LMFDB - Embedded newform 1900.4.c.b.1749.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1900,4,Mod(1749,1900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1900, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1900.1749");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1900 = 2^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1900.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$112.103629011$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{33})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 17x^{2} + 64 $ x^4 + 17*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 5^{2} $
Twist minimal:	no (minimal twist has level 76)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1749.1
Root			$-3.37228i$ of defining polynomial
Character	$\chi$	$=$	1900.1749
Dual form			1900.4.c.b.1749.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.37228i q^{3} +26.4891i q^{7} -1.86141 q^{9} +O(q^{10})$	$q-5.37228i q^{3} +26.4891i q^{7} -1.86141 q^{9} -49.8614 q^{11} -49.0951i q^{13} -17.2337i q^{17} +19.0000 q^{19} +142.307 q^{21} +166.965i q^{23} -135.052i q^{27} +109.198 q^{29} -273.783 q^{31} +267.870i q^{33} -167.022i q^{37} -263.753 q^{39} +15.1684 q^{41} +413.557i q^{43} +161.438i q^{47} -358.674 q^{49} -92.5842 q^{51} -490.791i q^{53} -102.073i q^{57} +335.970 q^{59} +725.149 q^{61} -49.3070i q^{63} -497.709i q^{67} +896.981 q^{69} -798.554 q^{71} -311.174i q^{73} -1320.79i q^{77} +665.723 q^{79} -775.793 q^{81} -372.440i q^{83} -586.644i q^{87} +673.783 q^{89} +1300.49 q^{91} +1470.84i q^{93} +960.505i q^{97} +92.8124 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 50 q^{9}+O(q^{10}) $ 4 * q + 50 * q^9	$ 4 q + 50 q^{9} - 142 q^{11} + 76 q^{19} + 282 q^{21} - 310 q^{29} - 176 q^{31} - 538 q^{39} - 284 q^{41} - 56 q^{49} - 198 q^{51} + 1746 q^{59} + 890 q^{61} + 1922 q^{69} - 3424 q^{71} + 2548 q^{79} - 116 q^{81} + 1776 q^{89} + 2502 q^{91} - 950 q^{99}+O(q^{100}) $ 4 * q + 50 * q^9 - 142 * q^11 + 76 * q^19 + 282 * q^21 - 310 * q^29 - 176 * q^31 - 538 * q^39 - 284 * q^41 - 56 * q^49 - 198 * q^51 + 1746 * q^59 + 890 * q^61 + 1922 * q^69 - 3424 * q^71 + 2548 * q^79 - 116 * q^81 + 1776 * q^89 + 2502 * q^91 - 950 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times$.

$n$	$77$	$401$	$951$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	− 5.37228i	− 1.03390i	−0.856017	−	0.516948i	$-0.827068\pi$
$3$	− 5.37228i	− 1.03390i	0.856017	−	0.516948i	$-0.172932\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	26.4891i	1.43028i	0.698982	+	0.715139i	$0.253637\pi$
$7$	26.4891i	1.43028i	−0.698982	+	0.715139i	$0.746363\pi$
$8$	0	0
$9$	−1.86141	−0.0689410
$10$	0	0
$11$	−49.8614	−1.36671	−0.683354	−	0.730088i	$-0.739479\pi$
$11$	−49.8614	−1.36671	−0.683354	+	0.730088i	$0.739479\pi$
$12$	0	0
$13$	− 49.0951i	− 1.04743i	−0.851895	−	0.523713i	$-0.824547\pi$
$13$	− 49.0951i	− 1.04743i	0.851895	−	0.523713i	$-0.175453\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 17.2337i	− 0.245870i	−0.992415	−	0.122935i	$-0.960769\pi$
$17$	− 17.2337i	− 0.245870i	0.992415	−	0.122935i	$-0.0392306\pi$
$18$	0	0
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	0	0
$21$	142.307	1.47876
$22$	0	0
$23$	166.965i	1.51368i	0.653603	+	0.756838i	$0.273257\pi$
$23$	166.965i	1.51368i	−0.653603	+	0.756838i	$0.726743\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 135.052i	− 0.962618i
$28$	0	0
$29$	109.198	0.699228	0.349614	−	0.936894i	$-0.386313\pi$
$29$	109.198	0.699228	0.349614	+	0.936894i	$0.386313\pi$
$30$	0	0
$31$	−273.783	−1.58622	−0.793110	−	0.609079i	$-0.791539\pi$
$31$	−273.783	−1.58622	−0.793110	+	0.609079i	$0.791539\pi$
$32$	0	0
$33$	267.870i	1.41303i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 167.022i	− 0.742114i	−0.928610	−	0.371057i	$-0.878995\pi$
$37$	− 167.022i	− 0.742114i	0.928610	−	0.371057i	$-0.121005\pi$
$38$	0	0
$39$	−263.753	−1.08293
$40$	0	0
$41$	15.1684	0.0577783	0.0288892	−	0.999583i	$-0.490803\pi$
$41$	15.1684	0.0577783	0.0288892	+	0.999583i	$0.490803\pi$
$42$	0	0
$43$	413.557i	1.46667i	0.679867	+	0.733335i	$0.262037\pi$
$43$	413.557i	1.46667i	−0.679867	+	0.733335i	$0.737963\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	161.438i	0.501023i	0.968114	+	0.250512i	$0.0805988\pi$
$47$	161.438i	0.501023i	−0.968114	+	0.250512i	$0.919401\pi$
$48$	0	0
$49$	−358.674	−1.04570
$50$	0	0
$51$	−92.5842	−0.254204
$52$	0	0
$53$	− 490.791i	− 1.27199i	−0.771695	−	0.635993i	$-0.780591\pi$
$53$	− 490.791i	− 1.27199i	0.771695	−	0.635993i	$-0.219409\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 102.073i	− 0.237192i
$58$	0	0
$59$	335.970	0.741349	0.370674	−	0.928763i	$-0.379127\pi$
$59$	335.970	0.741349	0.370674	+	0.928763i	$0.379127\pi$
$60$	0	0
$61$	725.149	1.52206	0.761032	−	0.648715i	$-0.224693\pi$
$61$	725.149	1.52206	0.761032	+	0.648715i	$0.224693\pi$
$62$	0	0
$63$	− 49.3070i	− 0.0986048i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 497.709i	− 0.907535i	−0.891120	−	0.453768i	$-0.850080\pi$
$67$	− 497.709i	− 0.907535i	0.891120	−	0.453768i	$-0.149920\pi$
$68$	0	0
$69$	896.981	1.56498
$70$	0	0
$71$	−798.554	−1.33480	−0.667401	−	0.744698i	$-0.732593\pi$
$71$	−798.554	−1.33480	−0.667401	+	0.744698i	$0.732593\pi$
$72$	0	0
$73$	− 311.174i	− 0.498906i	−0.968387	−	0.249453i	$-0.919749\pi$
$73$	− 311.174i	− 0.498906i	0.968387	−	0.249453i	$-0.0802509\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 1320.79i	− 1.95477i
$78$	0	0
$79$	665.723	0.948097	0.474049	−	0.880499i	$-0.342792\pi$
$79$	665.723	0.948097	0.474049	+	0.880499i	$0.342792\pi$
$80$	0	0
$81$	−775.793	−1.06419
$82$	0	0
$83$	− 372.440i	− 0.492537i	−0.969202	−	0.246269i	$-0.920795\pi$
$83$	− 372.440i	− 0.492537i	0.969202	−	0.246269i	$-0.0792046\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 586.644i	− 0.722929i
$88$	0	0
$89$	673.783	0.802481	0.401240	−	0.915973i	$-0.368579\pi$
$89$	673.783	0.802481	0.401240	+	0.915973i	$0.368579\pi$
$90$	0	0
$91$	1300.49	1.49811
$92$	0	0
$93$	1470.84i	1.63999i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	960.505i	1.00541i	0.864459	+	0.502704i	$0.167661\pi$
$97$	960.505i	1.00541i	−0.864459	+	0.502704i	$0.832339\pi$
$98$	0	0
$99$	92.8124	0.0942221
$100$	0	0
$101$	1889.68	1.86169	0.930845	−	0.365415i	$-0.119073\pi$
$101$	1889.68	1.86169	0.930845	+	0.365415i	$0.119073\pi$
$102$	0	0
$103$	760.217i	0.727247i	0.931546	+	0.363624i	$0.118461\pi$
$103$	760.217i	0.727247i	−0.931546	+	0.363624i	$0.881539\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 681.730i	− 0.615938i	−0.951396	−	0.307969i	$-0.900351\pi$
$107$	− 681.730i	− 0.615938i	0.951396	−	0.307969i	$-0.0996493\pi$
$108$	0	0
$109$	1310.76	1.15182	0.575910	−	0.817513i	$-0.304648\pi$
$109$	1310.76	1.15182	0.575910	+	0.817513i	$0.304648\pi$
$110$	0	0
$111$	−897.288	−0.767268
$112$	0	0
$113$	− 713.413i	− 0.593914i	−0.954891	−	0.296957i	$-0.904028\pi$
$113$	− 713.413i	− 0.593914i	0.954891	−	0.296957i	$-0.0959718\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	91.3859i	0.0722105i
$118$	0	0
$119$	456.505	0.351662
$120$	0	0
$121$	1155.16	0.867889
$122$	0	0
$123$	− 81.4891i	− 0.0597368i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	107.847i	0.0753535i	0.999290	+	0.0376768i	$0.0119957\pi$
$127$	107.847i	0.0753535i	−0.999290	+	0.0376768i	$0.988004\pi$
$128$	0	0
$129$	2221.74	1.51638
$130$	0	0
$131$	373.247	0.248937	0.124469	−	0.992224i	$-0.460277\pi$
$131$	373.247	0.248937	0.124469	+	0.992224i	$0.460277\pi$
$132$	0	0
$133$	503.293i	0.328128i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3118.25i	1.94460i	0.233732	+	0.972301i	$0.424906\pi$
$137$	3118.25i	1.94460i	−0.233732	+	0.972301i	$0.575094\pi$
$138$	0	0
$139$	56.3774	0.0344019	0.0172010	−	0.999852i	$-0.494524\pi$
$139$	56.3774	0.0344019	0.0172010	+	0.999852i	$0.494524\pi$
$140$	0	0
$141$	867.288	0.518006
$142$	0	0
$143$	2447.95i	1.43152i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1926.90i	1.08114i
$148$	0	0
$149$	1810.51	0.995457	0.497728	−	0.867333i	$-0.334168\pi$
$149$	1810.51	0.995457	0.497728	+	0.867333i	$0.334168\pi$
$150$	0	0
$151$	2894.72	1.56006	0.780029	−	0.625744i	$-0.215204\pi$
$151$	2894.72	1.56006	0.780029	+	0.625744i	$0.215204\pi$
$152$	0	0
$153$	32.0789i	0.0169505i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 1381.71i	− 0.702371i	−0.936306	−	0.351185i	$-0.885779\pi$
$157$	− 1381.71i	− 0.702371i	0.936306	−	0.351185i	$-0.114221\pi$
$158$	0	0
$159$	−2636.67	−1.31510
$160$	0	0
$161$	−4422.75	−2.16498
$162$	0	0
$163$	− 3740.83i	− 1.79757i	−0.438389	−	0.898785i	$-0.644451\pi$
$163$	− 3740.83i	− 1.79757i	0.438389	−	0.898785i	$-0.355549\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 3085.17i	− 1.42957i	−0.699347	−	0.714783i	$-0.746526\pi$
$167$	− 3085.17i	− 1.42957i	0.699347	−	0.714783i	$-0.253474\pi$
$168$	0	0
$169$	−213.328	−0.0970998
$170$	0	0
$171$	−35.3667	−0.0158161
$172$	0	0
$173$	293.000i	0.128765i	0.997925	+	0.0643827i	$0.0205078\pi$
$173$	293.000i	0.128765i	−0.997925	+	0.0643827i	$0.979492\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 1804.93i	− 0.766478i
$178$	0	0
$179$	2996.32	1.25115	0.625574	−	0.780165i	$-0.284865\pi$
$179$	2996.32	1.25115	0.625574	+	0.780165i	$0.284865\pi$
$180$	0	0
$181$	−3265.41	−1.34097	−0.670487	−	0.741922i	$-0.733915\pi$
$181$	−3265.41	−1.34097	−0.670487	+	0.741922i	$0.733915\pi$
$182$	0	0
$183$	− 3895.71i	− 1.57365i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	859.296i	0.336032i
$188$	0	0
$189$	3577.40	1.37681
$190$	0	0
$191$	−1502.27	−0.569113	−0.284556	−	0.958659i	$-0.591846\pi$
$191$	−1502.27	−0.569113	−0.284556	+	0.958659i	$0.591846\pi$
$192$	0	0
$193$	− 4103.30i	− 1.53037i	−0.643809	−	0.765186i	$-0.722647\pi$
$193$	− 4103.30i	− 1.53037i	0.643809	−	0.765186i	$-0.277353\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 5420.95i	− 1.96054i	−0.197658	−	0.980271i	$-0.563334\pi$
$197$	− 5420.95i	− 1.96054i	0.197658	−	0.980271i	$-0.436666\pi$
$198$	0	0
$199$	−1666.05	−0.593484	−0.296742	−	0.954958i	$-0.595900\pi$
$199$	−1666.05	−0.593484	−0.296742	+	0.954958i	$0.595900\pi$
$200$	0	0
$201$	−2673.83	−0.938297
$202$	0	0
$203$	2892.57i	1.00009i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 310.789i	− 0.104354i
$208$	0	0
$209$	−947.367	−0.313544
$210$	0	0
$211$	5865.90	1.91386	0.956931	−	0.290314i	$-0.0937598\pi$
$211$	5865.90	1.91386	0.956931	+	0.290314i	$0.0937598\pi$
$212$	0	0
$213$	4290.06i	1.38005i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 7252.26i	− 2.26873i
$218$	0	0
$219$	−1671.71	−0.515817
$220$	0	0
$221$	−846.090	−0.257530
$222$	0	0
$223$	− 5402.94i	− 1.62246i	−0.584729	−	0.811229i	$-0.698799\pi$
$223$	− 5402.94i	− 1.62246i	0.584729	−	0.811229i	$-0.301201\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2571.34i	0.751833i	0.926654	+	0.375916i	$0.122672\pi$
$227$	2571.34i	0.751833i	−0.926654	+	0.375916i	$0.877328\pi$
$228$	0	0
$229$	−3256.08	−0.939597	−0.469799	−	0.882774i	$-0.655673\pi$
$229$	−3256.08	−0.939597	−0.469799	+	0.882774i	$0.655673\pi$
$230$	0	0
$231$	−7095.63	−2.02103
$232$	0	0
$233$	1205.58i	0.338972i	0.985533	+	0.169486i	$0.0542107\pi$
$233$	1205.58i	0.338972i	−0.985533	+	0.169486i	$0.945789\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 3576.45i	− 0.980234i
$238$	0	0
$239$	3404.52	0.921424	0.460712	−	0.887550i	$-0.347594\pi$
$239$	3404.52	0.921424	0.460712	+	0.887550i	$0.347594\pi$
$240$	0	0
$241$	3301.51	0.882443	0.441221	−	0.897398i	$-0.354545\pi$
$241$	3301.51	0.882443	0.441221	+	0.897398i	$0.354545\pi$
$242$	0	0
$243$	521.386i	0.137642i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 932.807i	− 0.240296i
$248$	0	0
$249$	−2000.85	−0.509233
$250$	0	0
$251$	−5330.81	−1.34055	−0.670275	−	0.742113i	$-0.733824\pi$
$251$	−5330.81	−1.34055	−0.670275	+	0.742113i	$0.733824\pi$
$252$	0	0
$253$	− 8325.09i	− 2.06875i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 3812.14i	− 0.925272i	−0.886548	−	0.462636i	$-0.846904\pi$
$257$	− 3812.14i	− 0.925272i	0.886548	−	0.462636i	$-0.153096\pi$
$258$	0	0
$259$	4424.26	1.06143
$260$	0	0
$261$	−203.262	−0.0482055
$262$	0	0
$263$	899.159i	0.210816i	0.994429	+	0.105408i	$0.0336148\pi$
$263$	899.159i	0.210816i	−0.994429	+	0.105408i	$0.966385\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 3619.75i	− 0.829682i
$268$	0	0
$269$	−2645.77	−0.599685	−0.299842	−	0.953989i	$-0.596934\pi$
$269$	−2645.77	−0.599685	−0.299842	+	0.953989i	$0.596934\pi$
$270$	0	0
$271$	516.529	0.115782	0.0578909	−	0.998323i	$-0.481562\pi$
$271$	516.529	0.115782	0.0578909	+	0.998323i	$0.481562\pi$
$272$	0	0
$273$	− 6986.58i	− 1.54889i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 2365.14i	− 0.513023i	−0.966541	−	0.256512i	$-0.917427\pi$
$277$	− 2365.14i	− 0.513023i	0.966541	−	0.256512i	$-0.0825732\pi$
$278$	0	0
$279$	509.621	0.109356
$280$	0	0
$281$	−1526.54	−0.324077	−0.162038	−	0.986784i	$-0.551807\pi$
$281$	−1526.54	−0.324077	−0.162038	+	0.986784i	$0.551807\pi$
$282$	0	0
$283$	4764.01i	1.00067i	0.865831	+	0.500337i	$0.166791\pi$
$283$	4764.01i	1.00067i	−0.865831	+	0.500337i	$0.833209\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	401.799i	0.0826391i
$288$	0	0
$289$	4616.00	0.939548
$290$	0	0
$291$	5160.10	1.03949
$292$	0	0
$293$	3189.07i	0.635862i	0.948114	+	0.317931i	$0.102988\pi$
$293$	3189.07i	0.635862i	−0.948114	+	0.317931i	$0.897012\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	6733.86i	1.31562i
$298$	0	0
$299$	8197.14	1.58546
$300$	0	0
$301$	−10954.8	−2.09775
$302$	0	0
$303$	− 10151.9i	− 1.92479i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 5998.56i	− 1.11517i	−0.830121	−	0.557583i	$-0.811729\pi$
$307$	− 5998.56i	− 1.11517i	0.830121	−	0.557583i	$-0.188271\pi$
$308$	0	0
$309$	4084.10	0.751898
$310$	0	0
$311$	5114.96	0.932614	0.466307	−	0.884623i	$-0.345584\pi$
$311$	5114.96	0.932614	0.466307	+	0.884623i	$0.345584\pi$
$312$	0	0
$313$	2131.63i	0.384942i	0.981303	+	0.192471i	$0.0616501\pi$
$313$	2131.63i	0.384942i	−0.981303	+	0.192471i	$0.938350\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 4457.03i	− 0.789691i	−0.918748	−	0.394845i	$-0.870798\pi$
$317$	− 4457.03i	− 0.789691i	0.918748	−	0.394845i	$-0.129202\pi$
$318$	0	0
$319$	−5444.78	−0.955640
$320$	0	0
$321$	−3662.45	−0.636816
$322$	0	0
$323$	− 327.440i	− 0.0564064i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 7041.79i	− 1.19086i
$328$	0	0
$329$	−4276.34	−0.716602
$330$	0	0
$331$	−4143.68	−0.688088	−0.344044	−	0.938954i	$-0.611797\pi$
$331$	−4143.68	−0.688088	−0.344044	+	0.938954i	$0.611797\pi$
$332$	0	0
$333$	310.895i	0.0511620i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	5148.60i	0.832231i	0.909312	+	0.416115i	$0.136609\pi$
$337$	5148.60i	0.832231i	−0.909312	+	0.416115i	$0.863391\pi$
$338$	0	0
$339$	−3832.66	−0.614045
$340$	0	0
$341$	13651.2	2.16790
$342$	0	0
$343$	− 415.184i	− 0.0653581i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	5873.56i	0.908672i	0.890830	+	0.454336i	$0.150123\pi$
$347$	5873.56i	0.908672i	−0.890830	+	0.454336i	$0.849877\pi$
$348$	0	0
$349$	−5167.58	−0.792590	−0.396295	−	0.918123i	$-0.629704\pi$
$349$	−5167.58	−0.792590	−0.396295	+	0.918123i	$0.629704\pi$
$350$	0	0
$351$	−6630.37	−1.00827
$352$	0	0
$353$	7191.17i	1.08427i	0.840291	+	0.542135i	$0.182384\pi$
$353$	7191.17i	1.08427i	−0.840291	+	0.542135i	$0.817616\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 2452.47i	− 0.363582i
$358$	0	0
$359$	7049.62	1.03639	0.518196	−	0.855262i	$-0.326604\pi$
$359$	7049.62	1.03639	0.518196	+	0.855262i	$0.326604\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	0	0
$363$	− 6205.84i	− 0.897307i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 6935.53i	− 0.986463i	−0.869898	−	0.493231i	$-0.835816\pi$
$367$	− 6935.53i	− 0.986463i	0.869898	−	0.493231i	$-0.164184\pi$
$368$	0	0
$369$	−28.2346	−0.00398330
$370$	0	0
$371$	13000.6	1.81929
$372$	0	0
$373$	− 6346.92i	− 0.881048i	−0.897741	−	0.440524i	$-0.854793\pi$
$373$	− 6346.92i	− 0.881048i	0.897741	−	0.440524i	$-0.145207\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 5361.10i	− 0.732389i
$378$	0	0
$379$	−11363.0	−1.54004	−0.770022	−	0.638017i	$-0.779755\pi$
$379$	−11363.0	−1.54004	−0.770022	+	0.638017i	$0.779755\pi$
$380$	0	0
$381$	579.386	0.0779077
$382$	0	0
$383$	− 3848.09i	− 0.513390i	−0.966493	−	0.256695i	$-0.917366\pi$
$383$	− 3848.09i	− 0.513390i	0.966493	−	0.256695i	$-0.0826335\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 769.798i	− 0.101114i
$388$	0	0
$389$	−1936.75	−0.252435	−0.126217	−	0.992003i	$-0.540284\pi$
$389$	−1936.75	−0.252435	−0.126217	+	0.992003i	$0.540284\pi$
$390$	0	0
$391$	2877.42	0.372167
$392$	0	0
$393$	− 2005.19i	− 0.257375i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 3851.40i	− 0.486892i	−0.969914	−	0.243446i	$-0.921722\pi$
$397$	− 3851.40i	− 0.486892i	0.969914	−	0.243446i	$-0.0782778\pi$
$398$	0	0
$399$	2703.83	0.339251
$400$	0	0
$401$	−1381.47	−0.172038	−0.0860189	−	0.996294i	$-0.527415\pi$
$401$	−1381.47	−0.172038	−0.0860189	+	0.996294i	$0.527415\pi$
$402$	0	0
$403$	13441.4i	1.66145i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	8327.94i	1.01425i
$408$	0	0
$409$	−368.878	−0.0445962	−0.0222981	−	0.999751i	$-0.507098\pi$
$409$	−368.878	−0.0445962	−0.0222981	+	0.999751i	$0.507098\pi$
$410$	0	0
$411$	16752.1	2.01052
$412$	0	0
$413$	8899.56i	1.06034i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 302.875i	− 0.0355680i
$418$	0	0
$419$	1814.85	0.211602	0.105801	−	0.994387i	$-0.466259\pi$
$419$	1814.85	0.211602	0.105801	+	0.994387i	$0.466259\pi$
$420$	0	0
$421$	4614.01	0.534141	0.267070	−	0.963677i	$-0.413944\pi$
$421$	4614.01	0.534141	0.267070	+	0.963677i	$0.413944\pi$
$422$	0	0
$423$	− 300.501i	− 0.0345410i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	19208.6i	2.17697i
$428$	0	0
$429$	13151.1	1.48005
$430$	0	0
$431$	−2739.14	−0.306125	−0.153062	−	0.988217i	$-0.548914\pi$
$431$	−2739.14	−0.306125	−0.153062	+	0.988217i	$0.548914\pi$
$432$	0	0
$433$	− 10827.9i	− 1.20174i	−0.799345	−	0.600872i	$-0.794820\pi$
$433$	− 10827.9i	− 1.20174i	0.799345	−	0.600872i	$-0.205180\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3172.33i	0.347261i
$438$	0	0
$439$	10359.5	1.12627	0.563137	−	0.826364i	$-0.309594\pi$
$439$	10359.5	1.12627	0.563137	+	0.826364i	$0.309594\pi$
$440$	0	0
$441$	667.638	0.0720913
$442$	0	0
$443$	8040.36i	0.862322i	0.902275	+	0.431161i	$0.141896\pi$
$443$	8040.36i	0.862322i	−0.902275	+	0.431161i	$0.858104\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 9726.59i	− 1.02920i
$448$	0	0
$449$	14071.8	1.47904	0.739522	−	0.673132i	$-0.235052\pi$
$449$	14071.8	1.47904	0.739522	+	0.673132i	$0.235052\pi$
$450$	0	0
$451$	−756.320	−0.0789661
$452$	0	0
$453$	− 15551.2i	− 1.61294i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	4372.74i	0.447589i	0.974636	+	0.223795i	$0.0718445\pi$
$457$	4372.74i	0.447589i	−0.974636	+	0.223795i	$0.928155\pi$
$458$	0	0
$459$	−2327.44	−0.236679
$460$	0	0
$461$	−17819.4	−1.80029	−0.900146	−	0.435589i	$-0.856540\pi$
$461$	−17819.4	−1.80029	−0.900146	+	0.435589i	$0.856540\pi$
$462$	0	0
$463$	7114.51i	0.714124i	0.934081	+	0.357062i	$0.116222\pi$
$463$	7114.51i	0.714124i	−0.934081	+	0.357062i	$0.883778\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	8208.41i	0.813361i	0.913570	+	0.406681i	$0.133314\pi$
$467$	8208.41i	0.813361i	−0.913570	+	0.406681i	$0.866686\pi$
$468$	0	0
$469$	13183.9	1.29803
$470$	0	0
$471$	−7422.92	−0.726178
$472$	0	0
$473$	− 20620.5i	− 2.00451i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	913.561i	0.0876920i
$478$	0	0
$479$	11889.5	1.13413	0.567064	−	0.823674i	$-0.308079\pi$
$479$	11889.5	1.13413	0.567064	+	0.823674i	$0.308079\pi$
$480$	0	0
$481$	−8199.95	−0.777309
$482$	0	0
$483$	23760.2i	2.23836i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 13922.4i	− 1.29545i	−0.761875	−	0.647725i	$-0.775721\pi$
$487$	− 13922.4i	− 1.29545i	0.761875	−	0.647725i	$-0.224279\pi$
$488$	0	0
$489$	−20096.8	−1.85850
$490$	0	0
$491$	2718.22	0.249840	0.124920	−	0.992167i	$-0.460133\pi$
$491$	2718.22	0.249840	0.124920	+	0.992167i	$0.460133\pi$
$492$	0	0
$493$	− 1881.89i	− 0.171919i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 21153.0i	− 1.90914i
$498$	0	0
$499$	7830.51	0.702489	0.351244	−	0.936284i	$-0.385759\pi$
$499$	7830.51	0.702489	0.351244	+	0.936284i	$0.385759\pi$
$500$	0	0
$501$	−16574.4	−1.47802
$502$	0	0
$503$	− 6554.35i	− 0.581002i	−0.956875	−	0.290501i	$-0.906178\pi$
$503$	− 6554.35i	− 0.581002i	0.956875	−	0.290501i	$-0.0938219\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1146.06i	0.100391i
$508$	0	0
$509$	−12611.4	−1.09821	−0.549105	−	0.835753i	$-0.685031\pi$
$509$	−12611.4	−1.09821	−0.549105	+	0.835753i	$0.685031\pi$
$510$	0	0
$511$	8242.73	0.713575
$512$	0	0
$513$	− 2565.98i	− 0.220840i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 8049.50i	− 0.684752i
$518$	0	0
$519$	1574.08	0.133130
$520$	0	0
$521$	1695.01	0.142533	0.0712666	−	0.997457i	$-0.477296\pi$
$521$	1695.01	0.142533	0.0712666	+	0.997457i	$0.477296\pi$
$522$	0	0
$523$	− 21001.4i	− 1.75589i	−0.478764	−	0.877944i	$-0.658915\pi$
$523$	− 21001.4i	− 1.75589i	0.478764	−	0.877944i	$-0.341085\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4718.28i	0.390003i
$528$	0	0
$529$	−15710.2	−1.29121
$530$	0	0
$531$	−625.377	−0.0511093
$532$	0	0
$533$	− 744.696i	− 0.0605185i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 16097.1i	− 1.29356i
$538$	0	0
$539$	17884.0	1.42916
$540$	0	0
$541$	−555.994	−0.0441849	−0.0220925	−	0.999756i	$-0.507033\pi$
$541$	−555.994	−0.0441849	−0.0220925	+	0.999756i	$0.507033\pi$
$542$	0	0
$543$	17542.7i	1.38643i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	2987.93i	0.233555i	0.993158	+	0.116778i	$0.0372565\pi$
$547$	2987.93i	0.233555i	−0.993158	+	0.116778i	$0.962744\pi$
$548$	0	0
$549$	−1349.80	−0.104933
$550$	0	0
$551$	2074.77	0.160414
$552$	0	0
$553$	17634.4i	1.35604i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	4357.38i	0.331469i	0.986170	+	0.165735i	$0.0529995\pi$
$557$	4357.38i	0.331469i	−0.986170	+	0.165735i	$0.947001\pi$
$558$	0	0
$559$	20303.6	1.53623
$560$	0	0
$561$	4616.38	0.347422
$562$	0	0
$563$	1669.67i	0.124988i	0.998045	+	0.0624941i	$0.0199055\pi$
$563$	1669.67i	0.124988i	−0.998045	+	0.0624941i	$0.980095\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 20550.1i	− 1.52209i
$568$	0	0
$569$	20440.1	1.50596	0.752982	−	0.658041i	$-0.228615\pi$
$569$	20440.1	1.50596	0.752982	+	0.658041i	$0.228615\pi$
$570$	0	0
$571$	−6920.98	−0.507240	−0.253620	−	0.967304i	$-0.581621\pi$
$571$	−6920.98	−0.507240	−0.253620	+	0.967304i	$0.581621\pi$
$572$	0	0
$573$	8070.62i	0.588403i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	5021.59i	0.362307i	0.983455	+	0.181154i	$0.0579832\pi$
$577$	5021.59i	0.362307i	−0.983455	+	0.181154i	$0.942017\pi$
$578$	0	0
$579$	−22044.1	−1.58225
$580$	0	0
$581$	9865.61	0.704466
$582$	0	0
$583$	24471.5i	1.73843i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	6784.80i	0.477068i	0.971134	+	0.238534i	$0.0766668\pi$
$587$	6784.80i	0.477068i	−0.971134	+	0.238534i	$0.923333\pi$
$588$	0	0
$589$	−5201.87	−0.363904
$590$	0	0
$591$	−29122.9	−2.02700
$592$	0	0
$593$	22058.5i	1.52754i	0.645487	+	0.763771i	$0.276654\pi$
$593$	22058.5i	1.52754i	−0.645487	+	0.763771i	$0.723346\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	8950.51i	0.613601i
$598$	0	0
$599$	15616.7	1.06524	0.532620	−	0.846354i	$-0.321207\pi$
$599$	15616.7	1.06524	0.532620	+	0.846354i	$0.321207\pi$
$600$	0	0
$601$	26769.6	1.81690	0.908448	−	0.417998i	$-0.137268\pi$
$601$	26769.6	1.81690	0.908448	+	0.417998i	$0.137268\pi$
$602$	0	0
$603$	926.439i	0.0625664i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 5164.26i	− 0.345323i	−0.984981	−	0.172661i	$-0.944763\pi$
$607$	− 5164.26i	− 0.345323i	0.984981	−	0.172661i	$-0.0552367\pi$
$608$	0	0
$609$	15539.7	1.03399
$610$	0	0
$611$	7925.79	0.524784
$612$	0	0
$613$	4226.03i	0.278447i	0.990261	+	0.139223i	$0.0444606\pi$
$613$	4226.03i	0.278447i	−0.990261	+	0.139223i	$0.955539\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 14233.5i	− 0.928717i	−0.885647	−	0.464358i	$-0.846285\pi$
$617$	− 14233.5i	− 0.928717i	0.885647	−	0.464358i	$-0.153715\pi$
$618$	0	0
$619$	3737.82	0.242707	0.121353	−	0.992609i	$-0.461277\pi$
$619$	3737.82	0.242707	0.121353	+	0.992609i	$0.461277\pi$
$620$	0	0
$621$	22548.8	1.45709
$622$	0	0
$623$	17847.9i	1.14777i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	5089.52i	0.324172i
$628$	0	0
$629$	−2878.40	−0.182463
$630$	0	0
$631$	−2893.31	−0.182537	−0.0912684	−	0.995826i	$-0.529092\pi$
$631$	−2893.31	−0.182537	−0.0912684	+	0.995826i	$0.529092\pi$
$632$	0	0
$633$	− 31513.3i	− 1.97874i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	17609.1i	1.09529i
$638$	0	0
$639$	1486.43	0.0920226
$640$	0	0
$641$	6867.26	0.423152	0.211576	−	0.977362i	$-0.432140\pi$
$641$	6867.26	0.423152	0.211576	+	0.977362i	$0.432140\pi$
$642$	0	0
$643$	4975.32i	0.305144i	0.988292	+	0.152572i	$0.0487555\pi$
$643$	4975.32i	0.305144i	−0.988292	+	0.152572i	$0.951244\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	24033.2i	1.46035i	0.683262	+	0.730174i	$0.260561\pi$
$647$	24033.2i	1.46035i	−0.683262	+	0.730174i	$0.739439\pi$
$648$	0	0
$649$	−16751.9	−1.01321
$650$	0	0
$651$	−38961.2	−2.34564
$652$	0	0
$653$	− 25718.4i	− 1.54125i	−0.637288	−	0.770626i	$-0.719944\pi$
$653$	− 25718.4i	− 1.54125i	0.637288	−	0.770626i	$-0.280056\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	579.221i	0.0343951i
$658$	0	0
$659$	7970.84	0.471168	0.235584	−	0.971854i	$-0.424300\pi$
$659$	7970.84	0.471168	0.235584	+	0.971854i	$0.424300\pi$
$660$	0	0
$661$	14207.8	0.836038	0.418019	−	0.908438i	$-0.362725\pi$
$661$	14207.8	0.836038	0.418019	+	0.908438i	$0.362725\pi$
$662$	0	0
$663$	4545.43i	0.266259i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	18232.2i	1.05840i
$668$	0	0
$669$	−29026.1	−1.67745
$670$	0	0
$671$	−36157.0	−2.08021
$672$	0	0
$673$	17080.0i	0.978287i	0.872203	+	0.489143i	$0.162691\pi$
$673$	17080.0i	0.978287i	−0.872203	+	0.489143i	$0.837309\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	16410.9i	0.931641i	0.884879	+	0.465821i	$0.154241\pi$
$677$	16410.9i	0.931641i	−0.884879	+	0.465821i	$0.845759\pi$
$678$	0	0
$679$	−25442.9	−1.43801
$680$	0	0
$681$	13814.0	0.777317
$682$	0	0
$683$	− 183.169i	− 0.0102617i	−0.999987	−	0.00513086i	$-0.998367\pi$
$683$	− 183.169i	− 0.0102617i	0.999987	−	0.00513086i	$-0.00163321\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	17492.6i	0.971446i
$688$	0	0
$689$	−24095.4	−1.33231
$690$	0	0
$691$	14974.0	0.824366	0.412183	−	0.911101i	$-0.364766\pi$
$691$	14974.0	0.824366	0.412183	+	0.911101i	$0.364766\pi$
$692$	0	0
$693$	2458.52i	0.134764i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 261.408i	− 0.0142059i
$698$	0	0
$699$	6476.74	0.350462
$700$	0	0
$701$	4341.08	0.233895	0.116947	−	0.993138i	$-0.462689\pi$
$701$	4341.08	0.233895	0.116947	+	0.993138i	$0.462689\pi$
$702$	0	0
$703$	− 3173.41i	− 0.170253i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	50056.1i	2.66273i
$708$	0	0
$709$	−23931.9	−1.26767	−0.633837	−	0.773467i	$-0.718521\pi$
$709$	−23931.9	−1.26767	−0.633837	+	0.773467i	$0.718521\pi$
$710$	0	0
$711$	−1239.18	−0.0653627
$712$	0	0
$713$	− 45712.0i	− 2.40102i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 18290.1i	− 0.952656i
$718$	0	0
$719$	−16258.6	−0.843314	−0.421657	−	0.906756i	$-0.638551\pi$
$719$	−16258.6	−0.843314	−0.421657	+	0.906756i	$0.638551\pi$
$720$	0	0
$721$	−20137.5	−1.04017
$722$	0	0
$723$	− 17736.6i	− 0.912354i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	2608.95i	0.133096i	0.997783	+	0.0665479i	$0.0211985\pi$
$727$	2608.95i	0.133096i	−0.997783	+	0.0665479i	$0.978801\pi$
$728$	0	0
$729$	−18145.4	−0.921881
$730$	0	0
$731$	7127.11	0.360610
$732$	0	0
$733$	9843.63i	0.496020i	0.968757	+	0.248010i	$0.0797766\pi$
$733$	9843.63i	0.496020i	−0.968757	+	0.248010i	$0.920223\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	24816.5i	1.24033i
$738$	0	0
$739$	−2419.46	−0.120435	−0.0602174	−	0.998185i	$-0.519179\pi$
$739$	−2419.46	−0.120435	−0.0602174	+	0.998185i	$0.519179\pi$
$740$	0	0
$741$	−5011.30	−0.248441
$742$	0	0
$743$	10442.9i	0.515631i	0.966194	+	0.257816i	$0.0830027\pi$
$743$	10442.9i	0.515631i	−0.966194	+	0.257816i	$0.916997\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	693.262i	0.0339560i
$748$	0	0
$749$	18058.4	0.880963
$750$	0	0
$751$	29434.6	1.43021	0.715103	−	0.699019i	$-0.246380\pi$
$751$	29434.6	1.43021	0.715103	+	0.699019i	$0.246380\pi$
$752$	0	0
$753$	28638.6i	1.38599i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	20638.3i	0.990902i	0.868636	+	0.495451i	$0.164997\pi$
$757$	20638.3i	0.990902i	−0.868636	+	0.495451i	$0.835003\pi$
$758$	0	0
$759$	−44724.7	−2.13887
$760$	0	0
$761$	4103.09	0.195449	0.0977246	−	0.995213i	$-0.468844\pi$
$761$	4103.09	0.195449	0.0977246	+	0.995213i	$0.468844\pi$
$762$	0	0
$763$	34721.0i	1.64742i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 16494.5i	− 0.776508i
$768$	0	0
$769$	14020.0	0.657444	0.328722	−	0.944427i	$-0.393382\pi$
$769$	14020.0	0.657444	0.328722	+	0.944427i	$0.393382\pi$
$770$	0	0
$771$	−20479.9	−0.956635
$772$	0	0
$773$	32823.8i	1.52728i	0.645640	+	0.763642i	$0.276591\pi$
$773$	32823.8i	1.52728i	−0.645640	+	0.763642i	$0.723409\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 23768.4i	− 1.09741i
$778$	0	0
$779$	288.200	0.0132553
$780$	0	0
$781$	39817.0	1.82428
$782$	0	0
$783$	− 14747.4i	− 0.673090i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 33903.0i	− 1.53559i	−0.640694	−	0.767796i	$-0.721353\pi$
$787$	− 33903.0i	− 1.53559i	0.640694	−	0.767796i	$-0.278647\pi$
$788$	0	0
$789$	4830.54	0.217962
$790$	0	0
$791$	18897.7	0.849462
$792$	0	0
$793$	− 35601.3i	− 1.59425i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 20830.2i	− 0.925778i	−0.886416	−	0.462889i	$-0.846813\pi$
$797$	− 20830.2i	− 0.925778i	0.886416	−	0.462889i	$-0.153187\pi$
$798$	0	0
$799$	2782.16	0.123186
$800$	0	0
$801$	−1254.18	−0.0553238
$802$	0	0
$803$	15515.6i	0.681859i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	14213.8i	0.620012i
$808$	0	0
$809$	26505.7	1.15190	0.575952	−	0.817484i	$-0.304632\pi$
$809$	26505.7	1.15190	0.575952	+	0.817484i	$0.304632\pi$
$810$	0	0
$811$	−10829.0	−0.468876	−0.234438	−	0.972131i	$-0.575325\pi$
$811$	−10829.0	−0.468876	−0.234438	+	0.972131i	$0.575325\pi$
$812$	0	0
$813$	− 2774.94i	− 0.119706i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	7857.58i	0.336477i
$818$	0	0
$819$	−2420.73	−0.103281
$820$	0	0
$821$	42046.8	1.78738	0.893692	−	0.448681i	$-0.148106\pi$
$821$	42046.8	1.78738	0.893692	+	0.448681i	$0.148106\pi$
$822$	0	0
$823$	− 12949.0i	− 0.548448i	−0.961666	−	0.274224i	$-0.911579\pi$
$823$	− 12949.0i	− 0.548448i	0.961666	−	0.274224i	$-0.0884209\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	13874.0i	0.583368i	0.956515	+	0.291684i	$0.0942156\pi$
$827$	13874.0i	0.583368i	−0.956515	+	0.291684i	$0.905784\pi$
$828$	0	0
$829$	2387.20	0.100013	0.0500065	−	0.998749i	$-0.484076\pi$
$829$	2387.20	0.100013	0.0500065	+	0.998749i	$0.484076\pi$
$830$	0	0
$831$	−12706.2	−0.530412
$832$	0	0
$833$	6181.27i	0.257105i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	36974.8i	1.52692i
$838$	0	0
$839$	−47308.8	−1.94670	−0.973350	−	0.229324i	$-0.926349\pi$
$839$	−47308.8	−1.94670	−0.973350	+	0.229324i	$0.926349\pi$
$840$	0	0
$841$	−12464.7	−0.511080
$842$	0	0
$843$	8200.99i	0.335062i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	30599.2i	1.24132i
$848$	0	0
$849$	25593.6	1.03459
$850$	0	0
$851$	27886.7	1.12332
$852$	0	0
$853$	− 18211.3i	− 0.730998i	−0.930812	−	0.365499i	$-0.880898\pi$
$853$	− 18211.3i	− 0.730998i	0.930812	−	0.365499i	$-0.119102\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	14824.5i	0.590893i	0.955359	+	0.295447i	$0.0954685\pi$
$857$	14824.5i	0.590893i	−0.955359	+	0.295447i	$0.904532\pi$
$858$	0	0
$859$	44191.1	1.75527	0.877637	−	0.479326i	$-0.159119\pi$
$859$	44191.1	1.75527	0.877637	+	0.479326i	$0.159119\pi$
$860$	0	0
$861$	2158.58	0.0854403
$862$	0	0
$863$	− 34927.0i	− 1.37767i	−0.724918	−	0.688835i	$-0.758122\pi$
$863$	− 34927.0i	− 1.37767i	0.724918	−	0.688835i	$-0.241878\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 24798.5i	− 0.971395i
$868$	0	0
$869$	−33193.9	−1.29577
$870$	0	0
$871$	−24435.1	−0.950575
$872$	0	0
$873$	− 1787.89i	− 0.0693138i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	16914.8i	0.651278i	0.945494	+	0.325639i	$0.105579\pi$
$877$	16914.8i	0.651278i	−0.945494	+	0.325639i	$0.894421\pi$
$878$	0	0
$879$	17132.6	0.657415
$880$	0	0
$881$	12634.8	0.483177	0.241588	−	0.970379i	$-0.422332\pi$
$881$	12634.8	0.483177	0.241588	+	0.970379i	$0.422332\pi$
$882$	0	0
$883$	− 47260.3i	− 1.80117i	−0.434678	−	0.900586i	$-0.643138\pi$
$883$	− 47260.3i	− 1.80117i	0.434678	−	0.900586i	$-0.356862\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	4758.84i	0.180142i	0.995935	+	0.0900711i	$0.0287094\pi$
$887$	4758.84i	0.180142i	−0.995935	+	0.0900711i	$0.971291\pi$
$888$	0	0
$889$	−2856.78	−0.107777
$890$	0	0
$891$	38682.1	1.45443
$892$	0	0
$893$	3067.31i	0.114943i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 44037.4i	− 1.63920i
$898$	0	0
$899$	−29896.6	−1.10913
$900$	0	0
$901$	−8458.13	−0.312743
$902$	0	0
$903$	58852.1i	2.16885i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	4111.61i	0.150522i	0.997164	+	0.0752612i	$0.0239791\pi$
$907$	4111.61i	0.150522i	−0.997164	+	0.0752612i	$0.976021\pi$
$908$	0	0
$909$	−3517.47	−0.128347
$910$	0	0
$911$	−35113.1	−1.27700	−0.638501	−	0.769621i	$-0.720445\pi$
$911$	−35113.1	−1.27700	−0.638501	+	0.769621i	$0.720445\pi$
$912$	0	0
$913$	18570.4i	0.673155i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	9887.00i	0.356049i
$918$	0	0
$919$	−37509.1	−1.34637	−0.673184	−	0.739475i	$-0.735074\pi$
$919$	−37509.1	−1.34637	−0.673184	+	0.739475i	$0.735074\pi$
$920$	0	0
$921$	−32226.0	−1.15297
$922$	0	0
$923$	39205.1i	1.39811i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 1415.07i	− 0.0501371i
$928$	0	0
$929$	−18393.8	−0.649604	−0.324802	−	0.945782i	$-0.605298\pi$
$929$	−18393.8	−0.649604	−0.324802	+	0.945782i	$0.605298\pi$
$930$	0	0
$931$	−6814.80	−0.239899
$932$	0	0
$933$	− 27479.0i	− 0.964226i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	17106.7i	0.596427i	0.954499	+	0.298214i	$0.0963908\pi$
$937$	17106.7i	0.596427i	−0.954499	+	0.298214i	$0.903609\pi$
$938$	0	0
$939$	11451.7	0.397990
$940$	0	0
$941$	−38100.7	−1.31992	−0.659961	−	0.751300i	$-0.729427\pi$
$941$	−38100.7	−1.31992	−0.659961	+	0.751300i	$0.729427\pi$
$942$	0	0
$943$	2532.59i	0.0874576i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	52712.7i	1.80880i	0.426686	+	0.904400i	$0.359681\pi$
$947$	52712.7i	1.80880i	−0.426686	+	0.904400i	$0.640319\pi$
$948$	0	0
$949$	−15277.1	−0.522567
$950$	0	0
$951$	−23944.4	−0.816458
$952$	0	0
$953$	− 6744.38i	− 0.229246i	−0.993409	−	0.114623i	$-0.963434\pi$
$953$	− 6744.38i	− 0.229246i	0.993409	−	0.114623i	$-0.0365661\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	29250.9i	0.988032i
$958$	0	0
$959$	−82599.8	−2.78132
$960$	0	0
$961$	45165.9	1.51609
$962$	0	0
$963$	1268.98i	0.0424634i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	28449.8i	0.946104i	0.881034	+	0.473052i	$0.156848\pi$
$967$	28449.8i	0.946104i	−0.881034	+	0.473052i	$0.843152\pi$
$968$	0	0
$969$	−1759.10	−0.0583183
$970$	0	0
$971$	−13018.7	−0.430266	−0.215133	−	0.976585i	$-0.569019\pi$
$971$	−13018.7	−0.430266	−0.215133	+	0.976585i	$0.569019\pi$
$972$	0	0
$973$	1493.39i	0.0492043i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	20043.8i	0.656355i	0.944616	+	0.328178i	$0.106434\pi$
$977$	20043.8i	0.656355i	−0.944616	+	0.328178i	$0.893566\pi$
$978$	0	0
$979$	−33595.7	−1.09676
$980$	0	0
$981$	−2439.86	−0.0794076
$982$	0	0
$983$	− 1823.72i	− 0.0591735i	−0.999562	−	0.0295868i	$-0.990581\pi$
$983$	− 1823.72i	− 0.0591735i	0.999562	−	0.0295868i	$-0.00941913\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	22973.7i	0.740892i
$988$	0	0
$989$	−69049.4	−2.22006
$990$	0	0
$991$	−26635.4	−0.853786	−0.426893	−	0.904302i	$-0.640392\pi$
$991$	−26635.4	−0.853786	−0.426893	+	0.904302i	$0.640392\pi$
$992$	0	0
$993$	22261.0i	0.711411i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 50874.6i	− 1.61606i	−0.589139	−	0.808031i	$-0.700533\pi$
$997$	− 50874.6i	− 1.61606i	0.589139	−	0.808031i	$-0.299467\pi$
$998$	0	0
$999$	−22556.6	−0.714372

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1900.4.c.b.1749.1		4
5.2	odd	4		76.4.a.a.1.1	✓	2
5.3	odd	4		1900.4.a.b.1.2		2
5.4	even	2	inner	1900.4.c.b.1749.4		4
15.2	even	4		684.4.a.g.1.1		2
20.7	even	4		304.4.a.f.1.2		2
40.27	even	4		1216.4.a.h.1.1		2
40.37	odd	4		1216.4.a.o.1.2		2
95.37	even	4		1444.4.a.d.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.4.a.a.1.1	✓	2	5.2	odd	4
304.4.a.f.1.2		2	20.7	even	4
684.4.a.g.1.1		2	15.2	even	4
1216.4.a.h.1.1		2	40.27	even	4
1216.4.a.o.1.2		2	40.37	odd	4
1444.4.a.d.1.2		2	95.37	even	4
1900.4.a.b.1.2		2	5.3	odd	4
1900.4.c.b.1749.1		4	1.1	even	1	trivial
1900.4.c.b.1749.4		4	5.4	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:	\( N \)	\(=\)	\( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1900.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-5.37228i q^{3} +26.4891i q^{7} -1.86141 q^{9} +O(q^{10})\)	\(q-5.37228i q^{3} +26.4891i q^{7} -1.86141 q^{9} -49.8614 q^{11} -49.0951i q^{13} -17.2337i q^{17} +19.0000 q^{19} +142.307 q^{21} +166.965i q^{23} -135.052i q^{27} +109.198 q^{29} -273.783 q^{31} +267.870i q^{33} -167.022i q^{37} -263.753 q^{39} +15.1684 q^{41} +413.557i q^{43} +161.438i q^{47} -358.674 q^{49} -92.5842 q^{51} -490.791i q^{53} -102.073i q^{57} +335.970 q^{59} +725.149 q^{61} -49.3070i q^{63} -497.709i q^{67} +896.981 q^{69} -798.554 q^{71} -311.174i q^{73} -1320.79i q^{77} +665.723 q^{79} -775.793 q^{81} -372.440i q^{83} -586.644i q^{87} +673.783 q^{89} +1300.49 q^{91} +1470.84i q^{93} +960.505i q^{97} +92.8124 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 50 q^{9}+O(q^{10}) \) 4 * q + 50 * q^9	\( 4 q + 50 q^{9} - 142 q^{11} + 76 q^{19} + 282 q^{21} - 310 q^{29} - 176 q^{31} - 538 q^{39} - 284 q^{41} - 56 q^{49} - 198 q^{51} + 1746 q^{59} + 890 q^{61} + 1922 q^{69} - 3424 q^{71} + 2548 q^{79} - 116 q^{81} + 1776 q^{89} + 2502 q^{91} - 950 q^{99}+O(q^{100}) \) 4 * q + 50 * q^9 - 142 * q^11 + 76 * q^19 + 282 * q^21 - 310 * q^29 - 176 * q^31 - 538 * q^39 - 284 * q^41 - 56 * q^49 - 198 * q^51 + 1746 * q^59 + 890 * q^61 + 1922 * q^69 - 3424 * q^71 + 2548 * q^79 - 116 * q^81 + 1776 * q^89 + 2502 * q^91 - 950 * q^99

Introduction

L-functions

Modular forms

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Database

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