LMFDB - Embedded newform 400.4.c.b.49.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,4,Mod(49,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("400.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	400.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:	$23.6007640023$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 200)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	400.49
Dual form			400.4.c.b.49.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.00000i q^{3} -26.0000i q^{7} -54.0000 q^{9} +O(q^{10})$	$q+9.00000i q^{3} -26.0000i q^{7} -54.0000 q^{9} +59.0000 q^{11} -28.0000i q^{13} +5.00000i q^{17} +109.000 q^{19} +234.000 q^{21} -194.000i q^{23} -243.000i q^{27} +32.0000 q^{29} -10.0000 q^{31} +531.000i q^{33} -198.000i q^{37} +252.000 q^{39} +117.000 q^{41} +388.000i q^{43} +68.0000i q^{47} -333.000 q^{49} -45.0000 q^{51} +18.0000i q^{53} +981.000i q^{57} +392.000 q^{59} -710.000 q^{61} +1404.00i q^{63} +253.000i q^{67} +1746.00 q^{69} +612.000 q^{71} +549.000i q^{73} -1534.00i q^{77} +414.000 q^{79} +729.000 q^{81} -121.000i q^{83} +288.000i q^{87} +81.0000 q^{89} -728.000 q^{91} -90.0000i q^{93} -1502.00i q^{97} -3186.00 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 108 q^{9}+O(q^{10}) $ 2 * q - 108 * q^9	$ 2 q - 108 q^{9} + 118 q^{11} + 218 q^{19} + 468 q^{21} + 64 q^{29} - 20 q^{31} + 504 q^{39} + 234 q^{41} - 666 q^{49} - 90 q^{51} + 784 q^{59} - 1420 q^{61} + 3492 q^{69} + 1224 q^{71} + 828 q^{79} + 1458 q^{81} + 162 q^{89} - 1456 q^{91} - 6372 q^{99}+O(q^{100}) $ 2 * q - 108 * q^9 + 118 * q^11 + 218 * q^19 + 468 * q^21 + 64 * q^29 - 20 * q^31 + 504 * q^39 + 234 * q^41 - 666 * q^49 - 90 * q^51 + 784 * q^59 - 1420 * q^61 + 3492 * q^69 + 1224 * q^71 + 828 * q^79 + 1458 * q^81 + 162 * q^89 - 1456 * q^91 - 6372 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$.

$n$	$101$	$177$	$351$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	9.00000i	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$3$	9.00000i	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 26.0000i	− 1.40387i	−0.712242	−	0.701934i	$-0.752320\pi$
$7$	− 26.0000i	− 1.40387i	0.712242	−	0.701934i	$-0.247680\pi$
$8$	0	0
$9$	−54.0000	−2.00000
$10$	0	0
$11$	59.0000	1.61720	0.808599	−	0.588361i	$-0.200226\pi$
$11$	59.0000	1.61720	0.808599	+	0.588361i	$0.200226\pi$
$12$	0	0
$13$	− 28.0000i	− 0.597369i	−0.954352	−	0.298685i	$-0.903452\pi$
$13$	− 28.0000i	− 0.597369i	0.954352	−	0.298685i	$-0.0965479\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.00000i	0.0713340i	0.999364	+	0.0356670i	$0.0113556\pi$
$17$	5.00000i	0.0713340i	−0.999364	+	0.0356670i	$0.988644\pi$
$18$	0	0
$19$	109.000	1.31612	0.658061	−	0.752965i	$-0.271377\pi$
$19$	109.000	1.31612	0.658061	+	0.752965i	$0.271377\pi$
$20$	0	0
$21$	234.000	2.43157
$22$	0	0
$23$	− 194.000i	− 1.75877i	−0.476108	−	0.879387i	$-0.657953\pi$
$23$	− 194.000i	− 1.75877i	0.476108	−	0.879387i	$-0.342047\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 243.000i	− 1.73205i
$28$	0	0
$29$	32.0000	0.204905	0.102453	−	0.994738i	$-0.467331\pi$
$29$	32.0000	0.204905	0.102453	+	0.994738i	$0.467331\pi$
$30$	0	0
$31$	−10.0000	−0.0579372	−0.0289686	−	0.999580i	$-0.509222\pi$
$31$	−10.0000	−0.0579372	−0.0289686	+	0.999580i	$0.509222\pi$
$32$	0	0
$33$	531.000i	2.80107i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 198.000i	− 0.879757i	−0.898057	−	0.439878i	$-0.855022\pi$
$37$	− 198.000i	− 0.879757i	0.898057	−	0.439878i	$-0.144978\pi$
$38$	0	0
$39$	252.000	1.03467
$40$	0	0
$41$	117.000	0.445667	0.222833	−	0.974857i	$-0.428469\pi$
$41$	117.000	0.445667	0.222833	+	0.974857i	$0.428469\pi$
$42$	0	0
$43$	388.000i	1.37603i	0.725695	+	0.688017i	$0.241518\pi$
$43$	388.000i	1.37603i	−0.725695	+	0.688017i	$0.758482\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	68.0000i	0.211039i	0.994417	+	0.105519i	$0.0336505\pi$
$47$	68.0000i	0.211039i	−0.994417	+	0.105519i	$0.966350\pi$
$48$	0	0
$49$	−333.000	−0.970845
$50$	0	0
$51$	−45.0000	−0.123554
$52$	0	0
$53$	18.0000i	0.0466508i	0.999728	+	0.0233254i	$0.00742537\pi$
$53$	18.0000i	0.0466508i	−0.999728	+	0.0233254i	$0.992575\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	981.000i	2.27959i
$58$	0	0
$59$	392.000	0.864984	0.432492	−	0.901638i	$-0.357634\pi$
$59$	392.000	0.864984	0.432492	+	0.901638i	$0.357634\pi$
$60$	0	0
$61$	−710.000	−1.49027	−0.745133	−	0.666916i	$-0.767614\pi$
$61$	−710.000	−1.49027	−0.745133	+	0.666916i	$0.767614\pi$
$62$	0	0
$63$	1404.00i	2.80774i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	253.000i	0.461326i	0.973034	+	0.230663i	$0.0740896\pi$
$67$	253.000i	0.461326i	−0.973034	+	0.230663i	$0.925910\pi$
$68$	0	0
$69$	1746.00	3.04629
$70$	0	0
$71$	612.000	1.02297	0.511486	−	0.859292i	$-0.329095\pi$
$71$	612.000	1.02297	0.511486	+	0.859292i	$0.329095\pi$
$72$	0	0
$73$	549.000i	0.880214i	0.897945	+	0.440107i	$0.145059\pi$
$73$	549.000i	0.880214i	−0.897945	+	0.440107i	$0.854941\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 1534.00i	− 2.27033i
$78$	0	0
$79$	414.000	0.589603	0.294802	−	0.955559i	$-0.404746\pi$
$79$	414.000	0.589603	0.294802	+	0.955559i	$0.404746\pi$
$80$	0	0
$81$	729.000	1.00000
$82$	0	0
$83$	− 121.000i	− 0.160018i	−0.996794	−	0.0800089i	$-0.974505\pi$
$83$	− 121.000i	− 0.160018i	0.996794	−	0.0800089i	$-0.0254949\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	288.000i	0.354906i
$88$	0	0
$89$	81.0000	0.0964717	0.0482359	−	0.998836i	$-0.484640\pi$
$89$	81.0000	0.0964717	0.0482359	+	0.998836i	$0.484640\pi$
$90$	0	0
$91$	−728.000	−0.838628
$92$	0	0
$93$	− 90.0000i	− 0.100350i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 1502.00i	− 1.57222i	−0.618089	−	0.786108i	$-0.712093\pi$
$97$	− 1502.00i	− 1.57222i	0.618089	−	0.786108i	$-0.287907\pi$
$98$	0	0
$99$	−3186.00	−3.23439
$100$	0	0
$101$	−234.000	−0.230533	−0.115267	−	0.993335i	$-0.536772\pi$
$101$	−234.000	−0.230533	−0.115267	+	0.993335i	$0.536772\pi$
$102$	0	0
$103$	− 1172.00i	− 1.12117i	−0.828097	−	0.560585i	$-0.810576\pi$
$103$	− 1172.00i	− 1.12117i	0.828097	−	0.560585i	$-0.189424\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 1125.00i	− 1.01643i	−0.861231	−	0.508214i	$-0.830306\pi$
$107$	− 1125.00i	− 1.01643i	0.861231	−	0.508214i	$-0.169694\pi$
$108$	0	0
$109$	1234.00	1.08436	0.542182	−	0.840261i	$-0.317598\pi$
$109$	1234.00	1.08436	0.542182	+	0.840261i	$0.317598\pi$
$110$	0	0
$111$	1782.00	1.52378
$112$	0	0
$113$	− 567.000i	− 0.472025i	−0.971750	−	0.236013i	$-0.924159\pi$
$113$	− 567.000i	− 0.472025i	0.971750	−	0.236013i	$-0.0758407\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1512.00i	1.19474i
$118$	0	0
$119$	130.000	0.100144
$120$	0	0
$121$	2150.00	1.61533
$122$	0	0
$123$	1053.00i	0.771917i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	2358.00i	1.64755i	0.566918	+	0.823774i	$0.308136\pi$
$127$	2358.00i	1.64755i	−0.566918	+	0.823774i	$0.691864\pi$
$128$	0	0
$129$	−3492.00	−2.38336
$130$	0	0
$131$	1692.00	1.12848	0.564239	−	0.825611i	$-0.309169\pi$
$131$	1692.00	1.12848	0.564239	+	0.825611i	$0.309169\pi$
$132$	0	0
$133$	− 2834.00i	− 1.84766i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	229.000i	0.142809i	0.997447	+	0.0714043i	$0.0227481\pi$
$137$	229.000i	0.142809i	−0.997447	+	0.0714043i	$0.977252\pi$
$138$	0	0
$139$	2781.00	1.69699	0.848494	−	0.529205i	$-0.177510\pi$
$139$	2781.00	1.69699	0.848494	+	0.529205i	$0.177510\pi$
$140$	0	0
$141$	−612.000	−0.365530
$142$	0	0
$143$	− 1652.00i	− 0.966064i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 2997.00i	− 1.68155i
$148$	0	0
$149$	−1472.00	−0.809335	−0.404668	−	0.914464i	$-0.632613\pi$
$149$	−1472.00	−0.809335	−0.404668	+	0.914464i	$0.632613\pi$
$150$	0	0
$151$	−1322.00	−0.712469	−0.356235	−	0.934397i	$-0.615940\pi$
$151$	−1322.00	−0.712469	−0.356235	+	0.934397i	$0.615940\pi$
$152$	0	0
$153$	− 270.000i	− 0.142668i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	298.000i	0.151484i	0.997127	+	0.0757420i	$0.0241325\pi$
$157$	298.000i	0.151484i	−0.997127	+	0.0757420i	$0.975867\pi$
$158$	0	0
$159$	−162.000	−0.0808015
$160$	0	0
$161$	−5044.00	−2.46909
$162$	0	0
$163$	− 341.000i	− 0.163860i	−0.996638	−	0.0819300i	$-0.973892\pi$
$163$	− 341.000i	− 0.163860i	0.996638	−	0.0819300i	$-0.0261084\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 684.000i	− 0.316943i	−0.987364	−	0.158472i	$-0.949343\pi$
$167$	− 684.000i	− 0.316943i	0.987364	−	0.158472i	$-0.0506566\pi$
$168$	0	0
$169$	1413.00	0.643150
$170$	0	0
$171$	−5886.00	−2.63224
$172$	0	0
$173$	2344.00i	1.03012i	0.857154	+	0.515061i	$0.172231\pi$
$173$	2344.00i	1.03012i	−0.857154	+	0.515061i	$0.827769\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	3528.00i	1.49820i
$178$	0	0
$179$	−1111.00	−0.463911	−0.231955	−	0.972726i	$-0.574512\pi$
$179$	−1111.00	−0.463911	−0.231955	+	0.972726i	$0.574512\pi$
$180$	0	0
$181$	2042.00	0.838567	0.419284	−	0.907855i	$-0.362281\pi$
$181$	2042.00	0.838567	0.419284	+	0.907855i	$0.362281\pi$
$182$	0	0
$183$	− 6390.00i	− 2.58122i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	295.000i	0.115361i
$188$	0	0
$189$	−6318.00	−2.43157
$190$	0	0
$191$	−5270.00	−1.99646	−0.998230	−	0.0594735i	$-0.981058\pi$
$191$	−5270.00	−1.99646	−0.998230	+	0.0594735i	$0.981058\pi$
$192$	0	0
$193$	− 613.000i	− 0.228625i	−0.993445	−	0.114313i	$-0.963533\pi$
$193$	− 613.000i	− 0.228625i	0.993445	−	0.114313i	$-0.0364666\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1174.00i	0.424589i	0.977206	+	0.212295i	$0.0680936\pi$
$197$	1174.00i	0.424589i	−0.977206	+	0.212295i	$0.931906\pi$
$198$	0	0
$199$	3428.00	1.22113	0.610564	−	0.791967i	$-0.290943\pi$
$199$	3428.00	1.22113	0.610564	+	0.791967i	$0.290943\pi$
$200$	0	0
$201$	−2277.00	−0.799041
$202$	0	0
$203$	− 832.000i	− 0.287660i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	10476.0i	3.51755i
$208$	0	0
$209$	6431.00	2.12843
$210$	0	0
$211$	−2339.00	−0.763144	−0.381572	−	0.924339i	$-0.624617\pi$
$211$	−2339.00	−0.763144	−0.381572	+	0.924339i	$0.624617\pi$
$212$	0	0
$213$	5508.00i	1.77184i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	260.000i	0.0813362i
$218$	0	0
$219$	−4941.00	−1.52457
$220$	0	0
$221$	140.000	0.0426128
$222$	0	0
$223$	3932.00i	1.18075i	0.807131	+	0.590373i	$0.201019\pi$
$223$	3932.00i	1.18075i	−0.807131	+	0.590373i	$0.798981\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 6084.00i	− 1.77890i	−0.457037	−	0.889448i	$-0.651089\pi$
$227$	− 6084.00i	− 1.77890i	0.457037	−	0.889448i	$-0.348911\pi$
$228$	0	0
$229$	−4996.00	−1.44168	−0.720841	−	0.693101i	$-0.756244\pi$
$229$	−4996.00	−1.44168	−0.720841	+	0.693101i	$0.756244\pi$
$230$	0	0
$231$	13806.0	3.93233
$232$	0	0
$233$	− 3222.00i	− 0.905924i	−0.891530	−	0.452962i	$-0.850367\pi$
$233$	− 3222.00i	− 0.905924i	0.891530	−	0.452962i	$-0.149633\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	3726.00i	1.02122i
$238$	0	0
$239$	2736.00	0.740490	0.370245	−	0.928934i	$-0.379274\pi$
$239$	2736.00	0.740490	0.370245	+	0.928934i	$0.379274\pi$
$240$	0	0
$241$	1673.00	0.447168	0.223584	−	0.974685i	$-0.428224\pi$
$241$	1673.00	0.447168	0.223584	+	0.974685i	$0.428224\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 3052.00i	− 0.786211i
$248$	0	0
$249$	1089.00	0.277159
$250$	0	0
$251$	−5355.00	−1.34663	−0.673316	−	0.739355i	$-0.735131\pi$
$251$	−5355.00	−1.34663	−0.673316	+	0.739355i	$0.735131\pi$
$252$	0	0
$253$	− 11446.0i	− 2.84428i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 5490.00i	− 1.33252i	−0.745721	−	0.666258i	$-0.767895\pi$
$257$	− 5490.00i	− 1.33252i	0.745721	−	0.666258i	$-0.232105\pi$
$258$	0	0
$259$	−5148.00	−1.23506
$260$	0	0
$261$	−1728.00	−0.409810
$262$	0	0
$263$	3150.00i	0.738545i	0.929321	+	0.369272i	$0.120393\pi$
$263$	3150.00i	0.738545i	−0.929321	+	0.369272i	$0.879607\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	729.000i	0.167094i
$268$	0	0
$269$	−176.000	−0.0398919	−0.0199459	−	0.999801i	$-0.506349\pi$
$269$	−176.000	−0.0398919	−0.0199459	+	0.999801i	$0.506349\pi$
$270$	0	0
$271$	−2394.00	−0.536624	−0.268312	−	0.963332i	$-0.586466\pi$
$271$	−2394.00	−0.536624	−0.268312	+	0.963332i	$0.586466\pi$
$272$	0	0
$273$	− 6552.00i	− 1.45255i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 6256.00i	− 1.35699i	−0.734604	−	0.678496i	$-0.762632\pi$
$277$	− 6256.00i	− 1.35699i	0.734604	−	0.678496i	$-0.237368\pi$
$278$	0	0
$279$	540.000	0.115874
$280$	0	0
$281$	−4802.00	−1.01944	−0.509721	−	0.860340i	$-0.670251\pi$
$281$	−4802.00	−1.01944	−0.509721	+	0.860340i	$0.670251\pi$
$282$	0	0
$283$	2123.00i	0.445934i	0.974826	+	0.222967i	$0.0715742\pi$
$283$	2123.00i	0.445934i	−0.974826	+	0.222967i	$0.928426\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 3042.00i	− 0.625657i
$288$	0	0
$289$	4888.00	0.994911
$290$	0	0
$291$	13518.0	2.72316
$292$	0	0
$293$	8834.00i	1.76139i	0.473682	+	0.880696i	$0.342925\pi$
$293$	8834.00i	1.76139i	−0.473682	+	0.880696i	$0.657075\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 14337.0i	− 2.80107i
$298$	0	0
$299$	−5432.00	−1.05064
$300$	0	0
$301$	10088.0	1.93177
$302$	0	0
$303$	− 2106.00i	− 0.399296i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 1369.00i	− 0.254505i	−0.991870	−	0.127252i	$-0.959384\pi$
$307$	− 1369.00i	− 0.254505i	0.991870	−	0.127252i	$-0.0406158\pi$
$308$	0	0
$309$	10548.0	1.94192
$310$	0	0
$311$	10426.0	1.90098	0.950489	−	0.310758i	$-0.100583\pi$
$311$	10426.0	1.90098	0.950489	+	0.310758i	$0.100583\pi$
$312$	0	0
$313$	3574.00i	0.645413i	0.946499	+	0.322707i	$0.104593\pi$
$313$	3574.00i	0.645413i	−0.946499	+	0.322707i	$0.895407\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	9036.00i	1.60099i	0.599343	+	0.800493i	$0.295429\pi$
$317$	9036.00i	1.60099i	−0.599343	+	0.800493i	$0.704571\pi$
$318$	0	0
$319$	1888.00	0.331372
$320$	0	0
$321$	10125.0	1.76051
$322$	0	0
$323$	545.000i	0.0938842i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	11106.0i	1.87817i
$328$	0	0
$329$	1768.00	0.296271
$330$	0	0
$331$	−10233.0	−1.69926	−0.849632	−	0.527376i	$-0.823176\pi$
$331$	−10233.0	−1.69926	−0.849632	+	0.527376i	$0.823176\pi$
$332$	0	0
$333$	10692.0i	1.75951i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	4627.00i	0.747919i	0.927445	+	0.373960i	$0.122000\pi$
$337$	4627.00i	0.747919i	−0.927445	+	0.373960i	$0.878000\pi$
$338$	0	0
$339$	5103.00	0.817572
$340$	0	0
$341$	−590.000	−0.0936959
$342$	0	0
$343$	− 260.000i	− 0.0409291i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 4901.00i	− 0.758212i	−0.925353	−	0.379106i	$-0.876232\pi$
$347$	− 4901.00i	− 0.758212i	0.925353	−	0.379106i	$-0.123768\pi$
$348$	0	0
$349$	4482.00	0.687438	0.343719	−	0.939072i	$-0.388313\pi$
$349$	4482.00	0.687438	0.343719	+	0.939072i	$0.388313\pi$
$350$	0	0
$351$	−6804.00	−1.03467
$352$	0	0
$353$	1210.00i	0.182441i	0.995831	+	0.0912207i	$0.0290769\pi$
$353$	1210.00i	0.182441i	−0.995831	+	0.0912207i	$0.970923\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	1170.00i	0.173454i
$358$	0	0
$359$	−9882.00	−1.45279	−0.726396	−	0.687277i	$-0.758806\pi$
$359$	−9882.00	−1.45279	−0.726396	+	0.687277i	$0.758806\pi$
$360$	0	0
$361$	5022.00	0.732177
$362$	0	0
$363$	19350.0i	2.79783i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	11260.0i	1.60155i	0.598968	+	0.800773i	$0.295578\pi$
$367$	11260.0i	1.60155i	−0.598968	+	0.800773i	$0.704422\pi$
$368$	0	0
$369$	−6318.00	−0.891333
$370$	0	0
$371$	468.000	0.0654915
$372$	0	0
$373$	3230.00i	0.448373i	0.974546	+	0.224186i	$0.0719724\pi$
$373$	3230.00i	0.448373i	−0.974546	+	0.224186i	$0.928028\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 896.000i	− 0.122404i
$378$	0	0
$379$	11575.0	1.56878	0.784390	−	0.620268i	$-0.212976\pi$
$379$	11575.0	1.56878	0.784390	+	0.620268i	$0.212976\pi$
$380$	0	0
$381$	−21222.0	−2.85364
$382$	0	0
$383$	− 18.0000i	− 0.00240145i	−0.999999	−	0.00120073i	$-0.999618\pi$
$383$	− 18.0000i	− 0.00240145i	0.999999	−	0.00120073i	$-0.000382203\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 20952.0i	− 2.75207i
$388$	0	0
$389$	−10710.0	−1.39593	−0.697967	−	0.716130i	$-0.745912\pi$
$389$	−10710.0	−1.39593	−0.697967	+	0.716130i	$0.745912\pi$
$390$	0	0
$391$	970.000	0.125460
$392$	0	0
$393$	15228.0i	1.95458i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 3788.00i	− 0.478877i	−0.970912	−	0.239439i	$-0.923037\pi$
$397$	− 3788.00i	− 0.478877i	0.970912	−	0.239439i	$-0.0769634\pi$
$398$	0	0
$399$	25506.0	3.20024
$400$	0	0
$401$	−10539.0	−1.31245	−0.656225	−	0.754565i	$-0.727848\pi$
$401$	−10539.0	−1.31245	−0.656225	+	0.754565i	$0.727848\pi$
$402$	0	0
$403$	280.000i	0.0346099i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 11682.0i	− 1.42274i
$408$	0	0
$409$	−5581.00	−0.674725	−0.337363	−	0.941375i	$-0.609535\pi$
$409$	−5581.00	−0.674725	−0.337363	+	0.941375i	$0.609535\pi$
$410$	0	0
$411$	−2061.00	−0.247352
$412$	0	0
$413$	− 10192.0i	− 1.21432i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	25029.0i	2.93927i
$418$	0	0
$419$	5193.00	0.605476	0.302738	−	0.953074i	$-0.402099\pi$
$419$	5193.00	0.605476	0.302738	+	0.953074i	$0.402099\pi$
$420$	0	0
$421$	4788.00	0.554282	0.277141	−	0.960829i	$-0.410613\pi$
$421$	4788.00	0.554282	0.277141	+	0.960829i	$0.410613\pi$
$422$	0	0
$423$	− 3672.00i	− 0.422077i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	18460.0i	2.09214i
$428$	0	0
$429$	14868.0	1.67327
$430$	0	0
$431$	−8006.00	−0.894746	−0.447373	−	0.894348i	$-0.647640\pi$
$431$	−8006.00	−0.894746	−0.447373	+	0.894348i	$0.647640\pi$
$432$	0	0
$433$	2395.00i	0.265811i	0.991129	+	0.132906i	$0.0424308\pi$
$433$	2395.00i	0.265811i	−0.991129	+	0.132906i	$0.957569\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 21146.0i	− 2.31476i
$438$	0	0
$439$	1864.00	0.202651	0.101326	−	0.994853i	$-0.467692\pi$
$439$	1864.00	0.202651	0.101326	+	0.994853i	$0.467692\pi$
$440$	0	0
$441$	17982.0	1.94169
$442$	0	0
$443$	5463.00i	0.585903i	0.956127	+	0.292951i	$0.0946374\pi$
$443$	5463.00i	0.585903i	−0.956127	+	0.292951i	$0.905363\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 13248.0i	− 1.40181i
$448$	0	0
$449$	−12969.0	−1.36313	−0.681565	−	0.731758i	$-0.738700\pi$
$449$	−12969.0	−1.36313	−0.681565	+	0.731758i	$0.738700\pi$
$450$	0	0
$451$	6903.00	0.720731
$452$	0	0
$453$	− 11898.0i	− 1.23403i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	18313.0i	1.87450i	0.348659	+	0.937249i	$0.386637\pi$
$457$	18313.0i	1.87450i	−0.348659	+	0.937249i	$0.613363\pi$
$458$	0	0
$459$	1215.00	0.123554
$460$	0	0
$461$	12492.0	1.26206	0.631031	−	0.775758i	$-0.282632\pi$
$461$	12492.0	1.26206	0.631031	+	0.775758i	$0.282632\pi$
$462$	0	0
$463$	4428.00i	0.444464i	0.974994	+	0.222232i	$0.0713342\pi$
$463$	4428.00i	0.444464i	−0.974994	+	0.222232i	$0.928666\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1084.00i	0.107412i	0.998557	+	0.0537061i	$0.0171034\pi$
$467$	1084.00i	0.107412i	−0.998557	+	0.0537061i	$0.982897\pi$
$468$	0	0
$469$	6578.00	0.647641
$470$	0	0
$471$	−2682.00	−0.262378
$472$	0	0
$473$	22892.0i	2.22532i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 972.000i	− 0.0933015i
$478$	0	0
$479$	−13082.0	−1.24787	−0.623937	−	0.781474i	$-0.714468\pi$
$479$	−13082.0	−1.24787	−0.623937	+	0.781474i	$0.714468\pi$
$480$	0	0
$481$	−5544.00	−0.525540
$482$	0	0
$483$	− 45396.0i	− 4.27658i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	3014.00i	0.280446i	0.990120	+	0.140223i	$0.0447820\pi$
$487$	3014.00i	0.280446i	−0.990120	+	0.140223i	$0.955218\pi$
$488$	0	0
$489$	3069.00	0.283814
$490$	0	0
$491$	3564.00	0.327579	0.163789	−	0.986495i	$-0.447628\pi$
$491$	3564.00	0.327579	0.163789	+	0.986495i	$0.447628\pi$
$492$	0	0
$493$	160.000i	0.0146167i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 15912.0i	− 1.43612i
$498$	0	0
$499$	−15796.0	−1.41709	−0.708543	−	0.705667i	$-0.750647\pi$
$499$	−15796.0	−1.41709	−0.708543	+	0.705667i	$0.750647\pi$
$500$	0	0
$501$	6156.00	0.548962
$502$	0	0
$503$	− 10908.0i	− 0.966926i	−0.875365	−	0.483463i	$-0.839379\pi$
$503$	− 10908.0i	− 0.966926i	0.875365	−	0.483463i	$-0.160621\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	12717.0i	1.11397i
$508$	0	0
$509$	−21946.0	−1.91108	−0.955540	−	0.294863i	$-0.904726\pi$
$509$	−21946.0	−1.91108	−0.955540	+	0.294863i	$0.904726\pi$
$510$	0	0
$511$	14274.0	1.23570
$512$	0	0
$513$	− 26487.0i	− 2.27959i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	4012.00i	0.341291i
$518$	0	0
$519$	−21096.0	−1.78422
$520$	0	0
$521$	−6395.00	−0.537754	−0.268877	−	0.963174i	$-0.586653\pi$
$521$	−6395.00	−0.537754	−0.268877	+	0.963174i	$0.586653\pi$
$522$	0	0
$523$	− 5615.00i	− 0.469459i	−0.972061	−	0.234729i	$-0.924580\pi$
$523$	− 5615.00i	− 0.469459i	0.972061	−	0.234729i	$-0.0754204\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 50.0000i	− 0.00413289i
$528$	0	0
$529$	−25469.0	−2.09329
$530$	0	0
$531$	−21168.0	−1.72997
$532$	0	0
$533$	− 3276.00i	− 0.266228i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 9999.00i	− 0.803517i
$538$	0	0
$539$	−19647.0	−1.57005
$540$	0	0
$541$	−4112.00	−0.326781	−0.163391	−	0.986561i	$-0.552243\pi$
$541$	−4112.00	−0.326781	−0.163391	+	0.986561i	$0.552243\pi$
$542$	0	0
$543$	18378.0i	1.45244i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	2167.00i	0.169386i	0.996407	+	0.0846931i	$0.0269910\pi$
$547$	2167.00i	0.169386i	−0.996407	+	0.0846931i	$0.973009\pi$
$548$	0	0
$549$	38340.0	2.98053
$550$	0	0
$551$	3488.00	0.269680
$552$	0	0
$553$	− 10764.0i	− 0.827725i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 19444.0i	− 1.47912i	−0.673092	−	0.739559i	$-0.735034\pi$
$557$	− 19444.0i	− 1.47912i	0.673092	−	0.739559i	$-0.264966\pi$
$558$	0	0
$559$	10864.0	0.822000
$560$	0	0
$561$	−2655.00	−0.199811
$562$	0	0
$563$	20416.0i	1.52830i	0.645040	+	0.764149i	$0.276841\pi$
$563$	20416.0i	1.52830i	−0.645040	+	0.764149i	$0.723159\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 18954.0i	− 1.40387i
$568$	0	0
$569$	3127.00	0.230388	0.115194	−	0.993343i	$-0.463251\pi$
$569$	3127.00	0.230388	0.115194	+	0.993343i	$0.463251\pi$
$570$	0	0
$571$	22580.0	1.65489	0.827446	−	0.561545i	$-0.189793\pi$
$571$	22580.0	1.65489	0.827446	+	0.561545i	$0.189793\pi$
$572$	0	0
$573$	− 47430.0i	− 3.45797i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 829.000i	− 0.0598123i	−0.999553	−	0.0299062i	$-0.990479\pi$
$577$	− 829.000i	− 0.0598123i	0.999553	−	0.0299062i	$-0.00952085\pi$
$578$	0	0
$579$	5517.00	0.395991
$580$	0	0
$581$	−3146.00	−0.224644
$582$	0	0
$583$	1062.00i	0.0754435i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 7119.00i	− 0.500567i	−0.968173	−	0.250283i	$-0.919476\pi$
$587$	− 7119.00i	− 0.500567i	0.968173	−	0.250283i	$-0.0805238\pi$
$588$	0	0
$589$	−1090.00	−0.0762524
$590$	0	0
$591$	−10566.0	−0.735410
$592$	0	0
$593$	8217.00i	0.569025i	0.958672	+	0.284512i	$0.0918317\pi$
$593$	8217.00i	0.569025i	−0.958672	+	0.284512i	$0.908168\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	30852.0i	2.11506i
$598$	0	0
$599$	−90.0000	−0.00613907	−0.00306953	−	0.999995i	$-0.500977\pi$
$599$	−90.0000	−0.00613907	−0.00306953	+	0.999995i	$0.500977\pi$
$600$	0	0
$601$	−17117.0	−1.16176	−0.580879	−	0.813990i	$-0.697291\pi$
$601$	−17117.0	−1.16176	−0.580879	+	0.813990i	$0.697291\pi$
$602$	0	0
$603$	− 13662.0i	− 0.922653i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	15120.0i	1.01104i	0.862815	+	0.505520i	$0.168699\pi$
$607$	15120.0i	1.01104i	−0.862815	+	0.505520i	$0.831301\pi$
$608$	0	0
$609$	7488.00	0.498241
$610$	0	0
$611$	1904.00	0.126068
$612$	0	0
$613$	6570.00i	0.432887i	0.976295	+	0.216444i	$0.0694457\pi$
$613$	6570.00i	0.432887i	−0.976295	+	0.216444i	$0.930554\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	18846.0i	1.22968i	0.788653	+	0.614839i	$0.210779\pi$
$617$	18846.0i	1.22968i	−0.788653	+	0.614839i	$0.789221\pi$
$618$	0	0
$619$	16316.0	1.05944	0.529722	−	0.848172i	$-0.322296\pi$
$619$	16316.0	1.05944	0.529722	+	0.848172i	$0.322296\pi$
$620$	0	0
$621$	−47142.0	−3.04629
$622$	0	0
$623$	− 2106.00i	− 0.135434i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	57879.0i	3.68655i
$628$	0	0
$629$	990.000	0.0627566
$630$	0	0
$631$	−20170.0	−1.27251	−0.636256	−	0.771478i	$-0.719518\pi$
$631$	−20170.0	−1.27251	−0.636256	+	0.771478i	$0.719518\pi$
$632$	0	0
$633$	− 21051.0i	− 1.32180i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	9324.00i	0.579953i
$638$	0	0
$639$	−33048.0	−2.04594
$640$	0	0
$641$	−12726.0	−0.784160	−0.392080	−	0.919931i	$-0.628244\pi$
$641$	−12726.0	−0.784160	−0.392080	+	0.919931i	$0.628244\pi$
$642$	0	0
$643$	2196.00i	0.134684i	0.997730	+	0.0673420i	$0.0214519\pi$
$643$	2196.00i	0.134684i	−0.997730	+	0.0673420i	$0.978548\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	16884.0i	1.02593i	0.858409	+	0.512966i	$0.171453\pi$
$647$	16884.0i	1.02593i	−0.858409	+	0.512966i	$0.828547\pi$
$648$	0	0
$649$	23128.0	1.39885
$650$	0	0
$651$	−2340.00	−0.140878
$652$	0	0
$653$	4018.00i	0.240791i	0.992726	+	0.120395i	$0.0384163\pi$
$653$	4018.00i	0.240791i	−0.992726	+	0.120395i	$0.961584\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 29646.0i	− 1.76043i
$658$	0	0
$659$	19071.0	1.12732	0.563658	−	0.826009i	$-0.309394\pi$
$659$	19071.0	1.12732	0.563658	+	0.826009i	$0.309394\pi$
$660$	0	0
$661$	17424.0	1.02529	0.512644	−	0.858601i	$-0.328666\pi$
$661$	17424.0	1.02529	0.512644	+	0.858601i	$0.328666\pi$
$662$	0	0
$663$	1260.00i	0.0738075i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 6208.00i	− 0.360382i
$668$	0	0
$669$	−35388.0	−2.04511
$670$	0	0
$671$	−41890.0	−2.41005
$672$	0	0
$673$	5382.00i	0.308263i	0.988050	+	0.154131i	$0.0492579\pi$
$673$	5382.00i	0.308263i	−0.988050	+	0.154131i	$0.950742\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 4496.00i	− 0.255237i	−0.991823	−	0.127618i	$-0.959267\pi$
$677$	− 4496.00i	− 0.255237i	0.991823	−	0.127618i	$-0.0407333\pi$
$678$	0	0
$679$	−39052.0	−2.20718
$680$	0	0
$681$	54756.0	3.08114
$682$	0	0
$683$	3249.00i	0.182020i	0.995850	+	0.0910099i	$0.0290095\pi$
$683$	3249.00i	0.182020i	−0.995850	+	0.0910099i	$0.970991\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 44964.0i	− 2.49706i
$688$	0	0
$689$	504.000	0.0278677
$690$	0	0
$691$	13399.0	0.737658	0.368829	−	0.929497i	$-0.379759\pi$
$691$	13399.0	0.737658	0.368829	+	0.929497i	$0.379759\pi$
$692$	0	0
$693$	82836.0i	4.54066i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	585.000i	0.0317912i
$698$	0	0
$699$	28998.0	1.56911
$700$	0	0
$701$	−18148.0	−0.977804	−0.488902	−	0.872339i	$-0.662602\pi$
$701$	−18148.0	−0.977804	−0.488902	+	0.872339i	$0.662602\pi$
$702$	0	0
$703$	− 21582.0i	− 1.15787i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6084.00i	0.323638i
$708$	0	0
$709$	−4868.00	−0.257858	−0.128929	−	0.991654i	$-0.541154\pi$
$709$	−4868.00	−0.257858	−0.128929	+	0.991654i	$0.541154\pi$
$710$	0	0
$711$	−22356.0	−1.17921
$712$	0	0
$713$	1940.00i	0.101898i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	24624.0i	1.28257i
$718$	0	0
$719$	−17366.0	−0.900755	−0.450377	−	0.892838i	$-0.648711\pi$
$719$	−17366.0	−0.900755	−0.450377	+	0.892838i	$0.648711\pi$
$720$	0	0
$721$	−30472.0	−1.57398
$722$	0	0
$723$	15057.0i	0.774517i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 21824.0i	− 1.11335i	−0.830729	−	0.556676i	$-0.812076\pi$
$727$	− 21824.0i	− 1.11335i	0.830729	−	0.556676i	$-0.187924\pi$
$728$	0	0
$729$	19683.0	1.00000
$730$	0	0
$731$	−1940.00	−0.0981580
$732$	0	0
$733$	31428.0i	1.58366i	0.610744	+	0.791828i	$0.290870\pi$
$733$	31428.0i	1.58366i	−0.610744	+	0.791828i	$0.709130\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	14927.0i	0.746056i
$738$	0	0
$739$	−14292.0	−0.711420	−0.355710	−	0.934596i	$-0.615761\pi$
$739$	−14292.0	−0.711420	−0.355710	+	0.934596i	$0.615761\pi$
$740$	0	0
$741$	27468.0	1.36176
$742$	0	0
$743$	13950.0i	0.688797i	0.938824	+	0.344398i	$0.111917\pi$
$743$	13950.0i	0.688797i	−0.938824	+	0.344398i	$0.888083\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	6534.00i	0.320036i
$748$	0	0
$749$	−29250.0	−1.42693
$750$	0	0
$751$	38736.0	1.88215	0.941076	−	0.338194i	$-0.109816\pi$
$751$	38736.0	1.88215	0.941076	+	0.338194i	$0.109816\pi$
$752$	0	0
$753$	− 48195.0i	− 2.33243i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 3664.00i	− 0.175919i	−0.996124	−	0.0879593i	$-0.971965\pi$
$757$	− 3664.00i	− 0.175919i	0.996124	−	0.0879593i	$-0.0280345\pi$
$758$	0	0
$759$	103014.	4.92644
$760$	0	0
$761$	19557.0	0.931591	0.465795	−	0.884892i	$-0.345768\pi$
$761$	19557.0	0.931591	0.465795	+	0.884892i	$0.345768\pi$
$762$	0	0
$763$	− 32084.0i	− 1.52231i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 10976.0i	− 0.516715i
$768$	0	0
$769$	13283.0	0.622883	0.311442	−	0.950265i	$-0.399188\pi$
$769$	13283.0	0.622883	0.311442	+	0.950265i	$0.399188\pi$
$770$	0	0
$771$	49410.0	2.30799
$772$	0	0
$773$	− 24840.0i	− 1.15580i	−0.816108	−	0.577900i	$-0.803873\pi$
$773$	− 24840.0i	− 1.15580i	0.816108	−	0.577900i	$-0.196127\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 46332.0i	− 2.13919i
$778$	0	0
$779$	12753.0	0.586552
$780$	0	0
$781$	36108.0	1.65435
$782$	0	0
$783$	− 7776.00i	− 0.354906i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 18044.0i	− 0.817280i	−0.912696	−	0.408640i	$-0.866003\pi$
$787$	− 18044.0i	− 0.817280i	0.912696	−	0.408640i	$-0.133997\pi$
$788$	0	0
$789$	−28350.0	−1.27920
$790$	0	0
$791$	−14742.0	−0.662661
$792$	0	0
$793$	19880.0i	0.890239i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 6174.00i	− 0.274397i	−0.990544	−	0.137198i	$-0.956190\pi$
$797$	− 6174.00i	− 0.274397i	0.990544	−	0.137198i	$-0.0438098\pi$
$798$	0	0
$799$	−340.000	−0.0150542
$800$	0	0
$801$	−4374.00	−0.192943
$802$	0	0
$803$	32391.0i	1.42348i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 1584.00i	− 0.0690947i
$808$	0	0
$809$	1998.00	0.0868306	0.0434153	−	0.999057i	$-0.486176\pi$
$809$	1998.00	0.0868306	0.0434153	+	0.999057i	$0.486176\pi$
$810$	0	0
$811$	7156.00	0.309841	0.154921	−	0.987927i	$-0.450488\pi$
$811$	7156.00	0.309841	0.154921	+	0.987927i	$0.450488\pi$
$812$	0	0
$813$	− 21546.0i	− 0.929460i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	42292.0i	1.81103i
$818$	0	0
$819$	39312.0	1.67726
$820$	0	0
$821$	27922.0	1.18695	0.593474	−	0.804853i	$-0.297756\pi$
$821$	27922.0	1.18695	0.593474	+	0.804853i	$0.297756\pi$
$822$	0	0
$823$	22636.0i	0.958738i	0.877613	+	0.479369i	$0.159134\pi$
$823$	22636.0i	0.958738i	−0.877613	+	0.479369i	$0.840866\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 26559.0i	− 1.11674i	−0.829591	−	0.558372i	$-0.811426\pi$
$827$	− 26559.0i	− 1.11674i	0.829591	−	0.558372i	$-0.188574\pi$
$828$	0	0
$829$	−12580.0	−0.527046	−0.263523	−	0.964653i	$-0.584885\pi$
$829$	−12580.0	−0.527046	−0.263523	+	0.964653i	$0.584885\pi$
$830$	0	0
$831$	56304.0	2.35038
$832$	0	0
$833$	− 1665.00i	− 0.0692543i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	2430.00i	0.100350i
$838$	0	0
$839$	−11344.0	−0.466792	−0.233396	−	0.972382i	$-0.574984\pi$
$839$	−11344.0	−0.466792	−0.233396	+	0.972382i	$0.574984\pi$
$840$	0	0
$841$	−23365.0	−0.958014
$842$	0	0
$843$	− 43218.0i	− 1.76573i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 55900.0i	− 2.26771i
$848$	0	0
$849$	−19107.0	−0.772380
$850$	0	0
$851$	−38412.0	−1.54729
$852$	0	0
$853$	− 14786.0i	− 0.593509i	−0.954954	−	0.296754i	$-0.904096\pi$
$853$	− 14786.0i	− 0.593509i	0.954954	−	0.296754i	$-0.0959043\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 29259.0i	− 1.16624i	−0.812386	−	0.583120i	$-0.801832\pi$
$857$	− 29259.0i	− 1.16624i	0.812386	−	0.583120i	$-0.198168\pi$
$858$	0	0
$859$	−13651.0	−0.542219	−0.271109	−	0.962549i	$-0.587391\pi$
$859$	−13651.0	−0.542219	−0.271109	+	0.962549i	$0.587391\pi$
$860$	0	0
$861$	27378.0	1.08367
$862$	0	0
$863$	29016.0i	1.14451i	0.820074	+	0.572257i	$0.193932\pi$
$863$	29016.0i	1.14451i	−0.820074	+	0.572257i	$0.806068\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	43992.0i	1.72324i
$868$	0	0
$869$	24426.0	0.953504
$870$	0	0
$871$	7084.00	0.275582
$872$	0	0
$873$	81108.0i	3.14443i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	21412.0i	0.824438i	0.911085	+	0.412219i	$0.135246\pi$
$877$	21412.0i	0.824438i	−0.911085	+	0.412219i	$0.864754\pi$
$878$	0	0
$879$	−79506.0	−3.05082
$880$	0	0
$881$	−1170.00	−0.0447427	−0.0223713	−	0.999750i	$-0.507122\pi$
$881$	−1170.00	−0.0447427	−0.0223713	+	0.999750i	$0.507122\pi$
$882$	0	0
$883$	− 12655.0i	− 0.482304i	−0.970487	−	0.241152i	$-0.922475\pi$
$883$	− 12655.0i	− 0.482304i	0.970487	−	0.241152i	$-0.0775253\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 32764.0i	− 1.24026i	−0.784500	−	0.620128i	$-0.787081\pi$
$887$	− 32764.0i	− 1.24026i	0.784500	−	0.620128i	$-0.212919\pi$
$888$	0	0
$889$	61308.0	2.31294
$890$	0	0
$891$	43011.0	1.61720
$892$	0	0
$893$	7412.00i	0.277753i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 48888.0i	− 1.81976i
$898$	0	0
$899$	−320.000	−0.0118716
$900$	0	0
$901$	−90.0000	−0.00332779
$902$	0	0
$903$	90792.0i	3.34592i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	29844.0i	1.09256i	0.837602	+	0.546281i	$0.183957\pi$
$907$	29844.0i	1.09256i	−0.837602	+	0.546281i	$0.816043\pi$
$908$	0	0
$909$	12636.0	0.461067
$910$	0	0
$911$	−15628.0	−0.568363	−0.284182	−	0.958770i	$-0.591722\pi$
$911$	−15628.0	−0.568363	−0.284182	+	0.958770i	$0.591722\pi$
$912$	0	0
$913$	− 7139.00i	− 0.258780i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 43992.0i	− 1.58424i
$918$	0	0
$919$	42974.0	1.54253	0.771263	−	0.636517i	$-0.219625\pi$
$919$	42974.0	1.54253	0.771263	+	0.636517i	$0.219625\pi$
$920$	0	0
$921$	12321.0	0.440815
$922$	0	0
$923$	− 17136.0i	− 0.611092i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	63288.0i	2.24234i
$928$	0	0
$929$	−13342.0	−0.471191	−0.235596	−	0.971851i	$-0.575704\pi$
$929$	−13342.0	−0.471191	−0.235596	+	0.971851i	$0.575704\pi$
$930$	0	0
$931$	−36297.0	−1.27775
$932$	0	0
$933$	93834.0i	3.29259i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 3005.00i	− 0.104770i	−0.998627	−	0.0523848i	$-0.983318\pi$
$937$	− 3005.00i	− 0.104770i	0.998627	−	0.0523848i	$-0.0166822\pi$
$938$	0	0
$939$	−32166.0	−1.11789
$940$	0	0
$941$	16204.0	0.561355	0.280678	−	0.959802i	$-0.409441\pi$
$941$	16204.0	0.561355	0.280678	+	0.959802i	$0.409441\pi$
$942$	0	0
$943$	− 22698.0i	− 0.783827i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	30200.0i	1.03629i	0.855292	+	0.518146i	$0.173378\pi$
$947$	30200.0i	1.03629i	−0.855292	+	0.518146i	$0.826622\pi$
$948$	0	0
$949$	15372.0	0.525813
$950$	0	0
$951$	−81324.0	−2.77299
$952$	0	0
$953$	29583.0i	1.00555i	0.864418	+	0.502774i	$0.167687\pi$
$953$	29583.0i	1.00555i	−0.864418	+	0.502774i	$0.832313\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	16992.0i	0.573953i
$958$	0	0
$959$	5954.00	0.200485
$960$	0	0
$961$	−29691.0	−0.996643
$962$	0	0
$963$	60750.0i	2.03286i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	6480.00i	0.215494i	0.994178	+	0.107747i	$0.0343637\pi$
$967$	6480.00i	0.215494i	−0.994178	+	0.107747i	$0.965636\pi$
$968$	0	0
$969$	−4905.00	−0.162612
$970$	0	0
$971$	40171.0	1.32765	0.663825	−	0.747888i	$-0.268932\pi$
$971$	40171.0	1.32765	0.663825	+	0.747888i	$0.268932\pi$
$972$	0	0
$973$	− 72306.0i	− 2.38235i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	50801.0i	1.66353i	0.555129	+	0.831765i	$0.312669\pi$
$977$	50801.0i	1.66353i	−0.555129	+	0.831765i	$0.687331\pi$
$978$	0	0
$979$	4779.00	0.156014
$980$	0	0
$981$	−66636.0	−2.16873
$982$	0	0
$983$	− 58338.0i	− 1.89287i	−0.322891	−	0.946436i	$-0.604655\pi$
$983$	− 58338.0i	− 1.89287i	0.322891	−	0.946436i	$-0.395345\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	15912.0i	0.513156i
$988$	0	0
$989$	75272.0	2.42013
$990$	0	0
$991$	51202.0	1.64126	0.820628	−	0.571462i	$-0.193624\pi$
$991$	51202.0	1.64126	0.820628	+	0.571462i	$0.193624\pi$
$992$	0	0
$993$	− 92097.0i	− 2.94321i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	1764.00i	0.0560345i	0.999607	+	0.0280173i	$0.00891934\pi$
$997$	1764.00i	0.0560345i	−0.999607	+	0.0280173i	$0.991081\pi$
$998$	0	0
$999$	−48114.0	−1.52378

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.4.c.b.49.2		2
4.3	odd	2		200.4.c.b.49.1		2
5.2	odd	4		400.4.a.t.1.1		1
5.3	odd	4		400.4.a.a.1.1		1
5.4	even	2	inner	400.4.c.b.49.1		2
12.11	even	2		1800.4.f.w.649.2		2
20.3	even	4		200.4.a.j.1.1	yes	1
20.7	even	4		200.4.a.b.1.1	✓	1
20.19	odd	2		200.4.c.b.49.2		2
40.3	even	4		1600.4.a.c.1.1		1
40.13	odd	4		1600.4.a.by.1.1		1
40.27	even	4		1600.4.a.bz.1.1		1
40.37	odd	4		1600.4.a.b.1.1		1
60.23	odd	4		1800.4.a.bh.1.1		1
60.47	odd	4		1800.4.a.c.1.1		1
60.59	even	2		1800.4.f.w.649.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
200.4.a.b.1.1	✓	1	20.7	even	4
200.4.a.j.1.1	yes	1	20.3	even	4
200.4.c.b.49.1		2	4.3	odd	2
200.4.c.b.49.2		2	20.19	odd	2
400.4.a.a.1.1		1	5.3	odd	4
400.4.a.t.1.1		1	5.2	odd	4
400.4.c.b.49.1		2	5.4	even	2	inner
400.4.c.b.49.2		2	1.1	even	1	trivial
1600.4.a.b.1.1		1	40.37	odd	4
1600.4.a.c.1.1		1	40.3	even	4
1600.4.a.by.1.1		1	40.13	odd	4
1600.4.a.bz.1.1		1	40.27	even	4
1800.4.a.c.1.1		1	60.47	odd	4
1800.4.a.bh.1.1		1	60.23	odd	4
1800.4.f.w.649.1		2	60.59	even	2
1800.4.f.w.649.2		2	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	400.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+9.00000i q^{3} -26.0000i q^{7} -54.0000 q^{9} +O(q^{10})\)	\(q+9.00000i q^{3} -26.0000i q^{7} -54.0000 q^{9} +59.0000 q^{11} -28.0000i q^{13} +5.00000i q^{17} +109.000 q^{19} +234.000 q^{21} -194.000i q^{23} -243.000i q^{27} +32.0000 q^{29} -10.0000 q^{31} +531.000i q^{33} -198.000i q^{37} +252.000 q^{39} +117.000 q^{41} +388.000i q^{43} +68.0000i q^{47} -333.000 q^{49} -45.0000 q^{51} +18.0000i q^{53} +981.000i q^{57} +392.000 q^{59} -710.000 q^{61} +1404.00i q^{63} +253.000i q^{67} +1746.00 q^{69} +612.000 q^{71} +549.000i q^{73} -1534.00i q^{77} +414.000 q^{79} +729.000 q^{81} -121.000i q^{83} +288.000i q^{87} +81.0000 q^{89} -728.000 q^{91} -90.0000i q^{93} -1502.00i q^{97} -3186.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 108 q^{9}+O(q^{10}) \) 2 * q - 108 * q^9	\( 2 q - 108 q^{9} + 118 q^{11} + 218 q^{19} + 468 q^{21} + 64 q^{29} - 20 q^{31} + 504 q^{39} + 234 q^{41} - 666 q^{49} - 90 q^{51} + 784 q^{59} - 1420 q^{61} + 3492 q^{69} + 1224 q^{71} + 828 q^{79} + 1458 q^{81} + 162 q^{89} - 1456 q^{91} - 6372 q^{99}+O(q^{100}) \) 2 * q - 108 * q^9 + 118 * q^11 + 218 * q^19 + 468 * q^21 + 64 * q^29 - 20 * q^31 + 504 * q^39 + 234 * q^41 - 666 * q^49 - 90 * q^51 + 784 * q^59 - 1420 * q^61 + 3492 * q^69 + 1224 * q^71 + 828 * q^79 + 1458 * q^81 + 162 * q^89 - 1456 * q^91 - 6372 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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