LMFDB - Embedded newform 2475.4.a.y.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,4,Mod(1,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, 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("2475.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2475.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:	$146.029727264$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.3368.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 15x + 11 $ x^3 - x^2 - 15*x + 11
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 825)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.723686$ of defining polynomial
Character	$\chi$	$=$	2475.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.723686 q^{2} -7.47628 q^{4} +1.13288 q^{7} -11.2000 q^{8} -11.0000 q^{11} -21.8566 q^{13} +0.819851 q^{14} +51.7050 q^{16} -6.18033 q^{17} -92.8393 q^{19} -7.96054 q^{22} +36.7317 q^{23} -15.8173 q^{26} -8.46975 q^{28} -71.1375 q^{29} +186.522 q^{31} +127.018 q^{32} -4.47261 q^{34} -356.581 q^{37} -67.1865 q^{38} -271.940 q^{41} +155.780 q^{43} +82.2391 q^{44} +26.5822 q^{46} -234.566 q^{47} -341.717 q^{49} +163.406 q^{52} -195.018 q^{53} -12.6882 q^{56} -51.4812 q^{58} +455.930 q^{59} -441.278 q^{61} +134.983 q^{62} -321.719 q^{64} +133.005 q^{67} +46.2058 q^{68} +1041.68 q^{71} -160.119 q^{73} -258.053 q^{74} +694.093 q^{76} -12.4617 q^{77} -761.367 q^{79} -196.799 q^{82} -51.7170 q^{83} +112.736 q^{86} +123.200 q^{88} +1075.25 q^{89} -24.7609 q^{91} -274.616 q^{92} -169.752 q^{94} -703.238 q^{97} -247.295 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} + 7 q^{4} - 16 q^{7} - 3 q^{8} - 33 q^{11} - 45 q^{13} + 116 q^{14} - 85 q^{16} - 58 q^{17} - 169 q^{19} - 11 q^{22} + 155 q^{23} - 167 q^{26} - 100 q^{28} + 277 q^{29} - 173 q^{31} + 97 q^{32}+ \cdots - 2273 q^{98}+O(q^{100}) $ 3 * q + q^2 + 7 * q^4 - 16 * q^7 - 3 * q^8 - 33 * q^11 - 45 * q^13 + 116 * q^14 - 85 * q^16 - 58 * q^17 - 169 * q^19 - 11 * q^22 + 155 * q^23 - 167 * q^26 - 100 * q^28 + 277 * q^29 - 173 * q^31 + 97 * q^32 - 146 * q^34 - 60 * q^37 + 169 * q^38 - 44 * q^41 + 109 * q^43 - 77 * q^44 + 425 * q^46 + 270 * q^47 - 427 * q^49 - 45 * q^52 - 148 * q^53 - 168 * q^56 + 783 * q^58 + 684 * q^59 - 1038 * q^61 + 953 * q^62 - 1129 * q^64 - 314 * q^67 - 366 * q^68 + 1459 * q^71 + 1170 * q^73 + 1764 * q^74 + 211 * q^76 + 176 * q^77 - 506 * q^79 + 1040 * q^82 - 347 * q^83 + 527 * q^86 + 33 * q^88 + 607 * q^89 - 398 * q^91 + 687 * q^92 - 1822 * q^94 - 1263 * q^97 - 2273 * q^98

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 $

$2$	0.723686	0.255862	0.127931	−	0.991783i	$-0.459166\pi$
$2$	0.723686	0.255862	0.127931	+	0.991783i	$0.459166\pi$
$3$	0	0
$3$	0	0
$4$	−7.47628	−0.934535
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.13288	0.0611699	0.0305850	−	0.999532i	$-0.490263\pi$
$7$	1.13288	0.0611699	0.0305850	+	0.999532i	$0.490263\pi$
$8$	−11.2000	−0.494973
$9$	0	0
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	−21.8566	−0.466302	−0.233151	−	0.972441i	$-0.574904\pi$
$13$	−21.8566	−0.466302	−0.233151	+	0.972441i	$0.574904\pi$
$14$	0.819851	0.0156510
$15$	0	0
$16$	51.7050	0.807890
$17$	−6.18033	−0.0881735	−0.0440867	−	0.999028i	$-0.514038\pi$
$17$	−6.18033	−0.0881735	−0.0440867	+	0.999028i	$0.514038\pi$
$18$	0	0
$19$	−92.8393	−1.12099	−0.560495	−	0.828158i	$-0.689389\pi$
$19$	−92.8393	−1.12099	−0.560495	+	0.828158i	$0.689389\pi$
$20$	0	0
$21$	0	0
$22$	−7.96054	−0.0771452
$23$	36.7317	0.333004	0.166502	−	0.986041i	$-0.446753\pi$
$23$	36.7317	0.333004	0.166502	+	0.986041i	$0.446753\pi$
$24$	0	0
$25$	0	0
$26$	−15.8173	−0.119309
$27$	0	0
$28$	−8.46975	−0.0571654
$29$	−71.1375	−0.455514	−0.227757	−	0.973718i	$-0.573139\pi$
$29$	−71.1375	−0.455514	−0.227757	+	0.973718i	$0.573139\pi$
$30$	0	0
$31$	186.522	1.08065	0.540327	−	0.841455i	$-0.318301\pi$
$31$	186.522	1.08065	0.540327	+	0.841455i	$0.318301\pi$
$32$	127.018	0.701681
$33$	0	0
$34$	−4.47261	−0.0225602
$35$	0	0
$36$	0	0
$37$	−356.581	−1.58437	−0.792184	−	0.610282i	$-0.791056\pi$
$37$	−356.581	−1.58437	−0.792184	+	0.610282i	$0.791056\pi$
$38$	−67.1865	−0.286818
$39$	0	0
$40$	0	0
$41$	−271.940	−1.03585	−0.517925	−	0.855426i	$-0.673295\pi$
$41$	−271.940	−1.03585	−0.517925	+	0.855426i	$0.673295\pi$
$42$	0	0
$43$	155.780	0.552470	0.276235	−	0.961090i	$-0.410913\pi$
$43$	155.780	0.552470	0.276235	+	0.961090i	$0.410913\pi$
$44$	82.2391	0.281773
$45$	0	0
$46$	26.5822	0.0852028
$47$	−234.566	−0.727978	−0.363989	−	0.931403i	$-0.618585\pi$
$47$	−234.566	−0.727978	−0.363989	+	0.931403i	$0.618585\pi$
$48$	0	0
$49$	−341.717	−0.996258
$50$	0	0
$51$	0	0
$52$	163.406	0.435775
$53$	−195.018	−0.505429	−0.252715	−	0.967541i	$-0.581323\pi$
$53$	−195.018	−0.505429	−0.252715	+	0.967541i	$0.581323\pi$
$54$	0	0
$55$	0	0
$56$	−12.6882	−0.0302775
$57$	0	0
$58$	−51.4812	−0.116548
$59$	455.930	1.00605	0.503025	−	0.864272i	$-0.332220\pi$
$59$	455.930	1.00605	0.503025	+	0.864272i	$0.332220\pi$
$60$	0	0
$61$	−441.278	−0.926227	−0.463114	−	0.886299i	$-0.653268\pi$
$61$	−441.278	−0.926227	−0.463114	+	0.886299i	$0.653268\pi$
$62$	134.983	0.276498
$63$	0	0
$64$	−321.719	−0.628357
$65$	0	0
$66$	0	0
$67$	133.005	0.242524	0.121262	−	0.992621i	$-0.461306\pi$
$67$	133.005	0.242524	0.121262	+	0.992621i	$0.461306\pi$
$68$	46.2058	0.0824012
$69$	0	0
$70$	0	0
$71$	1041.68	1.74119	0.870596	−	0.491999i	$-0.163734\pi$
$71$	1041.68	1.74119	0.870596	+	0.491999i	$0.163734\pi$
$72$	0	0
$73$	−160.119	−0.256719	−0.128360	−	0.991728i	$-0.540971\pi$
$73$	−160.119	−0.256719	−0.128360	+	0.991728i	$0.540971\pi$
$74$	−258.053	−0.405379
$75$	0	0
$76$	694.093	1.04760
$77$	−12.4617	−0.0184434
$78$	0	0
$79$	−761.367	−1.08431	−0.542155	−	0.840278i	$-0.682392\pi$
$79$	−761.367	−1.08431	−0.542155	+	0.840278i	$0.682392\pi$
$80$	0	0
$81$	0	0
$82$	−196.799	−0.265034
$83$	−51.7170	−0.0683938	−0.0341969	−	0.999415i	$-0.510887\pi$
$83$	−51.7170	−0.0683938	−0.0341969	+	0.999415i	$0.510887\pi$
$84$	0	0
$85$	0	0
$86$	112.736	0.141356
$87$	0	0
$88$	123.200	0.149240
$89$	1075.25	1.28064	0.640318	−	0.768110i	$-0.278802\pi$
$89$	1075.25	1.28064	0.640318	+	0.768110i	$0.278802\pi$
$90$	0	0
$91$	−24.7609	−0.0285236
$92$	−274.616	−0.311203
$93$	0	0
$94$	−169.752	−0.186262
$95$	0	0
$96$	0	0
$97$	−703.238	−0.736114	−0.368057	−	0.929803i	$-0.619977\pi$
$97$	−703.238	−0.736114	−0.368057	+	0.929803i	$0.619977\pi$
$98$	−247.295	−0.254904
$99$	0	0
$100$	0	0
$101$	856.886	0.844192	0.422096	−	0.906551i	$-0.361295\pi$
$101$	856.886	0.844192	0.422096	+	0.906551i	$0.361295\pi$
$102$	0	0
$103$	−805.989	−0.771034	−0.385517	−	0.922701i	$-0.625977\pi$
$103$	−805.989	−0.771034	−0.385517	+	0.922701i	$0.625977\pi$
$104$	244.793	0.230807
$105$	0	0
$106$	−141.132	−0.129320
$107$	1608.55	1.45331	0.726654	−	0.687004i	$-0.241074\pi$
$107$	1608.55	1.45331	0.726654	+	0.687004i	$0.241074\pi$
$108$	0	0
$109$	−925.724	−0.813471	−0.406735	−	0.913546i	$-0.633333\pi$
$109$	−925.724	−0.813471	−0.406735	+	0.913546i	$0.633333\pi$
$110$	0	0
$111$	0	0
$112$	58.5757	0.0494186
$113$	−1214.81	−1.01133	−0.505663	−	0.862731i	$-0.668752\pi$
$113$	−1214.81	−1.01133	−0.505663	+	0.862731i	$0.668752\pi$
$114$	0	0
$115$	0	0
$116$	531.844	0.425693
$117$	0	0
$118$	329.950	0.257410
$119$	−7.00159	−0.00539357
$120$	0	0
$121$	121.000	0.0909091
$122$	−319.347	−0.236986
$123$	0	0
$124$	−1394.49	−1.00991
$125$	0	0
$126$	0	0
$127$	1128.96	0.788810	0.394405	−	0.918937i	$-0.370951\pi$
$127$	1128.96	0.788810	0.394405	+	0.918937i	$0.370951\pi$
$128$	−1248.97	−0.862454
$129$	0	0
$130$	0	0
$131$	1562.32	1.04199	0.520993	−	0.853561i	$-0.325562\pi$
$131$	1562.32	1.04199	0.520993	+	0.853561i	$0.325562\pi$
$132$	0	0
$133$	−105.176	−0.0685708
$134$	96.2535	0.0620525
$135$	0	0
$136$	69.2194	0.0436435
$137$	−44.9323	−0.0280206	−0.0140103	−	0.999902i	$-0.504460\pi$
$137$	−44.9323	−0.0280206	−0.0140103	+	0.999902i	$0.504460\pi$
$138$	0	0
$139$	−415.574	−0.253587	−0.126793	−	0.991929i	$-0.540468\pi$
$139$	−415.574	−0.253587	−0.126793	+	0.991929i	$0.540468\pi$
$140$	0	0
$141$	0	0
$142$	753.849	0.445504
$143$	240.422	0.140595
$144$	0	0
$145$	0	0
$146$	−115.876	−0.0656846
$147$	0	0
$148$	2665.90	1.48065
$149$	−678.402	−0.372999	−0.186499	−	0.982455i	$-0.559714\pi$
$149$	−678.402	−0.372999	−0.186499	+	0.982455i	$0.559714\pi$
$150$	0	0
$151$	616.083	0.332027	0.166014	−	0.986123i	$-0.446910\pi$
$151$	616.083	0.332027	0.166014	+	0.986123i	$0.446910\pi$
$152$	1039.80	0.554860
$153$	0	0
$154$	−9.01837	−0.00471896
$155$	0	0
$156$	0	0
$157$	−616.599	−0.313439	−0.156720	−	0.987643i	$-0.550092\pi$
$157$	−616.599	−0.313439	−0.156720	+	0.987643i	$0.550092\pi$
$158$	−550.991	−0.277433
$159$	0	0
$160$	0	0
$161$	41.6127	0.0203698
$162$	0	0
$163$	2031.15	0.976026	0.488013	−	0.872836i	$-0.337722\pi$
$163$	2031.15	0.976026	0.488013	+	0.872836i	$0.337722\pi$
$164$	2033.10	0.968038
$165$	0	0
$166$	−37.4269	−0.0174993
$167$	2114.90	0.979973	0.489987	−	0.871730i	$-0.337002\pi$
$167$	2114.90	0.979973	0.489987	+	0.871730i	$0.337002\pi$
$168$	0	0
$169$	−1719.29	−0.782563
$170$	0	0
$171$	0	0
$172$	−1164.65	−0.516302
$173$	15.5096	0.00681602	0.00340801	−	0.999994i	$-0.498915\pi$
$173$	15.5096	0.00681602	0.00340801	+	0.999994i	$0.498915\pi$
$174$	0	0
$175$	0	0
$176$	−568.755	−0.243588
$177$	0	0
$178$	778.146	0.327666
$179$	−487.991	−0.203766	−0.101883	−	0.994796i	$-0.532487\pi$
$179$	−487.991	−0.203766	−0.101883	+	0.994796i	$0.532487\pi$
$180$	0	0
$181$	1248.71	0.512795	0.256397	−	0.966571i	$-0.417464\pi$
$181$	1248.71	0.512795	0.256397	+	0.966571i	$0.417464\pi$
$182$	−17.9191	−0.00729810
$183$	0	0
$184$	−411.393	−0.164828
$185$	0	0
$186$	0	0
$187$	67.9836	0.0265853
$188$	1753.68	0.680321
$189$	0	0
$190$	0	0
$191$	2571.60	0.974213	0.487107	−	0.873342i	$-0.338052\pi$
$191$	2571.60	0.974213	0.487107	+	0.873342i	$0.338052\pi$
$192$	0	0
$193$	1221.80	0.455686	0.227843	−	0.973698i	$-0.426833\pi$
$193$	1221.80	0.455686	0.227843	+	0.973698i	$0.426833\pi$
$194$	−508.924	−0.188343
$195$	0	0
$196$	2554.77	0.931038
$197$	1428.32	0.516568	0.258284	−	0.966069i	$-0.416843\pi$
$197$	1428.32	0.516568	0.258284	+	0.966069i	$0.416843\pi$
$198$	0	0
$199$	−816.166	−0.290736	−0.145368	−	0.989378i	$-0.546437\pi$
$199$	−816.166	−0.290736	−0.145368	+	0.989378i	$0.546437\pi$
$200$	0	0
$201$	0	0
$202$	620.116	0.215996
$203$	−80.5904	−0.0278637
$204$	0	0
$205$	0	0
$206$	−583.283	−0.197278
$207$	0	0
$208$	−1130.09	−0.376721
$209$	1021.23	0.337991
$210$	0	0
$211$	−903.360	−0.294739	−0.147369	−	0.989082i	$-0.547081\pi$
$211$	−903.360	−0.294739	−0.147369	+	0.989082i	$0.547081\pi$
$212$	1458.01	0.472341
$213$	0	0
$214$	1164.08	0.371846
$215$	0	0
$216$	0	0
$217$	211.307	0.0661035
$218$	−669.934	−0.208136
$219$	0	0
$220$	0	0
$221$	135.081	0.0411154
$222$	0	0
$223$	533.966	0.160345	0.0801727	−	0.996781i	$-0.474453\pi$
$223$	533.966	0.160345	0.0801727	+	0.996781i	$0.474453\pi$
$224$	143.896	0.0429218
$225$	0	0
$226$	−879.142	−0.258760
$227$	−696.531	−0.203658	−0.101829	−	0.994802i	$-0.532469\pi$
$227$	−696.531	−0.203658	−0.101829	+	0.994802i	$0.532469\pi$
$228$	0	0
$229$	−796.794	−0.229929	−0.114964	−	0.993370i	$-0.536675\pi$
$229$	−796.794	−0.229929	−0.114964	+	0.993370i	$0.536675\pi$
$230$	0	0
$231$	0	0
$232$	796.737	0.225467
$233$	5657.08	1.59059	0.795295	−	0.606223i	$-0.207316\pi$
$233$	5657.08	1.59059	0.795295	+	0.606223i	$0.207316\pi$
$234$	0	0
$235$	0	0
$236$	−3408.66	−0.940190
$237$	0	0
$238$	−5.06695	−0.00138001
$239$	2023.06	0.547534	0.273767	−	0.961796i	$-0.411730\pi$
$239$	2023.06	0.547534	0.273767	+	0.961796i	$0.411730\pi$
$240$	0	0
$241$	−3513.86	−0.939203	−0.469601	−	0.882879i	$-0.655602\pi$
$241$	−3513.86	−0.939203	−0.469601	+	0.882879i	$0.655602\pi$
$242$	87.5660	0.0232601
$243$	0	0
$244$	3299.12	0.865592
$245$	0	0
$246$	0	0
$247$	2029.15	0.522719
$248$	−2089.03	−0.534895
$249$	0	0
$250$	0	0
$251$	−1814.32	−0.456250	−0.228125	−	0.973632i	$-0.573260\pi$
$251$	−1814.32	−0.456250	−0.228125	+	0.973632i	$0.573260\pi$
$252$	0	0
$253$	−404.048	−0.100404
$254$	817.011	0.201826
$255$	0	0
$256$	1669.89	0.407688
$257$	−1445.46	−0.350839	−0.175419	−	0.984494i	$-0.556128\pi$
$257$	−1445.46	−0.350839	−0.175419	+	0.984494i	$0.556128\pi$
$258$	0	0
$259$	−403.965	−0.0969157
$260$	0	0
$261$	0	0
$262$	1130.63	0.266604
$263$	4411.79	1.03438	0.517191	−	0.855870i	$-0.326978\pi$
$263$	4411.79	1.03438	0.517191	+	0.855870i	$0.326978\pi$
$264$	0	0
$265$	0	0
$266$	−76.1144	−0.0175446
$267$	0	0
$268$	−994.379	−0.226647
$269$	−3955.14	−0.896465	−0.448232	−	0.893917i	$-0.647946\pi$
$269$	−3955.14	−0.896465	−0.448232	+	0.893917i	$0.647946\pi$
$270$	0	0
$271$	6656.96	1.49218	0.746091	−	0.665844i	$-0.231928\pi$
$271$	6656.96	1.49218	0.746091	+	0.665844i	$0.231928\pi$
$272$	−319.554	−0.0712345
$273$	0	0
$274$	−32.5169	−0.00716941
$275$	0	0
$276$	0	0
$277$	8523.79	1.84890	0.924450	−	0.381304i	$-0.124525\pi$
$277$	8523.79	1.84890	0.924450	+	0.381304i	$0.124525\pi$
$278$	−300.745	−0.0648831
$279$	0	0
$280$	0	0
$281$	4210.99	0.893973	0.446986	−	0.894541i	$-0.352497\pi$
$281$	4210.99	0.893973	0.446986	+	0.894541i	$0.352497\pi$
$282$	0	0
$283$	3529.66	0.741400	0.370700	−	0.928753i	$-0.379118\pi$
$283$	3529.66	0.741400	0.370700	+	0.928753i	$0.379118\pi$
$284$	−7787.88	−1.62720
$285$	0	0
$286$	173.990	0.0359729
$287$	−308.076	−0.0633629
$288$	0	0
$289$	−4874.80	−0.992225
$290$	0	0
$291$	0	0
$292$	1197.09	0.239913
$293$	7110.96	1.41784	0.708919	−	0.705290i	$-0.249183\pi$
$293$	7110.96	1.41784	0.708919	+	0.705290i	$0.249183\pi$
$294$	0	0
$295$	0	0
$296$	3993.70	0.784220
$297$	0	0
$298$	−490.950	−0.0954361
$299$	−802.828	−0.155280
$300$	0	0
$301$	176.480	0.0337945
$302$	445.851	0.0849531
$303$	0	0
$304$	−4800.25	−0.905636
$305$	0	0
$306$	0	0
$307$	9474.72	1.76140	0.880702	−	0.473671i	$-0.157071\pi$
$307$	9474.72	1.76140	0.880702	+	0.473671i	$0.157071\pi$
$308$	93.1672	0.0172360
$309$	0	0
$310$	0	0
$311$	5210.73	0.950075	0.475038	−	0.879965i	$-0.342434\pi$
$311$	5210.73	0.950075	0.475038	+	0.879965i	$0.342434\pi$
$312$	0	0
$313$	1944.27	0.351108	0.175554	−	0.984470i	$-0.443828\pi$
$313$	1944.27	0.351108	0.175554	+	0.984470i	$0.443828\pi$
$314$	−446.224	−0.0801970
$315$	0	0
$316$	5692.20	1.01333
$317$	6852.73	1.21416	0.607079	−	0.794642i	$-0.292341\pi$
$317$	6852.73	1.21416	0.607079	+	0.794642i	$0.292341\pi$
$318$	0	0
$319$	782.512	0.137343
$320$	0	0
$321$	0	0
$322$	30.1145	0.00521185
$323$	573.777	0.0988415
$324$	0	0
$325$	0	0
$326$	1469.92	0.249727
$327$	0	0
$328$	3045.72	0.512718
$329$	−265.736	−0.0445304
$330$	0	0
$331$	−2019.85	−0.335411	−0.167706	−	0.985837i	$-0.553636\pi$
$331$	−2019.85	−0.335411	−0.167706	+	0.985837i	$0.553636\pi$
$332$	386.651	0.0639163
$333$	0	0
$334$	1530.52	0.250737
$335$	0	0
$336$	0	0
$337$	11738.7	1.89748	0.948739	−	0.316060i	$-0.102360\pi$
$337$	11738.7	1.89748	0.948739	+	0.316060i	$0.102360\pi$
$338$	−1244.23	−0.200228
$339$	0	0
$340$	0	0
$341$	−2051.74	−0.325829
$342$	0	0
$343$	−775.704	−0.122111
$344$	−1744.73	−0.273458
$345$	0	0
$346$	11.2241	0.00174396
$347$	−12692.2	−1.96355	−0.981777	−	0.190036i	$-0.939140\pi$
$347$	−12692.2	−1.96355	−0.981777	+	0.190036i	$0.939140\pi$
$348$	0	0
$349$	12026.8	1.84465	0.922324	−	0.386417i	$-0.126287\pi$
$349$	12026.8	1.84465	0.922324	+	0.386417i	$0.126287\pi$
$350$	0	0
$351$	0	0
$352$	−1397.20	−0.211565
$353$	8948.39	1.34922	0.674610	−	0.738174i	$-0.264312\pi$
$353$	8948.39	1.34922	0.674610	+	0.738174i	$0.264312\pi$
$354$	0	0
$355$	0	0
$356$	−8038.90	−1.19680
$357$	0	0
$358$	−353.152	−0.0521360
$359$	−3906.52	−0.574313	−0.287156	−	0.957884i	$-0.592710\pi$
$359$	−3906.52	−0.574313	−0.287156	+	0.957884i	$0.592710\pi$
$360$	0	0
$361$	1760.14	0.256617
$362$	903.673	0.131204
$363$	0	0
$364$	185.120	0.0266563
$365$	0	0
$366$	0	0
$367$	4225.04	0.600941	0.300471	−	0.953791i	$-0.402856\pi$
$367$	4225.04	0.600941	0.300471	+	0.953791i	$0.402856\pi$
$368$	1899.21	0.269030
$369$	0	0
$370$	0	0
$371$	−220.932	−0.0309171
$372$	0	0
$373$	4223.50	0.586285	0.293143	−	0.956069i	$-0.405299\pi$
$373$	4223.50	0.586285	0.293143	+	0.956069i	$0.405299\pi$
$374$	49.1988	0.00680216
$375$	0	0
$376$	2627.13	0.360330
$377$	1554.82	0.212407
$378$	0	0
$379$	−748.009	−0.101379	−0.0506895	−	0.998714i	$-0.516142\pi$
$379$	−748.009	−0.101379	−0.0506895	+	0.998714i	$0.516142\pi$
$380$	0	0
$381$	0	0
$382$	1861.03	0.249264
$383$	−4668.07	−0.622786	−0.311393	−	0.950281i	$-0.600796\pi$
$383$	−4668.07	−0.622786	−0.311393	+	0.950281i	$0.600796\pi$
$384$	0	0
$385$	0	0
$386$	884.203	0.116593
$387$	0	0
$388$	5257.61	0.687924
$389$	−6544.12	−0.852957	−0.426478	−	0.904498i	$-0.640246\pi$
$389$	−6544.12	−0.852957	−0.426478	+	0.904498i	$0.640246\pi$
$390$	0	0
$391$	−227.014	−0.0293621
$392$	3827.21	0.493121
$393$	0	0
$394$	1033.66	0.132170
$395$	0	0
$396$	0	0
$397$	6439.84	0.814121	0.407061	−	0.913401i	$-0.366554\pi$
$397$	6439.84	0.814121	0.407061	+	0.913401i	$0.366554\pi$
$398$	−590.648	−0.0743882
$399$	0	0
$400$	0	0
$401$	−7727.03	−0.962268	−0.481134	−	0.876647i	$-0.659775\pi$
$401$	−7727.03	−0.962268	−0.481134	+	0.876647i	$0.659775\pi$
$402$	0	0
$403$	−4076.72	−0.503911
$404$	−6406.32	−0.788927
$405$	0	0
$406$	−58.3222	−0.00712926
$407$	3922.40	0.477705
$408$	0	0
$409$	−2539.97	−0.307074	−0.153537	−	0.988143i	$-0.549066\pi$
$409$	−2539.97	−0.307074	−0.153537	+	0.988143i	$0.549066\pi$
$410$	0	0
$411$	0	0
$412$	6025.80	0.720558
$413$	516.515	0.0615401
$414$	0	0
$415$	0	0
$416$	−2776.17	−0.327195
$417$	0	0
$418$	739.052	0.0864789
$419$	−1170.17	−0.136436	−0.0682181	−	0.997670i	$-0.521731\pi$
$419$	−1170.17	−0.136436	−0.0682181	+	0.997670i	$0.521731\pi$
$420$	0	0
$421$	4009.01	0.464103	0.232052	−	0.972703i	$-0.425456\pi$
$421$	4009.01	0.464103	0.232052	+	0.972703i	$0.425456\pi$
$422$	−653.749	−0.0754123
$423$	0	0
$424$	2184.19	0.250174
$425$	0	0
$426$	0	0
$427$	−499.916	−0.0566573
$428$	−12025.9	−1.35817
$429$	0	0
$430$	0	0
$431$	2676.70	0.299146	0.149573	−	0.988751i	$-0.452210\pi$
$431$	2676.70	0.299146	0.149573	+	0.988751i	$0.452210\pi$
$432$	0	0
$433$	5655.83	0.627718	0.313859	−	0.949470i	$-0.398378\pi$
$433$	5655.83	0.627718	0.313859	+	0.949470i	$0.398378\pi$
$434$	152.920	0.0169133
$435$	0	0
$436$	6920.97	0.760217
$437$	−3410.14	−0.373294
$438$	0	0
$439$	−6305.94	−0.685572	−0.342786	−	0.939414i	$-0.611371\pi$
$439$	−6305.94	−0.685572	−0.342786	+	0.939414i	$0.611371\pi$
$440$	0	0
$441$	0	0
$442$	97.7560	0.0105199
$443$	6772.22	0.726315	0.363158	−	0.931728i	$-0.381699\pi$
$443$	6772.22	0.726315	0.363158	+	0.931728i	$0.381699\pi$
$444$	0	0
$445$	0	0
$446$	386.424	0.0410262
$447$	0	0
$448$	−364.470	−0.0384365
$449$	5049.16	0.530701	0.265351	−	0.964152i	$-0.414512\pi$
$449$	5049.16	0.530701	0.265351	+	0.964152i	$0.414512\pi$
$450$	0	0
$451$	2991.34	0.312321
$452$	9082.27	0.945120
$453$	0	0
$454$	−504.070	−0.0521083
$455$	0	0
$456$	0	0
$457$	3866.14	0.395733	0.197867	−	0.980229i	$-0.436599\pi$
$457$	3866.14	0.395733	0.197867	+	0.980229i	$0.436599\pi$
$458$	−576.629	−0.0588299
$459$	0	0
$460$	0	0
$461$	9108.85	0.920263	0.460132	−	0.887851i	$-0.347802\pi$
$461$	9108.85	0.920263	0.460132	+	0.887851i	$0.347802\pi$
$462$	0	0
$463$	−17421.8	−1.74873	−0.874363	−	0.485273i	$-0.838720\pi$
$463$	−17421.8	−1.74873	−0.874363	+	0.485273i	$0.838720\pi$
$464$	−3678.16	−0.368005
$465$	0	0
$466$	4093.95	0.406971
$467$	−2602.36	−0.257864	−0.128932	−	0.991653i	$-0.541155\pi$
$467$	−2602.36	−0.257864	−0.128932	+	0.991653i	$0.541155\pi$
$468$	0	0
$469$	150.679	0.0148352
$470$	0	0
$471$	0	0
$472$	−5106.40	−0.497968
$473$	−1713.58	−0.166576
$474$	0	0
$475$	0	0
$476$	52.3458	0.00504047
$477$	0	0
$478$	1464.06	0.140093
$479$	−10751.8	−1.02560	−0.512801	−	0.858508i	$-0.671392\pi$
$479$	−10751.8	−1.02560	−0.512801	+	0.858508i	$0.671392\pi$
$480$	0	0
$481$	7793.65	0.738794
$482$	−2542.93	−0.240306
$483$	0	0
$484$	−904.630	−0.0849577
$485$	0	0
$486$	0	0
$487$	−2115.37	−0.196831	−0.0984154	−	0.995145i	$-0.531377\pi$
$487$	−2115.37	−0.196831	−0.0984154	+	0.995145i	$0.531377\pi$
$488$	4942.30	0.458458
$489$	0	0
$490$	0	0
$491$	−4634.30	−0.425953	−0.212977	−	0.977057i	$-0.568316\pi$
$491$	−4634.30	−0.425953	−0.212977	+	0.977057i	$0.568316\pi$
$492$	0	0
$493$	439.653	0.0401642
$494$	1468.47	0.133744
$495$	0	0
$496$	9644.09	0.873049
$497$	1180.10	0.106509
$498$	0	0
$499$	2495.43	0.223870	0.111935	−	0.993716i	$-0.464295\pi$
$499$	2495.43	0.223870	0.111935	+	0.993716i	$0.464295\pi$
$500$	0	0
$501$	0	0
$502$	−1313.00	−0.116737
$503$	−18521.0	−1.64177	−0.820886	−	0.571092i	$-0.806520\pi$
$503$	−18521.0	−1.64177	−0.820886	+	0.571092i	$0.806520\pi$
$504$	0	0
$505$	0	0
$506$	−292.404	−0.0256896
$507$	0	0
$508$	−8440.41	−0.737170
$509$	6102.40	0.531403	0.265702	−	0.964055i	$-0.414396\pi$
$509$	6102.40	0.531403	0.265702	+	0.964055i	$0.414396\pi$
$510$	0	0
$511$	−181.396	−0.0157035
$512$	11200.2	0.966765
$513$	0	0
$514$	−1046.06	−0.0897661
$515$	0	0
$516$	0	0
$517$	2580.23	0.219494
$518$	−292.344	−0.0247970
$519$	0	0
$520$	0	0
$521$	2813.12	0.236555	0.118277	−	0.992981i	$-0.462263\pi$
$521$	2813.12	0.236555	0.118277	+	0.992981i	$0.462263\pi$
$522$	0	0
$523$	1563.27	0.130702	0.0653509	−	0.997862i	$-0.479183\pi$
$523$	1563.27	0.130702	0.0653509	+	0.997862i	$0.479183\pi$
$524$	−11680.3	−0.973772
$525$	0	0
$526$	3192.75	0.264659
$527$	−1152.76	−0.0952850
$528$	0	0
$529$	−10817.8	−0.889109
$530$	0	0
$531$	0	0
$532$	786.326	0.0640818
$533$	5943.67	0.483019
$534$	0	0
$535$	0	0
$536$	−1489.65	−0.120043
$537$	0	0
$538$	−2862.28	−0.229371
$539$	3758.88	0.300383
$540$	0	0
$541$	−5959.61	−0.473611	−0.236806	−	0.971557i	$-0.576100\pi$
$541$	−5959.61	−0.473611	−0.236806	+	0.971557i	$0.576100\pi$
$542$	4817.55	0.381792
$543$	0	0
$544$	−785.012	−0.0618697
$545$	0	0
$546$	0	0
$547$	18084.2	1.41357	0.706787	−	0.707426i	$-0.250144\pi$
$547$	18084.2	1.41357	0.706787	+	0.707426i	$0.250144\pi$
$548$	335.927	0.0261863
$549$	0	0
$550$	0	0
$551$	6604.35	0.510626
$552$	0	0
$553$	−862.540	−0.0663272
$554$	6168.55	0.473062
$555$	0	0
$556$	3106.95	0.236985
$557$	−18129.5	−1.37912	−0.689561	−	0.724227i	$-0.742197\pi$
$557$	−18129.5	−1.37912	−0.689561	+	0.724227i	$0.742197\pi$
$558$	0	0
$559$	−3404.81	−0.257618
$560$	0	0
$561$	0	0
$562$	3047.43	0.228733
$563$	17910.2	1.34072	0.670361	−	0.742035i	$-0.266139\pi$
$563$	17910.2	1.34072	0.670361	+	0.742035i	$0.266139\pi$
$564$	0	0
$565$	0	0
$566$	2554.36	0.189696
$567$	0	0
$568$	−11666.8	−0.861843
$569$	−24176.6	−1.78126	−0.890629	−	0.454731i	$-0.849735\pi$
$569$	−24176.6	−1.78126	−0.890629	+	0.454731i	$0.849735\pi$
$570$	0	0
$571$	−13167.8	−0.965073	−0.482537	−	0.875876i	$-0.660285\pi$
$571$	−13167.8	−0.965073	−0.482537	+	0.875876i	$0.660285\pi$
$572$	−1797.46	−0.131391
$573$	0	0
$574$	−222.950	−0.0162121
$575$	0	0
$576$	0	0
$577$	−722.513	−0.0521293	−0.0260646	−	0.999660i	$-0.508298\pi$
$577$	−722.513	−0.0521293	−0.0260646	+	0.999660i	$0.508298\pi$
$578$	−3527.83	−0.253872
$579$	0	0
$580$	0	0
$581$	−58.5893	−0.00418364
$582$	0	0
$583$	2145.19	0.152393
$584$	1793.33	0.127069
$585$	0	0
$586$	5146.10	0.362770
$587$	17038.8	1.19807	0.599034	−	0.800723i	$-0.295551\pi$
$587$	17038.8	1.19807	0.599034	+	0.800723i	$0.295551\pi$
$588$	0	0
$589$	−17316.5	−1.21140
$590$	0	0
$591$	0	0
$592$	−18437.0	−1.28000
$593$	−6206.74	−0.429815	−0.214908	−	0.976634i	$-0.568945\pi$
$593$	−6206.74	−0.429815	−0.214908	+	0.976634i	$0.568945\pi$
$594$	0	0
$595$	0	0
$596$	5071.92	0.348580
$597$	0	0
$598$	−580.996	−0.0397302
$599$	23050.7	1.57233	0.786164	−	0.618018i	$-0.212064\pi$
$599$	23050.7	1.57233	0.786164	+	0.618018i	$0.212064\pi$
$600$	0	0
$601$	−6323.69	−0.429199	−0.214600	−	0.976702i	$-0.568845\pi$
$601$	−6323.69	−0.429199	−0.214600	+	0.976702i	$0.568845\pi$
$602$	127.716	0.00864673
$603$	0	0
$604$	−4606.01	−0.310291
$605$	0	0
$606$	0	0
$607$	4625.14	0.309273	0.154636	−	0.987971i	$-0.450579\pi$
$607$	4625.14	0.309273	0.154636	+	0.987971i	$0.450579\pi$
$608$	−11792.3	−0.786577
$609$	0	0
$610$	0	0
$611$	5126.81	0.339457
$612$	0	0
$613$	−28075.3	−1.84984	−0.924919	−	0.380164i	$-0.875868\pi$
$613$	−28075.3	−1.84984	−0.924919	+	0.380164i	$0.875868\pi$
$614$	6856.72	0.450676
$615$	0	0
$616$	139.571	0.00912900
$617$	7264.85	0.474023	0.237011	−	0.971507i	$-0.423832\pi$
$617$	7264.85	0.474023	0.237011	+	0.971507i	$0.423832\pi$
$618$	0	0
$619$	−30583.2	−1.98585	−0.992927	−	0.118728i	$-0.962119\pi$
$619$	−30583.2	−1.98585	−0.992927	+	0.118728i	$0.962119\pi$
$620$	0	0
$621$	0	0
$622$	3770.93	0.243088
$623$	1218.14	0.0783365
$624$	0	0
$625$	0	0
$626$	1407.04	0.0898351
$627$	0	0
$628$	4609.86	0.292920
$629$	2203.79	0.139699
$630$	0	0
$631$	−16679.8	−1.05232	−0.526159	−	0.850386i	$-0.676368\pi$
$631$	−16679.8	−1.05232	−0.526159	+	0.850386i	$0.676368\pi$
$632$	8527.29	0.536705
$633$	0	0
$634$	4959.23	0.310656
$635$	0	0
$636$	0	0
$637$	7468.75	0.464557
$638$	566.293	0.0351407
$639$	0	0
$640$	0	0
$641$	−4225.31	−0.260358	−0.130179	−	0.991490i	$-0.541555\pi$
$641$	−4225.31	−0.260358	−0.130179	+	0.991490i	$0.541555\pi$
$642$	0	0
$643$	23703.5	1.45377	0.726886	−	0.686758i	$-0.240967\pi$
$643$	23703.5	1.45377	0.726886	+	0.686758i	$0.240967\pi$
$644$	−311.108	−0.0190363
$645$	0	0
$646$	415.234	0.0252898
$647$	10045.1	0.610374	0.305187	−	0.952292i	$-0.401281\pi$
$647$	10045.1	0.610374	0.305187	+	0.952292i	$0.401281\pi$
$648$	0	0
$649$	−5015.23	−0.303336
$650$	0	0
$651$	0	0
$652$	−15185.5	−0.912130
$653$	32314.2	1.93652	0.968262	−	0.249936i	$-0.0804096\pi$
$653$	32314.2	1.93652	0.968262	+	0.249936i	$0.0804096\pi$
$654$	0	0
$655$	0	0
$656$	−14060.6	−0.836853
$657$	0	0
$658$	−192.309	−0.0113936
$659$	12032.4	0.711253	0.355627	−	0.934628i	$-0.384267\pi$
$659$	12032.4	0.711253	0.355627	+	0.934628i	$0.384267\pi$
$660$	0	0
$661$	12864.5	0.756988	0.378494	−	0.925604i	$-0.376442\pi$
$661$	12864.5	0.756988	0.378494	+	0.925604i	$0.376442\pi$
$662$	−1461.74	−0.0858189
$663$	0	0
$664$	579.229	0.0338531
$665$	0	0
$666$	0	0
$667$	−2613.00	−0.151688
$668$	−15811.5	−0.915819
$669$	0	0
$670$	0	0
$671$	4854.06	0.279268
$672$	0	0
$673$	12007.5	0.687751	0.343876	−	0.939015i	$-0.388260\pi$
$673$	12007.5	0.687751	0.343876	+	0.939015i	$0.388260\pi$
$674$	8495.16	0.485492
$675$	0	0
$676$	12853.9	0.731332
$677$	10732.5	0.609284	0.304642	−	0.952467i	$-0.401463\pi$
$677$	10732.5	0.609284	0.304642	+	0.952467i	$0.401463\pi$
$678$	0	0
$679$	−796.687	−0.0450280
$680$	0	0
$681$	0	0
$682$	−1484.81	−0.0833672
$683$	−14126.0	−0.791383	−0.395692	−	0.918383i	$-0.629495\pi$
$683$	−14126.0	−0.791383	−0.395692	+	0.918383i	$0.629495\pi$
$684$	0	0
$685$	0	0
$686$	−561.366	−0.0312435
$687$	0	0
$688$	8054.59	0.446335
$689$	4262.42	0.235682
$690$	0	0
$691$	−18467.8	−1.01671	−0.508357	−	0.861147i	$-0.669747\pi$
$691$	−18467.8	−1.01671	−0.508357	+	0.861147i	$0.669747\pi$
$692$	−115.954	−0.00636981
$693$	0	0
$694$	−9185.17	−0.502398
$695$	0	0
$696$	0	0
$697$	1680.68	0.0913345
$698$	8703.66	0.471975
$699$	0	0
$700$	0	0
$701$	24876.0	1.34030	0.670152	−	0.742224i	$-0.266229\pi$
$701$	24876.0	1.34030	0.670152	+	0.742224i	$0.266229\pi$
$702$	0	0
$703$	33104.8	1.77606
$704$	3538.91	0.189457
$705$	0	0
$706$	6475.83	0.345214
$707$	970.752	0.0516391
$708$	0	0
$709$	21271.1	1.12673	0.563367	−	0.826207i	$-0.309506\pi$
$709$	21271.1	1.12673	0.563367	+	0.826207i	$0.309506\pi$
$710$	0	0
$711$	0	0
$712$	−12042.8	−0.633881
$713$	6851.25	0.359862
$714$	0	0
$715$	0	0
$716$	3648.36	0.190427
$717$	0	0
$718$	−2827.09	−0.146945
$719$	−19213.9	−0.996603	−0.498302	−	0.867004i	$-0.666043\pi$
$719$	−19213.9	−0.996603	−0.498302	+	0.867004i	$0.666043\pi$
$720$	0	0
$721$	−913.091	−0.0471641
$722$	1273.79	0.0656585
$723$	0	0
$724$	−9335.70	−0.479224
$725$	0	0
$726$	0	0
$727$	−27454.7	−1.40060	−0.700302	−	0.713846i	$-0.746951\pi$
$727$	−27454.7	−1.40060	−0.700302	+	0.713846i	$0.746951\pi$
$728$	277.322	0.0141184
$729$	0	0
$730$	0	0
$731$	−962.770	−0.0487132
$732$	0	0
$733$	−1182.16	−0.0595691	−0.0297846	−	0.999556i	$-0.509482\pi$
$733$	−1182.16	−0.0595691	−0.0297846	+	0.999556i	$0.509482\pi$
$734$	3057.61	0.153758
$735$	0	0
$736$	4665.58	0.233662
$737$	−1463.05	−0.0731237
$738$	0	0
$739$	25688.2	1.27869	0.639347	−	0.768918i	$-0.279205\pi$
$739$	25688.2	1.27869	0.639347	+	0.768918i	$0.279205\pi$
$740$	0	0
$741$	0	0
$742$	−159.886	−0.00791049
$743$	−32431.2	−1.60132	−0.800662	−	0.599116i	$-0.795519\pi$
$743$	−32431.2	−1.60132	−0.800662	+	0.599116i	$0.795519\pi$
$744$	0	0
$745$	0	0
$746$	3056.49	0.150008
$747$	0	0
$748$	−508.264	−0.0248449
$749$	1822.29	0.0888988
$750$	0	0
$751$	−32093.5	−1.55940	−0.779700	−	0.626154i	$-0.784628\pi$
$751$	−32093.5	−1.55940	−0.779700	+	0.626154i	$0.784628\pi$
$752$	−12128.2	−0.588127
$753$	0	0
$754$	1125.20	0.0543467
$755$	0	0
$756$	0	0
$757$	−1293.36	−0.0620977	−0.0310488	−	0.999518i	$-0.509885\pi$
$757$	−1293.36	−0.0620977	−0.0310488	+	0.999518i	$0.509885\pi$
$758$	−541.323	−0.0259390
$759$	0	0
$760$	0	0
$761$	−29013.1	−1.38203	−0.691014	−	0.722841i	$-0.742836\pi$
$761$	−29013.1	−1.38203	−0.691014	+	0.722841i	$0.742836\pi$
$762$	0	0
$763$	−1048.74	−0.0497599
$764$	−19226.0	−0.910436
$765$	0	0
$766$	−3378.22	−0.159347
$767$	−9965.06	−0.469123
$768$	0	0
$769$	−31304.1	−1.46795	−0.733976	−	0.679176i	$-0.762337\pi$
$769$	−31304.1	−1.46795	−0.733976	+	0.679176i	$0.762337\pi$
$770$	0	0
$771$	0	0
$772$	−9134.55	−0.425855
$773$	−41913.5	−1.95023	−0.975113	−	0.221710i	$-0.928836\pi$
$773$	−41913.5	−1.95023	−0.975113	+	0.221710i	$0.928836\pi$
$774$	0	0
$775$	0	0
$776$	7876.24	0.364357
$777$	0	0
$778$	−4735.89	−0.218239
$779$	25246.7	1.16118
$780$	0	0
$781$	−11458.5	−0.524989
$782$	−164.287	−0.00751263
$783$	0	0
$784$	−17668.4	−0.804867
$785$	0	0
$786$	0	0
$787$	−6648.00	−0.301113	−0.150556	−	0.988601i	$-0.548106\pi$
$787$	−6648.00	−0.301113	−0.150556	+	0.988601i	$0.548106\pi$
$788$	−10678.6	−0.482751
$789$	0	0
$790$	0	0
$791$	−1376.24	−0.0618628
$792$	0	0
$793$	9644.82	0.431901
$794$	4660.42	0.208302
$795$	0	0
$796$	6101.88	0.271703
$797$	−14798.4	−0.657701	−0.328850	−	0.944382i	$-0.606661\pi$
$797$	−14798.4	−0.657701	−0.328850	+	0.944382i	$0.606661\pi$
$798$	0	0
$799$	1449.69	0.0641884
$800$	0	0
$801$	0	0
$802$	−5591.94	−0.246207
$803$	1761.31	0.0774038
$804$	0	0
$805$	0	0
$806$	−2950.27	−0.128931
$807$	0	0
$808$	−9597.09	−0.417852
$809$	−4500.04	−0.195566	−0.0977831	−	0.995208i	$-0.531175\pi$
$809$	−4500.04	−0.195566	−0.0977831	+	0.995208i	$0.531175\pi$
$810$	0	0
$811$	−1909.49	−0.0826773	−0.0413386	−	0.999145i	$-0.513162\pi$
$811$	−1909.49	−0.0826773	−0.0413386	+	0.999145i	$0.513162\pi$
$812$	602.517	0.0260396
$813$	0	0
$814$	2838.58	0.122226
$815$	0	0
$816$	0	0
$817$	−14462.5	−0.619313
$818$	−1838.14	−0.0785685
$819$	0	0
$820$	0	0
$821$	−10928.6	−0.464569	−0.232285	−	0.972648i	$-0.574620\pi$
$821$	−10928.6	−0.464569	−0.232285	+	0.972648i	$0.574620\pi$
$822$	0	0
$823$	29278.4	1.24007	0.620036	−	0.784573i	$-0.287118\pi$
$823$	29278.4	1.24007	0.620036	+	0.784573i	$0.287118\pi$
$824$	9027.05	0.381641
$825$	0	0
$826$	373.795	0.0157457
$827$	878.905	0.0369559	0.0184779	−	0.999829i	$-0.494118\pi$
$827$	878.905	0.0369559	0.0184779	+	0.999829i	$0.494118\pi$
$828$	0	0
$829$	16130.1	0.675779	0.337890	−	0.941186i	$-0.390287\pi$
$829$	16130.1	0.675779	0.337890	+	0.941186i	$0.390287\pi$
$830$	0	0
$831$	0	0
$832$	7031.67	0.293004
$833$	2111.92	0.0878436
$834$	0	0
$835$	0	0
$836$	−7635.02	−0.315864
$837$	0	0
$838$	−846.838	−0.0349088
$839$	39531.8	1.62669	0.813343	−	0.581784i	$-0.197645\pi$
$839$	39531.8	1.62669	0.813343	+	0.581784i	$0.197645\pi$
$840$	0	0
$841$	−19328.5	−0.792507
$842$	2901.27	0.118746
$843$	0	0
$844$	6753.77	0.275443
$845$	0	0
$846$	0	0
$847$	137.079	0.00556090
$848$	−10083.4	−0.408331
$849$	0	0
$850$	0	0
$851$	−13097.8	−0.527600
$852$	0	0
$853$	−5373.68	−0.215699	−0.107849	−	0.994167i	$-0.534396\pi$
$853$	−5373.68	−0.215699	−0.107849	+	0.994167i	$0.534396\pi$
$854$	−361.782	−0.0144964
$855$	0	0
$856$	−18015.7	−0.719349
$857$	45378.4	1.80875	0.904374	−	0.426740i	$-0.140338\pi$
$857$	45378.4	1.80875	0.904374	+	0.426740i	$0.140338\pi$
$858$	0	0
$859$	35336.8	1.40358	0.701791	−	0.712383i	$-0.252384\pi$
$859$	35336.8	1.40358	0.701791	+	0.712383i	$0.252384\pi$
$860$	0	0
$861$	0	0
$862$	1937.09	0.0765401
$863$	24595.3	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$863$	24595.3	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$864$	0	0
$865$	0	0
$866$	4093.05	0.160609
$867$	0	0
$868$	−1579.79	−0.0617760
$869$	8375.04	0.326932
$870$	0	0
$871$	−2907.02	−0.113089
$872$	10368.1	0.402646
$873$	0	0
$874$	−2467.87	−0.0955115
$875$	0	0
$876$	0	0
$877$	−8251.06	−0.317695	−0.158848	−	0.987303i	$-0.550778\pi$
$877$	−8251.06	−0.317695	−0.158848	+	0.987303i	$0.550778\pi$
$878$	−4563.52	−0.175412
$879$	0	0
$880$	0	0
$881$	3528.36	0.134930	0.0674651	−	0.997722i	$-0.478509\pi$
$881$	3528.36	0.134930	0.0674651	+	0.997722i	$0.478509\pi$
$882$	0	0
$883$	35459.1	1.35141	0.675703	−	0.737174i	$-0.263840\pi$
$883$	35459.1	1.35141	0.675703	+	0.737174i	$0.263840\pi$
$884$	−1009.90	−0.0384238
$885$	0	0
$886$	4900.96	0.185836
$887$	−32544.8	−1.23196	−0.615979	−	0.787763i	$-0.711239\pi$
$887$	−32544.8	−1.23196	−0.615979	+	0.787763i	$0.711239\pi$
$888$	0	0
$889$	1278.98	0.0482514
$890$	0	0
$891$	0	0
$892$	−3992.08	−0.149848
$893$	21777.0	0.816056
$894$	0	0
$895$	0	0
$896$	−1414.93	−0.0527562
$897$	0	0
$898$	3654.01	0.135786
$899$	−13268.7	−0.492253
$900$	0	0
$901$	1205.27	0.0445654
$902$	2164.79	0.0799108
$903$	0	0
$904$	13605.9	0.500579
$905$	0	0
$906$	0	0
$907$	20383.4	0.746219	0.373110	−	0.927787i	$-0.378291\pi$
$907$	20383.4	0.746219	0.373110	+	0.927787i	$0.378291\pi$
$908$	5207.46	0.190326
$909$	0	0
$910$	0	0
$911$	−7783.29	−0.283065	−0.141532	−	0.989934i	$-0.545203\pi$
$911$	−7783.29	−0.283065	−0.141532	+	0.989934i	$0.545203\pi$
$912$	0	0
$913$	568.887	0.0206215
$914$	2797.87	0.101253
$915$	0	0
$916$	5957.05	0.214876
$917$	1769.92	0.0637382
$918$	0	0
$919$	27294.9	0.979734	0.489867	−	0.871797i	$-0.337045\pi$
$919$	27294.9	0.979734	0.489867	+	0.871797i	$0.337045\pi$
$920$	0	0
$921$	0	0
$922$	6591.95	0.235460
$923$	−22767.5	−0.811920
$924$	0	0
$925$	0	0
$926$	−12607.9	−0.447432
$927$	0	0
$928$	−9035.73	−0.319625
$929$	46468.7	1.64111	0.820554	−	0.571568i	$-0.193665\pi$
$929$	46468.7	1.64111	0.820554	+	0.571568i	$0.193665\pi$
$930$	0	0
$931$	31724.7	1.11679
$932$	−42293.9	−1.48646
$933$	0	0
$934$	−1883.29	−0.0659776
$935$	0	0
$936$	0	0
$937$	6922.45	0.241352	0.120676	−	0.992692i	$-0.461494\pi$
$937$	6922.45	0.241352	0.120676	+	0.992692i	$0.461494\pi$
$938$	109.044	0.00379575
$939$	0	0
$940$	0	0
$941$	23772.1	0.823536	0.411768	−	0.911289i	$-0.364911\pi$
$941$	23772.1	0.823536	0.411768	+	0.911289i	$0.364911\pi$
$942$	0	0
$943$	−9988.80	−0.344942
$944$	23573.8	0.812779
$945$	0	0
$946$	−1240.09	−0.0426204
$947$	−7177.76	−0.246300	−0.123150	−	0.992388i	$-0.539300\pi$
$947$	−7177.76	−0.246300	−0.123150	+	0.992388i	$0.539300\pi$
$948$	0	0
$949$	3499.65	0.119709
$950$	0	0
$951$	0	0
$952$	78.4175	0.00266967
$953$	27530.2	0.935772	0.467886	−	0.883789i	$-0.345016\pi$
$953$	27530.2	0.935772	0.467886	+	0.883789i	$0.345016\pi$
$954$	0	0
$955$	0	0
$956$	−15124.9	−0.511690
$957$	0	0
$958$	−7780.94	−0.262412
$959$	−50.9031	−0.00171402
$960$	0	0
$961$	4999.29	0.167812
$962$	5640.15	0.189029
$963$	0	0
$964$	26270.6	0.877718
$965$	0	0
$966$	0	0
$967$	−55407.1	−1.84258	−0.921288	−	0.388881i	$-0.872862\pi$
$967$	−55407.1	−1.84258	−0.921288	+	0.388881i	$0.872862\pi$
$968$	−1355.20	−0.0449976
$969$	0	0
$970$	0	0
$971$	55100.7	1.82108	0.910539	−	0.413423i	$-0.135667\pi$
$971$	55100.7	1.82108	0.910539	+	0.413423i	$0.135667\pi$
$972$	0	0
$973$	−470.797	−0.0155119
$974$	−1530.86	−0.0503615
$975$	0	0
$976$	−22816.3	−0.748290
$977$	−44473.9	−1.45634	−0.728170	−	0.685396i	$-0.759629\pi$
$977$	−44473.9	−1.45634	−0.728170	+	0.685396i	$0.759629\pi$
$978$	0	0
$979$	−11827.8	−0.386127
$980$	0	0
$981$	0	0
$982$	−3353.78	−0.108985
$983$	28003.0	0.908605	0.454302	−	0.890848i	$-0.349889\pi$
$983$	28003.0	0.908605	0.454302	+	0.890848i	$0.349889\pi$
$984$	0	0
$985$	0	0
$986$	318.170	0.0102765
$987$	0	0
$988$	−15170.5	−0.488499
$989$	5722.06	0.183974
$990$	0	0
$991$	−40719.5	−1.30524	−0.652622	−	0.757684i	$-0.726331\pi$
$991$	−40719.5	−1.30524	−0.652622	+	0.757684i	$0.726331\pi$
$992$	23691.6	0.758274
$993$	0	0
$994$	854.022	0.0272514
$995$	0	0
$996$	0	0
$997$	18161.1	0.576898	0.288449	−	0.957495i	$-0.406860\pi$
$997$	18161.1	0.576898	0.288449	+	0.957495i	$0.406860\pi$
$998$	1805.91	0.0572796
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.4.a.y.1.2		3
3.2	odd	2		825.4.a.o.1.2	✓	3
5.4	even	2		2475.4.a.v.1.2		3
15.2	even	4		825.4.c.n.199.3		6
15.8	even	4		825.4.c.n.199.4		6
15.14	odd	2		825.4.a.q.1.2	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.4.a.o.1.2	✓	3	3.2	odd	2
825.4.a.q.1.2	yes	3	15.14	odd	2
825.4.c.n.199.3		6	15.2	even	4
825.4.c.n.199.4		6	15.8	even	4
2475.4.a.v.1.2		3	5.4	even	2
2475.4.a.y.1.2		3	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2475.a (trivial)

\(f(q)\)	\(=\)	\(q+0.723686 q^{2} -7.47628 q^{4} +1.13288 q^{7} -11.2000 q^{8} -11.0000 q^{11} -21.8566 q^{13} +0.819851 q^{14} +51.7050 q^{16} -6.18033 q^{17} -92.8393 q^{19} -7.96054 q^{22} +36.7317 q^{23} -15.8173 q^{26} -8.46975 q^{28} -71.1375 q^{29} +186.522 q^{31} +127.018 q^{32} -4.47261 q^{34} -356.581 q^{37} -67.1865 q^{38} -271.940 q^{41} +155.780 q^{43} +82.2391 q^{44} +26.5822 q^{46} -234.566 q^{47} -341.717 q^{49} +163.406 q^{52} -195.018 q^{53} -12.6882 q^{56} -51.4812 q^{58} +455.930 q^{59} -441.278 q^{61} +134.983 q^{62} -321.719 q^{64} +133.005 q^{67} +46.2058 q^{68} +1041.68 q^{71} -160.119 q^{73} -258.053 q^{74} +694.093 q^{76} -12.4617 q^{77} -761.367 q^{79} -196.799 q^{82} -51.7170 q^{83} +112.736 q^{86} +123.200 q^{88} +1075.25 q^{89} -24.7609 q^{91} -274.616 q^{92} -169.752 q^{94} -703.238 q^{97} -247.295 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} + 7 q^{4} - 16 q^{7} - 3 q^{8} - 33 q^{11} - 45 q^{13} + 116 q^{14} - 85 q^{16} - 58 q^{17} - 169 q^{19} - 11 q^{22} + 155 q^{23} - 167 q^{26} - 100 q^{28} + 277 q^{29} - 173 q^{31} + 97 q^{32}+ \cdots - 2273 q^{98}+O(q^{100}) \) 3 * q + q^2 + 7 * q^4 - 16 * q^7 - 3 * q^8 - 33 * q^11 - 45 * q^13 + 116 * q^14 - 85 * q^16 - 58 * q^17 - 169 * q^19 - 11 * q^22 + 155 * q^23 - 167 * q^26 - 100 * q^28 + 277 * q^29 - 173 * q^31 + 97 * q^32 - 146 * q^34 - 60 * q^37 + 169 * q^38 - 44 * q^41 + 109 * q^43 - 77 * q^44 + 425 * q^46 + 270 * q^47 - 427 * q^49 - 45 * q^52 - 148 * q^53 - 168 * q^56 + 783 * q^58 + 684 * q^59 - 1038 * q^61 + 953 * q^62 - 1129 * q^64 - 314 * q^67 - 366 * q^68 + 1459 * q^71 + 1170 * q^73 + 1764 * q^74 + 211 * q^76 + 176 * q^77 - 506 * q^79 + 1040 * q^82 - 347 * q^83 + 527 * q^86 + 33 * q^88 + 607 * q^89 - 398 * q^91 + 687 * q^92 - 1822 * q^94 - 1263 * q^97 - 2273 * q^98

Introduction

L-functions

Modular forms

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