LMFDB - Embedded newform 700.6.e.b.449.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [700,6,Mod(449,700)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(700, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("700.449");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 700 = 2^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	700.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$112.268673869$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 28)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	700.449
Dual form			700.6.e.b.449.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+26.0000i q^{3} +49.0000i q^{7} -433.000 q^{9} +O(q^{10})$	$q+26.0000i q^{3} +49.0000i q^{7} -433.000 q^{9} +8.00000 q^{11} +684.000i q^{13} +2218.00i q^{17} +2698.00 q^{19} -1274.00 q^{21} +3344.00i q^{23} -4940.00i q^{27} +3254.00 q^{29} +4788.00 q^{31} +208.000i q^{33} +11470.0i q^{37} -17784.0 q^{39} +13350.0 q^{41} -928.000i q^{43} -1212.00i q^{47} -2401.00 q^{49} -57668.0 q^{51} +13110.0i q^{53} +70148.0i q^{57} -34702.0 q^{59} -1032.00 q^{61} -21217.0i q^{63} -10108.0i q^{67} -86944.0 q^{69} +62720.0 q^{71} -18926.0i q^{73} +392.000i q^{77} -11400.0 q^{79} +23221.0 q^{81} +88958.0i q^{83} +84604.0i q^{87} -19722.0 q^{89} -33516.0 q^{91} +124488. i q^{93} -17062.0i q^{97} -3464.00 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 866 q^{9}+O(q^{10}) $ 2 * q - 866 * q^9	$ 2 q - 866 q^{9} + 16 q^{11} + 5396 q^{19} - 2548 q^{21} + 6508 q^{29} + 9576 q^{31} - 35568 q^{39} + 26700 q^{41} - 4802 q^{49} - 115336 q^{51} - 69404 q^{59} - 2064 q^{61} - 173888 q^{69} + 125440 q^{71} - 22800 q^{79} + 46442 q^{81} - 39444 q^{89} - 67032 q^{91} - 6928 q^{99}+O(q^{100}) $ 2 * q - 866 * q^9 + 16 * q^11 + 5396 * q^19 - 2548 * q^21 + 6508 * q^29 + 9576 * q^31 - 35568 * q^39 + 26700 * q^41 - 4802 * q^49 - 115336 * q^51 - 69404 * q^59 - 2064 * q^61 - 173888 * q^69 + 125440 * q^71 - 22800 * q^79 + 46442 * q^81 - 39444 * q^89 - 67032 * q^91 - 6928 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/700\mathbb{Z}\right)^\times$.

$n$	$101$	$351$	$477$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	26.0000i	1.66790i	0.551839	+	0.833950i	$0.313926\pi$
$3$	26.0000i	1.66790i	−0.551839	+	0.833950i	$0.686074\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	49.0000i	0.377964i
$7$	49.0000i	0.377964i
$8$	0	0
$9$	−433.000	−1.78189
$10$	0	0
$11$	8.00000	0.0199346	0.00996732	−	0.999950i	$-0.496827\pi$
$11$	8.00000	0.0199346	0.00996732	+	0.999950i	$0.496827\pi$
$12$	0	0
$13$	684.000i	1.12253i	0.827636	+	0.561265i	$0.189685\pi$
$13$	684.000i	1.12253i	−0.827636	+	0.561265i	$0.810315\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2218.00i	1.86140i	0.365786	+	0.930699i	$0.380800\pi$
$17$	2218.00i	1.86140i	−0.365786	+	0.930699i	$0.619200\pi$
$18$	0	0
$19$	2698.00	1.71458	0.857290	−	0.514833i	$-0.172146\pi$
$19$	2698.00	1.71458	0.857290	+	0.514833i	$0.172146\pi$
$20$	0	0
$21$	−1274.00	−0.630407
$22$	0	0
$23$	3344.00i	1.31809i	0.752101	+	0.659047i	$0.229040\pi$
$23$	3344.00i	1.31809i	−0.752101	+	0.659047i	$0.770960\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 4940.00i	− 1.30412i
$28$	0	0
$29$	3254.00	0.718493	0.359247	−	0.933243i	$-0.383034\pi$
$29$	3254.00	0.718493	0.359247	+	0.933243i	$0.383034\pi$
$30$	0	0
$31$	4788.00	0.894849	0.447425	−	0.894322i	$-0.352341\pi$
$31$	4788.00	0.894849	0.447425	+	0.894322i	$0.352341\pi$
$32$	0	0
$33$	208.000i	0.0332490i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	11470.0i	1.37740i	0.725048	+	0.688698i	$0.241818\pi$
$37$	11470.0i	1.37740i	−0.725048	+	0.688698i	$0.758182\pi$
$38$	0	0
$39$	−17784.0	−1.87227
$40$	0	0
$41$	13350.0	1.24029	0.620143	−	0.784489i	$-0.287075\pi$
$41$	13350.0	1.24029	0.620143	+	0.784489i	$0.287075\pi$
$42$	0	0
$43$	− 928.000i	− 0.0765380i	−0.999267	−	0.0382690i	$-0.987816\pi$
$43$	− 928.000i	− 0.0765380i	0.999267	−	0.0382690i	$-0.0121844\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 1212.00i	− 0.0800310i	−0.999199	−	0.0400155i	$-0.987259\pi$
$47$	− 1212.00i	− 0.0800310i	0.999199	−	0.0400155i	$-0.0127407\pi$
$48$	0	0
$49$	−2401.00	−0.142857
$50$	0	0
$51$	−57668.0	−3.10463
$52$	0	0
$53$	13110.0i	0.641081i	0.947235	+	0.320541i	$0.103865\pi$
$53$	13110.0i	0.641081i	−0.947235	+	0.320541i	$0.896135\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	70148.0i	2.85975i
$58$	0	0
$59$	−34702.0	−1.29785	−0.648925	−	0.760852i	$-0.724781\pi$
$59$	−34702.0	−1.29785	−0.648925	+	0.760852i	$0.724781\pi$
$60$	0	0
$61$	−1032.00	−0.0355104	−0.0177552	−	0.999842i	$-0.505652\pi$
$61$	−1032.00	−0.0355104	−0.0177552	+	0.999842i	$0.505652\pi$
$62$	0	0
$63$	− 21217.0i	− 0.673492i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 10108.0i	− 0.275092i	−0.990495	−	0.137546i	$-0.956078\pi$
$67$	− 10108.0i	− 0.275092i	0.990495	−	0.137546i	$-0.0439215\pi$
$68$	0	0
$69$	−86944.0	−2.19845
$70$	0	0
$71$	62720.0	1.47659	0.738295	−	0.674477i	$-0.235631\pi$
$71$	62720.0	1.47659	0.738295	+	0.674477i	$0.235631\pi$
$72$	0	0
$73$	− 18926.0i	− 0.415673i	−0.978164	−	0.207836i	$-0.933358\pi$
$73$	− 18926.0i	− 0.415673i	0.978164	−	0.207836i	$-0.0666422\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	392.000i	0.00753458i
$78$	0	0
$79$	−11400.0	−0.205512	−0.102756	−	0.994707i	$-0.532766\pi$
$79$	−11400.0	−0.205512	−0.102756	+	0.994707i	$0.532766\pi$
$80$	0	0
$81$	23221.0	0.393250
$82$	0	0
$83$	88958.0i	1.41739i	0.705514	+	0.708696i	$0.250716\pi$
$83$	88958.0i	1.41739i	−0.705514	+	0.708696i	$0.749284\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	84604.0i	1.19838i
$88$	0	0
$89$	−19722.0	−0.263922	−0.131961	−	0.991255i	$-0.542127\pi$
$89$	−19722.0	−0.263922	−0.131961	+	0.991255i	$0.542127\pi$
$90$	0	0
$91$	−33516.0	−0.424276
$92$	0	0
$93$	124488.i	1.49252i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 17062.0i	− 0.184120i	−0.995753	−	0.0920599i	$-0.970655\pi$
$97$	− 17062.0i	− 0.184120i	0.995753	−	0.0920599i	$-0.0293451\pi$
$98$	0	0
$99$	−3464.00	−0.0355214
$100$	0	0
$101$	45904.0	0.447762	0.223881	−	0.974617i	$-0.428127\pi$
$101$	45904.0	0.447762	0.223881	+	0.974617i	$0.428127\pi$
$102$	0	0
$103$	− 136012.i	− 1.26324i	−0.775280	−	0.631618i	$-0.782391\pi$
$103$	− 136012.i	− 1.26324i	0.775280	−	0.631618i	$-0.217609\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	69156.0i	0.583943i	0.956427	+	0.291971i	$0.0943112\pi$
$107$	69156.0i	0.583943i	−0.956427	+	0.291971i	$0.905689\pi$
$108$	0	0
$109$	146414.	1.18037	0.590183	−	0.807270i	$-0.299056\pi$
$109$	146414.	1.18037	0.590183	+	0.807270i	$0.299056\pi$
$110$	0	0
$111$	−298220.	−2.29736
$112$	0	0
$113$	− 80186.0i	− 0.590748i	−0.955382	−	0.295374i	$-0.904556\pi$
$113$	− 80186.0i	− 0.590748i	0.955382	−	0.295374i	$-0.0954443\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 296172.i	− 2.00023i
$118$	0	0
$119$	−108682.	−0.703542
$120$	0	0
$121$	−160987.	−0.999603
$122$	0	0
$123$	347100.i	2.06867i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 274800.i	− 1.51185i	−0.654661	−	0.755923i	$-0.727189\pi$
$127$	− 274800.i	− 1.51185i	0.654661	−	0.755923i	$-0.272811\pi$
$128$	0	0
$129$	24128.0	0.127658
$130$	0	0
$131$	180742.	0.920197	0.460099	−	0.887868i	$-0.347814\pi$
$131$	180742.	0.920197	0.460099	+	0.887868i	$0.347814\pi$
$132$	0	0
$133$	132202.i	0.648051i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	209678.i	0.954446i	0.878782	+	0.477223i	$0.158357\pi$
$137$	209678.i	0.954446i	−0.878782	+	0.477223i	$0.841643\pi$
$138$	0	0
$139$	−17242.0	−0.0756921	−0.0378461	−	0.999284i	$-0.512050\pi$
$139$	−17242.0	−0.0756921	−0.0378461	+	0.999284i	$0.512050\pi$
$140$	0	0
$141$	31512.0	0.133484
$142$	0	0
$143$	5472.00i	0.0223772i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 62426.0i	− 0.238272i
$148$	0	0
$149$	−59358.0	−0.219035	−0.109518	−	0.993985i	$-0.534931\pi$
$149$	−59358.0	−0.219035	−0.109518	+	0.993985i	$0.534931\pi$
$150$	0	0
$151$	−336344.	−1.20044	−0.600221	−	0.799834i	$-0.704921\pi$
$151$	−336344.	−1.20044	−0.600221	+	0.799834i	$0.704921\pi$
$152$	0	0
$153$	− 960394.i	− 3.31681i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 464588.i	− 1.50425i	−0.659023	−	0.752123i	$-0.729030\pi$
$157$	− 464588.i	− 1.50425i	0.659023	−	0.752123i	$-0.270970\pi$
$158$	0	0
$159$	−340860.	−1.06926
$160$	0	0
$161$	−163856.	−0.498193
$162$	0	0
$163$	314792.i	0.928014i	0.885831	+	0.464007i	$0.153589\pi$
$163$	314792.i	0.928014i	−0.885831	+	0.464007i	$0.846411\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 285724.i	− 0.792785i	−0.918081	−	0.396393i	$-0.870262\pi$
$167$	− 285724.i	− 0.792785i	0.918081	−	0.396393i	$-0.129738\pi$
$168$	0	0
$169$	−96563.0	−0.260072
$170$	0	0
$171$	−1.16823e6	−3.05520
$172$	0	0
$173$	− 709148.i	− 1.80145i	−0.434392	−	0.900724i	$-0.643037\pi$
$173$	− 709148.i	− 1.80145i	0.434392	−	0.900724i	$-0.356963\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 902252.i	− 2.16468i
$178$	0	0
$179$	617148.	1.43965	0.719825	−	0.694156i	$-0.244222\pi$
$179$	617148.	1.43965	0.719825	+	0.694156i	$0.244222\pi$
$180$	0	0
$181$	237828.	0.539593	0.269797	−	0.962917i	$-0.413044\pi$
$181$	237828.	0.539593	0.269797	+	0.962917i	$0.413044\pi$
$182$	0	0
$183$	− 26832.0i	− 0.0592278i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	17744.0i	0.0371063i
$188$	0	0
$189$	242060.	0.492911
$190$	0	0
$191$	−133512.	−0.264812	−0.132406	−	0.991196i	$-0.542270\pi$
$191$	−133512.	−0.264812	−0.132406	+	0.991196i	$0.542270\pi$
$192$	0	0
$193$	270446.i	0.522622i	0.965255	+	0.261311i	$0.0841547\pi$
$193$	270446.i	0.522622i	−0.965255	+	0.261311i	$0.915845\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 875102.i	− 1.60655i	−0.595611	−	0.803273i	$-0.703090\pi$
$197$	− 875102.i	− 1.60655i	0.595611	−	0.803273i	$-0.296910\pi$
$198$	0	0
$199$	347620.	0.622260	0.311130	−	0.950367i	$-0.399292\pi$
$199$	347620.	0.622260	0.311130	+	0.950367i	$0.399292\pi$
$200$	0	0
$201$	262808.	0.458826
$202$	0	0
$203$	159446.i	0.271565i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 1.44795e6i	− 2.34870i
$208$	0	0
$209$	21584.0	0.0341795
$210$	0	0
$211$	−425380.	−0.657765	−0.328883	−	0.944371i	$-0.606672\pi$
$211$	−425380.	−0.657765	−0.328883	+	0.944371i	$0.606672\pi$
$212$	0	0
$213$	1.63072e6i	2.46281i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	234612.i	0.338221i
$218$	0	0
$219$	492076.	0.693301
$220$	0	0
$221$	−1.51711e6	−2.08947
$222$	0	0
$223$	481592.i	0.648511i	0.945970	+	0.324255i	$0.105114\pi$
$223$	481592.i	0.648511i	−0.945970	+	0.324255i	$0.894886\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	6042.00i	0.00778245i	0.999992	+	0.00389122i	$0.00123862\pi$
$227$	6042.00i	0.00778245i	−0.999992	+	0.00389122i	$0.998761\pi$
$228$	0	0
$229$	−1804.00	−0.00227325	−0.00113663	−	0.999999i	$-0.500362\pi$
$229$	−1804.00	−0.00227325	−0.00113663	+	0.999999i	$0.500362\pi$
$230$	0	0
$231$	−10192.0	−0.0125669
$232$	0	0
$233$	− 1.61153e6i	− 1.94468i	−0.233576	−	0.972339i	$-0.575043\pi$
$233$	− 1.61153e6i	− 1.94468i	0.233576	−	0.972339i	$-0.424957\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 296400.i	− 0.342774i
$238$	0	0
$239$	987096.	1.11780	0.558901	−	0.829235i	$-0.311223\pi$
$239$	987096.	1.11780	0.558901	+	0.829235i	$0.311223\pi$
$240$	0	0
$241$	893510.	0.990962	0.495481	−	0.868619i	$-0.334992\pi$
$241$	893510.	0.990962	0.495481	+	0.868619i	$0.334992\pi$
$242$	0	0
$243$	− 596674.i	− 0.648219i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.84543e6i	1.92467i
$248$	0	0
$249$	−2.31291e6	−2.36407
$250$	0	0
$251$	365946.	0.366634	0.183317	−	0.983054i	$-0.441317\pi$
$251$	365946.	0.366634	0.183317	+	0.983054i	$0.441317\pi$
$252$	0	0
$253$	26752.0i	0.0262757i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 1.40459e6i	− 1.32653i	−0.748383	−	0.663266i	$-0.769170\pi$
$257$	− 1.40459e6i	− 1.32653i	0.748383	−	0.663266i	$-0.230830\pi$
$258$	0	0
$259$	−562030.	−0.520607
$260$	0	0
$261$	−1.40898e6	−1.28028
$262$	0	0
$263$	1.09968e6i	0.980341i	0.871627	+	0.490170i	$0.163065\pi$
$263$	1.09968e6i	0.980341i	−0.871627	+	0.490170i	$0.836935\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 512772.i	− 0.440196i
$268$	0	0
$269$	−814948.	−0.686672	−0.343336	−	0.939213i	$-0.611557\pi$
$269$	−814948.	−0.686672	−0.343336	+	0.939213i	$0.611557\pi$
$270$	0	0
$271$	−1.69906e6	−1.40535	−0.702675	−	0.711511i	$-0.748011\pi$
$271$	−1.69906e6	−1.40535	−0.702675	+	0.711511i	$0.748011\pi$
$272$	0	0
$273$	− 871416.i	− 0.707651i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1.36508e6i	1.06895i	0.845183	+	0.534477i	$0.179492\pi$
$277$	1.36508e6i	1.06895i	−0.845183	+	0.534477i	$0.820508\pi$
$278$	0	0
$279$	−2.07320e6	−1.59453
$280$	0	0
$281$	−715846.	−0.540821	−0.270411	−	0.962745i	$-0.587159\pi$
$281$	−715846.	−0.540821	−0.270411	+	0.962745i	$0.587159\pi$
$282$	0	0
$283$	217726.i	0.161601i	0.996730	+	0.0808005i	$0.0257477\pi$
$283$	217726.i	0.161601i	−0.996730	+	0.0808005i	$0.974252\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	654150.i	0.468784i
$288$	0	0
$289$	−3.49967e6	−2.46480
$290$	0	0
$291$	443612.	0.307094
$292$	0	0
$293$	1.50708e6i	1.02557i	0.858516	+	0.512787i	$0.171387\pi$
$293$	1.50708e6i	1.02557i	−0.858516	+	0.512787i	$0.828613\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 39520.0i	− 0.0259972i
$298$	0	0
$299$	−2.28730e6	−1.47960
$300$	0	0
$301$	45472.0	0.0289286
$302$	0	0
$303$	1.19350e6i	0.746822i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 12502.0i	− 0.00757066i	−0.999993	−	0.00378533i	$-0.998795\pi$
$307$	− 12502.0i	− 0.00757066i	0.999993	−	0.00378533i	$-0.00120491\pi$
$308$	0	0
$309$	3.53631e6	2.10695
$310$	0	0
$311$	−647432.	−0.379571	−0.189786	−	0.981826i	$-0.560779\pi$
$311$	−647432.	−0.379571	−0.189786	+	0.981826i	$0.560779\pi$
$312$	0	0
$313$	− 935978.i	− 0.540014i	−0.962858	−	0.270007i	$-0.912974\pi$
$313$	− 935978.i	− 0.540014i	0.962858	−	0.270007i	$-0.0870260\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 705942.i	− 0.394567i	−0.980346	−	0.197284i	$-0.936788\pi$
$317$	− 705942.i	− 0.394567i	0.980346	−	0.197284i	$-0.0632120\pi$
$318$	0	0
$319$	26032.0	0.0143229
$320$	0	0
$321$	−1.79806e6	−0.973959
$322$	0	0
$323$	5.98416e6i	3.19152i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	3.80676e6i	1.96873i
$328$	0	0
$329$	59388.0	0.0302489
$330$	0	0
$331$	−1.14304e6	−0.573445	−0.286722	−	0.958014i	$-0.592566\pi$
$331$	−1.14304e6	−0.573445	−0.286722	+	0.958014i	$0.592566\pi$
$332$	0	0
$333$	− 4.96651e6i	− 2.45437i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	2.36402e6i	1.13390i	0.823751	+	0.566952i	$0.191877\pi$
$337$	2.36402e6i	1.13390i	−0.823751	+	0.566952i	$0.808123\pi$
$338$	0	0
$339$	2.08484e6	0.985309
$340$	0	0
$341$	38304.0	0.0178385
$342$	0	0
$343$	− 117649.i	− 0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 726240.i	− 0.323785i	−0.986808	−	0.161892i	$-0.948240\pi$
$347$	− 726240.i	− 0.323785i	0.986808	−	0.161892i	$-0.0517598\pi$
$348$	0	0
$349$	−136180.	−0.0598480	−0.0299240	−	0.999552i	$-0.509527\pi$
$349$	−136180.	−0.0598480	−0.0299240	+	0.999552i	$0.509527\pi$
$350$	0	0
$351$	3.37896e6	1.46391
$352$	0	0
$353$	− 1.16907e6i	− 0.499349i	−0.968330	−	0.249674i	$-0.919676\pi$
$353$	− 1.16907e6i	− 0.499349i	0.968330	−	0.249674i	$-0.0803235\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 2.82573e6i	− 1.17344i
$358$	0	0
$359$	4280.00	0.00175270	0.000876350	−	1.00000i	$-0.499721\pi$
$359$	4280.00	0.00175270	0.000876350		1.00000i	$0.499721\pi$
$360$	0	0
$361$	4.80310e6	1.93979
$362$	0	0
$363$	− 4.18566e6i	− 1.66724i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 2.44796e6i	− 0.948722i	−0.880330	−	0.474361i	$-0.842679\pi$
$367$	− 2.44796e6i	− 0.948722i	0.880330	−	0.474361i	$-0.157321\pi$
$368$	0	0
$369$	−5.78055e6	−2.21006
$370$	0	0
$371$	−642390.	−0.242306
$372$	0	0
$373$	− 904514.i	− 0.336623i	−0.985734	−	0.168311i	$-0.946169\pi$
$373$	− 904514.i	− 0.336623i	0.985734	−	0.168311i	$-0.0538314\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.22574e6i	0.806530i
$378$	0	0
$379$	4.23034e6	1.51279	0.756393	−	0.654117i	$-0.226960\pi$
$379$	4.23034e6	1.51279	0.756393	+	0.654117i	$0.226960\pi$
$380$	0	0
$381$	7.14480e6	2.52161
$382$	0	0
$383$	4.55400e6i	1.58634i	0.609002	+	0.793169i	$0.291570\pi$
$383$	4.55400e6i	1.58634i	−0.609002	+	0.793169i	$0.708430\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	401824.i	0.136382i
$388$	0	0
$389$	3.98541e6	1.33536	0.667680	−	0.744448i	$-0.267287\pi$
$389$	3.98541e6	1.33536	0.667680	+	0.744448i	$0.267287\pi$
$390$	0	0
$391$	−7.41699e6	−2.45350
$392$	0	0
$393$	4.69929e6i	1.53480i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 552420.i	− 0.175911i	−0.996124	−	0.0879555i	$-0.971967\pi$
$397$	− 552420.i	− 0.175911i	0.996124	−	0.0879555i	$-0.0280333\pi$
$398$	0	0
$399$	−3.43725e6	−1.08088
$400$	0	0
$401$	38190.0	0.0118601	0.00593006	−	0.999982i	$-0.498112\pi$
$401$	38190.0	0.0118601	0.00593006	+	0.999982i	$0.498112\pi$
$402$	0	0
$403$	3.27499e6i	1.00449i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	91760.0i	0.0274579i
$408$	0	0
$409$	3.92475e6	1.16012	0.580062	−	0.814573i	$-0.303028\pi$
$409$	3.92475e6	1.16012	0.580062	+	0.814573i	$0.303028\pi$
$410$	0	0
$411$	−5.45163e6	−1.59192
$412$	0	0
$413$	− 1.70040e6i	− 0.490541i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 448292.i	− 0.126247i
$418$	0	0
$419$	−598386.	−0.166512	−0.0832562	−	0.996528i	$-0.526532\pi$
$419$	−598386.	−0.166512	−0.0832562	+	0.996528i	$0.526532\pi$
$420$	0	0
$421$	4.61597e6	1.26928	0.634641	−	0.772807i	$-0.281148\pi$
$421$	4.61597e6	1.26928	0.634641	+	0.772807i	$0.281148\pi$
$422$	0	0
$423$	524796.i	0.142607i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 50568.0i	− 0.0134217i
$428$	0	0
$429$	−142272.	−0.0373230
$430$	0	0
$431$	−61560.0	−0.0159627	−0.00798133	−	0.999968i	$-0.502541\pi$
$431$	−61560.0	−0.0159627	−0.00798133	+	0.999968i	$0.502541\pi$
$432$	0	0
$433$	− 3.79727e6i	− 0.973310i	−0.873594	−	0.486655i	$-0.838217\pi$
$433$	− 3.79727e6i	− 0.973310i	0.873594	−	0.486655i	$-0.161783\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	9.02211e6i	2.25998i
$438$	0	0
$439$	2.28852e6	0.566752	0.283376	−	0.959009i	$-0.408545\pi$
$439$	2.28852e6	0.566752	0.283376	+	0.959009i	$0.408545\pi$
$440$	0	0
$441$	1.03963e6	0.254556
$442$	0	0
$443$	− 4.75976e6i	− 1.15233i	−0.817335	−	0.576163i	$-0.804549\pi$
$443$	− 4.75976e6i	− 1.15233i	0.817335	−	0.576163i	$-0.195451\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 1.54331e6i	− 0.365329i
$448$	0	0
$449$	4.36715e6	1.02231	0.511155	−	0.859489i	$-0.329218\pi$
$449$	4.36715e6	1.02231	0.511155	+	0.859489i	$0.329218\pi$
$450$	0	0
$451$	106800.	0.0247246
$452$	0	0
$453$	− 8.74494e6i	− 2.00222i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 5.44994e6i	− 1.22068i	−0.792140	−	0.610339i	$-0.791033\pi$
$457$	− 5.44994e6i	− 1.22068i	0.792140	−	0.610339i	$-0.208967\pi$
$458$	0	0
$459$	1.09569e7	2.42749
$460$	0	0
$461$	1.66966e6	0.365911	0.182956	−	0.983121i	$-0.441434\pi$
$461$	1.66966e6	0.365911	0.182956	+	0.983121i	$0.441434\pi$
$462$	0	0
$463$	70768.0i	0.0153421i	0.999971	+	0.00767104i	$0.00244179\pi$
$463$	70768.0i	0.0153421i	−0.999971	+	0.00767104i	$0.997558\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 5.66083e6i	− 1.20112i	−0.799578	−	0.600562i	$-0.794944\pi$
$467$	− 5.66083e6i	− 1.20112i	0.799578	−	0.600562i	$-0.205056\pi$
$468$	0	0
$469$	495292.	0.103975
$470$	0	0
$471$	1.20793e7	2.50893
$472$	0	0
$473$	− 7424.00i	− 0.00152576i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 5.67663e6i	− 1.14234i
$478$	0	0
$479$	1.44948e6	0.288652	0.144326	−	0.989530i	$-0.453899\pi$
$479$	1.44948e6	0.288652	0.144326	+	0.989530i	$0.453899\pi$
$480$	0	0
$481$	−7.84548e6	−1.54617
$482$	0	0
$483$	− 4.26026e6i	− 0.830937i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 4.07504e6i	− 0.778591i	−0.921113	−	0.389296i	$-0.872718\pi$
$487$	− 4.07504e6i	− 0.778591i	0.921113	−	0.389296i	$-0.127282\pi$
$488$	0	0
$489$	−8.18459e6	−1.54784
$490$	0	0
$491$	986100.	0.184594	0.0922969	−	0.995732i	$-0.470579\pi$
$491$	986100.	0.184594	0.0922969	+	0.995732i	$0.470579\pi$
$492$	0	0
$493$	7.21737e6i	1.33740i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3.07328e6i	0.558099i
$498$	0	0
$499$	−5.98342e6	−1.07572	−0.537859	−	0.843035i	$-0.680767\pi$
$499$	−5.98342e6	−1.07572	−0.537859	+	0.843035i	$0.680767\pi$
$500$	0	0
$501$	7.42882e6	1.32229
$502$	0	0
$503$	− 3.49373e6i	− 0.615700i	−0.951435	−	0.307850i	$-0.900391\pi$
$503$	− 3.49373e6i	− 0.615700i	0.951435	−	0.307850i	$-0.0996095\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 2.51064e6i	− 0.433775i
$508$	0	0
$509$	−2.15711e6	−0.369043	−0.184522	−	0.982828i	$-0.559074\pi$
$509$	−2.15711e6	−0.369043	−0.184522	+	0.982828i	$0.559074\pi$
$510$	0	0
$511$	927374.	0.157110
$512$	0	0
$513$	− 1.33281e7i	− 2.23602i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 9696.00i	− 0.00159539i
$518$	0	0
$519$	1.84378e7	3.00464
$520$	0	0
$521$	−6.65817e6	−1.07463	−0.537317	−	0.843380i	$-0.680562\pi$
$521$	−6.65817e6	−1.07463	−0.537317	+	0.843380i	$0.680562\pi$
$522$	0	0
$523$	5.95223e6i	0.951537i	0.879571	+	0.475768i	$0.157830\pi$
$523$	5.95223e6i	0.951537i	−0.879571	+	0.475768i	$0.842170\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.06198e7i	1.66567i
$528$	0	0
$529$	−4.74599e6	−0.737374
$530$	0	0
$531$	1.50260e7	2.31263
$532$	0	0
$533$	9.13140e6i	1.39226i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	1.60458e7i	2.40119i
$538$	0	0
$539$	−19208.0	−0.00284780
$540$	0	0
$541$	−6.39681e6	−0.939659	−0.469830	−	0.882757i	$-0.655685\pi$
$541$	−6.39681e6	−0.939659	−0.469830	+	0.882757i	$0.655685\pi$
$542$	0	0
$543$	6.18353e6i	0.899988i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	5.51851e6i	0.788594i	0.918983	+	0.394297i	$0.129012\pi$
$547$	5.51851e6i	0.788594i	−0.918983	+	0.394297i	$0.870988\pi$
$548$	0	0
$549$	446856.	0.0632757
$550$	0	0
$551$	8.77929e6	1.23191
$552$	0	0
$553$	− 558600.i	− 0.0776762i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 2.02159e6i	− 0.276093i	−0.990426	−	0.138046i	$-0.955918\pi$
$557$	− 2.02159e6i	− 0.276093i	0.990426	−	0.138046i	$-0.0440823\pi$
$558$	0	0
$559$	634752.	0.0859161
$560$	0	0
$561$	−461344.	−0.0618896
$562$	0	0
$563$	8.14678e6i	1.08322i	0.840631	+	0.541608i	$0.182184\pi$
$563$	8.14678e6i	1.08322i	−0.840631	+	0.541608i	$0.817816\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.13783e6i	0.148634i
$568$	0	0
$569$	1.19824e7	1.55154	0.775772	−	0.631013i	$-0.217361\pi$
$569$	1.19824e7	1.55154	0.775772	+	0.631013i	$0.217361\pi$
$570$	0	0
$571$	1.39582e6	0.179159	0.0895793	−	0.995980i	$-0.471448\pi$
$571$	1.39582e6	0.179159	0.0895793	+	0.995980i	$0.471448\pi$
$572$	0	0
$573$	− 3.47131e6i	− 0.441679i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 1.96784e6i	− 0.246065i	−0.992403	−	0.123033i	$-0.960738\pi$
$577$	− 1.96784e6i	− 0.246065i	0.992403	−	0.123033i	$-0.0392620\pi$
$578$	0	0
$579$	−7.03160e6	−0.871681
$580$	0	0
$581$	−4.35894e6	−0.535724
$582$	0	0
$583$	104880.i	0.0127797i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	3.18897e6i	0.381993i	0.981591	+	0.190997i	$0.0611720\pi$
$587$	3.18897e6i	0.381993i	−0.981591	+	0.190997i	$0.938828\pi$
$588$	0	0
$589$	1.29180e7	1.53429
$590$	0	0
$591$	2.27527e7	2.67956
$592$	0	0
$593$	1.67500e6i	0.195604i	0.995206	+	0.0978022i	$0.0311813\pi$
$593$	1.67500e6i	0.195604i	−0.995206	+	0.0978022i	$0.968819\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	9.03812e6i	1.03787i
$598$	0	0
$599$	1.00635e7	1.14599	0.572994	−	0.819559i	$-0.305782\pi$
$599$	1.00635e7	1.14599	0.572994	+	0.819559i	$0.305782\pi$
$600$	0	0
$601$	1.72798e6	0.195143	0.0975713	−	0.995229i	$-0.468893\pi$
$601$	1.72798e6	0.195143	0.0975713	+	0.995229i	$0.468893\pi$
$602$	0	0
$603$	4.37676e6i	0.490185i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	1.69523e7i	1.86748i	0.357953	+	0.933740i	$0.383475\pi$
$607$	1.69523e7i	1.86748i	−0.357953	+	0.933740i	$0.616525\pi$
$608$	0	0
$609$	−4.14560e6	−0.452943
$610$	0	0
$611$	829008.	0.0898371
$612$	0	0
$613$	1.01942e7i	1.09572i	0.836569	+	0.547861i	$0.184558\pi$
$613$	1.01942e7i	1.09572i	−0.836569	+	0.547861i	$0.815442\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 1.57452e7i	− 1.66508i	−0.553965	−	0.832540i	$-0.686886\pi$
$617$	− 1.57452e7i	− 1.66508i	0.553965	−	0.832540i	$-0.313114\pi$
$618$	0	0
$619$	332690.	0.0348990	0.0174495	−	0.999848i	$-0.494445\pi$
$619$	332690.	0.0348990	0.0174495	+	0.999848i	$0.494445\pi$
$620$	0	0
$621$	1.65194e7	1.71895
$622$	0	0
$623$	− 966378.i	− 0.0997532i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	561184.i	0.0570081i
$628$	0	0
$629$	−2.54405e7	−2.56388
$630$	0	0
$631$	3.59720e6	0.359659	0.179830	−	0.983698i	$-0.442445\pi$
$631$	3.59720e6	0.359659	0.179830	+	0.983698i	$0.442445\pi$
$632$	0	0
$633$	− 1.10599e7i	− 1.09709i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 1.64228e6i	− 0.160361i
$638$	0	0
$639$	−2.71578e7	−2.63113
$640$	0	0
$641$	−1.46389e7	−1.40723	−0.703614	−	0.710583i	$-0.748431\pi$
$641$	−1.46389e7	−1.40723	−0.703614	+	0.710583i	$0.748431\pi$
$642$	0	0
$643$	− 1.38386e7i	− 1.31997i	−0.751277	−	0.659987i	$-0.770562\pi$
$643$	− 1.38386e7i	− 1.31997i	0.751277	−	0.659987i	$-0.229438\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1.40358e7i	1.31819i	0.752061	+	0.659093i	$0.229060\pi$
$647$	1.40358e7i	1.31819i	−0.752061	+	0.659093i	$0.770940\pi$
$648$	0	0
$649$	−277616.	−0.0258722
$650$	0	0
$651$	−6.09991e6	−0.564119
$652$	0	0
$653$	1.61063e7i	1.47813i	0.673635	+	0.739064i	$0.264732\pi$
$653$	1.61063e7i	1.47813i	−0.673635	+	0.739064i	$0.735268\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	8.19496e6i	0.740685i
$658$	0	0
$659$	−4.80075e6	−0.430622	−0.215311	−	0.976546i	$-0.569076\pi$
$659$	−4.80075e6	−0.430622	−0.215311	+	0.976546i	$0.569076\pi$
$660$	0	0
$661$	−1.76565e7	−1.57181	−0.785905	−	0.618347i	$-0.787803\pi$
$661$	−1.76565e7	−1.57181	−0.785905	+	0.618347i	$0.787803\pi$
$662$	0	0
$663$	− 3.94449e7i	− 3.48504i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.08814e7i	0.947042i
$668$	0	0
$669$	−1.25214e7	−1.08165
$670$	0	0
$671$	−8256.00	−0.000707886 0
$672$	0	0
$673$	6.59225e6i	0.561043i	0.959848	+	0.280521i	$0.0905074\pi$
$673$	6.59225e6i	0.561043i	−0.959848	+	0.280521i	$0.909493\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 9.77178e6i	− 0.819411i	−0.912218	−	0.409706i	$-0.865631\pi$
$677$	− 9.77178e6i	− 0.819411i	0.912218	−	0.409706i	$-0.134369\pi$
$678$	0	0
$679$	836038.	0.0695908
$680$	0	0
$681$	−157092.	−0.0129803
$682$	0	0
$683$	− 1.88663e7i	− 1.54752i	−0.633481	−	0.773758i	$-0.718374\pi$
$683$	− 1.88663e7i	− 1.54752i	0.633481	−	0.773758i	$-0.281626\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 46904.0i	− 0.00379156i
$688$	0	0
$689$	−8.96724e6	−0.719632
$690$	0	0
$691$	8.67018e6	0.690769	0.345385	−	0.938461i	$-0.387748\pi$
$691$	8.67018e6	0.690769	0.345385	+	0.938461i	$0.387748\pi$
$692$	0	0
$693$	− 169736.i	− 0.0134258i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	2.96103e7i	2.30866i
$698$	0	0
$699$	4.18997e7	3.24353
$700$	0	0
$701$	7.93482e6	0.609877	0.304938	−	0.952372i	$-0.401364\pi$
$701$	7.93482e6	0.609877	0.304938	+	0.952372i	$0.401364\pi$
$702$	0	0
$703$	3.09461e7i	2.36166i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.24930e6i	0.169238i
$708$	0	0
$709$	−2.62600e7	−1.96191	−0.980956	−	0.194228i	$-0.937780\pi$
$709$	−2.62600e7	−1.96191	−0.980956	+	0.194228i	$0.937780\pi$
$710$	0	0
$711$	4.93620e6	0.366200
$712$	0	0
$713$	1.60111e7i	1.17950i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	2.56645e7i	1.86438i
$718$	0	0
$719$	−2.20763e7	−1.59259	−0.796295	−	0.604909i	$-0.793210\pi$
$719$	−2.20763e7	−1.59259	−0.796295	+	0.604909i	$0.793210\pi$
$720$	0	0
$721$	6.66459e6	0.477458
$722$	0	0
$723$	2.32313e7i	1.65283i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 8.49245e6i	− 0.595933i	−0.954576	−	0.297966i	$-0.903692\pi$
$727$	− 8.49245e6i	− 0.595933i	0.954576	−	0.297966i	$-0.0963083\pi$
$728$	0	0
$729$	2.11562e7	1.47441
$730$	0	0
$731$	2.05830e6	0.142468
$732$	0	0
$733$	− 1.90713e7i	− 1.31105i	−0.755172	−	0.655526i	$-0.772447\pi$
$733$	− 1.90713e7i	− 1.31105i	0.755172	−	0.655526i	$-0.227553\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 80864.0i	− 0.00548386i
$738$	0	0
$739$	1.46832e7	0.989032	0.494516	−	0.869169i	$-0.335345\pi$
$739$	1.46832e7	0.989032	0.494516	+	0.869169i	$0.335345\pi$
$740$	0	0
$741$	−4.79812e7	−3.21015
$742$	0	0
$743$	1.64265e7i	1.09162i	0.837908	+	0.545812i	$0.183779\pi$
$743$	1.64265e7i	1.09162i	−0.837908	+	0.545812i	$0.816221\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 3.85188e7i	− 2.52564i
$748$	0	0
$749$	−3.38864e6	−0.220710
$750$	0	0
$751$	−2.44357e7	−1.58097	−0.790486	−	0.612479i	$-0.790172\pi$
$751$	−2.44357e7	−1.58097	−0.790486	+	0.612479i	$0.790172\pi$
$752$	0	0
$753$	9.51460e6i	0.611509i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	295566.i	0.0187463i	0.999956	+	0.00937313i	$0.00298360\pi$
$757$	295566.i	0.0187463i	−0.999956	+	0.00937313i	$0.997016\pi$
$758$	0	0
$759$	−695552.	−0.0438253
$760$	0	0
$761$	−473842.	−0.0296601	−0.0148300	−	0.999890i	$-0.504721\pi$
$761$	−473842.	−0.0296601	−0.0148300	+	0.999890i	$0.504721\pi$
$762$	0	0
$763$	7.17429e6i	0.446136i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 2.37362e7i	− 1.45687i
$768$	0	0
$769$	−2.33241e7	−1.42229	−0.711145	−	0.703045i	$-0.751823\pi$
$769$	−2.33241e7	−1.42229	−0.711145	+	0.703045i	$0.751823\pi$
$770$	0	0
$771$	3.65194e7	2.21253
$772$	0	0
$773$	1.55583e7i	0.936511i	0.883593	+	0.468255i	$0.155117\pi$
$773$	1.55583e7i	0.936511i	−0.883593	+	0.468255i	$0.844883\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 1.46128e7i	− 0.868321i
$778$	0	0
$779$	3.60183e7	2.12657
$780$	0	0
$781$	501760.	0.0294353
$782$	0	0
$783$	− 1.60748e7i	− 0.937001i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 6.66843e6i	− 0.383784i	−0.981416	−	0.191892i	$-0.938538\pi$
$787$	− 6.66843e6i	− 0.383784i	0.981416	−	0.191892i	$-0.0614623\pi$
$788$	0	0
$789$	−2.85917e7	−1.63511
$790$	0	0
$791$	3.92911e6	0.223282
$792$	0	0
$793$	− 705888.i	− 0.0398614i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.22461e7i	0.682892i	0.939901	+	0.341446i	$0.110917\pi$
$797$	1.22461e7i	0.682892i	−0.939901	+	0.341446i	$0.889083\pi$
$798$	0	0
$799$	2.68822e6	0.148969
$800$	0	0
$801$	8.53963e6	0.470281
$802$	0	0
$803$	− 151408.i	− 0.00828629i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 2.11886e7i	− 1.14530i
$808$	0	0
$809$	2.91495e7	1.56588	0.782941	−	0.622095i	$-0.213718\pi$
$809$	2.91495e7	1.56588	0.782941	+	0.622095i	$0.213718\pi$
$810$	0	0
$811$	7.58849e6	0.405138	0.202569	−	0.979268i	$-0.435071\pi$
$811$	7.58849e6	0.405138	0.202569	+	0.979268i	$0.435071\pi$
$812$	0	0
$813$	− 4.41755e7i	− 2.34398i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 2.50374e6i	− 0.131230i
$818$	0	0
$819$	1.45124e7	0.756015
$820$	0	0
$821$	−5.98849e6	−0.310070	−0.155035	−	0.987909i	$-0.549549\pi$
$821$	−5.98849e6	−0.310070	−0.155035	+	0.987909i	$0.549549\pi$
$822$	0	0
$823$	817960.i	0.0420952i	0.999778	+	0.0210476i	$0.00670015\pi$
$823$	817960.i	0.0420952i	−0.999778	+	0.0210476i	$0.993300\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 2.51963e6i	− 0.128107i	−0.997946	−	0.0640535i	$-0.979597\pi$
$827$	− 2.51963e6i	− 0.128107i	0.997946	−	0.0640535i	$-0.0204028\pi$
$828$	0	0
$829$	1.61006e7	0.813684	0.406842	−	0.913499i	$-0.366630\pi$
$829$	1.61006e7	0.813684	0.406842	+	0.913499i	$0.366630\pi$
$830$	0	0
$831$	−3.54921e7	−1.78291
$832$	0	0
$833$	− 5.32542e6i	− 0.265914i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 2.36527e7i	− 1.16699i
$838$	0	0
$839$	2.58167e7	1.26618	0.633091	−	0.774077i	$-0.281786\pi$
$839$	2.58167e7	1.26618	0.633091	+	0.774077i	$0.281786\pi$
$840$	0	0
$841$	−9.92263e6	−0.483768
$842$	0	0
$843$	− 1.86120e7i	− 0.902036i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 7.88836e6i	− 0.377814i
$848$	0	0
$849$	−5.66088e6	−0.269535
$850$	0	0
$851$	−3.83557e7	−1.81554
$852$	0	0
$853$	− 1.54270e7i	− 0.725954i	−0.931798	−	0.362977i	$-0.881760\pi$
$853$	− 1.54270e7i	− 0.725954i	0.931798	−	0.362977i	$-0.118240\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 3.60517e6i	− 0.167677i	−0.996479	−	0.0838384i	$-0.973282\pi$
$857$	− 3.60517e6i	− 0.167677i	0.996479	−	0.0838384i	$-0.0267180\pi$
$858$	0	0
$859$	−4.06995e6	−0.188194	−0.0940970	−	0.995563i	$-0.529996\pi$
$859$	−4.06995e6	−0.188194	−0.0940970	+	0.995563i	$0.529996\pi$
$860$	0	0
$861$	−1.70079e7	−0.781885
$862$	0	0
$863$	7.25111e6i	0.331419i	0.986175	+	0.165710i	$0.0529914\pi$
$863$	7.25111e6i	0.331419i	−0.986175	+	0.165710i	$0.947009\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 9.09913e7i	− 4.11105i
$868$	0	0
$869$	−91200.0	−0.00409681
$870$	0	0
$871$	6.91387e6	0.308799
$872$	0	0
$873$	7.38785e6i	0.328082i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 2.37414e6i	− 0.104233i	−0.998641	−	0.0521167i	$-0.983403\pi$
$877$	− 2.37414e6i	− 0.104233i	0.998641	−	0.0521167i	$-0.0165968\pi$
$878$	0	0
$879$	−3.91841e7	−1.71056
$880$	0	0
$881$	3.03558e7	1.31766	0.658828	−	0.752293i	$-0.271052\pi$
$881$	3.03558e7	1.31766	0.658828	+	0.752293i	$0.271052\pi$
$882$	0	0
$883$	1.53338e6i	0.0661832i	0.999452	+	0.0330916i	$0.0105353\pi$
$883$	1.53338e6i	0.0661832i	−0.999452	+	0.0330916i	$0.989465\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	2.92379e7i	1.24778i	0.781514	+	0.623888i	$0.214448\pi$
$887$	2.92379e7i	1.24778i	−0.781514	+	0.623888i	$0.785552\pi$
$888$	0	0
$889$	1.34652e7	0.571424
$890$	0	0
$891$	185768.	0.00783929
$892$	0	0
$893$	− 3.26998e6i	− 0.137220i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 5.94697e7i	− 2.46783i
$898$	0	0
$899$	1.55802e7	0.642943
$900$	0	0
$901$	−2.90780e7	−1.19331
$902$	0	0
$903$	1.18227e6i	0.0482501i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 4.48227e7i	− 1.80917i	−0.426289	−	0.904587i	$-0.640179\pi$
$907$	− 4.48227e7i	− 1.80917i	0.426289	−	0.904587i	$-0.359821\pi$
$908$	0	0
$909$	−1.98764e7	−0.797864
$910$	0	0
$911$	−3.62906e7	−1.44877	−0.724384	−	0.689397i	$-0.757876\pi$
$911$	−3.62906e7	−1.44877	−0.724384	+	0.689397i	$0.757876\pi$
$912$	0	0
$913$	711664.i	0.0282552i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	8.85636e6i	0.347802i
$918$	0	0
$919$	−3.25350e7	−1.27076	−0.635378	−	0.772201i	$-0.719156\pi$
$919$	−3.25350e7	−1.27076	−0.635378	+	0.772201i	$0.719156\pi$
$920$	0	0
$921$	325052.	0.0126271
$922$	0	0
$923$	4.29005e7i	1.65752i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	5.88932e7i	2.25095i
$928$	0	0
$929$	4.46676e7	1.69806	0.849030	−	0.528344i	$-0.177187\pi$
$929$	4.46676e7	1.69806	0.849030	+	0.528344i	$0.177187\pi$
$930$	0	0
$931$	−6.47790e6	−0.244940
$932$	0	0
$933$	− 1.68332e7i	− 0.633087i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 1.56680e7i	− 0.582995i	−0.956572	−	0.291498i	$-0.905846\pi$
$937$	− 1.56680e7i	− 0.582995i	0.956572	−	0.291498i	$-0.0941536\pi$
$938$	0	0
$939$	2.43354e7	0.900689
$940$	0	0
$941$	−2.01175e7	−0.740627	−0.370313	−	0.928907i	$-0.620750\pi$
$941$	−2.01175e7	−0.740627	−0.370313	+	0.928907i	$0.620750\pi$
$942$	0	0
$943$	4.46424e7i	1.63481i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	8.84518e6i	0.320503i	0.987076	+	0.160251i	$0.0512305\pi$
$947$	8.84518e6i	0.320503i	−0.987076	+	0.160251i	$0.948769\pi$
$948$	0	0
$949$	1.29454e7	0.466605
$950$	0	0
$951$	1.83545e7	0.658099
$952$	0	0
$953$	− 3.14364e7i	− 1.12124i	−0.828072	−	0.560622i	$-0.810562\pi$
$953$	− 3.14364e7i	− 1.12124i	0.828072	−	0.560622i	$-0.189438\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	676832.i	0.0238892i
$958$	0	0
$959$	−1.02742e7	−0.360747
$960$	0	0
$961$	−5.70421e6	−0.199245
$962$	0	0
$963$	− 2.99445e7i	− 1.04052i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	9.52158e6i	0.327449i	0.986506	+	0.163724i	$0.0523507\pi$
$967$	9.52158e6i	0.327449i	−0.986506	+	0.163724i	$0.947649\pi$
$968$	0	0
$969$	−1.55588e8	−5.32313
$970$	0	0
$971$	1.06520e7	0.362564	0.181282	−	0.983431i	$-0.441975\pi$
$971$	1.06520e7	0.362564	0.181282	+	0.983431i	$0.441975\pi$
$972$	0	0
$973$	− 844858.i	− 0.0286089i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 2.72931e7i	− 0.914779i	−0.889266	−	0.457389i	$-0.848785\pi$
$977$	− 2.72931e7i	− 0.914779i	0.889266	−	0.457389i	$-0.151215\pi$
$978$	0	0
$979$	−157776.	−0.00526119
$980$	0	0
$981$	−6.33973e7	−2.10328
$982$	0	0
$983$	1.04764e7i	0.345802i	0.984939	+	0.172901i	$0.0553140\pi$
$983$	1.04764e7i	0.345802i	−0.984939	+	0.172901i	$0.944686\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	1.54409e6i	0.0504521i
$988$	0	0
$989$	3.10323e6	0.100884
$990$	0	0
$991$	1.88230e6	0.0608843	0.0304422	−	0.999537i	$-0.490308\pi$
$991$	1.88230e6	0.0608843	0.0304422	+	0.999537i	$0.490308\pi$
$992$	0	0
$993$	− 2.97190e7i	− 0.956449i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 2.71518e7i	− 0.865090i	−0.901612	−	0.432545i	$-0.857616\pi$
$997$	− 2.71518e7i	− 0.865090i	0.901612	−	0.432545i	$-0.142384\pi$
$998$	0	0
$999$	5.66618e7	1.79629

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	700.6.e.b.449.2		2
5.2	odd	4		28.6.a.b.1.1	✓	1
5.3	odd	4		700.6.a.b.1.1		1
5.4	even	2	inner	700.6.e.b.449.1		2
15.2	even	4		252.6.a.a.1.1		1
20.7	even	4		112.6.a.b.1.1		1
35.2	odd	12		196.6.e.a.165.1		2
35.12	even	12		196.6.e.i.165.1		2
35.17	even	12		196.6.e.i.177.1		2
35.27	even	4		196.6.a.a.1.1		1
35.32	odd	12		196.6.e.a.177.1		2
40.27	even	4		448.6.a.o.1.1		1
40.37	odd	4		448.6.a.b.1.1		1
60.47	odd	4		1008.6.a.l.1.1		1
140.27	odd	4		784.6.a.m.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.6.a.b.1.1	✓	1	5.2	odd	4
112.6.a.b.1.1		1	20.7	even	4
196.6.a.a.1.1		1	35.27	even	4
196.6.e.a.165.1		2	35.2	odd	12
196.6.e.a.177.1		2	35.32	odd	12
196.6.e.i.165.1		2	35.12	even	12
196.6.e.i.177.1		2	35.17	even	12
252.6.a.a.1.1		1	15.2	even	4
448.6.a.b.1.1		1	40.37	odd	4
448.6.a.o.1.1		1	40.27	even	4
700.6.a.b.1.1		1	5.3	odd	4
700.6.e.b.449.1		2	5.4	even	2	inner
700.6.e.b.449.2		2	1.1	even	1	trivial
784.6.a.m.1.1		1	140.27	odd	4
1008.6.a.l.1.1		1	60.47	odd	4

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:	\( N \)	\(=\)	\( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	700.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+26.0000i q^{3} +49.0000i q^{7} -433.000 q^{9} +O(q^{10})\)	\(q+26.0000i q^{3} +49.0000i q^{7} -433.000 q^{9} +8.00000 q^{11} +684.000i q^{13} +2218.00i q^{17} +2698.00 q^{19} -1274.00 q^{21} +3344.00i q^{23} -4940.00i q^{27} +3254.00 q^{29} +4788.00 q^{31} +208.000i q^{33} +11470.0i q^{37} -17784.0 q^{39} +13350.0 q^{41} -928.000i q^{43} -1212.00i q^{47} -2401.00 q^{49} -57668.0 q^{51} +13110.0i q^{53} +70148.0i q^{57} -34702.0 q^{59} -1032.00 q^{61} -21217.0i q^{63} -10108.0i q^{67} -86944.0 q^{69} +62720.0 q^{71} -18926.0i q^{73} +392.000i q^{77} -11400.0 q^{79} +23221.0 q^{81} +88958.0i q^{83} +84604.0i q^{87} -19722.0 q^{89} -33516.0 q^{91} +124488. i q^{93} -17062.0i q^{97} -3464.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 866 q^{9}+O(q^{10}) \) 2 * q - 866 * q^9	\( 2 q - 866 q^{9} + 16 q^{11} + 5396 q^{19} - 2548 q^{21} + 6508 q^{29} + 9576 q^{31} - 35568 q^{39} + 26700 q^{41} - 4802 q^{49} - 115336 q^{51} - 69404 q^{59} - 2064 q^{61} - 173888 q^{69} + 125440 q^{71} - 22800 q^{79} + 46442 q^{81} - 39444 q^{89} - 67032 q^{91} - 6928 q^{99}+O(q^{100}) \) 2 * q - 866 * q^9 + 16 * q^11 + 5396 * q^19 - 2548 * q^21 + 6508 * q^29 + 9576 * q^31 - 35568 * q^39 + 26700 * q^41 - 4802 * q^49 - 115336 * q^51 - 69404 * q^59 - 2064 * q^61 - 173888 * q^69 + 125440 * q^71 - 22800 * q^79 + 46442 * q^81 - 39444 * q^89 - 67032 * q^91 - 6928 * q^99

Introduction

L-functions

Modular forms

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Database

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