$412$		0			0
$413$		−	5.10759i		−	0.251328i
$414$		0			0
$415$		−	8.56841i		−	0.420607i
$416$		0			0
$417$	30.6355			1.50023
$418$		0			0
$419$		−	8.63775i		−	0.421982i	−0.977488	−	0.210991i	$-0.932331\pi$
$419$		−	8.63775i		−	0.421982i	0.977488	−	0.210991i	$-0.0676690\pi$
$420$		0			0
$421$	−16.9313			−0.825183			−0.412591	−	0.910916i	$-0.635376\pi$
$421$	−16.9313			−0.825183			−0.412591	+	0.910916i	$0.635376\pi$
$422$		0			0
$423$	−1.31411			−0.0638942
$424$		0			0
$425$	−3.75881	+	13.3334i	−0.182329	+	0.646767i
$426$		0			0
$427$	−33.7421			−1.63289
$428$		0			0
$429$	20.5997			0.994564
$430$		0			0
$431$		−	30.2496i		−	1.45707i	−0.685008	−	0.728535i	$-0.740202\pi$
$431$		−	30.2496i		−	1.45707i	0.685008	−	0.728535i	$-0.259798\pi$
$432$		0			0
$433$	−0.289043			−0.0138905			−0.00694526	−	0.999976i	$-0.502211\pi$
$433$	−0.289043			−0.0138905			−0.00694526	+	0.999976i	$0.502211\pi$
$434$		0			0
$435$			15.0631i			0.722219i
$436$		0			0
$437$			36.9938i			1.76965i
$438$		0			0
$439$			4.82621i			0.230343i	0.993346	+	0.115171i	$0.0367417\pi$
$439$			4.82621i			0.230343i	−0.993346	+	0.115171i	$0.963258\pi$
$440$		0			0
$441$	8.77763			0.417982
$442$		0			0
$443$	−3.99453			−0.189786			−0.0948929	−	0.995487i	$-0.530251\pi$
$443$	−3.99453			−0.189786			−0.0948929	+	0.995487i	$0.530251\pi$
$444$		0			0
$445$			11.9185i			0.564992i
$446$		0			0
$447$		−	13.1090i		−	0.620035i
$448$		0			0
$449$			13.9627i			0.658943i	0.944166	+	0.329471i	$0.106870\pi$
$449$			13.9627i			0.658943i	−0.944166	+	0.329471i	$0.893130\pi$
$450$		0			0
$451$	40.5479			1.90932
$452$		0			0
$453$		−	36.9288i		−	1.73507i
$454$		0			0
$455$	−44.2721			−2.07551
$456$		0			0
$457$	0.395922			0.0185205			0.00926023	−	0.999957i	$-0.497052\pi$
$457$	0.395922			0.0185205			0.00926023	+	0.999957i	$0.497052\pi$
$458$		0			0
$459$	−18.7502	−	5.28583i	−0.875183	−	0.246721i
$460$		0			0
$461$	−28.6559			−1.33464			−0.667318	−	0.744772i	$-0.732558\pi$
$461$	−28.6559			−1.33464			−0.667318	+	0.744772i	$0.732558\pi$
$462$		0			0
$463$	−4.01729			−0.186699			−0.0933496	−	0.995633i	$-0.529757\pi$
$463$	−4.01729			−0.186699			−0.0933496	+	0.995633i	$0.529757\pi$
$464$		0			0
$465$			7.92825i			0.367664i
$466$		0			0
$467$	11.2678			0.521412			0.260706	−	0.965418i	$-0.416045\pi$
$467$	11.2678			0.521412			0.260706	+	0.965418i	$0.416045\pi$
$468$		0			0
$469$			41.8680i			1.93328i
$470$		0			0
$471$			41.1586i			1.89649i
$472$		0			0
$473$			20.1771i			0.927746i
$474$		0			0
$475$	−23.3554			−1.07162
$476$		0			0
$477$	−6.23400			−0.285435
$478$		0			0
$479$		−	23.1622i		−	1.05831i	−0.848525	−	0.529155i	$-0.822509\pi$
$479$		−	23.1622i		−	1.05831i	0.848525	−	0.529155i	$-0.177491\pi$
$480$		0			0
$481$		−	0.662489i		−	0.0302069i
$482$		0			0
$483$			50.5603i			2.30057i
$484$		0			0
$485$	1.38680			0.0629713
$486$		0			0
$487$			3.01963i			0.136833i	0.997657	+	0.0684164i	$0.0217946\pi$
$487$			3.01963i			0.136833i	−0.997657	+	0.0684164i	$0.978205\pi$
$488$		0			0
$489$	−4.14410			−0.187403
$490$		0			0
$491$	30.7368			1.38713			0.693566	−	0.720393i	$-0.256039\pi$
$491$	30.7368			1.38713			0.693566	+	0.720393i	$0.256039\pi$
$492$		0			0
$493$	11.1148	+	3.13336i	0.500586	+	0.141120i
$494$		0			0
$495$	4.91187			0.220772
$496$		0			0
$497$	38.5052			1.72719
$498$		0			0
$499$		−	32.7116i		−	1.46437i	−0.681104	−	0.732187i	$-0.738500\pi$
$499$		−	32.7116i		−	1.46437i	0.681104	−	0.732187i	$-0.261500\pi$
$500$		0			0
$501$	31.9612			1.42792
$502$		0			0
$503$		−	2.96150i		−	0.132047i	−0.997818	−	0.0660235i	$-0.978969\pi$
$503$		−	2.96150i		−	0.132047i	0.997818	−	0.0660235i	$-0.0210312\pi$
$504$		0			0
$505$		−	48.4358i		−	2.15537i
$506$		0			0
$507$			7.46394i			0.331485i
$508$		0			0
$509$	−1.52668			−0.0676689			−0.0338344	−	0.999427i	$-0.510772\pi$
$509$	−1.52668			−0.0676689			−0.0338344	+	0.999427i	$0.510772\pi$
$510$		0			0
$511$	−62.4849			−2.76417
$512$		0			0
$513$		−	32.8436i		−	1.45008i
$514$		0			0
$515$		−	32.2881i		−	1.42278i
$516$		0			0
$517$			10.5565i			0.464274i
$518$		0			0
$519$	−21.7481			−0.954637
$520$		0			0
$521$		−	9.11965i		−	0.399539i	−0.979843	−	0.199770i	$-0.935981\pi$
$521$		−	9.11965i		−	0.399539i	0.979843	−	0.199770i	$-0.0640194\pi$
$522$		0			0
$523$	−11.6805			−0.510753			−0.255377	−	0.966842i	$-0.582199\pi$
$523$	−11.6805			−0.510753			−0.255377	+	0.966842i	$0.582199\pi$
$524$		0			0
$525$	−31.9204			−1.39312
$526$		0			0
$527$	5.85014	+	1.64920i	0.254836	+	0.0718405i
$528$		0			0
$529$	−5.32231			−0.231405
$530$		0			0
$531$	0.459864			0.0199564
$532$		0			0
$533$		−	32.9052i		−	1.42528i
$534$		0			0
$535$	−21.0166			−0.908627
$536$		0			0
$537$			28.0607i			1.21091i
$538$		0			0
$539$		−	70.5124i		−	3.03718i
$540$		0			0
$541$			34.0156i			1.46245i	0.682139	+	0.731223i	$0.261050\pi$
$541$			34.0156i			1.46245i	−0.682139	+	0.731223i	$0.738950\pi$
$542$		0			0
$543$	−30.8068			−1.32204
$544$		0			0
$545$	−31.3411			−1.34250
$546$		0			0
$547$		−	5.85346i		−	0.250276i	−0.992139	−	0.125138i	$-0.960063\pi$
$547$		−	5.85346i		−	0.250276i	0.992139	−	0.125138i	$-0.0399373\pi$
$548$		0			0
$549$		−	3.03799i		−	0.129658i
$550$		0			0
$551$			19.4692i			0.829416i
$552$		0			0
$553$	3.61134			0.153570
$554$		0			0
$555$		−	1.18849i		−	0.0504484i
$556$		0			0
$557$	31.3388			1.32787			0.663934	−	0.747791i	$-0.268886\pi$
$557$	31.3388			1.32787			0.663934	+	0.747791i	$0.268886\pi$
$558$		0			0
$559$	16.3741			0.692549
$560$		0			0
$561$	7.68730	−	27.2688i	0.324558	−	1.15129i
$562$		0			0
$563$	−32.7560			−1.38050			−0.690251	−	0.723570i	$-0.742500\pi$
$563$	−32.7560			−1.38050			−0.690251	+	0.723570i	$0.742500\pi$
$564$		0			0
$565$	24.7259			1.04023
$566$		0			0
$567$		−	51.9345i		−	2.18105i
$568$		0			0
$569$	22.4064			0.939324			0.469662	−	0.882846i	$-0.344376\pi$
$569$	22.4064			0.939324			0.469662	+	0.882846i	$0.344376\pi$
$570$		0			0
$571$			14.1930i			0.593959i	0.954884	+	0.296979i	$0.0959793\pi$
$571$			14.1930i			0.593959i	−0.954884	+	0.296979i	$0.904021\pi$
$572$		0			0
$573$		−	31.6017i		−	1.32018i
$574$		0			0
$575$		−	17.8808i		−	0.745682i
$576$		0			0
$577$	−15.4796			−0.644425			−0.322213	−	0.946667i	$-0.604427\pi$
$577$	−15.4796			−0.644425			−0.322213	+	0.946667i	$0.604427\pi$
$578$		0			0
$579$	−21.0589			−0.875178
$580$		0			0
$581$		−	15.1362i		−	0.627954i
$582$		0			0
$583$			50.0789i			2.07406i
$584$		0			0
$585$		−	3.98606i		−	0.164803i
$586$		0			0
$587$	−16.7788			−0.692537			−0.346269	−	0.938135i	$-0.612551\pi$
$587$	−16.7788			−0.692537			−0.346269	+	0.938135i	$0.612551\pi$
$588$		0			0
$589$			10.2474i			0.422235i
$590$		0			0
$591$	−18.7552			−0.771486
$592$		0			0
$593$	−11.0281			−0.452868			−0.226434	−	0.974026i	$-0.572707\pi$
$593$	−11.0281			−0.452868			−0.226434	+	0.974026i	$0.572707\pi$
$594$		0			0
$595$	−16.5212	+	58.6050i	−0.677305	+	2.40257i
$596$		0			0
$597$	21.5553			0.882201
$598$		0			0
$599$	29.2486			1.19507			0.597533	−	0.801844i	$-0.296147\pi$
$599$	29.2486			1.19507			0.597533	+	0.801844i	$0.296147\pi$
$600$		0			0
$601$		−	20.6149i		−	0.840899i	−0.907316	−	0.420449i	$-0.861872\pi$
$601$		−	20.6149i		−	0.840899i	0.907316	−	0.420449i	$-0.138128\pi$
$602$		0			0
$603$	−3.76960			−0.153510
$604$		0			0
$605$		−	7.65321i		−	0.311147i
$606$		0			0
$607$			38.2772i			1.55363i	0.629732	+	0.776813i	$0.283165\pi$
$607$			38.2772i			1.55363i	−0.629732	+	0.776813i	$0.716835\pi$
$608$		0			0
$609$			26.6090i			1.07825i
$610$		0			0
$611$	8.56675			0.346574
$612$		0			0
$613$	−16.3710			−0.661219			−0.330609	−	0.943768i	$-0.607254\pi$
$613$	−16.3710			−0.661219			−0.330609	+	0.943768i	$0.607254\pi$
$614$		0			0
$615$		−	59.0310i		−	2.38036i
$616$		0			0
$617$			35.1061i			1.41332i	0.707554	+	0.706660i	$0.249799\pi$
$617$			35.1061i			1.41332i	−0.707554	+	0.706660i	$0.750201\pi$
$618$		0			0
$619$			19.8943i			0.799619i	0.916598	+	0.399809i	$0.130924\pi$
$619$			19.8943i			0.799619i	−0.916598	+	0.399809i	$0.869076\pi$
$620$		0			0
$621$	25.1450			1.00903
$622$		0			0
$623$			21.0542i			0.843517i
$624$		0			0
$625$	−30.5106			−1.22042
$626$		0			0
$627$	47.7652			1.90756
$628$		0			0
$629$	−0.876966	−	0.247224i	−0.0349669	−	0.00985748i
$630$		0			0
$631$	38.1079			1.51705			0.758526	−	0.651643i	$-0.225920\pi$
$631$	38.1079			1.51705			0.758526	+	0.651643i	$0.225920\pi$
$632$		0			0
$633$	−36.5713			−1.45358
$634$		0			0
$635$		−	37.8895i		−	1.50360i
$636$		0			0
$637$	−57.2219			−2.26721
$638$		0			0
$639$			3.46683i			0.137146i
$640$		0			0
$641$			0.113319i			0.00447584i	0.999997	+	0.00223792i	$0.000712353\pi$
$641$			0.113319i			0.00447584i	−0.999997	+	0.00223792i	$0.999288\pi$
$642$		0			0
$643$			39.8246i			1.57053i	0.619160	+	0.785265i	$0.287473\pi$
$643$			39.8246i			1.57053i	−0.619160	+	0.785265i	$0.712527\pi$
$644$		0			0
$645$	29.3746			1.15662
$646$		0			0
$647$	−5.81098			−0.228453			−0.114227	−	0.993455i	$-0.536439\pi$
$647$	−5.81098			−0.228453			−0.114227	+	0.993455i	$0.536439\pi$
$648$		0			0
$649$		−	3.69418i		−	0.145009i
$650$		0			0
$651$			14.0053i			0.548912i
$652$		0			0
$653$			0.495161i			0.0193771i	0.999953	+	0.00968857i	$0.00308402\pi$
$653$			0.495161i			0.0193771i	−0.999953	+	0.00968857i	$0.996916\pi$
$654$		0			0
$655$	65.6837			2.56648
$656$		0			0
$657$		−	5.62586i		−	0.219486i
$658$		0			0
$659$	−8.94328			−0.348381			−0.174190	−	0.984712i	$-0.555731\pi$
$659$	−8.94328			−0.348381			−0.174190	+	0.984712i	$0.555731\pi$
$660$		0			0
$661$	−1.72325			−0.0670266			−0.0335133	−	0.999438i	$-0.510670\pi$
$661$	−1.72325			−0.0670266			−0.0335133	+	0.999438i	$0.510670\pi$
$662$		0			0
$663$	−22.1290	−	6.23836i	−0.859420	−	0.242278i
$664$		0			0
$665$	−102.655			−3.98079
$666$		0			0
$667$	−14.9056			−0.577145
$668$		0			0
$669$		−	16.3430i		−	0.631858i
$670$		0			0
$671$	−24.4047			−0.942134
$672$		0			0
$673$			34.9098i			1.34568i	0.739790	+	0.672838i	$0.234925\pi$
$673$			34.9098i			1.34568i	−0.739790	+	0.672838i	$0.765075\pi$
$674$		0			0
$675$			15.8749i			0.611024i
$676$		0			0
$677$		−	38.3115i		−	1.47243i	−0.676747	−	0.736216i	$-0.736611\pi$
$677$		−	38.3115i		−	1.47243i	0.676747	−	0.736216i	$-0.263389\pi$
$678$		0			0
$679$	2.44979			0.0940143
$680$		0			0
$681$	−30.4813			−1.16805
$682$		0			0
$683$		−	25.0818i		−	0.959728i	−0.877343	−	0.479864i	$-0.840686\pi$
$683$		−	25.0818i		−	0.959728i	0.877343	−	0.479864i	$-0.159314\pi$
$684$		0			0
$685$		−	19.6772i		−	0.751827i
$686$		0			0
$687$			15.5954i			0.595001i
$688$		0			0
$689$	40.6398			1.54825
$690$		0			0
$691$			38.0500i			1.44749i	0.690068	+	0.723745i	$0.257581\pi$
$691$			38.0500i			1.44749i	−0.690068	+	0.723745i	$0.742419\pi$
$692$		0			0
$693$	8.67686			0.329607
$694$		0			0
$695$	−47.6207			−1.80636
$696$		0			0
$697$	−43.5581	−	12.2794i	−1.64988	−	0.465115i
$698$		0			0
$699$	−40.0697			−1.51557
$700$		0			0
$701$	5.28205			0.199500			0.0997501	−	0.995013i	$-0.468196\pi$
$701$	5.28205			0.199500			0.0997501	+	0.995013i	$0.468196\pi$
$702$		0			0
$703$		−	1.53613i		−	0.0579363i
$704$		0			0
$705$	15.3685			0.578811
$706$		0			0
$707$		−	85.5623i		−	3.21790i
$708$		0			0
$709$		−	5.57802i		−	0.209487i	−0.994499	−	0.104743i	$-0.966598\pi$
$709$		−	5.57802i		−	0.209487i	0.994499	−	0.104743i	$-0.0334022\pi$
$710$		0			0
$711$			0.325148i			0.0121940i
$712$		0			0
$713$	−7.84535			−0.293810
$714$		0			0
$715$	−32.0208			−1.19751
$716$		0			0
$717$			21.1544i			0.790026i
$718$		0			0
$719$			44.1175i			1.64530i	0.568546	+	0.822652i	$0.307506\pi$
$719$			44.1175i			1.64530i	−0.568546	+	0.822652i	$0.692494\pi$
$720$		0			0
$721$		−	57.0372i		−	2.12418i
$722$		0			0
$723$	−14.9030			−0.554247
$724$		0			0
$725$		−	9.41038i		−	0.349493i
$726$		0			0
$727$	43.6638			1.61940			0.809700	−	0.586844i	$-0.199630\pi$
$727$	43.6638			1.61940			0.809700	+	0.586844i	$0.199630\pi$
$728$		0			0
$729$	−21.6896			−0.803319
$730$		0			0
$731$	6.11039	−	21.6751i	0.226001	−	0.801682i
$732$		0			0
$733$	−44.6813			−1.65034			−0.825171	−	0.564883i	$-0.808921\pi$
$733$	−44.6813			−1.65034			−0.825171	+	0.564883i	$0.808921\pi$
$734$		0			0
$735$	−102.654			−3.78646
$736$		0			0
$737$			30.2819i			1.11545i
$738$		0			0
$739$	−18.5464			−0.682240			−0.341120	−	0.940020i	$-0.610806\pi$
$739$	−18.5464			−0.682240			−0.341120	+	0.940020i	$0.610806\pi$
$740$		0			0
$741$		−	38.7622i		−	1.42396i
$742$		0			0
$743$		−	33.0639i		−	1.21300i	−0.795084	−	0.606499i	$-0.792573\pi$
$743$		−	33.0639i		−	1.21300i	0.795084	−	0.606499i	$-0.207427\pi$
$744$		0			0
$745$			20.3770i			0.746555i
$746$		0			0
$747$	1.36279			0.0498620
$748$		0			0
$749$	−37.1260			−1.35656
$750$		0			0
$751$			45.1155i			1.64629i	0.567834	+	0.823143i	$0.307782\pi$
$751$			45.1155i			1.64629i	−0.567834	+	0.823143i	$0.692218\pi$
$752$		0			0
$753$		−	45.2977i		−	1.65074i
$754$		0			0
$755$			57.4031i			2.08911i
$756$		0			0
$757$	33.9707			1.23469			0.617343	−	0.786694i	$-0.288209\pi$
$757$	33.9707			1.23469			0.617343	+	0.786694i	$0.288209\pi$
$758$		0			0
$759$			36.5689i			1.32737i
$760$		0			0
$761$	−12.7173			−0.461002			−0.230501	−	0.973072i	$-0.574036\pi$
$761$	−12.7173			−0.461002			−0.230501	+	0.973072i	$0.574036\pi$
$762$		0			0
$763$	−55.3643			−2.00432
$764$		0			0
$765$	−5.27653	−	1.48750i	−0.190773	−	0.0537806i
$766$		0			0
$767$	−2.99788			−0.108247
$768$		0			0
$769$	3.61166			0.130240			0.0651199	−	0.997877i	$-0.479257\pi$
$769$	3.61166			0.130240			0.0651199	+	0.997877i	$0.479257\pi$
$770$		0			0
$771$			28.6452i			1.03163i
$772$		0			0
$773$	26.6077			0.957014			0.478507	−	0.878084i	$-0.341178\pi$
$773$	26.6077			0.957014			0.478507	+	0.878084i	$0.341178\pi$
$774$		0			0
$775$		−	4.95303i		−	0.177918i
$776$		0			0
$777$		−	2.09947i		−	0.0753180i
$778$		0			0
$779$		−	76.2983i		−	2.73367i
$780$		0			0
$781$	27.8497			0.996542
$782$		0			0
$783$	13.2334			0.472922
$784$		0			0
$785$		−	63.9781i		−	2.28348i
$786$		0			0
$787$		−	3.97052i		−	0.141534i	−0.997493	−	0.0707669i	$-0.977455\pi$
$787$		−	3.97052i		−	0.141534i	0.997493	−	0.0707669i	$-0.0225447\pi$
$788$		0			0
$789$		−	21.1036i		−	0.751310i
$790$		0			0
$791$	43.6785			1.55303
$792$		0			0
$793$			19.8048i			0.703289i
$794$		0			0
$795$	72.9066			2.58573
$796$		0			0
$797$	−49.6734			−1.75952			−0.879761	−	0.475417i	$-0.842297\pi$
$797$	−49.6734			−1.75952			−0.879761	+	0.475417i	$0.842297\pi$
$798$		0			0
$799$	3.19690	−	11.3402i	0.113098	−	0.401187i
$800$		0			0
$801$	−1.89562			−0.0669785
$802$		0			0
$803$	−45.1936			−1.59485
$804$		0			0
$805$		−	78.5923i		−	2.77001i
$806$		0			0
$807$	44.7533			1.57539
$808$		0			0
$809$			8.68560i			0.305369i	0.988275	+	0.152685i	$0.0487919\pi$
$809$			8.68560i			0.305369i	−0.988275	+	0.152685i	$0.951208\pi$
$810$		0			0
$811$			18.1011i			0.635615i	0.948155	+	0.317808i	$0.102947\pi$
$811$			18.1011i			0.635615i	−0.948155	+	0.317808i	$0.897053\pi$
$812$		0			0
$813$			14.1597i			0.496603i
$814$		0			0
$815$	6.44170			0.225643
$816$		0			0
$817$	37.9670			1.32830
$818$		0			0
$819$		−	7.04141i		−	0.246047i
$820$		0			0
$821$			1.64287i			0.0573364i	0.999589	+	0.0286682i	$0.00912663\pi$
$821$			1.64287i			0.0573364i	−0.999589	+	0.0286682i	$0.990873\pi$
$822$		0			0
$823$			30.2052i			1.05289i	0.850210	+	0.526444i	$0.176475\pi$
$823$			30.2052i			1.05289i	−0.850210	+	0.526444i	$0.823525\pi$
$824$		0			0
$825$	−23.0872			−0.803792
$826$		0			0
$827$		−	21.8966i		−	0.761421i	−0.924694	−	0.380711i	$-0.875679\pi$
$827$		−	21.8966i		−	0.761421i	0.924694	−	0.380711i	$-0.124321\pi$
$828$		0			0
$829$	−33.1253			−1.15049			−0.575244	−	0.817982i	$-0.695093\pi$
$829$	−33.1253			−1.15049			−0.575244	+	0.817982i	$0.695093\pi$
$830$		0			0
$831$	−20.3486			−0.705885
$832$		0			0
$833$	−21.3538	+	75.7472i	−0.739864	+	2.62448i
$834$		0			0
$835$	−49.6814			−1.71929
$836$		0			0
$837$	6.96521			0.240753
$838$		0			0
$839$			40.6890i			1.40474i	0.711813	+	0.702369i	$0.247875\pi$
$839$			40.6890i			1.40474i	−0.711813	+	0.702369i	$0.752125\pi$
$840$		0			0
$841$	21.1555			0.729499
$842$		0			0
$843$			7.78862i			0.268254i
$844$		0			0
$845$		−	11.6021i		−	0.399126i
$846$		0			0
$847$		−	13.5195i		−	0.464534i
$848$		0			0
$849$	29.2503			1.00387
$850$		0			0
$851$	1.17606			0.0403147
$852$		0			0
$853$		−	32.7331i		−	1.12076i	−0.828235	−	0.560380i	$-0.810655\pi$
$853$		−	32.7331i		−	1.12076i	0.828235	−	0.560380i	$-0.189345\pi$
$854$		0			0
$855$		−	9.24259i		−	0.316090i
$856$		0			0
$857$			14.4059i			0.492095i	0.969258	+	0.246047i	$0.0791319\pi$
$857$			14.4059i			0.492095i	−0.969258	+	0.246047i	$0.920868\pi$
$858$		0			0
$859$	22.4719			0.766733			0.383366	−	0.923596i	$-0.374765\pi$
$859$	22.4719			0.766733			0.383366	+	0.923596i	$0.374765\pi$
$860$		0			0
$861$		−	104.279i		−	3.55381i
$862$		0			0
$863$	6.66348			0.226827			0.113414	−	0.993548i	$-0.463821\pi$
$863$	6.66348			0.226827			0.113414	+	0.993548i	$0.463821\pi$
$864$		0			0
$865$	33.8059			1.14943
$866$		0			0
$867$	−16.5160	+	26.9652i	−0.560913	+	0.915786i
$868$		0			0
$869$	2.61198			0.0886053
$870$		0			0
$871$	24.5743			0.832667
$872$		0			0
$873$			0.220568i			0.00746510i
$874$		0			0
$875$	−24.2210			−0.818819
$876$		0			0
$877$		−	34.7134i		−	1.17219i	−0.810243	−	0.586095i	$-0.800665\pi$
$877$		−	34.7134i		−	1.17219i	0.810243	−	0.586095i	$-0.199335\pi$
$878$		0			0
$879$			6.86688i			0.231614i
$880$		0			0
$881$		−	7.64644i		−	0.257615i	−0.991670	−	0.128808i	$-0.958885\pi$
$881$		−	7.64644i		−	0.257615i	0.991670	−	0.128808i	$-0.0411149\pi$
$882$		0			0
$883$	−27.2536			−0.917157			−0.458578	−	0.888654i	$-0.651641\pi$
$883$	−27.2536			−0.917157			−0.458578	+	0.888654i	$0.651641\pi$
$884$		0			0
$885$	−5.37811			−0.180783
$886$		0			0
$887$		−	25.7306i		−	0.863949i	−0.901886	−	0.431974i	$-0.857817\pi$
$887$		−	25.7306i		−	0.863949i	0.901886	−	0.431974i	$-0.142183\pi$
$888$		0			0
$889$		−	66.9321i		−	2.24483i
$890$		0			0
$891$		−	37.5628i		−	1.25840i
$892$		0			0
$893$	19.8640			0.664723
$894$		0			0
$895$		−	43.6184i		−	1.45800i
$896$		0			0
$897$	29.6762			0.990859
$898$		0			0
$899$	−4.12887			−0.137706
$900$		0			0
$901$	15.1658	−	53.7968i	0.505245	−	1.79223i
$902$		0			0
$903$	51.8904			1.72681
$904$		0			0
$905$	47.8869			1.59181
$906$		0			0
$907$			59.3052i			1.96920i	0.174831	+	0.984598i	$0.444062\pi$
$907$			59.3052i			1.96920i	−0.174831	+	0.984598i	$0.555938\pi$
$908$		0			0
$909$	7.70364			0.255514
$910$		0			0
$911$		−	32.8083i		−	1.08699i	−0.839414	−	0.543493i	$-0.817101\pi$
$911$		−	32.8083i		−	1.08699i	0.839414	−	0.543493i	$-0.182899\pi$
$912$		0			0
$913$		−	10.9476i		−	0.362312i
$914$		0			0
$915$			35.5292i			1.17456i
$916$		0			0
$917$	116.031			3.83168
$918$		0			0
$919$	8.57747			0.282945			0.141472	−	0.989942i	$-0.454816\pi$
$919$	8.57747			0.282945			0.141472	+	0.989942i	$0.454816\pi$
$920$		0			0
$921$		−	26.6454i		−	0.877996i
$922$		0			0
$923$		−	22.6005i		−	0.743904i
$924$		0			0
$925$			0.742485i			0.0244128i
$926$		0			0
$927$	5.13537			0.168668
$928$		0			0
$929$			43.6036i			1.43059i	0.698825	+	0.715293i	$0.253707\pi$
$929$			43.6036i			1.43059i	−0.698825	+	0.715293i	$0.746293\pi$
$930$		0			0
$931$	−132.682			−4.34848
$932$		0			0
$933$	33.2858			1.08973
$934$		0			0
$935$	−11.9493	+	42.3873i	−0.390786	+	1.38621i
$936$		0			0
$937$	39.2782			1.28316			0.641582	−	0.767054i	$-0.278278\pi$
$937$	39.2782			1.28316			0.641582	+	0.767054i	$0.278278\pi$
$938$		0			0
$939$	13.3204			0.434693
$940$		0			0
$941$			22.0309i			0.718187i	0.933302	+	0.359094i	$0.116914\pi$
$941$			22.0309i			0.718187i	−0.933302	+	0.359094i	$0.883086\pi$
$942$		0			0
$943$	58.4137			1.90221
$944$		0			0
$945$			69.7754i			2.26979i
$946$		0			0
$947$			2.54862i			0.0828191i	0.999142	+	0.0414096i	$0.0131848\pi$
$947$			2.54862i			0.0828191i	−0.999142	+	0.0414096i	$0.986815\pi$
$948$		0			0
$949$			36.6753i			1.19053i
$950$		0			0
$951$	−41.5962			−1.34885
$952$		0			0
$953$	−4.44267			−0.143912			−0.0719560	−	0.997408i	$-0.522924\pi$
$953$	−4.44267			−0.143912			−0.0719560	+	0.997408i	$0.522924\pi$
$954$		0			0
$955$			49.1226i			1.58957i
$956$		0			0
$957$			19.2456i			0.622121i
$958$		0			0
$959$		−	34.7599i		−	1.12246i
$960$		0			0
$961$	28.8268			0.929897
$962$		0			0
$963$		−	3.34266i		−	0.107716i
$964$		0			0
$965$	32.7345			1.05376
$966$		0			0
$967$	−26.0144			−0.836568			−0.418284	−	0.908316i	$-0.637368\pi$
$967$	−26.0144			−0.836568			−0.418284	+	0.908316i	$0.637368\pi$
$968$		0			0
$969$	−51.3112	−	14.4651i	−1.64835	−	0.464685i
$970$		0			0
$971$	29.3284			0.941194			0.470597	−	0.882348i	$-0.344039\pi$
$971$	29.3284			0.941194			0.470597	+	0.882348i	$0.344039\pi$
$972$		0			0
$973$	−84.1224			−2.69684
$974$		0			0
$975$			18.7356i			0.600019i
$976$		0			0
$977$	39.9802			1.27908			0.639540	−	0.768758i	$-0.279125\pi$
$977$	39.9802			1.27908			0.639540	+	0.768758i	$0.279125\pi$
$978$		0			0
$979$			15.2279i			0.486686i
$980$		0			0
$981$		−	4.98475i		−	0.159151i
$982$		0			0
$983$			25.4659i			0.812236i	0.913821	+	0.406118i	$0.133118\pi$
$983$			25.4659i			0.812236i	−0.913821	+	0.406118i	$0.866882\pi$
$984$		0			0
$985$	29.1536			0.928911
$986$		0			0
$987$	27.1486			0.864149
$988$		0			0
$989$			29.0674i			0.924290i
$990$		0			0
$991$		−	23.8180i		−	0.756603i	−0.925682	−	0.378302i	$-0.876508\pi$
$991$		−	23.8180i		−	0.756603i	0.925682	−	0.378302i	$-0.123492\pi$
$992$		0			0
$993$			14.7517i			0.468130i
$994$		0			0
$995$	−33.5062			−1.06222
$996$		0			0
$997$		−	26.8301i		−	0.849717i	−0.905260	−	0.424858i	$-0.860324\pi$
$997$		−	26.8301i		−	0.849717i	0.905260	−	0.424858i	$-0.139676\pi$
$998$		0			0
$999$	−1.04412			−0.0330345

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.b.a.237.10	✓	40
17.16	even	2	inner	4012.2.b.a.237.31	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.b.a.237.10	✓	40	1.1	even	1	trivial
4012.2.b.a.237.31	yes	40	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4012.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.86007i q^{3} +2.89135i q^{5} +5.10759i q^{7} -0.459864 q^{9} +O(q^{10})\)	\(q-1.86007i q^{3} +2.89135i q^{5} +5.10759i q^{7} -0.459864 q^{9} +3.69418i q^{11} +2.99788 q^{13} +5.37811 q^{15} +(1.11873 - 3.96843i) q^{17} +6.95127 q^{19} +9.50047 q^{21} +5.32187i q^{23} -3.35988 q^{25} -4.72483i q^{27} +2.80081i q^{29} +1.47417i q^{31} +6.87143 q^{33} -14.7678 q^{35} -0.220986i q^{37} -5.57627i q^{39} -10.9762i q^{41} +5.46188 q^{43} -1.32963i q^{45} +2.85760 q^{47} -19.0874 q^{49} +(-7.38156 - 2.08092i) q^{51} +13.5562 q^{53} -10.6811 q^{55} -12.9299i q^{57} -1.00000 q^{59} +6.60627i q^{61} -2.34880i q^{63} +8.66791i q^{65} +8.19721 q^{67} +9.89906 q^{69} -7.53883i q^{71} +12.2337i q^{73} +6.24961i q^{75} -18.8683 q^{77} -0.707053i q^{79} -10.1681 q^{81} -2.96347 q^{83} +(11.4741 + 3.23465i) q^{85} +5.20971 q^{87} +4.12214 q^{89} +15.3119i q^{91} +2.74206 q^{93} +20.0985i q^{95} -0.479638i q^{97} -1.69882i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 36 q^{9}+O(q^{10}) \) 40 * q - 36 * q^9	\( 40 q - 36 q^{9} + 8 q^{13} + 20 q^{15} - 8 q^{17} + 20 q^{19} + 6 q^{21} - 24 q^{25} - 14 q^{33} - 52 q^{35} + 22 q^{43} - 10 q^{47} + 8 q^{49} - 6 q^{51} - 2 q^{53} - 12 q^{55} - 40 q^{59} - 24 q^{67} - 36 q^{69} - 38 q^{77} + 16 q^{81} + 32 q^{83} - 22 q^{85} - 18 q^{87} + 40 q^{89} + 22 q^{93}+O(q^{100}) \) 40 * q - 36 * q^9 + 8 * q^13 + 20 * q^15 - 8 * q^17 + 20 * q^19 + 6 * q^21 - 24 * q^25 - 14 * q^33 - 52 * q^35 + 22 * q^43 - 10 * q^47 + 8 * q^49 - 6 * q^51 - 2 * q^53 - 12 * q^55 - 40 * q^59 - 24 * q^67 - 36 * q^69 - 38 * q^77 + 16 * q^81 + 32 * q^83 - 22 * q^85 - 18 * q^87 + 40 * q^89 + 22 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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