LMFDB - Embedded newform 325.6.b.h.274.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [325,6,Mod(274,325)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(325, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("325.274");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 325 = 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	325.b (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:	$52.1247414392$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} + 378 x^{16} + 59175 x^{14} + 4956036 x^{12} + 239061247 x^{10} + 6629044458 x^{8} + \cdots + 134659641600 $ x^18 + 378x^16 + 59175x^14 + 4956036x^12 + 239061247x^10 + 6629044458x^8 + 98240243849x^6 + 625586966104x^4 + 584919029904x^2 + 134659641600
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{7}\cdot 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			274.8
Root			$-0.838151i$ of defining polynomial
Character	$\chi$	$=$	325.274
Dual form			325.6.b.h.274.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.83815i q^{2} +11.9262i q^{3} +28.6212 q^{4} +21.9221 q^{6} -112.158i q^{7} -111.431i q^{8} +100.766 q^{9} +O(q^{10})$	\(q-1.83815i q^{2} +11.9262i q^{3} +28.6212 q^{4} +21.9221 q^{6} -112.158i q^{7} -111.431i q^{8} +100.766 q^{9} +162.916 q^{11} +341.342i q^{12} -169.000i q^{13} -206.164 q^{14} +711.052 q^{16} -379.413i q^{17} -185.224i q^{18} +284.116 q^{19} +1337.62 q^{21} -299.464i q^{22} -4188.02i q^{23} +1328.94 q^{24} -310.647 q^{26} +4099.82i q^{27} -3210.10i q^{28} -6279.50 q^{29} -7559.74 q^{31} -4872.81i q^{32} +1942.96i q^{33} -697.419 q^{34} +2884.05 q^{36} -7060.39i q^{37} -522.249i q^{38} +2015.52 q^{39} +4166.01 q^{41} -2458.75i q^{42} -21505.5i q^{43} +4662.85 q^{44} -7698.22 q^{46} +22686.6i q^{47} +8480.13i q^{48} +4227.52 q^{49} +4524.95 q^{51} -4836.98i q^{52} -4774.01i q^{53} +7536.08 q^{54} -12497.9 q^{56} +3388.42i q^{57} +11542.7i q^{58} +48311.4 q^{59} +3650.37 q^{61} +13895.9i q^{62} -11301.8i q^{63} +13796.7 q^{64} +3571.46 q^{66} -15803.8i q^{67} -10859.3i q^{68} +49947.1 q^{69} -37247.8 q^{71} -11228.5i q^{72} +29323.4i q^{73} -12978.1 q^{74} +8131.75 q^{76} -18272.4i q^{77} -3704.84i q^{78} +44954.4 q^{79} -24409.0 q^{81} -7657.75i q^{82} -23704.5i q^{83} +38284.3 q^{84} -39530.3 q^{86} -74890.4i q^{87} -18153.9i q^{88} -561.767 q^{89} -18954.7 q^{91} -119866. i q^{92} -90158.8i q^{93} +41701.3 q^{94} +58114.0 q^{96} -31260.4i q^{97} -7770.82i q^{98} +16416.4 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - 182 q^{4} - 166 q^{6} - 1124 q^{9}+O(q^{10}) $ 18 * q - 182 * q^4 - 166 * q^6 - 1124 * q^9	$ 18 q - 182 q^{4} - 166 q^{6} - 1124 q^{9} - 2844 q^{11} + 684 q^{14} - 2122 q^{16} + 816 q^{19} - 7824 q^{21} + 16938 q^{24} - 1690 q^{26} + 17474 q^{29} + 1496 q^{31} + 35578 q^{34} + 1024 q^{36} + 3718 q^{39} - 57352 q^{41} + 61198 q^{44} - 24112 q^{46} + 81814 q^{49} - 62012 q^{51} + 205522 q^{54} - 46004 q^{56} + 176284 q^{59} + 56330 q^{61} + 201690 q^{64} + 85154 q^{66} + 363494 q^{69} - 141124 q^{71} + 271352 q^{74} + 92746 q^{76} + 328146 q^{79} - 139870 q^{81} + 691312 q^{84} - 589840 q^{86} + 505396 q^{89} + 4056 q^{91} + 1003212 q^{94} - 638362 q^{96} + 1515552 q^{99}+O(q^{100}) $ 18 * q - 182 * q^4 - 166 * q^6 - 1124 * q^9 - 2844 * q^11 + 684 * q^14 - 2122 * q^16 + 816 * q^19 - 7824 * q^21 + 16938 * q^24 - 1690 * q^26 + 17474 * q^29 + 1496 * q^31 + 35578 * q^34 + 1024 * q^36 + 3718 * q^39 - 57352 * q^41 + 61198 * q^44 - 24112 * q^46 + 81814 * q^49 - 62012 * q^51 + 205522 * q^54 - 46004 * q^56 + 176284 * q^59 + 56330 * q^61 + 201690 * q^64 + 85154 * q^66 + 363494 * q^69 - 141124 * q^71 + 271352 * q^74 + 92746 * q^76 + 328146 * q^79 - 139870 * q^81 + 691312 * q^84 - 589840 * q^86 + 505396 * q^89 + 4056 * q^91 + 1003212 * q^94 - 638362 * q^96 + 1515552 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/325\mathbb{Z}\right)^\times$.

$n$	$27$	$301$
$\chi(n)$	$-1$	$1$

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 $

$2$	− 1.83815i	− 0.324942i	−0.986713	−	0.162471i	$-0.948054\pi$
$2$	− 1.83815i	− 0.324942i	0.986713	−	0.162471i	$-0.0519464\pi$
$3$	11.9262i	0.765065i	0.923942	+	0.382532i	$0.124948\pi$
$3$	11.9262i	0.765065i	−0.923942	+	0.382532i	$0.875052\pi$
$4$	28.6212	0.894413
$5$	0	0
$5$	0	0
$6$	21.9221	0.248602
$7$	− 112.158i	− 0.865140i	−0.901600	−	0.432570i	$-0.857607\pi$
$7$	− 112.158i	− 0.865140i	0.901600	−	0.432570i	$-0.142393\pi$
$8$	− 111.431i	− 0.615575i
$9$	100.766	0.414676
$10$	0	0
$11$	162.916	0.405958	0.202979	−	0.979183i	$-0.434938\pi$
$11$	162.916	0.405958	0.202979	+	0.979183i	$0.434938\pi$
$12$	341.342i	0.684284i
$13$	− 169.000i	− 0.277350i
$13$	− 169.000i	− 0.277350i
$14$	−206.164	−0.281120
$15$	0	0
$16$	711.052	0.694386
$17$	− 379.413i	− 0.318413i	−0.987245	−	0.159206i	$-0.949107\pi$
$17$	− 379.413i	− 0.318413i	0.987245	−	0.159206i	$-0.0508935\pi$
$18$	− 185.224i	− 0.134746i
$19$	284.116	0.180556	0.0902781	−	0.995917i	$-0.471224\pi$
$19$	284.116	0.180556	0.0902781	+	0.995917i	$0.471224\pi$
$20$	0	0
$21$	1337.62	0.661888
$22$	− 299.464i	− 0.131913i
$23$	− 4188.02i	− 1.65078i	−0.564562	−	0.825391i	$-0.690955\pi$
$23$	− 4188.02i	− 1.65078i	0.564562	−	0.825391i	$-0.309045\pi$
$24$	1328.94	0.470954
$25$	0	0
$26$	−310.647	−0.0901227
$27$	4099.82i	1.08232i
$28$	− 3210.10i	− 0.773792i
$29$	−6279.50	−1.38653	−0.693266	−	0.720682i	$-0.743829\pi$
$29$	−6279.50	−1.38653	−0.693266	+	0.720682i	$0.743829\pi$
$30$	0	0
$31$	−7559.74	−1.41287	−0.706436	−	0.707777i	$-0.749698\pi$
$31$	−7559.74	−1.41287	−0.706436	+	0.707777i	$0.749698\pi$
$32$	− 4872.81i	− 0.841210i
$33$	1942.96i	0.310584i
$34$	−697.419	−0.103466
$35$	0	0
$36$	2884.05	0.370891
$37$	− 7060.39i	− 0.847860i	−0.905695	−	0.423930i	$-0.860650\pi$
$37$	− 7060.39i	− 0.847860i	0.905695	−	0.423930i	$-0.139350\pi$
$38$	− 522.249i	− 0.0586703i
$39$	2015.52	0.212191
$40$	0	0
$41$	4166.01	0.387044	0.193522	−	0.981096i	$-0.438009\pi$
$41$	4166.01	0.387044	0.193522	+	0.981096i	$0.438009\pi$
$42$	− 2458.75i	− 0.215075i
$43$	− 21505.5i	− 1.77369i	−0.462067	−	0.886845i	$-0.652892\pi$
$43$	− 21505.5i	− 1.77369i	0.462067	−	0.886845i	$-0.347108\pi$
$44$	4662.85	0.363094
$45$	0	0
$46$	−7698.22	−0.536409
$47$	22686.6i	1.49804i	0.662546	+	0.749021i	$0.269476\pi$
$47$	22686.6i	1.49804i	−0.662546	+	0.749021i	$0.730524\pi$
$48$	8480.13i	0.531251i
$49$	4227.52	0.251533
$50$	0	0
$51$	4524.95	0.243606
$52$	− 4836.98i	− 0.248065i
$53$	− 4774.01i	− 0.233450i	−0.993164	−	0.116725i	$-0.962760\pi$
$53$	− 4774.01i	− 0.233450i	0.993164	−	0.116725i	$-0.0372396\pi$
$54$	7536.08	0.351691
$55$	0	0
$56$	−12497.9	−0.532558
$57$	3388.42i	0.138137i
$58$	11542.7i	0.450543i
$59$	48311.4	1.80684	0.903420	−	0.428756i	$-0.141048\pi$
$59$	48311.4	1.80684	0.903420	+	0.428756i	$0.141048\pi$
$60$	0	0
$61$	3650.37	0.125606	0.0628032	−	0.998026i	$-0.479996\pi$
$61$	3650.37	0.125606	0.0628032	+	0.998026i	$0.479996\pi$
$62$	13895.9i	0.459102i
$63$	− 11301.8i	− 0.358753i
$64$	13796.7	0.421042
$65$	0	0
$66$	3571.46	0.100922
$67$	− 15803.8i	− 0.430106i	−0.976602	−	0.215053i	$-0.931008\pi$
$67$	− 15803.8i	− 0.430106i	0.976602	−	0.215053i	$-0.0689924\pi$
$68$	− 10859.3i	− 0.284792i
$69$	49947.1	1.26295
$70$	0	0
$71$	−37247.8	−0.876909	−0.438455	−	0.898753i	$-0.644474\pi$
$71$	−37247.8	−0.876909	−0.438455	+	0.898753i	$0.644474\pi$
$72$	− 11228.5i	− 0.255264i
$73$	29323.4i	0.644033i	0.946734	+	0.322016i	$0.104361\pi$
$73$	29323.4i	0.644033i	−0.946734	+	0.322016i	$0.895639\pi$
$74$	−12978.1	−0.275506
$75$	0	0
$76$	8131.75	0.161492
$77$	− 18272.4i	− 0.351211i
$78$	− 3704.84i	− 0.0689497i
$79$	44954.4	0.810409	0.405204	−	0.914226i	$-0.367200\pi$
$79$	44954.4	0.810409	0.405204	+	0.914226i	$0.367200\pi$
$80$	0	0
$81$	−24409.0	−0.413368
$82$	− 7657.75i	− 0.125767i
$83$	− 23704.5i	− 0.377690i	−0.982007	−	0.188845i	$-0.939526\pi$
$83$	− 23704.5i	− 0.377690i	0.982007	−	0.188845i	$-0.0604743\pi$
$84$	38284.3	0.592001
$85$	0	0
$86$	−39530.3	−0.576347
$87$	− 74890.4i	− 1.06079i
$88$	− 18153.9i	− 0.249898i
$89$	−561.767	−0.00751763	−0.00375882	−	0.999993i	$-0.501196\pi$
$89$	−561.767	−0.00751763	−0.00375882	+	0.999993i	$0.501196\pi$
$90$	0	0
$91$	−18954.7	−0.239947
$92$	− 119866.i	− 1.47648i
$93$	− 90158.8i	− 1.08094i
$94$	41701.3	0.486777
$95$	0	0
$96$	58114.0	0.643580
$97$	− 31260.4i	− 0.337338i	−0.985673	−	0.168669i	$-0.946053\pi$
$97$	− 31260.4i	− 0.337338i	0.985673	−	0.168669i	$-0.0539470\pi$
$98$	− 7770.82i	− 0.0817338i
$99$	16416.4	0.168341
$100$	0	0
$101$	104822.	1.02246	0.511231	−	0.859443i	$-0.329190\pi$
$101$	104822.	1.02246	0.511231	+	0.859443i	$0.329190\pi$
$102$	− 8317.54i	− 0.0791579i
$103$	− 17867.6i	− 0.165948i	−0.996552	−	0.0829742i	$-0.973558\pi$
$103$	− 17867.6i	− 0.165948i	0.996552	−	0.0829742i	$-0.0264419\pi$
$104$	−18831.8	−0.170730
$105$	0	0
$106$	−8775.35	−0.0758577
$107$	− 13003.9i	− 0.109803i	−0.998492	−	0.0549015i	$-0.982516\pi$
$107$	− 13003.9i	− 0.109803i	0.998492	−	0.0549015i	$-0.0174845\pi$
$108$	117342.i	0.968039i
$109$	205680.	1.65816	0.829080	−	0.559130i	$-0.188865\pi$
$109$	205680.	1.65816	0.829080	+	0.559130i	$0.188865\pi$
$110$	0	0
$111$	84203.5	0.648668
$112$	− 79750.3i	− 0.600741i
$113$	− 119392.i	− 0.879585i	−0.898100	−	0.439792i	$-0.855052\pi$
$113$	− 119392.i	− 0.879585i	0.898100	−	0.439792i	$-0.144948\pi$
$114$	6228.43	0.0448866
$115$	0	0
$116$	−179727.	−1.24013
$117$	− 17029.5i	− 0.115010i
$118$	− 88803.7i	− 0.587119i
$119$	−42554.3	−0.275471
$120$	0	0
$121$	−134509.	−0.835198
$122$	− 6709.92i	− 0.0408148i
$123$	49684.5i	0.296114i
$124$	−216369.	−1.26369
$125$	0	0
$126$	−20774.4	−0.116574
$127$	− 213727.i	− 1.17584i	−0.808918	−	0.587922i	$-0.799946\pi$
$127$	− 213727.i	− 1.17584i	0.808918	−	0.587922i	$-0.200054\pi$
$128$	− 181290.i	− 0.978024i
$129$	256478.	1.35699
$130$	0	0
$131$	176742.	0.899833	0.449916	−	0.893071i	$-0.351454\pi$
$131$	176742.	0.899833	0.449916	+	0.893071i	$0.351454\pi$
$132$	55609.9i	0.277791i
$133$	− 31866.0i	− 0.156206i
$134$	−29049.8	−0.139759
$135$	0	0
$136$	−42278.4	−0.196007
$137$	133457.i	0.607493i	0.952753	+	0.303746i	$0.0982375\pi$
$137$	133457.i	0.607493i	−0.952753	+	0.303746i	$0.901762\pi$
$138$	− 91810.3i	− 0.410387i
$139$	179302.	0.787132	0.393566	−	0.919296i	$-0.371241\pi$
$139$	179302.	0.787132	0.393566	+	0.919296i	$0.371241\pi$
$140$	0	0
$141$	−270564.	−1.14610
$142$	68467.0i	0.284945i
$143$	− 27532.8i	− 0.112593i
$144$	71650.0	0.287945
$145$	0	0
$146$	53900.9	0.209273
$147$	50418.2i	0.192439i
$148$	− 202077.i	− 0.758337i
$149$	−69993.7	−0.258281	−0.129141	−	0.991626i	$-0.541222\pi$
$149$	−69993.7	−0.258281	−0.129141	+	0.991626i	$0.541222\pi$
$150$	0	0
$151$	−360861.	−1.28795	−0.643973	−	0.765048i	$-0.722715\pi$
$151$	−360861.	−1.28795	−0.643973	+	0.765048i	$0.722715\pi$
$152$	− 31659.3i	− 0.111146i
$153$	− 38232.0i	− 0.132038i
$154$	−33587.3	−0.114123
$155$	0	0
$156$	57686.7	0.189786
$157$	− 74024.2i	− 0.239676i	−0.992793	−	0.119838i	$-0.961762\pi$
$157$	− 74024.2i	− 0.239676i	0.992793	−	0.119838i	$-0.0382375\pi$
$158$	− 82632.9i	− 0.263336i
$159$	56935.7	0.178604
$160$	0	0
$161$	−469722.	−1.42816
$162$	44867.3i	0.134321i
$163$	395684.i	1.16649i	0.812298	+	0.583243i	$0.198217\pi$
$163$	395684.i	1.16649i	−0.812298	+	0.583243i	$0.801783\pi$
$164$	119236.	0.346177
$165$	0	0
$166$	−43572.4	−0.122727
$167$	243331.i	0.675160i	0.941297	+	0.337580i	$0.109608\pi$
$167$	243331.i	0.675160i	−0.941297	+	0.337580i	$0.890392\pi$
$168$	− 149052.i	− 0.407441i
$169$	−28561.0	−0.0769231
$170$	0	0
$171$	28629.3	0.0748723
$172$	− 615512.i	− 1.58641i
$173$	12828.9i	0.0325892i	0.999867	+	0.0162946i	$0.00518697\pi$
$173$	12828.9i	0.0325892i	−0.999867	+	0.0162946i	$0.994813\pi$
$174$	−137660.	−0.344694
$175$	0	0
$176$	115842.	0.281892
$177$	576171.i	1.38235i
$178$	1032.61i	0.00244280i
$179$	−643719.	−1.50163	−0.750817	−	0.660510i	$-0.770340\pi$
$179$	−643719.	−1.50163	−0.750817	+	0.660510i	$0.770340\pi$
$180$	0	0
$181$	598308.	1.35746	0.678732	−	0.734386i	$-0.262530\pi$
$181$	598308.	1.35746	0.678732	+	0.734386i	$0.262530\pi$
$182$	34841.7i	0.0779688i
$183$	43534.9i	0.0960970i
$184$	−466675.	−1.01618
$185$	0	0
$186$	−165726.	−0.351242
$187$	− 61812.4i	− 0.129262i
$188$	649317.i	1.33987i
$189$	459829.	0.936357
$190$	0	0
$191$	364186.	0.722337	0.361169	−	0.932501i	$-0.382378\pi$
$191$	364186.	0.722337	0.361169	+	0.932501i	$0.382378\pi$
$192$	164542.i	0.322124i
$193$	533579.i	1.03111i	0.856856	+	0.515555i	$0.172414\pi$
$193$	533579.i	1.03111i	−0.856856	+	0.515555i	$0.827586\pi$
$194$	−57461.4	−0.109615
$195$	0	0
$196$	120997.	0.224975
$197$	295693.i	0.542845i	0.962460	+	0.271422i	$0.0874940\pi$
$197$	295693.i	0.542845i	−0.962460	+	0.271422i	$0.912506\pi$
$198$	− 30175.8i	− 0.0547012i
$199$	581653.	1.04119	0.520597	−	0.853803i	$-0.325710\pi$
$199$	581653.	1.04119	0.520597	+	0.853803i	$0.325710\pi$
$200$	0	0
$201$	188479.	0.329059
$202$	− 192678.i	− 0.332241i
$203$	704297.i	1.19954i
$204$	129509.	0.217884
$205$	0	0
$206$	−32843.3	−0.0539236
$207$	− 422012.i	− 0.684539i
$208$	− 120168.i	− 0.192588i
$209$	46287.1	0.0732983
$210$	0	0
$211$	452629.	0.699900	0.349950	−	0.936768i	$-0.386198\pi$
$211$	452629.	0.699900	0.349950	+	0.936768i	$0.386198\pi$
$212$	− 136638.i	− 0.208801i
$213$	− 444224.i	− 0.670892i
$214$	−23903.1	−0.0356796
$215$	0	0
$216$	456846.	0.666248
$217$	847888.i	1.22233i
$218$	− 378071.i	− 0.538806i
$219$	−349717.	−0.492727
$220$	0	0
$221$	−64120.8	−0.0883117
$222$	− 154779.i	− 0.210780i
$223$	− 430237.i	− 0.579356i	−0.957124	−	0.289678i	$-0.906452\pi$
$223$	− 430237.i	− 0.579356i	0.957124	−	0.289678i	$-0.0935482\pi$
$224$	−546526.	−0.727764
$225$	0	0
$226$	−219460.	−0.285814
$227$	694207.i	0.894178i	0.894489	+	0.447089i	$0.147539\pi$
$227$	694207.i	0.894178i	−0.894489	+	0.447089i	$0.852461\pi$
$228$	96980.7i	0.123552i
$229$	−1.29181e6	−1.62784	−0.813920	−	0.580977i	$-0.802670\pi$
$229$	−1.29181e6	−1.62784	−0.813920	+	0.580977i	$0.802670\pi$
$230$	0	0
$231$	217919.	0.268699
$232$	699730.i	0.853514i
$233$	604845.i	0.729885i	0.931030	+	0.364942i	$0.118911\pi$
$233$	604845.i	0.729885i	−0.931030	+	0.364942i	$0.881089\pi$
$234$	−31302.8	−0.0373717
$235$	0	0
$236$	1.38273e6	1.61606
$237$	536134.i	0.620015i
$238$	78221.3i	0.0895122i
$239$	182757.	0.206957	0.103478	−	0.994632i	$-0.467003\pi$
$239$	182757.	0.206957	0.103478	+	0.994632i	$0.467003\pi$
$240$	0	0
$241$	1.34821e6	1.49525	0.747627	−	0.664118i	$-0.231193\pi$
$241$	1.34821e6	1.49525	0.747627	+	0.664118i	$0.231193\pi$
$242$	247249.i	0.271391i
$243$	705150.i	0.766065i
$244$	104478.	0.112344
$245$	0	0
$246$	91327.6	0.0962198
$247$	− 48015.7i	− 0.0500773i
$248$	842389.i	0.869728i
$249$	282704.	0.288957
$250$	0	0
$251$	−597169.	−0.598291	−0.299146	−	0.954207i	$-0.596702\pi$
$251$	−597169.	−0.598291	−0.299146	+	0.954207i	$0.596702\pi$
$252$	− 323470.i	− 0.320873i
$253$	− 682295.i	− 0.670149i
$254$	−392862.	−0.382081
$255$	0	0
$256$	108255.	0.103240
$257$	− 230980.i	− 0.218144i	−0.994034	−	0.109072i	$-0.965212\pi$
$257$	− 230980.i	− 0.218144i	0.994034	−	0.109072i	$-0.0347878\pi$
$258$	− 471445.i	− 0.440943i
$259$	−791881.	−0.733518
$260$	0	0
$261$	−632761.	−0.574961
$262$	− 324879.i	− 0.292394i
$263$	1.89928e6i	1.69317i	0.532254	+	0.846585i	$0.321345\pi$
$263$	1.89928e6i	1.69317i	−0.532254	+	0.846585i	$0.678655\pi$
$264$	216506.	0.191188
$265$	0	0
$266$	−58574.5	−0.0507580
$267$	− 6699.73i	− 0.00575148i
$268$	− 452324.i	− 0.384692i
$269$	−2.30772e6	−1.94448	−0.972239	−	0.233991i	$-0.924821\pi$
$269$	−2.30772e6	−1.94448	−0.972239	+	0.233991i	$0.924821\pi$
$270$	0	0
$271$	−2.18852e6	−1.81020	−0.905101	−	0.425197i	$-0.860205\pi$
$271$	−2.18852e6	−1.81020	−0.905101	+	0.425197i	$0.860205\pi$
$272$	− 269782.i	− 0.221101i
$273$	− 226058.i	− 0.183575i
$274$	245315.	0.197400
$275$	0	0
$276$	1.42955e6	1.12960
$277$	− 24533.4i	− 0.0192114i	−0.999954	−	0.00960569i	$-0.996942\pi$
$277$	− 24533.4i	− 0.0192114i	0.999954	−	0.00960569i	$-0.00305763\pi$
$278$	− 329583.i	− 0.255772i
$279$	−761767.	−0.585884
$280$	0	0
$281$	−237327.	−0.179300	−0.0896501	−	0.995973i	$-0.528575\pi$
$281$	−237327.	−0.179300	−0.0896501	+	0.995973i	$0.528575\pi$
$282$	497338.i	0.372416i
$283$	− 1.21567e6i	− 0.902297i	−0.892449	−	0.451148i	$-0.851015\pi$
$283$	− 1.21567e6i	− 0.902297i	0.892449	−	0.451148i	$-0.148985\pi$
$284$	−1.06608e6	−0.784318
$285$	0	0
$286$	−50609.4	−0.0365861
$287$	− 467252.i	− 0.334847i
$288$	− 491015.i	− 0.348830i
$289$	1.27590e6	0.898613
$290$	0	0
$291$	372818.	0.258086
$292$	839272.i	0.576031i
$293$	2.18776e6i	1.48878i	0.667744	+	0.744391i	$0.267260\pi$
$293$	2.18776e6i	1.48878i	−0.667744	+	0.744391i	$0.732740\pi$
$294$	92676.2	0.0625316
$295$	0	0
$296$	−786746.	−0.521921
$297$	667925.i	0.439376i
$298$	128659.i	0.0839265i
$299$	−707776.	−0.457844
$300$	0	0
$301$	−2.41202e6	−1.53449
$302$	663317.i	0.418508i
$303$	1.25012e6i	0.782249i
$304$	202021.	0.125376
$305$	0	0
$306$	−70276.3	−0.0429047
$307$	− 460260.i	− 0.278713i	−0.990242	−	0.139357i	$-0.955497\pi$
$307$	− 460260.i	− 0.278713i	0.990242	−	0.139357i	$-0.0445034\pi$
$308$	− 522977.i	− 0.314127i
$309$	213092.	0.126961
$310$	0	0
$311$	−1.54256e6	−0.904362	−0.452181	−	0.891926i	$-0.649354\pi$
$311$	−1.54256e6	−0.904362	−0.452181	+	0.891926i	$0.649354\pi$
$312$	− 224592.i	− 0.130619i
$313$	1.20176e6i	0.693357i	0.937984	+	0.346679i	$0.112691\pi$
$313$	1.20176e6i	0.693357i	−0.937984	+	0.346679i	$0.887309\pi$
$314$	−136068.	−0.0778808
$315$	0	0
$316$	1.28665e6	0.724840
$317$	− 512204.i	− 0.286282i	−0.989702	−	0.143141i	$-0.954280\pi$
$317$	− 512204.i	− 0.286282i	0.989702	−	0.143141i	$-0.0457203\pi$
$318$	− 104656.i	− 0.0580361i
$319$	−1.02303e6	−0.562874
$320$	0	0
$321$	155087.	0.0840064
$322$	863419.i	0.464068i
$323$	− 107797.i	− 0.0574913i
$324$	−698614.	−0.369721
$325$	0	0
$326$	727327.	0.379041
$327$	2.45298e6i	1.26860i
$328$	− 464222.i	− 0.238254i
$329$	2.54449e6	1.29602
$330$	0	0
$331$	−135150.	−0.0678026	−0.0339013	−	0.999425i	$-0.510793\pi$
$331$	−135150.	−0.0678026	−0.0339013	+	0.999425i	$0.510793\pi$
$332$	− 678451.i	− 0.337811i
$333$	− 711449.i	− 0.351587i
$334$	447279.	0.219388
$335$	0	0
$336$	951117.	0.459606
$337$	1.40146e6i	0.672214i	0.941824	+	0.336107i	$0.109110\pi$
$337$	1.40146e6i	0.672214i	−0.941824	+	0.336107i	$0.890890\pi$
$338$	52499.4i	0.0249956i
$339$	1.42389e6	0.672939
$340$	0	0
$341$	−1.23160e6	−0.573567
$342$	− 52625.0i	− 0.0243292i
$343$	− 2.35920e6i	− 1.08275i
$344$	−2.39637e6	−1.09184
$345$	0	0
$346$	23581.4	0.0105896
$347$	59106.2i	0.0263517i	0.999913	+	0.0131759i	$0.00419413\pi$
$347$	59106.2i	0.0263517i	−0.999913	+	0.0131759i	$0.995806\pi$
$348$	− 2.14345e6i	− 0.948781i
$349$	−2.58914e6	−1.13787	−0.568934	−	0.822383i	$-0.692644\pi$
$349$	−2.58914e6	−1.13787	−0.568934	+	0.822383i	$0.692644\pi$
$350$	0	0
$351$	692869.	0.300181
$352$	− 793858.i	− 0.341496i
$353$	1.08860e6i	0.464978i	0.972599	+	0.232489i	$0.0746870\pi$
$353$	1.08860e6i	0.464978i	−0.972599	+	0.232489i	$0.925313\pi$
$354$	1.05909e6	0.449184
$355$	0	0
$356$	−16078.4	−0.00672387
$357$	− 507511.i	− 0.210753i
$358$	1.18325e6i	0.487944i
$359$	3.81776e6	1.56341	0.781705	−	0.623648i	$-0.214350\pi$
$359$	3.81776e6	1.56341	0.781705	+	0.623648i	$0.214350\pi$
$360$	0	0
$361$	−2.39538e6	−0.967399
$362$	− 1.09978e6i	− 0.441097i
$363$	− 1.60418e6i	− 0.638980i
$364$	−542508.	−0.214611
$365$	0	0
$366$	80023.7	0.0312260
$367$	− 4.74165e6i	− 1.83766i	−0.394658	−	0.918828i	$-0.629137\pi$
$367$	− 4.74165e6i	− 1.83766i	0.394658	−	0.918828i	$-0.370863\pi$
$368$	− 2.97790e6i	− 1.14628i
$369$	419793.	0.160498
$370$	0	0
$371$	−535445.	−0.201967
$372$	− 2.58045e6i	− 0.966805i
$373$	705178.i	0.262438i	0.991353	+	0.131219i	$0.0418891\pi$
$373$	705178.i	0.262438i	−0.991353	+	0.131219i	$0.958111\pi$
$374$	−113621.	−0.0420028
$375$	0	0
$376$	2.52798e6	0.922157
$377$	1.06123e6i	0.384555i
$378$	− 845234.i	− 0.304262i
$379$	−615406.	−0.220071	−0.110036	−	0.993928i	$-0.535097\pi$
$379$	−615406.	−0.220071	−0.110036	+	0.993928i	$0.535097\pi$
$380$	0	0
$381$	2.54894e6	0.899597
$382$	− 669429.i	− 0.234718i
$383$	3.77847e6i	1.31619i	0.752934	+	0.658096i	$0.228638\pi$
$383$	3.77847e6i	1.31619i	−0.752934	+	0.658096i	$0.771362\pi$
$384$	2.16210e6	0.748252
$385$	0	0
$386$	980798.	0.335051
$387$	− 2.16703e6i	− 0.735507i
$388$	− 894712.i	− 0.301720i
$389$	1.92639e6	0.645461	0.322730	−	0.946491i	$-0.395399\pi$
$389$	1.92639e6	0.645461	0.322730	+	0.946491i	$0.395399\pi$
$390$	0	0
$391$	−1.58899e6	−0.525630
$392$	− 471076.i	− 0.154838i
$393$	2.10786e6i	0.688430i
$394$	543528.	0.176393
$395$	0	0
$396$	469858.	0.150566
$397$	− 3.30989e6i	− 1.05399i	−0.849868	−	0.526996i	$-0.823318\pi$
$397$	− 3.30989e6i	− 1.05399i	0.849868	−	0.526996i	$-0.176682\pi$
$398$	− 1.06917e6i	− 0.338328i
$399$	380040.	0.119508
$400$	0	0
$401$	2.63190e6	0.817350	0.408675	−	0.912680i	$-0.365991\pi$
$401$	2.63190e6	0.817350	0.408675	+	0.912680i	$0.365991\pi$
$402$	− 346453.i	− 0.106925i
$403$	1.27760e6i	0.391860i
$404$	3.00012e6	0.914502
$405$	0	0
$406$	1.29460e6	0.389782
$407$	− 1.15025e6i	− 0.344196i
$408$	− 504219.i	− 0.149958i
$409$	−575874.	−0.170223	−0.0851116	−	0.996371i	$-0.527125\pi$
$409$	−575874.	−0.170223	−0.0851116	+	0.996371i	$0.527125\pi$
$410$	0	0
$411$	−1.59164e6	−0.464771
$412$	− 511392.i	− 0.148426i
$413$	− 5.41853e6i	− 1.56317i
$414$	−775721.	−0.222436
$415$	0	0
$416$	−823505.	−0.233310
$417$	2.13838e6i	0.602207i
$418$	− 85082.6i	− 0.0238177i
$419$	5.72386e6	1.59277	0.796386	−	0.604788i	$-0.206742\pi$
$419$	5.72386e6	1.59277	0.796386	+	0.604788i	$0.206742\pi$
$420$	0	0
$421$	−6.93158e6	−1.90602	−0.953009	−	0.302942i	$-0.902031\pi$
$421$	−6.93158e6	−1.90602	−0.953009	+	0.302942i	$0.902031\pi$
$422$	− 832000.i	− 0.227427i
$423$	2.28604e6i	0.621202i
$424$	−531972.	−0.143706
$425$	0	0
$426$	−816550.	−0.218001
$427$	− 409419.i	− 0.108667i
$428$	− 372187.i	− 0.0982092i
$429$	328361.	0.0861406
$430$	0	0
$431$	−215036.	−0.0557594	−0.0278797	−	0.999611i	$-0.508876\pi$
$431$	−215036.	−0.0557594	−0.0278797	+	0.999611i	$0.508876\pi$
$432$	2.91518e6i	0.751547i
$433$	− 5.17858e6i	− 1.32737i	−0.748013	−	0.663684i	$-0.768992\pi$
$433$	− 5.17858e6i	− 1.32737i	0.748013	−	0.663684i	$-0.231008\pi$
$434$	1.55855e6	0.397187
$435$	0	0
$436$	5.88682e6	1.48308
$437$	− 1.18989e6i	− 0.298059i
$438$	642832.i	0.160108i
$439$	7.60133e6	1.88247	0.941235	−	0.337753i	$-0.109667\pi$
$439$	7.60133e6	1.88247	0.941235	+	0.337753i	$0.109667\pi$
$440$	0	0
$441$	425991.	0.104305
$442$	117864.i	0.0286962i
$443$	7.13928e6i	1.72840i	0.503146	+	0.864201i	$0.332176\pi$
$443$	7.13928e6i	1.72840i	−0.503146	+	0.864201i	$0.667824\pi$
$444$	2.41000e6	0.580177
$445$	0	0
$446$	−790840.	−0.188257
$447$	− 834757.i	− 0.197602i
$448$	− 1.54741e6i	− 0.364260i
$449$	848355.	0.198592	0.0992960	−	0.995058i	$-0.468341\pi$
$449$	848355.	0.198592	0.0992960	+	0.995058i	$0.468341\pi$
$450$	0	0
$451$	678708.	0.157124
$452$	− 3.41713e6i	− 0.786712i
$453$	− 4.30369e6i	− 0.985362i
$454$	1.27606e6	0.290556
$455$	0	0
$456$	377575.	0.0850337
$457$	3.35404e6i	0.751238i	0.926774	+	0.375619i	$0.122570\pi$
$457$	3.35404e6i	0.751238i	−0.926774	+	0.375619i	$0.877430\pi$
$458$	2.37455e6i	0.528954i
$459$	1.55552e6	0.344624
$460$	0	0
$461$	−3.57354e6	−0.783152	−0.391576	−	0.920146i	$-0.628070\pi$
$461$	−3.57354e6	−0.783152	−0.391576	+	0.920146i	$0.628070\pi$
$462$	− 400569.i	− 0.0873116i
$463$	1.44187e6i	0.312590i	0.987710	+	0.156295i	$0.0499550\pi$
$463$	1.44187e6i	0.312590i	−0.987710	+	0.156295i	$0.950045\pi$
$464$	−4.46505e6	−0.962789
$465$	0	0
$466$	1.11180e6	0.237170
$467$	7.03571e6i	1.49285i	0.665471	+	0.746424i	$0.268231\pi$
$467$	7.03571e6i	1.49285i	−0.665471	+	0.746424i	$0.731769\pi$
$468$	− 487405.i	− 0.102867i
$469$	−1.77253e6	−0.372101
$470$	0	0
$471$	882825.	0.183368
$472$	− 5.38339e6i	− 1.11225i
$473$	− 3.50358e6i	− 0.720044i
$474$	985494.	0.201469
$475$	0	0
$476$	−1.21796e6	−0.246385
$477$	− 481059.i	− 0.0968061i
$478$	− 335935.i	− 0.0672489i
$479$	1.40107e6	0.279010	0.139505	−	0.990221i	$-0.455449\pi$
$479$	1.40107e6	0.279010	0.139505	+	0.990221i	$0.455449\pi$
$480$	0	0
$481$	−1.19321e6	−0.235154
$482$	− 2.47821e6i	− 0.485871i
$483$	− 5.60198e6i	− 1.09263i
$484$	−3.84982e6	−0.747011
$485$	0	0
$486$	1.29617e6	0.248927
$487$	8.45449e6i	1.61534i	0.589632	+	0.807672i	$0.299273\pi$
$487$	8.45449e6i	1.61534i	−0.589632	+	0.807672i	$0.700727\pi$
$488$	− 406764.i	− 0.0773201i
$489$	−4.71900e6	−0.892438
$490$	0	0
$491$	−7.81764e6	−1.46343	−0.731715	−	0.681611i	$-0.761280\pi$
$491$	−7.81764e6	−1.46343	−0.731715	+	0.681611i	$0.761280\pi$
$492$	1.42203e6i	0.264848i
$493$	2.38252e6i	0.441489i
$494$	−88260.0	−0.0162722
$495$	0	0
$496$	−5.37537e6	−0.981079
$497$	4.17765e6i	0.758649i
$498$	− 519652.i	− 0.0938944i
$499$	6.37828e6	1.14671	0.573353	−	0.819308i	$-0.305642\pi$
$499$	6.37828e6	1.14671	0.573353	+	0.819308i	$0.305642\pi$
$500$	0	0
$501$	−2.90201e6	−0.516541
$502$	1.09769e6i	0.194410i
$503$	− 2.99032e6i	− 0.526985i	−0.964661	−	0.263492i	$-0.915126\pi$
$503$	− 2.99032e6i	− 0.526985i	0.964661	−	0.263492i	$-0.0848744\pi$
$504$	−1.25937e6	−0.220839
$505$	0	0
$506$	−1.25416e6	−0.217760
$507$	− 340624.i	− 0.0588511i
$508$	− 6.11712e6i	− 1.05169i
$509$	5.75011e6	0.983743	0.491871	−	0.870668i	$-0.336313\pi$
$509$	5.75011e6	0.983743	0.491871	+	0.870668i	$0.336313\pi$
$510$	0	0
$511$	3.28887e6	0.557178
$512$	− 6.00028e6i	− 1.01157i
$513$	1.16483e6i	0.195419i
$514$	−424577.	−0.0708840
$515$	0	0
$516$	7.34071e6	1.21371
$517$	3.69600e6i	0.608143i
$518$	1.45560e6i	0.238351i
$519$	−153000.	−0.0249329
$520$	0	0
$521$	−1.06342e7	−1.71638	−0.858188	−	0.513336i	$-0.828409\pi$
$521$	−1.06342e7	−1.71638	−0.858188	+	0.513336i	$0.828409\pi$
$522$	1.16311e6i	0.186829i
$523$	− 5.82469e6i	− 0.931148i	−0.885009	−	0.465574i	$-0.845848\pi$
$523$	− 5.82469e6i	− 0.931148i	0.885009	−	0.465574i	$-0.154152\pi$
$524$	5.05857e6	0.804822
$525$	0	0
$526$	3.49117e6	0.550182
$527$	2.86827e6i	0.449876i
$528$	1.38155e6i	0.215666i
$529$	−1.11032e7	−1.72508
$530$	0	0
$531$	4.86816e6	0.749253
$532$	− 912043.i	− 0.139713i
$533$	− 704055.i	− 0.107347i
$534$	−12315.1	−0.00186890
$535$	0	0
$536$	−1.76103e6	−0.264762
$537$	− 7.67711e6i	− 1.14885i
$538$	4.24194e6i	0.631843i
$539$	688730.	0.102112
$540$	0	0
$541$	249946.	0.0367158	0.0183579	−	0.999831i	$-0.494156\pi$
$541$	249946.	0.0367158	0.0183579	+	0.999831i	$0.494156\pi$
$542$	4.02283e6i	0.588211i
$543$	7.13552e6i	1.03855i
$544$	−1.84881e6	−0.267852
$545$	0	0
$546$	−415528.	−0.0596512
$547$	− 9.11755e6i	− 1.30290i	−0.758693	−	0.651448i	$-0.774162\pi$
$547$	− 9.11755e6i	− 1.30290i	0.758693	−	0.651448i	$-0.225838\pi$
$548$	3.81971e6i	0.543349i
$549$	367834.	0.0520860
$550$	0	0
$551$	−1.78411e6	−0.250347
$552$	− 5.56565e6i	− 0.777443i
$553$	− 5.04200e6i	− 0.701117i
$554$	−45096.1	−0.00624259
$555$	0	0
$556$	5.13183e6	0.704020
$557$	6.02542e6i	0.822905i	0.911431	+	0.411452i	$0.134978\pi$
$557$	6.02542e6i	0.822905i	−0.911431	+	0.411452i	$0.865022\pi$
$558$	1.40024e6i	0.190378i
$559$	−3.63442e6	−0.491933
$560$	0	0
$561$	737186.	0.0988940
$562$	436242.i	0.0582622i
$563$	− 1.08154e6i	− 0.143804i	−0.997412	−	0.0719021i	$-0.977093\pi$
$563$	− 1.08154e6i	− 0.143804i	0.997412	−	0.0719021i	$-0.0229069\pi$
$564$	−7.74387e6	−1.02509
$565$	0	0
$566$	−2.23458e6	−0.293194
$567$	2.73767e6i	0.357621i
$568$	4.15055e6i	0.539803i
$569$	−745292.	−0.0965041	−0.0482520	−	0.998835i	$-0.515365\pi$
$569$	−745292.	−0.0965041	−0.0482520	+	0.998835i	$0.515365\pi$
$570$	0	0
$571$	4.58349e6	0.588309	0.294155	−	0.955758i	$-0.404962\pi$
$571$	4.58349e6	0.588309	0.294155	+	0.955758i	$0.404962\pi$
$572$	− 788021.i	− 0.100704i
$573$	4.34335e6i	0.552635i
$574$	−858880.	−0.108806
$575$	0	0
$576$	1.39024e6	0.174596
$577$	− 1.17307e7i	− 1.46685i	−0.679772	−	0.733424i	$-0.737921\pi$
$577$	− 1.17307e7i	− 1.46685i	0.679772	−	0.733424i	$-0.262079\pi$
$578$	− 2.34530e6i	− 0.291997i
$579$	−6.36355e6	−0.788866
$580$	0	0
$581$	−2.65865e6	−0.326754
$582$	− 685295.i	− 0.0838629i
$583$	− 777762.i	− 0.0947710i
$584$	3.26754e6	0.396450
$585$	0	0
$586$	4.02144e6	0.483768
$587$	8.72678e6i	1.04534i	0.852534	+	0.522672i	$0.175065\pi$
$587$	8.72678e6i	1.04534i	−0.852534	+	0.522672i	$0.824935\pi$
$588$	1.44303e6i	0.172120i
$589$	−2.14785e6	−0.255103
$590$	0	0
$591$	−3.52649e6	−0.415311
$592$	− 5.02030e6i	− 0.588743i
$593$	500982.i	0.0585040i	0.999572	+	0.0292520i	$0.00931252\pi$
$593$	500982.i	0.0585040i	−0.999572	+	0.0292520i	$0.990687\pi$
$594$	1.22775e6	0.142772
$595$	0	0
$596$	−2.00330e6	−0.231010
$597$	6.93690e6i	0.796580i
$598$	1.30100e6i	0.148773i
$599$	1.04449e6	0.118943	0.0594714	−	0.998230i	$-0.481058\pi$
$599$	1.04449e6	0.118943	0.0594714	+	0.998230i	$0.481058\pi$
$600$	0	0
$601$	3.03678e6	0.342948	0.171474	−	0.985189i	$-0.445147\pi$
$601$	3.03678e6	0.342948	0.171474	+	0.985189i	$0.445147\pi$
$602$	4.43365e6i	0.498620i
$603$	− 1.59249e6i	− 0.178354i
$604$	−1.03283e7	−1.15195
$605$	0	0
$606$	2.29791e6	0.254186
$607$	− 809264.i	− 0.0891494i	−0.999006	−	0.0445747i	$-0.985807\pi$
$607$	− 809264.i	− 0.0891494i	0.999006	−	0.0445747i	$-0.0141933\pi$
$608$	− 1.38444e6i	− 0.151886i
$609$	−8.39958e6	−0.917728
$610$	0	0
$611$	3.83403e6	0.415482
$612$	− 1.09425e6i	− 0.118096i
$613$	1.04363e7i	1.12175i	0.827900	+	0.560875i	$0.189535\pi$
$613$	1.04363e7i	1.12175i	−0.827900	+	0.560875i	$0.810465\pi$
$614$	−846028.	−0.0905657
$615$	0	0
$616$	−2.03611e6	−0.216196
$617$	2.56343e6i	0.271086i	0.990771	+	0.135543i	$0.0432780\pi$
$617$	2.56343e6i	0.271086i	−0.990771	+	0.135543i	$0.956722\pi$
$618$	− 391695.i	− 0.0412551i
$619$	−6.77494e6	−0.710687	−0.355344	−	0.934736i	$-0.615636\pi$
$619$	−6.77494e6	−0.710687	−0.355344	+	0.934736i	$0.615636\pi$
$620$	0	0
$621$	1.71701e7	1.78667
$622$	2.83547e6i	0.293865i
$623$	63006.8i	0.00650380i
$624$	1.43314e6	0.147342
$625$	0	0
$626$	2.20902e6	0.225301
$627$	552028.i	0.0560779i
$628$	− 2.11866e6i	− 0.214369i
$629$	−2.67881e6	−0.269969
$630$	0	0
$631$	1.07855e7	1.07837	0.539185	−	0.842187i	$-0.318732\pi$
$631$	1.07855e7	1.07837	0.539185	+	0.842187i	$0.318732\pi$
$632$	− 5.00930e6i	− 0.498867i
$633$	5.39814e6i	0.535469i
$634$	−941508.	−0.0930252
$635$	0	0
$636$	1.62957e6	0.159746
$637$	− 714451.i	− 0.0697628i
$638$	1.88048e6i	0.182902i
$639$	−3.75332e6	−0.363633
$640$	0	0
$641$	329201.	0.0316458	0.0158229	−	0.999875i	$-0.494963\pi$
$641$	329201.	0.0316458	0.0158229	+	0.999875i	$0.494963\pi$
$642$	− 285073.i	− 0.0272972i
$643$	1.85214e7i	1.76663i	0.468777	+	0.883317i	$0.344695\pi$
$643$	1.85214e7i	1.76663i	−0.468777	+	0.883317i	$0.655305\pi$
$644$	−1.34440e7	−1.27736
$645$	0	0
$646$	−198148.	−0.0186814
$647$	− 4.85885e6i	− 0.456324i	−0.973623	−	0.228162i	$-0.926728\pi$
$647$	− 4.85885e6i	− 0.456324i	0.973623	−	0.228162i	$-0.0732715\pi$
$648$	2.71991e6i	0.254459i
$649$	7.87069e6	0.733502
$650$	0	0
$651$	−1.01121e7	−0.935163
$652$	1.13250e7i	1.04332i
$653$	− 4.56792e6i	− 0.419214i	−0.977786	−	0.209607i	$-0.932782\pi$
$653$	− 4.56792e6i	− 0.419214i	0.977786	−	0.209607i	$-0.0672184\pi$
$654$	4.50895e6	0.412222
$655$	0	0
$656$	2.96225e6	0.268758
$657$	2.95481e6i	0.267065i
$658$	− 4.67715e6i	− 0.421130i
$659$	9.51262e6	0.853270	0.426635	−	0.904424i	$-0.359699\pi$
$659$	9.51262e6	0.853270	0.426635	+	0.904424i	$0.359699\pi$
$660$	0	0
$661$	−1.91227e6	−0.170233	−0.0851167	−	0.996371i	$-0.527126\pi$
$661$	−1.91227e6	−0.170233	−0.0851167	+	0.996371i	$0.527126\pi$
$662$	248426.i	0.0220319i
$663$	− 764716.i	− 0.0675642i
$664$	−2.64141e6	−0.232496
$665$	0	0
$666$	−1.30775e6	−0.114246
$667$	2.62987e7i	2.28886i
$668$	6.96443e6i	0.603872i
$669$	5.13108e6	0.443245
$670$	0	0
$671$	594702.	0.0509910
$672$	− 6.51797e6i	− 0.556787i
$673$	5.38551e6i	0.458341i	0.973386	+	0.229171i	$0.0736014\pi$
$673$	5.38551e6i	0.458341i	−0.973386	+	0.229171i	$0.926399\pi$
$674$	2.57610e6	0.218431
$675$	0	0
$676$	−817450.	−0.0688010
$677$	1.65232e7i	1.38555i	0.721154	+	0.692775i	$0.243612\pi$
$677$	1.65232e7i	1.38555i	−0.721154	+	0.692775i	$0.756388\pi$
$678$	− 2.61732e6i	− 0.218666i
$679$	−3.50612e6	−0.291845
$680$	0	0
$681$	−8.27923e6	−0.684104
$682$	2.26387e6i	0.186376i
$683$	8.83329e6i	0.724554i	0.932070	+	0.362277i	$0.118001\pi$
$683$	8.83329e6i	0.724554i	−0.932070	+	0.362277i	$0.881999\pi$
$684$	819406.	0.0669667
$685$	0	0
$686$	−4.33656e6	−0.351832
$687$	− 1.54064e7i	− 1.24540i
$688$	− 1.52915e7i	− 1.23163i
$689$	−806808.	−0.0647474
$690$	0	0
$691$	1.77549e7	1.41457	0.707284	−	0.706929i	$-0.249920\pi$
$691$	1.77549e7	1.41457	0.707284	+	0.706929i	$0.249920\pi$
$692$	367179.i	0.0291482i
$693$	− 1.84124e6i	− 0.145639i
$694$	108646.	0.00856279
$695$	0	0
$696$	−8.34510e6	−0.652993
$697$	− 1.58064e6i	− 0.123240i
$698$	4.75923e6i	0.369741i
$699$	−7.21349e6	−0.558409
$700$	0	0
$701$	1.32222e7	1.01627	0.508133	−	0.861279i	$-0.330336\pi$
$701$	1.32222e7	1.01627	0.508133	+	0.861279i	$0.330336\pi$
$702$	− 1.27360e6i	− 0.0975415i
$703$	− 2.00597e6i	− 0.153086i
$704$	2.24770e6	0.170925
$705$	0	0
$706$	2.00101e6	0.151091
$707$	− 1.17566e7i	− 0.884572i
$708$	1.64907e7i	1.23639i
$709$	−2.51312e7	−1.87758	−0.938790	−	0.344490i	$-0.888052\pi$
$709$	−2.51312e7	−1.87758	−0.938790	+	0.344490i	$0.888052\pi$
$710$	0	0
$711$	4.52988e6	0.336057
$712$	62598.2i	0.00462766i
$713$	3.16604e7i	2.33234i
$714$	−932881.	−0.0684827
$715$	0	0
$716$	−1.84240e7	−1.34308
$717$	2.17959e6i	0.158335i
$718$	− 7.01763e6i	− 0.508018i
$719$	−2.01931e6	−0.145674	−0.0728369	−	0.997344i	$-0.523205\pi$
$719$	−2.01931e6	−0.145674	−0.0728369	+	0.997344i	$0.523205\pi$
$720$	0	0
$721$	−2.00400e6	−0.143568
$722$	4.40306e6i	0.314349i
$723$	1.60790e7i	1.14397i
$724$	1.71243e7	1.21413
$725$	0	0
$726$	−2.94873e6	−0.207632
$727$	2.25046e6i	0.157919i	0.996878	+	0.0789597i	$0.0251598\pi$
$727$	2.25046e6i	0.157919i	−0.996878	+	0.0789597i	$0.974840\pi$
$728$	2.11214e6i	0.147705i
$729$	−1.43411e7	−0.999458
$730$	0	0
$731$	−8.15946e6	−0.564765
$732$	1.24602e6i	0.0859504i
$733$	− 5.24333e6i	− 0.360452i	−0.983625	−	0.180226i	$-0.942317\pi$
$733$	− 5.24333e6i	− 0.360452i	0.983625	−	0.180226i	$-0.0576829\pi$
$734$	−8.71586e6	−0.597132
$735$	0	0
$736$	−2.04074e7	−1.38865
$737$	− 2.57469e6i	− 0.174605i
$738$	− 771642.i	− 0.0521525i
$739$	1.56695e7	1.05547	0.527733	−	0.849411i	$-0.323042\pi$
$739$	1.56695e7	1.05547	0.527733	+	0.849411i	$0.323042\pi$
$740$	0	0
$741$	572643.	0.0383123
$742$	984228.i	0.0656275i
$743$	− 6.91028e6i	− 0.459223i	−0.973282	−	0.229611i	$-0.926254\pi$
$743$	− 6.91028e6i	− 0.459223i	0.973282	−	0.229611i	$-0.0737456\pi$
$744$	−1.00465e7	−0.665398
$745$	0	0
$746$	1.29622e6	0.0852772
$747$	− 2.38861e6i	− 0.156619i
$748$	− 1.76915e6i	− 0.115614i
$749$	−1.45850e6	−0.0949949
$750$	0	0
$751$	9.35145e6	0.605033	0.302517	−	0.953144i	$-0.402173\pi$
$751$	9.35145e6	0.605033	0.302517	+	0.953144i	$0.402173\pi$
$752$	1.61313e7i	1.04022i
$753$	− 7.12194e6i	− 0.457731i
$754$	1.95071e6	0.124958
$755$	0	0
$756$	1.31608e7	0.837489
$757$	1.84763e7i	1.17186i	0.810362	+	0.585929i	$0.199270\pi$
$757$	1.84763e7i	1.17186i	−0.810362	+	0.585929i	$0.800730\pi$
$758$	1.13121e6i	0.0715105i
$759$	8.13718e6	0.512707
$760$	0	0
$761$	−1.84001e7	−1.15175	−0.575876	−	0.817537i	$-0.695339\pi$
$761$	−1.84001e7	−1.15175	−0.575876	+	0.817537i	$0.695339\pi$
$762$	− 4.68534e6i	− 0.292317i
$763$	− 2.30688e7i	− 1.43454i
$764$	1.04234e7	0.646067
$765$	0	0
$766$	6.94540e6	0.427686
$767$	− 8.16463e6i	− 0.501127i
$768$	1.29107e6i	0.0789856i
$769$	1.14513e7	0.698295	0.349148	−	0.937068i	$-0.386471\pi$
$769$	1.14513e7	0.698295	0.349148	+	0.937068i	$0.386471\pi$
$770$	0	0
$771$	2.75471e6	0.166894
$772$	1.52717e7i	0.922238i
$773$	− 1.86929e7i	− 1.12519i	−0.826731	−	0.562597i	$-0.809802\pi$
$773$	− 1.86929e7i	− 1.12519i	0.826731	−	0.562597i	$-0.190198\pi$
$774$	−3.98332e6	−0.238997
$775$	0	0
$776$	−3.48338e6	−0.207657
$777$	− 9.44412e6i	− 0.561189i
$778$	− 3.54099e6i	− 0.209737i
$779$	1.18363e6	0.0698831
$780$	0	0
$781$	−6.06825e6	−0.355989
$782$	2.92081e6i	0.170799i
$783$	− 2.57448e7i	− 1.50067i
$784$	3.00599e6	0.174661
$785$	0	0
$786$	3.87456e6	0.223700
$787$	− 2.53432e7i	− 1.45856i	−0.684214	−	0.729281i	$-0.739855\pi$
$787$	− 2.53432e7i	− 1.45856i	0.684214	−	0.729281i	$-0.260145\pi$
$788$	8.46309e6i	0.485527i
$789$	−2.26512e7	−1.29538
$790$	0	0
$791$	−1.33908e7	−0.760964
$792$	− 1.82930e6i	− 0.103627i
$793$	− 616912.i	− 0.0348370i
$794$	−6.08408e6	−0.342487
$795$	0	0
$796$	1.66476e7	0.931256
$797$	2.38962e7i	1.33255i	0.745706	+	0.666276i	$0.232113\pi$
$797$	2.38962e7i	1.33255i	−0.745706	+	0.666276i	$0.767887\pi$
$798$	− 698570.i	− 0.0388332i
$799$	8.60758e6	0.476995
$800$	0	0
$801$	−56607.1	−0.00311738
$802$	− 4.83783e6i	− 0.265592i
$803$	4.77725e6i	0.261450i
$804$	5.39450e6	0.294314
$805$	0	0
$806$	2.34841e6	0.127332
$807$	− 2.75223e7i	− 1.48765i
$808$	− 1.16804e7i	− 0.629401i
$809$	1.73141e7	0.930096	0.465048	−	0.885285i	$-0.346037\pi$
$809$	1.73141e7	0.930096	0.465048	+	0.885285i	$0.346037\pi$
$810$	0	0
$811$	2.18577e7	1.16695	0.583474	−	0.812131i	$-0.301693\pi$
$811$	2.18577e7	1.16695	0.583474	+	0.812131i	$0.301693\pi$
$812$	2.01578e7i	1.07289i
$813$	− 2.61007e7i	− 1.38492i
$814$	−2.11433e6	−0.111844
$815$	0	0
$816$	3.21747e6	0.169157
$817$	− 6.11006e6i	− 0.320251i
$818$	1.05854e6i	0.0553127i
$819$	−1.91000e6	−0.0995001
$820$	0	0
$821$	7.12188e6	0.368754	0.184377	−	0.982856i	$-0.440973\pi$
$821$	7.12188e6	0.368754	0.184377	+	0.982856i	$0.440973\pi$
$822$	2.92567e6i	0.151024i
$823$	− 2.08032e7i	− 1.07061i	−0.844660	−	0.535303i	$-0.820197\pi$
$823$	− 2.08032e7i	− 1.07061i	0.844660	−	0.535303i	$-0.179803\pi$
$824$	−1.99100e6	−0.102154
$825$	0	0
$826$	−9.96007e6	−0.507940
$827$	1.96032e6i	0.0996698i	0.998757	+	0.0498349i	$0.0158695\pi$
$827$	1.96032e6i	0.0996698i	−0.998757	+	0.0498349i	$0.984130\pi$
$828$	− 1.20785e7i	− 0.612261i
$829$	−1.66913e7	−0.843535	−0.421768	−	0.906704i	$-0.638590\pi$
$829$	−1.66913e7	−0.843535	−0.421768	+	0.906704i	$0.638590\pi$
$830$	0	0
$831$	292590.	0.0146979
$832$	− 2.33164e6i	− 0.116776i
$833$	− 1.60398e6i	− 0.0800914i
$834$	3.93067e6	0.195682
$835$	0	0
$836$	1.32479e6	0.0655589
$837$	− 3.09936e7i	− 1.52918i
$838$	− 1.05213e7i	− 0.517559i
$839$	2.47847e7	1.21557	0.607783	−	0.794103i	$-0.292059\pi$
$839$	2.47847e7	1.21557	0.607783	+	0.794103i	$0.292059\pi$
$840$	0	0
$841$	1.89209e7	0.922470
$842$	1.27413e7i	0.619346i
$843$	− 2.83040e6i	− 0.137176i
$844$	1.29548e7	0.626000
$845$	0	0
$846$	4.20209e6	0.201855
$847$	1.50863e7i	0.722563i
$848$	− 3.39457e6i	− 0.162104i
$849$	1.44983e7	0.690315
$850$	0	0
$851$	−2.95691e7	−1.39963
$852$	− 1.27142e7i	− 0.600054i
$853$	1.16070e7i	0.546193i	0.961987	+	0.273096i	$0.0880478\pi$
$853$	1.16070e7i	0.546193i	−0.961987	+	0.273096i	$0.911952\pi$
$854$	−752573.	−0.0353105
$855$	0	0
$856$	−1.44904e6	−0.0675919
$857$	2.85621e7i	1.32843i	0.747543	+	0.664214i	$0.231233\pi$
$857$	2.85621e7i	1.32843i	−0.747543	+	0.664214i	$0.768767\pi$
$858$	− 603577.i	− 0.0279907i
$859$	2.72393e6	0.125955	0.0629773	−	0.998015i	$-0.479940\pi$
$859$	2.72393e6	0.125955	0.0629773	+	0.998015i	$0.479940\pi$
$860$	0	0
$861$	5.57253e6	0.256180
$862$	395269.i	0.0181186i
$863$	4.18556e7i	1.91305i	0.291651	+	0.956525i	$0.405795\pi$
$863$	4.18556e7i	1.91305i	−0.291651	+	0.956525i	$0.594205\pi$
$864$	1.99776e7	0.910457
$865$	0	0
$866$	−9.51902e6	−0.431318
$867$	1.52166e7i	0.687497i
$868$	2.42676e7i	1.09327i
$869$	7.32378e6	0.328992
$870$	0	0
$871$	−2.67085e6	−0.119290
$872$	− 2.29191e7i	− 1.02072i
$873$	− 3.15000e6i	− 0.139886i
$874$	−2.18719e6	−0.0968519
$875$	0	0
$876$	−1.00093e7	−0.440701
$877$	− 3.60953e7i	− 1.58472i	−0.610055	−	0.792359i	$-0.708853\pi$
$877$	− 3.60953e7i	− 1.58472i	0.610055	−	0.792359i	$-0.291147\pi$
$878$	− 1.39724e7i	− 0.611694i
$879$	−2.60917e7	−1.13901
$880$	0	0
$881$	3.31393e7	1.43848	0.719240	−	0.694761i	$-0.244490\pi$
$881$	3.31393e7	1.43848	0.719240	+	0.694761i	$0.244490\pi$
$882$	− 783036.i	− 0.0338930i
$883$	− 9.64225e6i	− 0.416176i	−0.978110	−	0.208088i	$-0.933276\pi$
$883$	− 9.64225e6i	− 0.416176i	0.978110	−	0.208088i	$-0.0667240\pi$
$884$	−1.83522e6	−0.0789871
$885$	0	0
$886$	1.31231e7	0.561631
$887$	− 3.59387e6i	− 0.153375i	−0.997055	−	0.0766873i	$-0.975566\pi$
$887$	− 3.59387e6i	− 0.153375i	0.997055	−	0.0766873i	$-0.0244343\pi$
$888$	− 9.38287e6i	− 0.399304i
$889$	−2.39712e7	−1.01727
$890$	0	0
$891$	−3.97661e6	−0.167810
$892$	− 1.23139e7i	− 0.518183i
$893$	6.44563e6i	0.270481i
$894$	−1.53441e6	−0.0642092
$895$	0	0
$896$	−2.03332e7	−0.846128
$897$	− 8.44106e6i	− 0.350281i
$898$	− 1.55940e6i	− 0.0645309i
$899$	4.74714e7	1.95899
$900$	0	0
$901$	−1.81132e6	−0.0743334
$902$	− 1.24757e6i	− 0.0510561i
$903$	− 2.87661e7i	− 1.17398i
$904$	−1.33039e7	−0.541450
$905$	0	0
$906$	−7.91083e6	−0.320186
$907$	− 4.23884e7i	− 1.71092i	−0.517872	−	0.855458i	$-0.673276\pi$
$907$	− 4.23884e7i	− 1.71092i	0.517872	−	0.855458i	$-0.326724\pi$
$908$	1.98690e7i	0.799764i
$909$	1.05625e7	0.423990
$910$	0	0
$911$	−2.39933e7	−0.957841	−0.478921	−	0.877858i	$-0.658972\pi$
$911$	−2.39933e7	−0.957841	−0.478921	+	0.877858i	$0.658972\pi$
$912$	2.40934e6i	0.0959205i
$913$	− 3.86184e6i	− 0.153326i
$914$	6.16522e6	0.244109
$915$	0	0
$916$	−3.69733e7	−1.45596
$917$	− 1.98231e7i	− 0.778481i
$918$	− 2.85929e6i	− 0.111983i
$919$	2.53438e7	0.989883	0.494941	−	0.868926i	$-0.335190\pi$
$919$	2.53438e7	0.989883	0.494941	+	0.868926i	$0.335190\pi$
$920$	0	0
$921$	5.48915e6	0.213234
$922$	6.56870e6i	0.254479i
$923$	6.29488e6i	0.243211i
$924$	6.23712e6	0.240328
$925$	0	0
$926$	2.65038e6	0.101574
$927$	− 1.80045e6i	− 0.0688148i
$928$	3.05988e7i	1.16636i
$929$	2.04548e7	0.777601	0.388800	−	0.921322i	$-0.372890\pi$
$929$	2.04548e7	0.777601	0.388800	+	0.921322i	$0.372890\pi$
$930$	0	0
$931$	1.20111e6	0.0454159
$932$	1.73114e7i	0.652818i
$933$	− 1.83969e7i	− 0.691896i
$934$	1.29327e7	0.485089
$935$	0	0
$936$	−1.89761e6	−0.0707975
$937$	− 1.98261e7i	− 0.737716i	−0.929486	−	0.368858i	$-0.879749\pi$
$937$	− 1.98261e7i	− 0.737716i	0.929486	−	0.368858i	$-0.120251\pi$
$938$	3.25818e6i	0.120911i
$939$	−1.43324e7	−0.530463
$940$	0	0
$941$	−3.32679e7	−1.22476	−0.612380	−	0.790564i	$-0.709788\pi$
$941$	−3.32679e7	−1.22476	−0.612380	+	0.790564i	$0.709788\pi$
$942$	− 1.62277e6i	− 0.0595839i
$943$	− 1.74473e7i	− 0.638925i
$944$	3.43519e7	1.25465
$945$	0	0
$946$	−6.44011e6	−0.233973
$947$	− 4.75820e7i	− 1.72412i	−0.506806	−	0.862060i	$-0.669174\pi$
$947$	− 4.75820e7i	− 1.72412i	0.506806	−	0.862060i	$-0.330826\pi$
$948$	1.53448e7i	0.554549i
$949$	4.95566e6	0.178623
$950$	0	0
$951$	6.10863e6	0.219025
$952$	4.74187e6i	0.169573i
$953$	4.16316e7i	1.48488i	0.669913	+	0.742440i	$0.266331\pi$
$953$	4.16316e7i	1.48488i	−0.669913	+	0.742440i	$0.733669\pi$
$954$	−884259.	−0.0314564
$955$	0	0
$956$	5.23073e6	0.185105
$957$	− 1.22008e7i	− 0.430635i
$958$	− 2.57537e6i	− 0.0906623i
$959$	1.49683e7	0.525566
$960$	0	0
$961$	2.85206e7	0.996207
$962$	2.19329e6i	0.0764115i
$963$	− 1.31035e6i	− 0.0455327i
$964$	3.85874e7	1.33737
$965$	0	0
$966$	−1.02973e7	−0.355042
$967$	− 3.70062e7i	− 1.27265i	−0.771422	−	0.636324i	$-0.780454\pi$
$967$	− 3.70062e7i	− 1.27265i	0.771422	−	0.636324i	$-0.219546\pi$
$968$	1.49885e7i	0.514127i
$969$	1.28561e6	0.0439846
$970$	0	0
$971$	846545.	0.0288139	0.0144070	−	0.999896i	$-0.495414\pi$
$971$	846545.	0.0288139	0.0144070	+	0.999896i	$0.495414\pi$
$972$	2.01822e7i	0.685179i
$973$	− 2.01102e7i	− 0.680979i
$974$	1.55406e7	0.524894
$975$	0	0
$976$	2.59560e6	0.0872194
$977$	5.86464e7i	1.96564i	0.184554	+	0.982822i	$0.440916\pi$
$977$	5.86464e7i	1.96564i	−0.184554	+	0.982822i	$0.559084\pi$
$978$	8.67423e6i	0.289991i
$979$	−91520.7	−0.00305185
$980$	0	0
$981$	2.07256e7	0.687599
$982$	1.43700e7i	0.475530i
$983$	1.17394e7i	0.387492i	0.981052	+	0.193746i	$0.0620637\pi$
$983$	1.17394e7i	0.387492i	−0.981052	+	0.193746i	$0.937936\pi$
$984$	5.53639e6	0.182280
$985$	0	0
$986$	4.37944e6	0.143458
$987$	3.03460e7i	0.991536i
$988$	− 1.37427e6i	− 0.0447897i
$989$	−9.00654e7	−2.92798
$990$	0	0
$991$	−2.34339e7	−0.757984	−0.378992	−	0.925400i	$-0.623729\pi$
$991$	−2.34339e7	−0.757984	−0.378992	+	0.925400i	$0.623729\pi$
$992$	3.68372e7i	1.18852i
$993$	− 1.61182e6i	− 0.0518734i
$994$	7.67914e6	0.246517
$995$	0	0
$996$	8.09133e6	0.258447
$997$	− 3.77262e7i	− 1.20200i	−0.799248	−	0.601001i	$-0.794769\pi$
$997$	− 3.77262e7i	− 1.20200i	0.799248	−	0.601001i	$-0.205231\pi$
$998$	− 1.17242e7i	− 0.372613i
$999$	2.89463e7	0.917655

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.6.b.h.274.8		18
5.2	odd	4		325.6.a.i.1.6	yes	9
5.3	odd	4		325.6.a.h.1.4	✓	9
5.4	even	2	inner	325.6.b.h.274.11		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
325.6.a.h.1.4	✓	9	5.3	odd	4
325.6.a.i.1.6	yes	9	5.2	odd	4
325.6.b.h.274.8		18	1.1	even	1	trivial
325.6.b.h.274.11		18	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:	\( N \)	\(=\)	\( 325 = 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	325.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.83815i q^{2} +11.9262i q^{3} +28.6212 q^{4} +21.9221 q^{6} -112.158i q^{7} -111.431i q^{8} +100.766 q^{9} +O(q^{10})\)	\(q-1.83815i q^{2} +11.9262i q^{3} +28.6212 q^{4} +21.9221 q^{6} -112.158i q^{7} -111.431i q^{8} +100.766 q^{9} +162.916 q^{11} +341.342i q^{12} -169.000i q^{13} -206.164 q^{14} +711.052 q^{16} -379.413i q^{17} -185.224i q^{18} +284.116 q^{19} +1337.62 q^{21} -299.464i q^{22} -4188.02i q^{23} +1328.94 q^{24} -310.647 q^{26} +4099.82i q^{27} -3210.10i q^{28} -6279.50 q^{29} -7559.74 q^{31} -4872.81i q^{32} +1942.96i q^{33} -697.419 q^{34} +2884.05 q^{36} -7060.39i q^{37} -522.249i q^{38} +2015.52 q^{39} +4166.01 q^{41} -2458.75i q^{42} -21505.5i q^{43} +4662.85 q^{44} -7698.22 q^{46} +22686.6i q^{47} +8480.13i q^{48} +4227.52 q^{49} +4524.95 q^{51} -4836.98i q^{52} -4774.01i q^{53} +7536.08 q^{54} -12497.9 q^{56} +3388.42i q^{57} +11542.7i q^{58} +48311.4 q^{59} +3650.37 q^{61} +13895.9i q^{62} -11301.8i q^{63} +13796.7 q^{64} +3571.46 q^{66} -15803.8i q^{67} -10859.3i q^{68} +49947.1 q^{69} -37247.8 q^{71} -11228.5i q^{72} +29323.4i q^{73} -12978.1 q^{74} +8131.75 q^{76} -18272.4i q^{77} -3704.84i q^{78} +44954.4 q^{79} -24409.0 q^{81} -7657.75i q^{82} -23704.5i q^{83} +38284.3 q^{84} -39530.3 q^{86} -74890.4i q^{87} -18153.9i q^{88} -561.767 q^{89} -18954.7 q^{91} -119866. i q^{92} -90158.8i q^{93} +41701.3 q^{94} +58114.0 q^{96} -31260.4i q^{97} -7770.82i q^{98} +16416.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 182 q^{4} - 166 q^{6} - 1124 q^{9}+O(q^{10}) \) 18 * q - 182 * q^4 - 166 * q^6 - 1124 * q^9	\( 18 q - 182 q^{4} - 166 q^{6} - 1124 q^{9} - 2844 q^{11} + 684 q^{14} - 2122 q^{16} + 816 q^{19} - 7824 q^{21} + 16938 q^{24} - 1690 q^{26} + 17474 q^{29} + 1496 q^{31} + 35578 q^{34} + 1024 q^{36} + 3718 q^{39} - 57352 q^{41} + 61198 q^{44} - 24112 q^{46} + 81814 q^{49} - 62012 q^{51} + 205522 q^{54} - 46004 q^{56} + 176284 q^{59} + 56330 q^{61} + 201690 q^{64} + 85154 q^{66} + 363494 q^{69} - 141124 q^{71} + 271352 q^{74} + 92746 q^{76} + 328146 q^{79} - 139870 q^{81} + 691312 q^{84} - 589840 q^{86} + 505396 q^{89} + 4056 q^{91} + 1003212 q^{94} - 638362 q^{96} + 1515552 q^{99}+O(q^{100}) \) 18 * q - 182 * q^4 - 166 * q^6 - 1124 * q^9 - 2844 * q^11 + 684 * q^14 - 2122 * q^16 + 816 * q^19 - 7824 * q^21 + 16938 * q^24 - 1690 * q^26 + 17474 * q^29 + 1496 * q^31 + 35578 * q^34 + 1024 * q^36 + 3718 * q^39 - 57352 * q^41 + 61198 * q^44 - 24112 * q^46 + 81814 * q^49 - 62012 * q^51 + 205522 * q^54 - 46004 * q^56 + 176284 * q^59 + 56330 * q^61 + 201690 * q^64 + 85154 * q^66 + 363494 * q^69 - 141124 * q^71 + 271352 * q^74 + 92746 * q^76 + 328146 * q^79 - 139870 * q^81 + 691312 * q^84 - 589840 * q^86 + 505396 * q^89 + 4056 * q^91 + 1003212 * q^94 - 638362 * q^96 + 1515552 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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