LMFDB - Embedded newform 4012.2.a.h.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4012,2,Mod(1,4012)]

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, 0, 0]))

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

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

chi := DirichletCharacter("4012.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4012 = 2^{2} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4012.a (trivial)

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:	yes
Analytic conductor:	$32.0359812909$
Analytic rank:	$1$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 5 x^{14} - 16 x^{13} + 123 x^{12} - 76 x^{11} - 630 x^{10} + 937 x^{9} + 1083 x^{8} - 2602 x^{7} + \cdots - 2 $ x^15 - 5x^14 - 16x^13 + 123x^12 - 76x^11 - 630x^10 + 937x^9 + 1083x^8 - 2602x^7 - 249x^6 + 2736x^5 - 801x^4 - 900x^3 + 429x^2 - 36x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.20072$ of defining polynomial
Character	$\chi$	$=$	4012.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.21367 q^{3} -3.70336 q^{5} -3.08925 q^{7} +1.90033 q^{9} +O(q^{10})$	$q-2.21367 q^{3} -3.70336 q^{5} -3.08925 q^{7} +1.90033 q^{9} -5.39193 q^{11} -1.54767 q^{13} +8.19802 q^{15} +1.00000 q^{17} +5.15336 q^{19} +6.83858 q^{21} +6.37839 q^{23} +8.71491 q^{25} +2.43430 q^{27} +4.84013 q^{29} -8.51907 q^{31} +11.9360 q^{33} +11.4406 q^{35} +2.87078 q^{37} +3.42603 q^{39} -6.72655 q^{41} +11.4323 q^{43} -7.03762 q^{45} -6.97477 q^{47} +2.54347 q^{49} -2.21367 q^{51} +4.82147 q^{53} +19.9683 q^{55} -11.4078 q^{57} -1.00000 q^{59} +1.29328 q^{61} -5.87060 q^{63} +5.73159 q^{65} -1.85819 q^{67} -14.1196 q^{69} -3.42281 q^{71} +14.2021 q^{73} -19.2919 q^{75} +16.6570 q^{77} -6.18394 q^{79} -11.0897 q^{81} -15.5741 q^{83} -3.70336 q^{85} -10.7145 q^{87} -3.28727 q^{89} +4.78115 q^{91} +18.8584 q^{93} -19.0848 q^{95} -17.0490 q^{97} -10.2465 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - q^{3} + q^{5} - 11 q^{7} + 16 q^{9}+O(q^{10}) $ 15 * q - q^3 + q^5 - 11 * q^7 + 16 * q^9	$ 15 q - q^{3} + q^{5} - 11 q^{7} + 16 q^{9} - 12 q^{11} - 10 q^{13} - 13 q^{15} + 15 q^{17} - 4 q^{21} - 21 q^{23} + 4 q^{25} - 37 q^{27} + 23 q^{29} - 31 q^{31} - 11 q^{33} - 35 q^{35} - 10 q^{37} - 2 q^{39} - 15 q^{41} - 3 q^{43} + 15 q^{45} - 47 q^{47} + 18 q^{49} - q^{51} - 7 q^{53} - 20 q^{55} - 30 q^{57} - 15 q^{59} + q^{61} + 19 q^{63} + 11 q^{65} - 20 q^{67} - 24 q^{69} - 13 q^{71} - 24 q^{73} - 10 q^{75} + 21 q^{77} - 34 q^{79} - 21 q^{81} - 40 q^{83} + q^{85} - 60 q^{87} - 46 q^{89} - 19 q^{91} + 33 q^{93} - 4 q^{95} - 5 q^{97} - 74 q^{99}+O(q^{100}) $ 15 * q - q^3 + q^5 - 11 * q^7 + 16 * q^9 - 12 * q^11 - 10 * q^13 - 13 * q^15 + 15 * q^17 - 4 * q^21 - 21 * q^23 + 4 * q^25 - 37 * q^27 + 23 * q^29 - 31 * q^31 - 11 * q^33 - 35 * q^35 - 10 * q^37 - 2 * q^39 - 15 * q^41 - 3 * q^43 + 15 * q^45 - 47 * q^47 + 18 * q^49 - q^51 - 7 * q^53 - 20 * q^55 - 30 * q^57 - 15 * q^59 + q^61 + 19 * q^63 + 11 * q^65 - 20 * q^67 - 24 * q^69 - 13 * q^71 - 24 * q^73 - 10 * q^75 + 21 * q^77 - 34 * q^79 - 21 * q^81 - 40 * q^83 + q^85 - 60 * q^87 - 46 * q^89 - 19 * q^91 + 33 * q^93 - 4 * q^95 - 5 * q^97 - 74 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−2.21367	−1.27806	−0.639031	−	0.769181i	$-0.720665\pi$
$3$	−2.21367	−1.27806	−0.639031	+	0.769181i	$0.720665\pi$
$4$	0	0
$5$	−3.70336	−1.65619	−0.828097	−	0.560584i	$-0.810577\pi$
$5$	−3.70336	−1.65619	−0.828097	+	0.560584i	$0.810577\pi$
$6$	0	0
$7$	−3.08925	−1.16763	−0.583814	−	0.811888i	$-0.698440\pi$
$7$	−3.08925	−1.16763	−0.583814	+	0.811888i	$0.698440\pi$
$8$	0	0
$9$	1.90033	0.633444
$10$	0	0
$11$	−5.39193	−1.62573	−0.812864	−	0.582453i	$-0.802093\pi$
$11$	−5.39193	−1.62573	−0.812864	+	0.582453i	$0.802093\pi$
$12$	0	0
$13$	−1.54767	−0.429247	−0.214623	−	0.976697i	$-0.568852\pi$
$13$	−1.54767	−0.429247	−0.214623	+	0.976697i	$0.568852\pi$
$14$	0	0
$15$	8.19802	2.11672
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	5.15336	1.18226	0.591131	−	0.806576i	$-0.298682\pi$
$19$	5.15336	1.18226	0.591131	+	0.806576i	$0.298682\pi$
$20$	0	0
$21$	6.83858	1.49230
$22$	0	0
$23$	6.37839	1.32999	0.664993	−	0.746849i	$-0.268434\pi$
$23$	6.37839	1.32999	0.664993	+	0.746849i	$0.268434\pi$
$24$	0	0
$25$	8.71491	1.74298
$26$	0	0
$27$	2.43430	0.468481
$28$	0	0
$29$	4.84013	0.898790	0.449395	−	0.893333i	$-0.351640\pi$
$29$	4.84013	0.898790	0.449395	+	0.893333i	$0.351640\pi$
$30$	0	0
$31$	−8.51907	−1.53007	−0.765035	−	0.643989i	$-0.777278\pi$
$31$	−8.51907	−1.53007	−0.765035	+	0.643989i	$0.777278\pi$
$32$	0	0
$33$	11.9360	2.07778
$34$	0	0
$35$	11.4406	1.93382
$36$	0	0
$37$	2.87078	0.471953	0.235977	−	0.971759i	$-0.424171\pi$
$37$	2.87078	0.471953	0.235977	+	0.971759i	$0.424171\pi$
$38$	0	0
$39$	3.42603	0.548604
$40$	0	0
$41$	−6.72655	−1.05051	−0.525256	−	0.850944i	$-0.676030\pi$
$41$	−6.72655	−1.05051	−0.525256	+	0.850944i	$0.676030\pi$
$42$	0	0
$43$	11.4323	1.74341	0.871706	−	0.490030i	$-0.163014\pi$
$43$	11.4323	1.74341	0.871706	+	0.490030i	$0.163014\pi$
$44$	0	0
$45$	−7.03762	−1.04911
$46$	0	0
$47$	−6.97477	−1.01738	−0.508688	−	0.860951i	$-0.669869\pi$
$47$	−6.97477	−1.01738	−0.508688	+	0.860951i	$0.669869\pi$
$48$	0	0
$49$	2.54347	0.363353
$50$	0	0
$51$	−2.21367	−0.309976
$52$	0	0
$53$	4.82147	0.662280	0.331140	−	0.943582i	$-0.392567\pi$
$53$	4.82147	0.662280	0.331140	+	0.943582i	$0.392567\pi$
$54$	0	0
$55$	19.9683	2.69252
$56$	0	0
$57$	−11.4078	−1.51100
$58$	0	0
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	1.29328	0.165588	0.0827940	−	0.996567i	$-0.473616\pi$
$61$	1.29328	0.165588	0.0827940	+	0.996567i	$0.473616\pi$
$62$	0	0
$63$	−5.87060	−0.739627
$64$	0	0
$65$	5.73159	0.710917
$66$	0	0
$67$	−1.85819	−0.227014	−0.113507	−	0.993537i	$-0.536208\pi$
$67$	−1.85819	−0.227014	−0.113507	+	0.993537i	$0.536208\pi$
$68$	0	0
$69$	−14.1196	−1.69981
$70$	0	0
$71$	−3.42281	−0.406213	−0.203107	−	0.979157i	$-0.565104\pi$
$71$	−3.42281	−0.406213	−0.203107	+	0.979157i	$0.565104\pi$
$72$	0	0
$73$	14.2021	1.66223	0.831117	−	0.556098i	$-0.187702\pi$
$73$	14.2021	1.66223	0.831117	+	0.556098i	$0.187702\pi$
$74$	0	0
$75$	−19.2919	−2.22764
$76$	0	0
$77$	16.6570	1.89824
$78$	0	0
$79$	−6.18394	−0.695747	−0.347874	−	0.937541i	$-0.613096\pi$
$79$	−6.18394	−0.695747	−0.347874	+	0.937541i	$0.613096\pi$
$80$	0	0
$81$	−11.0897	−1.23219
$82$	0	0
$83$	−15.5741	−1.70948	−0.854740	−	0.519057i	$-0.826283\pi$
$83$	−15.5741	−1.70948	−0.854740	+	0.519057i	$0.826283\pi$
$84$	0	0
$85$	−3.70336	−0.401686
$86$	0	0
$87$	−10.7145	−1.14871
$88$	0	0
$89$	−3.28727	−0.348450	−0.174225	−	0.984706i	$-0.555742\pi$
$89$	−3.28727	−0.348450	−0.174225	+	0.984706i	$0.555742\pi$
$90$	0	0
$91$	4.78115	0.501200
$92$	0	0
$93$	18.8584	1.95553
$94$	0	0
$95$	−19.0848	−1.95805
$96$	0	0
$97$	−17.0490	−1.73106	−0.865530	−	0.500856i	$-0.833018\pi$
$97$	−17.0490	−1.73106	−0.865530	+	0.500856i	$0.833018\pi$
$98$	0	0
$99$	−10.2465	−1.02981
$100$	0	0
$101$	15.6967	1.56188	0.780940	−	0.624605i	$-0.214740\pi$
$101$	15.6967	1.56188	0.780940	+	0.624605i	$0.214740\pi$
$102$	0	0
$103$	−1.19248	−0.117499	−0.0587495	−	0.998273i	$-0.518711\pi$
$103$	−1.19248	−0.117499	−0.0587495	+	0.998273i	$0.518711\pi$
$104$	0	0
$105$	−25.3258	−2.47154
$106$	0	0
$107$	−3.29445	−0.318486	−0.159243	−	0.987239i	$-0.550905\pi$
$107$	−3.29445	−0.318486	−0.159243	+	0.987239i	$0.550905\pi$
$108$	0	0
$109$	18.8987	1.81017	0.905086	−	0.425229i	$-0.139806\pi$
$109$	18.8987	1.81017	0.905086	+	0.425229i	$0.139806\pi$
$110$	0	0
$111$	−6.35496	−0.603186
$112$	0	0
$113$	1.43252	0.134760	0.0673800	−	0.997727i	$-0.478536\pi$
$113$	1.43252	0.134760	0.0673800	+	0.997727i	$0.478536\pi$
$114$	0	0
$115$	−23.6215	−2.20272
$116$	0	0
$117$	−2.94109	−0.271904
$118$	0	0
$119$	−3.08925	−0.283191
$120$	0	0
$121$	18.0729	1.64299
$122$	0	0
$123$	14.8904	1.34262
$124$	0	0
$125$	−13.7577	−1.23052
$126$	0	0
$127$	−6.69642	−0.594212	−0.297106	−	0.954845i	$-0.596021\pi$
$127$	−6.69642	−0.594212	−0.297106	+	0.954845i	$0.596021\pi$
$128$	0	0
$129$	−25.3074	−2.22819
$130$	0	0
$131$	9.07445	0.792839	0.396419	−	0.918070i	$-0.370253\pi$
$131$	9.07445	0.792839	0.396419	+	0.918070i	$0.370253\pi$
$132$	0	0
$133$	−15.9200	−1.38044
$134$	0	0
$135$	−9.01510	−0.775897
$136$	0	0
$137$	−10.5458	−0.900992	−0.450496	−	0.892778i	$-0.648753\pi$
$137$	−10.5458	−0.900992	−0.450496	+	0.892778i	$0.648753\pi$
$138$	0	0
$139$	−8.88352	−0.753490	−0.376745	−	0.926317i	$-0.622957\pi$
$139$	−8.88352	−0.753490	−0.376745	+	0.926317i	$0.622957\pi$
$140$	0	0
$141$	15.4398	1.30027
$142$	0	0
$143$	8.34494	0.697839
$144$	0	0
$145$	−17.9248	−1.48857
$146$	0	0
$147$	−5.63041	−0.464388
$148$	0	0
$149$	20.3513	1.66725	0.833623	−	0.552334i	$-0.186263\pi$
$149$	20.3513	1.66725	0.833623	+	0.552334i	$0.186263\pi$
$150$	0	0
$151$	20.2124	1.64486	0.822432	−	0.568863i	$-0.192617\pi$
$151$	20.2124	1.64486	0.822432	+	0.568863i	$0.192617\pi$
$152$	0	0
$153$	1.90033	0.153633
$154$	0	0
$155$	31.5492	2.53409
$156$	0	0
$157$	7.60864	0.607236	0.303618	−	0.952794i	$-0.401805\pi$
$157$	7.60864	0.607236	0.303618	+	0.952794i	$0.401805\pi$
$158$	0	0
$159$	−10.6731	−0.846435
$160$	0	0
$161$	−19.7045	−1.55293
$162$	0	0
$163$	1.42893	0.111923	0.0559614	−	0.998433i	$-0.482178\pi$
$163$	1.42893	0.111923	0.0559614	+	0.998433i	$0.482178\pi$
$164$	0	0
$165$	−44.2032	−3.44121
$166$	0	0
$167$	2.27723	0.176217	0.0881086	−	0.996111i	$-0.471918\pi$
$167$	2.27723	0.176217	0.0881086	+	0.996111i	$0.471918\pi$
$168$	0	0
$169$	−10.6047	−0.815747
$170$	0	0
$171$	9.79309	0.748896
$172$	0	0
$173$	8.62476	0.655728	0.327864	−	0.944725i	$-0.393671\pi$
$173$	8.62476	0.655728	0.327864	+	0.944725i	$0.393671\pi$
$174$	0	0
$175$	−26.9225	−2.03515
$176$	0	0
$177$	2.21367	0.166390
$178$	0	0
$179$	7.64586	0.571479	0.285739	−	0.958307i	$-0.407761\pi$
$179$	7.64586	0.571479	0.285739	+	0.958307i	$0.407761\pi$
$180$	0	0
$181$	6.06970	0.451157	0.225579	−	0.974225i	$-0.427573\pi$
$181$	6.06970	0.451157	0.225579	+	0.974225i	$0.427573\pi$
$182$	0	0
$183$	−2.86290	−0.211632
$184$	0	0
$185$	−10.6315	−0.781647
$186$	0	0
$187$	−5.39193	−0.394297
$188$	0	0
$189$	−7.52017	−0.547012
$190$	0	0
$191$	21.6673	1.56779	0.783897	−	0.620891i	$-0.213229\pi$
$191$	21.6673	1.56779	0.783897	+	0.620891i	$0.213229\pi$
$192$	0	0
$193$	5.99906	0.431822	0.215911	−	0.976413i	$-0.430728\pi$
$193$	5.99906	0.431822	0.215911	+	0.976413i	$0.430728\pi$
$194$	0	0
$195$	−12.6879	−0.908596
$196$	0	0
$197$	−22.5954	−1.60986	−0.804928	−	0.593373i	$-0.797796\pi$
$197$	−22.5954	−1.60986	−0.804928	+	0.593373i	$0.797796\pi$
$198$	0	0
$199$	3.27228	0.231965	0.115983	−	0.993251i	$-0.462998\pi$
$199$	3.27228	0.231965	0.115983	+	0.993251i	$0.462998\pi$
$200$	0	0
$201$	4.11342	0.290138
$202$	0	0
$203$	−14.9524	−1.04945
$204$	0	0
$205$	24.9109	1.73985
$206$	0	0
$207$	12.1211	0.842472
$208$	0	0
$209$	−27.7865	−1.92204
$210$	0	0
$211$	−14.3443	−0.987504	−0.493752	−	0.869603i	$-0.664375\pi$
$211$	−14.3443	−0.987504	−0.493752	+	0.869603i	$0.664375\pi$
$212$	0	0
$213$	7.57698	0.519166
$214$	0	0
$215$	−42.3380	−2.88743
$216$	0	0
$217$	26.3175	1.78655
$218$	0	0
$219$	−31.4388	−2.12444
$220$	0	0
$221$	−1.54767	−0.104108
$222$	0	0
$223$	22.0379	1.47577	0.737884	−	0.674927i	$-0.235825\pi$
$223$	22.0379	1.47577	0.737884	+	0.674927i	$0.235825\pi$
$224$	0	0
$225$	16.5612	1.10408
$226$	0	0
$227$	−6.09459	−0.404512	−0.202256	−	0.979333i	$-0.564827\pi$
$227$	−6.09459	−0.404512	−0.202256	+	0.979333i	$0.564827\pi$
$228$	0	0
$229$	20.4441	1.35098	0.675492	−	0.737367i	$-0.263931\pi$
$229$	20.4441	1.35098	0.675492	+	0.737367i	$0.263931\pi$
$230$	0	0
$231$	−36.8732	−2.42608
$232$	0	0
$233$	16.8733	1.10541	0.552703	−	0.833379i	$-0.313597\pi$
$233$	16.8733	1.10541	0.552703	+	0.833379i	$0.313597\pi$
$234$	0	0
$235$	25.8301	1.68497
$236$	0	0
$237$	13.6892	0.889209
$238$	0	0
$239$	−25.6298	−1.65785	−0.828926	−	0.559358i	$-0.811048\pi$
$239$	−25.6298	−1.65785	−0.828926	+	0.559358i	$0.811048\pi$
$240$	0	0
$241$	0.726187	0.0467778	0.0233889	−	0.999726i	$-0.492554\pi$
$241$	0.726187	0.0467778	0.0233889	+	0.999726i	$0.492554\pi$
$242$	0	0
$243$	17.2461	1.10634
$244$	0	0
$245$	−9.41941	−0.601784
$246$	0	0
$247$	−7.97570	−0.507482
$248$	0	0
$249$	34.4759	2.18482
$250$	0	0
$251$	−4.55029	−0.287212	−0.143606	−	0.989635i	$-0.545870\pi$
$251$	−4.55029	−0.287212	−0.143606	+	0.989635i	$0.545870\pi$
$252$	0	0
$253$	−34.3918	−2.16220
$254$	0	0
$255$	8.19802	0.513380
$256$	0	0
$257$	−12.2902	−0.766640	−0.383320	−	0.923616i	$-0.625219\pi$
$257$	−12.2902	−0.766640	−0.383320	+	0.923616i	$0.625219\pi$
$258$	0	0
$259$	−8.86856	−0.551066
$260$	0	0
$261$	9.19786	0.569333
$262$	0	0
$263$	−27.3571	−1.68691	−0.843456	−	0.537198i	$-0.819483\pi$
$263$	−27.3571	−1.68691	−0.843456	+	0.537198i	$0.819483\pi$
$264$	0	0
$265$	−17.8557	−1.09686
$266$	0	0
$267$	7.27694	0.445341
$268$	0	0
$269$	1.10661	0.0674712	0.0337356	−	0.999431i	$-0.489260\pi$
$269$	1.10661	0.0674712	0.0337356	+	0.999431i	$0.489260\pi$
$270$	0	0
$271$	10.6373	0.646171	0.323085	−	0.946370i	$-0.395280\pi$
$271$	10.6373	0.646171	0.323085	+	0.946370i	$0.395280\pi$
$272$	0	0
$273$	−10.5839	−0.640565
$274$	0	0
$275$	−46.9902	−2.83361
$276$	0	0
$277$	10.5196	0.632062	0.316031	−	0.948749i	$-0.397650\pi$
$277$	10.5196	0.632062	0.316031	+	0.948749i	$0.397650\pi$
$278$	0	0
$279$	−16.1891	−0.969214
$280$	0	0
$281$	−15.0655	−0.898733	−0.449366	−	0.893348i	$-0.648350\pi$
$281$	−15.0655	−0.898733	−0.449366	+	0.893348i	$0.648350\pi$
$282$	0	0
$283$	−0.569914	−0.0338779	−0.0169389	−	0.999857i	$-0.505392\pi$
$283$	−0.569914	−0.0338779	−0.0169389	+	0.999857i	$0.505392\pi$
$284$	0	0
$285$	42.2473	2.50252
$286$	0	0
$287$	20.7800	1.22661
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	37.7408	2.21240
$292$	0	0
$293$	10.4612	0.611152	0.305576	−	0.952168i	$-0.401151\pi$
$293$	10.4612	0.611152	0.305576	+	0.952168i	$0.401151\pi$
$294$	0	0
$295$	3.70336	0.215618
$296$	0	0
$297$	−13.1256	−0.761624
$298$	0	0
$299$	−9.87165	−0.570893
$300$	0	0
$301$	−35.3173	−2.03565
$302$	0	0
$303$	−34.7473	−1.99618
$304$	0	0
$305$	−4.78950	−0.274246
$306$	0	0
$307$	−5.26995	−0.300772	−0.150386	−	0.988627i	$-0.548052\pi$
$307$	−5.26995	−0.300772	−0.150386	+	0.988627i	$0.548052\pi$
$308$	0	0
$309$	2.63977	0.150171
$310$	0	0
$311$	−1.88205	−0.106721	−0.0533607	−	0.998575i	$-0.516993\pi$
$311$	−1.88205	−0.106721	−0.0533607	+	0.998575i	$0.516993\pi$
$312$	0	0
$313$	−6.53598	−0.369435	−0.184718	−	0.982792i	$-0.559137\pi$
$313$	−6.53598	−0.369435	−0.184718	+	0.982792i	$0.559137\pi$
$314$	0	0
$315$	21.7410	1.22497
$316$	0	0
$317$	−24.7628	−1.39082	−0.695409	−	0.718614i	$-0.744777\pi$
$317$	−24.7628	−1.39082	−0.695409	+	0.718614i	$0.744777\pi$
$318$	0	0
$319$	−26.0977	−1.46119
$320$	0	0
$321$	7.29282	0.407045
$322$	0	0
$323$	5.15336	0.286740
$324$	0	0
$325$	−13.4878	−0.748170
$326$	0	0
$327$	−41.8356	−2.31351
$328$	0	0
$329$	21.5468	1.18791
$330$	0	0
$331$	−1.17453	−0.0645579	−0.0322790	−	0.999479i	$-0.510276\pi$
$331$	−1.17453	−0.0645579	−0.0322790	+	0.999479i	$0.510276\pi$
$332$	0	0
$333$	5.45544	0.298956
$334$	0	0
$335$	6.88155	0.375979
$336$	0	0
$337$	4.04484	0.220337	0.110168	−	0.993913i	$-0.464861\pi$
$337$	4.04484	0.220337	0.110168	+	0.993913i	$0.464861\pi$
$338$	0	0
$339$	−3.17112	−0.172232
$340$	0	0
$341$	45.9342	2.48748
$342$	0	0
$343$	13.7673	0.743366
$344$	0	0
$345$	52.2902	2.81521
$346$	0	0
$347$	−28.6885	−1.54008	−0.770041	−	0.637995i	$-0.779764\pi$
$347$	−28.6885	−1.54008	−0.770041	+	0.637995i	$0.779764\pi$
$348$	0	0
$349$	21.2613	1.13809	0.569046	−	0.822306i	$-0.307313\pi$
$349$	21.2613	1.13809	0.569046	+	0.822306i	$0.307313\pi$
$350$	0	0
$351$	−3.76750	−0.201094
$352$	0	0
$353$	1.85332	0.0986424	0.0493212	−	0.998783i	$-0.484294\pi$
$353$	1.85332	0.0986424	0.0493212	+	0.998783i	$0.484294\pi$
$354$	0	0
$355$	12.6759	0.672768
$356$	0	0
$357$	6.83858	0.361936
$358$	0	0
$359$	−23.6079	−1.24598	−0.622988	−	0.782231i	$-0.714082\pi$
$359$	−23.6079	−1.24598	−0.622988	+	0.782231i	$0.714082\pi$
$360$	0	0
$361$	7.55708	0.397741
$362$	0	0
$363$	−40.0075	−2.09985
$364$	0	0
$365$	−52.5957	−2.75298
$366$	0	0
$367$	−26.7814	−1.39798	−0.698988	−	0.715134i	$-0.746366\pi$
$367$	−26.7814	−1.39798	−0.698988	+	0.715134i	$0.746366\pi$
$368$	0	0
$369$	−12.7827	−0.665440
$370$	0	0
$371$	−14.8947	−0.773296
$372$	0	0
$373$	−32.9974	−1.70854	−0.854271	−	0.519829i	$-0.825996\pi$
$373$	−32.9974	−1.70854	−0.854271	+	0.519829i	$0.825996\pi$
$374$	0	0
$375$	30.4549	1.57269
$376$	0	0
$377$	−7.49094	−0.385803
$378$	0	0
$379$	24.5906	1.26313	0.631566	−	0.775322i	$-0.282412\pi$
$379$	24.5906	1.26313	0.631566	+	0.775322i	$0.282412\pi$
$380$	0	0
$381$	14.8237	0.759440
$382$	0	0
$383$	−36.7160	−1.87610	−0.938050	−	0.346501i	$-0.887370\pi$
$383$	−36.7160	−1.87610	−0.938050	+	0.346501i	$0.887370\pi$
$384$	0	0
$385$	−61.6870	−3.14386
$386$	0	0
$387$	21.7252	1.10435
$388$	0	0
$389$	26.5665	1.34697	0.673487	−	0.739199i	$-0.264796\pi$
$389$	26.5665	1.34697	0.673487	+	0.739199i	$0.264796\pi$
$390$	0	0
$391$	6.37839	0.322569
$392$	0	0
$393$	−20.0878	−1.01330
$394$	0	0
$395$	22.9014	1.15229
$396$	0	0
$397$	−9.52153	−0.477872	−0.238936	−	0.971035i	$-0.576799\pi$
$397$	−9.52153	−0.477872	−0.238936	+	0.971035i	$0.576799\pi$
$398$	0	0
$399$	35.2416	1.76429
$400$	0	0
$401$	4.63267	0.231344	0.115672	−	0.993287i	$-0.463098\pi$
$401$	4.63267	0.231344	0.115672	+	0.993287i	$0.463098\pi$
$402$	0	0
$403$	13.1847	0.656778
$404$	0	0
$405$	41.0693	2.04075
$406$	0	0
$407$	−15.4790	−0.767268
$408$	0	0
$409$	31.2502	1.54522	0.772612	−	0.634879i	$-0.218950\pi$
$409$	31.2502	1.54522	0.772612	+	0.634879i	$0.218950\pi$
$410$	0	0
$411$	23.3450	1.15152
$412$	0	0
$413$	3.08925	0.152012
$414$	0	0
$415$	57.6766	2.83123
$416$	0	0
$417$	19.6652	0.963008
$418$	0	0
$419$	24.2208	1.18326	0.591631	−	0.806209i	$-0.298484\pi$
$419$	24.2208	1.18326	0.591631	+	0.806209i	$0.298484\pi$
$420$	0	0
$421$	−26.0905	−1.27157	−0.635787	−	0.771864i	$-0.719325\pi$
$421$	−26.0905	−1.27157	−0.635787	+	0.771864i	$0.719325\pi$
$422$	0	0
$423$	−13.2544	−0.644450
$424$	0	0
$425$	8.71491	0.422735
$426$	0	0
$427$	−3.99528	−0.193345
$428$	0	0
$429$	−18.4729	−0.891882
$430$	0	0
$431$	30.1468	1.45212	0.726060	−	0.687631i	$-0.241349\pi$
$431$	30.1468	1.45212	0.726060	+	0.687631i	$0.241349\pi$
$432$	0	0
$433$	25.0737	1.20497	0.602483	−	0.798132i	$-0.294178\pi$
$433$	25.0737	1.20497	0.602483	+	0.798132i	$0.294178\pi$
$434$	0	0
$435$	39.6795	1.90249
$436$	0	0
$437$	32.8701	1.57239
$438$	0	0
$439$	−11.7437	−0.560496	−0.280248	−	0.959928i	$-0.590417\pi$
$439$	−11.7437	−0.560496	−0.280248	+	0.959928i	$0.590417\pi$
$440$	0	0
$441$	4.83344	0.230164
$442$	0	0
$443$	−14.2063	−0.674963	−0.337482	−	0.941332i	$-0.609575\pi$
$443$	−14.2063	−0.674963	−0.337482	+	0.941332i	$0.609575\pi$
$444$	0	0
$445$	12.1740	0.577102
$446$	0	0
$447$	−45.0511	−2.13085
$448$	0	0
$449$	28.4218	1.34131	0.670654	−	0.741771i	$-0.266014\pi$
$449$	28.4218	1.34131	0.670654	+	0.741771i	$0.266014\pi$
$450$	0	0
$451$	36.2691	1.70785
$452$	0	0
$453$	−44.7436	−2.10224
$454$	0	0
$455$	−17.7063	−0.830086
$456$	0	0
$457$	3.80502	0.177991	0.0889957	−	0.996032i	$-0.471634\pi$
$457$	3.80502	0.177991	0.0889957	+	0.996032i	$0.471634\pi$
$458$	0	0
$459$	2.43430	0.113623
$460$	0	0
$461$	−11.9530	−0.556708	−0.278354	−	0.960479i	$-0.589789\pi$
$461$	−11.9530	−0.556708	−0.278354	+	0.960479i	$0.589789\pi$
$462$	0	0
$463$	21.9697	1.02102	0.510509	−	0.859873i	$-0.329457\pi$
$463$	21.9697	1.02102	0.510509	+	0.859873i	$0.329457\pi$
$464$	0	0
$465$	−69.8395	−3.23873
$466$	0	0
$467$	−26.1861	−1.21175	−0.605875	−	0.795560i	$-0.707177\pi$
$467$	−26.1861	−1.21175	−0.605875	+	0.795560i	$0.707177\pi$
$468$	0	0
$469$	5.74041	0.265068
$470$	0	0
$471$	−16.8430	−0.776085
$472$	0	0
$473$	−61.6422	−2.83431
$474$	0	0
$475$	44.9110	2.06066
$476$	0	0
$477$	9.16239	0.419517
$478$	0	0
$479$	5.70074	0.260473	0.130237	−	0.991483i	$-0.458426\pi$
$479$	5.70074	0.260473	0.130237	+	0.991483i	$0.458426\pi$
$480$	0	0
$481$	−4.44303	−0.202585
$482$	0	0
$483$	43.6191	1.98474
$484$	0	0
$485$	63.1386	2.86697
$486$	0	0
$487$	20.9374	0.948763	0.474382	−	0.880319i	$-0.342672\pi$
$487$	20.9374	0.948763	0.474382	+	0.880319i	$0.342672\pi$
$488$	0	0
$489$	−3.16319	−0.143044
$490$	0	0
$491$	−17.7801	−0.802406	−0.401203	−	0.915989i	$-0.631408\pi$
$491$	−17.7801	−0.802406	−0.401203	+	0.915989i	$0.631408\pi$
$492$	0	0
$493$	4.84013	0.217989
$494$	0	0
$495$	37.9464	1.70556
$496$	0	0
$497$	10.5739	0.474306
$498$	0	0
$499$	30.7452	1.37635	0.688173	−	0.725547i	$-0.258413\pi$
$499$	30.7452	1.37635	0.688173	+	0.725547i	$0.258413\pi$
$500$	0	0
$501$	−5.04103	−0.225217
$502$	0	0
$503$	−28.2999	−1.26183	−0.630915	−	0.775852i	$-0.717320\pi$
$503$	−28.2999	−1.26183	−0.630915	+	0.775852i	$0.717320\pi$
$504$	0	0
$505$	−58.1306	−2.58678
$506$	0	0
$507$	23.4753	1.04258
$508$	0	0
$509$	24.9533	1.10604	0.553019	−	0.833169i	$-0.313476\pi$
$509$	24.9533	1.10604	0.553019	+	0.833169i	$0.313476\pi$
$510$	0	0
$511$	−43.8740	−1.94087
$512$	0	0
$513$	12.5448	0.553867
$514$	0	0
$515$	4.41620	0.194601
$516$	0	0
$517$	37.6075	1.65398
$518$	0	0
$519$	−19.0924	−0.838062
$520$	0	0
$521$	−27.7028	−1.21368	−0.606841	−	0.794823i	$-0.707564\pi$
$521$	−27.7028	−1.21368	−0.606841	+	0.794823i	$0.707564\pi$
$522$	0	0
$523$	5.66513	0.247719	0.123859	−	0.992300i	$-0.460473\pi$
$523$	5.66513	0.247719	0.123859	+	0.992300i	$0.460473\pi$
$524$	0	0
$525$	59.5976	2.60105
$526$	0	0
$527$	−8.51907	−0.371097
$528$	0	0
$529$	17.6839	0.768864
$530$	0	0
$531$	−1.90033	−0.0824674
$532$	0	0
$533$	10.4105	0.450929
$534$	0	0
$535$	12.2005	0.527475
$536$	0	0
$537$	−16.9254	−0.730386
$538$	0	0
$539$	−13.7142	−0.590714
$540$	0	0
$541$	30.1702	1.29712	0.648558	−	0.761165i	$-0.275372\pi$
$541$	30.1702	1.29712	0.648558	+	0.761165i	$0.275372\pi$
$542$	0	0
$543$	−13.4363	−0.576607
$544$	0	0
$545$	−69.9890	−2.99800
$546$	0	0
$547$	−39.1122	−1.67232	−0.836158	−	0.548488i	$-0.815204\pi$
$547$	−39.1122	−1.67232	−0.836158	+	0.548488i	$0.815204\pi$
$548$	0	0
$549$	2.45767	0.104891
$550$	0	0
$551$	24.9429	1.06260
$552$	0	0
$553$	19.1037	0.812374
$554$	0	0
$555$	23.5347	0.998993
$556$	0	0
$557$	−1.83661	−0.0778196	−0.0389098	−	0.999243i	$-0.512389\pi$
$557$	−1.83661	−0.0778196	−0.0389098	+	0.999243i	$0.512389\pi$
$558$	0	0
$559$	−17.6935	−0.748354
$560$	0	0
$561$	11.9360	0.503936
$562$	0	0
$563$	−40.6932	−1.71502	−0.857508	−	0.514471i	$-0.827988\pi$
$563$	−40.6932	−1.71502	−0.857508	+	0.514471i	$0.827988\pi$
$564$	0	0
$565$	−5.30514	−0.223189
$566$	0	0
$567$	34.2590	1.43874
$568$	0	0
$569$	11.6279	0.487465	0.243733	−	0.969842i	$-0.421628\pi$
$569$	11.6279	0.487465	0.243733	+	0.969842i	$0.421628\pi$
$570$	0	0
$571$	6.95396	0.291014	0.145507	−	0.989357i	$-0.453519\pi$
$571$	6.95396	0.291014	0.145507	+	0.989357i	$0.453519\pi$
$572$	0	0
$573$	−47.9643	−2.00374
$574$	0	0
$575$	55.5871	2.31814
$576$	0	0
$577$	−28.1763	−1.17300	−0.586498	−	0.809950i	$-0.699494\pi$
$577$	−28.1763	−1.17300	−0.586498	+	0.809950i	$0.699494\pi$
$578$	0	0
$579$	−13.2799	−0.551895
$580$	0	0
$581$	48.1123	1.99603
$582$	0	0
$583$	−25.9970	−1.07669
$584$	0	0
$585$	10.8919	0.450326
$586$	0	0
$587$	1.64411	0.0678598	0.0339299	−	0.999424i	$-0.489198\pi$
$587$	1.64411	0.0678598	0.0339299	+	0.999424i	$0.489198\pi$
$588$	0	0
$589$	−43.9018	−1.80894
$590$	0	0
$591$	50.0188	2.05750
$592$	0	0
$593$	17.2262	0.707395	0.353698	−	0.935360i	$-0.384924\pi$
$593$	17.2262	0.707395	0.353698	+	0.935360i	$0.384924\pi$
$594$	0	0
$595$	11.4406	0.469020
$596$	0	0
$597$	−7.24374	−0.296466
$598$	0	0
$599$	−3.80393	−0.155424	−0.0777122	−	0.996976i	$-0.524762\pi$
$599$	−3.80393	−0.155424	−0.0777122	+	0.996976i	$0.524762\pi$
$600$	0	0
$601$	−36.5698	−1.49171	−0.745856	−	0.666107i	$-0.767959\pi$
$601$	−36.5698	−1.49171	−0.745856	+	0.666107i	$0.767959\pi$
$602$	0	0
$603$	−3.53118	−0.143801
$604$	0	0
$605$	−66.9306	−2.72112
$606$	0	0
$607$	0.600623	0.0243785	0.0121893	−	0.999926i	$-0.496120\pi$
$607$	0.600623	0.0243785	0.0121893	+	0.999926i	$0.496120\pi$
$608$	0	0
$609$	33.0996	1.34126
$610$	0	0
$611$	10.7947	0.436705
$612$	0	0
$613$	−18.5123	−0.747703	−0.373851	−	0.927489i	$-0.621963\pi$
$613$	−18.5123	−0.747703	−0.373851	+	0.927489i	$0.621963\pi$
$614$	0	0
$615$	−55.1445	−2.22364
$616$	0	0
$617$	7.80043	0.314034	0.157017	−	0.987596i	$-0.449812\pi$
$617$	7.80043	0.314034	0.157017	+	0.987596i	$0.449812\pi$
$618$	0	0
$619$	−15.8729	−0.637986	−0.318993	−	0.947757i	$-0.603345\pi$
$619$	−15.8729	−0.637986	−0.318993	+	0.947757i	$0.603345\pi$
$620$	0	0
$621$	15.5269	0.623074
$622$	0	0
$623$	10.1552	0.406860
$624$	0	0
$625$	7.37509	0.295004
$626$	0	0
$627$	61.5102	2.45648
$628$	0	0
$629$	2.87078	0.114465
$630$	0	0
$631$	−30.8883	−1.22965	−0.614823	−	0.788665i	$-0.710772\pi$
$631$	−30.8883	−1.22965	−0.614823	+	0.788665i	$0.710772\pi$
$632$	0	0
$633$	31.7536	1.26209
$634$	0	0
$635$	24.7993	0.984130
$636$	0	0
$637$	−3.93646	−0.155968
$638$	0	0
$639$	−6.50448	−0.257313
$640$	0	0
$641$	42.7574	1.68882	0.844408	−	0.535700i	$-0.179952\pi$
$641$	42.7574	1.68882	0.844408	+	0.535700i	$0.179952\pi$
$642$	0	0
$643$	−20.2768	−0.799637	−0.399819	−	0.916594i	$-0.630927\pi$
$643$	−20.2768	−0.799637	−0.399819	+	0.916594i	$0.630927\pi$
$644$	0	0
$645$	93.7224	3.69031
$646$	0	0
$647$	35.4832	1.39499	0.697495	−	0.716590i	$-0.254298\pi$
$647$	35.4832	1.39499	0.697495	+	0.716590i	$0.254298\pi$
$648$	0	0
$649$	5.39193	0.211652
$650$	0	0
$651$	−58.2583	−2.28332
$652$	0	0
$653$	34.4674	1.34881	0.674407	−	0.738360i	$-0.264399\pi$
$653$	34.4674	1.34881	0.674407	+	0.738360i	$0.264399\pi$
$654$	0	0
$655$	−33.6060	−1.31310
$656$	0	0
$657$	26.9888	1.05293
$658$	0	0
$659$	7.64906	0.297965	0.148983	−	0.988840i	$-0.452400\pi$
$659$	7.64906	0.297965	0.148983	+	0.988840i	$0.452400\pi$
$660$	0	0
$661$	27.7853	1.08072	0.540361	−	0.841434i	$-0.318288\pi$
$661$	27.7853	1.08072	0.540361	+	0.841434i	$0.318288\pi$
$662$	0	0
$663$	3.42603	0.133056
$664$	0	0
$665$	58.9576	2.28628
$666$	0	0
$667$	30.8722	1.19538
$668$	0	0
$669$	−48.7847	−1.88612
$670$	0	0
$671$	−6.97329	−0.269201
$672$	0	0
$673$	33.4818	1.29063	0.645314	−	0.763917i	$-0.276726\pi$
$673$	33.4818	1.29063	0.645314	+	0.763917i	$0.276726\pi$
$674$	0	0
$675$	21.2147	0.816555
$676$	0	0
$677$	−8.36225	−0.321387	−0.160694	−	0.987004i	$-0.551373\pi$
$677$	−8.36225	−0.321387	−0.160694	+	0.987004i	$0.551373\pi$
$678$	0	0
$679$	52.6686	2.02123
$680$	0	0
$681$	13.4914	0.516992
$682$	0	0
$683$	0.726598	0.0278025	0.0139012	−	0.999903i	$-0.495575\pi$
$683$	0.726598	0.0278025	0.0139012	+	0.999903i	$0.495575\pi$
$684$	0	0
$685$	39.0551	1.49222
$686$	0	0
$687$	−45.2565	−1.72664
$688$	0	0
$689$	−7.46205	−0.284282
$690$	0	0
$691$	−4.63046	−0.176151	−0.0880754	−	0.996114i	$-0.528072\pi$
$691$	−4.63046	−0.176151	−0.0880754	+	0.996114i	$0.528072\pi$
$692$	0	0
$693$	31.6539	1.20243
$694$	0	0
$695$	32.8989	1.24793
$696$	0	0
$697$	−6.72655	−0.254786
$698$	0	0
$699$	−37.3519	−1.41278
$700$	0	0
$701$	5.97221	0.225567	0.112784	−	0.993620i	$-0.464023\pi$
$701$	5.97221	0.225567	0.112784	+	0.993620i	$0.464023\pi$
$702$	0	0
$703$	14.7942	0.557972
$704$	0	0
$705$	−57.1794	−2.15350
$706$	0	0
$707$	−48.4911	−1.82369
$708$	0	0
$709$	−29.5144	−1.10844	−0.554219	−	0.832371i	$-0.686983\pi$
$709$	−29.5144	−1.10844	−0.554219	+	0.832371i	$0.686983\pi$
$710$	0	0
$711$	−11.7515	−0.440717
$712$	0	0
$713$	−54.3380	−2.03497
$714$	0	0
$715$	−30.9043	−1.15576
$716$	0	0
$717$	56.7359	2.11884
$718$	0	0
$719$	−21.9739	−0.819487	−0.409743	−	0.912201i	$-0.634382\pi$
$719$	−21.9739	−0.819487	−0.409743	+	0.912201i	$0.634382\pi$
$720$	0	0
$721$	3.68388	0.137195
$722$	0	0
$723$	−1.60754	−0.0597850
$724$	0	0
$725$	42.1813	1.56657
$726$	0	0
$727$	0.447816	0.0166086	0.00830429	−	0.999966i	$-0.497357\pi$
$727$	0.447816	0.0166086	0.00830429	+	0.999966i	$0.497357\pi$
$728$	0	0
$729$	−4.90797	−0.181777
$730$	0	0
$731$	11.4323	0.422839
$732$	0	0
$733$	−45.7314	−1.68913	−0.844563	−	0.535456i	$-0.820140\pi$
$733$	−45.7314	−1.68913	−0.844563	+	0.535456i	$0.820140\pi$
$734$	0	0
$735$	20.8515	0.769118
$736$	0	0
$737$	10.0192	0.369063
$738$	0	0
$739$	9.18528	0.337886	0.168943	−	0.985626i	$-0.445965\pi$
$739$	9.18528	0.337886	0.168943	+	0.985626i	$0.445965\pi$
$740$	0	0
$741$	17.6556	0.648594
$742$	0	0
$743$	21.1136	0.774583	0.387292	−	0.921957i	$-0.373411\pi$
$743$	21.1136	0.774583	0.387292	+	0.921957i	$0.373411\pi$
$744$	0	0
$745$	−75.3684	−2.76128
$746$	0	0
$747$	−29.5960	−1.08286
$748$	0	0
$749$	10.1774	0.371873
$750$	0	0
$751$	10.5281	0.384175	0.192088	−	0.981378i	$-0.438474\pi$
$751$	10.5281	0.384175	0.192088	+	0.981378i	$0.438474\pi$
$752$	0	0
$753$	10.0728	0.367075
$754$	0	0
$755$	−74.8540	−2.72422
$756$	0	0
$757$	15.6458	0.568656	0.284328	−	0.958727i	$-0.408230\pi$
$757$	15.6458	0.568656	0.284328	+	0.958727i	$0.408230\pi$
$758$	0	0
$759$	76.1322	2.76342
$760$	0	0
$761$	−47.1670	−1.70980	−0.854901	−	0.518792i	$-0.826382\pi$
$761$	−47.1670	−1.70980	−0.854901	+	0.518792i	$0.826382\pi$
$762$	0	0
$763$	−58.3830	−2.11361
$764$	0	0
$765$	−7.03762	−0.254446
$766$	0	0
$767$	1.54767	0.0558832
$768$	0	0
$769$	26.5785	0.958446	0.479223	−	0.877693i	$-0.340918\pi$
$769$	26.5785	0.958446	0.479223	+	0.877693i	$0.340918\pi$
$770$	0	0
$771$	27.2064	0.979813
$772$	0	0
$773$	−2.90822	−0.104601	−0.0523007	−	0.998631i	$-0.516655\pi$
$773$	−2.90822	−0.104601	−0.0523007	+	0.998631i	$0.516655\pi$
$774$	0	0
$775$	−74.2429	−2.66688
$776$	0	0
$777$	19.6321	0.704296
$778$	0	0
$779$	−34.6643	−1.24198
$780$	0	0
$781$	18.4556	0.660392
$782$	0	0
$783$	11.7823	0.421066
$784$	0	0
$785$	−28.1776	−1.00570
$786$	0	0
$787$	−28.8972	−1.03007	−0.515037	−	0.857168i	$-0.672222\pi$
$787$	−28.8972	−1.03007	−0.515037	+	0.857168i	$0.672222\pi$
$788$	0	0
$789$	60.5596	2.15598
$790$	0	0
$791$	−4.42541	−0.157350
$792$	0	0
$793$	−2.00158	−0.0710781
$794$	0	0
$795$	39.5265	1.40186
$796$	0	0
$797$	−9.76360	−0.345845	−0.172922	−	0.984935i	$-0.555321\pi$
$797$	−9.76360	−0.345845	−0.172922	+	0.984935i	$0.555321\pi$
$798$	0	0
$799$	−6.97477	−0.246750
$800$	0	0
$801$	−6.24691	−0.220724
$802$	0	0
$803$	−76.5769	−2.70234
$804$	0	0
$805$	72.9728	2.57195
$806$	0	0
$807$	−2.44967	−0.0862324
$808$	0	0
$809$	−45.2115	−1.58955	−0.794777	−	0.606902i	$-0.792412\pi$
$809$	−45.2115	−1.58955	−0.794777	+	0.606902i	$0.792412\pi$
$810$	0	0
$811$	−24.9238	−0.875194	−0.437597	−	0.899171i	$-0.644170\pi$
$811$	−24.9238	−0.875194	−0.437597	+	0.899171i	$0.644170\pi$
$812$	0	0
$813$	−23.5475	−0.825847
$814$	0	0
$815$	−5.29186	−0.185366
$816$	0	0
$817$	58.9148	2.06117
$818$	0	0
$819$	9.08577	0.317482
$820$	0	0
$821$	−15.9283	−0.555900	−0.277950	−	0.960596i	$-0.589655\pi$
$821$	−15.9283	−0.555900	−0.277950	+	0.960596i	$0.589655\pi$
$822$	0	0
$823$	14.6208	0.509648	0.254824	−	0.966987i	$-0.417982\pi$
$823$	14.6208	0.509648	0.254824	+	0.966987i	$0.417982\pi$
$824$	0	0
$825$	104.021	3.62154
$826$	0	0
$827$	−17.5149	−0.609054	−0.304527	−	0.952504i	$-0.598498\pi$
$827$	−17.5149	−0.609054	−0.304527	+	0.952504i	$0.598498\pi$
$828$	0	0
$829$	44.8738	1.55853	0.779265	−	0.626695i	$-0.215593\pi$
$829$	44.8738	1.55853	0.779265	+	0.626695i	$0.215593\pi$
$830$	0	0
$831$	−23.2869	−0.807815
$832$	0	0
$833$	2.54347	0.0881261
$834$	0	0
$835$	−8.43341	−0.291850
$836$	0	0
$837$	−20.7380	−0.716809
$838$	0	0
$839$	−1.33810	−0.0461962	−0.0230981	−	0.999733i	$-0.507353\pi$
$839$	−1.33810	−0.0461962	−0.0230981	+	0.999733i	$0.507353\pi$
$840$	0	0
$841$	−5.57313	−0.192177
$842$	0	0
$843$	33.3500	1.14864
$844$	0	0
$845$	39.2731	1.35104
$846$	0	0
$847$	−55.8318	−1.91840
$848$	0	0
$849$	1.26160	0.0432981
$850$	0	0
$851$	18.3110	0.627691
$852$	0	0
$853$	−39.3511	−1.34735	−0.673677	−	0.739026i	$-0.735286\pi$
$853$	−39.3511	−1.34735	−0.673677	+	0.739026i	$0.735286\pi$
$854$	0	0
$855$	−36.2674	−1.24032
$856$	0	0
$857$	47.3658	1.61798	0.808992	−	0.587819i	$-0.200013\pi$
$857$	47.3658	1.61798	0.808992	+	0.587819i	$0.200013\pi$
$858$	0	0
$859$	22.3436	0.762354	0.381177	−	0.924502i	$-0.375519\pi$
$859$	22.3436	0.762354	0.381177	+	0.924502i	$0.375519\pi$
$860$	0	0
$861$	−46.0001	−1.56768
$862$	0	0
$863$	8.38982	0.285593	0.142796	−	0.989752i	$-0.454391\pi$
$863$	8.38982	0.285593	0.142796	+	0.989752i	$0.454391\pi$
$864$	0	0
$865$	−31.9406	−1.08601
$866$	0	0
$867$	−2.21367	−0.0751802
$868$	0	0
$869$	33.3434	1.13110
$870$	0	0
$871$	2.87587	0.0974451
$872$	0	0
$873$	−32.3987	−1.09653
$874$	0	0
$875$	42.5009	1.43679
$876$	0	0
$877$	7.13401	0.240898	0.120449	−	0.992719i	$-0.461566\pi$
$877$	7.13401	0.240898	0.120449	+	0.992719i	$0.461566\pi$
$878$	0	0
$879$	−23.1577	−0.781090
$880$	0	0
$881$	−25.5526	−0.860890	−0.430445	−	0.902617i	$-0.641643\pi$
$881$	−25.5526	−0.860890	−0.430445	+	0.902617i	$0.641643\pi$
$882$	0	0
$883$	−18.1015	−0.609163	−0.304581	−	0.952486i	$-0.598517\pi$
$883$	−18.1015	−0.609163	−0.304581	+	0.952486i	$0.598517\pi$
$884$	0	0
$885$	−8.19802	−0.275574
$886$	0	0
$887$	−42.8031	−1.43719	−0.718593	−	0.695431i	$-0.755214\pi$
$887$	−42.8031	−1.43719	−0.718593	+	0.695431i	$0.755214\pi$
$888$	0	0
$889$	20.6869	0.693818
$890$	0	0
$891$	59.7951	2.00321
$892$	0	0
$893$	−35.9435	−1.20280
$894$	0	0
$895$	−28.3154	−0.946480
$896$	0	0
$897$	21.8526	0.729636
$898$	0	0
$899$	−41.2334	−1.37521
$900$	0	0
$901$	4.82147	0.160626
$902$	0	0
$903$	78.1808	2.60169
$904$	0	0
$905$	−22.4783	−0.747204
$906$	0	0
$907$	43.2041	1.43457	0.717284	−	0.696781i	$-0.245385\pi$
$907$	43.2041	1.43457	0.717284	+	0.696781i	$0.245385\pi$
$908$	0	0
$909$	29.8290	0.989364
$910$	0	0
$911$	33.0228	1.09409	0.547047	−	0.837102i	$-0.315752\pi$
$911$	33.0228	1.09409	0.547047	+	0.837102i	$0.315752\pi$
$912$	0	0
$913$	83.9744	2.77915
$914$	0	0
$915$	10.6024	0.350503
$916$	0	0
$917$	−28.0333	−0.925740
$918$	0	0
$919$	46.7449	1.54197	0.770987	−	0.636851i	$-0.219764\pi$
$919$	46.7449	1.54197	0.770987	+	0.636851i	$0.219764\pi$
$920$	0	0
$921$	11.6659	0.384406
$922$	0	0
$923$	5.29739	0.174366
$924$	0	0
$925$	25.0186	0.822606
$926$	0	0
$927$	−2.26612	−0.0744290
$928$	0	0
$929$	−6.62636	−0.217404	−0.108702	−	0.994074i	$-0.534669\pi$
$929$	−6.62636	−0.217404	−0.108702	+	0.994074i	$0.534669\pi$
$930$	0	0
$931$	13.1074	0.429578
$932$	0	0
$933$	4.16624	0.136397
$934$	0	0
$935$	19.9683	0.653033
$936$	0	0
$937$	−37.0761	−1.21122	−0.605612	−	0.795760i	$-0.707072\pi$
$937$	−37.0761	−1.21122	−0.605612	+	0.795760i	$0.707072\pi$
$938$	0	0
$939$	14.4685	0.472161
$940$	0	0
$941$	−42.1120	−1.37281	−0.686407	−	0.727218i	$-0.740813\pi$
$941$	−42.1120	−1.37281	−0.686407	+	0.727218i	$0.740813\pi$
$942$	0	0
$943$	−42.9046	−1.39717
$944$	0	0
$945$	27.8499	0.905958
$946$	0	0
$947$	−10.3996	−0.337941	−0.168970	−	0.985621i	$-0.554044\pi$
$947$	−10.3996	−0.337941	−0.168970	+	0.985621i	$0.554044\pi$
$948$	0	0
$949$	−21.9802	−0.713509
$950$	0	0
$951$	54.8167	1.77755
$952$	0	0
$953$	17.9059	0.580029	0.290014	−	0.957022i	$-0.406340\pi$
$953$	17.9059	0.580029	0.290014	+	0.957022i	$0.406340\pi$
$954$	0	0
$955$	−80.2421	−2.59657
$956$	0	0
$957$	57.7716	1.86749
$958$	0	0
$959$	32.5787	1.05202
$960$	0	0
$961$	41.5745	1.34111
$962$	0	0
$963$	−6.26054	−0.201743
$964$	0	0
$965$	−22.2167	−0.715181
$966$	0	0
$967$	−16.1695	−0.519975	−0.259987	−	0.965612i	$-0.583718\pi$
$967$	−16.1695	−0.519975	−0.259987	+	0.965612i	$0.583718\pi$
$968$	0	0
$969$	−11.4078	−0.366472
$970$	0	0
$971$	−35.9065	−1.15230	−0.576148	−	0.817345i	$-0.695445\pi$
$971$	−35.9065	−1.15230	−0.576148	+	0.817345i	$0.695445\pi$
$972$	0	0
$973$	27.4434	0.879796
$974$	0	0
$975$	29.8576	0.956208
$976$	0	0
$977$	31.5568	1.00959	0.504796	−	0.863239i	$-0.331568\pi$
$977$	31.5568	1.00959	0.504796	+	0.863239i	$0.331568\pi$
$978$	0	0
$979$	17.7247	0.566485
$980$	0	0
$981$	35.9139	1.14664
$982$	0	0
$983$	−31.9423	−1.01880	−0.509401	−	0.860529i	$-0.670133\pi$
$983$	−31.9423	−1.01880	−0.509401	+	0.860529i	$0.670133\pi$
$984$	0	0
$985$	83.6790	2.66624
$986$	0	0
$987$	−47.6975	−1.51823
$988$	0	0
$989$	72.9197	2.31871
$990$	0	0
$991$	61.7996	1.96313	0.981565	−	0.191128i	$-0.0612144\pi$
$991$	61.7996	1.96313	0.981565	+	0.191128i	$0.0612144\pi$
$992$	0	0
$993$	2.60002	0.0825091
$994$	0	0
$995$	−12.1184	−0.384180
$996$	0	0
$997$	18.1489	0.574782	0.287391	−	0.957813i	$-0.407212\pi$
$997$	18.1489	0.574782	0.287391	+	0.957813i	$0.407212\pi$
$998$	0	0
$999$	6.98834	0.221101

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.a.h.1.5	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.a.h.1.5	✓	15	1.1	even	1	trivial

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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.a (trivial)

\(f(q)\)	\(=\)	\(q-2.21367 q^{3} -3.70336 q^{5} -3.08925 q^{7} +1.90033 q^{9} +O(q^{10})\)	\(q-2.21367 q^{3} -3.70336 q^{5} -3.08925 q^{7} +1.90033 q^{9} -5.39193 q^{11} -1.54767 q^{13} +8.19802 q^{15} +1.00000 q^{17} +5.15336 q^{19} +6.83858 q^{21} +6.37839 q^{23} +8.71491 q^{25} +2.43430 q^{27} +4.84013 q^{29} -8.51907 q^{31} +11.9360 q^{33} +11.4406 q^{35} +2.87078 q^{37} +3.42603 q^{39} -6.72655 q^{41} +11.4323 q^{43} -7.03762 q^{45} -6.97477 q^{47} +2.54347 q^{49} -2.21367 q^{51} +4.82147 q^{53} +19.9683 q^{55} -11.4078 q^{57} -1.00000 q^{59} +1.29328 q^{61} -5.87060 q^{63} +5.73159 q^{65} -1.85819 q^{67} -14.1196 q^{69} -3.42281 q^{71} +14.2021 q^{73} -19.2919 q^{75} +16.6570 q^{77} -6.18394 q^{79} -11.0897 q^{81} -15.5741 q^{83} -3.70336 q^{85} -10.7145 q^{87} -3.28727 q^{89} +4.78115 q^{91} +18.8584 q^{93} -19.0848 q^{95} -17.0490 q^{97} -10.2465 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - q^{3} + q^{5} - 11 q^{7} + 16 q^{9}+O(q^{10}) \) 15 * q - q^3 + q^5 - 11 * q^7 + 16 * q^9	\( 15 q - q^{3} + q^{5} - 11 q^{7} + 16 q^{9} - 12 q^{11} - 10 q^{13} - 13 q^{15} + 15 q^{17} - 4 q^{21} - 21 q^{23} + 4 q^{25} - 37 q^{27} + 23 q^{29} - 31 q^{31} - 11 q^{33} - 35 q^{35} - 10 q^{37} - 2 q^{39} - 15 q^{41} - 3 q^{43} + 15 q^{45} - 47 q^{47} + 18 q^{49} - q^{51} - 7 q^{53} - 20 q^{55} - 30 q^{57} - 15 q^{59} + q^{61} + 19 q^{63} + 11 q^{65} - 20 q^{67} - 24 q^{69} - 13 q^{71} - 24 q^{73} - 10 q^{75} + 21 q^{77} - 34 q^{79} - 21 q^{81} - 40 q^{83} + q^{85} - 60 q^{87} - 46 q^{89} - 19 q^{91} + 33 q^{93} - 4 q^{95} - 5 q^{97} - 74 q^{99}+O(q^{100}) \) 15 * q - q^3 + q^5 - 11 * q^7 + 16 * q^9 - 12 * q^11 - 10 * q^13 - 13 * q^15 + 15 * q^17 - 4 * q^21 - 21 * q^23 + 4 * q^25 - 37 * q^27 + 23 * q^29 - 31 * q^31 - 11 * q^33 - 35 * q^35 - 10 * q^37 - 2 * q^39 - 15 * q^41 - 3 * q^43 + 15 * q^45 - 47 * q^47 + 18 * q^49 - q^51 - 7 * q^53 - 20 * q^55 - 30 * q^57 - 15 * q^59 + q^61 + 19 * q^63 + 11 * q^65 - 20 * q^67 - 24 * q^69 - 13 * q^71 - 24 * q^73 - 10 * q^75 + 21 * q^77 - 34 * q^79 - 21 * q^81 - 40 * q^83 + q^85 - 60 * q^87 - 46 * q^89 - 19 * q^91 + 33 * q^93 - 4 * q^95 - 5 * q^97 - 74 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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