LMFDB - Embedded newform 2312.4.a.k.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2312,4,Mod(1,2312)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2312, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2312.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2312 = 2^{3} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2312.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:	$136.412415933$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 95x^{6} + 756x^{4} - 1780x^{2} + 1152 $ x^8 - 95x^6 + 756x^4 - 1780*x^2 + 1152
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{13} $
Twist minimal:	no (minimal twist has level 136)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.60125$ of defining polynomial
Character	$\chi$	$=$	2312.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.95309 q^{3} -4.89575 q^{5} -4.46235 q^{7} -18.2793 q^{9} +O(q^{10})$	$q-2.95309 q^{3} -4.89575 q^{5} -4.46235 q^{7} -18.2793 q^{9} +60.3865 q^{11} -56.1938 q^{13} +14.4576 q^{15} +134.845 q^{19} +13.1777 q^{21} +39.1633 q^{23} -101.032 q^{25} +133.714 q^{27} +113.700 q^{29} -306.516 q^{31} -178.326 q^{33} +21.8465 q^{35} +61.9621 q^{37} +165.945 q^{39} -317.272 q^{41} -122.660 q^{43} +89.4907 q^{45} +303.233 q^{47} -323.087 q^{49} -133.743 q^{53} -295.637 q^{55} -398.210 q^{57} +130.660 q^{59} -772.618 q^{61} +81.5685 q^{63} +275.111 q^{65} +378.907 q^{67} -115.653 q^{69} +465.116 q^{71} -664.801 q^{73} +298.355 q^{75} -269.466 q^{77} -925.763 q^{79} +98.6724 q^{81} +723.561 q^{83} -335.766 q^{87} +889.439 q^{89} +250.757 q^{91} +905.170 q^{93} -660.168 q^{95} -1506.43 q^{97} -1103.82 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 132 q^{9}+O(q^{10}) $ 8 * q + 132 * q^9	$ 8 q + 132 q^{9} + 44 q^{13} - 24 q^{15} - 48 q^{19} + 308 q^{21} + 520 q^{25} + 812 q^{33} - 1064 q^{35} - 8 q^{43} + 312 q^{47} + 1124 q^{49} - 472 q^{53} + 1416 q^{55} + 72 q^{59} - 624 q^{67} - 180 q^{69} - 1660 q^{77} + 3156 q^{81} - 2472 q^{83} + 6664 q^{87} + 68 q^{89} + 4036 q^{93}+O(q^{100}) $ 8 * q + 132 * q^9 + 44 * q^13 - 24 * q^15 - 48 * q^19 + 308 * q^21 + 520 * q^25 + 812 * q^33 - 1064 * q^35 - 8 * q^43 + 312 * q^47 + 1124 * q^49 - 472 * q^53 + 1416 * q^55 + 72 * q^59 - 624 * q^67 - 180 * q^69 - 1660 * q^77 + 3156 * q^81 - 2472 * q^83 + 6664 * q^87 + 68 * q^89 + 4036 * q^93

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
$3$	−2.95309	−0.568322	−0.284161	−	0.958777i	$-0.591715\pi$
$3$	−2.95309	−0.568322	−0.284161	+	0.958777i	$0.591715\pi$
$4$	0	0
$5$	−4.89575	−0.437889	−0.218944	−	0.975737i	$-0.570261\pi$
$5$	−4.89575	−0.437889	−0.218944	+	0.975737i	$0.570261\pi$
$6$	0	0
$7$	−4.46235	−0.240944	−0.120472	−	0.992717i	$-0.538441\pi$
$7$	−4.46235	−0.240944	−0.120472	+	0.992717i	$0.538441\pi$
$8$	0	0
$9$	−18.2793	−0.677010
$10$	0	0
$11$	60.3865	1.65520	0.827600	−	0.561318i	$-0.189706\pi$
$11$	60.3865	1.65520	0.827600	+	0.561318i	$0.189706\pi$
$12$	0	0
$13$	−56.1938	−1.19887	−0.599437	−	0.800422i	$-0.704609\pi$
$13$	−56.1938	−1.19887	−0.599437	+	0.800422i	$0.704609\pi$
$14$	0	0
$15$	14.4576	0.248862
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	134.845	1.62819	0.814095	−	0.580731i	$-0.197233\pi$
$19$	134.845	1.62819	0.814095	+	0.580731i	$0.197233\pi$
$20$	0	0
$21$	13.1777	0.136934
$22$	0	0
$23$	39.1633	0.355049	0.177524	−	0.984116i	$-0.443191\pi$
$23$	39.1633	0.355049	0.177524	+	0.984116i	$0.443191\pi$
$24$	0	0
$25$	−101.032	−0.808253
$26$	0	0
$27$	133.714	0.953082
$28$	0	0
$29$	113.700	0.728053	0.364027	−	0.931389i	$-0.381402\pi$
$29$	113.700	0.728053	0.364027	+	0.931389i	$0.381402\pi$
$30$	0	0
$31$	−306.516	−1.77587	−0.887935	−	0.459969i	$-0.847861\pi$
$31$	−306.516	−1.77587	−0.887935	+	0.459969i	$0.847861\pi$
$32$	0	0
$33$	−178.326	−0.940686
$34$	0	0
$35$	21.8465	0.105507
$36$	0	0
$37$	61.9621	0.275311	0.137656	−	0.990480i	$-0.456043\pi$
$37$	61.9621	0.275311	0.137656	+	0.990480i	$0.456043\pi$
$38$	0	0
$39$	165.945	0.681346
$40$	0	0
$41$	−317.272	−1.20853	−0.604263	−	0.796785i	$-0.706532\pi$
$41$	−317.272	−1.20853	−0.604263	+	0.796785i	$0.706532\pi$
$42$	0	0
$43$	−122.660	−0.435009	−0.217505	−	0.976059i	$-0.569792\pi$
$43$	−122.660	−0.435009	−0.217505	+	0.976059i	$0.569792\pi$
$44$	0	0
$45$	89.4907	0.296455
$46$	0	0
$47$	303.233	0.941086	0.470543	−	0.882377i	$-0.344058\pi$
$47$	303.233	0.941086	0.470543	+	0.882377i	$0.344058\pi$
$48$	0	0
$49$	−323.087	−0.941946
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−133.743	−0.346623	−0.173311	−	0.984867i	$-0.555447\pi$
$53$	−133.743	−0.346623	−0.173311	+	0.984867i	$0.555447\pi$
$54$	0	0
$55$	−295.637	−0.724794
$56$	0	0
$57$	−398.210	−0.925336
$58$	0	0
$59$	130.660	0.288312	0.144156	−	0.989555i	$-0.453953\pi$
$59$	130.660	0.288312	0.144156	+	0.989555i	$0.453953\pi$
$60$	0	0
$61$	−772.618	−1.62170	−0.810849	−	0.585256i	$-0.800994\pi$
$61$	−772.618	−1.62170	−0.810849	+	0.585256i	$0.800994\pi$
$62$	0	0
$63$	81.5685	0.163122
$64$	0	0
$65$	275.111	0.524974
$66$	0	0
$67$	378.907	0.690909	0.345455	−	0.938435i	$-0.387725\pi$
$67$	378.907	0.690909	0.345455	+	0.938435i	$0.387725\pi$
$68$	0	0
$69$	−115.653	−0.201782
$70$	0	0
$71$	465.116	0.777452	0.388726	−	0.921353i	$-0.372915\pi$
$71$	465.116	0.777452	0.388726	+	0.921353i	$0.372915\pi$
$72$	0	0
$73$	−664.801	−1.06588	−0.532939	−	0.846154i	$-0.678913\pi$
$73$	−664.801	−1.06588	−0.532939	+	0.846154i	$0.678913\pi$
$74$	0	0
$75$	298.355	0.459348
$76$	0	0
$77$	−269.466	−0.398811
$78$	0	0
$79$	−925.763	−1.31844	−0.659218	−	0.751952i	$-0.729113\pi$
$79$	−925.763	−1.31844	−0.659218	+	0.751952i	$0.729113\pi$
$80$	0	0
$81$	98.6724	0.135353
$82$	0	0
$83$	723.561	0.956882	0.478441	−	0.878120i	$-0.341202\pi$
$83$	723.561	0.956882	0.478441	+	0.878120i	$0.341202\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−335.766	−0.413769
$88$	0	0
$89$	889.439	1.05933	0.529665	−	0.848207i	$-0.322318\pi$
$89$	889.439	1.05933	0.529665	+	0.848207i	$0.322318\pi$
$90$	0	0
$91$	250.757	0.288862
$92$	0	0
$93$	905.170	1.00927
$94$	0	0
$95$	−660.168	−0.712967
$96$	0	0
$97$	−1506.43	−1.57686	−0.788428	−	0.615127i	$-0.789105\pi$
$97$	−1506.43	−1.57686	−0.788428	+	0.615127i	$0.789105\pi$
$98$	0	0
$99$	−1103.82	−1.12059
$100$	0	0
$101$	606.815	0.597825	0.298912	−	0.954281i	$-0.403376\pi$
$101$	606.815	0.597825	0.298912	+	0.954281i	$0.403376\pi$
$102$	0	0
$103$	582.633	0.557364	0.278682	−	0.960383i	$-0.410102\pi$
$103$	582.633	0.557364	0.278682	+	0.960383i	$0.410102\pi$
$104$	0	0
$105$	−64.5147	−0.0599619
$106$	0	0
$107$	−14.1057	−0.0127444	−0.00637221	−	0.999980i	$-0.502028\pi$
$107$	−14.1057	−0.0127444	−0.00637221	+	0.999980i	$0.502028\pi$
$108$	0	0
$109$	1599.48	1.40553	0.702765	−	0.711422i	$-0.251949\pi$
$109$	1599.48	1.40553	0.702765	+	0.711422i	$0.251949\pi$
$110$	0	0
$111$	−182.980	−0.156465
$112$	0	0
$113$	−620.655	−0.516693	−0.258347	−	0.966052i	$-0.583178\pi$
$113$	−620.655	−0.516693	−0.258347	+	0.966052i	$0.583178\pi$
$114$	0	0
$115$	−191.734	−0.155472
$116$	0	0
$117$	1027.18	0.811650
$118$	0	0
$119$	0	0
$120$	0	0
$121$	2315.52	1.73969
$122$	0	0
$123$	936.932	0.686832
$124$	0	0
$125$	1106.59	0.791814
$126$	0	0
$127$	970.396	0.678021	0.339011	−	0.940783i	$-0.389908\pi$
$127$	970.396	0.678021	0.339011	+	0.940783i	$0.389908\pi$
$128$	0	0
$129$	362.224	0.247225
$130$	0	0
$131$	−705.717	−0.470678	−0.235339	−	0.971913i	$-0.575620\pi$
$131$	−705.717	−0.470678	−0.235339	+	0.971913i	$0.575620\pi$
$132$	0	0
$133$	−601.727	−0.392303
$134$	0	0
$135$	−654.628	−0.417344
$136$	0	0
$137$	−2470.01	−1.54035	−0.770173	−	0.637835i	$-0.779830\pi$
$137$	−2470.01	−1.54035	−0.770173	+	0.637835i	$0.779830\pi$
$138$	0	0
$139$	−435.484	−0.265736	−0.132868	−	0.991134i	$-0.542419\pi$
$139$	−435.484	−0.265736	−0.132868	+	0.991134i	$0.542419\pi$
$140$	0	0
$141$	−895.473	−0.534840
$142$	0	0
$143$	−3393.35	−1.98438
$144$	0	0
$145$	−556.646	−0.318806
$146$	0	0
$147$	954.105	0.535328
$148$	0	0
$149$	2428.83	1.33542	0.667709	−	0.744422i	$-0.267275\pi$
$149$	2428.83	1.33542	0.667709	+	0.744422i	$0.267275\pi$
$150$	0	0
$151$	−2608.09	−1.40558	−0.702792	−	0.711396i	$-0.748063\pi$
$151$	−2608.09	−1.40558	−0.702792	+	0.711396i	$0.748063\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1500.63	0.777634
$156$	0	0
$157$	164.006	0.0833703	0.0416851	−	0.999131i	$-0.486727\pi$
$157$	164.006	0.0833703	0.0416851	+	0.999131i	$0.486727\pi$
$158$	0	0
$159$	394.955	0.196993
$160$	0	0
$161$	−174.760	−0.0855469
$162$	0	0
$163$	289.416	0.139072	0.0695362	−	0.997579i	$-0.477848\pi$
$163$	289.416	0.139072	0.0695362	+	0.997579i	$0.477848\pi$
$164$	0	0
$165$	873.041	0.411916
$166$	0	0
$167$	2147.52	0.995090	0.497545	−	0.867438i	$-0.334235\pi$
$167$	2147.52	0.995090	0.497545	+	0.867438i	$0.334235\pi$
$168$	0	0
$169$	960.747	0.437299
$170$	0	0
$171$	−2464.87	−1.10230
$172$	0	0
$173$	4383.99	1.92664	0.963320	−	0.268357i	$-0.0864805\pi$
$173$	4383.99	1.92664	0.963320	+	0.268357i	$0.0864805\pi$
$174$	0	0
$175$	450.839	0.194744
$176$	0	0
$177$	−385.849	−0.163854
$178$	0	0
$179$	−3793.10	−1.58385	−0.791926	−	0.610617i	$-0.790921\pi$
$179$	−3793.10	−1.58385	−0.791926	+	0.610617i	$0.790921\pi$
$180$	0	0
$181$	1726.75	0.709108	0.354554	−	0.935036i	$-0.384633\pi$
$181$	1726.75	0.709108	0.354554	+	0.935036i	$0.384633\pi$
$182$	0	0
$183$	2281.61	0.921646
$184$	0	0
$185$	−303.351	−0.120556
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−596.677	−0.229640
$190$	0	0
$191$	1286.01	0.487184	0.243592	−	0.969878i	$-0.421674\pi$
$191$	1286.01	0.487184	0.243592	+	0.969878i	$0.421674\pi$
$192$	0	0
$193$	−1682.07	−0.627349	−0.313674	−	0.949531i	$-0.601560\pi$
$193$	−1682.07	−0.627349	−0.313674	+	0.949531i	$0.601560\pi$
$194$	0	0
$195$	−812.426	−0.298354
$196$	0	0
$197$	415.578	0.150298	0.0751489	−	0.997172i	$-0.476057\pi$
$197$	415.578	0.150298	0.0751489	+	0.997172i	$0.476057\pi$
$198$	0	0
$199$	2761.31	0.983637	0.491819	−	0.870698i	$-0.336332\pi$
$199$	2761.31	0.983637	0.491819	+	0.870698i	$0.336332\pi$
$200$	0	0
$201$	−1118.95	−0.392659
$202$	0	0
$203$	−507.369	−0.175420
$204$	0	0
$205$	1553.28	0.529200
$206$	0	0
$207$	−715.877	−0.240372
$208$	0	0
$209$	8142.83	2.69498
$210$	0	0
$211$	−2713.80	−0.885431	−0.442715	−	0.896662i	$-0.645985\pi$
$211$	−2713.80	−0.885431	−0.442715	+	0.896662i	$0.645985\pi$
$212$	0	0
$213$	−1373.53	−0.441843
$214$	0	0
$215$	600.510	0.190486
$216$	0	0
$217$	1367.78	0.427886
$218$	0	0
$219$	1963.22	0.605762
$220$	0	0
$221$	0	0
$222$	0	0
$223$	6395.76	1.92059	0.960296	−	0.278984i	$-0.0899978\pi$
$223$	6395.76	1.92059	0.960296	+	0.278984i	$0.0899978\pi$
$224$	0	0
$225$	1846.79	0.547196
$226$	0	0
$227$	3844.30	1.12403	0.562016	−	0.827127i	$-0.310026\pi$
$227$	3844.30	1.12403	0.562016	+	0.827127i	$0.310026\pi$
$228$	0	0
$229$	4849.35	1.39936	0.699682	−	0.714455i	$-0.253325\pi$
$229$	4849.35	1.39936	0.699682	+	0.714455i	$0.253325\pi$
$230$	0	0
$231$	795.755	0.226653
$232$	0	0
$233$	−2719.86	−0.764738	−0.382369	−	0.924010i	$-0.624892\pi$
$233$	−2719.86	−0.764738	−0.382369	+	0.924010i	$0.624892\pi$
$234$	0	0
$235$	−1484.55	−0.412091
$236$	0	0
$237$	2733.86	0.749296
$238$	0	0
$239$	2524.92	0.683362	0.341681	−	0.939816i	$-0.389004\pi$
$239$	2524.92	0.683362	0.341681	+	0.939816i	$0.389004\pi$
$240$	0	0
$241$	−1943.29	−0.519413	−0.259706	−	0.965688i	$-0.583626\pi$
$241$	−1943.29	−0.519413	−0.259706	+	0.965688i	$0.583626\pi$
$242$	0	0
$243$	−3901.66	−1.03001
$244$	0	0
$245$	1581.75	0.412468
$246$	0	0
$247$	−7577.47	−1.95200
$248$	0	0
$249$	−2136.74	−0.543817
$250$	0	0
$251$	2332.58	0.586578	0.293289	−	0.956024i	$-0.405250\pi$
$251$	2332.58	0.586578	0.293289	+	0.956024i	$0.405250\pi$
$252$	0	0
$253$	2364.93	0.587676
$254$	0	0
$255$	0	0
$256$	0	0
$257$	807.810	0.196069	0.0980346	−	0.995183i	$-0.468744\pi$
$257$	807.810	0.196069	0.0980346	+	0.995183i	$0.468744\pi$
$258$	0	0
$259$	−276.497	−0.0663347
$260$	0	0
$261$	−2078.35	−0.492900
$262$	0	0
$263$	6420.82	1.50542	0.752708	−	0.658354i	$-0.228747\pi$
$263$	6420.82	1.50542	0.752708	+	0.658354i	$0.228747\pi$
$264$	0	0
$265$	654.772	0.151782
$266$	0	0
$267$	−2626.59	−0.602040
$268$	0	0
$269$	6335.06	1.43589	0.717947	−	0.696097i	$-0.245082\pi$
$269$	6335.06	1.43589	0.717947	+	0.696097i	$0.245082\pi$
$270$	0	0
$271$	−1429.35	−0.320394	−0.160197	−	0.987085i	$-0.551213\pi$
$271$	−1429.35	−0.320394	−0.160197	+	0.987085i	$0.551213\pi$
$272$	0	0
$273$	−740.506	−0.164167
$274$	0	0
$275$	−6100.94	−1.33782
$276$	0	0
$277$	−7878.89	−1.70901	−0.854506	−	0.519441i	$-0.826140\pi$
$277$	−7878.89	−1.70901	−0.854506	+	0.519441i	$0.826140\pi$
$278$	0	0
$279$	5602.90	1.20228
$280$	0	0
$281$	−2967.99	−0.630090	−0.315045	−	0.949077i	$-0.602020\pi$
$281$	−2967.99	−0.630090	−0.315045	+	0.949077i	$0.602020\pi$
$282$	0	0
$283$	401.204	0.0842725	0.0421363	−	0.999112i	$-0.486584\pi$
$283$	401.204	0.0842725	0.0421363	+	0.999112i	$0.486584\pi$
$284$	0	0
$285$	1949.53	0.405195
$286$	0	0
$287$	1415.78	0.291188
$288$	0	0
$289$	0	0
$290$	0	0
$291$	4448.63	0.896162
$292$	0	0
$293$	7390.62	1.47360	0.736800	−	0.676111i	$-0.236336\pi$
$293$	7390.62	1.47360	0.736800	+	0.676111i	$0.236336\pi$
$294$	0	0
$295$	−639.676	−0.126249
$296$	0	0
$297$	8074.49	1.57754
$298$	0	0
$299$	−2200.74	−0.425659
$300$	0	0
$301$	547.350	0.104813
$302$	0	0
$303$	−1791.98	−0.339757
$304$	0	0
$305$	3782.54	0.710123
$306$	0	0
$307$	−50.2688	−0.00934524	−0.00467262	−	0.999989i	$-0.501487\pi$
$307$	−50.2688	−0.00934524	−0.00467262	+	0.999989i	$0.501487\pi$
$308$	0	0
$309$	−1720.57	−0.316762
$310$	0	0
$311$	−215.028	−0.0392062	−0.0196031	−	0.999808i	$-0.506240\pi$
$311$	−215.028	−0.0392062	−0.0196031	+	0.999808i	$0.506240\pi$
$312$	0	0
$313$	4639.52	0.837831	0.418915	−	0.908025i	$-0.362410\pi$
$313$	4639.52	0.837831	0.418915	+	0.908025i	$0.362410\pi$
$314$	0	0
$315$	−399.339	−0.0714292
$316$	0	0
$317$	3965.63	0.702624	0.351312	−	0.936259i	$-0.385736\pi$
$317$	3965.63	0.702624	0.351312	+	0.936259i	$0.385736\pi$
$318$	0	0
$319$	6865.94	1.20507
$320$	0	0
$321$	41.6554	0.00724293
$322$	0	0
$323$	0	0
$324$	0	0
$325$	5677.36	0.968994
$326$	0	0
$327$	−4723.41	−0.798793
$328$	0	0
$329$	−1353.13	−0.226749
$330$	0	0
$331$	−7497.94	−1.24509	−0.622544	−	0.782585i	$-0.713901\pi$
$331$	−7497.94	−1.24509	−0.622544	+	0.782585i	$0.713901\pi$
$332$	0	0
$333$	−1132.62	−0.186388
$334$	0	0
$335$	−1855.03	−0.302541
$336$	0	0
$337$	798.500	0.129071	0.0645357	−	0.997915i	$-0.479443\pi$
$337$	798.500	0.129071	0.0645357	+	0.997915i	$0.479443\pi$
$338$	0	0
$339$	1832.85	0.293648
$340$	0	0
$341$	−18509.4	−2.93942
$342$	0	0
$343$	2972.32	0.467901
$344$	0	0
$345$	566.206	0.0883581
$346$	0	0
$347$	4014.43	0.621054	0.310527	−	0.950564i	$-0.399494\pi$
$347$	4014.43	0.621054	0.310527	+	0.950564i	$0.399494\pi$
$348$	0	0
$349$	−4747.53	−0.728165	−0.364083	−	0.931367i	$-0.618617\pi$
$349$	−4747.53	−0.728165	−0.364083	+	0.931367i	$0.618617\pi$
$350$	0	0
$351$	−7513.88	−1.14262
$352$	0	0
$353$	9653.02	1.45546	0.727731	−	0.685862i	$-0.240575\pi$
$353$	9653.02	1.45546	0.727731	+	0.685862i	$0.240575\pi$
$354$	0	0
$355$	−2277.09	−0.340438
$356$	0	0
$357$	0	0
$358$	0	0
$359$	6641.04	0.976325	0.488162	−	0.872753i	$-0.337667\pi$
$359$	6641.04	0.976325	0.488162	+	0.872753i	$0.337667\pi$
$360$	0	0
$361$	11324.2	1.65100
$362$	0	0
$363$	−6837.94	−0.988703
$364$	0	0
$365$	3254.70	0.466736
$366$	0	0
$367$	3305.96	0.470217	0.235108	−	0.971969i	$-0.424455\pi$
$367$	3305.96	0.470217	0.235108	+	0.971969i	$0.424455\pi$
$368$	0	0
$369$	5799.50	0.818185
$370$	0	0
$371$	596.808	0.0835168
$372$	0	0
$373$	238.635	0.0331261	0.0165631	−	0.999863i	$-0.494728\pi$
$373$	238.635	0.0331261	0.0165631	+	0.999863i	$0.494728\pi$
$374$	0	0
$375$	−3267.87	−0.450005
$376$	0	0
$377$	−6389.23	−0.872844
$378$	0	0
$379$	11081.6	1.50191	0.750955	−	0.660353i	$-0.229593\pi$
$379$	11081.6	1.50191	0.750955	+	0.660353i	$0.229593\pi$
$380$	0	0
$381$	−2865.66	−0.385334
$382$	0	0
$383$	−7956.45	−1.06150	−0.530752	−	0.847527i	$-0.678090\pi$
$383$	−7956.45	−1.06150	−0.530752	+	0.847527i	$0.678090\pi$
$384$	0	0
$385$	1319.24	0.174635
$386$	0	0
$387$	2242.13	0.294506
$388$	0	0
$389$	−5128.06	−0.668388	−0.334194	−	0.942504i	$-0.608464\pi$
$389$	−5128.06	−0.668388	−0.334194	+	0.942504i	$0.608464\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	2084.04	0.267496
$394$	0	0
$395$	4532.30	0.577329
$396$	0	0
$397$	−2933.96	−0.370910	−0.185455	−	0.982653i	$-0.559376\pi$
$397$	−2933.96	−0.370910	−0.185455	+	0.982653i	$0.559376\pi$
$398$	0	0
$399$	1776.95	0.222955
$400$	0	0
$401$	5278.07	0.657293	0.328646	−	0.944453i	$-0.393408\pi$
$401$	5278.07	0.657293	0.328646	+	0.944453i	$0.393408\pi$
$402$	0	0
$403$	17224.3	2.12904
$404$	0	0
$405$	−483.075	−0.0592696
$406$	0	0
$407$	3741.67	0.455695
$408$	0	0
$409$	−6880.00	−0.831770	−0.415885	−	0.909417i	$-0.636528\pi$
$409$	−6880.00	−0.831770	−0.415885	+	0.909417i	$0.636528\pi$
$410$	0	0
$411$	7294.16	0.875412
$412$	0	0
$413$	−583.049	−0.0694672
$414$	0	0
$415$	−3542.37	−0.419008
$416$	0	0
$417$	1286.02	0.151024
$418$	0	0
$419$	−3048.43	−0.355431	−0.177715	−	0.984082i	$-0.556871\pi$
$419$	−3048.43	−0.355431	−0.177715	+	0.984082i	$0.556871\pi$
$420$	0	0
$421$	3607.83	0.417660	0.208830	−	0.977952i	$-0.433034\pi$
$421$	3607.83	0.417660	0.208830	+	0.977952i	$0.433034\pi$
$422$	0	0
$423$	−5542.88	−0.637125
$424$	0	0
$425$	0	0
$426$	0	0
$427$	3447.69	0.390739
$428$	0	0
$429$	10020.8	1.12776
$430$	0	0
$431$	8866.93	0.990962	0.495481	−	0.868619i	$-0.334992\pi$
$431$	8866.93	0.990962	0.495481	+	0.868619i	$0.334992\pi$
$432$	0	0
$433$	11353.2	1.26005	0.630024	−	0.776575i	$-0.283045\pi$
$433$	11353.2	1.26005	0.630024	+	0.776575i	$0.283045\pi$
$434$	0	0
$435$	1643.82	0.181185
$436$	0	0
$437$	5280.99	0.578087
$438$	0	0
$439$	2786.53	0.302947	0.151474	−	0.988461i	$-0.451598\pi$
$439$	2786.53	0.302947	0.151474	+	0.988461i	$0.451598\pi$
$440$	0	0
$441$	5905.80	0.637707
$442$	0	0
$443$	15244.4	1.63495	0.817474	−	0.575965i	$-0.195374\pi$
$443$	15244.4	1.63495	0.817474	+	0.575965i	$0.195374\pi$
$444$	0	0
$445$	−4354.47	−0.463869
$446$	0	0
$447$	−7172.54	−0.758947
$448$	0	0
$449$	−17473.1	−1.83654	−0.918269	−	0.395958i	$-0.870413\pi$
$449$	−17473.1	−1.83654	−0.918269	+	0.395958i	$0.870413\pi$
$450$	0	0
$451$	−19158.9	−2.00035
$452$	0	0
$453$	7701.91	0.798824
$454$	0	0
$455$	−1227.64	−0.126489
$456$	0	0
$457$	16422.8	1.68102	0.840508	−	0.541800i	$-0.182257\pi$
$457$	16422.8	1.68102	0.840508	+	0.541800i	$0.182257\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	12802.9	1.29347	0.646737	−	0.762713i	$-0.276133\pi$
$461$	12802.9	1.29347	0.646737	+	0.762713i	$0.276133\pi$
$462$	0	0
$463$	−2291.00	−0.229961	−0.114980	−	0.993368i	$-0.536680\pi$
$463$	−2291.00	−0.229961	−0.114980	+	0.993368i	$0.536680\pi$
$464$	0	0
$465$	−4431.48	−0.441946
$466$	0	0
$467$	14464.0	1.43322	0.716610	−	0.697474i	$-0.245693\pi$
$467$	14464.0	1.43322	0.716610	+	0.697474i	$0.245693\pi$
$468$	0	0
$469$	−1690.82	−0.166471
$470$	0	0
$471$	−484.325	−0.0473812
$472$	0	0
$473$	−7406.97	−0.720028
$474$	0	0
$475$	−13623.6	−1.31599
$476$	0	0
$477$	2444.72	0.234667
$478$	0	0
$479$	−6988.15	−0.666590	−0.333295	−	0.942823i	$-0.608161\pi$
$479$	−6988.15	−0.666590	−0.333295	+	0.942823i	$0.608161\pi$
$480$	0	0
$481$	−3481.89	−0.330063
$482$	0	0
$483$	516.083	0.0486182
$484$	0	0
$485$	7375.11	0.690488
$486$	0	0
$487$	8494.76	0.790420	0.395210	−	0.918591i	$-0.370672\pi$
$487$	8494.76	0.790420	0.395210	+	0.918591i	$0.370672\pi$
$488$	0	0
$489$	−854.670	−0.0790379
$490$	0	0
$491$	5195.82	0.477565	0.238782	−	0.971073i	$-0.423252\pi$
$491$	5195.82	0.477565	0.238782	+	0.971073i	$0.423252\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	5404.03	0.490693
$496$	0	0
$497$	−2075.51	−0.187323
$498$	0	0
$499$	−8790.68	−0.788627	−0.394313	−	0.918976i	$-0.629018\pi$
$499$	−8790.68	−0.788627	−0.394313	+	0.918976i	$0.629018\pi$
$500$	0	0
$501$	−6341.81	−0.565531
$502$	0	0
$503$	7764.74	0.688295	0.344148	−	0.938916i	$-0.388168\pi$
$503$	7764.74	0.688295	0.344148	+	0.938916i	$0.388168\pi$
$504$	0	0
$505$	−2970.81	−0.261781
$506$	0	0
$507$	−2837.17	−0.248527
$508$	0	0
$509$	9919.29	0.863782	0.431891	−	0.901926i	$-0.357847\pi$
$509$	9919.29	0.863782	0.431891	+	0.901926i	$0.357847\pi$
$510$	0	0
$511$	2966.58	0.256817
$512$	0	0
$513$	18030.6	1.55180
$514$	0	0
$515$	−2852.42	−0.244064
$516$	0	0
$517$	18311.2	1.55769
$518$	0	0
$519$	−12946.3	−1.09495
$520$	0	0
$521$	4230.44	0.355737	0.177868	−	0.984054i	$-0.443080\pi$
$521$	4230.44	0.355737	0.177868	+	0.984054i	$0.443080\pi$
$522$	0	0
$523$	10566.8	0.883465	0.441732	−	0.897147i	$-0.354364\pi$
$523$	10566.8	0.883465	0.441732	+	0.897147i	$0.354364\pi$
$524$	0	0
$525$	−1331.37	−0.110677
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−10633.2	−0.873941
$530$	0	0
$531$	−2388.36	−0.195190
$532$	0	0
$533$	17828.7	1.44887
$534$	0	0
$535$	69.0581	0.00558064
$536$	0	0
$537$	11201.3	0.900138
$538$	0	0
$539$	−19510.1	−1.55911
$540$	0	0
$541$	1980.86	0.157419	0.0787094	−	0.996898i	$-0.474920\pi$
$541$	1980.86	0.157419	0.0787094	+	0.996898i	$0.474920\pi$
$542$	0	0
$543$	−5099.25	−0.403001
$544$	0	0
$545$	−7830.67	−0.615466
$546$	0	0
$547$	−3940.22	−0.307992	−0.153996	−	0.988071i	$-0.549214\pi$
$547$	−3940.22	−0.307992	−0.153996	+	0.988071i	$0.549214\pi$
$548$	0	0
$549$	14122.9	1.09791
$550$	0	0
$551$	15331.9	1.18541
$552$	0	0
$553$	4131.08	0.317670
$554$	0	0
$555$	895.822	0.0685145
$556$	0	0
$557$	3709.12	0.282155	0.141077	−	0.989999i	$-0.454943\pi$
$557$	3709.12	0.282155	0.141077	+	0.989999i	$0.454943\pi$
$558$	0	0
$559$	6892.71	0.521522
$560$	0	0
$561$	0	0
$562$	0	0
$563$	3393.34	0.254018	0.127009	−	0.991902i	$-0.459462\pi$
$563$	3393.34	0.254018	0.127009	+	0.991902i	$0.459462\pi$
$564$	0	0
$565$	3038.57	0.226254
$566$	0	0
$567$	−440.311	−0.0326125
$568$	0	0
$569$	2927.65	0.215700	0.107850	−	0.994167i	$-0.465603\pi$
$569$	2927.65	0.215700	0.107850	+	0.994167i	$0.465603\pi$
$570$	0	0
$571$	−19781.8	−1.44981	−0.724907	−	0.688846i	$-0.758117\pi$
$571$	−19781.8	−1.44981	−0.724907	+	0.688846i	$0.758117\pi$
$572$	0	0
$573$	−3797.69	−0.276877
$574$	0	0
$575$	−3956.74	−0.286969
$576$	0	0
$577$	17520.1	1.26407	0.632037	−	0.774938i	$-0.282219\pi$
$577$	17520.1	1.26407	0.632037	+	0.774938i	$0.282219\pi$
$578$	0	0
$579$	4967.31	0.356536
$580$	0	0
$581$	−3228.79	−0.230555
$582$	0	0
$583$	−8076.26	−0.573730
$584$	0	0
$585$	−5028.83	−0.355413
$586$	0	0
$587$	−7846.90	−0.551748	−0.275874	−	0.961194i	$-0.588967\pi$
$587$	−7846.90	−0.551748	−0.275874	+	0.961194i	$0.588967\pi$
$588$	0	0
$589$	−41332.3	−2.89145
$590$	0	0
$591$	−1227.24	−0.0854175
$592$	0	0
$593$	−10572.9	−0.732173	−0.366086	−	0.930581i	$-0.619303\pi$
$593$	−10572.9	−0.732173	−0.366086	+	0.930581i	$0.619303\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−8154.38	−0.559023
$598$	0	0
$599$	−27156.5	−1.85240	−0.926198	−	0.377038i	$-0.876943\pi$
$599$	−27156.5	−1.85240	−0.926198	+	0.377038i	$0.876943\pi$
$600$	0	0
$601$	11180.6	0.758849	0.379424	−	0.925223i	$-0.376122\pi$
$601$	11180.6	0.758849	0.379424	+	0.925223i	$0.376122\pi$
$602$	0	0
$603$	−6926.15	−0.467752
$604$	0	0
$605$	−11336.2	−0.761790
$606$	0	0
$607$	−19444.2	−1.30019	−0.650097	−	0.759851i	$-0.725272\pi$
$607$	−19444.2	−1.30019	−0.650097	+	0.759851i	$0.725272\pi$
$608$	0	0
$609$	1498.30	0.0996952
$610$	0	0
$611$	−17039.8	−1.12824
$612$	0	0
$613$	−16630.9	−1.09579	−0.547893	−	0.836548i	$-0.684570\pi$
$613$	−16630.9	−1.09579	−0.547893	+	0.836548i	$0.684570\pi$
$614$	0	0
$615$	−4586.98	−0.300756
$616$	0	0
$617$	−8035.03	−0.524276	−0.262138	−	0.965030i	$-0.584428\pi$
$617$	−8035.03	−0.524276	−0.262138	+	0.965030i	$0.584428\pi$
$618$	0	0
$619$	−16161.6	−1.04942	−0.524709	−	0.851282i	$-0.675826\pi$
$619$	−16161.6	−1.04942	−0.524709	+	0.851282i	$0.675826\pi$
$620$	0	0
$621$	5236.67	0.338390
$622$	0	0
$623$	−3968.99	−0.255239
$624$	0	0
$625$	7211.35	0.461527
$626$	0	0
$627$	−24046.5	−1.53162
$628$	0	0
$629$	0	0
$630$	0	0
$631$	10043.9	0.633665	0.316833	−	0.948481i	$-0.397381\pi$
$631$	10043.9	0.633665	0.316833	+	0.948481i	$0.397381\pi$
$632$	0	0
$633$	8014.09	0.503210
$634$	0	0
$635$	−4750.81	−0.296898
$636$	0	0
$637$	18155.5	1.12927
$638$	0	0
$639$	−8501.98	−0.526343
$640$	0	0
$641$	15316.1	0.943757	0.471879	−	0.881664i	$-0.343576\pi$
$641$	15316.1	0.943757	0.471879	+	0.881664i	$0.343576\pi$
$642$	0	0
$643$	24684.6	1.51394	0.756970	−	0.653449i	$-0.226679\pi$
$643$	24684.6	1.51394	0.756970	+	0.653449i	$0.226679\pi$
$644$	0	0
$645$	−1773.36	−0.108257
$646$	0	0
$647$	7774.11	0.472383	0.236192	−	0.971707i	$-0.424101\pi$
$647$	7774.11	0.472383	0.236192	+	0.971707i	$0.424101\pi$
$648$	0	0
$649$	7890.07	0.477214
$650$	0	0
$651$	−4039.18	−0.243177
$652$	0	0
$653$	−752.623	−0.0451032	−0.0225516	−	0.999746i	$-0.507179\pi$
$653$	−752.623	−0.0451032	−0.0225516	+	0.999746i	$0.507179\pi$
$654$	0	0
$655$	3455.01	0.206105
$656$	0	0
$657$	12152.1	0.721611
$658$	0	0
$659$	−4887.68	−0.288918	−0.144459	−	0.989511i	$-0.546144\pi$
$659$	−4887.68	−0.288918	−0.144459	+	0.989511i	$0.546144\pi$
$660$	0	0
$661$	7122.91	0.419136	0.209568	−	0.977794i	$-0.432794\pi$
$661$	7122.91	0.419136	0.209568	+	0.977794i	$0.432794\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2945.90	0.171785
$666$	0	0
$667$	4452.87	0.258494
$668$	0	0
$669$	−18887.2	−1.09151
$670$	0	0
$671$	−46655.6	−2.68423
$672$	0	0
$673$	−32249.6	−1.84715	−0.923574	−	0.383421i	$-0.874746\pi$
$673$	−32249.6	−1.84715	−0.923574	+	0.383421i	$0.874746\pi$
$674$	0	0
$675$	−13509.3	−0.770331
$676$	0	0
$677$	−17447.9	−0.990511	−0.495255	−	0.868748i	$-0.664925\pi$
$677$	−17447.9	−0.990511	−0.495255	+	0.868748i	$0.664925\pi$
$678$	0	0
$679$	6722.23	0.379934
$680$	0	0
$681$	−11352.6	−0.638812
$682$	0	0
$683$	13935.3	0.780701	0.390351	−	0.920666i	$-0.372354\pi$
$683$	13935.3	0.780701	0.390351	+	0.920666i	$0.372354\pi$
$684$	0	0
$685$	12092.6	0.674500
$686$	0	0
$687$	−14320.6	−0.795289
$688$	0	0
$689$	7515.53	0.415557
$690$	0	0
$691$	22358.6	1.23092	0.615458	−	0.788170i	$-0.288971\pi$
$691$	22358.6	1.23092	0.615458	+	0.788170i	$0.288971\pi$
$692$	0	0
$693$	4925.64	0.269999
$694$	0	0
$695$	2132.02	0.116363
$696$	0	0
$697$	0	0
$698$	0	0
$699$	8031.99	0.434617
$700$	0	0
$701$	30456.5	1.64098	0.820491	−	0.571660i	$-0.193700\pi$
$701$	30456.5	1.64098	0.820491	+	0.571660i	$0.193700\pi$
$702$	0	0
$703$	8355.30	0.448259
$704$	0	0
$705$	4384.01	0.234201
$706$	0	0
$707$	−2707.82	−0.144042
$708$	0	0
$709$	−36945.7	−1.95702	−0.978508	−	0.206210i	$-0.933887\pi$
$709$	−36945.7	−1.95702	−0.978508	+	0.206210i	$0.933887\pi$
$710$	0	0
$711$	16922.3	0.892595
$712$	0	0
$713$	−12004.2	−0.630520
$714$	0	0
$715$	16613.0	0.868937
$716$	0	0
$717$	−7456.31	−0.388369
$718$	0	0
$719$	12442.4	0.645375	0.322687	−	0.946506i	$-0.395414\pi$
$719$	12442.4	0.645375	0.322687	+	0.946506i	$0.395414\pi$
$720$	0	0
$721$	−2599.91	−0.134294
$722$	0	0
$723$	5738.71	0.295194
$724$	0	0
$725$	−11487.3	−0.588451
$726$	0	0
$727$	−29943.3	−1.52756	−0.763780	−	0.645477i	$-0.776659\pi$
$727$	−29943.3	−1.52756	−0.763780	+	0.645477i	$0.776659\pi$
$728$	0	0
$729$	8857.78	0.450022
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−19615.5	−0.988423	−0.494212	−	0.869342i	$-0.664543\pi$
$733$	−19615.5	−0.988423	−0.494212	+	0.869342i	$0.664543\pi$
$734$	0	0
$735$	−4671.06	−0.234414
$736$	0	0
$737$	22880.9	1.14359
$738$	0	0
$739$	3459.26	0.172194	0.0860968	−	0.996287i	$-0.472561\pi$
$739$	3459.26	0.172194	0.0860968	+	0.996287i	$0.472561\pi$
$740$	0	0
$741$	22376.9	1.10936
$742$	0	0
$743$	37392.9	1.84632	0.923158	−	0.384421i	$-0.125599\pi$
$743$	37392.9	1.84632	0.923158	+	0.384421i	$0.125599\pi$
$744$	0	0
$745$	−11890.9	−0.584765
$746$	0	0
$747$	−13226.2	−0.647819
$748$	0	0
$749$	62.9447	0.00307069
$750$	0	0
$751$	34379.7	1.67048	0.835241	−	0.549884i	$-0.185328\pi$
$751$	34379.7	1.67048	0.835241	+	0.549884i	$0.185328\pi$
$752$	0	0
$753$	−6888.31	−0.333365
$754$	0	0
$755$	12768.5	0.615489
$756$	0	0
$757$	20120.9	0.966057	0.483028	−	0.875605i	$-0.339537\pi$
$757$	20120.9	0.966057	0.483028	+	0.875605i	$0.339537\pi$
$758$	0	0
$759$	−6983.86	−0.333989
$760$	0	0
$761$	−21739.7	−1.03556	−0.517781	−	0.855513i	$-0.673242\pi$
$761$	−21739.7	−1.03556	−0.517781	+	0.855513i	$0.673242\pi$
$762$	0	0
$763$	−7137.46	−0.338654
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−7342.26	−0.345650
$768$	0	0
$769$	26884.9	1.26072	0.630360	−	0.776303i	$-0.282907\pi$
$769$	26884.9	1.26072	0.630360	+	0.776303i	$0.282907\pi$
$770$	0	0
$771$	−2385.53	−0.111430
$772$	0	0
$773$	21927.0	1.02026	0.510128	−	0.860099i	$-0.329598\pi$
$773$	21927.0	1.02026	0.510128	+	0.860099i	$0.329598\pi$
$774$	0	0
$775$	30967.9	1.43535
$776$	0	0
$777$	816.519	0.0376994
$778$	0	0
$779$	−42782.6	−1.96771
$780$	0	0
$781$	28086.7	1.28684
$782$	0	0
$783$	15203.2	0.693894
$784$	0	0
$785$	−802.934	−0.0365069
$786$	0	0
$787$	28401.4	1.28641	0.643203	−	0.765695i	$-0.277605\pi$
$787$	28401.4	1.28641	0.643203	+	0.765695i	$0.277605\pi$
$788$	0	0
$789$	−18961.2	−0.855561
$790$	0	0
$791$	2769.58	0.124494
$792$	0	0
$793$	43416.3	1.94421
$794$	0	0
$795$	−1933.60	−0.0862612
$796$	0	0
$797$	19462.7	0.864999	0.432499	−	0.901634i	$-0.357632\pi$
$797$	19462.7	0.864999	0.432499	+	0.901634i	$0.357632\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−16258.3	−0.717177
$802$	0	0
$803$	−40145.0	−1.76424
$804$	0	0
$805$	855.583	0.0374601
$806$	0	0
$807$	−18708.0	−0.816050
$808$	0	0
$809$	11073.0	0.481220	0.240610	−	0.970622i	$-0.422653\pi$
$809$	11073.0	0.481220	0.240610	+	0.970622i	$0.422653\pi$
$810$	0	0
$811$	8512.57	0.368578	0.184289	−	0.982872i	$-0.441002\pi$
$811$	8512.57	0.368578	0.184289	+	0.982872i	$0.441002\pi$
$812$	0	0
$813$	4220.99	0.182087
$814$	0	0
$815$	−1416.91	−0.0608983
$816$	0	0
$817$	−16540.1	−0.708278
$818$	0	0
$819$	−4583.65	−0.195562
$820$	0	0
$821$	−26291.2	−1.11762	−0.558812	−	0.829294i	$-0.688743\pi$
$821$	−26291.2	−1.11762	−0.558812	+	0.829294i	$0.688743\pi$
$822$	0	0
$823$	−4379.37	−0.185486	−0.0927432	−	0.995690i	$-0.529564\pi$
$823$	−4379.37	−0.185486	−0.0927432	+	0.995690i	$0.529564\pi$
$824$	0	0
$825$	18016.6	0.760313
$826$	0	0
$827$	−2994.98	−0.125932	−0.0629659	−	0.998016i	$-0.520056\pi$
$827$	−2994.98	−0.125932	−0.0629659	+	0.998016i	$0.520056\pi$
$828$	0	0
$829$	−33497.1	−1.40338	−0.701690	−	0.712482i	$-0.747571\pi$
$829$	−33497.1	−1.40338	−0.701690	+	0.712482i	$0.747571\pi$
$830$	0	0
$831$	23267.0	0.971269
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−10513.7	−0.435739
$836$	0	0
$837$	−40985.4	−1.69255
$838$	0	0
$839$	−25588.9	−1.05295	−0.526477	−	0.850189i	$-0.676487\pi$
$839$	−25588.9	−1.05295	−0.526477	+	0.850189i	$0.676487\pi$
$840$	0	0
$841$	−11461.3	−0.469938
$842$	0	0
$843$	8764.73	0.358094
$844$	0	0
$845$	−4703.57	−0.191489
$846$	0	0
$847$	−10332.7	−0.419168
$848$	0	0
$849$	−1184.79	−0.0478939
$850$	0	0
$851$	2426.64	0.0977488
$852$	0	0
$853$	−31609.1	−1.26879	−0.634394	−	0.773010i	$-0.718750\pi$
$853$	−31609.1	−1.26879	−0.634394	+	0.773010i	$0.718750\pi$
$854$	0	0
$855$	12067.4	0.482686
$856$	0	0
$857$	−7519.34	−0.299715	−0.149858	−	0.988708i	$-0.547881\pi$
$857$	−7519.34	−0.299715	−0.149858	+	0.988708i	$0.547881\pi$
$858$	0	0
$859$	3923.30	0.155834	0.0779169	−	0.996960i	$-0.475173\pi$
$859$	3923.30	0.155834	0.0779169	+	0.996960i	$0.475173\pi$
$860$	0	0
$861$	−4180.92	−0.165488
$862$	0	0
$863$	32254.8	1.27227	0.636133	−	0.771580i	$-0.280533\pi$
$863$	32254.8	1.27227	0.636133	+	0.771580i	$0.280533\pi$
$864$	0	0
$865$	−21462.9	−0.843654
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−55903.5	−2.18228
$870$	0	0
$871$	−21292.3	−0.828313
$872$	0	0
$873$	27536.5	1.06755
$874$	0	0
$875$	−4938.01	−0.190783
$876$	0	0
$877$	−9587.97	−0.369171	−0.184585	−	0.982816i	$-0.559094\pi$
$877$	−9587.97	−0.369171	−0.184585	+	0.982816i	$0.559094\pi$
$878$	0	0
$879$	−21825.2	−0.837479
$880$	0	0
$881$	10243.5	0.391726	0.195863	−	0.980631i	$-0.437249\pi$
$881$	10243.5	0.391726	0.195863	+	0.980631i	$0.437249\pi$
$882$	0	0
$883$	−17802.2	−0.678474	−0.339237	−	0.940701i	$-0.610169\pi$
$883$	−17802.2	−0.678474	−0.339237	+	0.940701i	$0.610169\pi$
$884$	0	0
$885$	1889.02	0.0717499
$886$	0	0
$887$	−5409.67	−0.204779	−0.102389	−	0.994744i	$-0.532649\pi$
$887$	−5409.67	−0.204779	−0.102389	+	0.994744i	$0.532649\pi$
$888$	0	0
$889$	−4330.25	−0.163365
$890$	0	0
$891$	5958.47	0.224036
$892$	0	0
$893$	40889.5	1.53227
$894$	0	0
$895$	18570.1	0.693551
$896$	0	0
$897$	6498.97	0.241911
$898$	0	0
$899$	−34850.9	−1.29293
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−1616.37	−0.0595675
$904$	0	0
$905$	−8453.74	−0.310510
$906$	0	0
$907$	27596.6	1.01029	0.505143	−	0.863036i	$-0.331440\pi$
$907$	27596.6	1.01029	0.505143	+	0.863036i	$0.331440\pi$
$908$	0	0
$909$	−11092.1	−0.404733
$910$	0	0
$911$	−47306.9	−1.72047	−0.860235	−	0.509898i	$-0.829683\pi$
$911$	−47306.9	−1.72047	−0.860235	+	0.509898i	$0.829683\pi$
$912$	0	0
$913$	43693.3	1.58383
$914$	0	0
$915$	−11170.2	−0.403579
$916$	0	0
$917$	3149.16	0.113407
$918$	0	0
$919$	40635.5	1.45859	0.729293	−	0.684202i	$-0.239849\pi$
$919$	40635.5	1.45859	0.729293	+	0.684202i	$0.239849\pi$
$920$	0	0
$921$	148.448	0.00531111
$922$	0	0
$923$	−26136.6	−0.932067
$924$	0	0
$925$	−6260.14	−0.222521
$926$	0	0
$927$	−10650.1	−0.377341
$928$	0	0
$929$	−34143.2	−1.20582	−0.602908	−	0.797811i	$-0.705991\pi$
$929$	−34143.2	−1.20582	−0.602908	+	0.797811i	$0.705991\pi$
$930$	0	0
$931$	−43566.8	−1.53367
$932$	0	0
$933$	634.997	0.0222817
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−4097.67	−0.142865	−0.0714327	−	0.997445i	$-0.522757\pi$
$937$	−4097.67	−0.142865	−0.0714327	+	0.997445i	$0.522757\pi$
$938$	0	0
$939$	−13700.9	−0.476157
$940$	0	0
$941$	−32801.4	−1.13634	−0.568169	−	0.822912i	$-0.692348\pi$
$941$	−32801.4	−1.13634	−0.568169	+	0.822912i	$0.692348\pi$
$942$	0	0
$943$	−12425.4	−0.429086
$944$	0	0
$945$	2921.18	0.100557
$946$	0	0
$947$	32528.3	1.11618	0.558092	−	0.829779i	$-0.311534\pi$
$947$	32528.3	1.11618	0.558092	+	0.829779i	$0.311534\pi$
$948$	0	0
$949$	37357.7	1.27785
$950$	0	0
$951$	−11710.8	−0.399316
$952$	0	0
$953$	14335.6	0.487276	0.243638	−	0.969866i	$-0.421659\pi$
$953$	14335.6	0.487276	0.243638	+	0.969866i	$0.421659\pi$
$954$	0	0
$955$	−6295.96	−0.213332
$956$	0	0
$957$	−20275.7	−0.684870
$958$	0	0
$959$	11022.1	0.371138
$960$	0	0
$961$	64161.3	2.15371
$962$	0	0
$963$	257.842	0.00862810
$964$	0	0
$965$	8235.01	0.274709
$966$	0	0
$967$	688.081	0.0228823	0.0114412	−	0.999935i	$-0.496358\pi$
$967$	688.081	0.0228823	0.0114412	+	0.999935i	$0.496358\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−49985.2	−1.65201	−0.826005	−	0.563663i	$-0.809391\pi$
$971$	−49985.2	−1.65201	−0.826005	+	0.563663i	$0.809391\pi$
$972$	0	0
$973$	1943.28	0.0640276
$974$	0	0
$975$	−16765.7	−0.550701
$976$	0	0
$977$	−38752.4	−1.26899	−0.634493	−	0.772929i	$-0.718791\pi$
$977$	−38752.4	−1.26899	−0.634493	+	0.772929i	$0.718791\pi$
$978$	0	0
$979$	53710.1	1.75340
$980$	0	0
$981$	−29237.4	−0.951558
$982$	0	0
$983$	21516.4	0.698135	0.349067	−	0.937098i	$-0.386498\pi$
$983$	21516.4	0.698135	0.349067	+	0.937098i	$0.386498\pi$
$984$	0	0
$985$	−2034.56	−0.0658138
$986$	0	0
$987$	3995.92	0.128867
$988$	0	0
$989$	−4803.75	−0.154449
$990$	0	0
$991$	6941.00	0.222491	0.111245	−	0.993793i	$-0.464516\pi$
$991$	6941.00	0.222491	0.111245	+	0.993793i	$0.464516\pi$
$992$	0	0
$993$	22142.1	0.707611
$994$	0	0
$995$	−13518.7	−0.430724
$996$	0	0
$997$	−44135.5	−1.40199	−0.700996	−	0.713165i	$-0.747261\pi$
$997$	−44135.5	−1.40199	−0.700996	+	0.713165i	$0.747261\pi$
$998$	0	0
$999$	8285.18	0.262394

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2312.4.a.k.1.3		8
17.4	even	4		136.4.b.b.33.6	yes	8
17.13	even	4		136.4.b.b.33.3	✓	8
17.16	even	2	inner	2312.4.a.k.1.6		8
51.38	odd	4		1224.4.c.e.577.4		8
51.47	odd	4		1224.4.c.e.577.5		8
68.47	odd	4		272.4.b.f.33.6		8
68.55	odd	4		272.4.b.f.33.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.4.b.b.33.3	✓	8	17.13	even	4
136.4.b.b.33.6	yes	8	17.4	even	4
272.4.b.f.33.3		8	68.55	odd	4
272.4.b.f.33.6		8	68.47	odd	4
1224.4.c.e.577.4		8	51.38	odd	4
1224.4.c.e.577.5		8	51.47	odd	4
2312.4.a.k.1.3		8	1.1	even	1	trivial
2312.4.a.k.1.6		8	17.16	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2312 = 2^{3} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2312.a (trivial)

\(f(q)\)	\(=\)	\(q-2.95309 q^{3} -4.89575 q^{5} -4.46235 q^{7} -18.2793 q^{9} +O(q^{10})\)	\(q-2.95309 q^{3} -4.89575 q^{5} -4.46235 q^{7} -18.2793 q^{9} +60.3865 q^{11} -56.1938 q^{13} +14.4576 q^{15} +134.845 q^{19} +13.1777 q^{21} +39.1633 q^{23} -101.032 q^{25} +133.714 q^{27} +113.700 q^{29} -306.516 q^{31} -178.326 q^{33} +21.8465 q^{35} +61.9621 q^{37} +165.945 q^{39} -317.272 q^{41} -122.660 q^{43} +89.4907 q^{45} +303.233 q^{47} -323.087 q^{49} -133.743 q^{53} -295.637 q^{55} -398.210 q^{57} +130.660 q^{59} -772.618 q^{61} +81.5685 q^{63} +275.111 q^{65} +378.907 q^{67} -115.653 q^{69} +465.116 q^{71} -664.801 q^{73} +298.355 q^{75} -269.466 q^{77} -925.763 q^{79} +98.6724 q^{81} +723.561 q^{83} -335.766 q^{87} +889.439 q^{89} +250.757 q^{91} +905.170 q^{93} -660.168 q^{95} -1506.43 q^{97} -1103.82 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 132 q^{9}+O(q^{10}) \) 8 * q + 132 * q^9	\( 8 q + 132 q^{9} + 44 q^{13} - 24 q^{15} - 48 q^{19} + 308 q^{21} + 520 q^{25} + 812 q^{33} - 1064 q^{35} - 8 q^{43} + 312 q^{47} + 1124 q^{49} - 472 q^{53} + 1416 q^{55} + 72 q^{59} - 624 q^{67} - 180 q^{69} - 1660 q^{77} + 3156 q^{81} - 2472 q^{83} + 6664 q^{87} + 68 q^{89} + 4036 q^{93}+O(q^{100}) \) 8 * q + 132 * q^9 + 44 * q^13 - 24 * q^15 - 48 * q^19 + 308 * q^21 + 520 * q^25 + 812 * q^33 - 1064 * q^35 - 8 * q^43 + 312 * q^47 + 1124 * q^49 - 472 * q^53 + 1416 * q^55 + 72 * q^59 - 624 * q^67 - 180 * q^69 - 1660 * q^77 + 3156 * q^81 - 2472 * q^83 + 6664 * q^87 + 68 * q^89 + 4036 * q^93

Introduction

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Modular forms

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