LMFDB - Embedded newform 378.5.b.a.323.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,5,Mod(323,378)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(378, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("378.323");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	378.b (of order $2$, degree $1$, 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:	$39.0738460457$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.5443747577856.29
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 24x^{6} + 180x^{4} + 488x^{2} + 324 $ x^8 + 24x^6 + 180x^4 + 488*x^2 + 324
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{9}\cdot 3^{4}\cdot 7^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			323.2
Root			$2.14697i$ of defining polynomial
Character	$\chi$	$=$	378.323
Dual form			378.5.b.a.323.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.82843i q^{2} -8.00000 q^{4} -9.19100i q^{5} +18.5203 q^{7} +22.6274i q^{8} +O(q^{10})$	\(q-2.82843i q^{2} -8.00000 q^{4} -9.19100i q^{5} +18.5203 q^{7} +22.6274i q^{8} -25.9961 q^{10} -68.1448i q^{11} +212.413 q^{13} -52.3832i q^{14} +64.0000 q^{16} +340.636i q^{17} -176.519 q^{19} +73.5280i q^{20} -192.743 q^{22} +241.733i q^{23} +540.526 q^{25} -600.795i q^{26} -148.162 q^{28} -549.225i q^{29} +980.834 q^{31} -181.019i q^{32} +963.463 q^{34} -170.220i q^{35} +743.228 q^{37} +499.271i q^{38} +207.968 q^{40} -1964.59i q^{41} -1353.97 q^{43} +545.158i q^{44} +683.725 q^{46} +1321.73i q^{47} +343.000 q^{49} -1528.84i q^{50} -1699.30 q^{52} -4062.59i q^{53} -626.318 q^{55} +419.066i q^{56} -1553.44 q^{58} -3523.36i q^{59} +5207.28 q^{61} -2774.22i q^{62} -512.000 q^{64} -1952.29i q^{65} +2249.24 q^{67} -2725.08i q^{68} -481.454 q^{70} -7437.67i q^{71} -1051.46 q^{73} -2102.17i q^{74} +1412.15 q^{76} -1262.06i q^{77} +3149.66 q^{79} -588.224i q^{80} -5556.71 q^{82} +467.561i q^{83} +3130.78 q^{85} +3829.62i q^{86} +1541.94 q^{88} +2865.60i q^{89} +3933.94 q^{91} -1933.87i q^{92} +3738.42 q^{94} +1622.38i q^{95} -15884.4 q^{97} -970.151i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 64 q^{4}+O(q^{10}) $ 8 * q - 64 * q^4	$ 8 q - 64 q^{4} - 224 q^{10} - 376 q^{13} + 512 q^{16} + 1120 q^{19} - 1120 q^{22} + 792 q^{25} - 880 q^{31} - 1792 q^{34} + 1576 q^{37} + 1792 q^{40} - 5768 q^{43} - 160 q^{46} + 2744 q^{49} + 3008 q^{52} + 488 q^{55} + 7552 q^{58} - 2560 q^{61} - 4096 q^{64} + 23784 q^{67} + 1568 q^{70} - 13176 q^{73} - 8960 q^{76} - 9592 q^{79} - 3360 q^{82} - 39880 q^{85} + 8960 q^{88} + 5096 q^{91} + 4608 q^{94} - 14016 q^{97}+O(q^{100}) $ 8 * q - 64 * q^4 - 224 * q^10 - 376 * q^13 + 512 * q^16 + 1120 * q^19 - 1120 * q^22 + 792 * q^25 - 880 * q^31 - 1792 * q^34 + 1576 * q^37 + 1792 * q^40 - 5768 * q^43 - 160 * q^46 + 2744 * q^49 + 3008 * q^52 + 488 * q^55 + 7552 * q^58 - 2560 * q^61 - 4096 * q^64 + 23784 * q^67 + 1568 * q^70 - 13176 * q^73 - 8960 * q^76 - 9592 * q^79 - 3360 * q^82 - 39880 * q^85 + 8960 * q^88 + 5096 * q^91 + 4608 * q^94 - 14016 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$.

$n$	$29$	$325$
$\chi(n)$	$-1$	$1$

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 $

$2$	− 2.82843i	− 0.707107i
$2$	− 2.82843i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−8.00000	−0.500000
$5$	− 9.19100i	− 0.367640i	−0.982960	−	0.183820i	$-0.941154\pi$
$5$	− 9.19100i	− 0.367640i	0.982960	−	0.183820i	$-0.0588463\pi$
$6$	0	0
$7$	18.5203	0.377964
$7$	18.5203	0.377964
$8$	22.6274i	0.353553i
$9$	0	0
$10$	−25.9961	−0.259961
$11$	− 68.1448i	− 0.563180i	−0.959535	−	0.281590i	$-0.909138\pi$
$11$	− 68.1448i	− 0.563180i	0.959535	−	0.281590i	$-0.0908618\pi$
$12$	0	0
$13$	212.413	1.25688	0.628441	−	0.777857i	$-0.283693\pi$
$13$	212.413	1.25688	0.628441	+	0.777857i	$0.283693\pi$
$14$	− 52.3832i	− 0.267261i
$15$	0	0
$16$	64.0000	0.250000
$17$	340.636i	1.17867i	0.807889	+	0.589335i	$0.200610\pi$
$17$	340.636i	1.17867i	−0.807889	+	0.589335i	$0.799390\pi$
$18$	0	0
$19$	−176.519	−0.488972	−0.244486	−	0.969653i	$-0.578619\pi$
$19$	−176.519	−0.488972	−0.244486	+	0.969653i	$0.578619\pi$
$20$	73.5280i	0.183820i
$21$	0	0
$22$	−192.743	−0.398228
$23$	241.733i	0.456963i	0.973548	+	0.228481i	$0.0733760\pi$
$23$	241.733i	0.456963i	−0.973548	+	0.228481i	$0.926624\pi$
$24$	0	0
$25$	540.526	0.864841
$26$	− 600.795i	− 0.888750i
$27$	0	0
$28$	−148.162	−0.188982
$29$	− 549.225i	− 0.653062i	−0.945186	−	0.326531i	$-0.894120\pi$
$29$	− 549.225i	− 0.653062i	0.945186	−	0.326531i	$-0.105880\pi$
$30$	0	0
$31$	980.834	1.02064	0.510319	−	0.859985i	$-0.329527\pi$
$31$	980.834	1.02064	0.510319	+	0.859985i	$0.329527\pi$
$32$	− 181.019i	− 0.176777i
$33$	0	0
$34$	963.463	0.833445
$35$	− 170.220i	− 0.138955i
$36$	0	0
$37$	743.228	0.542899	0.271449	−	0.962453i	$-0.412497\pi$
$37$	743.228	0.542899	0.271449	+	0.962453i	$0.412497\pi$
$38$	499.271i	0.345755i
$39$	0	0
$40$	207.968	0.129980
$41$	− 1964.59i	− 1.16871i	−0.811500	−	0.584353i	$-0.801348\pi$
$41$	− 1964.59i	− 1.16871i	0.811500	−	0.584353i	$-0.198652\pi$
$42$	0	0
$43$	−1353.97	−0.732274	−0.366137	−	0.930561i	$-0.619320\pi$
$43$	−1353.97	−0.732274	−0.366137	+	0.930561i	$0.619320\pi$
$44$	545.158i	0.281590i
$45$	0	0
$46$	683.725	0.323121
$47$	1321.73i	0.598339i	0.954200	+	0.299169i	$0.0967095\pi$
$47$	1321.73i	0.598339i	−0.954200	+	0.299169i	$0.903290\pi$
$48$	0	0
$49$	343.000	0.142857
$50$	− 1528.84i	− 0.611535i
$51$	0	0
$52$	−1699.30	−0.628441
$53$	− 4062.59i	− 1.44628i	−0.690703	−	0.723139i	$-0.742699\pi$
$53$	− 4062.59i	− 1.44628i	0.690703	−	0.723139i	$-0.257301\pi$
$54$	0	0
$55$	−626.318	−0.207047
$56$	419.066i	0.133631i
$57$	0	0
$58$	−1553.44	−0.461785
$59$	− 3523.36i	− 1.01217i	−0.862484	−	0.506085i	$-0.831092\pi$
$59$	− 3523.36i	− 1.01217i	0.862484	−	0.506085i	$-0.168908\pi$
$60$	0	0
$61$	5207.28	1.39943	0.699715	−	0.714423i	$-0.253311\pi$
$61$	5207.28	1.39943	0.699715	+	0.714423i	$0.253311\pi$
$62$	− 2774.22i	− 0.721700i
$63$	0	0
$64$	−512.000	−0.125000
$65$	− 1952.29i	− 0.462080i
$66$	0	0
$67$	2249.24	0.501056	0.250528	−	0.968109i	$-0.419396\pi$
$67$	2249.24	0.501056	0.250528	+	0.968109i	$0.419396\pi$
$68$	− 2725.08i	− 0.589335i
$69$	0	0
$70$	−481.454	−0.0982559
$71$	− 7437.67i	− 1.47544i	−0.675109	−	0.737718i	$-0.735903\pi$
$71$	− 7437.67i	− 1.47544i	0.675109	−	0.737718i	$-0.264097\pi$
$72$	0	0
$73$	−1051.46	−0.197308	−0.0986542	−	0.995122i	$-0.531454\pi$
$73$	−1051.46	−0.197308	−0.0986542	+	0.995122i	$0.531454\pi$
$74$	− 2102.17i	− 0.383887i
$75$	0	0
$76$	1412.15	0.244486
$77$	− 1262.06i	− 0.212862i
$78$	0	0
$79$	3149.66	0.504673	0.252336	−	0.967640i	$-0.418801\pi$
$79$	3149.66	0.504673	0.252336	+	0.967640i	$0.418801\pi$
$80$	− 588.224i	− 0.0919100i
$81$	0	0
$82$	−5556.71	−0.826400
$83$	467.561i	0.0678707i	0.999424	+	0.0339354i	$0.0108040\pi$
$83$	467.561i	0.0678707i	−0.999424	+	0.0339354i	$0.989196\pi$
$84$	0	0
$85$	3130.78	0.433326
$86$	3829.62i	0.517796i
$87$	0	0
$88$	1541.94	0.199114
$89$	2865.60i	0.361773i	0.983504	+	0.180886i	$0.0578966\pi$
$89$	2865.60i	0.361773i	−0.983504	+	0.180886i	$0.942103\pi$
$90$	0	0
$91$	3933.94	0.475057
$92$	− 1933.87i	− 0.228481i
$93$	0	0
$94$	3738.42	0.423089
$95$	1622.38i	0.179766i
$96$	0	0
$97$	−15884.4	−1.68821	−0.844107	−	0.536175i	$-0.819869\pi$
$97$	−15884.4	−1.68821	−0.844107	+	0.536175i	$0.819869\pi$
$98$	− 970.151i	− 0.101015i
$99$	0	0
$100$	−4324.20	−0.432420
$101$	− 8321.92i	− 0.815795i	−0.913028	−	0.407897i	$-0.866262\pi$
$101$	− 8321.92i	− 0.815795i	0.913028	−	0.407897i	$-0.133738\pi$
$102$	0	0
$103$	−2251.19	−0.212196	−0.106098	−	0.994356i	$-0.533836\pi$
$103$	−2251.19	−0.212196	−0.106098	+	0.994356i	$0.533836\pi$
$104$	4806.36i	0.444375i
$105$	0	0
$106$	−11490.8	−1.02267
$107$	− 21923.5i	− 1.91488i	−0.288629	−	0.957441i	$-0.593200\pi$
$107$	− 21923.5i	− 1.91488i	0.288629	−	0.957441i	$-0.406800\pi$
$108$	0	0
$109$	−3095.26	−0.260522	−0.130261	−	0.991480i	$-0.541582\pi$
$109$	−3095.26	−0.260522	−0.130261	+	0.991480i	$0.541582\pi$
$110$	1771.50i	0.146405i
$111$	0	0
$112$	1185.30	0.0944911
$113$	4295.01i	0.336362i	0.985756	+	0.168181i	$0.0537893\pi$
$113$	4295.01i	0.336362i	−0.985756	+	0.168181i	$0.946211\pi$
$114$	0	0
$115$	2221.77	0.167998
$116$	4393.80i	0.326531i
$117$	0	0
$118$	−9965.57	−0.715712
$119$	6308.66i	0.445495i
$120$	0	0
$121$	9997.29	0.682828
$122$	− 14728.4i	− 0.989546i
$123$	0	0
$124$	−7846.67	−0.510319
$125$	− 10712.3i	− 0.685590i
$126$	0	0
$127$	3845.70	0.238434	0.119217	−	0.992868i	$-0.461962\pi$
$127$	3845.70	0.238434	0.119217	+	0.992868i	$0.461962\pi$
$128$	1448.15i	0.0883883i
$129$	0	0
$130$	−5521.90	−0.326740
$131$	− 12992.3i	− 0.757083i	−0.925584	−	0.378542i	$-0.876426\pi$
$131$	− 12992.3i	− 0.757083i	0.925584	−	0.378542i	$-0.123574\pi$
$132$	0	0
$133$	−3269.17	−0.184814
$134$	− 6361.81i	− 0.354300i
$135$	0	0
$136$	−7707.70	−0.416723
$137$	7048.28i	0.375528i	0.982214	+	0.187764i	$0.0601240\pi$
$137$	7048.28i	0.375528i	−0.982214	+	0.187764i	$0.939876\pi$
$138$	0	0
$139$	13159.3	0.681089	0.340544	−	0.940228i	$-0.389389\pi$
$139$	13159.3	0.681089	0.340544	+	0.940228i	$0.389389\pi$
$140$	1361.76i	0.0694774i
$141$	0	0
$142$	−21036.9	−1.04329
$143$	− 14474.8i	− 0.707851i
$144$	0	0
$145$	−5047.93	−0.240092
$146$	2973.97i	0.139518i
$147$	0	0
$148$	−5945.83	−0.271449
$149$	26816.8i	1.20791i	0.797018	+	0.603955i	$0.206409\pi$
$149$	26816.8i	1.20791i	−0.797018	+	0.603955i	$0.793591\pi$
$150$	0	0
$151$	24578.6	1.07796	0.538980	−	0.842319i	$-0.318810\pi$
$151$	24578.6	1.07796	0.538980	+	0.842319i	$0.318810\pi$
$152$	− 3994.17i	− 0.172878i
$153$	0	0
$154$	−3569.64	−0.150516
$155$	− 9014.84i	− 0.375227i
$156$	0	0
$157$	31474.8	1.27692	0.638460	−	0.769655i	$-0.279572\pi$
$157$	31474.8	1.27692	0.638460	+	0.769655i	$0.279572\pi$
$158$	− 8908.59i	− 0.356858i
$159$	0	0
$160$	−1663.75	−0.0649902
$161$	4476.96i	0.172716i
$162$	0	0
$163$	−32656.3	−1.22911	−0.614556	−	0.788873i	$-0.710665\pi$
$163$	−32656.3	−1.22911	−0.614556	+	0.788873i	$0.710665\pi$
$164$	15716.8i	0.584353i
$165$	0	0
$166$	1322.46	0.0479918
$167$	30734.2i	1.10202i	0.834499	+	0.551010i	$0.185757\pi$
$167$	30734.2i	1.10202i	−0.834499	+	0.551010i	$0.814243\pi$
$168$	0	0
$169$	16558.3	0.579752
$170$	− 8855.18i	− 0.306408i
$171$	0	0
$172$	10831.8	0.366137
$173$	32068.3i	1.07148i	0.844383	+	0.535740i	$0.179967\pi$
$173$	32068.3i	1.07148i	−0.844383	+	0.535740i	$0.820033\pi$
$174$	0	0
$175$	10010.7	0.326879
$176$	− 4361.27i	− 0.140795i
$177$	0	0
$178$	8105.14	0.255812
$179$	18927.5i	0.590729i	0.955385	+	0.295364i	$0.0954410\pi$
$179$	18927.5i	0.590729i	−0.955385	+	0.295364i	$0.904559\pi$
$180$	0	0
$181$	16502.4	0.503720	0.251860	−	0.967764i	$-0.418958\pi$
$181$	16502.4	0.503720	0.251860	+	0.967764i	$0.418958\pi$
$182$	− 11126.9i	− 0.335916i
$183$	0	0
$184$	−5469.80	−0.161561
$185$	− 6831.01i	− 0.199591i
$186$	0	0
$187$	23212.5	0.663803
$188$	− 10573.8i	− 0.299169i
$189$	0	0
$190$	4588.79	0.127113
$191$	33728.4i	0.924548i	0.886737	+	0.462274i	$0.152966\pi$
$191$	33728.4i	0.924548i	−0.886737	+	0.462274i	$0.847034\pi$
$192$	0	0
$193$	29108.7	0.781462	0.390731	−	0.920505i	$-0.372222\pi$
$193$	29108.7	0.781462	0.390731	+	0.920505i	$0.372222\pi$
$194$	44927.9i	1.19375i
$195$	0	0
$196$	−2744.00	−0.0714286
$197$	− 61088.1i	− 1.57407i	−0.616908	−	0.787035i	$-0.711615\pi$
$197$	− 61088.1i	− 1.57407i	0.616908	−	0.787035i	$-0.288385\pi$
$198$	0	0
$199$	22664.7	0.572328	0.286164	−	0.958181i	$-0.407620\pi$
$199$	22664.7	0.572328	0.286164	+	0.958181i	$0.407620\pi$
$200$	12230.7i	0.305767i
$201$	0	0
$202$	−23538.0	−0.576854
$203$	− 10171.8i	− 0.246834i
$204$	0	0
$205$	−18056.6	−0.429663
$206$	6367.33i	0.150045i
$207$	0	0
$208$	13594.4	0.314220
$209$	12028.8i	0.275379i
$210$	0	0
$211$	−47754.5	−1.07263	−0.536314	−	0.844018i	$-0.680184\pi$
$211$	−47754.5	−1.07263	−0.536314	+	0.844018i	$0.680184\pi$
$212$	32500.8i	0.723139i
$213$	0	0
$214$	−62009.0	−1.35403
$215$	12444.4i	0.269213i
$216$	0	0
$217$	18165.3	0.385765
$218$	8754.73i	0.184217i
$219$	0	0
$220$	5010.55	0.103524
$221$	72355.4i	1.48145i
$222$	0	0
$223$	40317.3	0.810741	0.405370	−	0.914153i	$-0.367143\pi$
$223$	40317.3	0.810741	0.405370	+	0.914153i	$0.367143\pi$
$224$	− 3352.53i	− 0.0668153i
$225$	0	0
$226$	12148.1	0.237844
$227$	69798.9i	1.35455i	0.735728	+	0.677277i	$0.236840\pi$
$227$	69798.9i	1.35455i	−0.735728	+	0.677277i	$0.763160\pi$
$228$	0	0
$229$	58230.3	1.11040	0.555199	−	0.831718i	$-0.312642\pi$
$229$	58230.3	1.11040	0.555199	+	0.831718i	$0.312642\pi$
$230$	− 6284.11i	− 0.118792i
$231$	0	0
$232$	12427.5	0.230892
$233$	36975.2i	0.681081i	0.940230	+	0.340540i	$0.110610\pi$
$233$	36975.2i	0.681081i	−0.940230	+	0.340540i	$0.889390\pi$
$234$	0	0
$235$	12148.0	0.219973
$236$	28186.9i	0.506085i
$237$	0	0
$238$	17843.6	0.315013
$239$	44554.3i	0.779999i	0.920815	+	0.390000i	$0.127525\pi$
$239$	44554.3i	0.779999i	−0.920815	+	0.390000i	$0.872475\pi$
$240$	0	0
$241$	79999.2	1.37737	0.688686	−	0.725060i	$-0.258188\pi$
$241$	79999.2	1.37737	0.688686	+	0.725060i	$0.258188\pi$
$242$	− 28276.6i	− 0.482833i
$243$	0	0
$244$	−41658.2	−0.699715
$245$	− 3152.51i	− 0.0525200i
$246$	0	0
$247$	−37494.9	−0.614580
$248$	22193.7i	0.360850i
$249$	0	0
$250$	−30299.1	−0.484785
$251$	− 47272.6i	− 0.750346i	−0.926955	−	0.375173i	$-0.877583\pi$
$251$	− 47272.6i	− 0.750346i	0.926955	−	0.375173i	$-0.122417\pi$
$252$	0	0
$253$	16472.9	0.257352
$254$	− 10877.3i	− 0.168598i
$255$	0	0
$256$	4096.00	0.0625000
$257$	53489.4i	0.809845i	0.914351	+	0.404922i	$0.132701\pi$
$257$	53489.4i	0.809845i	−0.914351	+	0.404922i	$0.867299\pi$
$258$	0	0
$259$	13764.8	0.205196
$260$	15618.3i	0.231040i
$261$	0	0
$262$	−36747.8	−0.535339
$263$	59690.1i	0.862961i	0.902122	+	0.431480i	$0.142009\pi$
$263$	59690.1i	0.862961i	−0.902122	+	0.431480i	$0.857991\pi$
$264$	0	0
$265$	−37339.3	−0.531709
$266$	9246.62i	0.130683i
$267$	0	0
$268$	−17993.9	−0.250528
$269$	− 76612.4i	− 1.05875i	−0.848387	−	0.529376i	$-0.822426\pi$
$269$	− 76612.4i	− 1.05875i	0.848387	−	0.529376i	$-0.177574\pi$
$270$	0	0
$271$	13224.9	0.180076	0.0900379	−	0.995938i	$-0.471301\pi$
$271$	13224.9	0.180076	0.0900379	+	0.995938i	$0.471301\pi$
$272$	21800.7i	0.294667i
$273$	0	0
$274$	19935.6	0.265538
$275$	− 36834.0i	− 0.487061i
$276$	0	0
$277$	−2728.19	−0.0355562	−0.0177781	−	0.999842i	$-0.505659\pi$
$277$	−2728.19	−0.0355562	−0.0177781	+	0.999842i	$0.505659\pi$
$278$	− 37220.2i	− 0.481602i
$279$	0	0
$280$	3851.63	0.0491279
$281$	92335.3i	1.16938i	0.811258	+	0.584689i	$0.198783\pi$
$281$	92335.3i	1.16938i	−0.811258	+	0.584689i	$0.801217\pi$
$282$	0	0
$283$	−117175.	−1.46306	−0.731528	−	0.681812i	$-0.761192\pi$
$283$	−117175.	−1.46306	−0.731528	+	0.681812i	$0.761192\pi$
$284$	59501.4i	0.737718i
$285$	0	0
$286$	−40941.0	−0.500526
$287$	− 36384.8i	− 0.441729i
$288$	0	0
$289$	−32511.6	−0.389262
$290$	14277.7i	0.169770i
$291$	0	0
$292$	8411.65	0.0986542
$293$	− 67218.3i	− 0.782982i	−0.920182	−	0.391491i	$-0.871959\pi$
$293$	− 67218.3i	− 0.782982i	0.920182	−	0.391491i	$-0.128041\pi$
$294$	0	0
$295$	−32383.2	−0.372114
$296$	16817.3i	0.191944i
$297$	0	0
$298$	75849.4	0.854122
$299$	51347.3i	0.574348i
$300$	0	0
$301$	−25075.9	−0.276773
$302$	− 69518.6i	− 0.762232i
$303$	0	0
$304$	−11297.2	−0.122243
$305$	− 47860.0i	− 0.514486i
$306$	0	0
$307$	−108355.	−1.14966	−0.574832	−	0.818271i	$-0.694933\pi$
$307$	−108355.	−1.14966	−0.574832	+	0.818271i	$0.694933\pi$
$308$	10096.5i	0.106431i
$309$	0	0
$310$	−25497.8	−0.265326
$311$	− 95709.1i	− 0.989538i	−0.869024	−	0.494769i	$-0.835253\pi$
$311$	− 95709.1i	− 0.989538i	0.869024	−	0.494769i	$-0.164747\pi$
$312$	0	0
$313$	158112.	1.61390	0.806951	−	0.590619i	$-0.201116\pi$
$313$	158112.	1.61390	0.806951	+	0.590619i	$0.201116\pi$
$314$	− 89024.2i	− 0.902919i
$315$	0	0
$316$	−25197.3	−0.252336
$317$	91133.8i	0.906903i	0.891281	+	0.453451i	$0.149807\pi$
$317$	91133.8i	0.906903i	−0.891281	+	0.453451i	$0.850193\pi$
$318$	0	0
$319$	−37426.8	−0.367791
$320$	4705.79i	0.0459550i
$321$	0	0
$322$	12662.8	0.122128
$323$	− 60128.6i	− 0.576336i
$324$	0	0
$325$	114815.	1.08700
$326$	92365.9i	0.869113i
$327$	0	0
$328$	44453.7	0.413200
$329$	24478.8i	0.226151i
$330$	0	0
$331$	−129014.	−1.17755	−0.588777	−	0.808296i	$-0.700390\pi$
$331$	−129014.	−1.17755	−0.588777	+	0.808296i	$0.700390\pi$
$332$	− 3740.49i	− 0.0339354i
$333$	0	0
$334$	86929.6	0.779246
$335$	− 20672.8i	− 0.184208i
$336$	0	0
$337$	−166293.	−1.46425	−0.732124	−	0.681172i	$-0.761471\pi$
$337$	−166293.	−1.46425	−0.732124	+	0.681172i	$0.761471\pi$
$338$	− 46833.9i	− 0.409947i
$339$	0	0
$340$	−25046.2	−0.216663
$341$	− 66838.7i	− 0.574803i
$342$	0	0
$343$	6352.45	0.0539949
$344$	− 30636.9i	− 0.258898i
$345$	0	0
$346$	90702.9	0.757650
$347$	229085.i	1.90256i	0.308335	+	0.951278i	$0.400228\pi$
$347$	229085.i	1.90256i	−0.308335	+	0.951278i	$0.599772\pi$
$348$	0	0
$349$	97978.4	0.804413	0.402207	−	0.915549i	$-0.368243\pi$
$349$	97978.4	0.804413	0.402207	+	0.915549i	$0.368243\pi$
$350$	− 28314.5i	− 0.231138i
$351$	0	0
$352$	−12335.5	−0.0995571
$353$	− 212359.i	− 1.70420i	−0.523380	−	0.852100i	$-0.675329\pi$
$353$	− 212359.i	− 1.70420i	0.523380	−	0.852100i	$-0.324671\pi$
$354$	0	0
$355$	−68359.6	−0.542429
$356$	− 22924.8i	− 0.180886i
$357$	0	0
$358$	53535.2	0.417708
$359$	− 72001.6i	− 0.558668i	−0.960194	−	0.279334i	$-0.909886\pi$
$359$	− 72001.6i	− 0.558668i	0.960194	−	0.279334i	$-0.0901136\pi$
$360$	0	0
$361$	−99162.1	−0.760907
$362$	− 46675.8i	− 0.356184i
$363$	0	0
$364$	−31471.6	−0.237528
$365$	9663.93i	0.0725384i
$366$	0	0
$367$	−75888.0	−0.563432	−0.281716	−	0.959498i	$-0.590904\pi$
$367$	−75888.0	−0.563432	−0.281716	+	0.959498i	$0.590904\pi$
$368$	15470.9i	0.114241i
$369$	0	0
$370$	−19321.0	−0.141132
$371$	− 75240.3i	− 0.546642i
$372$	0	0
$373$	−141578.	−1.01760	−0.508800	−	0.860885i	$-0.669911\pi$
$373$	−141578.	−1.01760	−0.508800	+	0.860885i	$0.669911\pi$
$374$	− 65654.9i	− 0.469380i
$375$	0	0
$376$	−29907.3	−0.211545
$377$	− 116663.i	− 0.820822i
$378$	0	0
$379$	81013.3	0.563999	0.281999	−	0.959415i	$-0.409002\pi$
$379$	81013.3	0.563999	0.281999	+	0.959415i	$0.409002\pi$
$380$	− 12979.1i	− 0.0898828i
$381$	0	0
$382$	95398.4	0.653754
$383$	− 81113.4i	− 0.552961i	−0.961019	−	0.276481i	$-0.910832\pi$
$383$	− 81113.4i	− 0.552961i	0.961019	−	0.276481i	$-0.0891682\pi$
$384$	0	0
$385$	−11599.6	−0.0782565
$386$	− 82331.8i	− 0.552577i
$387$	0	0
$388$	127075.	0.844107
$389$	77440.9i	0.511766i	0.966708	+	0.255883i	$0.0823662\pi$
$389$	77440.9i	0.511766i	−0.966708	+	0.255883i	$0.917634\pi$
$390$	0	0
$391$	−82342.9	−0.538608
$392$	7761.20i	0.0505076i
$393$	0	0
$394$	−172783.	−1.11304
$395$	− 28948.5i	− 0.185538i
$396$	0	0
$397$	18460.7	0.117130	0.0585650	−	0.998284i	$-0.481348\pi$
$397$	18460.7	0.117130	0.0585650	+	0.998284i	$0.481348\pi$
$398$	− 64105.6i	− 0.404697i
$399$	0	0
$400$	34593.6	0.216210
$401$	− 201249.i	− 1.25154i	−0.780008	−	0.625769i	$-0.784785\pi$
$401$	− 201249.i	− 1.25154i	0.780008	−	0.625769i	$-0.215215\pi$
$402$	0	0
$403$	208342.	1.28282
$404$	66575.4i	0.407897i
$405$	0	0
$406$	−28770.2	−0.174538
$407$	− 50647.1i	− 0.305750i
$408$	0	0
$409$	5492.43	0.0328335	0.0164168	−	0.999865i	$-0.494774\pi$
$409$	5492.43	0.0328335	0.0164168	+	0.999865i	$0.494774\pi$
$410$	51071.7i	0.303818i
$411$	0	0
$412$	18009.5	0.106098
$413$	− 65253.6i	− 0.382564i
$414$	0	0
$415$	4297.36	0.0249520
$416$	− 38450.9i	− 0.222187i
$417$	0	0
$418$	34022.7	0.194722
$419$	− 84105.3i	− 0.479066i	−0.970888	−	0.239533i	$-0.923006\pi$
$419$	− 84105.3i	− 0.479066i	0.970888	−	0.239533i	$-0.0769943\pi$
$420$	0	0
$421$	−49470.9	−0.279117	−0.139558	−	0.990214i	$-0.544568\pi$
$421$	−49470.9	−0.279117	−0.139558	+	0.990214i	$0.544568\pi$
$422$	135070.i	0.758463i
$423$	0	0
$424$	91926.0	0.511336
$425$	184122.i	1.01936i
$426$	0	0
$427$	96440.1	0.528934
$428$	175388.i	0.957441i
$429$	0	0
$430$	35198.0	0.190362
$431$	193441.i	1.04135i	0.853756	+	0.520673i	$0.174319\pi$
$431$	193441.i	1.04135i	−0.853756	+	0.520673i	$0.825681\pi$
$432$	0	0
$433$	−153071.	−0.816425	−0.408213	−	0.912887i	$-0.633848\pi$
$433$	−153071.	−0.816425	−0.408213	+	0.912887i	$0.633848\pi$
$434$	− 51379.2i	− 0.272777i
$435$	0	0
$436$	24762.1	0.130261
$437$	− 42670.5i	− 0.223442i
$438$	0	0
$439$	−138512.	−0.718717	−0.359359	−	0.933200i	$-0.617005\pi$
$439$	−138512.	−0.718717	−0.359359	+	0.933200i	$0.617005\pi$
$440$	− 14172.0i	− 0.0732023i
$441$	0	0
$442$	204652.	1.04754
$443$	32183.4i	0.163993i	0.996633	+	0.0819963i	$0.0261296\pi$
$443$	32183.4i	0.163993i	−0.996633	+	0.0819963i	$0.973870\pi$
$444$	0	0
$445$	26337.7	0.133002
$446$	− 114035.i	− 0.573280i
$447$	0	0
$448$	−9482.37	−0.0472456
$449$	− 100743.i	− 0.499713i	−0.968283	−	0.249857i	$-0.919617\pi$
$449$	− 100743.i	− 0.499713i	0.968283	−	0.249857i	$-0.0803835\pi$
$450$	0	0
$451$	−133877.	−0.658192
$452$	− 34360.1i	− 0.168181i
$453$	0	0
$454$	197421.	0.957815
$455$	− 36156.9i	− 0.174650i
$456$	0	0
$457$	249340.	1.19387	0.596937	−	0.802288i	$-0.296384\pi$
$457$	249340.	1.19387	0.596937	+	0.802288i	$0.296384\pi$
$458$	− 164700.i	− 0.785169i
$459$	0	0
$460$	−17774.2	−0.0839989
$461$	− 340783.i	− 1.60353i	−0.597640	−	0.801764i	$-0.703895\pi$
$461$	− 340783.i	− 1.60353i	0.597640	−	0.801764i	$-0.296105\pi$
$462$	0	0
$463$	−86247.9	−0.402334	−0.201167	−	0.979557i	$-0.564473\pi$
$463$	−86247.9	−0.402334	−0.201167	+	0.979557i	$0.564473\pi$
$464$	− 35150.4i	− 0.163266i
$465$	0	0
$466$	104582.	0.481597
$467$	− 21803.1i	− 0.0999734i	−0.998750	−	0.0499867i	$-0.984082\pi$
$467$	− 21803.1i	− 0.0999734i	0.998750	−	0.0499867i	$-0.0159179\pi$
$468$	0	0
$469$	41656.5	0.189381
$470$	− 34359.8i	− 0.155544i
$471$	0	0
$472$	79724.5	0.357856
$473$	92266.2i	0.412402i
$474$	0	0
$475$	−95412.9	−0.422883
$476$	− 50469.3i	− 0.222748i
$477$	0	0
$478$	126019.	0.551543
$479$	− 71881.6i	− 0.313290i	−0.987655	−	0.156645i	$-0.949932\pi$
$479$	− 71881.6i	− 0.313290i	0.987655	−	0.156645i	$-0.0500679\pi$
$480$	0	0
$481$	157871.	0.682359
$482$	− 226272.i	− 0.973949i
$483$	0	0
$484$	−79978.3	−0.341414
$485$	145994.i	0.620655i
$486$	0	0
$487$	−319320.	−1.34638	−0.673191	−	0.739469i	$-0.735077\pi$
$487$	−319320.	−1.34638	−0.673191	+	0.739469i	$0.735077\pi$
$488$	117827.i	0.494773i
$489$	0	0
$490$	−8916.65	−0.0371372
$491$	− 250141.i	− 1.03758i	−0.854902	−	0.518790i	$-0.826383\pi$
$491$	− 250141.i	− 1.03758i	0.854902	−	0.518790i	$-0.173617\pi$
$492$	0	0
$493$	187086.	0.769744
$494$	106052.i	0.434574i
$495$	0	0
$496$	62773.4	0.255160
$497$	− 137748.i	− 0.557662i
$498$	0	0
$499$	−154356.	−0.619899	−0.309950	−	0.950753i	$-0.600312\pi$
$499$	−154356.	−0.619899	−0.309950	+	0.950753i	$0.600312\pi$
$500$	85698.7i	0.342795i
$501$	0	0
$502$	−133707.	−0.530575
$503$	− 344575.i	− 1.36191i	−0.732325	−	0.680955i	$-0.761565\pi$
$503$	− 344575.i	− 1.36191i	0.732325	−	0.680955i	$-0.238435\pi$
$504$	0	0
$505$	−76486.8	−0.299919
$506$	− 46592.3i	− 0.181976i
$507$	0	0
$508$	−30765.6	−0.119217
$509$	127372.i	0.491632i	0.969317	+	0.245816i	$0.0790559\pi$
$509$	127372.i	0.491632i	−0.969317	+	0.245816i	$0.920944\pi$
$510$	0	0
$511$	−19473.3	−0.0745756
$512$	− 11585.2i	− 0.0441942i
$513$	0	0
$514$	151291.	0.572647
$515$	20690.7i	0.0780118i
$516$	0	0
$517$	90069.0	0.336972
$518$	− 38932.7i	− 0.145096i
$519$	0	0
$520$	44175.2	0.163370
$521$	310653.i	1.14446i	0.820094	+	0.572229i	$0.193921\pi$
$521$	310653.i	1.14446i	−0.820094	+	0.572229i	$0.806079\pi$
$522$	0	0
$523$	200802.	0.734116	0.367058	−	0.930198i	$-0.380365\pi$
$523$	200802.	0.734116	0.367058	+	0.930198i	$0.380365\pi$
$524$	103938.i	0.378542i
$525$	0	0
$526$	168829.	0.610205
$527$	334107.i	1.20300i
$528$	0	0
$529$	221406.	0.791185
$530$	105611.i	0.375975i
$531$	0	0
$532$	26153.4	0.0924070
$533$	− 417306.i	− 1.46893i
$534$	0	0
$535$	−201499.	−0.703987
$536$	50894.5i	0.177150i
$537$	0	0
$538$	−216693.	−0.748651
$539$	− 23373.7i	− 0.0804543i
$540$	0	0
$541$	286252.	0.978033	0.489016	−	0.872275i	$-0.337356\pi$
$541$	286252.	0.978033	0.489016	+	0.872275i	$0.337356\pi$
$542$	− 37405.8i	− 0.127333i
$543$	0	0
$544$	61661.6	0.208361
$545$	28448.6i	0.0957783i
$546$	0	0
$547$	78649.5	0.262858	0.131429	−	0.991326i	$-0.458043\pi$
$547$	78649.5	0.262858	0.131429	+	0.991326i	$0.458043\pi$
$548$	− 56386.3i	− 0.187764i
$549$	0	0
$550$	−104182.	−0.344404
$551$	96948.6i	0.319329i
$552$	0	0
$553$	58332.6	0.190748
$554$	7716.48i	0.0251420i
$555$	0	0
$556$	−105275.	−0.340544
$557$	580730.i	1.87182i	0.352241	+	0.935909i	$0.385420\pi$
$557$	580730.i	1.87182i	−0.352241	+	0.935909i	$0.614580\pi$
$558$	0	0
$559$	−287602.	−0.920381
$560$	− 10894.1i	− 0.0347387i
$561$	0	0
$562$	261164.	0.826875
$563$	− 293483.i	− 0.925906i	−0.886383	−	0.462953i	$-0.846790\pi$
$563$	− 293483.i	− 0.925906i	0.886383	−	0.462953i	$-0.153210\pi$
$564$	0	0
$565$	39475.4	0.123660
$566$	331420.i	1.03454i
$567$	0	0
$568$	168295.	0.521645
$569$	− 149286.i	− 0.461100i	−0.973060	−	0.230550i	$-0.925947\pi$
$569$	− 149286.i	− 0.461100i	0.973060	−	0.230550i	$-0.0740526\pi$
$570$	0	0
$571$	−522954.	−1.60395	−0.801976	−	0.597356i	$-0.796218\pi$
$571$	−522954.	−1.60395	−0.801976	+	0.597356i	$0.796218\pi$
$572$	115799.i	0.353925i
$573$	0	0
$574$	−102912.	−0.312350
$575$	130663.i	0.395200i
$576$	0	0
$577$	−351985.	−1.05724	−0.528619	−	0.848859i	$-0.677290\pi$
$577$	−351985.	−1.05724	−0.528619	+	0.848859i	$0.677290\pi$
$578$	91956.6i	0.275250i
$579$	0	0
$580$	40383.4	0.120046
$581$	8659.36i	0.0256527i
$582$	0	0
$583$	−276845.	−0.814515
$584$	− 23791.7i	− 0.0697591i
$585$	0	0
$586$	−190122.	−0.553652
$587$	− 256546.i	− 0.744541i	−0.928124	−	0.372270i	$-0.878579\pi$
$587$	− 256546.i	− 0.744541i	0.928124	−	0.372270i	$-0.121421\pi$
$588$	0	0
$589$	−173136.	−0.499064
$590$	91593.5i	0.263124i
$591$	0	0
$592$	47566.6	0.135725
$593$	469980.i	1.33650i	0.743935	+	0.668252i	$0.232957\pi$
$593$	469980.i	1.33650i	−0.743935	+	0.668252i	$0.767043\pi$
$594$	0	0
$595$	57982.9	0.163782
$596$	− 214535.i	− 0.603955i
$597$	0	0
$598$	145232.	0.406126
$599$	667523.i	1.86043i	0.367021	+	0.930213i	$0.380378\pi$
$599$	667523.i	1.86043i	−0.367021	+	0.930213i	$0.619622\pi$
$600$	0	0
$601$	578943.	1.60283	0.801414	−	0.598109i	$-0.204081\pi$
$601$	578943.	1.60283	0.801414	+	0.598109i	$0.204081\pi$
$602$	70925.5i	0.195708i
$603$	0	0
$604$	−196628.	−0.538980
$605$	− 91885.1i	− 0.251035i
$606$	0	0
$607$	−393712.	−1.06857	−0.534283	−	0.845306i	$-0.679418\pi$
$607$	−393712.	−1.06857	−0.534283	+	0.845306i	$0.679418\pi$
$608$	31953.3i	0.0864388i
$609$	0	0
$610$	−135369.	−0.363796
$611$	280753.i	0.752041i
$612$	0	0
$613$	−112288.	−0.298823	−0.149412	−	0.988775i	$-0.547738\pi$
$613$	−112288.	−0.298823	−0.149412	+	0.988775i	$0.547738\pi$
$614$	306473.i	0.812935i
$615$	0	0
$616$	28557.1	0.0752581
$617$	− 106565.i	− 0.279927i	−0.990157	−	0.139964i	$-0.955301\pi$
$617$	− 106565.i	− 0.279927i	0.990157	−	0.139964i	$-0.0446986\pi$
$618$	0	0
$619$	−115118.	−0.300444	−0.150222	−	0.988652i	$-0.547999\pi$
$619$	−115118.	−0.300444	−0.150222	+	0.988652i	$0.547999\pi$
$620$	72118.7i	0.187614i
$621$	0	0
$622$	−270706.	−0.699709
$623$	53071.7i	0.136737i
$624$	0	0
$625$	239371.	0.612791
$626$	− 447209.i	− 1.14120i
$627$	0	0
$628$	−251798.	−0.638460
$629$	253170.i	0.639898i
$630$	0	0
$631$	−776896.	−1.95121	−0.975606	−	0.219530i	$-0.929547\pi$
$631$	−776896.	−1.95121	−0.975606	+	0.219530i	$0.929547\pi$
$632$	71268.7i	0.178429i
$633$	0	0
$634$	257765.	0.641277
$635$	− 35345.8i	− 0.0876577i
$636$	0	0
$637$	72857.7	0.179555
$638$	105859.i	0.260068i
$639$	0	0
$640$	13310.0	0.0324951
$641$	574512.i	1.39824i	0.715002	+	0.699122i	$0.246426\pi$
$641$	574512.i	1.39824i	−0.715002	+	0.699122i	$0.753574\pi$
$642$	0	0
$643$	702571.	1.69929	0.849647	−	0.527352i	$-0.176815\pi$
$643$	702571.	1.69929	0.849647	+	0.527352i	$0.176815\pi$
$644$	− 35815.7i	− 0.0863578i
$645$	0	0
$646$	−170069.	−0.407531
$647$	51929.4i	0.124052i	0.998075	+	0.0620262i	$0.0197562\pi$
$647$	51929.4i	0.124052i	−0.998075	+	0.0620262i	$0.980244\pi$
$648$	0	0
$649$	−240099.	−0.570033
$650$	− 324745.i	− 0.768627i
$651$	0	0
$652$	261250.	0.614556
$653$	608223.i	1.42638i	0.700969	+	0.713192i	$0.252751\pi$
$653$	608223.i	1.42638i	−0.700969	+	0.713192i	$0.747249\pi$
$654$	0	0
$655$	−119412.	−0.278334
$656$	− 125734.i	− 0.292176i
$657$	0	0
$658$	69236.4	0.159913
$659$	556649.i	1.28177i	0.767636	+	0.640886i	$0.221433\pi$
$659$	556649.i	1.28177i	−0.767636	+	0.640886i	$0.778567\pi$
$660$	0	0
$661$	211725.	0.484584	0.242292	−	0.970203i	$-0.422101\pi$
$661$	211725.	0.484584	0.242292	+	0.970203i	$0.422101\pi$
$662$	364907.i	0.832656i
$663$	0	0
$664$	−10579.7	−0.0239959
$665$	30047.0i	0.0679450i
$666$	0	0
$667$	132766.	0.298425
$668$	− 245874.i	− 0.551010i
$669$	0	0
$670$	−58471.4	−0.130255
$671$	− 354849.i	− 0.788130i
$672$	0	0
$673$	−476473.	−1.05198	−0.525991	−	0.850490i	$-0.676306\pi$
$673$	−476473.	−1.05198	−0.525991	+	0.850490i	$0.676306\pi$
$674$	470348.i	1.03538i
$675$	0	0
$676$	−132466.	−0.289876
$677$	582092.i	1.27003i	0.772499	+	0.635016i	$0.219006\pi$
$677$	582092.i	1.27003i	−0.772499	+	0.635016i	$0.780994\pi$
$678$	0	0
$679$	−294183.	−0.638085
$680$	70841.5i	0.153204i
$681$	0	0
$682$	−189048.	−0.406447
$683$	685947.i	1.47045i	0.677825	+	0.735223i	$0.262923\pi$
$683$	685947.i	1.47045i	−0.677825	+	0.735223i	$0.737077\pi$
$684$	0	0
$685$	64780.8	0.138059
$686$	− 17967.4i	− 0.0381802i
$687$	0	0
$688$	−86654.3	−0.183068
$689$	− 862948.i	− 1.81780i
$690$	0	0
$691$	−244679.	−0.512437	−0.256219	−	0.966619i	$-0.582477\pi$
$691$	−244679.	−0.512437	−0.256219	+	0.966619i	$0.582477\pi$
$692$	− 256546.i	− 0.535740i
$693$	0	0
$694$	647950.	1.34531
$695$	− 120947.i	− 0.250395i
$696$	0	0
$697$	669211.	1.37752
$698$	− 277125.i	− 0.568806i
$699$	0	0
$700$	−80085.4	−0.163440
$701$	− 275489.i	− 0.560619i	−0.959910	−	0.280310i	$-0.909563\pi$
$701$	− 275489.i	− 0.560619i	0.959910	−	0.280310i	$-0.0904371\pi$
$702$	0	0
$703$	−131194.	−0.265462
$704$	34890.1i	0.0703975i
$705$	0	0
$706$	−600641.	−1.20505
$707$	− 154124.i	− 0.308341i
$708$	0	0
$709$	732370.	1.45693	0.728464	−	0.685084i	$-0.240234\pi$
$709$	732370.	1.45693	0.728464	+	0.685084i	$0.240234\pi$
$710$	193350.i	0.383555i
$711$	0	0
$712$	−64841.2	−0.127906
$713$	237100.i	0.466394i
$714$	0	0
$715$	−133038.	−0.260234
$716$	− 151420.i	− 0.295364i
$717$	0	0
$718$	−203651.	−0.395038
$719$	900565.i	1.74204i	0.491251	+	0.871018i	$0.336540\pi$
$719$	900565.i	1.74204i	−0.491251	+	0.871018i	$0.663460\pi$
$720$	0	0
$721$	−41692.6	−0.0802026
$722$	280473.i	0.538042i
$723$	0	0
$724$	−132019.	−0.251860
$725$	− 296870.i	− 0.564795i
$726$	0	0
$727$	681602.	1.28962	0.644811	−	0.764342i	$-0.276936\pi$
$727$	681602.	1.28962	0.644811	+	0.764342i	$0.276936\pi$
$728$	89015.0i	0.167958i
$729$	0	0
$730$	27333.7	0.0512924
$731$	− 461212.i	− 0.863109i
$732$	0	0
$733$	369951.	0.688552	0.344276	−	0.938869i	$-0.388124\pi$
$733$	369951.	0.688552	0.344276	+	0.938869i	$0.388124\pi$
$734$	214644.i	0.398406i
$735$	0	0
$736$	43758.4	0.0807804
$737$	− 153274.i	− 0.282185i
$738$	0	0
$739$	318069.	0.582416	0.291208	−	0.956660i	$-0.405943\pi$
$739$	318069.	0.582416	0.291208	+	0.956660i	$0.405943\pi$
$740$	54648.1i	0.0997956i
$741$	0	0
$742$	−212812.	−0.386534
$743$	− 531939.i	− 0.963571i	−0.876289	−	0.481786i	$-0.839988\pi$
$743$	− 531939.i	− 0.963571i	0.876289	−	0.481786i	$-0.160012\pi$
$744$	0	0
$745$	246473.	0.444076
$746$	400442.i	0.719552i
$747$	0	0
$748$	−185700.	−0.331902
$749$	− 406029.i	− 0.723757i
$750$	0	0
$751$	411990.	0.730477	0.365239	−	0.930914i	$-0.380987\pi$
$751$	411990.	0.730477	0.365239	+	0.930914i	$0.380987\pi$
$752$	84590.7i	0.149585i
$753$	0	0
$754$	−329972.	−0.580409
$755$	− 225901.i	− 0.396301i
$756$	0	0
$757$	1.11562e6	1.94681	0.973407	−	0.229081i	$-0.0735722\pi$
$757$	1.11562e6	1.94681	0.973407	+	0.229081i	$0.0735722\pi$
$758$	− 229140.i	− 0.398807i
$759$	0	0
$760$	−36710.4	−0.0635567
$761$	− 251236.i	− 0.433822i	−0.976191	−	0.216911i	$-0.930402\pi$
$761$	− 251236.i	− 0.433822i	0.976191	−	0.216911i	$-0.0695982\pi$
$762$	0	0
$763$	−57325.1	−0.0984681
$764$	− 269827.i	− 0.462274i
$765$	0	0
$766$	−229423.	−0.391003
$767$	− 748408.i	− 1.27218i
$768$	0	0
$769$	215136.	0.363799	0.181899	−	0.983317i	$-0.441776\pi$
$769$	215136.	0.363799	0.181899	+	0.983317i	$0.441776\pi$
$770$	32808.6i	0.0553357i
$771$	0	0
$772$	−232870.	−0.390731
$773$	377669.i	0.632051i	0.948751	+	0.316026i	$0.102349\pi$
$773$	377669.i	0.632051i	−0.948751	+	0.316026i	$0.897651\pi$
$774$	0	0
$775$	530166.	0.882690
$776$	− 359423.i	− 0.596874i
$777$	0	0
$778$	219036.	0.361873
$779$	346788.i	0.571464i
$780$	0	0
$781$	−506838.	−0.830936
$782$	232901.i	0.380853i
$783$	0	0
$784$	21952.0	0.0357143
$785$	− 289285.i	− 0.469447i
$786$	0	0
$787$	−610887.	−0.986306	−0.493153	−	0.869943i	$-0.664156\pi$
$787$	−610887.	−0.986306	−0.493153	+	0.869943i	$0.664156\pi$
$788$	488705.i	0.787035i
$789$	0	0
$790$	−81878.8	−0.131195
$791$	79544.7i	0.127133i
$792$	0	0
$793$	1.10609e6	1.75892
$794$	− 52214.9i	− 0.0828234i
$795$	0	0
$796$	−181318.	−0.286164
$797$	144095.i	0.226847i	0.993547	+	0.113423i	$0.0361817\pi$
$797$	144095.i	0.226847i	−0.993547	+	0.113423i	$0.963818\pi$
$798$	0	0
$799$	−450228.	−0.705243
$800$	− 97845.6i	− 0.152884i
$801$	0	0
$802$	−569217.	−0.884971
$803$	71651.3i	0.111120i
$804$	0	0
$805$	41147.8	0.0634972
$806$	− 589280.i	− 0.907092i
$807$	0	0
$808$	188304.	0.288427
$809$	469089.i	0.716735i	0.933581	+	0.358367i	$0.116666\pi$
$809$	469089.i	0.716735i	−0.933581	+	0.358367i	$0.883334\pi$
$810$	0	0
$811$	−651439.	−0.990449	−0.495224	−	0.868765i	$-0.664914\pi$
$811$	−651439.	−0.990449	−0.495224	+	0.868765i	$0.664914\pi$
$812$	81374.3i	0.123417i
$813$	0	0
$814$	−143252.	−0.216198
$815$	300144.i	0.451870i
$816$	0	0
$817$	239002.	0.358061
$818$	− 15534.9i	− 0.0232168i
$819$	0	0
$820$	144453.	0.214831
$821$	− 344638.i	− 0.511301i	−0.966769	−	0.255650i	$-0.917710\pi$
$821$	− 344638.i	− 0.511301i	0.966769	−	0.255650i	$-0.0822896\pi$
$822$	0	0
$823$	−77588.8	−0.114551	−0.0572755	−	0.998358i	$-0.518241\pi$
$823$	−77588.8	−0.114551	−0.0572755	+	0.998358i	$0.518241\pi$
$824$	− 50938.6i	− 0.0750227i
$825$	0	0
$826$	−184565.	−0.270514
$827$	− 449172.i	− 0.656752i	−0.944547	−	0.328376i	$-0.893499\pi$
$827$	− 449172.i	− 0.656752i	0.944547	−	0.328376i	$-0.106501\pi$
$828$	0	0
$829$	−1.21343e6	−1.76565	−0.882824	−	0.469704i	$-0.844361\pi$
$829$	−1.21343e6	−1.76565	−0.882824	+	0.469704i	$0.844361\pi$
$830$	− 12154.8i	− 0.0176437i
$831$	0	0
$832$	−108755.	−0.157110
$833$	116838.i	0.168381i
$834$	0	0
$835$	282478.	0.405147
$836$	− 96230.7i	− 0.137690i
$837$	0	0
$838$	−237886.	−0.338751
$839$	290391.i	0.412534i	0.978496	+	0.206267i	$0.0661316\pi$
$839$	290391.i	0.412534i	−0.978496	+	0.206267i	$0.933868\pi$
$840$	0	0
$841$	405633.	0.573510
$842$	139925.i	0.197365i
$843$	0	0
$844$	382036.	0.536314
$845$	− 152187.i	− 0.213140i
$846$	0	0
$847$	185152.	0.258085
$848$	− 260006.i	− 0.361569i
$849$	0	0
$850$	520776.	0.720798
$851$	179663.i	0.248084i
$852$	0	0
$853$	1.13718e6	1.56291	0.781453	−	0.623964i	$-0.214479\pi$
$853$	1.13718e6	1.56291	0.781453	+	0.623964i	$0.214479\pi$
$854$	− 272774.i	− 0.374013i
$855$	0	0
$856$	496072.	0.677013
$857$	− 1.10331e6i	− 1.50223i	−0.660174	−	0.751113i	$-0.729517\pi$
$857$	− 1.10331e6i	− 1.50223i	0.660174	−	0.751113i	$-0.270483\pi$
$858$	0	0
$859$	−57100.5	−0.0773844	−0.0386922	−	0.999251i	$-0.512319\pi$
$859$	−57100.5	−0.0773844	−0.0386922	+	0.999251i	$0.512319\pi$
$860$	− 99554.9i	− 0.134606i
$861$	0	0
$862$	547135.	0.736342
$863$	787528.i	1.05741i	0.848805	+	0.528706i	$0.177323\pi$
$863$	787528.i	1.05741i	−0.848805	+	0.528706i	$0.822677\pi$
$864$	0	0
$865$	294740.	0.393919
$866$	432949.i	0.577300i
$867$	0	0
$868$	−145322.	−0.192883
$869$	− 214633.i	− 0.284222i
$870$	0	0
$871$	477768.	0.629768
$872$	− 70037.8i	− 0.0921085i
$873$	0	0
$874$	−120690.	−0.157997
$875$	− 198395.i	− 0.259129i
$876$	0	0
$877$	−804114.	−1.04549	−0.522743	−	0.852490i	$-0.675091\pi$
$877$	−804114.	−1.04549	−0.522743	+	0.852490i	$0.675091\pi$
$878$	391771.i	0.508210i
$879$	0	0
$880$	−40084.4	−0.0517618
$881$	− 824622.i	− 1.06244i	−0.847235	−	0.531218i	$-0.821734\pi$
$881$	− 824622.i	− 1.06244i	0.847235	−	0.531218i	$-0.178266\pi$
$882$	0	0
$883$	−78126.7	−0.100202	−0.0501012	−	0.998744i	$-0.515954\pi$
$883$	−78126.7	−0.100202	−0.0501012	+	0.998744i	$0.515954\pi$
$884$	− 578843.i	− 0.740724i
$885$	0	0
$886$	91028.4	0.115960
$887$	− 372894.i	− 0.473956i	−0.971515	−	0.236978i	$-0.923843\pi$
$887$	− 372894.i	− 0.473956i	0.971515	−	0.236978i	$-0.0761570\pi$
$888$	0	0
$889$	71223.3	0.0901195
$890$	− 74494.3i	− 0.0940466i
$891$	0	0
$892$	−322539.	−0.405370
$893$	− 233310.i	− 0.292571i
$894$	0	0
$895$	173963.	0.217175
$896$	26820.2i	0.0334077i
$897$	0	0
$898$	−284943.	−0.353351
$899$	− 538699.i	− 0.666540i
$900$	0	0
$901$	1.38386e6	1.70468
$902$	378661.i	0.465412i
$903$	0	0
$904$	−97184.9	−0.118922
$905$	− 151673.i	− 0.185188i
$906$	0	0
$907$	−915350.	−1.11269	−0.556343	−	0.830953i	$-0.687796\pi$
$907$	−915350.	−1.11269	−0.556343	+	0.830953i	$0.687796\pi$
$908$	− 558391.i	− 0.677277i
$909$	0	0
$910$	−102267.	−0.123496
$911$	− 1.09746e6i	− 1.32237i	−0.750224	−	0.661184i	$-0.770054\pi$
$911$	− 1.09746e6i	− 1.32237i	0.750224	−	0.661184i	$-0.229946\pi$
$912$	0	0
$913$	31861.9	0.0382234
$914$	− 705239.i	− 0.844197i
$915$	0	0
$916$	−465843.	−0.555199
$917$	− 240621.i	− 0.286151i
$918$	0	0
$919$	916021.	1.08461	0.542306	−	0.840181i	$-0.317551\pi$
$919$	916021.	1.08461	0.542306	+	0.840181i	$0.317551\pi$
$920$	50272.9i	0.0593962i
$921$	0	0
$922$	−963881.	−1.13387
$923$	− 1.57986e6i	− 1.85445i
$924$	0	0
$925$	401734.	0.469521
$926$	243946.i	0.284493i
$927$	0	0
$928$	−99420.4	−0.115446
$929$	977241.i	1.13232i	0.824294	+	0.566161i	$0.191572\pi$
$929$	977241.i	1.13232i	−0.824294	+	0.566161i	$0.808428\pi$
$930$	0	0
$931$	−60546.0	−0.0698531
$932$	− 295802.i	− 0.340540i
$933$	0	0
$934$	−61668.5	−0.0706919
$935$	− 213346.i	− 0.244040i
$936$	0	0
$937$	−211902.	−0.241355	−0.120678	−	0.992692i	$-0.538507\pi$
$937$	−211902.	−0.241355	−0.120678	+	0.992692i	$0.538507\pi$
$938$	− 117822.i	− 0.133913i
$939$	0	0
$940$	−97184.1	−0.109987
$941$	886510.i	1.00116i	0.865690	+	0.500581i	$0.166880\pi$
$941$	886510.i	1.00116i	−0.865690	+	0.500581i	$0.833120\pi$
$942$	0	0
$943$	474908.	0.534055
$944$	− 225495.i	− 0.253042i
$945$	0	0
$946$	260968.	0.291612
$947$	− 391263.i	− 0.436283i	−0.975917	−	0.218142i	$-0.930001\pi$
$947$	− 391263.i	− 0.436283i	0.975917	−	0.218142i	$-0.0699995\pi$
$948$	0	0
$949$	−223343.	−0.247993
$950$	269869.i	0.299023i
$951$	0	0
$952$	−142749.	−0.157506
$953$	445329.i	0.490337i	0.969480	+	0.245169i	$0.0788433\pi$
$953$	445329.i	0.490337i	−0.969480	+	0.245169i	$0.921157\pi$
$954$	0	0
$955$	309998.	0.339901
$956$	− 356435.i	− 0.390000i
$957$	0	0
$958$	−203312.	−0.221530
$959$	130536.i	0.141936i
$960$	0	0
$961$	38513.8	0.0417032
$962$	− 446528.i	− 0.482501i
$963$	0	0
$964$	−639993.	−0.688686
$965$	− 267538.i	− 0.287297i
$966$	0	0
$967$	−1.31391e6	−1.40511	−0.702557	−	0.711627i	$-0.747959\pi$
$967$	−1.31391e6	−1.40511	−0.702557	+	0.711627i	$0.747959\pi$
$968$	226213.i	0.241416i
$969$	0	0
$970$	412932.	0.438869
$971$	− 1.12629e6i	− 1.19457i	−0.802029	−	0.597285i	$-0.796246\pi$
$971$	− 1.12629e6i	− 1.19457i	0.802029	−	0.597285i	$-0.203754\pi$
$972$	0	0
$973$	243714.	0.257427
$974$	903174.i	0.952036i
$975$	0	0
$976$	333266.	0.349857
$977$	1.70721e6i	1.78854i	0.447528	+	0.894270i	$0.352304\pi$
$977$	1.70721e6i	1.78854i	−0.447528	+	0.894270i	$0.647696\pi$
$978$	0	0
$979$	195276.	0.203743
$980$	25220.1i	0.0262600i
$981$	0	0
$982$	−707505.	−0.733680
$983$	438109.i	0.453394i	0.973965	+	0.226697i	$0.0727926\pi$
$983$	438109.i	0.453394i	−0.973965	+	0.226697i	$0.927207\pi$
$984$	0	0
$985$	−561460.	−0.578691
$986$	− 529158.i	− 0.544292i
$987$	0	0
$988$	299959.	0.307290
$989$	− 327301.i	− 0.334622i
$990$	0	0
$991$	−1.40134e6	−1.42690	−0.713452	−	0.700704i	$-0.752870\pi$
$991$	−1.40134e6	−1.42690	−0.713452	+	0.700704i	$0.752870\pi$
$992$	− 177550.i	− 0.180425i
$993$	0	0
$994$	−389609.	−0.394327
$995$	− 208312.i	− 0.210410i
$996$	0	0
$997$	−971355.	−0.977209	−0.488605	−	0.872505i	$-0.662494\pi$
$997$	−971355.	−0.977209	−0.488605	+	0.872505i	$0.662494\pi$
$998$	436583.i	0.438335i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.5.b.a.323.2	✓	8
3.2	odd	2	inner	378.5.b.a.323.7	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.5.b.a.323.2	✓	8	1.1	even	1	trivial
378.5.b.a.323.7	yes	8	3.2	odd	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:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	378.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.82843i q^{2} -8.00000 q^{4} -9.19100i q^{5} +18.5203 q^{7} +22.6274i q^{8} +O(q^{10})\)	\(q-2.82843i q^{2} -8.00000 q^{4} -9.19100i q^{5} +18.5203 q^{7} +22.6274i q^{8} -25.9961 q^{10} -68.1448i q^{11} +212.413 q^{13} -52.3832i q^{14} +64.0000 q^{16} +340.636i q^{17} -176.519 q^{19} +73.5280i q^{20} -192.743 q^{22} +241.733i q^{23} +540.526 q^{25} -600.795i q^{26} -148.162 q^{28} -549.225i q^{29} +980.834 q^{31} -181.019i q^{32} +963.463 q^{34} -170.220i q^{35} +743.228 q^{37} +499.271i q^{38} +207.968 q^{40} -1964.59i q^{41} -1353.97 q^{43} +545.158i q^{44} +683.725 q^{46} +1321.73i q^{47} +343.000 q^{49} -1528.84i q^{50} -1699.30 q^{52} -4062.59i q^{53} -626.318 q^{55} +419.066i q^{56} -1553.44 q^{58} -3523.36i q^{59} +5207.28 q^{61} -2774.22i q^{62} -512.000 q^{64} -1952.29i q^{65} +2249.24 q^{67} -2725.08i q^{68} -481.454 q^{70} -7437.67i q^{71} -1051.46 q^{73} -2102.17i q^{74} +1412.15 q^{76} -1262.06i q^{77} +3149.66 q^{79} -588.224i q^{80} -5556.71 q^{82} +467.561i q^{83} +3130.78 q^{85} +3829.62i q^{86} +1541.94 q^{88} +2865.60i q^{89} +3933.94 q^{91} -1933.87i q^{92} +3738.42 q^{94} +1622.38i q^{95} -15884.4 q^{97} -970.151i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 64 q^{4}+O(q^{10}) \) 8 * q - 64 * q^4	\( 8 q - 64 q^{4} - 224 q^{10} - 376 q^{13} + 512 q^{16} + 1120 q^{19} - 1120 q^{22} + 792 q^{25} - 880 q^{31} - 1792 q^{34} + 1576 q^{37} + 1792 q^{40} - 5768 q^{43} - 160 q^{46} + 2744 q^{49} + 3008 q^{52} + 488 q^{55} + 7552 q^{58} - 2560 q^{61} - 4096 q^{64} + 23784 q^{67} + 1568 q^{70} - 13176 q^{73} - 8960 q^{76} - 9592 q^{79} - 3360 q^{82} - 39880 q^{85} + 8960 q^{88} + 5096 q^{91} + 4608 q^{94} - 14016 q^{97}+O(q^{100}) \) 8 * q - 64 * q^4 - 224 * q^10 - 376 * q^13 + 512 * q^16 + 1120 * q^19 - 1120 * q^22 + 792 * q^25 - 880 * q^31 - 1792 * q^34 + 1576 * q^37 + 1792 * q^40 - 5768 * q^43 - 160 * q^46 + 2744 * q^49 + 3008 * q^52 + 488 * q^55 + 7552 * q^58 - 2560 * q^61 - 4096 * q^64 + 23784 * q^67 + 1568 * q^70 - 13176 * q^73 - 8960 * q^76 - 9592 * q^79 - 3360 * q^82 - 39880 * q^85 + 8960 * q^88 + 5096 * q^91 + 4608 * q^94 - 14016 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more