LMFDB - Embedded newform 2352.4.a.bu.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2352,4,Mod(1,2352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2352, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("2352.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2352 = 2^{4} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2352.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:	$138.772492334$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 294)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	2352.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} +4.58579 q^{5} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} +4.58579 q^{5} +9.00000 q^{9} +6.48528 q^{11} +45.2132 q^{13} -13.7574 q^{15} +81.5563 q^{17} +5.05382 q^{19} -106.250 q^{23} -103.971 q^{25} -27.0000 q^{27} -268.132 q^{29} -292.368 q^{31} -19.4558 q^{33} +114.558 q^{37} -135.640 q^{39} -161.605 q^{41} +471.294 q^{43} +41.2721 q^{45} -346.004 q^{47} -244.669 q^{51} +405.529 q^{53} +29.7401 q^{55} -15.1615 q^{57} +253.436 q^{59} -751.217 q^{61} +207.338 q^{65} -11.6468 q^{67} +318.749 q^{69} +681.661 q^{71} +685.457 q^{73} +311.912 q^{75} -0.264069 q^{79} +81.0000 q^{81} +437.137 q^{83} +374.000 q^{85} +804.396 q^{87} -58.5126 q^{89} +877.103 q^{93} +23.1758 q^{95} -1280.09 q^{97} +58.3675 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} + 12 q^{5} + 18 q^{9}+O(q^{10}) $ 2 * q - 6 * q^3 + 12 * q^5 + 18 * q^9	$ 2 q - 6 q^{3} + 12 q^{5} + 18 q^{9} - 4 q^{11} + 48 q^{13} - 36 q^{15} + 132 q^{17} - 120 q^{19} + 76 q^{23} - 174 q^{25} - 54 q^{27} - 112 q^{29} - 432 q^{31} + 12 q^{33} - 280 q^{37} - 144 q^{39} + 36 q^{41} + 128 q^{43} + 108 q^{45} + 264 q^{47} - 396 q^{51} + 268 q^{53} - 48 q^{55} + 360 q^{57} - 336 q^{59} - 504 q^{61} + 228 q^{65} + 384 q^{67} - 228 q^{69} + 396 q^{71} - 312 q^{73} + 522 q^{75} + 848 q^{79} + 162 q^{81} + 648 q^{83} + 748 q^{85} + 336 q^{87} - 612 q^{89} + 1296 q^{93} - 904 q^{95} - 2184 q^{97} - 36 q^{99}+O(q^{100}) $ 2 * q - 6 * q^3 + 12 * q^5 + 18 * q^9 - 4 * q^11 + 48 * q^13 - 36 * q^15 + 132 * q^17 - 120 * q^19 + 76 * q^23 - 174 * q^25 - 54 * q^27 - 112 * q^29 - 432 * q^31 + 12 * q^33 - 280 * q^37 - 144 * q^39 + 36 * q^41 + 128 * q^43 + 108 * q^45 + 264 * q^47 - 396 * q^51 + 268 * q^53 - 48 * q^55 + 360 * q^57 - 336 * q^59 - 504 * q^61 + 228 * q^65 + 384 * q^67 - 228 * q^69 + 396 * q^71 - 312 * q^73 + 522 * q^75 + 848 * q^79 + 162 * q^81 + 648 * q^83 + 748 * q^85 + 336 * q^87 - 612 * q^89 + 1296 * q^93 - 904 * q^95 - 2184 * q^97 - 36 * 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
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	4.58579	0.410165	0.205083	−	0.978745i	$-0.434254\pi$
$5$	4.58579	0.410165	0.205083	+	0.978745i	$0.434254\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	6.48528	0.177762	0.0888812	−	0.996042i	$-0.471671\pi$
$11$	6.48528	0.177762	0.0888812	+	0.996042i	$0.471671\pi$
$12$	0	0
$13$	45.2132	0.964607	0.482303	−	0.876004i	$-0.339800\pi$
$13$	45.2132	0.964607	0.482303	+	0.876004i	$0.339800\pi$
$14$	0	0
$15$	−13.7574	−0.236809
$16$	0	0
$17$	81.5563	1.16355	0.581774	−	0.813350i	$-0.302359\pi$
$17$	81.5563	1.16355	0.581774	+	0.813350i	$0.302359\pi$
$18$	0	0
$19$	5.05382	0.0610225	0.0305112	−	0.999534i	$-0.490286\pi$
$19$	5.05382	0.0610225	0.0305112	+	0.999534i	$0.490286\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−106.250	−0.963244	−0.481622	−	0.876379i	$-0.659952\pi$
$23$	−106.250	−0.963244	−0.481622	+	0.876379i	$0.659952\pi$
$24$	0	0
$25$	−103.971	−0.831765
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−268.132	−1.71693	−0.858463	−	0.512875i	$-0.828580\pi$
$29$	−268.132	−1.71693	−0.858463	+	0.512875i	$0.828580\pi$
$30$	0	0
$31$	−292.368	−1.69390	−0.846948	−	0.531676i	$-0.821562\pi$
$31$	−292.368	−1.69390	−0.846948	+	0.531676i	$0.821562\pi$
$32$	0	0
$33$	−19.4558	−0.102631
$34$	0	0
$35$	0	0
$36$	0	0
$37$	114.558	0.509008	0.254504	−	0.967072i	$-0.418088\pi$
$37$	114.558	0.509008	0.254504	+	0.967072i	$0.418088\pi$
$38$	0	0
$39$	−135.640	−0.556916
$40$	0	0
$41$	−161.605	−0.615573	−0.307786	−	0.951456i	$-0.599588\pi$
$41$	−161.605	−0.615573	−0.307786	+	0.951456i	$0.599588\pi$
$42$	0	0
$43$	471.294	1.67143	0.835716	−	0.549162i	$-0.185053\pi$
$43$	471.294	1.67143	0.835716	+	0.549162i	$0.185053\pi$
$44$	0	0
$45$	41.2721	0.136722
$46$	0	0
$47$	−346.004	−1.07383	−0.536914	−	0.843637i	$-0.680410\pi$
$47$	−346.004	−1.07383	−0.536914	+	0.843637i	$0.680410\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−244.669	−0.671775
$52$	0	0
$53$	405.529	1.05101	0.525507	−	0.850790i	$-0.323876\pi$
$53$	405.529	1.05101	0.525507	+	0.850790i	$0.323876\pi$
$54$	0	0
$55$	29.7401	0.0729119
$56$	0	0
$57$	−15.1615	−0.0352313
$58$	0	0
$59$	253.436	0.559229	0.279614	−	0.960112i	$-0.409793\pi$
$59$	253.436	0.559229	0.279614	+	0.960112i	$0.409793\pi$
$60$	0	0
$61$	−751.217	−1.57678	−0.788390	−	0.615176i	$-0.789085\pi$
$61$	−751.217	−1.57678	−0.788390	+	0.615176i	$0.789085\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	207.338	0.395648
$66$	0	0
$67$	−11.6468	−0.0212370	−0.0106185	−	0.999944i	$-0.503380\pi$
$67$	−11.6468	−0.0212370	−0.0106185	+	0.999944i	$0.503380\pi$
$68$	0	0
$69$	318.749	0.556129
$70$	0	0
$71$	681.661	1.13941	0.569706	−	0.821848i	$-0.307057\pi$
$71$	681.661	1.13941	0.569706	+	0.821848i	$0.307057\pi$
$72$	0	0
$73$	685.457	1.09900	0.549498	−	0.835495i	$-0.314819\pi$
$73$	685.457	1.09900	0.549498	+	0.835495i	$0.314819\pi$
$74$	0	0
$75$	311.912	0.480219
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−0.264069	−0.000376077 0	−0.000188038	−	1.00000i	$-0.500060\pi$
$79$	−0.264069	−0.000376077 0	−0.000188038		1.00000i	$0.500060\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	437.137	0.578097	0.289048	−	0.957314i	$-0.406661\pi$
$83$	437.137	0.578097	0.289048	+	0.957314i	$0.406661\pi$
$84$	0	0
$85$	374.000	0.477247
$86$	0	0
$87$	804.396	0.991268
$88$	0	0
$89$	−58.5126	−0.0696891	−0.0348445	−	0.999393i	$-0.511094\pi$
$89$	−58.5126	−0.0696891	−0.0348445	+	0.999393i	$0.511094\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	877.103	0.977971
$94$	0	0
$95$	23.1758	0.0250293
$96$	0	0
$97$	−1280.09	−1.33993	−0.669966	−	0.742391i	$-0.733692\pi$
$97$	−1280.09	−1.33993	−0.669966	+	0.742391i	$0.733692\pi$
$98$	0	0
$99$	58.3675	0.0592541
$100$	0	0
$101$	−1306.39	−1.28704	−0.643518	−	0.765431i	$-0.722526\pi$
$101$	−1306.39	−1.28704	−0.643518	+	0.765431i	$0.722526\pi$
$102$	0	0
$103$	758.975	0.726058	0.363029	−	0.931778i	$-0.381743\pi$
$103$	758.975	0.726058	0.363029	+	0.931778i	$0.381743\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1262.51	−1.14067	−0.570336	−	0.821412i	$-0.693187\pi$
$107$	−1262.51	−1.14067	−0.570336	+	0.821412i	$0.693187\pi$
$108$	0	0
$109$	−2105.53	−1.85021	−0.925105	−	0.379711i	$-0.876023\pi$
$109$	−2105.53	−1.85021	−0.925105	+	0.379711i	$0.876023\pi$
$110$	0	0
$111$	−343.675	−0.293876
$112$	0	0
$113$	1535.76	1.27852	0.639258	−	0.768992i	$-0.279241\pi$
$113$	1535.76	1.27852	0.639258	+	0.768992i	$0.279241\pi$
$114$	0	0
$115$	−487.239	−0.395089
$116$	0	0
$117$	406.919	0.321536
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1288.94	−0.968401
$122$	0	0
$123$	484.815	0.355401
$124$	0	0
$125$	−1050.01	−0.751326
$126$	0	0
$127$	−24.1749	−0.0168911	−0.00844557	−	0.999964i	$-0.502688\pi$
$127$	−24.1749	−0.0168911	−0.00844557	+	0.999964i	$0.502688\pi$
$128$	0	0
$129$	−1413.88	−0.965002
$130$	0	0
$131$	−1581.53	−1.05480	−0.527400	−	0.849617i	$-0.676833\pi$
$131$	−1581.53	−1.05480	−0.527400	+	0.849617i	$0.676833\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−123.816	−0.0789363
$136$	0	0
$137$	−745.188	−0.464713	−0.232357	−	0.972631i	$-0.574644\pi$
$137$	−745.188	−0.464713	−0.232357	+	0.972631i	$0.574644\pi$
$138$	0	0
$139$	1373.60	0.838179	0.419090	−	0.907945i	$-0.362349\pi$
$139$	1373.60	0.838179	0.419090	+	0.907945i	$0.362349\pi$
$140$	0	0
$141$	1038.01	0.619975
$142$	0	0
$143$	293.220	0.171471
$144$	0	0
$145$	−1229.60	−0.704224
$146$	0	0
$147$	0	0
$148$	0	0
$149$	620.530	0.341180	0.170590	−	0.985342i	$-0.445433\pi$
$149$	620.530	0.341180	0.170590	+	0.985342i	$0.445433\pi$
$150$	0	0
$151$	1939.26	1.04513	0.522567	−	0.852598i	$-0.324975\pi$
$151$	1939.26	1.04513	0.522567	+	0.852598i	$0.324975\pi$
$152$	0	0
$153$	734.007	0.387849
$154$	0	0
$155$	−1340.74	−0.694777
$156$	0	0
$157$	−412.843	−0.209863	−0.104931	−	0.994479i	$-0.533462\pi$
$157$	−412.843	−0.209863	−0.104931	+	0.994479i	$0.533462\pi$
$158$	0	0
$159$	−1216.59	−0.606803
$160$	0	0
$161$	0	0
$162$	0	0
$163$	3907.44	1.87763	0.938817	−	0.344417i	$-0.111923\pi$
$163$	3907.44	1.87763	0.938817	+	0.344417i	$0.111923\pi$
$164$	0	0
$165$	−89.2203	−0.0420957
$166$	0	0
$167$	1286.41	0.596082	0.298041	−	0.954553i	$-0.403667\pi$
$167$	1286.41	0.596082	0.298041	+	0.954553i	$0.403667\pi$
$168$	0	0
$169$	−152.766	−0.0695340
$170$	0	0
$171$	45.4844	0.0203408
$172$	0	0
$173$	−1251.26	−0.549892	−0.274946	−	0.961460i	$-0.588660\pi$
$173$	−1251.26	−0.549892	−0.274946	+	0.961460i	$0.588660\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−760.307	−0.322871
$178$	0	0
$179$	−3623.51	−1.51304	−0.756520	−	0.653971i	$-0.773102\pi$
$179$	−3623.51	−1.51304	−0.756520	+	0.653971i	$0.773102\pi$
$180$	0	0
$181$	181.727	0.0746280	0.0373140	−	0.999304i	$-0.488120\pi$
$181$	181.727	0.0746280	0.0373140	+	0.999304i	$0.488120\pi$
$182$	0	0
$183$	2253.65	0.910354
$184$	0	0
$185$	525.341	0.208777
$186$	0	0
$187$	528.916	0.206835
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1481.28	−0.561160	−0.280580	−	0.959831i	$-0.590527\pi$
$191$	−1481.28	−0.561160	−0.280580	+	0.959831i	$0.590527\pi$
$192$	0	0
$193$	−356.708	−0.133038	−0.0665192	−	0.997785i	$-0.521189\pi$
$193$	−356.708	−0.133038	−0.0665192	+	0.997785i	$0.521189\pi$
$194$	0	0
$195$	−622.014	−0.228428
$196$	0	0
$197$	4890.53	1.76871	0.884355	−	0.466816i	$-0.154599\pi$
$197$	4890.53	1.76871	0.884355	+	0.466816i	$0.154599\pi$
$198$	0	0
$199$	−3542.85	−1.26204	−0.631020	−	0.775766i	$-0.717364\pi$
$199$	−3542.85	−1.26204	−0.631020	+	0.775766i	$0.717364\pi$
$200$	0	0
$201$	34.9403	0.0122612
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−741.087	−0.252486
$206$	0	0
$207$	−956.248	−0.321081
$208$	0	0
$209$	32.7755	0.0108475
$210$	0	0
$211$	4289.50	1.39953	0.699765	−	0.714373i	$-0.253288\pi$
$211$	4289.50	1.39953	0.699765	+	0.714373i	$0.253288\pi$
$212$	0	0
$213$	−2044.98	−0.657840
$214$	0	0
$215$	2161.25	0.685563
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−2056.37	−0.634505
$220$	0	0
$221$	3687.42	1.12237
$222$	0	0
$223$	−5795.73	−1.74041	−0.870204	−	0.492692i	$-0.836013\pi$
$223$	−5795.73	−1.74041	−0.870204	+	0.492692i	$0.836013\pi$
$224$	0	0
$225$	−935.735	−0.277255
$226$	0	0
$227$	−4104.04	−1.19998	−0.599989	−	0.800008i	$-0.704828\pi$
$227$	−4104.04	−1.19998	−0.599989	+	0.800008i	$0.704828\pi$
$228$	0	0
$229$	−1296.83	−0.374223	−0.187111	−	0.982339i	$-0.559913\pi$
$229$	−1296.83	−0.374223	−0.187111	+	0.982339i	$0.559913\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1478.33	−0.415660	−0.207830	−	0.978165i	$-0.566640\pi$
$233$	−1478.33	−0.415660	−0.207830	+	0.978165i	$0.566640\pi$
$234$	0	0
$235$	−1586.70	−0.440447
$236$	0	0
$237$	0.792206	0.000217128 0
$238$	0	0
$239$	3776.92	1.02221	0.511106	−	0.859518i	$-0.329236\pi$
$239$	3776.92	1.02221	0.511106	+	0.859518i	$0.329236\pi$
$240$	0	0
$241$	−3996.38	−1.06817	−0.534086	−	0.845430i	$-0.679344\pi$
$241$	−3996.38	−1.06817	−0.534086	+	0.845430i	$0.679344\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	228.500	0.0588627
$248$	0	0
$249$	−1311.41	−0.333764
$250$	0	0
$251$	−5423.58	−1.36388	−0.681939	−	0.731409i	$-0.738863\pi$
$251$	−5423.58	−1.36388	−0.681939	+	0.731409i	$0.738863\pi$
$252$	0	0
$253$	−689.060	−0.171229
$254$	0	0
$255$	−1122.00	−0.275539
$256$	0	0
$257$	5964.22	1.44762	0.723809	−	0.690000i	$-0.242389\pi$
$257$	5964.22	1.44762	0.723809	+	0.690000i	$0.242389\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−2413.19	−0.572309
$262$	0	0
$263$	−5166.01	−1.21122	−0.605608	−	0.795763i	$-0.707070\pi$
$263$	−5166.01	−1.21122	−0.605608	+	0.795763i	$0.707070\pi$
$264$	0	0
$265$	1859.67	0.431089
$266$	0	0
$267$	175.538	0.0402350
$268$	0	0
$269$	3883.29	0.880180	0.440090	−	0.897954i	$-0.354946\pi$
$269$	3883.29	0.880180	0.440090	+	0.897954i	$0.354946\pi$
$270$	0	0
$271$	−5527.65	−1.23904	−0.619522	−	0.784979i	$-0.712674\pi$
$271$	−5527.65	−1.23904	−0.619522	+	0.784979i	$0.712674\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−674.278	−0.147856
$276$	0	0
$277$	2268.12	0.491979	0.245989	−	0.969273i	$-0.420887\pi$
$277$	2268.12	0.491979	0.245989	+	0.969273i	$0.420887\pi$
$278$	0	0
$279$	−2631.31	−0.564632
$280$	0	0
$281$	725.656	0.154053	0.0770267	−	0.997029i	$-0.475457\pi$
$281$	725.656	0.154053	0.0770267	+	0.997029i	$0.475457\pi$
$282$	0	0
$283$	−4237.00	−0.889976	−0.444988	−	0.895536i	$-0.646792\pi$
$283$	−4237.00	−0.889976	−0.444988	+	0.895536i	$0.646792\pi$
$284$	0	0
$285$	−69.5273	−0.0144507
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1738.44	0.353845
$290$	0	0
$291$	3840.27	0.773611
$292$	0	0
$293$	−4373.78	−0.872079	−0.436039	−	0.899928i	$-0.643619\pi$
$293$	−4373.78	−0.872079	−0.436039	+	0.899928i	$0.643619\pi$
$294$	0	0
$295$	1162.20	0.229376
$296$	0	0
$297$	−175.103	−0.0342104
$298$	0	0
$299$	−4803.89	−0.929152
$300$	0	0
$301$	0	0
$302$	0	0
$303$	3919.17	0.743070
$304$	0	0
$305$	−3444.92	−0.646740
$306$	0	0
$307$	−4133.47	−0.768435	−0.384217	−	0.923243i	$-0.625529\pi$
$307$	−4133.47	−0.768435	−0.384217	+	0.923243i	$0.625529\pi$
$308$	0	0
$309$	−2276.92	−0.419190
$310$	0	0
$311$	−5063.23	−0.923182	−0.461591	−	0.887093i	$-0.652721\pi$
$311$	−5063.23	−0.923182	−0.461591	+	0.887093i	$0.652721\pi$
$312$	0	0
$313$	7411.56	1.33842	0.669211	−	0.743073i	$-0.266632\pi$
$313$	7411.56	1.33842	0.669211	+	0.743073i	$0.266632\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−6737.05	−1.19366	−0.596831	−	0.802367i	$-0.703574\pi$
$317$	−6737.05	−1.19366	−0.596831	+	0.802367i	$0.703574\pi$
$318$	0	0
$319$	−1738.91	−0.305205
$320$	0	0
$321$	3787.54	0.658567
$322$	0	0
$323$	412.171	0.0710026
$324$	0	0
$325$	−4700.84	−0.802326
$326$	0	0
$327$	6316.58	1.06822
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−11175.9	−1.85585	−0.927923	−	0.372771i	$-0.878408\pi$
$331$	−11175.9	−1.85585	−0.927923	+	0.372771i	$0.878408\pi$
$332$	0	0
$333$	1031.03	0.169669
$334$	0	0
$335$	−53.4095	−0.00871067
$336$	0	0
$337$	9379.78	1.51617	0.758085	−	0.652156i	$-0.226135\pi$
$337$	9379.78	1.51617	0.758085	+	0.652156i	$0.226135\pi$
$338$	0	0
$339$	−4607.29	−0.738152
$340$	0	0
$341$	−1896.09	−0.301111
$342$	0	0
$343$	0	0
$344$	0	0
$345$	1461.72	0.228105
$346$	0	0
$347$	−5681.46	−0.878953	−0.439476	−	0.898254i	$-0.644836\pi$
$347$	−5681.46	−0.878953	−0.439476	+	0.898254i	$0.644836\pi$
$348$	0	0
$349$	−704.250	−0.108016	−0.0540080	−	0.998541i	$-0.517200\pi$
$349$	−704.250	−0.108016	−0.0540080	+	0.998541i	$0.517200\pi$
$350$	0	0
$351$	−1220.76	−0.185639
$352$	0	0
$353$	−4284.96	−0.646078	−0.323039	−	0.946386i	$-0.604705\pi$
$353$	−4284.96	−0.646078	−0.323039	+	0.946386i	$0.604705\pi$
$354$	0	0
$355$	3125.95	0.467347
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−4661.27	−0.685272	−0.342636	−	0.939468i	$-0.611320\pi$
$359$	−4661.27	−0.685272	−0.342636	+	0.939468i	$0.611320\pi$
$360$	0	0
$361$	−6833.46	−0.996276
$362$	0	0
$363$	3866.82	0.559106
$364$	0	0
$365$	3143.36	0.450770
$366$	0	0
$367$	6935.30	0.986430	0.493215	−	0.869907i	$-0.335822\pi$
$367$	6935.30	0.986430	0.493215	+	0.869907i	$0.335822\pi$
$368$	0	0
$369$	−1454.45	−0.205191
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−3081.10	−0.427704	−0.213852	−	0.976866i	$-0.568601\pi$
$373$	−3081.10	−0.427704	−0.213852	+	0.976866i	$0.568601\pi$
$374$	0	0
$375$	3150.03	0.433778
$376$	0	0
$377$	−12123.1	−1.65616
$378$	0	0
$379$	−941.827	−0.127647	−0.0638237	−	0.997961i	$-0.520330\pi$
$379$	−941.827	−0.127647	−0.0638237	+	0.997961i	$0.520330\pi$
$380$	0	0
$381$	72.5247	0.00975210
$382$	0	0
$383$	677.526	0.0903915	0.0451958	−	0.998978i	$-0.485609\pi$
$383$	677.526	0.0903915	0.0451958	+	0.998978i	$0.485609\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4241.64	0.557144
$388$	0	0
$389$	−11865.4	−1.54653	−0.773266	−	0.634082i	$-0.781378\pi$
$389$	−11865.4	−1.54653	−0.773266	+	0.634082i	$0.781378\pi$
$390$	0	0
$391$	−8665.34	−1.12078
$392$	0	0
$393$	4744.58	0.608989
$394$	0	0
$395$	−1.21096	−0.000154254 0
$396$	0	0
$397$	−5140.76	−0.649893	−0.324947	−	0.945732i	$-0.605346\pi$
$397$	−5140.76	−0.649893	−0.324947	+	0.945732i	$0.605346\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−12382.0	−1.54196	−0.770981	−	0.636858i	$-0.780234\pi$
$401$	−12382.0	−1.54196	−0.770981	+	0.636858i	$0.780234\pi$
$402$	0	0
$403$	−13218.9	−1.63394
$404$	0	0
$405$	371.449	0.0455739
$406$	0	0
$407$	742.944	0.0904824
$408$	0	0
$409$	−15875.6	−1.91931	−0.959657	−	0.281173i	$-0.909277\pi$
$409$	−15875.6	−1.91931	−0.959657	+	0.281173i	$0.909277\pi$
$410$	0	0
$411$	2235.56	0.268302
$412$	0	0
$413$	0	0
$414$	0	0
$415$	2004.62	0.237115
$416$	0	0
$417$	−4120.79	−0.483923
$418$	0	0
$419$	16111.9	1.87857	0.939283	−	0.343145i	$-0.111492\pi$
$419$	16111.9	1.87857	0.939283	+	0.343145i	$0.111492\pi$
$420$	0	0
$421$	8691.58	1.00618	0.503090	−	0.864234i	$-0.332197\pi$
$421$	8691.58	1.00618	0.503090	+	0.864234i	$0.332197\pi$
$422$	0	0
$423$	−3114.04	−0.357943
$424$	0	0
$425$	−8479.46	−0.967798
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−879.661	−0.0989987
$430$	0	0
$431$	4195.10	0.468842	0.234421	−	0.972135i	$-0.424681\pi$
$431$	4195.10	0.468842	0.234421	+	0.972135i	$0.424681\pi$
$432$	0	0
$433$	−5426.54	−0.602270	−0.301135	−	0.953582i	$-0.597365\pi$
$433$	−5426.54	−0.602270	−0.301135	+	0.953582i	$0.597365\pi$
$434$	0	0
$435$	3688.79	0.406584
$436$	0	0
$437$	−536.968	−0.0587795
$438$	0	0
$439$	3771.24	0.410003	0.205002	−	0.978762i	$-0.434280\pi$
$439$	3771.24	0.410003	0.205002	+	0.978762i	$0.434280\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	5930.30	0.636020	0.318010	−	0.948087i	$-0.396985\pi$
$443$	5930.30	0.636020	0.318010	+	0.948087i	$0.396985\pi$
$444$	0	0
$445$	−268.326	−0.0285840
$446$	0	0
$447$	−1861.59	−0.196980
$448$	0	0
$449$	−529.065	−0.0556083	−0.0278041	−	0.999613i	$-0.508851\pi$
$449$	−529.065	−0.0556083	−0.0278041	+	0.999613i	$0.508851\pi$
$450$	0	0
$451$	−1048.05	−0.109426
$452$	0	0
$453$	−5817.79	−0.603408
$454$	0	0
$455$	0	0
$456$	0	0
$457$	10057.2	1.02944	0.514721	−	0.857358i	$-0.327896\pi$
$457$	10057.2	1.02944	0.514721	+	0.857358i	$0.327896\pi$
$458$	0	0
$459$	−2202.02	−0.223925
$460$	0	0
$461$	−5010.31	−0.506190	−0.253095	−	0.967441i	$-0.581448\pi$
$461$	−5010.31	−0.506190	−0.253095	+	0.967441i	$0.581448\pi$
$462$	0	0
$463$	7124.38	0.715114	0.357557	−	0.933891i	$-0.383610\pi$
$463$	7124.38	0.715114	0.357557	+	0.933891i	$0.383610\pi$
$464$	0	0
$465$	4022.21	0.401130
$466$	0	0
$467$	−7501.44	−0.743309	−0.371654	−	0.928371i	$-0.621209\pi$
$467$	−7501.44	−0.743309	−0.371654	+	0.928371i	$0.621209\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	1238.53	0.121164
$472$	0	0
$473$	3056.47	0.297118
$474$	0	0
$475$	−525.449	−0.0507563
$476$	0	0
$477$	3649.76	0.350338
$478$	0	0
$479$	8173.80	0.779688	0.389844	−	0.920881i	$-0.372529\pi$
$479$	8173.80	0.779688	0.389844	+	0.920881i	$0.372529\pi$
$480$	0	0
$481$	5179.55	0.490992
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−5870.22	−0.549594
$486$	0	0
$487$	11968.8	1.11367	0.556835	−	0.830623i	$-0.312015\pi$
$487$	11968.8	1.11367	0.556835	+	0.830623i	$0.312015\pi$
$488$	0	0
$489$	−11722.3	−1.08405
$490$	0	0
$491$	−2079.96	−0.191176	−0.0955878	−	0.995421i	$-0.530473\pi$
$491$	−2079.96	−0.191176	−0.0955878	+	0.995421i	$0.530473\pi$
$492$	0	0
$493$	−21867.9	−1.99773
$494$	0	0
$495$	267.661	0.0243040
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−12834.4	−1.15140	−0.575699	−	0.817662i	$-0.695270\pi$
$499$	−12834.4	−1.15140	−0.575699	+	0.817662i	$0.695270\pi$
$500$	0	0
$501$	−3859.24	−0.344148
$502$	0	0
$503$	−16808.8	−1.48999	−0.744997	−	0.667068i	$-0.767549\pi$
$503$	−16808.8	−1.48999	−0.744997	+	0.667068i	$0.767549\pi$
$504$	0	0
$505$	−5990.82	−0.527897
$506$	0	0
$507$	458.299	0.0401455
$508$	0	0
$509$	−4270.26	−0.371859	−0.185929	−	0.982563i	$-0.559530\pi$
$509$	−4270.26	−0.371859	−0.185929	+	0.982563i	$0.559530\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−136.453	−0.0117438
$514$	0	0
$515$	3480.50	0.297804
$516$	0	0
$517$	−2243.93	−0.190886
$518$	0	0
$519$	3753.77	0.317480
$520$	0	0
$521$	15283.3	1.28517	0.642584	−	0.766215i	$-0.277862\pi$
$521$	15283.3	1.28517	0.642584	+	0.766215i	$0.277862\pi$
$522$	0	0
$523$	4499.33	0.376180	0.188090	−	0.982152i	$-0.439770\pi$
$523$	4499.33	0.376180	0.188090	+	0.982152i	$0.439770\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−23844.4	−1.97093
$528$	0	0
$529$	−877.984	−0.0721611
$530$	0	0
$531$	2280.92	0.186410
$532$	0	0
$533$	−7306.69	−0.593785
$534$	0	0
$535$	−5789.62	−0.467864
$536$	0	0
$537$	10870.5	0.873554
$538$	0	0
$539$	0	0
$540$	0	0
$541$	3970.82	0.315561	0.157781	−	0.987474i	$-0.449566\pi$
$541$	3970.82	0.315561	0.157781	+	0.987474i	$0.449566\pi$
$542$	0	0
$543$	−545.181	−0.0430865
$544$	0	0
$545$	−9655.50	−0.758892
$546$	0	0
$547$	2703.90	0.211353	0.105677	−	0.994401i	$-0.466299\pi$
$547$	2703.90	0.211353	0.105677	+	0.994401i	$0.466299\pi$
$548$	0	0
$549$	−6760.96	−0.525593
$550$	0	0
$551$	−1355.09	−0.104771
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−1576.02	−0.120538
$556$	0	0
$557$	790.824	0.0601585	0.0300793	−	0.999548i	$-0.490424\pi$
$557$	790.824	0.0601585	0.0300793	+	0.999548i	$0.490424\pi$
$558$	0	0
$559$	21308.7	1.61227
$560$	0	0
$561$	−1586.75	−0.119416
$562$	0	0
$563$	−7517.15	−0.562718	−0.281359	−	0.959603i	$-0.590785\pi$
$563$	−7517.15	−0.562718	−0.281359	+	0.959603i	$0.590785\pi$
$564$	0	0
$565$	7042.68	0.524403
$566$	0	0
$567$	0	0
$568$	0	0
$569$	13945.4	1.02746	0.513728	−	0.857953i	$-0.328264\pi$
$569$	13945.4	1.02746	0.513728	+	0.857953i	$0.328264\pi$
$570$	0	0
$571$	2118.49	0.155265	0.0776323	−	0.996982i	$-0.475264\pi$
$571$	2118.49	0.155265	0.0776323	+	0.996982i	$0.475264\pi$
$572$	0	0
$573$	4443.84	0.323986
$574$	0	0
$575$	11046.8	0.801192
$576$	0	0
$577$	22857.7	1.64918	0.824592	−	0.565728i	$-0.191405\pi$
$577$	22857.7	1.64918	0.824592	+	0.565728i	$0.191405\pi$
$578$	0	0
$579$	1070.12	0.0768098
$580$	0	0
$581$	0	0
$582$	0	0
$583$	2629.97	0.186831
$584$	0	0
$585$	1866.04	0.131883
$586$	0	0
$587$	−23955.3	−1.68440	−0.842199	−	0.539167i	$-0.818739\pi$
$587$	−23955.3	−1.68440	−0.842199	+	0.539167i	$0.818739\pi$
$588$	0	0
$589$	−1477.57	−0.103366
$590$	0	0
$591$	−14671.6	−1.02116
$592$	0	0
$593$	−10778.0	−0.746373	−0.373186	−	0.927756i	$-0.621735\pi$
$593$	−10778.0	−0.746373	−0.373186	+	0.927756i	$0.621735\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	10628.6	0.728639
$598$	0	0
$599$	−7597.58	−0.518245	−0.259122	−	0.965845i	$-0.583433\pi$
$599$	−7597.58	−0.518245	−0.259122	+	0.965845i	$0.583433\pi$
$600$	0	0
$601$	19956.1	1.35445	0.677225	−	0.735776i	$-0.263182\pi$
$601$	19956.1	1.35445	0.677225	+	0.735776i	$0.263182\pi$
$602$	0	0
$603$	−104.821	−0.00707899
$604$	0	0
$605$	−5910.81	−0.397204
$606$	0	0
$607$	236.311	0.0158016	0.00790079	−	0.999969i	$-0.497485\pi$
$607$	236.311	0.0158016	0.00790079	+	0.999969i	$0.497485\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−15644.0	−1.03582
$612$	0	0
$613$	26414.9	1.74044	0.870219	−	0.492664i	$-0.163977\pi$
$613$	26414.9	1.74044	0.870219	+	0.492664i	$0.163977\pi$
$614$	0	0
$615$	2223.26	0.145773
$616$	0	0
$617$	18473.6	1.20538	0.602689	−	0.797976i	$-0.294096\pi$
$617$	18473.6	1.20538	0.602689	+	0.797976i	$0.294096\pi$
$618$	0	0
$619$	−16047.9	−1.04204	−0.521018	−	0.853546i	$-0.674448\pi$
$619$	−16047.9	−1.04204	−0.521018	+	0.853546i	$0.674448\pi$
$620$	0	0
$621$	2868.74	0.185376
$622$	0	0
$623$	0	0
$624$	0	0
$625$	8181.20	0.523597
$626$	0	0
$627$	−98.3264	−0.00626280
$628$	0	0
$629$	9342.97	0.592255
$630$	0	0
$631$	15065.7	0.950487	0.475243	−	0.879854i	$-0.342360\pi$
$631$	15065.7	0.950487	0.475243	+	0.879854i	$0.342360\pi$
$632$	0	0
$633$	−12868.5	−0.808020
$634$	0	0
$635$	−110.861	−0.00692816
$636$	0	0
$637$	0	0
$638$	0	0
$639$	6134.95	0.379804
$640$	0	0
$641$	−31198.4	−1.92241	−0.961203	−	0.275842i	$-0.911043\pi$
$641$	−31198.4	−1.92241	−0.961203	+	0.275842i	$0.911043\pi$
$642$	0	0
$643$	12497.9	0.766517	0.383259	−	0.923641i	$-0.374802\pi$
$643$	12497.9	0.766517	0.383259	+	0.923641i	$0.374802\pi$
$644$	0	0
$645$	−6483.75	−0.395810
$646$	0	0
$647$	9929.72	0.603366	0.301683	−	0.953408i	$-0.402452\pi$
$647$	9929.72	0.603366	0.301683	+	0.953408i	$0.402452\pi$
$648$	0	0
$649$	1643.60	0.0994099
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−8145.58	−0.488149	−0.244075	−	0.969756i	$-0.578484\pi$
$653$	−8145.58	−0.488149	−0.244075	+	0.969756i	$0.578484\pi$
$654$	0	0
$655$	−7252.55	−0.432642
$656$	0	0
$657$	6169.11	0.366332
$658$	0	0
$659$	16975.8	1.00347	0.501733	−	0.865022i	$-0.332696\pi$
$659$	16975.8	1.00347	0.501733	+	0.865022i	$0.332696\pi$
$660$	0	0
$661$	20637.8	1.21440	0.607199	−	0.794550i	$-0.292293\pi$
$661$	20637.8	1.21440	0.607199	+	0.794550i	$0.292293\pi$
$662$	0	0
$663$	−11062.3	−0.647999
$664$	0	0
$665$	0	0
$666$	0	0
$667$	28489.0	1.65382
$668$	0	0
$669$	17387.2	1.00482
$670$	0	0
$671$	−4871.86	−0.280292
$672$	0	0
$673$	−2150.29	−0.123161	−0.0615807	−	0.998102i	$-0.519614\pi$
$673$	−2150.29	−0.123161	−0.0615807	+	0.998102i	$0.519614\pi$
$674$	0	0
$675$	2807.21	0.160073
$676$	0	0
$677$	−27783.4	−1.57726	−0.788628	−	0.614871i	$-0.789208\pi$
$677$	−27783.4	−1.57726	−0.788628	+	0.614871i	$0.789208\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	12312.1	0.692807
$682$	0	0
$683$	−18181.8	−1.01860	−0.509302	−	0.860588i	$-0.670096\pi$
$683$	−18181.8	−1.01860	−0.509302	+	0.860588i	$0.670096\pi$
$684$	0	0
$685$	−3417.27	−0.190609
$686$	0	0
$687$	3890.49	0.216058
$688$	0	0
$689$	18335.3	1.01381
$690$	0	0
$691$	−23935.1	−1.31771	−0.658853	−	0.752272i	$-0.728958\pi$
$691$	−23935.1	−1.31771	−0.658853	+	0.752272i	$0.728958\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	6299.02	0.343792
$696$	0	0
$697$	−13179.9	−0.716249
$698$	0	0
$699$	4435.00	0.239982
$700$	0	0
$701$	−20627.2	−1.11138	−0.555691	−	0.831389i	$-0.687546\pi$
$701$	−20627.2	−1.11138	−0.555691	+	0.831389i	$0.687546\pi$
$702$	0	0
$703$	578.958	0.0310609
$704$	0	0
$705$	4760.10	0.254292
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−6584.64	−0.348789	−0.174394	−	0.984676i	$-0.555797\pi$
$709$	−6584.64	−0.348789	−0.174394	+	0.984676i	$0.555797\pi$
$710$	0	0
$711$	−2.37662	−0.000125359 0
$712$	0	0
$713$	31064.0	1.63163
$714$	0	0
$715$	1344.65	0.0703313
$716$	0	0
$717$	−11330.8	−0.590174
$718$	0	0
$719$	−170.886	−0.00886366	−0.00443183	−	0.999990i	$-0.501411\pi$
$719$	−170.886	−0.00886366	−0.00443183	+	0.999990i	$0.501411\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	11989.1	0.616710
$724$	0	0
$725$	27877.8	1.42808
$726$	0	0
$727$	11127.5	0.567671	0.283836	−	0.958873i	$-0.408393\pi$
$727$	11127.5	0.567671	0.283836	+	0.958873i	$0.408393\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	38437.0	1.94479
$732$	0	0
$733$	−22575.9	−1.13760	−0.568800	−	0.822476i	$-0.692592\pi$
$733$	−22575.9	−1.13760	−0.568800	+	0.822476i	$0.692592\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−75.5325	−0.00377513
$738$	0	0
$739$	22936.4	1.14172	0.570860	−	0.821048i	$-0.306610\pi$
$739$	22936.4	1.14172	0.570860	+	0.821048i	$0.306610\pi$
$740$	0	0
$741$	−685.499	−0.0339844
$742$	0	0
$743$	−16973.4	−0.838081	−0.419041	−	0.907967i	$-0.637634\pi$
$743$	−16973.4	−0.838081	−0.419041	+	0.907967i	$0.637634\pi$
$744$	0	0
$745$	2845.62	0.139940
$746$	0	0
$747$	3934.23	0.192699
$748$	0	0
$749$	0	0
$750$	0	0
$751$	21197.9	1.02999	0.514994	−	0.857194i	$-0.327794\pi$
$751$	21197.9	1.02999	0.514994	+	0.857194i	$0.327794\pi$
$752$	0	0
$753$	16270.8	0.787436
$754$	0	0
$755$	8893.05	0.428677
$756$	0	0
$757$	−7962.24	−0.382289	−0.191144	−	0.981562i	$-0.561220\pi$
$757$	−7962.24	−0.382289	−0.191144	+	0.981562i	$0.561220\pi$
$758$	0	0
$759$	2067.18	0.0988588
$760$	0	0
$761$	26856.2	1.27928	0.639642	−	0.768673i	$-0.279083\pi$
$761$	26856.2	1.27928	0.639642	+	0.768673i	$0.279083\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	3366.00	0.159082
$766$	0	0
$767$	11458.6	0.539436
$768$	0	0
$769$	−12183.6	−0.571331	−0.285666	−	0.958329i	$-0.592215\pi$
$769$	−12183.6	−0.571331	−0.285666	+	0.958329i	$0.592215\pi$
$770$	0	0
$771$	−17892.7	−0.835783
$772$	0	0
$773$	−17455.2	−0.812187	−0.406093	−	0.913832i	$-0.633109\pi$
$773$	−17455.2	−0.812187	−0.406093	+	0.913832i	$0.633109\pi$
$774$	0	0
$775$	30397.6	1.40892
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−816.724	−0.0375638
$780$	0	0
$781$	4420.76	0.202545
$782$	0	0
$783$	7239.56	0.330423
$784$	0	0
$785$	−1893.21	−0.0860785
$786$	0	0
$787$	−30981.0	−1.40324	−0.701622	−	0.712549i	$-0.747540\pi$
$787$	−30981.0	−1.40324	−0.701622	+	0.712549i	$0.747540\pi$
$788$	0	0
$789$	15498.0	0.699296
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−33964.9	−1.52097
$794$	0	0
$795$	−5579.01	−0.248889
$796$	0	0
$797$	−10517.6	−0.467446	−0.233723	−	0.972303i	$-0.575091\pi$
$797$	−10517.6	−0.467446	−0.233723	+	0.972303i	$0.575091\pi$
$798$	0	0
$799$	−28218.8	−1.24945
$800$	0	0
$801$	−526.614	−0.0232297
$802$	0	0
$803$	4445.38	0.195360
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−11649.9	−0.508172
$808$	0	0
$809$	−41778.4	−1.81564	−0.907819	−	0.419361i	$-0.862254\pi$
$809$	−41778.4	−1.81564	−0.907819	+	0.419361i	$0.862254\pi$
$810$	0	0
$811$	12935.4	0.560079	0.280039	−	0.959988i	$-0.409652\pi$
$811$	12935.4	0.560079	0.280039	+	0.959988i	$0.409652\pi$
$812$	0	0
$813$	16583.0	0.715363
$814$	0	0
$815$	17918.7	0.770140
$816$	0	0
$817$	2381.83	0.101995
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−14499.2	−0.616354	−0.308177	−	0.951329i	$-0.599719\pi$
$821$	−14499.2	−0.616354	−0.308177	+	0.951329i	$0.599719\pi$
$822$	0	0
$823$	11977.1	0.507287	0.253643	−	0.967298i	$-0.418371\pi$
$823$	11977.1	0.507287	0.253643	+	0.967298i	$0.418371\pi$
$824$	0	0
$825$	2022.84	0.0853649
$826$	0	0
$827$	−27613.5	−1.16108	−0.580541	−	0.814231i	$-0.697159\pi$
$827$	−27613.5	−1.16108	−0.580541	+	0.814231i	$0.697159\pi$
$828$	0	0
$829$	−677.836	−0.0283983	−0.0141992	−	0.999899i	$-0.504520\pi$
$829$	−677.836	−0.0283983	−0.0141992	+	0.999899i	$0.504520\pi$
$830$	0	0
$831$	−6804.36	−0.284044
$832$	0	0
$833$	0	0
$834$	0	0
$835$	5899.22	0.244492
$836$	0	0
$837$	7893.92	0.325990
$838$	0	0
$839$	−42209.6	−1.73687	−0.868436	−	0.495801i	$-0.834874\pi$
$839$	−42209.6	−1.73687	−0.868436	+	0.495801i	$0.834874\pi$
$840$	0	0
$841$	47505.8	1.94784
$842$	0	0
$843$	−2176.97	−0.0889428
$844$	0	0
$845$	−700.553	−0.0285204
$846$	0	0
$847$	0	0
$848$	0	0
$849$	12711.0	0.513828
$850$	0	0
$851$	−12171.8	−0.490299
$852$	0	0
$853$	−2796.45	−0.112249	−0.0561247	−	0.998424i	$-0.517874\pi$
$853$	−2796.45	−0.112249	−0.0561247	+	0.998424i	$0.517874\pi$
$854$	0	0
$855$	208.582	0.00834310
$856$	0	0
$857$	23181.1	0.923979	0.461989	−	0.886886i	$-0.347136\pi$
$857$	23181.1	0.923979	0.461989	+	0.886886i	$0.347136\pi$
$858$	0	0
$859$	11897.5	0.472569	0.236285	−	0.971684i	$-0.424070\pi$
$859$	11897.5	0.472569	0.236285	+	0.971684i	$0.424070\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	29815.9	1.17607	0.588033	−	0.808837i	$-0.299902\pi$
$863$	29815.9	1.17607	0.588033	+	0.808837i	$0.299902\pi$
$864$	0	0
$865$	−5737.99	−0.225546
$866$	0	0
$867$	−5215.31	−0.204292
$868$	0	0
$869$	−1.71256	−6.68523e−5 0
$870$	0	0
$871$	−526.587	−0.0204853
$872$	0	0
$873$	−11520.8	−0.446644
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−10338.8	−0.398082	−0.199041	−	0.979991i	$-0.563783\pi$
$877$	−10338.8	−0.398082	−0.199041	+	0.979991i	$0.563783\pi$
$878$	0	0
$879$	13121.3	0.503495
$880$	0	0
$881$	24140.2	0.923160	0.461580	−	0.887099i	$-0.347283\pi$
$881$	24140.2	0.923160	0.461580	+	0.887099i	$0.347283\pi$
$882$	0	0
$883$	−12997.6	−0.495361	−0.247681	−	0.968842i	$-0.579668\pi$
$883$	−12997.6	−0.495361	−0.247681	+	0.968842i	$0.579668\pi$
$884$	0	0
$885$	−3486.61	−0.132430
$886$	0	0
$887$	−45266.6	−1.71353	−0.856766	−	0.515705i	$-0.827530\pi$
$887$	−45266.6	−1.71353	−0.856766	+	0.515705i	$0.827530\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	525.308	0.0197514
$892$	0	0
$893$	−1748.64	−0.0655276
$894$	0	0
$895$	−16616.7	−0.620596
$896$	0	0
$897$	14411.7	0.536446
$898$	0	0
$899$	78393.1	2.90829
$900$	0	0
$901$	33073.5	1.22290
$902$	0	0
$903$	0	0
$904$	0	0
$905$	833.361	0.0306098
$906$	0	0
$907$	27766.5	1.01651	0.508253	−	0.861208i	$-0.330292\pi$
$907$	27766.5	1.01651	0.508253	+	0.861208i	$0.330292\pi$
$908$	0	0
$909$	−11757.5	−0.429012
$910$	0	0
$911$	−18531.2	−0.673948	−0.336974	−	0.941514i	$-0.609403\pi$
$911$	−18531.2	−0.673948	−0.336974	+	0.941514i	$0.609403\pi$
$912$	0	0
$913$	2834.96	0.102764
$914$	0	0
$915$	10334.8	0.373395
$916$	0	0
$917$	0	0
$918$	0	0
$919$	18093.4	0.649452	0.324726	−	0.945808i	$-0.394728\pi$
$919$	18093.4	0.649452	0.324726	+	0.945808i	$0.394728\pi$
$920$	0	0
$921$	12400.4	0.443656
$922$	0	0
$923$	30820.1	1.09908
$924$	0	0
$925$	−11910.7	−0.423375
$926$	0	0
$927$	6830.77	0.242019
$928$	0	0
$929$	10622.5	0.375149	0.187574	−	0.982250i	$-0.439937\pi$
$929$	10622.5	0.375149	0.187574	+	0.982250i	$0.439937\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	15189.7	0.532999
$934$	0	0
$935$	2425.50	0.0848366
$936$	0	0
$937$	16057.6	0.559851	0.279925	−	0.960022i	$-0.409690\pi$
$937$	16057.6	0.559851	0.279925	+	0.960022i	$0.409690\pi$
$938$	0	0
$939$	−22234.7	−0.772738
$940$	0	0
$941$	55153.3	1.91068	0.955338	−	0.295516i	$-0.0954915\pi$
$941$	55153.3	1.91068	0.955338	+	0.295516i	$0.0954915\pi$
$942$	0	0
$943$	17170.5	0.592947
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−18785.7	−0.644619	−0.322309	−	0.946634i	$-0.604459\pi$
$947$	−18785.7	−0.644619	−0.322309	+	0.946634i	$0.604459\pi$
$948$	0	0
$949$	30991.7	1.06010
$950$	0	0
$951$	20211.2	0.689161
$952$	0	0
$953$	−36499.4	−1.24064	−0.620321	−	0.784348i	$-0.712998\pi$
$953$	−36499.4	−1.24064	−0.620321	+	0.784348i	$0.712998\pi$
$954$	0	0
$955$	−6792.83	−0.230168
$956$	0	0
$957$	5216.74	0.176210
$958$	0	0
$959$	0	0
$960$	0	0
$961$	55687.8	1.86928
$962$	0	0
$963$	−11362.6	−0.380224
$964$	0	0
$965$	−1635.79	−0.0545677
$966$	0	0
$967$	−26059.9	−0.866627	−0.433314	−	0.901243i	$-0.642656\pi$
$967$	−26059.9	−0.866627	−0.433314	+	0.901243i	$0.642656\pi$
$968$	0	0
$969$	−1236.51	−0.0409934
$970$	0	0
$971$	−21689.0	−0.716820	−0.358410	−	0.933564i	$-0.616681\pi$
$971$	−21689.0	−0.716820	−0.358410	+	0.933564i	$0.616681\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	14102.5	0.463223
$976$	0	0
$977$	2119.19	0.0693950	0.0346975	−	0.999398i	$-0.488953\pi$
$977$	2119.19	0.0693950	0.0346975	+	0.999398i	$0.488953\pi$
$978$	0	0
$979$	−379.471	−0.0123881
$980$	0	0
$981$	−18949.7	−0.616737
$982$	0	0
$983$	48504.1	1.57380	0.786898	−	0.617083i	$-0.211686\pi$
$983$	48504.1	1.57380	0.786898	+	0.617083i	$0.211686\pi$
$984$	0	0
$985$	22426.9	0.725463
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−50074.8	−1.61000
$990$	0	0
$991$	−2043.84	−0.0655144	−0.0327572	−	0.999463i	$-0.510429\pi$
$991$	−2043.84	−0.0655144	−0.0327572	+	0.999463i	$0.510429\pi$
$992$	0	0
$993$	33527.8	1.07147
$994$	0	0
$995$	−16246.8	−0.517645
$996$	0	0
$997$	−26738.9	−0.849378	−0.424689	−	0.905339i	$-0.639617\pi$
$997$	−26738.9	−0.849378	−0.424689	+	0.905339i	$0.639617\pi$
$998$	0	0
$999$	−3093.08	−0.0979586

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.4.a.bu.1.1		2
4.3	odd	2		294.4.a.o.1.1	yes	2
7.6	odd	2		2352.4.a.bw.1.2		2
12.11	even	2		882.4.a.t.1.2		2
28.3	even	6		294.4.e.m.79.1		4
28.11	odd	6		294.4.e.k.79.2		4
28.19	even	6		294.4.e.m.67.1		4
28.23	odd	6		294.4.e.k.67.2		4
28.27	even	2		294.4.a.l.1.2	✓	2
84.11	even	6		882.4.g.bk.667.1		4
84.23	even	6		882.4.g.bk.361.1		4
84.47	odd	6		882.4.g.be.361.2		4
84.59	odd	6		882.4.g.be.667.2		4
84.83	odd	2		882.4.a.bb.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
294.4.a.l.1.2	✓	2	28.27	even	2
294.4.a.o.1.1	yes	2	4.3	odd	2
294.4.e.k.67.2		4	28.23	odd	6
294.4.e.k.79.2		4	28.11	odd	6
294.4.e.m.67.1		4	28.19	even	6
294.4.e.m.79.1		4	28.3	even	6
882.4.a.t.1.2		2	12.11	even	2
882.4.a.bb.1.1		2	84.83	odd	2
882.4.g.be.361.2		4	84.47	odd	6
882.4.g.be.667.2		4	84.59	odd	6
882.4.g.bk.361.1		4	84.23	even	6
882.4.g.bk.667.1		4	84.11	even	6
2352.4.a.bu.1.1		2	1.1	even	1	trivial
2352.4.a.bw.1.2		2	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2352.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +4.58579 q^{5} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} +4.58579 q^{5} +9.00000 q^{9} +6.48528 q^{11} +45.2132 q^{13} -13.7574 q^{15} +81.5563 q^{17} +5.05382 q^{19} -106.250 q^{23} -103.971 q^{25} -27.0000 q^{27} -268.132 q^{29} -292.368 q^{31} -19.4558 q^{33} +114.558 q^{37} -135.640 q^{39} -161.605 q^{41} +471.294 q^{43} +41.2721 q^{45} -346.004 q^{47} -244.669 q^{51} +405.529 q^{53} +29.7401 q^{55} -15.1615 q^{57} +253.436 q^{59} -751.217 q^{61} +207.338 q^{65} -11.6468 q^{67} +318.749 q^{69} +681.661 q^{71} +685.457 q^{73} +311.912 q^{75} -0.264069 q^{79} +81.0000 q^{81} +437.137 q^{83} +374.000 q^{85} +804.396 q^{87} -58.5126 q^{89} +877.103 q^{93} +23.1758 q^{95} -1280.09 q^{97} +58.3675 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} + 12 q^{5} + 18 q^{9}+O(q^{10}) \) 2 * q - 6 * q^3 + 12 * q^5 + 18 * q^9	\( 2 q - 6 q^{3} + 12 q^{5} + 18 q^{9} - 4 q^{11} + 48 q^{13} - 36 q^{15} + 132 q^{17} - 120 q^{19} + 76 q^{23} - 174 q^{25} - 54 q^{27} - 112 q^{29} - 432 q^{31} + 12 q^{33} - 280 q^{37} - 144 q^{39} + 36 q^{41} + 128 q^{43} + 108 q^{45} + 264 q^{47} - 396 q^{51} + 268 q^{53} - 48 q^{55} + 360 q^{57} - 336 q^{59} - 504 q^{61} + 228 q^{65} + 384 q^{67} - 228 q^{69} + 396 q^{71} - 312 q^{73} + 522 q^{75} + 848 q^{79} + 162 q^{81} + 648 q^{83} + 748 q^{85} + 336 q^{87} - 612 q^{89} + 1296 q^{93} - 904 q^{95} - 2184 q^{97} - 36 q^{99}+O(q^{100}) \) 2 * q - 6 * q^3 + 12 * q^5 + 18 * q^9 - 4 * q^11 + 48 * q^13 - 36 * q^15 + 132 * q^17 - 120 * q^19 + 76 * q^23 - 174 * q^25 - 54 * q^27 - 112 * q^29 - 432 * q^31 + 12 * q^33 - 280 * q^37 - 144 * q^39 + 36 * q^41 + 128 * q^43 + 108 * q^45 + 264 * q^47 - 396 * q^51 + 268 * q^53 - 48 * q^55 + 360 * q^57 - 336 * q^59 - 504 * q^61 + 228 * q^65 + 384 * q^67 - 228 * q^69 + 396 * q^71 - 312 * q^73 + 522 * q^75 + 848 * q^79 + 162 * q^81 + 648 * q^83 + 748 * q^85 + 336 * q^87 - 612 * q^89 + 1296 * q^93 - 904 * q^95 - 2184 * q^97 - 36 * q^99

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